Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis

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NUMERICAL MATHEMATICS AND SCIENTIFIC COMPUTATION Series Editors A. M. STUART

¨ E. SULI

NUMERICAL MATHEMATICS AND SCIENTIFIC COMPUTATION Books in the series Monographs marked with an asterisk (∗ ) appeared in the series ‘Monographs in Numerical Analysis’ which is continued by the current series. For a full list of titles please visit http://www.oup.co.uk/academic/science/maths/series/nmsc ∗ ∗ ∗ ∗

J. H. Wilkinson: The algebraic eigenvalue problem I. Duﬀ, A. Erisman, and J. Reid: Direct methods for sparse matrices M. J. Baines: Moving ﬁnite elements J. D. Pryce: Numerical solution of Sturm-Liouville problems

C. Schwab: p- and hp- ﬁnite element methods: theory and applications to solid and ﬂuid mechanics J. W. Jerome: Modelling and computation for applications in mathematics, science, and engineering A. Quarteroni and A. Valli: Domain decomposition methods for partial diﬀerential equations G. Em Karniadakis and S. J. Sherwin: Spectral/hp element methods for CFD I. Babuˇska and T. Strouboulis: The ﬁnite element method and its reliability B. Mohammadi and O. Pironneau: Applied shape optimization for ﬂuids S. Succi: The lattice Boltzmann equation: for ﬂuid dynamics and beyond P. Monk: Finite element methods for Maxwell’s equations A. Bellen and M. Zennaro: Numerical methods for delay diﬀerential equations J. Modersitzki: Numerical methods for image registration M. Feistauer, J. Felcman, and I. Straˇskraba: Mathematical and computational methods for compressible ﬂow W. Gautschi: Orthogonal polynomials: computation and approximation M. K. Ng: Iterative methods for Toeplitz systems M. Metcalf, J. Reid, and M. Cohen: Fortran 95/2003 explained G. Em Karniadakis and S. Sherwin: Spectral/hp element methods for computational ﬂuid dynamics, second edition D. A. Bini, G. Latouche, and B. Meini: Numerical methods for structured Markov chains H. Elman, D. Silvester, and A. Wathen: Finite elements and fast iterative solvers: with applications in incompressible ﬂuid dynamics M. Chu and G. Golub: Inverse eigenvalue problems: theory, algorithms, and applications J.-F. Gerbeau, C. Le Bris, and T. Leli`evre: Mathematical methods for the magnetohydrodynamics of liquid metals G. Allaire and A. Craig: Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation K. Urban: Wavelet methods for elliptic partial diﬀerential equations B. Mohammadi and O. Pironneau: Applied shape optimization for ﬂuids, second edition K. B¨ ohmer: Numerical methods for nonlinear elliptic diﬀerential equations: a synopsis M. Metcalf, J. Reid, and M. Cohen: Modern Fortran Explained

Numerical Methods for Nonlinear Elliptic Diﬀerential Equations A Synopsis Klaus B¨ohmer University of Marburg

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Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Klaus B¨ ohmer 2010 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2010 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2009943744 Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by CPI Antony Rowe, Chippenham, Wiltshire ISBN 978–0–19–957704–0 1 3 5 7 9 10 8 6 4 2

With love to my late wife Inge, my daughters Annette and Christine, and my grandchildren Joachim, Dorothea, Leon, Luca, and Rebekka ∗

To God Alone Be The Glory, Organ St. Michaelis, L¨ uneburg

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Contents xiv

Preface PART I 1

ANALYTICAL RESULTS

From linear to nonlinear equations, fundamental results 1.1 Introduction 1.2 Linear versus nonlinear models 1.3 Examples for nonlinear partial diﬀerential equations 1.4 Fundamental results 1.4.1 Linear operators and functionals in Banach spaces 1.4.2 Inequalities and Lp (Ω) spaces 1.4.3 H¨ older and Sobolev spaces and more 1.4.4 Derivatives in Banach spaces

2 Elements of analysis for linear and nonlinear partial elliptic diﬀerential equations and systems 2.1 Introduction 2.2 Linear elliptic diﬀerential operators of second order, bilinear forms and solution concepts 2.3 Bilinear forms and induced linear operators 2.4 Linear elliptic diﬀerential operators, Fredholm alternative and regular solutions 2.4.1 Introduction 2.4.2 Linear operators of order 2m with C ∞ coeﬃcients 2.4.3 Linear operators of order 2 under C k conditions 2.4.4 Weak elliptic equation of order 2m in Hilbert spaces 2.5 Nonlinear elliptic equations 2.5.1 Introduction 2.5.2 Deﬁnitions for nonlinear elliptic operators 2.5.3 Special semilinear and quasilinear operators 2.5.4 Quasilinear elliptic equations of order 2 2.5.5 General nonlinear and Nemyckii operators 2.5.6 Divergent quasilinear elliptic equations of order 2m 2.5.7 Fully nonlinear elliptic equations of orders 2, m and 2m 2.6 Linear and nonlinear elliptic systems 2.6.1 Introduction 2.6.2 General systems of elliptic diﬀerential equations 2.6.3 Linear elliptic systems of order 2

3 3 3 10 13 13 18 20 27 32 32 36 45 54 54 58 64 69 77 77 79 81 88 96 100 108 113 113 114 118

viii

Contents

2.6.4

2.7

2.8

PART II

Quasilinear elliptic systems of order 2 and variational methods 2.6.5 Linear elliptic systems of order 2m, m ≥ 1 2.6.6 Divergent quasilinear elliptic systems of order 2m 2.6.7 Nemyckii operators and quasilinear divergent systems of order 2m 2.6.8 Fully nonlinear elliptic systems of orders 2 and 2m Linearization of nonlinear operators 2.7.1 Introduction 2.7.2 Special semilinear and quasilinear equations 2.7.3 Divergent quasilinear and fully nonlinear equations 2.7.4 Quasilinear elliptic systems of orders 2 and 2m 2.7.5 Linearizing general divergent quasilinear and fully nonlinear systems The Navier–Stokes equation 2.8.1 Introduction 2.8.2 The Stokes operator and saddle point problems 2.8.3 The Navier–Stokes operator and its linearization

125 132 137 140 146 147 147 149 151 157 158 163 163 163 167

NUMERICAL METHODS

3 A general discretization theory 3.1 Introduction 3.2 Petrov–Galerkin and general discretization methods 3.3 Variational and classical consistency 3.4 Stability and consistency yield convergence 3.5 Techniques for proving stability 3.6 Stability implies invertibility 3.7 Solving nonlinear systems: Continuation and Newton’s method based upon the mesh independence principle (MIP) 3.7.1 Continuation methods 3.7.2 MIP for nonlinear systems

173 173 175 185 189 194 203

4 Conforming ﬁnite element methods (FEMs) 4.1 Introduction 4.2 Approximation theory for ﬁnite elements 4.2.1 Subdivisions and ﬁnite elements 4.2.2 Polynomial ﬁnite elements, triangular and rectangular K 4.2.3 Interpolation in ﬁnite element spaces, an example 4.2.4 Interpolation errors and inverse estimates 4.2.5 Inverse estimates on nonquasiuniform triangulations 4.2.6 Smooth FEs on polyhedral domains, with O. Davydov 4.2.7 Curved boundaries 4.3 FEMs for linear problems

209 209 212 212

205 205 206

214 221 229 233 238 250 257

Contents

4.3.1 4.3.2

4.4 4.5 4.6

4.7

Finite element methods: a simple example, essential tools Finite element methods for general linear equations and systems of orders 2 and 2m 4.3.3 General convergence theory for conforming FEMs Finite element methods for divergent quasilinear elliptic equations and systems General convergence theory for monotone and quasilinear operators Mixed FEMs for Navier–Stokes and saddle point equations 4.6.1 Navier–Stokes and saddle point equations 4.6.2 Mixed FEMs for Stokes and saddle point equations 4.6.3 Mixed FEMs for the Navier–Stokes operator Variational methods for eigenvalue problems 4.7.1 Introduction 4.7.2 Theory for eigenvalue problems 4.7.3 Diﬀerent variational methods for eigenvalue problems

5 Nonconforming ﬁnite element methods 5.1 Introduction 5.2 Finite element methods for fully nonlinear elliptic problems 5.2.1 Introduction 5.2.2 Main ideas and results for the new FEM: An extended summary 5.2.3 Fully nonlinear and general quasilinear elliptic equations 5.2.4 Existence and convergence for semiconforming FEMs 5.2.5 Deﬁnition of nonconforming FEMs 5.2.6 Consistency for nonconforming FEMs 5.2.7 Stability for the linearized operator and convergence 5.2.8 Discretization of equations and systems of order 2m 5.2.9 Consistency, stability and convergence for m, q ≥ 1 5.2.10 Numerical solution of the FE equations with Newton’s method 5.3 FE and other methods for nonlinear boundary conditions 5.4 Quadrature approximate FEMs 5.4.1 Introduction 5.4.2 Quadrature and cubature formulas 5.4.3 Quadrature for second order linear problems 5.4.4 Quadrature for second order fully nonlinear equations 5.4.5 Quadrature FEMs for equations and systems of order 2m 5.4.6 Two useful propositions 5.5 Consistency, stability and convergence for FEMs with variational crimes 5.5.1 Introduction 5.5.2 Variational crimes for our standard example

ix 258 264 266 273 277 281 281 282 286 288 288 289 292 296 296 298 298 299 305 308 311 317 319 332 336 341 345 346 346 348 350 357 361 367 368 368 370

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Contents

5.5.3

FEMs with crimes for linear and quasilinear problems 5.5.4 Discrete coercivity and consistency 5.5.5 High order quadrature on edges 5.5.6 Violated boundary conditions 5.5.7 Violated continuity 5.5.8 Stability for nonconforming FEMs 5.5.9 Convergence, quadrature and solution of FEMs with crimes 5.5.10 Isoparametric FEMs 6 Adaptive ﬁnite element methods, by W. D¨ orﬂer 6.1 Introduction 6.1.1 The model problem 6.1.2 Singular solutions 6.1.3 A priori error bounds 6.1.4 Necessity of nonuniform mesh reﬁnement 6.1.5 Optimal meshes – A heuristic argument 6.1.6 Optimal meshes for 2D corner singularities 6.1.7 The ﬁnite element method–Notation and requirements 6.2 The residual error estimator for the Poisson problem 6.2.1 Upper a posteriori bound 6.2.2 Lower a posteriori bound 6.2.3 The a posteriori error estimate 6.2.4 The adaptive ﬁnite element method 6.2.5 Stable reﬁnement methods for triangulations in R2 6.2.6 Convergence of the adaptive ﬁnite element method 6.2.7 Optimality 6.2.8 Other types of estimators 6.2.9 hp ﬁnite element method 6.3 Estimation of quantities of interest 6.3.1 Quantities of interest 6.3.2 Error estimates for point errors 6.3.3 Optimal meshes–A heuristic argument 6.3.4 The general approach 7

Discontinuous Galerkin methods (DCGMs), with V. Dolejˇ s´ı 7.1 Introduction 7.2 The model problem 7.3 Discretization of the problem 7.3.1 Triangulations 7.3.2 Broken Sobolev spaces 7.3.3 Extended variational formulation of the problem 7.3.4 Discretization 7.4 General linear elliptic problems

380 387 390 392 399 406 411 414 420 420 421 421 423 425 425 427 428 430 430 432 433 434 436 438 442 447 448 449 449 449 451 451 455 455 459 461 461 462 463 469 472

Contents

7.5

7.6 7.7

7.8 7.9

7.10

7.11

7.12 7.13

7.14

7.15

8

Semilinear and quasilinear elliptic problems 7.5.1 Semilinear elliptic problems 7.5.2 Variational formulation and discretization of the problem 7.5.3 Quasilinear elliptic systems 7.5.4 Discretization of the quasilinear systems DCGMs are general discretization methods Geometry of the mesh, error and inverse estimates 7.7.1 Geometry of the mesh 7.7.2 Inverse and interpolation error estimates Penalty norms and consistency of the Jhσ Coercive linearized principal parts 7.9.1 Coercivity of the original linearized principal parts 7.9.2 Coercivity and boundedness in V h for the Laplacian 7.9.3 Coercivity and boundedness in V h for the general linear and the semilinear case 7.9.4 V h -coercivity and boundedness for quasilinear problems Consistency results for the ch , bh , h 7.10.1 Consistency of the ch and bh 7.10.2 Consistency of the h Consistency properties of the ah 7.11.1 Consistency of the ah for the Laplacian 7.11.2 Consistency of the ah for general linear problems 7.11.3 Consistency of the semilinear ah 7.11.4 Consistency of the quasilinear ah for systems 7.11.5 Consistency of the quasilinear ah for the equations of Houston, Robson, S¨ uli, and for systems Convergence for DCGMs Solving nonlinear equations in DCGMs 7.13.1 Introduction 7.13.2 Discretized linearized quasilinear system and diﬀerentiable consistency hp-variants of DCGM 7.14.1 hp-ﬁnite element spaces 7.14.2 hp-DCGMs 7.14.3 hp-inverse and approximation error estimates 7.14.4 Consistency and convergence of hp-DCGMs Numerical experiences 7.15.1 Scalar quasilinear equation 7.15.2 System of the steady compressible Navier–Stokes equations

Finite diﬀerence methods 8.1 Introduction 8.2 Diﬀerence methods for simple examples, notation

xi 474 474 475 477 478 482 486 487 487 491 494 494 495 499 502 503 503 505 507 507 511 514 518 523 527 532 532 532 538 539 540 540 542 546 546 554 560 560 562

xii

Contents

8.3

8.4

8.5

8.6

8.7 8.8

8.9

Discrete Sobolev spaces 8.3.1 Notation and deﬁnitions 8.3.2 Discrete Sobolev spaces General elliptic problems with Dirichlet conditions, and their diﬀerence methods 8.4.1 General elliptic problems 8.4.2 Second order linear elliptic diﬀerence equations 8.4.3 Symmetric diﬀerence methods 8.4.4 Linear equations of order 2m 8.4.5 Quasilinear elliptic equations of orders 2, and 2m 8.4.6 Systems of linear and quasilinear elliptic equations 8.4.7 Fully nonlinear elliptic equations and systems Convergence for diﬀerence methods 8.5.1 Discretization concepts in discrete Sobolev spaces 8.5.2 The operators P h , Q h 8.5.3 Consistency for diﬀerence equations 8.5.4 Vbh -coercivity for linear(ized) elliptic diﬀerence equations 8.5.5 Stability and convergence for general elliptic diﬀerence equations Natural boundary value problems of order 2 8.6.1 Analysis for natural boundary value problems 8.6.2 Diﬀerence methods for natural boundary value problems Other diﬀerence methods on curved boundaries 8.7.1 The Shortley–Weller–Collatz method for linear equations Asymptotic expansions, extrapolation, and defect corrections 8.8.1 A diﬀerence method based on polynomial interpolation for linear, and semilinear equations 8.8.2 Asymptotic expansions for other methods Numerical experiments for the von K´ arm´an equations, with C.S. Chien

9 Variational methods for wavelets, with S. Dahlke 9.1 Introduction 9.2 The scope of problems 9.3 Wavelet analysis 9.3.1 The discrete wavelet transform 9.3.2 Biorthogonal bases 9.3.3 Wavelets and function spaces 9.3.4 Wavelets on domains 9.3.5 Evaluation of nonlinear functionals 9.4 Stable discretizations and preconditioning 9.5 Applications to elliptic equations 9.6 Saddle point and (Navier–)Stokes equations 9.6.1 Saddle point equations 9.6.2 Navier–Stokes equations

566 566 569 572 572 574 581 583 584 586 587 588 589 591 594 600 604 610 611 613 622 623 626 627 630 633 635 635 637 639 640 644 646 647 652 653 659 664 664 666

Contents

9.7

Adaptive wavelet methods, by T. Raasch 9.7.1 Nonlinear approximation with wavelet systems 9.7.2 Wavelet matrix compression 9.7.3 Adaptive wavelet–Galerkin methods 9.7.4 Adaptive descent iterations 9.7.5 Nonlinear stationary problems

xiii 669 672 675 678 680 683

Bibliography

686

Index

733

Preface Nonlinear problems play an increasingly important role in mathematics, science and engineering. Thus, an exciting interplay originates between the diﬀerent disciplines, stimulated by the insight that linear problems often provide only poor approximations for the often nonlinear original problems. This applies particularly for all the bifurcation and induced dynamical regimes in nonlinear diﬀerential equations. In fact, this book, A ‘Synopsis’, and the following Numerical Methods for Bifurcation and Center Manifolds in Nonlinear Elliptic and Parabolic Equations [120], are motivated by these interrelations and their numerical realization. A major part of these phenomena is governed by elliptic and the corresponding parabolic partial diﬀerential equations and systems of orders 2 and 2m. For these nonlinear equations, bifurcation and the related, hence local, dynamics are mainly governed by the underlying elliptic partial diﬀerential equations and a small dimensional time dependent model, the center manifold. So we can avoid most of the problems of time discretization of parabolic partial diﬀerential equations and concentrate on the space discretization of elliptic partial diﬀerential equations. The problem of convergence of the solutions of space discretized elliptic problems to those of the original nonlinear problems is the core of this book. Astonishing is the fact that in the many excellent books on numerical methods for elliptic problems, see, e.g. below, nonlinear phenomena are either totally omitted or only treated by one or two important examples. Most books concentrate on just one of the many available discretization methods. Certainly, one of the reasons is the fact that by linearization many of the nonlinear equations and their numerical methods can be reduced to the linear case. However, for many problems, e.g. all types of bifurcation scenarios, a closer relation between the original nonlinear problem and its nonlinear discretization is mandatory. For linear problems the existence, uniqueness and regularity of solutions are well known. For nonlinear problems this is no longer correct and the results are widely spread in a huge number of textbooks, monographs and much more original papers. They are obtained by very diﬀerent methods. So we have to summarize and formulate those analytical results, necessary for the following proofs for and the convergence rates of the numerical methods. In particular, we need a systematic listing of conditions guaranteeing the existence, uniqueness and regularity for the solutions of these nonlinear elliptic problems, and exact conditions for the linearization. We include hints to available textbooks and monographs for this area and to special problems, not covered here. With a few exceptions, we do not refer to nor list the results of original papers. In contrast to other books, the goal here is a systematic study, in particular of the joint structure, of the discretization methods for linear and nonlinear elliptic problems.

Preface

xv

Necessarily this has to start with the linear foundations but has to proceed to fully nonlinear problems, where the highest derivatives occur nonlinearly. In particular, this last highlight is possible, since recently the author, probably for the ﬁrst time, has published a ﬁnite element method and proved its convergence for this wide class of general nonlinear problems, cf. B¨ ohmer [118, 119]. So, we prove stability and convergence for the numerical methods below and for all the linear and nonlinear problems studied in Chapter 2. It cannot be the goal of this book to provide an encyclopedia of all available space discretization methods for elliptic problems, since new methods are emerging all the time. Rather we have chosen the most important examples of diﬀerent types. New methods will probably ﬁt into our framework, certainly with appropriate generalizations and modiﬁcations. It would be pretty unrealistic to assume that a standard reader would study the whole book. Rather, a scientist would look for the class of elliptic problems and the corresponding numerical method most interesting for him. Therefore this preface gives a much extended survey of the content of each chapter and thus allows an appropriate choice. We formulate two examples at the end of this preface for a goal-oriented selection of material from the book. The book presents the result that these types of short studies are possible. To give a feeling for the type of our elliptic problems, we formulate a hierarchy n of increasingly nonlinear model problems, generalizing the Laplacian g independent of the solutions u. Its obvious linΔu = i=1 ∂ i ∂ i u = g, with n j i ear generalization in Rn is i,j=0 (−1)j>0 ∂ (aij ∂ u) = g. We use the notation j 0 j n 2 ∂ u := ∂u/∂xj , ∂ u := u, ∇u := (∂ u)j=1 ∇ u := (∂ i ∂ j u)ni,j=1 , and (−1)j>0 = −1 for j > 0 and else = 1. Special semilinear forms are Δu + f (u) = 0 or, the more n general, Δu + f (x, u, ∇u) = 0. Quasilinear equations in divergence form are − i=0 ∂ i (ai (x, u, ∇u)) = 0. Fully nonlinear problems are the most diﬃcult: a(x, u, ∇u, ∇2 u) = 0. These second order equations are generalized in Chapter 2 to equations of order 2m and to systems of order 2m, m ≥ 1, and the generalized Laplacian. We study them on polygonal or curved domains mainly with generalized Dirichlet boundary conditions. For m = 1 we admit (nonlinear) boundary conditions, induced by the (nonlinear) diﬀerential operators. The goal for our discretization approach is the proof of the unique existence of discrete solutions for all these previous problems, their computability and convergence to the exact solutions. We reach this goal for ﬁnite element methods, FEMs, and adaptive forms, diﬀerence, mesh-free and spectral methods. For discontinuous Galerkin methods it allows linear to quasilinear equations and systems of order 2. Due to the open problem of calculating complicated nonlinear operators, wavelet methods are restricted to semilinear equations and systems of order 2m, m ≥ 1. So this monograph is essentially oriented towards theoretical numerical mathematics. The only problems computed here are quasilinear and steady compressible Navier– Stokes equations in Section 7.15 and the van K´ arm´an equations in Section 8.9. All the other case studies, e.g. the dew drops on a spider’s web, the local dynamics in the earth mantel, and weather models, and the beginning chaos in Navier–Stokes equations require the tools of numerical symmetry breaking. They are delayed to [120].

xvi

Preface

Linearization techniques are the main tools for both books. The trick which makes it work relies upon the bounded invertibility of the derivative of the operator in a neighborhood of a solution. If this is assumed, it may be shown that even an equibounded invertibility, hence stability, holds for the chosen discretizations. Most previous approaches for quasilinear problems rely upon monotonicity results for the equation at hand. This limits the nonlinearities and also the discretizations which may be used. The new approach permits a uniﬁed convergence theory for the previously mentioned problems and discretizations. Nevertheless, each of these combinations requires a speciﬁc theoretical “lifting” to establish convergence and stability. The author is pretty sure that by modifying this lifting the techniques in this book can be applied to the other space discretization methods as well. Indeed, it has worked very well starting with FEMs and proceeding to all our other methods. The linearization technique for the previous FEMs and all the following methods does not allow convergence results for monotone and quasilinear operators in W m,p (Ω) for 1 < p < 2, cf. Subsection 4.3.3 and Lemma 2.77, Section 2.7, and Chapters 4 and 5. However it yields, for 2 ≤ p < ∞, convergence of the expected order with respect to the discrete H m (Ω) norms for all types of elliptic problems. On the other hand, the monotone operator techniques yield convergence results with diﬀerent orders for 1 < p < ∞ with respect to discrete W m,p (Ω) norms, cf. Theorem 4.67. Strongly monotone operators or quasilinear operators in H 2 (Ω) allow speciﬁc results. In fact, many important applications are modeled by monotone operators. So we embed the monotone operator approach, independent of linearization, into our general discretization theory. We do not aim for a full proof of these results. Rather we combine the excellent presentation in Zeidler [678], with Chapter 3. We formulate the results in Section 4.5 such that the other discretization methods, in particular the nonconforming methods in Chapters 5 and 7, are included. These results seem to extend the state of the art in several directions. These two independent approaches, here linearization and monotone operator techniques, complement each other very appropriately. Both yield complementing results for the existence, uniqueness and convergence of the discrete solutions for all previous discretizations and for quasilinear to fully nonlinear problems. Back to the linearized problems. These are split into a coercive part and a compact perturbation, thus allowing the highly successful perturbation techniques. Many methods are nonconforming, so the approximating functions violate the boundary and/or smoothness properties of the exact solutions. Then the relations between the strong and the weak form of the (linearized) problem are essential. A so-called anticrime transformation masters these diﬃculties for the convergence theory. For the adaptive FEMs in Chapter 6, D¨ orﬂer uses related techniques. These techniques allow the above-mentioned formulation of a new class of FEMs, diﬀerence, mesh-free and spectral methods for fully nonlinear problems, cf. [118–120] This is diﬀerent from the earlier handling of quasilinear problems via monotonicity arguments. They do not seem to be applicable to fully nonlinear problems. The next book [120] is the motivation for this book. There all the bifurcation and local dynamical scenarios are again governed by linearization of the corresponding problems. So the possibility of linearization has to be carefully studied. The class of

Preface

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nonlinear problems amenable to these techniques is characterized in Section 2.7. It essentially requires the above-mentioned boundedly invertible linearized operator. Finally, many methods for solving nonlinear discrete problems are based upon the corresponding linear problems, e.g. all types of Newton and continuation techniques. Furthermore, we obtain the Fredholm alternative and can prove stability and convergence in a uniﬁed way for the diﬀerent types of space discretization methods and the large class of linear and nonlinear problems characterized before. Despite this uniﬁed convergence theory, the diﬀerent methods presented here, cf. the following paragraphs, and in [120], require carefully distinguishing their diﬀerent approximation characteristics. So we have many good reasons for linearization as the main tool. It is a fascinating phenomenon that an appropriate structure allows very similar arguments for a wide range of problems. The most frequently used method are FEMs, including nonconforming FEMs. Some of the techniques in FEMs can be modiﬁed for the other methods as well. Adaptive FEMs employ a posteriori estimates for determining problem-adapted FEMs. Discontinuous Galerkin methods, DCGMs, are closely related to and modify FEMs by penalty terms. The oldest, the ﬁnite diﬀerence method, is still a favorite in bifurcation problems. Wavelets are one of the quite recent methods with still many open technical problems. Only spectral methods allow the numerical realization of inﬁnite symmetry groups. The mesh-free or radial basis methods allow a very ﬂexible positioning of their centers, again very advantageous for maintaining the symmetry of problems in their discretization. Therefore both methods are omitted here and used in B¨ ohmer [120] as demonstration examples for the general discretization theory employed in both books, cf. Chapter 3 and the new approach in [120]. We omit several interesting methods, e.g. some generalizations of ﬁnite elements, the ﬁnite volume, cf. Bey [90] and Knabner and Angermann [445]. For boundary integral methods, several interesting books are available by Steinbach and Rjasanov [556, 595] and Hsiao and Wendland [413]. They have been applied to nonlinear problems by linearizing the original problem and combining Newton–Kantorowich methods and continuation. The corresponding linear equations can then be solved with these methods. The same techniques have recently been used for wavelet and mesh-free methods, now included in our convergence theory. For boundary integral methods we did not see this possibility. Originally this book was planned as part of [120]. As a basis for numerical methods in bifurcation of nonlinear diﬀerential equations, we had to summarize the state of the art of numerical methods here for elliptic problems. But, for most of the methods studied here, these results turned out to be rather incomplete towards nonlinearity, in particular full nonlinearity. So we had to extend many known results for nonlinear problems into diﬀerent and missing directions. This was possible by systematically using the above general discretization theory. It even allowed the proof of the convergence results for nonlinear boundary operators. These are problems which seem to be not much discussed in the literature. Forced by this wide range of problems, we split the presentation into two essentially independent books.

xviii Preface As mentioned, most of the many excellent books on numerical methods for PDEs are concentrated on linear problems. In particular, they intensively discuss the corresponding linear solvers. Many standard solvers for large nonlinear problems are essentially based upon linear solvers. The discretization methods presented here are so-called “linear” methods. This is the essential condition for numerical methods adapted to bifurcation and for the mesh independence principle (MIP). This states that the Newton method applied to the discrete problem converges, for a small enough “step size”, h, essentially as fast and independent of h, as that applied to the analytic problem. This implies quadratic and, e.g. linear convergence for the original and modiﬁed Newton’s method. So the convergence of these iteration methods to discrete solutions is guaranteed for our wide range of discretization methods and nonlinear problems. The other tools are continuation methods. Both methods solve sequences of linear problems. The nonlinearity only enters via the computation of the defects or residuals of the approximate discrete solution in the nonlinear problem. A combination of Newton with continuation techniques provides eﬃcient solvers for nonlinear discrete systems. Usually this is combined with available multilevel methods. Since these linear solvers are pretty diﬀerent for the diﬀerent types of the above methods and problems, a synopsis of the linear solvers does not make too much sense. Hence, we do not discuss linear solvers. Modiﬁcations for bifurcation and approximations for the eigenvalues and invariant subspaces for the linearized operators are delayed to [120]. So this synopsis on numerical methods for nonlinear elliptic problems essentially studies stability, consistency, convergence, and computability. The following references, here and throughout the book, are chosen to support speciﬁc points in the text. It would be totally impossible to list all relevant books and papers. The diﬀerent discretization methods are studied in hundreds or even thousands of papers. So we usually only cite books, unless speciﬁc points have to be made. This implies that the contributions of many scientists represent extremely important contributions to the area in this book without being mentioned here. One example are the numerical methods for the Navier–Stokes equations, with 3,731 citations in October 2009 in the Zentralblatt. According to the previous discussion, this book is organized as follows. Part I is devoted to analytical results. Chapter 1 demonstrates, for the simple mechanical example of a bent rod, the change in the character from linear to nonlinear regimes. This is followed by several examples for diﬀerent types of nonlinear elliptic diﬀerential equations, e.g. in mathematics – the Monge–Amp`ere equation – in science – the reaction–diﬀusion systems – and in engineering – the von K´ arm´an and the Navier– Stokes equations. These problems and methods require the analytical results necessary for our (space-) discretization methods. The necessary tools from functional analysis and calculus in Banach spaces are listed in Section 1.4. In Chapter 2, we summarize the indicated analytical results. We include general linear, special semilinear, semilinear, quasilinear and fully nonlinear elliptic diﬀerential equations and systems of orders 2 and 2m, e.g. the Navier–Stokes equations. An early form of such a collection of results for engineering problems are the two volumes of Ames [26,27]. Numerical methods are studied there as well. Extensions to science and

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the necessary analytic and group theoretic tools are collected in Rogers and Ames [557]. The essentials are existence, uniqueness and regularity of their solutions and properties of the linearizations. This linearization is applicable to nearly all the nonlinear elliptic problems discussed before (see Section 2.7). Excluded are nonlinear problems that are not elliptic in the standard sense, but monotone. Their FEMs are discussed in Section 4.5. The linearization and its bounded invertibility for the included problems yield the Fredholm alternative and the stability of space discretization methods, respectively. Linearization is not always appropriate for the analysis of nonlinear problems. Diﬀerent techniques have to be used for other aspects of nonlinear problems: existence, uniqueness and other properties of solutions are obtained by methods, usually independent of linearization. They are appropriately studied via monotone and Nemyckii operators, systematically described, e.g. in Zeidler’s impressive “multigraph” [675]–[678]. Other examples are discussed, e.g. by Haslinger et al. [393], for unilateral boundary value problems, by Feistauer and Zen´ısek [316, 317], and by Zen´ısek [681], with monotone and quasimonotone methods and variational crimes. Haslinger et al. [393] consider mainly mechanical contact problems via inequalities. Often maximum principles, and diﬀerent inequalities are employed, cf. Gilbarg and Trudinger, [346]. In a fascinating series of papers, [215–228] Crandall, Lions and colleagues study viscosity solutions, which we do not consider here. In Caffarelli and Glowinski, [154], they are applied to numerical methods for the Pucci equation. Another example is the huge area of practically relevant problems, e.g. in ﬂuid and solid mechanics and in modeling. In solid mechanics the nonlinearity is complicated by additional corner and edge singularities and diﬀerent scales for the interesting scenarios. Here and in related problems, adaptive mixed FEMs are mandatory, cf. e.g., Braess, [135–137] and with Carstensen, and Reddy, [138], Johannsen et al. and Kawohl et al. [421, 438] Stein/Sagar, [594]. Essential phenomena, based upon a posteriori error estimates, employ the nonlinearity directly. Some of these problems are indicated in Chapter 6, but we do not study them explicitly. The question how well these problems can be solved with lineraization techniques is still open. We turn to Part II numerical methods. Most standard monographs or textbooks, present the convergence theory of space discretizations only for one speciﬁc discretization method and only linear problems. In other books, e.g., Brenner and Scott, [142], FEMs for nonlinear problems are essentially described by a few examples. The books of Glowinski, partially with Lions and Tremolieres, and surveys [350, 351, 353, 354], systematically study numerical methods for variational inequalities and nonlinear variational problems. Again nonlinear problems play an important role in Glowinski’s many proceedings edited, e.g. with Absi, Angrand, Bertin, Bristeau, Chen, Dervieux, Desideri, Fortin, Larrouturou, Lascaux, Le Tallec, Lions, Liou, Mantel, Neittanm¨ aki, Pekka, P´eriaux, Pouletty, Tezduyar, Tong, Tremolieres, Veysseyre, Zhang. We only list the ﬁrst and last one [355, 356]. Glowinski discusses special fully nonlinear problems with Dean and recently with Caﬀarelli and Guidoboni, Juarez, Pan [154,272–276,352].

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The last two papers use viscosity solutions. These allow other problems than our linearization, e.g. the Pucci equation. Numerical methods for nonlinear problems are intensively studied in China as well, cf. e.g. the proceedings edited by Chen, Chow, Glowinski, Kako, Li, P´eriaux, Shi, Tong, Ushijima, Widlund, Zhang. e.g. [586, 642]. Diﬀerent methods are discussed, e.g. in Temam, [621–624], Feistauer, Felcman and Straˇskraba, [315]. [621–624] considers for the Navi´er–Stokes problem, ﬁnite diﬀerence and ﬁnite element methods. [315] discuss, for compressible ﬂow problems, in addition ﬁnite volume and DCGMs, with proofs in Novotn´ y, Straˇskraba, [515]. Finite diﬀerence and ﬁnite element methods for linear problems are discussed by Hackbusch [387]. Diﬀerent strategies are presented for proving convergence, e.g. Mmatrix techniques for ﬁnite diﬀerences versus the standard bilinear forms for FEMs. Grossmann et al. [374–376] mainly study linear problems, approximated by diﬀerence, FE and boundary integral methods, cf. Sloan, S¨ uli and Vandewalle [592]. Convergence results for nonlinear boundary operators seem to be nearly totally missing. In the earlier approaches, studying general convergence theories, e.g. as in [28– 30, 32, 371, 441, 528–532, 547, 596, 675–678], spectral or FEMs with variational crimes, DCGMs, wavelet and mesh-free methods are necessarily missing. Sometimes ﬁnite diﬀerence and FEMs are studied via outer and inner approximation methods, e.g. in [528–532, 678]. Conforming FEMs are considered as inner approximation schemes, and classical diﬀerence methods as external approximation schemes, cf. Temam, Vainikko, Schumann and Zeidler [574, 622, 645, 647, 678]. More complicated methods, e.g. nonconforming FEMs, and systems and fully nonlinear equations are not included. Here we employ a new theory, combining generalized Petrov–Galerkin methods and one of the classical theories, here that by Stetter [596]. This yields, in Chapter 3, the necessary existence, stability, convergence and computing results for all these problems and discretization methods in a uniﬁed way. In particular, it avoids distinguishing inner and external approximation schemes. The standard “consistency and stability imply convergence” result is proved. Chapter 3, with its general discretization theory, strongly depends upon linearization and requires discussing its limitations and advantages. As previously mentioned, linearization for quasilinear problems, deﬁned on Sobolev spaces W m,p (Ω), is limited to 1 ≤ p < ∞. We obtain stability and convergence results for 1 ≤ p ≤ 2 discrete W m,p (Ω) norms, but for 2 ≤ p < ∞ only with respect to discrete H m (Ω) norms. This is complemented by the monotone convergence approach mentioned above. Based upon the previous analytical existence, uniqueness, and regularity results, obtained by any other method, linearization allows proving, in a uniﬁed way and for a major part of these problems, the desired stability, convergence, and Fredholm results. In fact, the stability of the nonlinear system is a consequence of the stability of the linearized discrete equation. For general fully nonlinear problems this seems, until now, the only possible way at all. Finally, we have already mentioned the appropriate combination of Newton and continuation techniques in the MIP. Most problems considered in this book can be studied in their weak form. Then the spaces of ansatz functions, including the exact solution, and of test functions are identical. For the exceptional nondivergent quasilinear and the fully nonlinear

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problems this is no longer correct, cf. Subsection 2.5.7 and Sections 5.2 and 5.3. So Chapter 3 is formulated to include these weak cases. Additionally, we assume that the derivative of the nonlinear operator, evaluated in the exact (isolated) solution, is boundedly invertible. We do have to defend this condition: The arguments are all closely related to the numerically necessary condition of a (locally) well conditioned problem. This is often guaranteed by monotonicity arguments, cf. Subsections 2.5.5 and 2.5.6. For an existing locally unique solution, Frechet diﬀerentiability is satisﬁed for many practically relevant problems. Exceptions might be unilateral problems, which seem to have been studied until now mainly for linear problems, see e.g. Haslinger, Hlav´ acek and Neˇcas [393]. As a consequence of the Taylor formula and the Fredholm alternative for these operators, the bounded invertibility of the derivative is usually, but not necessarily, satisﬁed as well. Otherwise a nontrivial kernel indicates bifurcation, see below, or the second derivative has to be positive or negative deﬁnite, or other combinations with higher derivatives have to be satisﬁed to ﬁt the above monotonicity arguments. Thus there will be (exceptional) operators with locally unique solutions violating the condition of bounded invertibility. For our numerical approaches we are luckily able to monitor this danger. If this exceptional case is detected, the convergence for the following numerical methods simply is not proved. Furthermore, except for monotonicity and iteration techniques, cf. Zeidler [678], Section 26.4, or small dimensional systems, there are not too many numerical methods for solving the nonlinear discrete equation without requiring a boundedly invertible derivative in one form or the other. The latter, in particular, has to be correct for linear solvers and the Newton method for the discrete problem. Unless the derivative is nearly singular, the corresponding discrete linear equation is stable, hence boundedly invertible. This fact is often monitored automatically. There is another important reason for assuming the bounded invertibility of the derivative, strongly related to the Fredholm alternative for the linearized elliptic operator. If it were violated, small perturbations of the original problem, e.g. corresponding to round-oﬀ errors, could be embedded into a parameter-dependent problem. This would usually have a bifurcation point or another singularity. The standard numerical methods would have to be replaced by numerical methods for bifurcation, see e.g. B¨ohmer [113–115, 120] Caloz and Rappaz [156], and Cliﬀe, Spence and Tavener [180]. Conversely, the bounded invertibility of the linearized original operator as a consequence of the stability of the linearized discrete operator is proved in Section 3.6. This last result is particularly interesting: For the more complicated elliptic problems the available analytical results become more and more lucanary. Then continuation techniques can be combined with the results in Section 3.6, relating the stability of the linearized Ah with the invertibility of A, and with methods similar to the classical existence results, obtained e.g. by Courant, Friedrichs and Lewy [213]. This allows research trips into analytically unknown areas, e.g. systems of the fully nonlinear Monge–Amp`ere equations. All these results apply to the above wide class of nonlinear problems and the chosen important types of space discretization methods. So there are enough good and numerically necessary reasons for assuming a boundedly invertible derivative.

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There is a totally diﬀerent other reason for this assumption in several discretization methods, see above. For example, for boundary integral methods a boundary integral formulation of nonlinear problems seems to be diﬃcult. For wavelet and mesh-free methods stability and convergence results for nonlinear problems have not been available. So the diﬀerent Newton–Kantorowich and continuation techniques are applied to the analytical problem. For these originating linear problems the above-mentioned discretization methods yield convergence. However, most scientists in the numerical community seem to prefer a discretization of the nonlinear problem compared to this linear detour. For wavelet and mesh-free methods we succeeded in proving convergence for a wide range of nonlinear problems in Chapter 9 and in [120]. We want to return to stability and convergence of these discretizations. Stability is reduced in Chapter 3 to a few well-known basic concepts from functional analysis and approximation theory. We combine coercive bilinear forms or monotone operators, their compact perturbations, bordered systems (for bifurcation and center manifolds), exact and approximate projectors with interpolation, best approximation and inverse estimates for approximating spaces. This allows a uniﬁed proof for stability for the above discretization methods and operator equations. Stability for the linear operator suﬃces. Its principal part is essentially coercive, hence its discretization is stable. The stability of the linear operator Ah itself, with A, a compact perturbation of its principal part, turns out to be a consequence of the bounded invertibility of A. Examples are the von K´ arm´an or the Navier–Stokes equations. Sections 2.7 and 2.8 show that our elliptic linear and nonlinear equations and systems do satisfy this condition. Consistency is simple for conforming FE, wavelet and spectral methods. For the nonconforming cases we have to employ appropriate tricks. Often the relation between the weak and strong form is essential. The necessary conditions and results, including the mesh independence principle, are formulated. Following this rather extended discussion of stability and convergence for discretization methods and their diﬀerences compared to other approaches, we present short summaries of the results for our diﬀerent methods in the following chapters. The corresponding introductions formulate extended summaries. It is certainly justiﬁed considering the nowadays most important FEMs more extensively than the others. In Chapter 4 we start the discussion with conforming FEMs. Thus the ansatz and test functions satisfy the appropriate regularity and boundary conditions. We start by summarizing the well-known approximation theory of FEs. Linear to quasilinear problems allow strong and weak formulations. The original and discrete forms have to be compared. In fact, the weak, simultaneously variational forms are directly used for these problems to yield the conforming FEMs. Based upon Chapter 3 the proof of consistency, stability and convergence is pretty simple. Consistency is nearly obvious. Stability is proved by compactly perturbing the coercive principal parts, or for the Navier–Stokes problem, the Stokes operator. We do not study super convergence results, cf. Bank, Xu and Griewank, Reddien and Jiang, Shi, Xue, [66, 368, 417]. However, the strong requirements for conforming FEMs with respect to ansatz and test functions, exactly evaluated test integrals, and restrictions to weak forms is a severe drawback. This is solved by considering nonconforming FEMs in Chapter 5.

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Only to these nonconforming FEMs do we apply and prove convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. Weak forms are no longer possible for fully nonlinear problems. So speciﬁc techniques are applied. They relate the strong form of the fully nonlinear system to the weak form of the linearized equations with stability properties. We present, with O. Davydov, special diﬀerentiable FEs, needed for these problems, cf. Subsection 4.2.6. So new FEMs are formulated for general, nondivergent quasilinear and fully nonlinear equations and systems, see B¨ohmer [118, 119]. In particular for nonlinear problems and nonconforming FEMs, the relation between classical and variational consistency has to be discussed. This technique can even be used for proving convergence for nonlinear boundary operators. The necessary quadrature approximations for all FEMs are described. Variational crimes for FEs violating regularity and boundary conditions are studied in Section 5.5 in R2 for linear and quasilinear problems. This allows high order conforming and nonconforming versions, particularly important for bifurcation, and yields several new results. An essential tool is the anticrime transformation. As a consequence of the dominant role of FEMs, the presentation of numerical solutions of ﬁve classes of problems is restricted to FEMs: the variational methods for eigenvalue problems are presented in Section 4.7. The convergence theory for monotone operators as applied to quasilinear problems is described in Section 4.5. Section 5.2 is devoted to FEMs for fully nonlinear elliptic problems. In Section 5.3 we study FEMs for nonlinear boundary conditions. Convergence results for these problems are hardly, if at all, discussed in the literature. Quadrature approximate FEMs are the topic in Section 5.4. We thus close several gaps in the literature. The reason for this restriction to FEMs is very simple. These stability, convergence and computing results are valid essentially for all the other methods as well. In fact, the proofs for all these results remain correct for all the other methods. This could be shown by only slightly and obviously modifying our uniﬁed approach on the basis of the speciﬁc consistency and stability results available for each method. This claim has to be slightly restricted for DCGMs and wavelet methods. Our DCGM approach is available only for problems of order 2, and is not possible for fully nonlinear elliptic problems. Wavelet methods are limited to semilinear problems. In the meantime, for many of the diﬃcult problems in applications, e.g. the Navier– Stokes problem in ﬂuid dynamics, or in elastomechanics and electromagnetism, mixed FEMs are combined with nonconforming FEs, cf. Braess [135, 136] and Brezzi and Fortin [144]. Totally new potentials seem to develop within the mesh-free methods, cf. [120]. In Chapter 6, W. D¨ orﬂer describes the idea of a posteriori error estimation and adaptive mesh reﬁnement. The main points are the description of the adaptive ﬁnite element algorithm and its convergence and the sketch of the optimal complexity of this algorithm with respect to an approximation class of solutions. The techniques in this chapter, presented essentially for linear elliptic problems, can be combined with those for nonlinear problems in the previous chapters. As indicated in the last subsection, this might allow generalizing many results to nonlinear problems. Interesting results for nonlinear problems are due to Verf¨ uhrt [657].

xxiv Preface Discontinuous Galerkin methods (DCGMs) with V. Dolejˇs´ı in Chapter 7 are another type of nonconforming FEMs. For avoiding rather highly technical conditions and proofs we restricted DCGMs to quasilinear equations and systems of order 2. DCGMs violate boundary conditions and continuity of the piecewise polynomials. These violations are compensated by additional penalty functions in the discrete weak form. The above technique in FEMs applied to fully nonlinear problems seems to be impossible for DCGMs. In Chapter 8, diﬀerence methods are treated, as the other methods, by applying our new approach to the diﬀerence equations in weak form. This is diﬀerent from most earlier approaches, either based upon the strong forms or studied as external approximation schemes. There weak forms have been considered in many papers and books, cf. [574, 622, 647, 678], and in lectures, cf. Bank [62]. Our results again are formulated for linear to quasilinear equations and systems of orders 2m, m ≥ 1. In contrast to the earlier M -matrix approach, the discretization errors are here estimated with respect to a discrete Sobolev norm of order m for a problem of order 2m, similar to our FEMs results. This allows eﬃcient defect correction methods as an appropriate strategy for formulating high order methods as in B¨ ohmer [105–109] and Hackbusch [384]. Particularly interesting are symmetric forms, again for equations and systems of order 2m, m ≥ 1. For cuboidal domains with faces parallel to the axes the boundary condition are exactly reproduced. This implies convergence of order 2 and 2k for k defect corrections, compared to order 1 and k for the unsymmetric forms. Only partially do we prove the results for the diﬀerence methods for two reasons. The techniques are very similar to FEMs, including the case of violated boundary conditions in Sections 8.2–8.6. The special methods for curved boundaries in Section 8.7 are known only for special cases and require highly technical proofs based upon totally diﬀerent techniques. They do not seem to be applied too much these days. We ﬁnally apply, with C.S. Chien, diﬀerence, combined with extrapolation methods, to the von K´ arm´an equation. Wavelet methods are applied in Chapter 9 with S. Dahlke for the ﬁrst time in this generality and appropriate for nonlinear problems and bifurcation. This presents an obvious extension of the ﬁrst papers on wavelets for nonlinear elliptic problems, B¨ohmer and Dahlke [121, 122]. This contrasts with the recent books on wavelets, combined with elliptic PDEs, by Cohen [195] and Urban [640, 641]. The most farreaching results for linear problems until now were obtained for self-adjoint and saddle point problems requiring a nonsingular linear operator A. For these cases proofs for convergence are available, directly employing the special properties of wavelets. As a consequence of the diﬃculties with evaluating nonlinear functions and operators with wavelet arguments, cf. Subsection 9.3.5, general quasilinear and fully nonlinear problems have to be excluded. With this exception, the whole spectrum of corresponding wavelet methods is shown to be stable and convergent. Again the corresponding linearized operator has to be boundedly invertible. T. Raasch ﬁnishes this chapter with adaptive wavelet methods and presents some numerical results. Spectral methods in B¨ ohmer [120] inherit, for correct choices of approximating spaces, even Γ-equivariance for inﬁnite symmetry groups, e.g. O(3) × D4 , from the

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original to its discrete bifurcation problem. This is the key condition for reproducing bifurcation scenarios by discretization. Mesh-free methods have been successfully applied to nonlinear problems. Convergence proofs have been totally missing for more than 20 years. So our linearization approach again shows its power by proving, only very recently, the missing convergence for these problems and the standard mesh-free methods in [120]. Due to the rather free placement of their “centers”, they are very appropriate for problems with symmetries as well. So we delay the presentation of spectral and mesh-free methods to [120]. We discuss dicretization methods for problems in Rn , n ≥ 2. The analysis for FEMs works well in Rn . The methods in Section 5.5 seem to be the only type of FEMs, analytically restricted to R2 , losing accuracy in possible generalizations to R3 . However, there is a severe restriction of FEMs on triangulations essentially to Rn , n ≤ 3, caused by the tremendously increasing complexity of the software, in particular for adaptive FEMs. This situations is maintained for DCGMs. FEMs and diﬀerence methods on cuboidal subdivisions allow general Rn , but do not admit adaptivity. The wavelet methods in Chapter 9 have not yet reached the level of FEMs and DCGMs. For the spectral methods in [120] we have excluded domain decomposition techniques, usually not necessary for equivariant problems. So all these methods in our presentation allow convergence in Rn , but the problem of highly technical software is not yet solved. The situation is totally diﬀerent for the mesh-free methods in [120]. The analysis is possible for Rn , n ≥ 1, but the approximation quality strongly increases with increasing n. The software is nearly independent of n. So for many cases mesh-free methods might be the future methods of choice for n ≥ 3. In [120] we give a proof for convergence of these methods. This is an approach totally diﬀerent from Chapter 3 in this book. For the ﬁrst time it shows convergence of meshfree methods applied to nonlinear problems. This can be applied to the other methods as well and allows modiﬁcations and adaptivity for some of them. Summarizing, we obtain stability and convergence for the diﬀerent types of space discretization methods and all the elliptic linear and nonlinear equations and systems of orders 2 and 2m. This synopsis allows the extensions and generalizations described in the previous paragraphs, partially much beyond the present state of the art. Thus it ﬁlls a gap in the available literature. This book is aimed at three groups of people. Firstly, graduate students and scientists who want to study and to numerically analyze nonlinear elliptic problems in mathematics, science and engineering. They will ﬁnd the necessary tools and convergence results. This simultaneously serves as a solid foundation for many in the huge number of papers in scientiﬁc computing in this area. Finally, the bounded invertibility of the linearized original operator as a consequence of the stability of discrete operator in Section 3.6 can be combined with methods similar to the classical existence results as in Courant, Friedrichs and Lewy [213], for indicating new existence results. All these methods apply to the above wide class of nonlinear and even fully nonlinear problems and the important types of space discretization methods. In particular, many of the existence, uniqueness and regularity results are only known for the case H0m (Ω). Generalizations to the W0m,p (Ω, Rq ) situation would be worthwhile. In fact, results as in Theorem 2.122, form the basis for numerical methods for the

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W0m,p (Ω, Rq ) setting, opening the possibility for research into new problems outside the previous scope. Secondly, we have previously indicated, and update that later on, several gaps in our numerical results. These might stimulate future research. For example, FEMs violating continuity and boundary conditions and DCGMs are restricted here to order 2, these FEMs even to R2 . Possible tools might be complementary boundary conditions and higher order cubature formulas. Asymptotic expansions for the discretization error and defect correction methods on this basis can probably be generalized with our techniques to FE and the other methods as well. Extensions of our results to other methods, e.g. the excluded ﬁnite volume methods, might be worthwhile. The complicated problems in elastomechanics and electromagnetism, approximated by mixed FEMs combined with nonconforming FEs, might ﬁnd some new tools in combination with adaptive FEMs. These adaptive FEMs are presented here essentially for the Laplacian with some hints for generalizations. So there are still interesting areas for further research. Thirdly, material for many diﬀerent graduate courses advanced seminars or research projects can be chosen. The starting point is the motivation for nonlinear problems in Chapter 1. The next step would lead to the existence, uniqueness and regularity results for the desired class of elliptic problems in Chapter 2. The next Chapter 3 would be mandatory for all elliptic problems and their numerical methods. Then e.g. a DCGM for a quasilinear operator, F , is chosen. If the Frechet derivative, F , of this operator is coercive, then stability and convergence of the DCGM is completely proved in Chapter 7. For a more general, only boundedly invertible F , the proof for stability for the nonconforming FEM has to be updated to the DCGM. This straight forward job is left as a (not quite trivial) task to the interested reader. Or the new numerical methods for fully nonlinear problems allow diﬀerent new research projects. In geometry a surface with a rather general prescribed Gauss curvature, closely related to the Monge–Amp`ere equation, could be numerically solved for the ﬁrst time even in higher dimensions. It would be desirable to extend the deep results for Monge–Amp`ere equations to systems. Again our numerical methods for fully nonlinear problems can be combined with continuation methods indicating new possibilities. These examples show the wide range of applications from pure mathematics to science and engineering. So the author hopes that this book will stimulate further research, e.g., elliptic problems on manifolds. Concentrating my work for decades on numerical methods for general defect corrections and bifurcation of PDEs, and only recently on FEMs, certainly some important results for the diﬀerent discretization methods will have escaped me. In a huge project of ca. 1500 pages in both books, I will have missed a lot of misprints and hopefully only a few minor errors. So I will gratefully accept critics, suggestions, or remarks. Acknowledgements: I am very grateful to many people who have strongly supported and inﬂuenced my books or encouraged me in their genesis. First of all my co-authors of this volume, C.S. Chien, S. Dahlke, O, Davydov, W. D¨ orﬂer, V. Dolejˇs´ı, and T. Raasch have done a great job. It was always fascinating how initial ideas developed to the ﬁnal forms. Then my good friends and cooperators

Preface

xxvii

E. Allgower and Z. Mei, cf. the Bibliogrphy, were particularly inﬂuential for many results in this and the next book. The co-authors of [120], E. Allgower, P. Beltram, P. Chossat, A. Cliﬀe, G. Dangelmayr, C. Geiger, K. Georg(†) , N. Jangle, D. Janovska, V. Janovski, Z. Mei, R. Schaback, K.-H. Schild, B. Schmitt, J. Tausch and my other friends and colleagues P. Antonietti, M. Buhmann, E. Doedel, P. DuChateau, B. Fiedler, J. Gwinner, S. Feistauer, W. Hackbusch, R. OMalley, K. Nickel(†) , L. Schumaker, Z.-C. Shi, A. Spence, W. Wendland, G. Wittum, E. Zeidler, stimulated the progress with many discussions. Last but not least most of my Ph.D. students, J. Boˇsek, U. Garbotz, N. Sassmannshausen, A. Schwarzer, R. Sebastian, and additional guests in Marburg, P. Ashwin, W. Govaerts, A. Hoy, and S. F. McCormick provided essential results. The Deutsche Forsungsgemeinschaft (DFG), and the Deutscher Akademischer Austauschdienst (DAAD), provided the necessary additional ﬁnancial support. The Philipps University Marburg oﬀered a stimulating atmosphere for teaching and research. Essential parts of this book had been used in year long lectures. Without the help of my many masters and Ph.D. students in particular [120] would have been impossible. N. Sassmannshausen contributed good ideas to Chapter 3. Several colleagues in our Department, W. Gromes, V. Mammitzsch, C. Portenier, G. Schumacher, H. Upmeier, were always ready to answer my many questions. Most important was the support by our group numerical mathematics, mainly by S. Dahlke, J. Kappei, T. Raasch and B. Schmitt. F. Muth has done a great job. She has typed approximately half of the early forms of Chapters 1, 2, 4, 5. Very supporting was Oxford University Press’ highly professional, very constructive, kind and personal advice by S. Adlung, E. S¨ uli, A. Warman, M. Johnstone, P. Hendry and many others. Finally, the love, caring, and encouragement of my late wife through all the many years with successes and intrigues is the basis for this book. Marburg May 2010 Klaus B¨ohmer

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Part I Analytical Results

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1 From linear to nonlinear equations, fundamental results 1.1

Introduction

Why do linear models describe nonlinear reality sometimes correctly and sometimes totally insuﬃciently? To get a feeling and foster the insight for this diﬀerence the ﬁrst two examples are chosen as very simple mechanical problems, deduced by physical reasoning. In this chapter we do not intend to give a rigorous analysis of the problems, but rather motivate, by the following examples, the rigorous analysis in later chapters. So we do not care at the moment to present the mathematical equations in the appropriate function spaces, nor do we refer much to physics, chemistry or biology. Usually varying parameters play a dominant role. The ﬁrst examples even indicate bifurcation eﬀects in diﬀerential equations. These phenomena are the goal of B¨ohmer [120]. We end this chapter by summarizing the necessary basic results for this book.

1.2

Linear versus nonlinear models

To answer the introductory question we show why the following linear model cannot describe the situation correctly. For more explicit mechanical arguments, see, e.g. Troger and Steindl [637]. Example 1.1. Bending rod with perpendicular load Let a vertically positioned rod be clamped at the lower end and be free at the upper end, see Figure 1.1. A load P at the end of the rod, originally perpendicular to the rod, forces it to bend sideways. Its displacement x is proportional to P , for small P , i.e. the following linear relationship holds: x = cP

for an appropriate c.

(1.1)

However, in a simple experiment we observe that the rod will deform nonlinearly or even break when the load P passes a certain critical value. In other words, for large P , the linear law (1.1) does not represent the “reality” appropriately. So we have to incorporate the “nonlinear” eﬀects of deformation. We want to derive a nonlinear mathematical/physical model for the stationary position of the bent rod. There is a strong analogy between the relatively unknown strain energy of a rod and

4 1. From linear to nonlinear equations, fundamental results P=0

P>0

j (s)

Figure 1.1 Rod with load perpendicular to the rod.

the well-known kinetic energy E of a moved body. This is described by the formula E = m/2 (du(t)/dt)2 with the mass m, the velocity, v = du(t)/dt, the distance traversed, u(t), and the time, t. Analogously, we obtain for the (local) strain energy, U, at a point with arc length s on the deformed rod the rule 2 α dϕ (s) . U= 2 ds Here α is the bending stiﬀness and ϕ(s) represents the angle between the vertical axis and the tangent in the point with arc length s (see Figure 1.1). The total bending energy, Ubend , of the deformed rod is given by 2 α L dϕ ds for ϕ ∈ C 1 [0, L], Ubend = 2 0 ds with the total length of the rod, L. The potential energy, due to the moving of the tip of the rod in the direction P , is given by −P d(L), where d(L) indicates the displacement at the end of the rod. Figure 1.1 shows that L sin ϕ(s)ds. d(L) = 0

Therefore, the total energy, i.e. the so-called potential, is the sum of these two terms, see Troger and Steindl [637]: V (ϕ) :=

α 2

0

L

dϕ ds

2

ds − P

L

sin ϕ ds

for

ϕ ∈ C 1 [0, L].

(1.2)

0

One of the fundamental principles of mechanics states that a position of equilibrium of a mechanical system, here the equilibrium of the rod bent by ϕ, is characterized by a minimum of its potential, here V , i.e. for every small function ψ we have necessarily V (ϕ + ψ) ≥ V (ϕ).

(1.3)

1.2. Linear versus nonlinear models

5

We claim that this minimizing function ϕ is characterized by a nonlinear boundary value problem with the diﬀerential equation G(ϕ, λ) :=

d2 ϕ + λ cos ϕ = 0, ds2

for λ := P/α,

and the boundary conditions, indicated by b in the corresponding subspace, dϕ (L) = 0 . Cb2 [0, L] := ϕ ∈ C 2 [0, L]; ϕ(0) = ds

(1.4)

(1.5)

This represents a nonlinear model for the equilibrium or stationary position of the bent rod. Now we ﬁnd, by the standard linearization of (1.2), for small ψ and sin ψ = ψ + O(ψ2 ), compare Deﬁnition 1.41 and Theorem 1.42, L L dϕ dψ ds − P 0 ≤ V (ϕ + ψ) − V (ϕ) = α ψ cos ϕds + O(ψ2 ). ds ds 0 0 Here we use the standard notation g(ψ) = O(ψα ) is equivalent to g(ψ)/ψ

α

(1.6)

remains bounded for ψ → 0, and

g(ψ) = o(ψ) ⇐⇒ lim = 0, ψ→0

with ψ indicating any norm of ψ. By partial integration we obtain L 2 d ϕ dϕ L ψ α 2 + P cos ϕ ds + O(ψ2 ). 0 ≤ V (ϕ + ψ) − V (ϕ) = α ψ − ds 0 ds 0 (1.7) For the situation in Figure 1.1, ϕ has to satisfy the boundary conditions, see (1.5), ϕ(0) =

dϕ (L) = 0, ds

(1.8)

since the rod at the endpoint is not bent. The ﬁrst term in (1.7) vanishes if we require, according to (1.8), ψ(0) = (dϕ/ds)(L) = 0. This has to be satisﬁed for every small perturbation, ψ, for a rod, ϕ + ψ, still clamped at s = 0. Thus L 2 d ϕ ψ α 2 + P cos ϕ ds + O(ψ2 ). (1.9) 0 ≤ V (ϕ + ψ) − V (ϕ) = − ds 0 We started with ϕ, ψ ∈ C 2 [0, L], but ﬁnd ψ ∈ C[0, L] with ψ(0) = 0 suﬃcient for (1.9). These changes in diﬀerentiability will play an increasingly important role for more complicated existence results, see e.g. Example 1.7 below. We have split the increment V (ϕ + ψ) − V (ϕ) into a linear operator (here a functional), acting upon ψ, and a higher order remainder term O(ψ2 ), sometimes denoted as h.o.t. We have to show that the minimizing solution ϕ of (1.3) indeed solves the nonlinear diﬀerential equation (1.4). Using the standard argument of the fundamental lemma of

6 1. From linear to nonlinear equations, fundamental results the calculus of variations, we assume, contrary to (1.4), that d2 ϕ + λ cos ϕ ≡ 0, ds2

for

ϕ ∈ C 2 [0, L] and λ := P/α.

Choosing dψ0 := −(d2 ϕ/ds2 + λ cos ϕ) ∈ C 0 [0, L] leads to L 2 L d ϕ ψ0 α 2 + P cos ϕ ds = α (ψ0 )2 ds > 0. c0 := − ds 0 0 The remainder term in (1.9) satisﬁes O(μψ0 2 ) = μ2 c1 ,

with c1 := O(ψ0 2 ).

Hence, for small enough μ we have |μc1 | < c0 /2, hence c0 + μc1 > c0 /2, and V (ϕ + μψ0 ) − V (ϕ) > μc0 /2 for μ > 0, V (ϕ + μψ0 ) − V (ϕ) < μc0 /2 for μ < 0. This contradicts the fact that ϕ is a minimum for V , thus showing (1.4), indeed a nonlinear model for the stationary position of the bent rod. To study the relationship of (1.4) and (1.5) to (1.1), we start the discussion with the case of small angles ϕ. From cos ϕ ≈ 1 and (1.4) we obtain d2 ϕ d2 ϕ + λ cos ϕ ≈ 2 + λ = 0. 2 ds ds Hence ϕ ≈ −λs2 /2 + μs + ν and, by (1.5), we ﬁnd ν = 0, μ = Lλ = LP/α, and 0 ≤ ϕ(s) = λ(−s2 /2 + Ls) ≤ ϕ(L) = P

L2 1 for small P. 2α

This implies in particular, that a nontrivial solution exists for every λ = 0. Similarly we may use sin ϕ ≈ ϕ and ﬁnd L 3 L L s2 s L3 ; sin ϕ(s)ds ≈ ϕ(s)ds = λ − + L = P d(L) = 6 2 0 3α 0 0

(1.10)

thus we have reproduced the form (1.1). When we proceed to larger values of P , and hence ϕ, the solution of the boundary value problem (1.4), (1.5) requires the use of numerical methods or elliptic functions. Physical arguments indicate that the rod will break at a distance s, where, for increasing P , for the ﬁrst time, the tension σ = σ(s) on the surface of the rod becomes larger than the critical tension σcrit of the material used. Furthermore, it is well known that the curvature κ = dϕ/ds is proportional to σ = cdϕ/ds. If we assume |ϕ(s)| < π/2, we derive from (1.4) and (1.5) that d2 ϕ (s) = −λ cos ϕ < 0. ds2

1.2. Linear versus nonlinear models

7

Hence dϕ dϕ dϕ (0) > (s) > (L) = 0 for 0 < s < L. ds ds ds So the rod will break for s = 0 if P is so large that cdϕ/ds(0) > σcrit . Certainly the linear relation (1.1) can no longer describe this situation correctly. In particular, the linear model (1.1) is not appropriate, since we have assumed at diﬀerent stages that either ϕ or ψ is small. Only in this case does neglecting the h.o.t.s makes sense, otherwise they will strongly inﬂuence the solutions for a nonlinear model. In Example 1.1, showing the need for nonlinear models, the solutions for the linear and nonlinear problem were pretty similar for small values of P . The next Example 1.2 deals with a qualitative diﬀerence in the behavior of solutions of linear and nonlinear problems. Example 1.2. Rod with axial load Again, we consider deformations of a thin inextensible rod of length L. This time, we study an axial load P . The rod is pinned at one end (for a horizontally positioned rod this is denoted as a simply supported end); the other end moves freely in the vertical direction. So, the rod tends to deform in a plane. We perform the experiment indicated in Figure 1.2. If we slowly increase P , starting with P = 0, the rod stays unchanged. For a further increase of P beyond a critical P0 , deviations of the rod from the vertical position are observable. These deviations increase in size with P. For modeling the new situation we can use the same bending energy as in Example 1.1. The total length equals the total arc length, s = L , of the rod. The potential energy is again given as −P d(L) with the displacement at the upper end of the rod, d(L), here in the vertical direction. Figure 1.2 shows that, instead of the original L, P

P

j (s)

Figure 1.2 Rod with axial load.

8 1. From linear to nonlinear equations, fundamental results contrasting to (1.10), the upper end has come down by L (1 − cos ϕ)ds. d(L) = 0

With this d(L) we obtain the total energy V (ϕ) and the diﬀerence V (ϕ + ψ) − V (ϕ), see also (1.2), as 2 L dϕ (s) ds/2 − P (1 − cos ϕ)ds for ϕ ∈ C 2 [0, L], (1.11) ds 0 0 L 2 dϕ L d ϕ V (ϕ + ψ) − V (ϕ) = α ψ ψ α 2 + P sin ϕ ds + O(ψ2 ). − ds 0 ds 0

L

V (ϕ) := α

An analogous argument as in Example 1.1 shows that the bent rod in the stationary position necessarily minimizes V (ϕ) in (1.11). The ﬁrst term in V (ϕ + ψ) − V (ϕ) ≥ 0 vanishes, typically, for the following boundary conditions. dϕ dϕ (0) = (L) = 0. (1.12) ds ds Equation (1.12) represent the so-called natural or Neumann boundary conditions as in Figure 1.2: the rod is not bent at the endpoints, but the lower endpoint is ﬁxed, and the upper moves along the vertical axis. Imposing (1.12), the minimized V (ϕ) in (1.11) yields L 2 d ϕ ψ α 2 + P sin ϕ ds + O(ψ2 ). 0 ≤ V (ϕ + ψ) − V (ϕ) = − ds 0 The minimizing function is characterized by a now diﬀerent boundary value problem dϕ dϕ d2 ϕ (0) = (L) = 0. (1.13) + λ sin ϕ = 0 with λ := P/α, and ds2 ds ds Obviously, ϕ ≡ 0 represents a trivial solution for (1.13), independent of λ, L. This G(ϕ, λ) = 0 in (1.13) is a nonlinear equation for ϕ. The question of whether (1.13) has a (unique) solution or not has to be answered by mathematical results. For the case of elliptic partial diﬀerential equations we will present those results in Chapter 2. If a (unique) solution ϕ of (1.13) exists, similarly for (1.4), (1.5) in Example 1.1, then an analogous argument shows that ϕ minimizes V (ϕ) and thus solves our problem. G(ϕ, λ) :=

These two examples show totally diﬀerent behavior, as a consequence of the diﬀerent nonlinearities in the diﬀerential equations, diﬀerent boundary conditions, and loads. In Example 1.1 a solution ϕ ≡ 0 is possible only for λ = 0 = P . Already, a very small P causes displacements d(L) > 0. The experiments in Example 1.2 show that up to a certain P = P0 we only have the trivial solution ϕ(s) ≡ 0. If P increases beyond P0 , the rod starts to buckle in the form of a, at the beginning, sinusoidal function. If we were to ﬁx the rod at its middle point, a much higher P would be necessary to deform the rod. So why and when are nontrivial solutions generated? To explain these observations, we again study the nontrivial solutions for (1.13), originating from

1.2. Linear versus nonlinear models

9

the trivial state ϕ(s) ≡ 0. We start with a very small function ϕ and consider the approximation sin ϕ = ϕ − ϕ3 /3! + ϕ5 /5! + · · · = ϕ + h.o.t. The small nontrivial solutions ϕ of (1.13) have to satisfy the linearized problem d2 ϕ + λϕ = 0, ds2

dϕ dϕ (0) = (L) = 0, λ = P/α, ds ds

(1.14)

see (1.4). √ This linear boundary eigenvalue problem has nontrivial solutions of the form ϕ = c cos λ s if and only if √ √ dϕ dϕ (L) = −c λ sin λL = 0 = (0), (1.15) ds ds √ hence if and only if λL = nπ, λ = (nπ/L)2 and ϕ(s) = c cos nπs/L. In this case the linearized operator has a nontrivial kernel or null space. This observation shows that only for certain discrete values of P = α(πn/L)2 may the linearized problem (1.14) have a nontrivial solution, originating in the trivial solution. This seems to be and in fact is a necessary condition for a nontrivial solution “bifurcating from the trivial solution” of the original problem, see Figure 1.3. We will study “bifurcation problems”, their local dynamical behavior, and their numerical approximation for PDEs in B¨ ohmer [120]. These very simple examples indicate two features, essential for further studies. For a nonlinear operator its linearized form provides essential information. If “bifurcation phenomena” of the original nonlinear problem are interesting the null space of its linearized operator plays a major role. This is one of the reasons why in the following chapters the problem of linearization and its corresponding Fredholm alternative, hence results about the null space of its linearized operator, are so important. As we indicated already in the Preface, linearization is an essential feature of our later convergence proofs for all the numerical methods studied in the two books. Nevertheless, there are diﬀerent opinions about combining linearization and numerical ± ⎟ ⎢j⎟ ⎢ L /2

λ

−L /2

Figure 1.3 Global structure of bifurcating solution branches.

10

1. From linear to nonlinear equations, fundamental results

methods. Some use linearizations for the original analytical problems. The corresponding numerical methods are only applied to linear problems. Hence a nonlinear convergence theory is unnecessary. The many papers on nonlinear convergence theory show that most mathematicians disagree with this idea. On the other hand, adaptive mixed FEMs, e.g. for mechanical problems, still seem to require the full nonlinear techniques.

1.3

Examples for nonlinear partial diﬀerential equations

We consider mathematical models for important problems in diﬀerent sciences. We ﬁnd a spectrum of parameter-dependent semi- to fully nonlinear partial elliptic diﬀerential equations. As usual in mathematics and physics, they are appropriately simpliﬁed models. We do not go into details, but only give the corresponding equations and the meaning of the occurring functions and usually omit the exact function spaces in this introduction. Most of these examples will be more intensively discussed in the later case studies in this book and in B¨ ohmer [120]. For many other interseting, however mostly linear applications, cf. Dautray and Lions [256, 257]. Example 1.3. Chladny sound ﬁgures Assume that we have a thin ﬂexible square plate, horizontally ﬁxed at its center [13–16]. On the surface of the plate we uniformly distribute tiny particles, for example, sand. Above the plate we ﬁx a source of sound waves with variable frequency related to the λ in (1.16). We want to examine the reaction of the thin plate for varying λ. The pressure of the sound waves acts as a load on the surface of the plate. In the case of resonance, a vibration with (unmoved) nodal lines is induced on the plate. There the tiny particles are collected. Thus, the distribution of these particles with respect to varying λ illustrates the vibration of the plate. Let Ω := [0, L] × [0, L] represent the plate and u = u(x, y) denote the maximal deviation of the vibration from the trivial ﬂat position of the plate at the point (x, y). Then u(x, y) satisﬁes the strongly simpliﬁed mathematical model equation, compare (1.13) and Theorem 2.55, G(u, λ) := Δu + λ sin u = 0 in Ω = [0, L] × [0, L] deﬁned on ∂u = 0 on ∂Ω . Cb2 (Ω) := u ∈ C 2 (Ω); ∂n

(1.16) (1.17)

Here Δ := ∂ 2 /∂x2 + ∂ 2 /∂y 2 is the Laplacian operator and ∂u/∂n the normal derivative in the outer direction of ∂Ω. For arbitrary λ the trivial ﬂat state u(x, y) ≡ 0 represents a solution of (1.16), (1.17). As in Example 1.2 we linearize Equation (1.16) with respect to u at (u, λ) = (0, λ), cf. Deﬁnition 1.41 and Theorem 1.42, and obtain, similarly to (1.14), (1.15), with the trivial solution v = 0 for arbitrary λ, ∂u G(0, λ)v = Δv + λv = 0 ⇐⇒ Δv = −λv

in

Ω for v ∈ Cb2 (Ω).

(1.18)

1.3. Examples for nonlinear partial diﬀerential equations

11

The −λ, allowing nontrivial solutions in the kernel of ∂u G(0, λ), are the (1.19) eigenvalues λ0 = −λ = −(m2 + n2 )π 2 /L2 of Δ for all m, n ∈ N0 s.t. n m (1.20) v(x, y) = vm,n (x, y) := cos π x cos πy, and v(x, y) = vn,m (x, y) are L L

m n m2 + n2 2 the eigenfunctions of Δ, hence, Δv = Δ cos π x cos π y = − π v. L L L2 Therefore, we obtain, e.g. for (m, n) = (1, 1), (m, n) = (2, 3) and (m, n) = (5, 5), (m, n) = (1, 7) the eigenvalues λ0 = −2π 2 /L2 , −13π 2 /L2 , and −50π 2 /L2 with one, two and three linearly independent solutions vm,n , vn,m . All their linear combinations solve (1.18) as well, for ﬁxed λ0 . Hence, these λ0 for the sound well are candidates for bifurcations to nontrivial u, and for the generation of various Chladny ﬁgures. Figure 1.4 shows v1,3 − v3,1 , one of the many possible linear combinations with speciﬁc symmetry properties. This Chladny problem is a typical example of multiple bifurcations in nonlinear problems with symmetries. Example 1.4. Buckling of a plate Consider a plate P under compression of the form Ω ∈ R2 . It is deﬁned by the Airy stress function w(x, y) of the plate P at the point (x, y) and the deviation or deﬂection u(x, y) for P from its ﬂat trivial state. They are modeled by the von K´ arm´an equations.

0

0.5

cos (x)*cos (3*y)-cos (3*x)*cos (y) x 1 1.5 2 2.5

3 3 2.5 2 1.5

y

1 0.5 0

Figure 1.4 Chladny sound ﬁgures on [0, π] × [0, π]. The straight (!) lines through the origin are nodal lines (ﬁxed, elevation 0); the curves show elevation lines. Notice the high symmetry, and neglect the wriggles near the origin.

12

1. From linear to nonlinear equations, fundamental results

A mathematical justiﬁcation for this model is given by Berger and Five [82–84], and, as the ﬁrst terms in an asymptotic expansion, by Ciarlet [175]: ⎞ ⎛ ⎞ ⎛ 2 Δ u −[u, w] f ⎠ = ⎝ ⎠ in Ω ⊂ R2 . Gs (u, w) := ⎝ (1.21) 0 Δ2 w + 12 [u, u] Here λ is the external load, Δ2 = ΔΔ is the two-dimensional biharmonic operator, Ω the shape of the plate, and the bracket operator [·, ·] is deﬁned by [u, v] = uxx vyy − 2uxy vxy + uyy vxx . This semilinear system will be discussed in Subsection 2.6.6, cf. (2.403) ﬀ. Bifurcation regimes should be avoided in realistic constructions. But the mode jumping is perhaps one of the most noteworthy features of experimental and mathematical studies about the buckling of plates. Equation (1.21) is taken up as a case study in Chapter 8. Example 1.5. Reaction–diﬀusion equations Semilinear systems of reaction–diﬀusion equations are typical models in nuclear reactor physics, and in chemical and biological reactions. In particular, many chemical reactions, ecological systems and mechanisms of pattern formation are described mathematically as systems of semilinear diﬀerential equations. Here we study stationary equations with ∂ u/∂t = 0 of the form ∂ u = G( u, α) = DΔ u + f ( u, α) = 0 in Ω × (0, T ). (1.22) ∂t In this process the domain Ω ⊂ Rn represents the area where the chemical reaction occurs or a biological/ecological system lives. u : Rn → Rq represents various (intermediate) components in a chemical reaction or species of a biological system. It is a smooth vector function representing the reaction eﬀect of the system. The vector α ∈ Rp deﬁnes the control parameters in the system. The components of D ∈ Rq×q , a diagonal matrix, describe the diﬀusion rate of each species and f : Rn × Rp → Rq their reactions. Appropriate boundary conditions are combined with (1.22) for each given situation. They can be any of Dirichlet, Neumann, or Robin type, or even in some dynamical forms. The interaction of diﬀusion and reaction, measured by D and f ( u, α), can yield rich and interesting phenomena. This is demonstrated in case studies in B¨ohmer [120]. An example used in many numerical studies is D = 1 = q, u = u, f (u, α) = αf (u), see Theorem 2.58. Example 1.6. Three quasilinear and fully nonlinear geometric problems The solutions of these problems are functions u : Ω → R with prescribed boundary conditions u = φ on ∂Ω. The minimal surface equation, see Example 2.72 and Gilbarg and Trudinger [346], p. 339, has the quasilinear form Gs u = Gu = (1 + |Du| )Δu + 2

n i,j=1

∂ i u∂ j u∂ i ∂ j u = 0 on Ω.

(1.23)

1.4. Fundamental results

13

The Monge–Amp`ere equation is a fully nonlinear problem. For general n and for n = 2, see Example 2.78 and [346], pp. 441, 471, it has the form 0 = G(u) := det D2 u − f (x, u, Du) = uxx uyy − u2xy − f (x, u, Du) for n = 2. (1.24) These G in (1.24) and in (1.25) are uniformly elliptic in an u1 ∈ C 2 (Ω) if D2 u1 is strictly positive deﬁnite and symmetric in Ω. Equation for a surface with prescribed Gauss curvature: now let K = K(x) at x ∈ Ω be the prescribed Gauss curvature. Then (1.24) has the special form, see Example 2.78 and [346], p. 442, G(u) := det D2 u − K(1 + |Du|2 )(n+2)/2 = 0, K(x) > 0.

(1.25)

Example 1.7. Navier–Stokes equations One of the most stimulating equations for research in mathematics and physics is the Navier–Stokes equation. It models, e.g. the ﬂow of water around a moving vessel, and the air around the wing of an airplane. In this context (1.26) with small values of 0 < ν are interesting. Then boundary layers and turbulence develop, modeling the up-wind properties. Mathematically, it was the ﬁrst extremely important system of nonlinear diﬀerential equations. It required an update of the deﬁnition of elliptic or parabolic equations, and for small 0 < ν the boundaries between parabolic and hyperbolic problems are no longer well deﬁned. Furthermore, by varying 0 < ν and other parameters, extremely interesting regimes of bifurcation and the corresponding dynamics and chaotic phenomena originate, cf. [120]. Still, for a lot of these problems satisfactory answers are missing. ⎛ ⎞ n ∂ u −νΔ u + u ∂ u + ∇ p f i i ⎝ ⎠ = G( u, p) : = = in Ω i=1 0 ∂t div u

pdx = 0,

u = 0 on Γ = ∂Ω.

(1.26)

Ω

Here u, p and f denote velocity, pressure and forcing term of an incompressible medium, respectively, u = (u1 , . . . , un )T , f = (f1 , . . . , fn )T : Ω ⊂ Rn → Rn , p : Ω → R, n ≤ 3, and the divergence and gradient are div u := ∂u1 /∂x1 + · · · + ∂un /∂xn and grad p = T ∇ p = (∂p/∂x1 , . . . , ∂p/∂xn ) . The equation Ω pdx = 0 guarantees a unique p. Here we only study the stationary form with ∂ u/∂t ≡ 0. We will discuss its so-called center manifold approximations as a case study in B¨ ohmer [120].

1.4 1.4.1

Fundamental results Linear operators and functionals in Banach spaces

Continuous or bounded and compact linear operators are among the most important concepts in functional analysis. We use them in both books very intensively.

14

1. From linear to nonlinear equations, fundamental results

Deﬁnition 1.8. Dense, compact sets, linear bounded, compact operators: 1. A subset U of a Banach space X (a linear space, closed with respect to its norm) is called dense, if every x ∈ X is the limit of a sequence {xn }n∈N ⊂ U. A U is called precompact (or relatively compact) and compact, if every sequence {xn }n∈N ⊂ U contains a convergent subsequence {xni }i,ni ∈N and additionally satisﬁes limi→∞ xni ∈ U, respectively. 2. The linear operator L : X → Y is called bounded (or equivalently continuous) and compact if the image of the unit sphere {x ∈ X : x ≤ 1} is bounded and precompact, respectively. Hence, for a bounded L there exists C ∈ R+ such that LxY ≤ CxX for all x ∈ X and L := LY←X := inf{C}

(1.27)

is called (and is) a norm for L. The set of all these continuous linear operators is denoted as L(X , Y). Furthermore, L = 0 ⇔ ∀x ∈ X : Lx = 0, and for L ∈ L(X , Y) we call N (L) := {x ∈ X : Lx = 0} and

(1.28)

R(L) := {y ∈ Y : ∃ x ∈ X : y = Lx} the kernel or null space and range of L, respectively. 3. Analogously we deﬁne bounded linear and bilinear forms, l : X → R, x → l, x := l(x) and b : X × Y → R, l = 0, b = 0, and their norms. We deﬁne the dual space as X := {l : X → R : l is a bounded linear form}.

(1.29)

A subset U ⊂ X is precompact, if its closure, U, is compact. The compactness of a subset U is equivalent to the following property: Any open covering, U ⊂ ∪i∈I Ui , Ui open in X , I an index set, contains a ﬁnite subcovering. For ﬁnite-dimensional Banach spaces, X , compactness is equivalent to U being a closed, bounded set. For inﬁnitedimensional Banach spaces compactness only implies, but is not equivalent to this property. A change from · X , · Y to equivalent norms does not change L(X , Y), but L. Now we have the following Proposition 1.9. Simultaneously with Y, L(X , Y) is a Banach space. The functional in (1.27) is indeed a norm for linear operators, and for Y = R for linear funcionals. It satisﬁes L = sup LxY , L2 ◦ L1 ≤ L2 L1 ∀ L, L1 ∈ L(X , Y), L2 ∈ L(Y, Z). xX =1

Analogously for bilinear forms b : X × Y → R we use b(x, y)R ≤ Cb xX · yY to introduce b := inf {Cb }. The kernel, N (L), is a closed subspace, the range, R(L), is usually only a subspace, see Theorem 1.19. Example 1.10. Special linear operators 1. Linear operators for dim X < ∞: Obviously, in this case all linear operators are bounded and have closed kernel and range.

1.4. Fundamental results

15

2. Rod with axial load: For small ϕ we have sin ϕ ≈ ϕ, hence, see Example 1.2 and its linear counterpart in (1.14), we study L : C 2 [0, l] → C[0, l],

Lϕ := ϕ¨ + λϕ,

(1.30)

with the norms ˙ ∞ + ϕ ¨ ∞ ϕ∞ := sup |ϕ(s)| and ϕ2,∞ := ϕ∞ + ϕ 0≤s≤l

in C[0, l] and C 2 [0, l], respectively. Then L is a bounded linear operator, independent of the boundary conditions, since ¨ ∞ ≤ ϕ2,∞ max{1, |λ|} with L ≤ max{1, |λ|}. Lϕ ≤ |λ|ϕ∞ + ϕ Above we have introduced the linear functionals on X yielding the space X . In a totally analogous way it is possible to deﬁne the linear functionals on X , thus introducing X := (X ) . Since l, x is linear in both arguments, ·, x ∈ (X ) =: X

for every x ∈ X

it is possible to identify X with a subset of X , sometimes even with X itself. Deﬁnition 1.11. The mapping ·, · =: ·, ·X ∗×X : X × X → R, {x , x} → x , x ∈ R (or ∈ C) is called a pairing (for X and X ). X is called reﬂexive if there exists a bijection b : X → X such that for the pairing ·, · : X × X → R, x , lX ×X = l, b(x )X ×X = l, xX ×X =: l, x =: l(x) ∀ l ∈ X , x = b(x ) implying x = b(x ) = x

∀ x ∈ X .

(1.31)

x is indeed a norm, and we use the notation X = X for reﬂexive X . Sometimes, the analogous xd := sup{|l, x| : ∀ l with lX = 1}

(1.32)

is called the dual norm of x and we have xd = x. The compactness of operators and their products is a powerful property. We introduce Deﬁnition 1.12. Weak convergence: A sequence {xn }∞ n=1 ⊂ X , a Banach space, is weakly convergent towards x0 ∈ X iﬀ ∀l ∈ X we have limn→∞ l(xn ) = l(x0 ). Theorem 1.13. Properties of compact operators: 1. For Banach spaces X , Y, Z let L1 ∈ L(X , Y), L2 ∈ L(Y, Z). If one of the L1 , L2 is compact, then L2 ◦ L1 is compact as well. Furthermore, L1 ∈ L(X , Y) is compact if and only if (L1 )d ∈ L(Y , X ) is compact; for Ld see (1.33) below.

16

1. From linear to nonlinear equations, fundamental results

2. For compact L1 , and equibounded Lh2 ∈ L(Y, Z), Lh2 ◦ L1 is compact. 1 3. A compact L1 ∈ L(X , Y) maps every weakly convergent convergent sequence into a strongly convergent sequence. Among the many important theorems on linear operators we particularly need theorems on extensions, open mappings and closed ranges and some consequences. We summarize the existence of extensions of bounded linear and bilinear forms and operators, extending the Hahn–Banach extension theorem, and their inverses: Theorem 1.14. Extension theorem for bounded forms and linear operators: 1. Let X0 ⊂ X , Y0 ⊂ Y be linear subspaces of the Banach spaces X , Y. Let l0 : X0 → R, L0 : X0 → Y and b0 : X0 × Y0 → R be bounded linear functionals, operators and bilinear forms with respect to the norms in X , Y, respectively. Then l0 , L0 and b0 can be extended as l, L, b to X and X × Y with the same norms. They are well deﬁned bounded linear functionals, operators and bilinear forms on X , Y with lX = l0 X , LY←X = L0 Y←X , bX ×Y = b0 X ×Y , respectively. 2. If X0 , Y0 are dense subsets of X , Y these l, L, b are deﬁned by the corresponding limits: For x ∈ X , y ∈ Y, choose sequences xn ∈ X0 ,

x = limn→∞ xn and deﬁne

yn ∈ Y0 ,

y = limn→∞ yn

lx := limxn →x l0 xn Lx := limxn →x L0 xn b(x, y) := limxn →x,yn →y b0 (xn , yn ).

Then the l, L, b are independent of the choice of the convergent sequences {xn }n∈N , {yn }n∈N . If L0 : X0 ⊂ X → Y is compact, then so is L0 : X → Y. Theorem 1.15. Banach open mapping theorem: Let L ∈ L(X , Y) be surjective: L(X ) = Y. Then the image, L(U), of every open subset, U ⊂ X , is open in Y. Corollary 1.16. Bijective linear operators are invertible: L ∈ L(X , Y) is bijective, often called an isomorphism, iﬀ it is surjective and injective, that is Lx1 = Lx2 only for x1 = x2 . If L is bijective, then L−1 exists and L−1 ∈ L(Y, X ). For the next theorem we need the concepts of dual operators, generalizing transpose matrices, and generalized orthogonal complements. For L ∈ L(X , Y) the dual operator Ld : Y → X (to distinguish later derivatives, L , we use Ld for the dual operator), is uniquely deﬁned for every l ∈ Y by Ld ∈ L(Y , X ) with l ∈ Y → Ld l ∈ X : l, LxY ×Y = Ld l, xX ×X ∀ x ∈ X

(1.33)

with the pairings in Y , Y and X , X , often denoted by the same ·, ·. Only if the speciﬁc combination is emphasized do we use the index of the pairing as in 1 For the later applications to discretizations with parameter h, we allow Lh , cf. Hackbusch [387], 2 Lemma 6.4.5, and Wloka [666], Satz 2.

1.4. Fundamental results

17

(1.31), (1.33). The above Ld is unique, linear and continuous. Furthermore, for L, L1 ∈ L(X , Y), L2 ∈ L(Y, Z), α ∈ C, Ld = L, (L1 ◦ L2 )d = Ld2 ◦ Ld1 , (αL)d = αLd ; Ldd := (Ld )d = L;

(1.34)

the last Ldd = L is valid only in reﬂexive Banach spaces X = X , Y = Y . We denote the dual space of a Hilbert space as X ∗ := X . We obtain the following representation and natural embedding: Theorem 1.17. Riesz representation theorem: Let X be a Hilbert space and l ∈ X = X ∗ . Then there exists a unique xl ∈ X such that l(x) = (x, xl ) ∀ x ∈ X and xl X = lX .

(1.35)

Corollary 1.18. There exists a unique (Riesz-)isomorphism JX ∈ L(X , X ∗ = X ), see Corollary 1.16, such that JX xl = l, JX−1 l = xl , JX = JX−1 = 1. X ∗ = X is again a Hilbert space with scalar product (x , y )X ∗ := (JX−1 x , JX−1 y )X and lX ∗ = xl X . X and (X ∗ )∗ = X can be identiﬁed by xl = JX−1 l. For two Hilbert spaces X , Y with scalar products (·, ·)X , (·, ·)Y , let L ∈ L(X , Y). Besides the dual operator, the adjoint operator L∗ : Y → X is deﬁned by L∗ ∈ L(Y, X ) and ∀ y ∈ Y : (y, Lx)Y = (L∗ y, x)X ∀ x ∈ X .

(1.36)

The dual and adjoint operators are related via the Riesz isomorphism as L∗ = (JX )−1 Ld JY ∈ L(Y, X ).

(1.37)

Only if X , Y are identiﬁed with X , Y , do we ﬁnd L∗ = Ld . In particular, we introduce self-adjoint or symmetric operators and orthogonal projectors as L ∈ L(X , X ) is called a self-adjoint or symmetric operator and an orthogonal projector, if, L = L∗ and L = L2 , L = L∗ , respectively. Pairings may be used to generalize orthogonal complements: let subsets or subspaces U ⊂ X and V ⊂ X be given. Then ⊥

V ⊥

(V ) := (V

U := {x ∈ X : x , uX ×X = 0 ∀ u ∈ U } ⊂ X , ⊥

(1.38)

:= {x ∈ X : v , xX ×X = 0 ∀ v ∈ V } ⊂ X ,

⊥

) := {x ∈ X : x , v X ×X = 0 ∀ v ∈ V } ⊂ X ,

with X = X for reﬂexive Banach spaces. The last space (V ⊥ ) is used only very rarely. For continuous L (and ·, ·), the subspaces ⊥ U, V ⊥ and N (L) are closed. For R(L) we have

18

1. From linear to nonlinear equations, fundamental results

Theorem 1.19. Closed range theorem: For L ∈ L(X , Y) the following four statements are equivalent (⇐⇒): R(L) is closed ⇐⇒ R(Ld ) is closed ⇐⇒ R(L) = N (Ld )⊥ ⇐⇒ R(Ld ) =⊥ N (L). Deﬁnition 1.20. For L ∈ L (X , Y), let 1. N (L) = {x ∈ X : Lx = 0},

dim N (L) < ∞, and

(1.39)

2. R(L) be closed, hence, a subspace N ⊂ Y exists, with Y = N ⊕ R(L) and let codim R(L) := dim N < ∞. Then L is called a Fredholm operator and the number i(L) := dim N (L) − codim R(L) its Fredholm index. Deﬁnition 1.21. Complexiﬁcation of Banach spaces and operators: We extend a real X to a complex Banach space Xc cf. R to C, as, cf. [676] Xc := {x + iy : x, y ∈ X },

i2 = −1, (a + ib)(x + iy) := (ax − by) + i(bx + ay),

(x1 + iy1 )(x2 + iy2 ) = (x1 x2 − y1 y2 ) + i(y1 x2 + x2 y1 ), ∀a, b ∈ R, x1 , x2 , y1 , y2 ∈ X , and x + iyXc := max x(cos φ) + y(sin φ).

(1.40)

0≤φ≤2π

Similarly we extend an A ∈ L(X , Y) to Ac (x + iy) := Ax + iAy, ∀x, y ∈ X , thus Ac ∈ L(Xc , Yc ).

(1.41)

Theorem 1.22. Complexiﬁed Banach space L(X , Y) and linear operator: Assume Banach spaces X , Y and A ∈ L(X , Y). Deﬁne Xc , · Xc , Ac as in (1.40), (1.41). Then Xc , Yc are Banach spaces and Ac ∈ L(Xc , Yc ) is linear and bounded with AY←X ≤ Ac Yc ←Xc ≤ 2AY←X . 1.4.2

Inequalities and Lp (Ω) spaces

We start with some inequalities for ﬁnite sums. The discrete Chebyshev inequality holds for ordered sums. Then we obtain, e.g. [396], p 99, for the two sets of f1 ≥ f2 ≥ . . . ≥ fN ∈ R and g1 ≥ g2 ≥ . . . ≥ gN ∈ R, N j=1

fj /N

N j=1

gj /N ≤

N

fj gj /N.

(1.42)

j=1

This implies for arbitrary f1 , f2 , . . . , fN ∈ R that N N N fj ≤ fj ≤ N fj2 . j=1 j=1 j=1

(1.43)

1.4. Fundamental results

19

Often we need the discrete H¨ older inequalities. We always choose a pair of 1 ≤ p, q ≤ N N |gj |q |1/q := max |gj | for q = ∞ [396,397] ∞ with 1/p + 1/q = 1. Then we ﬁnd, with j=1

j=1

for 1 < p, q < ∞, and for 1 = p < q = ∞ as in (1.46), ⎛ ⎞1/p ⎛ ⎞1/q N N N fj gj ≤ ⎝ |fj |p ⎠ ⎝ |gj |q ⎠ for 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1. j=1 j=1 j=1 (1.44) n

n

For cuboids Q := j=1 [aj , bj ] ∈ Rn , aj ≤ bj we have μ(Q) := j=1 (bj − aj ). For the set Ω, we choose two sets Ai and Ao , the union of countably many inner and outer They have to Ao := ∪Qok ⊃ Ω and cuboids Qil and Qok with disjoint interiors. satisfy i i o o i i A := ∪Ql ⊂ Ω. For μ(A ) := k μ(Qk ) and μ(A ) := l μ(Ql ), let inf Ao {μ(Ao )} = supAi {μ(Ai )}. Then Ω is called μ-measurable, and μ(Ω) := inf Ao {μ(Ao )}. For Ω measurable, a function, f : Ω → R ∪ ∞ ∪ −∞, is called measurable, if for every α ∈ R the set {x ∈ Ω : f (x) ≥ α} is measurable. For any (measurable) Ω ⊂ Rn we deﬁne and estimate ⎛

f Lp (Ω) := ⎝

⎞1/p |f (x)|p dx⎠

, 1 ≤ p < ∞, and f L∞ (Ω) := ess sup |f (x)| x∈Ω

Ω

Lp (Ω) := {f : Ω → R : f Lp (Ω) < ∞}, Ω ⊂ Rn , f gL1 (Ω) =

⎛

|f (x)g(x)|dx ≤ ⎝

Ω

(1.45)

⎞1/p ⎛ |f (x)|p dx⎠

Ω

⎝

⎞1/q |g(x)|q dx⎠

Ω

= f Lp (Ω) gLq (Ω) ∀ f ∈ Lp (Ω), g ∈ Lq (Ω) ∀ 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, the H¨ older inequalities. The usually missing case 1 = p, q = ∞ in (1.44), (1.45) is directly obtainable: f ∈ L1 (Ω), g ∈ L∞ (Ω) imply f g ∈ L1 (Ω), and similarly for (1.44), f gL1 (Ω) = |f (x)g(x)|dx (1.46) Ω

⎛

≤ gL∞ (Ω) ⎝

⎞ |f (x)|dx⎠ = f L1 (Ω) gL∞ (Ω) .

(1.47)

Ω

In particular we obtain for the case p = q = 2 the Cauchy–Schwarz inequality |(u, v)L2 (Ω) | ≤ uL2 (Ω) · vL2 (Ω) .

(1.48)

20

1. From linear to nonlinear equations, fundamental results

For later discussions of nonlinearities we need the extension ⎛ ⎞1/pj d d d ⎝ |fj (x)|pj dx⎠ fj fj dx ≤ = 1 j=1 j=1 L (Ω) j=1 Ω

=

d

(1.49)

Ω

fj Lpj (Ω) ∀ fj ∈ Lpj (Ω)∀ 1 ≤ pj ≤ ∞ with

d

j=1

1.4.3

1/pj = 1.

j=1

H¨ older and Sobolev spaces and more

For a function u ∈ C0∞ (Ω, R) the following partial derivatives are well deﬁned, with α := (α1 , . . . , αn ) ∈ Nn0 , |α| := α1 + · · · + αn , α ! := (α1 )! · · · (αn )! ∂ i u :=

(1.50)

∂u ∂ |α| , ∂u := (∂ 1 u, . . . , ∂ n u), ∂ α u := ∂1α1 . . . ∂nαn u := u. α1 n ∂xi ∂x1 . . . ∂xα n

The C k , C k,1 and C k+μ indicate the usual classical k-times continuously diﬀerentiable functions u : D → R, with appropriate D ⊂ Rn . For u ∈ C k,1 [and u ∈ C k+μ ], respecolder] continuous, hence tively, all derivatives ∂ α u for |α| ≤ k are Lipschitz [and H¨ |∂ α u(x) − ∂ α u(y)| ≤ C|x − y|μ for all x, y ∈ D for μ = 1 [and 0 < μ < 1]. For Sobolev spaces we have to generalize this concept: Deﬁnition 1.23. For a function u ∈ Lp (Ω), 1 ≤ p ≤ ∞, a function v := ∂ α u ∈ Lp (Ω) is called a weak α derivative of u, iﬀ (w, v)Lp (Ω) = (−1)|α| (wα , u)Lp (Ω) for all w ∈ C0∞ (Ω).

(1.51)

The Sobolev space W k,p (Ω), k ∈ N0 , 1 ≤ p ≤ ∞, is the set of functions with weak derivatives up to order k: W k,p (Ω) := {u ∈ Lp (Ω) : ∀α with |α| ≤ k : ∂ α u ∈ Lp (Ω)}.

(1.52)

For u ∈ Lp (Ω) let the weak derivative ∂ α u ∈ Lp (Ω), and in Ω ⊂ Ω let the classical derivative ∂ α u ∈ Lp (Ω ) exist. Then the weak and the classical derivative coincide almost everywhere (a.e.) in Ω . Theorem 1.24. Sobolev spaces are Banach spaces: The Sobolev spaces W k,p (Ω), k ∈ N0 , 1 ≤ p ≤ ∞, are Banach spaces with respect to the norm ⎞1/p ⎛ ∂ α upLp (Ω) ⎠ . (1.53) uW k,p (Ω) := uW k,p := uk,p := ⎝ |α|≤k

For the special case p = 2 we have the Hilbert space H k (Ω) with 1/2 (u, v)k := (u, v)H k (Ω) := (∂ α u, ∂ α v)L2 (Ω) , uH k (Ω) := (u, v)H k (Ω) ; |α|≤k

(1.54)

1.4. Fundamental results

21

sometimes we use uk := uH k (Ω) . The spaces W0k,p (Ω) and H0k (Ω), deﬁned as closure of C0∞ (Ω) with respect to the corresponding norm, e.g. · H k (Ω) , W0k,p (Ω) := C0∞ (Ω)

·k,p

, H0k (Ω) := C0∞ (Ω)

·H k (Ω)

,

(1.55)

are again Banach and Hilbert spaces with the same norm (1.53) and (1.54), respectively. Sometimes, we need the following seminorms ⎛ |u|W k,p (Ω) := ⎝

⎞1/p

∂ α upLp (Ω) ⎠

, |u|H k (Ω) := |u|W k,p=2 (Ω) .

(1.56)

|α|=k

If Ω is bounded, then · W k,p (Ω) := | · |W k,p (Ω) W k,p (Ω) and · W k,p (Ω) are equivalent 0

0

norms on W0k,p (Ω); similarly, for p = 2, on H0k (Ω). In particular, we obtain with Poincar´e’s constant CP (Ω) |u|W k,p (Ω) ≤ uW k,p (Ω) ≤ CP (Ω)|u|W k,p (Ω) ∀u ∈ W0k,p (Ω).

(1.57)

Proposition 1.25. cf. Zeidler, [678], p. 1032: For Ω ∈ C 0,1 the standard norm uW k,p (Ω) for the Sobolev spaces W k,p (Ω) with k ∈ N is equivalent to the norms ⎛

⎝

Ω |α|=k

⎛

⎝

Ω |α|=k

⎛

⎝

⎞ p 1/p |∂ α u|p dx + udx ⎠

|∂ α u|p dx +

∂Ω

⎞ p 1/p u ds ⎠ ⎞1/p

|u|p ds⎠

|∂ α u|p dx +

Ω |α|=k

(1.58)

Ω

,

∂Ω

where ds indicates the surface element on ∂Ω. For n = 1 and Ω = (a, b) we deﬁne g ds := g(b) − g(a). ∂Ω For X ⊂ Y we denote the mapping I : X → Y, x → I x := x ∈ Y for X ⊂ Y as an inclusion or embedding, often abbreviated as X → Y. If I is a continuous and compact operator, then X is called continuously and compactly embedded into Y, respectively, so xY ≤ CxX ∀x ∈ X . For the compact embedding, additionally each bounded sequence {xn } in X has a subsequence {xnk } converging in Y, cf. [677], pp. 1026 ﬀ., [676], p. 248, [387], Example 6.4.7:

22

1. From linear to nonlinear equations, fundamental results

Theorem 1.26. Embedding theorem for Sobolev spaces: Let Ω be a bounded Lipschitz domain, j ∈ N0 , k ∈ N, 0 ≤ j < k, 1 ≤ p, q < ∞, 0 < α < 1 and deﬁne d := 1/p − (k − j)/n. Then the following embedding results are valid. 2 1. W k,p (Ω) → W j,q (Ω) and W0k,p (Ω) → W0j,q (Ω) are continuous for d ≤ 1/q, j ≤ k, and compact for d < 1/q, j < k. These results remain valid for W0k,p (Ω) and arbitrary open Ω. 2. In particular, W k,p (Ω) → W k,q (Ω) is continuous for 1 ≤ q ≤ p ≤ ∞, 0 ≤ k. 3. W k,p (Ω) → C j (Ω) are compact for d < 0, i.e. k − j > n/p, and W k,p (Ω) → C j,α (Ω) → C j (Ω) for d + α/n < 0 or d < 0. This inclusion might require changing u ∈ W k,p (Ω) on a subset of Ω of n-dimensional measure 0. 4. If we replace the above bounded Ω by Rn we ﬁnd: W k,p (Rn ) → C j (Rn ) for d < 0, i.e. k − j > n/p, and W k,p (Rn ) → W j,q (Rn ) for d ≤ 1/q and j < k are continuous. ¯ → Lp (Ω) and C 0,1 (Ω) ¯ → Lp (Ω) are continuous for 0 ≤ α ≤ 1 and 1 ≤ 5. C α (Ω) p ≤ ∞. 6. For 0 < α ≤ 1, j < k, the embeddings C k,α (Ω) → C k (Ω) → C j (Ω) are compact. The previous compact embedding result implies, e.g. Proposition 1.27. Under the conditions of Theorem 1.26, for every ε > 0 a Cε > 0 exists such that xW j,q (Ω) ≤ εxW k,p (Ω) + Cε xLs (Ω)

for 0 ≤ j < k,

1 ≤ p, q, s < ∞, 1/p − (k − j)/n < 1/q, 1/q − k/n ≤ 1/s. In particular, we ﬁnd for p = q = s and 0 < j < k: xW j.p (Ω) ≤ εxW k,p (Ω) + Cε xLp (Ω) . Theorem 1.28. Sobolev–Stein extension theorem [141], p. 31: Let Ω ⊂ Rn have a Lipschitz boundary, let k ∈ N0 and p ∈ R with 1 ≤ p ≤ ∞. Then there exists a bounded operator E : W k,p (Ω) → W k,p (Rn ) and a constant, C, independent of v, such that Ev|Ω = v and EvW k,p (Rn ) ≤ CvW k,p (Ω) ∀v ∈ W k,p (Ω).

(1.59)

Further bounded extension operators for an open set with Ω ⊂ Ω of the form k,α (Ω) → C0k,α (Ω ), with 0 and D indicating trivial and nontrivial Dirichlet ED : CD boundary conditions are discussed in [344], p. 136, [677], pp. 305–306. In Chapter 5, for example, we need Theorem 1.28. Theorem 1.29. Extension of a boundary function, [344], p 137: Let Ω ⊂ Rn be a C k,α domain, let k ∈ N and let Ω be an open set with Ω ⊂ Ω . Let v ∈ C k,α (∂Ω) Then there exists a function w ∈ C0k,α (Ω ), such that v = w on ∂Ω. Trace operators for Sobolev spaces of fractional order, s ∈ R+ , are necessary in diﬀerent contexts, in particular for nonconforming, adaptive FE and discontinuous Galerkin methods. We formulate the following results in H s (Ω), and W s,p (Ω) form, 2

Note that Lq (Ω) = W 0,q (Ω) and d is independent of n for k = j.

1.4. Fundamental results

23

cf. Zeidler [678], pp. 1031 ﬀ., Hackbusch [387], Subsections 6.2.4 and 6.2.5. They represent an integral variant of the H¨ older spaces C k,μ (Ω). We deﬁne for these H s (Ω), W s,p (Ω) for ∂Ω ∈ C 0,1 , s = k + μ, k = 0, 1, . . . , 0 < μ < 1, 1 < p < ∞, (1.60) the corresponding scalar products and Sobolev–Slobodeckij norms as 1/2 (u, v)H s (Ω) := (u, v)H k (Ω) + f (∂ α u, ∂ α v), uH s (Ω) := (u, u)H s (Ω) , (1.61) |α|≤k

with

[u(x) − u(y)][v(x) − v(y)] dxdy |x − y|n+2μ Ω×Ω |u(x) − u(y)|p and for 1 ≤ p ≤ ∞, fp (u) := dxdy, n+pμ Ω×Ω |x − y| uW s,p (Ω) := (upW k,p (Ω) + fp (∂ α u))1/p . f (u, v) :=

|α|≤k

For H s (∂Ω), W s,p (∂Ω) replace in (1.61) the Ω by ∂Ω and dxdy by dsds, and n by n − 1. Theorem 1.30. Sobolev spaces of fractional order: The Sobolev spaces H s (Ω), H s (∂Ω), and W s,p (Ω), W s,p (∂Ω), s = k + μ, k = 0, 1, . . . , 0 < μ < 1, 1 < p < ∞, in (1.60) are Hilbert and reﬂexive Banach spaces, respectively, with respect to the scalar product and norms in (1.61). For considering the boundary and functions deﬁned there, we introduce the concept of an atlas. We achieve that by introducing a parametrization, hence, local bijections between neighborhoods of boundary points and the unit sphere along the boundary. This allows formulating the necessary diﬀerentiability conditions for ∂Ω and a parametrization for the boundary. Deﬁnition 1.31. Let k ∈ N0 [and t with 0 < t ≤ ∞, t = k + λ, k ∈ N0 , 0 ≤ λ < 1] be given. [Here and below we indicate by [...] this case 0 < t ≤ ∞,. . . and its consequences.] For each x ∈ ∂Ω we require a neighborhood U = U(x) ⊂ Rn and a bijection φ : U → B1 (0) := B1n (0) = {x ∈ Rn : |x| ≤ 1}, the unit ball in Rn , such that ¯ φ−1 ∈ C k,1 ( B1 (0) ) 1. φ ∈ C k,1 (U), ¯ φ−1 ∈ C t ( B1 (0) )], [and φ ∈ C t (U), 2. φ(U ∩ ∂Ω) = {ξ ∈ B1 (0) : ξn = 0}, 3. φ(U ∩ Ω) = {ξ ∈ B1 (0) : ξn > 0}, 4. φ(U ∩ (Rn \Ω)) = {ξ ∈ B1 (0) : ξn < 0}.

(1.62)

Then we denote Ω as (k, 1) [and t] smooth, indicated by Ω ∈ C k,1 [and ∈ C t ]. Ω ∈ C (0,1) is called a Lipschitz domain. These (U, φ) are denoted as an atlas.

24

1. From linear to nonlinear equations, fundamental results

U2

U1

U3

U0

Figure 1.5 Atlas for a circle.

Proposition 1.32. Let Ω ∈ C k,1 , k ∈ N0 [and Ω ∈ C t , 0 < t ≤ ∞] be open and bounded. Then we need only ﬁnitely many neighborhoods, U i , bijections, φi , 0 ≤ i ≤ N, and parametrizations of the boundary, αi , 1 ≤ i ≤ N , in (1.62), such that cf. Figures 1.5 and 1.6, 1. U i is open, bounded, 0 ≤ i ≤ N,

N

¯ U 0 ⊂ Ω, ∪ U i ⊃ Ω,

i=0

2. φi : U i → B1n (0) is bijective, φi ∈ C k,1 U i [and ∈ C t U i ], 0 ≤ i ≤ N, 3. Ui := U i ∩ ∂Ω, αi := φi |Ui , 1 ≤ i ≤ N,

N

∪ Ui = ∂Ω,

(1.63)

i=1

4. αi : Ui → αi (Ui ) = B1n−1 (0) ⊂ Rn−1 is bijective, 1 ≤ i ≤ N, 5. αi ◦ αj−1 ∈ C k,1 αj (Ui ∩ Uj ) [and αi ◦ αj−1 ∈ C t αj (Ui ∩ Uj ) ].

U1

U2

U3

U0 U4

Figure 1.6 Atlas for a square.

1.4. Fundamental results

25

These U i , φi , 0 ≤ i ≤ N , deﬁne an atlas, hence satisfy (1.62). The system (Ui , αi ), 1 ≤ i ≤ N , is denoted as a parametrization or coordinate system for ∂Ω. In particular, Ω is sometimes called a smooth manifold, if in (1.63) the φi , αj and their inverses are ∈ C ∞. The above boundary operators have to be applied to functions u ∈ Wpm (Ω) and will m−1/p

(∂Ω) . Using the above φi , αi , it is possible to reduce yield functions u|∂Ω ∈ Wp the discussion to functions deﬁned on the U i , Ui or the B1n (0), B1n−1 (0) or even to Rn , Rn−1 . We use the following partition of unity: Proposition 1.33. Let U i , 0 ≤ i ≤ N, satisfy (1.63). Then functions σi ∈ C0∞ (Rn ), 0 ≤ i ≤ N, exist such that carr (σi ) := closure {x ∈ Rn : σi (x) = 0} = supp (σi ) ⊂ U , i

N

¯ σi2 (x) = 1 for all x ∈ Ω.

(1.64)

i=0

This representation is called a partition of unity. N N Since, by (1.64), i=0 σi2 (x)u(x) = i=0 u(x)σi2 (x) = u(x), we may consider each of the σi u separately. In particular, we will use the above bijections αi : Ui → B1n−1 (0). Then we study the local parametrizations (σi u) ◦ (αi )−1 : B1n−1 (0) → R and sum over all appropriate terms. With carr (σi ) ⊂ U i , we extend (σi u) ◦ (αi )−1 to (σi u) ◦ (αi )−1 : Rn−1 → R. Deﬁnition 1.34. Let Ω be bounded and choose for Ω ∈ C k,1 [and Ω ∈ C t], an s with s ≤ k + 1 [and s < t for t ∈ / N, t > 1 or s ≤ t ∈ N]. Furthermore, let the U i , φi , Ui , αi and σi , 0 ≤ i ≤ N, satisfy (1.63), (1.64). Then the Sobolev space W s,p (∂Ω) is deﬁned as n−1 s W s,p (∂Ω) := u : ∂Ω → R : (σi u) ◦ αi−1 ∈ Wp,0 B1 (0) , 1 ≤ i ≤ N . By U 0 ⊂ Ω we may restrict the discussion essentially to 1 ≤ i ≤ N. Theorem 1.35. Sobolev spaces W s,p (∂Ω) are Banach spaces: For bounded Ω ∈ C k,1 [and C t] and s as in Deﬁnition 1.34, let the U i , φi , Ui , αi and σi , 0 ≤ i ≤ N, satisfy (1.63), (1.64). Then the above Sobolev space W s,p (∂Ω) is a Banach and a Hilbert space, for 1 ≤ p < ∞, p = 2 and p = 2, respectively, with the obvious modiﬁcation of the norms in (1.61), and replacing Ω by ∂Ω. The W s,p (∂Ω) are reﬂexive for 1 < p < ∞. For p = 2 the scalar product is (u, v)H s (∂Ω) :=

N

(σi u) ◦ αi−1 , (σi v) ◦ αi−1

H s (Rn−1 )

.

(1.65)

i=1

A diﬀerent choice of atlas and partition of unity yields the same Sobolev spaces. ˜ , again ˜ i , σ˜i , 0 ≤ i ≤ N More precisely, replace the Ui , αi , σi chosen above by U˜i , α k,1 t [and C ] coordinate system and satisfying (1.63), (1.64). This deﬁnes another C another partition of unity. The norms, as in Therem 1.24 and the modiﬁed (1.61),

26

1. From linear to nonlinear equations, fundamental results

˜ s,p (∂Ω) contain the same are equivalent and the two Sobolev spaces W s,p (∂Ω) and W elements, so can be identiﬁed. Now we are able to deﬁne the restriction operators for functions deﬁned on Ω to the boundary ∂Ω. Let Ω ⊂ Ω0 . Then a restriction to the boundary is uniquely determined for suﬃciently smooth functions. So γ0 : C0∞ (Ω0 ) → C τ (∂Ω), (γ0 u)(x) := u(x)∀x ∈ ∂Ω,

(1.66)

for appropriate τ ∈ R, is well deﬁned. Deﬁnition 1.36. Trace or restriction operator: Let an extension γ of γ0 in (1.66) exist such that γp = γ ∈ L(W s,p (Ω), W t,p (∂Ω)) for appropriate s, t, with γ|C0∞ (Ω) = γ0 , and denote it as u|∂Ω := γu. Then this γ is called a trace operator or a restriction operator. The following theorem shows that these trace operators exist, if Ω ∈ C 0,1 and the smoothness in W s,p (Ω) is better than in W t,p (∂Ω), cf. [677], p. 1030. Theorem 1.37. Properties of the trace operator: Let Ω be open and bounded, Ω ∈ C 0,1 , n ≥ 2, 1 ≤ p < ∞, and γ as in Deﬁnition 1.36. Then 1. For p < ∞ the γ = γp : W 1,p (Ω) → W 1−1/p,p (∂Ω) is linear, surjective, and continuous: γp uW 1−1/p (∂Ω) ≤ CuW 1,p (Ω) ∀u ∈ W 1,p (Ω). 2. u ∈ W01,p (Ω) is equivalent to u ∈ W 1,p (Ω) and γ u = 0 on ∂Ω. 3. For Ω ∈ C m−1,1 the u ∈ W0m,p (Ω) is equivalent to u ∈ W m,p (Ω) and Dα u ∈ W01,p (Ω) or γ Dα u = 0 ∀ |α| ≤ m − 1. 4. For p < ∞ there exists a linear continuous extension operator Ep : W 1−1/p,p (∂Ω) → W 1,p (Ω) such that the following diagram is commutative with the identity I : W 1,p (Ω) → W 1,p (Ω) W 1,p (Ω) I

γp

−→ W 1,p (Ω)

W 1−1/p,p (∂Ω) Ep

Theorem 1.38. Trace theorem: Let Ω ⊂ Rn have a Lipschitz boundary, Ω ∈ C 0,1 , and let 1 ≤ p ≤ ∞. Then there exist constants, C, such that 1−1/p

1/p

vLp (∂Ω) ≤ CvLp (Ω) · vW 1,p (Ω) ∀v ∈ W 1,p (Ω) and 1/2

(1.67)

1/2

vLp (∂Ω) ≤ CvL2 (Ω) vH 1 (Ω) ≤ CvH 1 (Ω) for p = 2. The vW k,p (Ω) satisfy a trace estimate as well: 1−1/p

1/p

vW k,p (∂Ω) ≤ CvW k,p (Ω) · vW k+1 (Ω) ∀v ∈ Wpk+1 (Ω). p

(1.68)

1.4. Fundamental results

27

Theorem 1.39. Trace operators for Sobolev spaces of fractional order: For Ω ∈ C k,1 , choose s such that 1/2 < s = k + 1 ∈ N [and for Ω ∈ C t choose s such that 1/2 < s ≤ t ∈ N or 1/2 < s < t ∈ N] and s > |α| + 1/2, |α| ≥ 0. We formulate separately for |α| = 0 and |α| > 0. Then the following trace and extension statements are correct: the trace operators γ∂ α : H s (Ω) → H s−|α|−1/2 (∂Ω) exist. 1. γ ∈ L(H s (Ω), H s−1/2 (∂Ω)), so u ∈ H s (Ω) implies γu = u|∂Ω ∈ H s−1/2 (∂Ω). 2. γ∂ α ∈ L(H s (Ω), H s−|α|−1/2 (∂Ω)), hence, for u ∈ H s (Ω) the restriction implies γ∂ α u = ∂ α u|∂Ω ∈ H s−|α|−1/2 (∂Ω) and ∂ α u|∂Ω H s−|α|−1/2 (∂Ω) ≤ CuH s (Ω) . In particular the normal derivatives ∂ l u/∂nl exist, e.g. for t > s > l + 1/2 and Ω ∈ C t , and u ∈ H0s (Ω) with s > l + 1/2 implies γ∂ α u = ∂ α u|∂Ω = 0, if 0 ≤ |α| ≤ l. 3. For each w ∈ H s−1/2 (∂Ω) there exists u ∈ H s (Ω) such that w = γu and uH s (Ω) ≤ Cs wH s−1/2 (∂Ω) . For the Gauss integral theorem, we use the notation F = (F1 , · · · , Fn ) ∈ W 1,1 (Ω) divF =

n

∂Fi /∂xi .

i=1

Theorem 1.40. Gauss integral theorem: Let Ω ⊂ Rn be a C 0,1 compact domain with outer normal vector ν = (ν1 , · · · , νn ) along ∂Ω, and F ∈ W 1,1 (Ω, Rn ). Then divF dx = F, νn ds and ∂Fi /∂xi dx = Fi νi ds, i = 1, . . . , n. (1.69) Ω

1.4.4

∂Ω

Ω

∂Ω

Derivatives in Banach spaces

We introduce two types of derivatives in Banach spaces. We motivate it by the nonlinear operator, cf. (1.13), for the loaded rod. G : C 2 [0, l] → C[0, l], G(ϕ) = d2 ϕ/ds2 + λ sin ϕ.

(1.70)

(1) Choosing functions ϕ, ψ ∈ C 2 [0, l] with small ˙ ∞ + ψ ¨ ∞, ψ2,∞ := ψ∞ + ψ we obtain the relation, with ﬁxed parameter λ, G(ϕ + ψ) − G(ϕ) = d2 ψ/ds2 + λ(sin(ϕ + ψ) − sin ϕ) = d2 ψ/ds2 + (λ cos ϕ)ψ + o(ψ) =: L(ϕ)ψ + o(ψ) =: G (ϕ)ψ + o(ψ). The notation L(ϕ) = G (ϕ), called the Frech´et derivative, is formalized in (1.71) below: G (ϕ) : C 2 [0, l] → C[0, l], G (ϕ)ψ := d2 ψ/ds2 + (λ cos ϕ)ψ := ψ¨ + (λ cos ϕ)ψ.

28

1. From linear to nonlinear equations, fundamental results

This operator G (ϕ) is obviously linear and it is bounded, see Example 1.2, ¨ ∞ ≤ max{1, |λ|}ψ2,∞ or G (ϕ) ≤ max{1, |λ|}. G (ϕ)ψ ≤ |λ cos ϕ|ψ∞ + ψ For appropriate ϕ and ψ we even have equality. (2) To determine the directional derivative, we choose ﬁxed ϕ, ψ ∈ C 2 [0, l], τ ∈ [0, l]. We introduce a function of a real variable t ∈ [0, l], as g : R → R, g(t) := G(ϕ(τ ) + tψ(τ )) = (d2 (ϕ + tψ)/ds2 )(τ ) + λ sin(ϕ(τ ) + tψ(τ )). We can use the classical diﬀerentiation rules, e.g. the chain rule for a function of a real variable to obtain dg/dtt=0 = (d2 ψ/ds2 )(τ ) + (λ cos ϕ(τ ))ψ(τ ). By considering this relation for variable τ ∈ [0, l], and ﬁxed functions ϕ, ψ, we could equally well have deﬁned the corresponding g¯ : R → C[0, l], g¯(t) := d2 (ϕ + tψ)/ds2 + λ sin(ϕ + tψ). Similarly to the above, we can linearize g(t) − g(0) = td2 ψ/ds2 + λ(sin(ϕ + tψ) − sin ϕ) = td2 ψ/ds2 + t(λ cos ϕ)ψ + t o(ψ) = t(L(ϕ)ψ) + o(tψ) = t G (ϕ)ψ + o(tψ). This L(ϕ)ψ = G (ϕ)ψ will be called the Gateaux derivative in (1.72). Obviously, by ﬁxing the functions and variables, e.g. the ϕ, ψ, τ above, the computation of these directional derivatives only uses classical calculus. The linearization is more complicated, but it has advantages for the bifurcation in [120]. We need a simple criterion which guarantees a directional derivative to be a Frech´et derivative, see Deﬁnition 1.41 and Theorem 1.42. Deﬁnition 1.41. Frech´et (F) and Gateaux (G) derivative: 1. Let H : D ⊂ X → Y, x0 ∈ U := U(x0 ) ⊂ D and U be open. Then F is called Frech´et (F) diﬀerentiable in x0 if there is an LF ∈ L(X , Y), depending on x0 but not on h, such that H(x0 + h) − H(x0 ) = LF h + o(h)

for

h → 0, x0 + h ∈ U.

(1.71)

Then HF (x0 ) := H (x0 ) := LF is called the Frech´et derivative of H in x0 . 2. H is called Gateaux (G) diﬀerentiable in x0 ∈ U, if for every h (such that h = (x0 , h) ∈ Y such that 1) there is a HG (x0 , h) + o(t) for t → 0, x0 + th ∈ U; H(x0 + th) − H(x0 ) = tHG

(1.72)

this o(t) = oh (t) depends on h. If (1.72) should be valid only for a ﬁxed h (with h = 1), then H is called Gateaux diﬀerentiable in x0 in the direction of h.

1.4. Fundamental results

29

(x0 , h) in (1.72) is called the Gateaux (G) derivative of H Furthermore, the HG in x0 in the direction of h.

These Frech´et and Gateaux derivatives generalize the total and directional derivatives for F : D ⊂ Rn → Rm , respectively. We will mainly apply Frech´et derivatives. However, by G(t) := H(x0 + th) we have reduced H : D ⊂ X → Y to a G : I ⊂ R → Y mapping for a real variable t. Thus, all the rules, e.g. sum, product and chain rule, obtained by limits of divided diﬀerences, remain correct for Gateaux derivatives as well, if they are interpreted as limits. This requires that x0 and h are kept ﬁxed during this diﬀerentiation. This allows an easier and more straightforward approach. Under appropriate conditions, which essentially cover all our applications, Gateaux derivatives yield a simple tool to determine Frech´et derivatives as well. F and G derivatives are uniquely determined. Sometimes, “one-sided” derivatives are appropriate, in particular, if U(x0 ) is not open and, hence, not all x0 + h and x0 + th are in U(x0 ), for suﬃciently small h and t. An operator which is F diﬀerentiable in x0 is continuous in x0 . That is not true, if it is only G diﬀerentiable. This situation is well known for real valued functions of two variables. It is obvious that both derivatives are generated by linearizing the diﬀerence of operator values: In (1.71) we have linearized with respect to h ∈ X , in (1.72) with respect to t ∈ R. In particular we have as a consequence of (1.72) (x0 , h) = lim HG

t→0

d H(x0 + th) − H(x0 ) = H(x0 + th)|t=0 . t dt

For some examples we have the situation that LG ∈ L(X , Y) exists such that for all (x0 , h) = LG h. Sometimes this is even part of the deﬁnition of a Gateaux h = 1, HG derivative, see [677]. The o(t) term in (1.72) does depends on h. We have Theorem 1.42. Frech´et (F) and Gateaux (G) derivatives and their relations: 1. The Frech´et and Gateaux derivatives are uniquely determined. 2. If H is Frech´et diﬀerentiable in x0 ∈ U(x0 ) ⊂ D, it is continuous in x0 , Gateaux (x0 , h) = LF h. diﬀerentiable and HG 3. Let H be Gateaux diﬀerentiable in x0 , such that (x0 ) ∈ L(X , Y) exists, such that LG = LG (x0 ) = HG H(x0 + th) − H(x0 ) = tHG (x0 )h + oh (t) and

(1.73)

oh (t) = o(t)C(h) with C(h) ≤ C for h = 1. Then H is Frech´et diﬀerentiable in x0 and for all h with h = 1 we have (x0 , h) = HG (x0 )h = LG h = LF h, hence LF = HG (x0 ). HG 4. Let H be Gateaux diﬀerentiable in a neighborhood U(x0 ) of x0 and such that (x)h + oh (t) s.t. H(x+ t h) − H(x) = t HG (x) ∈ L(X , Y) is continuous w.r.t x for all x ∈ U(x0 ). LG (x) = HG

Then H is Frech´et diﬀerentiable in U(x0 ) and LF (x) = LG (x).

(1.74)

30

1. From linear to nonlinear equations, fundamental results

Theorem 1.43. Mean value theorem: Let F : D ⊂ X → Y and a, b ∈ D given with ab = {a + t(b − a); 0 ≤ t ≤ 1} ⊂ D, such that F is F diﬀerentiable on ab (or in D) and F (a + t(b − a)) ≤ g (t), 0 ≤ t ≤ 1, were g and f (t) := F (a + t(b − a))(b − a) are integrable in [0, 1]. Then the following relations are valid, prefeably with x0 near ab, 1 F (b) − F (a) = F (a + t(b − a))dt(b − a), (1.75) 0

F (b) − F (a) ≤ (g(1) − g(0))b − a,

(1.76)

F (b) − F (a) ≤ b − a sup F (x),

(1.77)

x∈ab

F (b) − F (a) − F (x0 )(b − a) ≤ b − a sup F (x) − F (x0 ).

(1.78)

x∈ab

Modiﬁcations for G derivatives into the direction of b − a are straightforward. Partial derivatives are deﬁned by splitting the variable into components. Let F := (F1 (x), . . . , Fm (x)) : D ⊂ X =

n i=1

Xi → Y =

m

Yj

(1.79)

j=1

with diﬀerent arguments x = (x, . . . , xu ) ∈ X and components, (F1 (x), . . . , Fm (x)). For Banach spaces Xi and Yj and the norm ||x||X := maxni=1 {||xi ||Xi }, the X , Y are Banach spaces as well. We deﬁne partially diﬀerentiable F : Deﬁnition 1.44. Choose x0 ∈ U(x0 ) ⊂ D, U(x0 ) a neighborhood of x0 . Assume there exists a Li ∈ L(Xi , Y) and a Lji ∈ L(Xi , Yj ), respectively, such that for all small hi ∈ Xi , F (x0 + hi ) − F (x0 ) − Li hi = o(hi ) and Fj (x0 + hi ) − Fj (x0 ) − Lji hi = o(hi ).

(1.80)

Then the operator F and its components Fj , are called partially (Frechet) diﬀerentiable in x0 with respect to xi and the (∂F/∂xi )(x0 ) := (∂xi F )(x0 ) := Li and Lji are called its partial derivatives. By replacing hi by thi and t → 0 we obtain the partial Gateaux diﬀerentiability. For Xi = Yj = R and hi = tei with the i-th unit vector, ei , and small t ∈ R, we obtain the classical partial derivatives. Theorem 1.45. Continuously partially diﬀerentiable operators are (Frechet) diﬀerentiable: 1. Let F in (1.79) be continuously partially diﬀerentiable in U(x0 ), that is ∂F /∂xi ∈ C(U(x0 )). Then F is (Frechet) diﬀerentiable in x0 . 2. Let F in (1.79) be diﬀerentiable in x0 . Then n n m n m ∂F 0 ∂Fj 0 (x ) = (x ) = Lji . F (x0 ) = ∂xi ∂xi i=1,j=1 i=1 i=1,j=1

31

1.4. Fundamental results

With h = (h1 , . . . , hn ) this yields the result n ∂F (x0 )hi = F (x + h) − F (x ) + o(h) = F (x )h = ∂xi i=1 0

0

0

n ∂Fj i=1

∂xi

m 0

(x )hi

. j=1

Theorem 1.46. Implicit function theorem: Let U ⊂ X , and V ⊂ Y be open sets, F : U¯ × V → Z, F ∈ C 1 (U¯ × V), and let (x0 , y0 ) ∈ U × V be a solution of F (x0 , y0 ) = 0. Furthermore, let ∂x F (x0 , y0 ) ∈ L(X , Z) be boundedly invertible. Then there exist neighborhoods U(x0 ) ⊂ U(x0 ) ⊂ U, V(y0 ) ⊂ V for x0 , y0 , and an operator, g : V(y0 ) → U(x0 ) such that F (g(y), y) = 0 ∀y ∈ V(y0 ).

(1.81)

Furthermore, g ∈ C (V(y0 )) and g is unique in the following sense: For every (x, y) ∈ U(x0 ) × V(y0 ) such that F (x, y) = 0 this x = g(y), and limy→y0 g(y) = g(y0 ). 1

Deﬁnition 1.47. C 1 diﬀeomorphism: Let M, N be arbitrary subsets of Banach spaces X , Y and let F : M → N . 1. F : M → N is called a C 1 diﬀeomorphism iﬀ F is bijective and F and F −1 are C 1 mappings. 2. F is called a C 1 local diﬀeomorphism at x0 ∈ M iﬀ U(x0 ) ⊂ M and F (U(x0 )) ⊂ N are neighborhoods of x0 and F (x0 ), respectively, and F : U(x0 ) → F (U(x0 )) is a C 1 diﬀeomorphism. Theorem 1.48. Local inverse function theorem [675], Theorem 4.F: Let the mapping F : U(x0 ) ⊂ M → N be C 1 with Banach spaces X , Y. Then F is a local C 1 diﬀeomorphism at x0 iﬀ F (x0 ) : U(x0 ) → F (x0 )(U(x0 )) is bijective. Corollary 1.49. Derivative of inverse functions [676], Corollary 4.37: Let F : U(x0 ) ⊂ M → N be in C 1 , and let F (x0 ) : M → N be bijective. Then F is a local C 1 diﬀeomorphism at x0 and, in a small neighborhood U(x0 ) of x0 : (F −1 ) (y) = (F (x))−1 with y = F (x), x ∈ U(x0 ).

(1.82)

Higher order derivatives of order l are obtained by applying the Gateaux deﬁnition for l independent h1 , . . . , hl . Then results are valid analogously to Theorem 1.42. Since they are hardly used in this book, but intensely in [120]. Starting with Theorem 1.46, we could then replace the C 1 by C r , r ≥ 1.

2 Elements of analysis for linear and nonlinear partial elliptic diﬀerential equations and systems 2.1

Introduction

We attempt to give a necessarily incomplete survey of this huge area. The presentation is restricted to facts which seem to be most important for our context. In particular, it is totally impossible to refer to all the many highly interesting results available in this area. Nonlinear elliptic diﬀerential equations or systems and their numerical solutions, including bifurcation and center manifolds in B¨ ohmer [120] are our main goal. Accordingly, we select the references essentially from among textbooks. The study of numerical solutions and bifurcations is based upon linearization and the Fredholm alternative for the linearized problem. 3 So we study linear and nonlinear elliptic diﬀerential equations and systems and their linearization of order 2 and 2m. The Fredholm alternative for the linearized problem shows that, in some sense, these systems behave similarly to ﬁnite-dimensional systems: the kernel and co-range of the linearized operator, A, have the same ﬁnite dimension unless parameters extend the kernel. Solutions for Au = f exist if and only if f is perpendicular to the kernel of the dual operator Ad . We summarize the existence and uniqueness results for solutions, and the Fredholm alternative for the bifurcation context. Regularity results for the solutions are necessary for good convergence of numerical methods. We only formulate proofs for results that are straightforward generalizations of the reviewed books. However, we omit the results related to the diﬀerent types of maximum principles and oscillations and the concept of viscosity solutions. Monotony and (nonlinear) coercivity techniques yield a lot of results, summarized here for our context. Good references for these important results are the books, by e.g. Gilbarg and Trudinger [346], Dong [299], Chen and Wu [170], Evans [310], Kreiss and Lorenz [454], Showalter [589], Taylor [618–620], and Zeidler [675–678]. The transition from order 2 to 2m, from one to several space variables, from linear to nonlinear, and from one elliptic diﬀerential equation to systems complicates the situation. Correspondingly, the results become less strong. In particular, not many regularity results for quasilinear and nonlinear equations and systems of higher order 3 Linearized problem always means the derivative of a problem and diﬀerentiable is always Frech´ et diﬀerentiable, unless stated otherwise.

2.1. Introduction

33

seem to be available in the textbooks and monographs used in this survey. For a fascinating survey of the historic development of the analysis of partial diﬀerential equations, see Br´ezis and Browder [143]. As motivation we consider the following simple semilinear model problem A u = −Δu + f (u) = g ∈ C(Ω), u|∂Ω = 0,

(2.1)

with the Laplacian Δ and continuous functions f, g. A straightforward way to solve it would be to determine a “classical solution” u0 ∈ C 2 (Ω) satisfying (2.1). In fact this is still one of the (necessary) solution concepts for nonlinear problems, which we will summarize in Section 2.5. For linear problems with f (u) = cu, c ∈ R, and nonlinear problems other important concepts have been developed. In particular, the nonlinear f (u) must not grow too fast for |u| → ∞, so growth conditions have to be imposed for f (·). Often a solution u0 ∈ C 2 (Ω) does not exist. So we have to generalize the “classical solution” u0 ∈ C 2 (Ω), Au0 ∈ C(Ω) in two steps into u0 ∈ H 2 (Ω), Au0 ∈ L2 (Ω), the “strong solution”, and u0 ∈ H 1 (Ω), Au0 ∈ H −1 (Ω), the “weak solution”. Accordingly, the boundary condition u0 |∂Ω = 0 has to be replaced by u0 ∈ H01 (Ω), cf. Theorem 1.24 and Remark 2.3. For linear problems the bilinear form a(·, ·) induced by A is an essential tool. Sobolev spaces H 1 (Ω), H01 (Ω), H −1 (Ω), and their generalizations W m,p (Ω), W0m,p (Ω), W −m,q (Ω), 1/p + 1/q = 1, cf. Subsection 1.4.3, the necessary boundary analysis for ∂Ω, and the trace operators are summarized in Section 1.4. We start in Sections 2.2–2.4 with linear problems. In Section 2.2, motivated by the linear version (2.2) of our above model problem (2.1). We determine appropriate concepts for generalized solutions in Hilbert spaces for a second order elliptic operator and the corresponding bilinear form. In this motivating section, we sometimes refer to deﬁnitions and properties introduced later on. Section 2.3 studies, in the Banach space setting, bilinear forms a(·, ·) and the induced linear operators A and vice versa. The norms, duals and invertibility of A are characterized by properties of a(·, ·). The concepts of Gelfand triples, coercive and elliptic bilinear forms a(·, ·) are introduced and relations to extremal problems are discussed. Finally the Riesz–Schauder theory, the Fredholm alternative and conditions for elliptic bilinear forms are studied. These results provide a solid foundation for the later linearization. General 2m-th order elliptic operators and the corresponding bilinear forms are the topic of Section 2.4. We formulate Fredholm alternatives, estimate the solutions, summarize the essential regularity results in H¨ older and Sobolev (Banach) spaces for 2m-th order elliptic operators with C ∞ coeﬃcients. C k coeﬃcients are allowed for order 2. Finally, weak solutions in Sobolev (Hilbert) spaces are studied. In Section 2.5 we introduce nonlinear elliptic problems. Most of them live in Banach spaces, often in H¨older or Sobolev spaces. Usually, growth conditions for the coeﬃcients have to be imposed. We start with the diﬀerent deﬁnitions and types of solutions and study semilinear, quasilinear and fully nonlinear problems. Examples and some existence, uniqueness, regularity and Fredholm-type results are added. In Section 2.6 we turn to linear and nonlinear systems of elliptic equations. Again we distinguish between elliptic and coercive systems, their compact perturbations, and the corresponding bilinear forms.

34

2. Analysis for linear and nonlinear elliptic problems

We do know that the linearized operators play a dominant role for the stability of numerical methods and for bifurcation. So we study in Section 2.7 the linearized forms of the nonlinear operators presented in Sections 2.5 and 2.6. It turns out that, under slightly stronger conditions, Fredholm-type results are available for all of them. One of the most challenging examples of nonlinear systems of elliptic equations are the Navier- Stokes equations, studied in Section 2.8. We start with the Stokes operator. It represents a (noncoercive) saddle point problem. Its extension to the nonlinear version requires trilinear forms and bilinear operators. Linearization of the nonlinear problems in this book yields linear operators, A, inducing bilinear forms, a(u, v), sums of Ω a∂ α u∂ β vdx, |α|, |β| ≤ m. For a ∈ L∞ (Ω), these are well deﬁned only for functions u, v ∈ W m,p (Ω) with 2 ≤ p ≤ ∞, but not for 1 ≤ p < 2, cf. Sections 2.2, 2.3, and Subsection 2.4.4. The A are compact perturbations E.g. the linearized Navier-Stokes operator compactly perturbs the invertible Stokes operator (with stable discretization). Similarly operators B with H m (Ω)-coercive bilinear forms are induced by their principal parts. For 1 ≤ p < 2, we would obtain W m,p (Ω)-coercive bilinear forms; unfortunately they are deﬁned no longer. We will show in Chapters 3 ﬀ. that operators with H m (Ω)-coercive bilinear forms are stable with respect to discrete H m (Ω) norms. Furthermore, linear boundedly invertible operators with stable consistent discretization essentially inherit this stability to their compactly perturbed invertible operators. Consistency results will be available for this large class of linear and nonlinear equations and systems as well. Stability of the linearized problem implies stability of the nonlinear problem as well. Therefore, linear and nonlinear operators A : W m,p (Ω) → W −m,q (Ω), 1/p + 1/q = 1, are particularly important, cf. Corollary 2.44. These results yield the key for a complete convergence theory for the following equations, systems and the main discretization methods for W m,p (Ω), 2 ≤ p ≤ ∞ w.r.t the H m (Ω) norm for 2 < p. This allows in many areas strong extensions of the state of the art, e.g. for wavelet and radial basis or mesh-free methods. The excluded case 1 ≤ p < 2 allows a complete convergence theory as well, however based upon monotone operators, cf. Section 4.5. Many problems in science, e.g. in mechanics, are related to variational problems deﬁned via integrals. We want to touch on some aspects. In many cases (systems of) elliptic equations are the so-called Euler systems of a variational problem, e.g. in Theorem 2.97. This is a main topic in the calculus of variations, see, e.g. the monographs by Giaquinta and Hildebrandt [341, 342]. The ﬁrst variation of these integrals applied to a test function yields the weak form, the so-called Euler system, and weak solutions of a PDE. A transformation of this equation into the strong form yields a PDE in divergence form, sometimes known as a divergent PDE. This transition from a variational problem to its ﬁrst variation or the weak, often divergence form is discussed in Subsection 2.6.4. Many of these problems are related to monotone operators. They are systematically discussed by Zeidler [677, 678] and partially summarized in Subsections 2.5.5, 2.5.6 and 2.6.4, Theorems 2.97, 2.98. For linear problems the divergence form and the general form can easily be exchanged, if only appropriate regularity properties for the coeﬃcients are guaranteed. Therefore in most of the literature nonlinear problems are studied in divergence form. We too concentrate our presentation mainly on this form. However, the exceptions in

2.1. Introduction

35

Subsections 2.5.3, 2.5.7 and 2.6.8 are important. For the general case we refer, e.g. to Ladyˇzenskaja and Uralceva [463, 464, 466]. For fully nonlinear elliptic problems, there is no divergence form corresponding to the strong form, and, hence, no weak form or solution of a PDE in strong form. This is relevant for their dicretizations, formulated for the ﬁrst time for the general case in this book, mainly as an FEM. In the preface we have already mentioned the viscosity solutions. We do not discuss them here. Most of the following results are valid only for uniformly elliptic equations. The case of nonuniformly elliptic equations is indicated in Subsection 2.5.3, and in, e.g. Ivanov [415]. For this chapter we consulted only a few of the original papers, but mainly refer to the textbooks and monographs cited in the text. We begin with a short list, not aiming for completeness. Elliptic PDEs of second order, including quasilinear and fully nonlinear equations are studied by Ladyˇzenskaja and Uralceva [464], Gilbarg and Trudinger [346], and Chen and Wu [170]. More general PDEs essentially for the linear case, are discussed by Garabedian [336], H¨ ormander [402], Neˇcas [509,510] and Wloka [667]. PDEs, now including nonlinear problems, are further studied by Evans [310] and Dong [299]. Evans [310] extensively discusses diﬀerent representations for solutions for ﬁrst and second order equations. His, and some of the other, theoretically oriented books extend the theory for linear equations to nonlinear problems. Usually the H¨ older, Sobolev and Schauder results are complemented with maximum principles. Dong [299] concentrates on second order equations and presents several important examples mainly of parabolic type. Two impressive “multi-graphs” on linear and nonlinear problems are due to Zeidler [675–678], and Taylor [618–620]. Zeidler’s main interest is nonlinear functional analysis with applications to PDEs. He gives a large number of results for linear and nonlinear PDEs. Taylor more directly aims for equations and systems of PDEs. Both combine many of the preceding techniques for elliptic equations and systems, Taylor even on manifolds. Triebel [634,635] initiated and obtained a new hierarchy of spaces, the so-called Besov–Triebel–Lizorkin spaces and many results for nonlinear equations. Based upon these results, Runst and Sickel [561] develop a very general theory for nonlinear PDEs. Polyanin and Zaitsev [535] collect the exact solutions for many nonlinear elliptic (parabolic and hyperbolic) equations. Necessarily, in our context, the maximum principle, lower and upper solutions and oscillation are neglected compared to the preceding monographs. Morrey [500] and Giaquinta and Hildebrandt [341, 342] study mainly quasilinear diﬀerential equations related to variational problems, a main topic in the calculus of variations. For higher order PDEs nonhomogeneous boundary conditions are far more complicated, in contrast to second order; in fact they are a serious problem. They are presented by Lions and Magenes [478–480]. For more literature see the following sections. In this book, we mainly consider isolated solutions u0 with boundedly invertible linearized operator G (u0 ). Extensions are described by B¨ohmer [120]. For isolated solutions, Theorem 1.46 and the results from Section 2.4 usually allow us the restriction to homogeneous boundary conditions. But for fully nonlinear problems, we discuss the separate role of diﬀerential and boundary operators. So, in general, it is possible to deﬁne an exact extension of the boundary function and use the standard substitution trick. Alternatively, an approximating function, extended to the full domain, yields a small error at the boundary, causing only a small change in the exact solution.

36

2.2

2. Analysis for linear and nonlinear elliptic problems

Linear elliptic diﬀerential operators of second order, bilinear forms and solution concepts

In this section we motivate diﬀerent solution concepts for the easiest case of second order linear equations. The linear form of our above model problem (2.1) is A u = −Δu + cu = f ∈ C(Ω), u|∂Ω = 0.

(2.2)

But even for c > 0, f ∈ C(Ω), and a smooth enough ∂Ω, a “classical solution” u0 ∈ C 2 (Ω) with u0 C 2 (Ω) ≤ Cf C(Ω) does not always exist, hence A : C 2 (Ω) → C(Ω) is not boundedly invertible. For a counterexample, see Hackbusch [387] and for appropriate modiﬁcations see, e.g. our Theorem 2.37. Linear functionals allow us a way out of this dilemma. From Deﬁnition 1.8 we do know that for a Hilbert space V and its dual V , f ∈ V

with

f, vV ×V = 0 ∀ v ∈ V

is equivalent to f = 0.

(2.3)

So we use linear functionals for testing A u = −Δu + cu = f . We relax A u ∈ C(Ω) in two steps into A u ∈ L2 (Ω) and then A u ∈ H −1 (Ω). We have to relate operators, bilinear forms and boundary terms. We use the standard notations for partial derivatives ∂u ∂2u , ∂i∂j u = , ∂ 0 u := u, Δu := (∂ i )2 u. ∂ u := ∂xi ∂xi ∂xj i=1 n

i

(2.4)

Unless stated otherwise, we require throughout this chapter: Ω ⊂ Rn is open, bounded and ∂Ω is Lipschitz-continuous.

(2.5)

For this Ω the corresponding spaces are deﬁned with respect to their Sobolev (Hilbert) space inner products, norms, and seminorms, with H 0 (Ω) = L2 (Ω), (u, v)H 1 (Ω) :=

n

n

(∂ i u, ∂ i v)L2 (Ω) , (u, v)H 2 (Ω) :=

i=0

(∂ j ∂ i u, ∂ j ∂ i v)L2 (Ω) , (2.6)

i,j=0 1/2

uH i (Ω) := (u, u)H i (Ω) , i = 0, 1, 2, |u|H 1 (Ω) :=

n

∂ i u2L2 (Ω)

1/2 .

i=1

As usual, H −i (Ω), i = 1, 2, denote the dual spaces of bounded linear functionals on H i (Ω), i = 1, 2. We start testing (2.2) with v ∈ L2 (Ω) in the sense of (2.3) and for u ∈ H 2 (Ω) and introduce classical, strong and weak operators and bilinear forms. In contrast to the standard literature we use in this section diﬀerent notations Ac , for the strong forms deﬁned on C 2 and As , as (·, ·), on H 2 , sometimes called the Friedrichs extension, and

2.2. Linear elliptic diﬀerential operators of second order

37

A, a(·, ·) for the weak forms in (2.11): Ac : C 2 (Ω) → C(Ω), Ac u = −Δu + cu,

(2.7)

As : H (Ω) → L (Ω), As u = −Δu + cu, and 2

2

as (·, ·) : H 2 (Ω) × L2 (Ω) → R deﬁned by as (u, v) := (As u, v)L2 (Ω) = (−Δ u + cu) vdx ∀v ∈ L2 (Ω).

(2.8)

Ω

The condition u ∈ H (Ω) still is too restrictive. We apply (2.9). Zeidler [677], pp. 19 and 20, calls this one of the cornerstones of modern analysis. For v, w ∈ H 1 (Ω) we b b generalize the classical integration by parts formula, a v wdx = vw|ba − a vw dx. With the outer normal ν = (ν1 , . . . , νn )T and a Lipschitz-continuous manifold ∂Ω, we obtain the fundamental Green’s formula, cf. Ciarlet [174], p. 14, w∂v/∂xi dx + v∂w/∂xi dx = ∂(vw)/∂xi dx = vwνi ds, (2.9) 2

Ω

Ω

Ω

∂Ω

∂Ω ∈ CL , ∀w, v ∈ H 1 (Ω). We apply this to the last equation in (2.8) to get the ﬁrst Green’s formula ∂u vds, (−Δ u + cu) vdx = as (u, v) = (∇ u, ∇ v)n + cuvdx − ∂Ω ∂ν Ω

(2.10)

Ω

with the Euclidean product and norm (·, ·)n and | · |n in Rn . Note that ∂Ω ∂u/∂ν vds = 0 if ∀∂u/∂ν ∈ L2 (∂Ω) and ∀v ∈ H 1 (Ω) with v = 0 on ∂Ω. By Theorems 1.37, 1.38, we denote this as ∀v ∈ H01 (Ω) in the trace sense. This allows us to reduce the diﬀerentiability conditions for u and the solution u0 . We introduce the weak bilinear form and operator, a(·, ·), A sometimes called the energy extension, as a(·, ·) : H 1 (Ω) × H 1 (Ω) → R and A : H 1 (Ω) → H −1 (Ω) deﬁned by (2.11) a(u, v) := ∇ u∇ v + cuvdx =: Au, vH −1 (Ω)×H 1 (Ω) , thus, for “regular” u, Ω

∂u vds and = as (u, v), ∂Ω ∂ν ∂u ∂u vds = 0 for ∀v|∂Ω1 = 0 and ∀ = 0, ∂ν ∂ν ∂Ω2

∀u ∈ H 2 (Ω), v ∈ H 1 (Ω) : a(u, v) = as (u, v) + if ∂Ω1 ∪ ∂Ω2 = Ω and ∂Ω

often with ∂Ω1 = ∂Ω or ∂Ω2 = ∂Ω. The restriction v|∂Ω1 for v ∈ H 1 (Ω) yields v|∂Ω ∈ H 1/2 (Ω) in the trace sense. For our Ω in (2.5), this is an immediate consequence of Theorems 1.37, 1.39. Remark 2.1. 1. For the weak form a(u, v) we need u, v ∈ H 1 (Ω), and for the strong form as (u, v) we need u ∈ H 2 (Ω), v ∈ L2 (Ω). The relation a(u, v) = as (u, v) + ∂Ω ∂u/∂ν vds,

38

2. Analysis for linear and nonlinear elliptic problems

shows that a(u, v) = as (u, v) if ∂Ω ∂u/∂ν vds = 0. In (2.11), u ∈ H 2 (Ω) implies ∂u/∂ν|∂Ω2 ∈ H 1/2 (Ω), and a well-deﬁned ∂Ω ∂u/∂ν vds. Under appropriate conditions even for u ∈ H 1 (Ω) a ∂u/∂ν|∂Ω ∈ H −1/2(Ω) is possible, cf. Costabel [211]. With v ∈ H 1 (Ω), hence v|∂Ω ∈ H 1/2 (Ω) the ∂Ω ∂u/∂ν vds is still deﬁned. Under those conditions, the previous Neumann boundary value, but not the strong form as (u, v), in (2.11) is well deﬁned for u ∈ H 1 (Ω). Other results for this problem of boundary traces are discussed, e.g. in Taylor [618], Chapter 4, Proposition 4.5, and Jonsson and Wallin [425], Jerison and Kenig [419], Schwab [575] and Grisvard [373]. We do not consider these extensions here in more detail. 2. For the later nonconforming discretization methods in Chapters 5 and 7, and adaptive FEMs in Chapter 6, we will consider (2.11) on subtriangles T ⊂ Ω, indicated as aT (u, v) = as,T (u, v) + ∂T ∂u/∂ν vds. This will turn out to be one of the deciding tools for estimating the consistency errors in the discrete generalization of the a(u, v). The piecewise functions, uh |T , usually polynomials, in FEMs and DCGMs always satisfy uh ∈ H k (T ), k ≥ 2, hence aT (uh , v h ) = as,T (uh , v h ) + ∂T ∂uh /∂ν v h ds. For the corresponding consistency estimates, we will require u as close as possible to the u ∈ H 1 (Ω) in the weak form. Therefore motivated by Theorems 1.37, 1.39 and 2.48, we replace the above u ∈ H 1 (Ω) either by u ∈ H 2 (Ω) as in (2.11), or require, in DCGMs, u ∈ H 3/2+ (Ω) with any > 0. Then ∂u/∂ν|∂Ω2 ∈ H (∂Ω) ⊂ L2 (∂Ω) is valid and ∂Ω ∂u/∂ν vds is well deﬁned, cf. Remarks 5.51 and 7.1. 3. For simplicity, we assume, here and for analogous cases, unless stated otherwise: For natural or Neumann boundary conditions let u ∈ H 2 (Ω) or u ∈ H 3/2+ (Ω), hence ∂u/∂ν|∂Ω2 ∈ L2 (∂Ω). Obviously the two operators and bilinear forms deﬁned in (2.8) and (2.11) are diﬀerent. We could even go one step further: By an additional partial integration, again using (2.9), one would end up with the distributional bilinear form u(−Δv + cv)dx ∀ v ∈ C0∞ (Ω), ∀ u ∈ (C0∞ (Ω)) . (2.12) ad (u, v) := Ω

This allows us to formulate solutions u0 ∈ (C0∞ (Ω)) , the space of distributions. We do not study this possibility in this book; see, e.g. Lions and Magenes [478], Runst and Sickel [561] Showalter [589]. Based upon (2.8) and (2.11), we are now able to introduce the concepts of classical and strong and weak solutions u0 = uc,0 and u0 = us,0 and u0 = uw,0 , respectively, of Au = f. We use uc,0 , us,0 or uw,0 and Ac , As , A = Aw only if the context does not directly show the chosen solution concept. The existence, uniqueness and regularity of the solutions are studied in the following sections. These regularity conditions will show that the type of solution uc,0 , us,0 or uw,0 is essentially a consequence of the smoothness of the problem. Note that the deﬁnition of the linear operators and bilinear forms does not require boundary conditions, but the following deﬁnition of the solutions does require boundary conditions, e.g. u0 ∈ H01 (Ω). The

2.2. Linear elliptic diﬀerential operators of second order

39

classical solution u0 = uc,0 and the strong solution us,0 are then deﬁned, e.g. for Dirichlet boundary conditions, by u0 = uc,0 ∈ C 2 (Ω), u0 |∂Ω = 0 : As u0 = −Δu0 + cu0 = f ∈ C(Ω),

(2.13)

u0 = us,0 ∈ H 2 (Ω) ∩ H01 (Ω) : As u0 = −Δu0 + cu0 = f ∈ L2 (Ω) ⇔ (2.14) f vdx = (As u0 , v)L2 (Ω) = (−Δ u0 + cu0 )vdx = as (u0 , v) ∀ v ∈ L2 (Ω).

Ω

Ω

The weak solution u0 is deﬁned, with f ∈ H −1 (Ω) instead of f ∈ L2 (Ω), by u0 = uw,0 ∈ H01 (Ω) ⇔ A u0 = f ∈ H −1 (Ω) ⇔ Au0 , vH −1 (Ω)×H 1 (Ω) = (∇ u0 , ∇ v)n + c u0 v dx = a(u0 , v) 0

(2.15)

Ω

f0 v +

= Ω

n

fj ∂ j vdx

j=1

= f, vH −1 (Ω)×H 1 (Ω) ∀ v ∈ H01 (Ω), cf. (2.107). 0

As a special case of Theorem 2.43 and Corollary 2.44, we obtain Theorem 2.2. For c ≥ 0 and f ∈ H −1 (Ω) or f = f0 ∈ L2 (Ω) the problem (2.15) has a unique solution u0 ∈ H01 (Ω). In (2.15) we required f ∈ H −1 (Ω) and tested with v ∈ H01 (Ω), indicating the need for Au0 = f ∈ H −1 (Ω). In fact, there are reasons for distinguishing and for identifying H −1 (Ω) and H0−1 (Ω). Since we need this discussion for all the following more general cases, we formulate it here. Remark 2.3. We assume a bounded Ω as in (2.5). Then, according to Proposition 2.34, cf. Adams [1], pp. 49 ﬀ. W0m,p (Ω) ⊂ W m,p (Ω) = W0m,p (Ω). On the other hand, Taylor [618], Chapter 4, Proposition 5.1, proves, for a smooth Ω, hence ∂Ω ∈ C ∞ , cf. Proposition 1.32, and m ∈ N0 , that there is a natural and norm-preserving isomorphism H0−m (Ω) = dual space of H0m (Ω) ≈ H −m (Ω). This fact is one of the reasons why sometimes in the literature H −m (Ω) and H0−m (Ω), and W −m,q (Ω), 1/p + 1/q = 1, and W0−m,q (Ω) are not distinguished. On the other hand, often only f ∈ W −m,q (Ω) are used for the f ∈ W0−m,q (Ω), e.g. in Proposition 2.34. That H −m (Ω) and H0−m (Ω) are in fact diﬀerent is demonstrated by (2.107) versus (2.110). Now we extend (2.15) to more general boundary and elliptic operators of second order. The strong operator As can be directly deﬁned and is related to its strong

40

2. Analysis for linear and nonlinear elliptic problems

bilinear form as (·, ·) , with real valued coeﬃcients as As : H 2 (Ω) → L2 (Ω), as (·, ·) : H 2 (Ω) × L2 (Ω) → R and As u :=

n

(2.16)

(−1)j>0 ∂ j (aij ∂ i u) with (−1)j>0 := 1 for j = 0 else := −1,

i,j=0

as (u, v) := (As u, v)L2 (Ω) , ∀u ∈ H 2 (Ω), v ∈ L2 (Ω), aij ∈ W 1,∞ (Ω) and (2.17) Ac : C 2 (Ω) → C(Ω), Ac := As |

C 2 (Ω) ,

aij ∈ C 1 (Ω).

Even for Ac : C 2 (Ω) → C(Ω) test spaces can be deﬁned as functions of bounded variation deﬁning a bilinear form for uc,0 ∈ C 2 (Ω). We do not consider this here, nor generalize the above distributional bilinear form ad (·, ·) in (2.12). For the (standard) weak problem we directly deﬁne the weak bilinear form a(·, ·) and the induced A by partial integration of (2.16) and the Green’s formula with vanishing boundary terms as a(·, ·) : H 1 (Ω) × H 1 (Ω) → R and A : H01 (Ω) → H −1 (Ω), with n aij ∂ i u∂ j vdx a(u, v) :=

(2.18)

Ω i,j=0

=: Au, vH −1 (Ω)×H 1 (Ω) , ∀u, v ∈ H01 (Ω), aij ∈ L∞ (Ω). 0

Deﬁnition 2.4. Principal part and ellipticity: The principal parts Ap and ap (u, v) of the operator, A, and its bilinear form, a(u, v), in (2.18) are Ap u :=

n

(aij ∂ i u)∂ j ∈ H −1 (Ω) and

i,j=1

ap (u, v) := Ω

⎛ ⎝

n

i,j=1

⎞ aij ∂ i u ∂ j v ⎠ dx =

(Ap u) ◦ vdx,

(2.19)

Ω

and similarly for As , Ac and as (·, ·). For ϑ = (ϑ1 , . . . , ϑn )T ∈ Rn let the characteristic polynomial of Ap satisfy; 4 n

aij (x)ϑi ϑj ≥ 0 and ≥ |ϑ|2n , for an > 0, ∀x ∈ Ω, ϑi ∈ R, respectively,

i,j=1

(2.20) the operator A (or As , Ac ) is called with the Euclidean norm |ϑ| = |ϑ|n in Rn . Then elliptic (≥ 0), and strongly elliptic ≥ |ϑ|2n . For aij ∈ L∞ (Ω) (or aij ∈ W 1,∞ (Ω)) this condition is only required almost everywhere (a.e.) in Ω. 4

It is always possible to use ≥ instead of ≤ in (2.20) by multiplying A by −1.

2.2. Linear elliptic diﬀerential operators of second order

41

Obviously, the above bilinear forms and linear operators in (2.16), (2.18) are continuous or bounded, that is, positive constants, C, exist, such that |a(u, v)|

< CuH 1 (Ω) vH 1 (Ω) ∀u, v ∈ H 1 (Ω),

|as (u, v)|

< CuH 2 (Ω) vL2 (Ω) ∀u ∈ H 2 (Ω), v ∈ L2 (Ω),

(2.21)

AuH −1 (Ω) < CuH 1 (Ω) ∀u ∈ H 1 (Ω), As uL2 (Ω) < CuH 2 (Ω) ∀u ∈ H 2 (Ω) and Ac uC(Ω)

< CuC 2 (Ω) ∀u ∈ C 2 (Ω).

As an immediate consequence of (2.19), (2.20), we obtain, for a strongly elliptic diﬀerential operator of second order, the so-called H01 (Ω) coercivity of the principal part: for aij ∈ L∞ (Ω), (2.20) implies

ap (u, u) ≥ |u|2H 1 (Ω) .

(2.22)

For u ∈ H01 (Ω) the seminorm |u|H 1 (Ω) is equivalent to the standard Sobolev norm uH 1 (Ω) , see (2.6). In fact, (2.21), (2.22) show the equivalence of · H 1 (Ω) and (ap (·, ·))1/2 in H01 (Ω), for As = −Δ, in general, called the energy norm. A generalization of (2.21) to A : W 1,p (Ω) → W −1,q (Ω), 1 ≤ p ≤ ∞, 1/p + 1/q = 1, with test spaces Vb = W 1,p (Ω) and to As : W 2,p (Ω) → Lq (Ω) with test spaces Lp (Ω), is possible as well; see Sections 2.4 and 2.5, and, e.g. [561, 618–620, 634, 635]. For higher order problems we will consider A : W m,p (Ω) → W −m,q (Ω). However, generalizations to A : C 2 (Ω) → C(Ω), and a corresponding theory is not appropriate, but only for A : C 2,γ (Ω) → C γ (Ω), 0 < γ < 1, see Subsections 2.4.2 and 2.4.3. Again the relation between the weak and strong bilinear forms is based upon the so-called natural boundary conditions. With the outer normal ν = (ν1 , . . . , νn )T on a C 1 manifold ∂Ω, we obtain from (2.9) or the Gauss integral theorem, v∂ j (aij ∂ i u)dx = − (∂ j v)aij ∂ i u + vaij ∂ i uνj ds. (2.23) Ω

Ω

∂Ω

This is applied to a(·, ·) in (2.18) yielding a generalized ﬁrst Green’s formula (Ba u)vds = as (u, v) + (Ba u)vds a(u, v) = (As u)vdx + Ω

∂Ω

(2.24)

∂Ω

with Ba u :=

j=1,...,n

νj aij ∂ i u,

the natural boundary operator.

(2.25)

i=0,...,n

Sometimes, instead of the boundary operator, Ba , a more general form is used: Bu :=

n i=1

bi ∂ i u + b0 u

with

b = (b1 , . . . , bn ) ∈ Rn , (ν, b)n = 0.

(2.26)

42

2. Analysis for linear and nonlinear elliptic problems

Ba u|∂Ω and Bu|∂Ω ∈ H −1/2 (∂Ω) or Bu|∂Ω ∈ L2 (∂Ω) are well deﬁned for u ∈ H 3/2+ (Ω), cf. Remark 2.1 and Theorem 1.38. In contrast to B we call Ba the natural, or induced boundary operator, since it is “induced” by a(·, ·). For the special case of (2.2) we obtain Neumann boundary conditions As u = −Δu + cu induces Ba u = ∂u/∂ν.

(2.27) n

The last condition, (ν, b)n = 0, in (2.26) excludes tangential derivatives i=1 bi ∂ i u of u along ∂Ω. By rescaling Bu we can obtain (ν, b)n = ν T b = ν T Aν = 0 for a strongly elliptic operator A, see Proposition 2.5, Remark 2.6, and Hackbusch [387], Remark 7.4.10. The diﬀerential operator Au = f and on ∂Ω Dirichlet and natural boundary conditions Bu := u = ϕ and Ba u = ϕ are tested by Vb , and are realized for u0 ∈ H 1 (Ω) with Vb = H01 (Ω) for Dirichlet boundary conditions: H 1/2 (Ω) u0 |∂Ω = 0 and Vb = V = H 1 (Ω) for natural boundary conditions: we require u ∈ H (Ω), implying H 2

1/2

(2.28)

(∂Ω) Ba u0 |∂Ω = ϕ

in the trace sense, cf. Remark 2.1 and Theorems 1.38, 1.37 and 2.43. Since v|∂Ω ∈ H 1/2 (∂Ω) for v ∈ H 1 (Ω), the ∂Ω (Ba u)vds is again well deﬁned in this case, cf. textbooks on PDEs, e.g. [135, 141, 387]. By the standard trick u := u − u ¯ with u ¯ = ϕ on ∂Ω we obtain homogeneous Dirichlet boundary conditions. In fact we may use Theorem 1.29 for extending the boundary function, ϕ to Ω, yielding ϕ¯ and thus u ¯ := ϕ, ¯ see Remark 2.7. An extension to parts of ∂Ω with Dirichlet and natural boundary conditions as in (2.11) is presented in most textbooks, e.g. [141]. Therefore we only indicate it here; ¯ = ϕ on an open subset ∂Ω1 of the compare Remark 2.7 below. For u = ϕ or Ba u ¯ on Ω and again use the boundary ∂Ω we ﬁrst extend ϕ from ∂Ω1 to ∂Ω and then to u above trick, applicable in later sections as well. Then we use as test space Vb := {u ∈ H 2 (Ω) : u|∂Ω1 = 0}.

(2.29)

We determine the weak and strong solution u0 = uw,0 and u0 = us,0 of the Dirichlet boundary value problem with the bilinear forms in (2.18) and (2.16) by uw,0 = u0 ∈ H01 (Ω) : a(u0 , v) = f, vH −1 (Ω)×H 1 (Ω) ∀v ∈ H01 (Ω), and

(2.30)

us,0 = u0 ∈ H01 (Ω) ∩ H 2 (Ω) : as (u0 , v) = (f, v)L2 (Ω) ∀v ∈ L2 (Ω), hence Au0 = Auw,0 = f ∈ H −1 (Ω) and As u0 = As us,0 = f ∈ L2 (Ω), respectively In Section 2.4 we will return to classical solutions, As us,0 = f ∈ C(Ω). For a smoother u0 ∈ H 2 (Ω) ∩ H01 (Ω) and u0 |∂Ω = 0, (2.24) and (2.25) imply uw,0 = u0 = us,0 , since H 1 (Ω) is dense in L2 (Ω), so 2 1 f vdx ∀v ∈ H01 . (2.31) u0 ∈ H ∩ H0 (Ω) : a(u0 , v) = (As u0 )vdx = as (u0 , v) = Ω

Ω

2.2. Linear elliptic diﬀerential operators of second order

43

The natural boundary operator Ba was “induced” by a(·, ·). By partial integration of as (·, ·) we obtained a(·, ·) and a boundary term deﬁned via Ba . But a(·, ·) does not necessarily uniquely determine As and Ba and vice versa. So the question naturally arises whether a boundary value problem with a general boundary operator B as in (2.26) uniquely determines a bilinear form a(·, ·) and vice versa. Let, cf. (2.16), (, )n be the Euclidean scalar product in Rn , As u :=

n

(−1)j>0 ∂ j (aij ∂ i u) and with b(x) := (b1 (x), · · · , bn (x)) ∈ Rn , (2.32)

i,j=0

B u :=

n

bi ∂ i u assume (b(x), ν(x))n = 0 ∀x ∈ ∂Ω.

(2.33)

i=0

n i If (b(x), ν(x))n = 0 for a speciﬁc x ∈ ∂Ω were allowed, i=1 bi (x)∂ u would be a tangential derivative of u in x. This sometimes does not make sense, however see Wloka [667] p. 161 ﬀ., where b0 ≡ 0, (b(x), ν(x)) = 0 is discussed. Hackbusch’s [387] Theorem 7.4.11 and Remark 7.4.12 show Proposition 2.5. Let As and B be given as in (2.32), (2.33). Then there exists a modiﬁed bilinear form a1 (·, ·) : H 1 (Ω) × H 1 (Ω) as in (2.18), such that the variational formulation as f vdx for u0 ∈ H 2 (Ω) ∀v ∈ H 1 (Ω), Ba1 u0 = Bu0 |∂Ω = ϕ (2.34) a1 (u0 , v) = Ω

corresponds to the classical formulation As u = f in Ω and Bu = ϕ on ∂Ω. If a(·, ·) in (2.18), corresponding to As in (2.16), is H 1 (Ω) elliptic, cf. Deﬁnition 2.17, and the coeﬃcients of As and Ω are C 1 , this 5 modiﬁed a1 (·, ·) is again H 1 (Ω) elliptic. Remark 2.6. 1. This proposition allows us to restrict the discussion for m = 1 to the Dirichlet or the natural boundary condition, or a combination, see below. 2. It is important to realize the impact of the boundary conditions, cf. Treves [632], Chapter 37. In fact, for f ∈ L2 (Ω), we obtain identical strong and weak solutions, u0 = us,0 = uw,0 , in (2.31) if one of the following conditions is satisﬁed, see (2.24). Either u0 ∈ H 2 (Ω) satisﬁes the natural boundary condition Ba (u0 )|∂Ω = 0, for Ba see (2.25). Then the boundary term drops out

5 Hackbusch indicates that it might be advantageous to employ additional boundary integral terms. This would avoid a smooth extension of a function, deﬁned on ∂Ω, to Ω. Theorem 1.29 and its proof in Gilbarg and Trudinger [346] presents a constructive way to solve this problem. Thus a1 (·, ·) can be constructed in the form (2.18).

44

2. Analysis for linear and nonlinear elliptic problems

∀v ∈ H 1 (Ω). Or u0 satisﬁes the Dirichlet boundary condition u0 |∂Ω = 0. Then the test functions are chosen as v ∈ H01 (Ω) = Vb . Again the boundary term drops out. Finally, Vb has to be appropriately modiﬁed, cf. (2.143), for a combination as in (2.11). For natural boundary conditions Ba u = ϕ on ∂Ω the reduction to Ba u = 0 as for Dirichlet conditions, is usually not considered. Instead, we choose u0 ∈ U = V = H 1 (Ω) and solve Au0 = f ∈ H −1 (Ω).

(2.35)

Note that for u0 ∈ H 2 (Ω) and (2.5), the strong forms As u0 ∈ L2 (Ω) and Ba u0 ∈ H 1/2 (∂Ω) are well deﬁned. Even for u0 ∈ H 1 (Ω), Ba u0 ∈ H −1/2 (∂Ω), the dual of H 1/2(∂Ω), might make sense, cf. Remark 2.1 and Theorem 1.37. So for v ∈ H 1/2 (∂Ω), the ∂Ω Ba u v ds are well deﬁned. So the strong solution u0 = us,0 of the natural boundary value problem solves us,0 = u0 ∈ H 2 (Ω) : As u0 = f ∈ L2 (Ω), Ba u0 = ϕ ∈ H 1/2 (Ω).

(2.36)

Returning to (2.24), we obtain instead of (2.31), with the smoother u0 ∈ H 2 (Ω), a(u0 , v) = (As u0 )vdx + Ba u0 v ds = as (u0 , v) + Ba u0 v ds (2.37) Ω

=

f vdx +

Ω

∂Ω

∂Ω

∂Ω

ϕ v ds or, more generally, = f, vV ×Vb + b

ϕ v ds, ∂Ω

with Vb = H01 (Ω), cf. Proposition 2.34 and Remark 7.1. Remark 2.7. 1. Equation (2.37) is used in two steps: Firstly, we insert v ∈ H01 (Ω) and delete the boundary integral term. This yields Au0 − f = 0 ∈ H −1 (Ω) or even Au0 − f = 0 ∈ L2 (Ω). For the previous Dirichlet conditions with u0 , v ∈ H01 (Ω) = Vb = Ub we would be done. Natural boundary conditions require a second step: By the , v) = f, v or = f vdx we are left with B u v ds = previous a(u 0 V ×V Ω ∂Ω a 0 1 ϕ v ds ∀v ∈ H (Ω), yielding B u | = ϕ. a 0 ∂Ω ∂Ω 2. An extension to parts of ∂Ω with Dirichlet and natural boundary conditions as in (2.11) is presented in most textbooks, e.g. Treves [632], Chapter 37, and Hackbusch [386, 387], Section 7.4 and Brenner and Scott [141]. Therefore we only indicate it here. For Dirichlet and natural boundary conditions let ∂Ω = ∂Ω1 ∪ ∂Ω2 be the union of two disjoint “regular” subsets. For u = ϕ on ∂Ω1 ⊂⊂ ¯ on Ω and again use ∂Ω = ∂Ω1 we ﬁrst extend ϕ from ∂Ω1 to ∂Ω and then to u ¯. Then the above trick for enforcing homogeneous Dirichlet conditions for u0 − u we use as test and solution space Vb := {v ∈ H 1 (Ω) : v|∂Ω1 = 0}

(2.38)

and, cf. (2.11), (2.24), (2.31), deﬁne ˆ f vdx + ϕ v ds or = f, vV ×Vb + f (v) := Ω

∂Ω2

b

∂Ω2

ϕ v ds.

2.3. Bilinear forms and induced linear operators

2.3

45

Bilinear forms and induced linear operators

This presentation is strongly inﬂuenced by Hackbusch [386, 387]. His proofs for the Hilbert space situation can usually be simply modiﬁed to get the corresponding results for Banach spaces, see e.g. Proposition 2.18. However, motivated by the discussion at the beginning of this chapter, Hilbert spaces are still the most important for this book. We use the notation of many authors in numerical analysis, and exchange, in contrast to [387] ellipticity to coercivity and vice versa. Hence coercive bilinear forms induce (generalized energy) norms and elliptic bilinear forms are induced by elliptic diﬀerential operators. We have seen in the last section that the linear diﬀerential operators which we want to consider in this context are closely related to bilinear forms. In fact they essentially determine each other uniquely, however see Proposition 2.5. So we start in this section by studying bilinear forms and the properties of linear operators induced by them. We will apply and extend this in the following sections to linear and nonlinear elliptic diﬀerential operators of order 2m in Sobolev spaces, see Adams, [1]. We assume V is a real Banach space,

(2.39)

and indicate whenever we need reﬂexivity. Linear and nonlinear (weak) diﬀerential operators related to bilinear or nonlinear forms are mostly studied as A : V → V , tested with v ∈ V. Often forms As : V → Y are used, e.g. for H¨older spaces in the next section. The discussion of bilinear forms in this section allows us to apply these results to linear and nonlinear operators and systems in Sections 2.4–2.6 and their linearization in Section 2.7. In the numerical context, they are often appropriately deﬁned in Banach spaces with diﬀerent spaces, U, for solutions and, V, for test functions. For our theoretical goals in this section, this is hardly ever relevant. So we mainly discuss the case U = V. For many of the following results, e.g. coercivity and the inf–sup condition, boundary conditions are mandatory. If we want to emphasize this we use the notation Vb for V, where the index b indicates the boundary conditions. These Vb are closed subspaces of the original V, hence again are Banach spaces. So we formulate most results in this section with respect to V. Deﬁnition 2.8. Properties of bilinear forms: Let a bilinear form a(·, ·) be deﬁned on U × V → R, e.g. V = H 1 (Ω) or V = W0m,p (Ω). This a(·, ·) is called continuous or bounded, if there exists a constant Cb ∈ R+ such that |a(u, v)| ≤ Cb uU vV ∀ u ∈ U, v ∈ V.

(2.40)

If in (2.41)–(2.43), U = V, this a(·, ·) is called V-coercive, if it is bounded and there exists α ∈ R+ such that a(u, u) > αu2V ∀u ∈ V.

(2.41)

The dual bilinear form ad (·, ·) : V × V → R for a(·, ·) is deﬁned as ad (u, v) := a(v, u) ∀ u, v ∈ V.

(2.42)

46

2. Analysis for linear and nonlinear elliptic problems

The a(·, ·) is called symmetric if a(u, v) = ad (u, v) = a(v, u) ∀ u, v ∈ V.

(2.43)

(For complex Banach spaces, a(u, u) and a(v, u) in (2.41) and (2.42), (2.43) have to be replaced by a(u, u) and a(v, u), respectively.) Thus a V-coercive bounded bilinear form introduces a norm, · a . Indeed, see (2.40), (2.41), this · a is equivalent to the norm · V on V. So, positive constants, γ1 = (α)1/2 , γ2 = (Cb )1/2 , exist such that 0 < γ1 < γ2 and γ1 uV < ua := (a(u, u))1/2 < γ2 uV ∀u ∈ V.

(2.44)

For the special case As u = −Δu or aij = δij , a00 = 0 the corresponding ua = ((∇u, ∇u)L2 (Ω) )1/2 = ∇uL2 (Ω) is denoted as the energy norm. Remark 2.9. Often the bilinear form a(·, ·) is deﬁned, as in Section 2.2 and Subsection 2.4.4, via an elliptic diﬀerential operator. Then the standard choice for V is H m (Ω). An extension to V = W m,p (Ω) is possible for p ≥ 1. For Ω in (2.5), W m,p (Ω) ⊂ W m,q (Ω), p ≥ q, is densely and continuously embedded, cf. Theorem 1.26. For the linearization of quasilinear and fully nonlinear equations, see e.g. Section 2.7.3, Theorem 2.122, 1 ≤ p ≤ ∞, is required, cf. the beginning of Subsection 2.4.4. Only for 2 ≤ p ≤ ∞ do we obtain the full convergence results for all discretizations with respect to the discrete H m (Ω)norm. We use, for linear operators, A : U → V , with U, V, Banach spaces, the notation L(U, V ) := {A : U → V } for bounded A,

(2.45)

N (A) := {u ∈ U : Au = 0} = ker(A), R(A) := {v ∈ V : ∃u ∈ U : v = Au} = range (A). The notation of a dual operator in (1.33) for a linear operator A ∈ L(U, V ) in a reﬂexive Banach space, V = V , is slightly updated here into Ad ∈ L(V, U ) : u, Ad vU ×U = Au, vV ×V ∀ u, v ∈ V.

(2.46)

Proposition 2.10. Let a(·, ·) be a continuous bilinear form, see (2.40). 1. Then there exists a unique operator A ∈ L(U, V ) such that a(u, v) = Au, vV ×V ∀ u ∈ U, v ∈ V, AV ←U ≤ Cb .

(2.47)

A is called the operator induced by the bilinear form. In particular, AV ←U =

sup

{

sup

{|a(u, v)|}

}

u∈U ,uU =1 v∈V,vV =1

= sup{|a(u, v)| : u ∈ U, v ∈ V, uU = vV = 1}.

(2.48)

2.3. Bilinear forms and induced linear operators

47

2. Similarly, for U = V, the ad (u, v) = a(v, u) induces the special dual operator Ad ∈ L(V, V ) by Ad u, vV ×V = ad (u, v) = a(v, u) = Av, uV ×V

(2.49)

∀ u, v ∈ V ⇒ Ad ∈ L(V, V ). This Ad is the dual operator to A, see (1.33), and Ad V ←V = AV ←V . 3. Let V1 be a dense subspace of V and let a(·, ·) be deﬁned on V1 × V1 , satisfying (2.40) for u, v ∈ V1 with the norm · V . Then there is a unique extension a(·, ·) of a(·, ·) to V × V such that (2.40) is still satisﬁed with the same Cb and norms ∀ u, v ∈ V. 4. If the (real) Hilbert space V is identiﬁed with V , then A ∈ L(V, V) induces the adjoint operator A∗ ∈ L(V, V), in the sense of (1.36) by replacing (2.49) by (A∗ u, v)V = ad (u, v) = a(v, u) = (u, Av)V ∀ v, u ∈ V.

(2.50)

Finally, see (1.37), A∗ V←V = AV←V and A∗ = A for a symmetric bilinear form a(·, ·). 5. An a(·, ·) : U × V → R, with U = W m,p1 , V = W m,p2 , with coeﬃcients ai,j ∈ W m,p3 , has to satisfy 1/p1 + 1/p2 + 1/p3 = 1. This case plays a certain role in the linearization of nonlinear elliptic equations. Remark 2.11. We have restricted the discussion here to bilinear forms a(·, ·) and their uniquely induced operator A ∈ L(U, V ), and its dual operator Ad ∈ L(V = V , U ). If, more generally, A ∈ L(X , Y) for two normed spaces X , Y, then Ad ∈ L(Y , X ) is, for every y ∈ Y , deﬁned by the following unique x ∈ X , such that Ax, y Y×Y = x, x X ×X ∀ x ∈ X ⇒ Ad ∈ L(Y , X ), Ad y := x ∀ y ∈ Y . In many situations it is important to characterize the invertibility of an induced A by appropriate properties of the bilinear form a(·, ·), in particular, the following inf–sup condition. This and the invertibility of A usually require boundary conditions to be incorporated into U, V, see below and [386], p. 127, [135], p. 117. The following theorem is often applied to numerical methods. So it is sometimes important to distinguish the u ∈ U = V v. It is straightforward to generalize Hackbusch’s proof for his Lemma 6.5.3 [387], from his A ∈ L(U, U ) form for Hilbert spaces U, U to our Banach space setting A ∈ L(U, V ). Theorem 2.12. Brezzi–Babuˇska condition: Let U, V be reﬂexive Banach spaces and A ∈ L(U, V ) be induced by the bounded a(·, ·) : U × V → R, see (2.47). Then the four

48

2. Analysis for linear and nonlinear elliptic problems

statements (2.51), (2.52), (2.53) and (2.54) are equivalent: A−1 ∈ L(V , U) exists, hence C exists, s.t. A−1 U ←V ≤ C,

(2.51)

∃ , > 0 s.t.

(2.52)

sup |a(u, v)|/vV ≥ uU ∀u ∈ U

0 =v∈V

sup |a(u, v)|/uU ≥ vV ∀v ∈ V,

and

0 =u∈U

∃ > 0 s.t

sup |a(u, v)|/vV ≥ uU ∀u ∈ U

(2.53)

0 =v∈V

sup |a(u, v)|/uU > 0 ∀v ∈ V, 0 = v

and

0 =u∈U

∃ > 0 s.t and

sup |a(u, v)|/vV ≥ uU ∀u ∈ U

(2.54)

0 =v∈V

∀ v ∈ V, v = 0 ∃ u ∈ U : a(u, v) = 0.

Each of the conditions (2.51)–(2.54) implies A−1 U ←V ≤ 1/. If , are chosen as the suprema in (2.52)–(2.54), then A−1 U ←V = 1/. If one of these (2.51)–(2.54) holds, then ∀ f ∈ V ∃1 u0 ∈ U such that Au0 = f ⇔ a(u0 , v) = Au0 , vV ×V = f, vV ×V ∀ v ∈ V.

(2.55)

We do not give a proof, see [386], p. 127, [135], p. 117. But it is straightforward to show that (2.52) or (2.53) are necessary for (2.51): If 0 = w ∈ R(A)⊥ exists, choose 0 = v ⊥ w. Then a(w, v) = 0, contradicting (2.52), (2.53). Remark 2.13. Injective A and the inf–sup condition: 1. The notation inf-sup condition is due to the equivalence of, e.g. the ﬁrst line of (2.52) with sup {|a(u, v)|/vV }/uU ≥ . inf 0 =u∈U

0 =v∈V

Some authors replace the above ≥ by = . 2. Sometimes we refer to the second line of (2.53) as sup{|a(u, v)| : u ∈ U, uU = 1} > 0

for all

0 = v ∈ V.

(2.56)

3. The ﬁrst line of the above condition (2.52) is directly related to the dual norm of Au as uU ≤ sup{|Au, vV ×V | : v ∈ V, vV = 1} = AuV .

(2.57)

This inequality is necessary for an injective A, and for the existence of A−1 . A combination with the second condition of the second line of (2.52) yields the bijectivity.

2.3. Bilinear forms and induced linear operators

49

4. Often Theorem 2.12 is applied to a discrete Ah : U h → V h . Then the last estimate in (2.51) applied to (Ah )−1 U h ←V h ≤ C yields the stability of Ah . 5. Under appropriate conditions, the ﬁrst and second line of (2.52) are equivalent, see Proposition 2.19. Sometimes a corollary of Theorem 2.12 is useful, obtained simply by rescaling. Corollary 2.14. Let U, V be reﬂexive Banach spaces and A ∈ L(U, V ) be induced by the bounded a(·, ·) : U × V → R, see (2.47). Then the two statements (2.58) and (2.59) are equivalent: A−1 ∈ L(V , U) exists, hence C exists, s.t. A−1 V ←U ≤ C,

(2.58)

∃ , > 0 s.t.

(2.59)

sup v∈V,vV =1

and

sup u∈U ,uU =1

|a(u, v)| ≥ ∀u ∈ U, uU = 1,

|a(u, v)| ≥ ∀v ∈ V, vV = 1.

Obviously a V-coercive, hence bounded bilinear form a(·, ·) satisﬁes each of (2.52)– (2.54). Hence a combination of (1.33) and (2.52) shows: Theorem 2.15. Coercivity implies invertibility; invertible A, Ad : 1. Let V be reﬂexive and the bounded a(·, ·) be either V-coercive, see (2.40), (2.41), or satisfy one of (2.52)–(2.54), thus A−1 ∈ L(V , V) exists, see Theorem 2.12. Then A and Ad are simultaneously invertible and A, Ad ∈ L(V, V ), AV ←V = Ad V ←V ≤ C ≤ Cb in (2.40), −1

A

d −1

, (A )

−1

∈ L(V , V), A

d −1

V←V = (A )

(2.60)

V←V ≤ C .

2. If a(·, ·) is V-coercive with α as in (2.41), then the previous C ≤ 1/α, or if a(·, ·) satisﬁes one of (2.52)–(2.54), then C ≤ 1/. 3. In particular, Au0 = f in (2.55) and its dual counterpart are uniquely solvable. In the remainder of this section we study V-coercive and V-elliptic bilinear forms. Sometimes, Equation (2.55) is equivalent to a minimum problem: Theorem 2.16. Extremal problem: 1. Let a(·, ·) be V-coercive, hence bounded, and symmetric, and f ∈ V be given. Then there is a unique minimum u0 ∈ U = V for the extremal problem J(u0 ) = inf{J(u) := a(u, u) − 2f (u) : ∀ u ∈ V}.

(2.61)

2. This minimum is attained for the (unique) solution u0 of (2.55). The elliptic diﬀerential operators in Sections 2.2 and 2.4 do not correspond to coercive, but to more general, so-called elliptic bilinear forms. So we need:

50

2. Analysis for linear and nonlinear elliptic problems

Deﬁnition 2.17. Embeddings, Gelfand triple, elliptic bilinear forms: 1. Let V ⊂ W be Banach spaces and I : V → W, Iv := v ∈ W ∀v ∈ V. This embedding I : V → W is called dense and continuous or compact if V is dense in W and I is continuous or compact, respectively 2. Let V be a Banach and W a Hilbert space, with 6 W = W identiﬁed. Then V ⊂ W ⊂ V is called a Gelfand triple if V → W = W → V , V → W is dense and continuous.

(2.62)

3. Under this condition, the bounded bilinear form a(·, ·) : V × V → R is called V-elliptic if there exist constants Cc , α ∈ R such that a(u, u) ≥ αu2V − Cc u2W ∀ u ∈ V with α > 0.

(2.63)

The second inequality is a so-called G˚ arding inequality. For Cc = 0 we obtain again a V-coercive bilinear form a(·, ·). We will show in Subsection 2.4.4 that elliptic operators induce, under standard conditions, H m (Ω)-elliptic bilinear forms. In (2.62) only embedding properties of V ⊂ W are listed. Relation (2.62) implies W to be continuously embedded into V for Banach spaces V, W, but V, W densely and continuously into V for Hilbert spaces, see [387], Corollary 6.3.10. Therefore, see (2.62), with V → W = W dense and continuous, we obtain for Hilbert spaces V, W : (2.62) ⇒ V → W = W → V dense and continuous.

(2.64)

For applications of partial diﬀerential equations the most important Gelfand triples are those with compact embeddings. To apply the following Riesz–Schauder theory we need criteria for compact operators, cf. Hackbusch [387], Theorem 6.4.10. Proposition 2.18. In the Gelfand triple (2.62) let the embedding V → W be compact and T ∈ L(V , V) be given. Then the dense embeddings in (2.62) allow us to modify this T ∈ L(V , V), e.g. A−1 = T , yielding compact T1 ∈ L(V , V ), T2 ∈ L(V , W), T3 ∈ L(V, V), T4 ∈ L(W, V), T5 ∈ L(W, W). Proof. The compact embedding I : V → W implies the embedding I1 : W = W → V to be compact as well. If in a product of continuous operators at least one factor is compact, the product is compact as well, see [387], Lemma 6.4.5. Now T1 = I1 IT : V → V → W → V , similarly T2 = IT : V → V → W = W , T3 = IT I1 : W = W → V → V → W = W , T4 = T I1 : W = W → V → V, T5 = T I1 I : V → W = W → V → V. It is interesting that for the most important Gelfand triples we can reduce the conditions in Theorem 2.12, cf. [387], Lemma 6.5.17: Proposition 2.19. In a Gelfand triple, see (2.62), with reﬂexive Banach space V let the embedding V → W be compact. Then in Theorem 2.12 the two conditions in 6

A Hilbert space W can always be identiﬁed with its dual W .

2.3. Bilinear forms and induced linear operators

51

(2.52) are equivalent. In particular, A−1 ∈ L(V , V) is guaranteed by one of the two equivalent lines of (2.52). Later we will apply the Liapunov–Schmidt method. It requires results concerning kernels and ranges of linear(ized) operators. Important candidates are compact perturbations of boundedly invertible operators and operators induced by elliptic bilinear forms. This includes essentially all the following problems, e.g. Navier–Stokes operators. The following Riesz–Schauder theory and its modiﬁcation show that these operators behave in some sense similarly to ﬁnite dimensional linear operators. In fact, the Fredholm alternative is 7 valid. We formulate it for the two cases indicated above. The form in Theorem 2.20 for the Riesz–Schauder theory is a combination of Hackbusch’s Theorem 6.4.12 and the Closed Range Theorem 1.19. For the many cases of operators induced by elliptic bilinear forms we modify this result in the following Theorem 2.21. Theorem 2.20. Riesz–Schauder theory: Let V be a Banach space, I ∈ L(V, V) denote the identity and let T ∈ L(V, V) be compact, for T d ∈ L(V , V ) cf. Remark 2.11. 1. For every λ ∈ C \ {0}, one of the following alternatives is valid: (i) (T − λI)−1 ∈ L(V, V); (ii) λ is an eigenvalue for T ⇔ ∃x = 0 : T x = λx. For (i) the (T − λI)x = y is uniquely solvable in V for every y ∈ V and ¯ −1 ∈ L(V , V ), (T − λI)−1 ∈ L(V, V) ⇐⇒ (T d − λI) that is, both are simultaneously bijective and boundedly invertible, so ¯ = V , R(T − λI) = V, R(T d − λI) ¯ = {0} ⊂ V . N (T − λI) = {0} ⊂ V, N (T d − λI)

(2.65)

¯ is an eigenvalue for T . Then For (ii) we ﬁnd: λ is an eigenvalue for T ⇐⇒ λ there exist ﬁnite dimensional eigenspaces d

¯ = N (T d − λI) ¯ = {0} E(T, λ) = N (T − λI) = {0}, and E(T d , λ) ¯ < ∞ and T x = λx ∀x ∈ E(T, λ). Then T x − with dim E(T, λ) = dim E(T d , λ) λx = y has at least one, and hence inﬁnitely many solutions x ∈ V if and only if ¯ y, x V×V = 0 ∀ x ∈ E(T d , λ). 2. The spectrum of T is deﬁned as the set σ(T ) := {λ eigenvalue for T } ∪ {0, if T −1 ∈ L(V, V)}. This σ(T ) is at most countable and the only possible limit point in C is 0. This ¯: is the reason for excepting λ = 0 here and in 1. We ﬁnd σ(T d ) = σ(T ) := {λ λ ∈ σ(T )}. 7 A Fredholm operator, compare Deﬁnition 1.20, is a bounded linear operator A ∈ L(V, Y) with ¯ for λ ∈ C and its inverses ﬁnite dimensional kernel and co-range. The following A − λI and Ad − λI usually requires the complexiﬁcation of V, V and the operators A, Ad , see Theorem 1.22.

52

2. Analysis for linear and nonlinear elliptic problems

¯ 3. For (i), (ii) and λ = 0, the closed ranges, R(T − λI), R(T d − λI), and kernels, ¯ = E(T d , λ), ¯ are subspaces of the same ﬁnite N (T − λI) = E(T, λ), N (T d − λI) co-dimension and dimension, respectively. They are related as ¯ ⊥ , and R((T − λI)d ) =⊥ (N (T − λI)) R(T − λI) = (N (T d − λI)) (2.66) ¯ equivalent to y, x V×V = 0 ∀ y ∈ R(T − λI), x ∈ E(T d , λ), and x, y V×V = 0 ∀ x ∈ E(T, λ), y ∈ R((T − λI)d ), respectively. Hence, T − λI are Fredholm operators of index 0. The last part of this theorem is a special case of the Closed Range Theorem 1.19 and usually not included in the Riesz–Schauder theory. The relation between this and the next theorem is mainly based upon the compact embedding V → W. This implies a compact embedding I : V → V as well. Let A be induced by the above bounded elliptic bilinear form a(·, ·). Then for large enough CK > 0 the A + CK I is boundedly invertible. So, T := (A + CK I)−1 I : V → V is compact by Theorem 1.13. Thus Theorem 2.20 can be applied to this T − μI : V → V and some parts only have to be slightly changed to yield, cf. [388], Theorem 6.5.15: Theorem 2.21. Fredholm alternative: Theorem 2.20 remains correct with the following modiﬁcations: Let V ⊂ W = W ⊂ V be a Gelfand triple, see (2.62), with a Hilbert space W, a compact embedding V → W (hence V → V is compact as well) and with a reﬂexive Banach space V = V . Let a(·, ·) be bounded elliptic, and let A ∈ L(V, V ) and I ∈ L(V, V ) be the induced operator and the embedding (hence I is compact), respectively, and Ad ∈ L(V, V ) be the dual for A. 1. For every λ ∈ C (including λ = 0!) one of the following alternatives is valid: (i) (A − λI)−1 ∈ L(V , V); (ii) λ is an eigenvalue for A. For (i), (A − λI)u0 = f is uniquely solvable in V for every f ∈ V with ¯ −1 ∈ L(V , V), (A − λI)−1 ∈ L(V , V) ⇐⇒ (Ad − λI) that is, both are simultaneously bijective and boundedly invertible, so ¯ = V , N (A − λI) = N (Ad − λI) ¯ = {0} ⊂ V. R(A − λI) = R(Ad − λI)

(2.67)

¯ many solution For (ii), Au0 − λu0 = f has at least one, and in fact dimE(Ad , λ) d d d ¯ u0 ∈ V,, if and only if f, u V ×V = 0 ∀ u ∈ E(A , λ) ⊂ V, and (2.67) is updated according to (2.66). 2. The spectrum of A is at most countable and admits no limit points in C, and is deﬁned, diﬀerently from above, as ¯ ∈ σ(Ad ). σ(A) := {λ eigenvalue for A}, and λ ∈ σ(A) ⇔ λ 3. For (i) and (ii) the closed ranges and kernels again satisfy (2.66) and A − λI ¯ are Fredholm operators of index 0. and Ad − λI

2.3. Bilinear forms and induced linear operators

53

Remark 2.22. 1. With A − λI, I ∈ L(V, V ), I compact and (A − λI)−1 ∈ L(V , V) the operator (A − λI)−1 I ∈ L(V, V) is compact by Theorem 1.13. 2. Solving Au0 = f and (A − λI)u0 = f the above two cases have to be distinguished. The unique existence of a solution u0 for the problem Au0 = f and (A − λI)u0 = f is correct if and only if λ = 0 and λ is not an eigenvalue of the operator A. If (A − λI)u0 = f has to be solved for an eigenvalue λ, ﬁnitely many, n := dimN (A − λI), conditions have to be imposed for existence and uniqueness, cf. Theorem 1.19: ¯ ⊥ ⇐⇒ f, yj f ∈ R(A − λI) = (N (Ad − λI)) V ×V = 0, j = 1, . . . , n , with

¯ ⊂ V, u0 ∈ V ⊥ , V = N (A − λI) ⊕ V ⊥ . span {y1 , . . . , yn } = N (Ad − λI) If a special solution us for (A − λI)u0 = f is known, the set of general solutions d ¯ ¯ u0 is u0 ∈ {us + N (A − λI)}, similarly for (Ad − λI)u 0 = f and N (A − λI). 3. Not all linearizations of the quasilinear equations, see Section 2.7, satisfy (2.63), particularly in W m,p (Ω) for p > 2. Then Theorem 2.21 is applicable with appropriate modiﬁcations. The following two propositions are often useful, cf. [387], Lemmas 6.4.13 and 6.5.18: Proposition 2.23. Let X → Y → Z be continuously embedded Banach spaces, and X → Y be compactly embedded. Then for every > 0 there exists C such that uY ≤ uX + C uZ ∀ u ∈ X .

(2.68)

Proposition 2.24. Let a(·, ·), b(·, ·) be bilinear forms and a(·, ·) be V-elliptic, with the Gelfand triple V ⊂ W = W ⊂ V and a reﬂexive Banach space V, see (2.63). Then a(·, ·) + b(·, ·) is again V-elliptic if b(·, ·) satisﬁes one of the following conditions: 1. For every > 0 there exists C such that, compare Proposition 2.23, |b(u, u)| ≤ u2V + C u2W ∀ u ∈ V.

(2.69)

2. Let the embeddings V → X , V → Y be continuous and at least one of them be compact and let |b(u, u)| ≤ Cb uX · uY ∀ u ∈ V.

(2.70)

3. Let the embeddings V → X , V → Y be continuous and (2.70) be satisﬁed. For · X or · Y , we assume: For every > 0 a C exists such that, cf. (2.68), uX ≤ uV + C uW or uY ≤ uV + C uW ∀u ∈ V.

(2.71)

This condition is guaranteed by Proposition 1.27. We will come back to these bilinear forms in the case of Hilbert spaces, and linear operators A of order 2m, see the following (2.133) ﬀ., in Section 2.4.4, and to the

54

2. Analysis for linear and nonlinear elliptic problems

general case of Banach spaces in Section 2.7, particularly in Subsection 2.7.4, where the linearization of nonlinear diﬀerential operators is studied.

2.4 2.4.1

Linear elliptic diﬀerential operators, Fredholm alternative and regular solutions Introduction

The aim of this section is the deﬁnition of suﬃciently general concepts of linear elliptic diﬀerential operators of order 2m in the H¨ older and Sobolev space settings. We need existence, uniqueness, regularity results and Fredholm alternatives for the solutions, which can be combined with the nonlinear problems in Sections 2.5 ﬀ. and numerical methods in the following chapters. A linear diﬀerential operator A of order 2m, directly generalizing Section 2.2, is A : V → V with V = H m (Ω) ⊂ W = L2 (Ω) = W ⊂ V = H −m (Ω).

(2.72)

But it makes sense to consider competing very successful approaches in H¨older and general Sobolev spaces. We will present them in this section. In Section 2.2 we introduced the classical, strong and weak operators Ac : C 2 (Ω) → C(Ω), As : H 2 (Ω) → L2 (Ω) and A : H 1 (Ω) → H −1 (Ω). We will generalize them to H¨older and Sobolev spaces and study elliptic operators Ac = As : C 2m+k,γ (Ω) → C k,γ (Ω) and As : W 2m+k,p (Ω) → W k,p (Ω) in Subsection 2.4.2 under C ∞ -conditions for the coefﬁcients of the diﬀerential and boundary operators and for Ω. In contrast to Section 2.2 we do no longer use these nonstandard indices in Ac , As , A. The regularity results in this section show that the operators are essentially the same; only the spaces and the smoothness conditions vary. We use the As , A only if necessary, e.g. when partial integration and the Green’s formula require this notation. It is worthwhile realizing that for second order equations the C ∞ -conditions can be relaxed to C k -conditions and still get A : C 2+k,γ (Ω) → C k,γ (Ω) results, summarized in Subsection 2.4.3. Finally, in Subsection 2.4.4 we study the direct generalization of Section 2.2 to elliptic operators of order 2m and formulate the necessary results for A : H m (Ω) → H −m (Ω). In Lions and Magenes [478] and Oden and Reddy [518], further generalizations are discussed. An essential diﬀerence between elliptic operators of order 2 and 2m is caused by the fact that only for order 2 is it always possible to transform the problem to homogeneous boundary conditions. Lemma 2.26 shows that for order 2m problems, m > 1, this is only possible for Dirichlet boundary systems. General boundary operators for 2m are more complicated and induce surprising eﬀects, see Lions and Magenes [478], Oden and Reddy [518], and Runst and Sickel [561]. For deﬁning linear elliptic operators of order 2m we use the standard notation for multi-indices, α, reals or vectors ϑα or ϑ and (partial) derivatives ∂ α u. The notation for the wide range of problems in the present chapter is a nightmare. Many diﬀerent possibilities have been suggested in the literature. We choose relatively similar symbols, ∂ i u, ∂ α u, ∂u, for partials, and ϑi , ϑα , ϑ, for reals or vectors. Let

2.4. Linear elliptic diﬀerential operators

55

α := (α1 , . . . , αn ) ∈ Nn0 , |α| := α1 + · · · + αn , α ! := (α1 )! · · · (αn )! (2.73) ∂ i u :=

∂u ∂ |α| , ∂u := (∂ 1 u, . . . , ∂ n u), ∂ α u := ∂1α1 . . . ∂nαn u := u, α1 n ∂xi ∂x1 . . . ∂xα n

∇k u := (∂ α u)|α|=k , ∇≤k u := (∂ α u)|α|≤k , ∂ 0 u := ∇0 u := u, ∇u := ∇1 u, with u(x), ∂ i u(x), ∂ α u(x) ∈ R, ∂u(x) ∈ Rn , ∇k u(x) ∈ Rnk , ∇≤k u(x) ∈ RNk , where nk :=

1, Nk :=

|α|=k

k

nj , and Θk ∈ Rnk , Θ≤k ∈ RNk , Θ0 ∈ R, Θ := Θ1 ,

j=0

ϑ := (ϑ1 , . . . , ϑn ) ∈ Rn , ϑα := (ϑ1 )α1 · · · (ϑn )αn , ϑi ∈ R, i ≥ 0, ϑ0 = 1, and sometimes Dk u := ∇k u, D≤k u := ∇≤k u, D0 u := ∇0 u := D0 u := u, Du := ∇1 u. Only in Subsections 2.5.4 and 2.5.7, 2.6.4 do we appropriately replace our notation (x, Θ0 , Θ1 ) and (x, Θ0 , Θ1 , Θ2 ) by (x, z, p) and (x, z, p, r), respectively. The standard Sobolev inner products, norms, and seminorms are deﬁned as (∂ α u, ∂ α v)L2 (Ω) , and vH m (Ω) := ((u, u)H m (Ω) )1/2 , (u, v)H m (Ω) := ⎞1/2 ⎞1/p ⎛ |α|≤m ⎛ |v|H m (Ω) := ⎝ ∂ α v2L2 (Ω) ⎠ , vW m,p (Ω) := ⎝ ∂ α vpLp (Ω) ⎠ (2.74) . |α|=m

|α|≤m

As a consequence of the polynomial formula the inequalities 8 m! m m ≤ Cm,n ∀m ∈ N, α ∈ Nn0 , ϑ ∈ Rn , := 1≤ := α α1 · · · αn α1 ! · · · αn ! 2m imply |ϑ|2m (ϑ)2α n /Cm,n ≤ |ϑ|n,α := =

|α|=m

1 2α1

(ϑ )

· · · (ϑn )2αn

|α|=m

≤

|α|=m

=

n

|ϑ|2m n

=

(ϑ1 )2α1 · · · (ϑn )2αn

= |ϑ|2m

(ϑi )2

i=1

=

m α1 · · · αn m

(2.75)

n

m i 2

(ϑ )

i=1

= ≤ Cm,n

(ϑ)2α , |ϑ|n Euclidean norm in Rn .

|α|=m 8 As a consequence of (2.75), we can use, in (2.79), here and throughout, either |ϑ| n,α or |ϑ|n with appropriately modiﬁed λ(x), Λ(x).

56

2. Analysis for linear and nonlinear elliptic problems

Deﬁnition 2.25. Linear elliptic diﬀerential operators of order 2m and diﬀerent boundary conditions: 1. A linear diﬀerential operator of order 2m has the strong form aα ∂ α u with real valued coeﬃcients aα ; Au :=

(2.76)

|α|≤2m

for the weak form A we prefer (2.102). The symbol S for A aα (x)ϑα , SA (x, ϑ) := S(x, ϑ) :=

(2.77)

|α|≤2m

is often called the characteristic polynomial for A. The principal part Ap is Ap u := aα (x)∂ α u. (2.78) |α|=2m

2. Generalizing (2.20), the principal part of the symbol of an elliptic A (or of −A) has to satisfy, in D ⊂ Ω and ∀ϑ ∈ Rn , x ∈ D, m n aα (x)ϑα ≤ Λ(x)|ϑ|2m λ(x)|ϑ|2m n ≤ (−1) n ∀ϑ ∈ R , x ∈ D. (2.79) |α|=2m

3. This A is called elliptic and strongly elliptic and uniformly elliptic, in D, if 0 ≤ λ(x) and 0 < ≤ λ(x) and 0 < ≤ λ(x), Λ(x)/λ(x) bounded ∀ x ∈ D, respectively. For aα ∈ L∞ (D) or aα,β ∈ W |β|,∞ (D) in (2.102), and aα,β ∈ L∞ (D) in (2.110) this condition is only required almost everywhere in D. 4. The most important types of boundary operators, in the trace sense, see Remark 2.1 and Theorem 1.37, are imposed ∀x ∈ ∂Ω or in complementing subsets ∂Ω = ∂Ω1 ∪ ∂Ω2 , and are denoted as Dirichlet operator BD u := (∂ i u/∂ν i )m−1 i=0

(2.80)

Neumann operator ∂ u/∂ν , m ≤ i ≤ 2m − 1, and (2.81) α General operator Bj u := bj,α ∂ u of order mj < 2m, j = 1, · · · , m, i

i

|α|≤mj

with real bj,α , e.g. the induced Ba or, for m = 1 : u,

(2.82) ∂u ∂u , b0 u + . ∂ν ∂ν

(2.83)

5. The system (2.82) is called a Dirichlet system if mj = j − 1, j = 1, . . . , m. It is called a normal system if 0 ≤ m1 < · · · < mm and SBj , the symbol of Bj , satisﬁes SBj (x, ν) = 0, ν = ν(x) the normal vector ∀x ∈ ∂Ω, j = 1, . . . , m. A boundary value problem is deﬁned by the corresponding pair (A, B = (Bj )m j=1 ), with one of . the previous B = (Bj )m j=1

2.4. Linear elliptic diﬀerential operators

57

Here and for our numerical methods we essentially require the case of uniformly elliptic equations, hence 0 < ≤ λ(x), Λ(x)/λ(x) ≤ ∞ ∀x ∈ Ω.

(2.84)

If (2.84) is violated, then Tychonov regularization, here with (−1)m δ |α|=2m ∂ α u, δ > 0, may be used for δ → 0. Inequality (2.84) is violated usually at boundary points. Then for numerical methods, these violating boundary points may be cut oﬀ and approximate boundary conditions may be imposed. The case of nonuniformly elliptic equations is indicated in Subsection 2.5.3, and systematically studied by, e.g. Ivanov [415]. By a standard trick, inhomogeneous boundary conditions can be transformed into homogeneous boundary conditions. If u1 and u2 satisfy the prescribed inhomogeneous linear conditions, then the diﬀerence u1 − u2 satisﬁes the homogeneous conditions. For a Dirichlet system this transformation is possible for m ≥ 1 due to the following lemma. In fact, the proof for Lemma 5.1 in Oden and Reddy [518] can be slightly generalized and extended, yielding in particular, the estimate in (2.86): Lemma 2.26. Transformation into homogeneous boundary conditions: Let (2.82) be a Dirichlet system with bj,α ∈ C m−j+1 (Ω), ∂Ω ∈ C m and (gj )m j=1 with gj ∈ W m−j+1−1/p,p (∂Ω), j = 1, . . . , m. Then there exists u1 ∈ W m,p (Ω) such that in the trace sense, see Remark 2.1 and Theorem 1.37 , Bj u1 − gj = 0 in ∂Ω, j = 1, · · · , m, and

(2.85)

u1 W m,p (Ω) ≤ C(g1 , . . . , gm )Πm m−j+1−1/p,p (∂Ω) . j=1 W

(2.86)

With any u2 ∈ W m,p (Ω), such that Bj u2 = gj Bj (u1 − u2 ) = 0 in ∂Ω, j = 1, . . . , m.

in ∂Ω, j = 1, · · · , m, we obtain

As a consequence of Theorem 1.37, we can extend g0 ∈ W k−1/p,p (∂Ω) to g0 ∈ W k,p (Ω). This allows us, for the most important special case, m = 1, to relax the previous condition ∂Ω ∈ C 1 . Lemma 2.27. Transformation to homogeneous conditions for m = 1: Choose m = 1, a Lipschitz boundary ∂Ω ∈ C 0,1 , and the Dirichlet condition with u = g0 on ∂Ω, with p < ∞. Then for g0 ∈ W k−1/p,p (∂Ω), a u1 ∈ W 1,p (Ω) exists, such that in the trace sense, u1 − g0 = 0 in ∂Ω. Then again (u1 − u2 ) − g0 = 0 in ∂Ω. Corresponding lemmas for Dirichlet systems for systems of equations are valid as well, so we study mainly trivial Dirichlet boundary conditions. For operators A of higher order m > 1, the problem of deﬁning induced and well ﬁtting boundary operators Ba is much more complicated than for m = 1. We will come back to this in Subsection 2.4.2 (2.92) ﬀ. and Subsection 2.4.4. In particular, for m > 1 the transformation of inhomogeneous into homogeneous boundary conditions is only possible in exceptional cases. So nonhomogeneous boundary conditions are studied by Lions and Magenes, [478–480].

58

2. Analysis for linear and nonlinear elliptic problems

2.4.2

Linear operators of order 2m with C ∞ coeﬃcients

The topic of this subsection are strong operators of the form (2.76) with C ∞ coeﬃcients and boundary. An adequate existence, uniqueness and regularity theory is not known for A : C 2 (Ω) → C(Ω), but only for A : C 2+s,γ (Ω) → C s,γ (Ω), 0 < γ < 1, s ∈ N0 . We essentially formulate the results in H¨ older and Sobolev spaces, C 2m+s,γ (Ω) and 2m+s,p (Ω). For extensions we refer to Runst and Sickel [561]. Instead of the W older spaces C k,γ (Ω), where standard C 2m (Ω) spaces we use the corresponding H¨ k = 2m + s, with the norms sup |∂ α u(x)| with k ∈ N0 , γ ∈ R, 0 < γ < 1, and uC k (Ω) := |α|≤k

x∈Ω

uC k,γ (Ω) := uC k (Ω) +

|α|=k

|∂ α u(x) − ∂ α u(y)| with |x − y|γn y =x,y∈Ω sup

(2.87)

C k (Ω) := {u : Ω → R : uC k (Ω) < ∞}, C k,γ (Ω) := {u s.t. uC k,γ (Ω) < ∞}. We study BVPs of the form (A, B), see (2.76)–(2.82), with the solution u0 ¯ → C γ (Ω) ¯ f, Bj : C 2m,γ (Ω) ¯ → C 2m−mj ,γ (∂Ω) gj , A : C 2m,γ (Ω) (2.88) Au := aα ∂ α u, B := (Bj )m bjα ∂ α u, mj < 2m, j=1 , Bj u := |α|≤2m

|α|≤mj

Au0 (x) = f (x)∀x ∈ Ω, Bu0 (x) = 0 or = g(x) = (gj (x))m j=1 ∀x ∈ ∂Ω,

(2.89)

¯ or u, u0 ∈ C 2m+k,γ (Ω), ¯ compare (2.93). for u, u0 ∈ C 2m,γ (Ω) Instead of this complicated situation one certainly would prefer a simpler setting. For the case 2m = 2 and Dirichlet boundary conditions we would aim for a solution ¯ u0 |∂Ω = 0 for, e.g. u0 ∈ C 2 (Ω) ∩ C(Ω), 2m = 2 : A := −Δ : C 2 (Ω) → C(Ω), − Δu0 (x) = f (x)∀x ∈ Ω.

(2.90)

It is known that this solution u0 is unique and it exists, if the Green function exists for Ω, see Hackbusch [387], Theorems 3.1.2, 3.2.11, 3.2.13. This u0 does not depend continuously upon f with respect to the sup norms in C 2 (Ω) and C(Ω). So older norms, see e.g. Theorem u0 C 2,γ (Ω) ≤ Cf C γ (Ω) , 0 < γ < 1 is correct in the H¨ 2.30 below, but it is wrong for the norms u0 C 2 (Ω) , f C(Ω) , compare Hackbusch’s Theorem 3.2.15. We get for (2.90) an estimate with the “unusual” norm u0 C(Ω) instead of u0 C 2 (Ω) . More generally, let A be strongly elliptic on a bounded Ω with coeﬃcients aα ∈ C(Ω), a0 ≤ 0 in Ω. Then there exists a C ∈ R+ , see [387], Theorem 5.1.9 and Exercise 5.1.10d, such that u0 C(Ω) ≤ Cf C(Ω) ∀f ∈ C(Ω).

(2.91)

These are two reasons for studying and returning to (2.88) and (2.89). For the following results we require A to be a uniformly elliptic diﬀerential operator in Ω. The B have to “ﬁt” to A, a condition much more complicated for m > 1 than for

2.4. Linear elliptic diﬀerential operators

59

m = 1. The coeﬃcients of A, B and the boundary have to be in C ∞ , more precisely, ¯ bjα ∈ C ∞ (∂Ω) ∀|α| ≤ 2m, A, B in (2.88) satisfy aα ∈ C ∞ (Ω),

(2.92)

∞

j = 1 . . . , m, ∂Ω ∈ C , A is strongly (so uniformly) elliptic, (Bj )m j=1 satisﬁes the complementary condition, e.g. Dirichlet boundary conditions for m ≥ 1, see (2.80), or for m = 1, let b0 (x) = 0, see (2.83), compare Theorem 2.50 below. These complementary conditions and the related concept of normal boundary value problems 9 are intensely studied, e.g. by Lions and Magenes [478], pp. 113 ﬀ., and Oden and Reddy [518], pp. 152 ﬀ. Equivalent (nonalgebraic) formulations via initial value problems of ODEs are described by Wloka [667], pp. 150 ﬀ. and Taylor [620], pp. 379 ﬀ. They are often handled more easily. As an example Wloka [667] shows that for a strongly elliptic operator A as in (2.76) with m = 1 the Dirichlet and natural boundary conditions are complementary and normal. The same is correct for the Laplacian −Δ combined with (2.83). For the biharmonic operator (Δ)2 , hence m = 2, several cases are considered, see [667] and (2.88). So m1 = m2 = 0 is neither normal nor complementary. m1 = 0, m2 = 1 is normal and complementary if b10 × ((b1α )|α|=1 , ν)n (x) = 0∀x ∈ ∂Ω. This includes the Dirichlet boundary conditions u|∂Ω = g1 , (∂u/∂ν)|∂Ω = g2 . For the following regularity results we choose smoother Banach spaces ¯ Y := C k,γ (Ω), ¯ Z := C(Ω), ¯ X := C 2m+k,γ (Ω), with

(2.93)

X0 := {u ∈ X : Bu|∂Ω = 0} for k ∈ N0 , γ ∈ R, 0 < γ < 1. So this X0 with homogeneous boundary conditions corresponds to our V above. Com¯ pared to (2.88) we change the spaces in A : X → Y, Bj : X → Yj := C 2m+k−mj ,γ (Ω), but use the same notation A, B. The following results in Theorems 2.28–2.33 and the last statement in 2.35 are due to Zeidler [676], Chapter 6.3, and Problems 6.8, p. 259. They remain valid for the A, B in (2.98), (2.101). Theorem 2.28. Fredholm alternative, regularity: 1. Under the conditions (2.92), (2.93) the elliptic operator A ∈ L(X0 , Y) is a Fredholm operator of index i = d − c := dimN (A) − codimR(A) < ∞. This i, determined by the pair A, B, is independent of k. This remains unchanged if in A and Bj terms of order < 2m and < mj are changed, respectively. This remains correct for all the combinations in this section. For many 9 Complementary condition: Let S (x, ϑ) and S A Bj (x, ϑ), see (2.77), denote the principal parts of the symbols for A and Bj , respectively. These polynomials correspond to the terms with the highest derivatives. For all x ∈ ∂Ω let ν = ν(x) denote the outer normal in x ∈ ∂Ω, let ϑ = 0 lie in the tangent plane of ∂Ω in x and let A be an elliptic operator with real coeﬃcients. Then SA (x, ϑ + zν(x)) is a polynomial in z ∈ C of degree 2m. It can be shown that it admits exactly m solutions zk ∈ C with positive imaginary part ∀x ∈ ∂Ω. Let P (z) := Πm k=1 (z − zk ). Then the system A, Bj , j = 1, . . . , m satisﬁes the complementary condition if the polynomials SBj (x, ϑ + zν(x)) in z are linearly r (x, ϑ + independent modulo P (z), i.e. divide the SBj (x, ϑ + zν(x)) by P (z). Then the remainders SB j zν(x)) of degree < m of this division are linearly independent.

60

2. Analysis for linear and nonlinear elliptic problems

important cases i vanishes, e.g. compare the conditions in Theorem 2.50 below. 2. If index(A) = i = 0 and dimN (A) = 0 then A ∈ L(X0 , Y) is boundedly invertible, hence ∃1 u0 for Au0 = f and u0 X0 ≤ Cf Y . 3. Finally, the Fredholm alternative in Theorem 2.21 for Au = f has to be modiﬁed: There exist c = codimR(A) and d = dimN (A) linearly independent elements fi∗ ∈ Y ∗ ⊥ R(A), i = 1, . . . , c and ui ∈ N (A) ⊂ X0 , i = 1, . . . , d such that: Either fi∗ (f ) = 0 for at least one i ∈ {1, . . . , c}. Then Au = f is not solvable. Otherwise Au = f is solvable. If us is a special solution, the general solution u0 is obtained as u0 = u s +

d

ci ui with ci ∈ R.

i=1

Remark 2.29. 1. Compared to Theorem 2.21 the eigenvalue formulation does not make sense here. For d > 0 we have N (A) = 0 ∀ λ. So the unique existence of an exact solution u0 for any f ∈ Y, g ≡ 0 ∈ Πm j=1 Yj in (2.89) is correct if and only if N (A|X0 ) = N (A) = {0} and index i = 0. 2. The above Remark 2.22 has to be updated for the new situation. The linear space V in Theorem 2.21 usually corresponds to our X0 with homogeneous boundary conditions, see above and Lions and Magenes [478], Oden and Reddy [518], Wloka [667], and Runst and Sickel [561], Section 3.5.3; cf. Section 8.6 in this book as well. 3. Some results for the index i can be found in these books as well. The next theorems summarize a priori estimates and regularity results for arbitrary ¯ In part this is related to the last line functions u ∈ X0 and for solutions u0 ∈ C 2m (Ω). of Theorem 2.21 or to Theorem 2.45: Theorem 2.30. Estimates in X0 : For (2.92), (2.93) any u ∈ X0 satisﬁes uX ≤ C(AuY + uZ ) and uX ≤ CAuY for N (A) = {0}.

(2.94)

¯ → R(A) is boundedly invertible. If addiFor N (A) = {0} the A : X0 = C 2m+k,γ (Ω)) ¯ with A−1 ∈ L(Y, X0 ) and A−1 ∈ tionally index i = 0 then R(A) = Y = C k,γ (Ω) L(Y, Y) is compact. Theorem 2.31. Regularity in X0 : For (2.92), (2.93) with g = 0 in (2.89) any ¯ For f ∈ Y in (2.93), let the classical solution u0 for Au0 = 0 satisﬁes u0 ∈ C ∞ (Ω). ¯ Then again u0 ∈ X0 . solution u0 of (2.89) be in C 2m (Ω).

2.4. Linear elliptic diﬀerential operators

61

Now we turn to problems with inhomogeneous boundary conditions. Choose X , Y, Z, as in (2.88), (2.93), Yj := C k+2m−mj ,γ (∂Ω), and

(2.95)

Au(x) = f (x)∀x ∈ Ω, Bu(x) = g(x) = (gj (x))m j=1 ∀ x ∈ ∂Ω.

(2.96)

Theorem 2.32. Estimates, regularity and Fredholm operator in X : 1. Let (A, B) ∈ L X , Y × Πm j=1 Yj be a normal boundary value problem and satisfy (2.92), (2.95), (2.96). Then it deﬁnes a Fredholm operator of index i. It allows us the a priori estimate ⎞ ⎛ m Bj uYj ⎠ ∀u ∈ X and uX ≤ C ⎝AuY + uZ + j=1

⎛ uX ≤ C ⎝AuY +

m

⎞ Bj uYj ⎠ for N (A, B) = {0}.

(2.97)

j=1

For N (A, B) = {0} and i = 0, the (A, B) : X → R(A, B) is boundedly invertible and we ﬁnd R(A, B) = Y × Πm j=1 Yj . ¯ and f ∈ Y, gj ∈ Yj , j = 2. If the classical solution u0 of (2.96) is in C 2m,γ (Ω), 1, . . . , m with k > 0 in (2.95), then u0 ∈ X with k in (2.93),(2.95). Now we replace the operators A := A, B := (Bj )m older spaces j=1 in (2.88), with the H¨ in (2.93), (2.95), by the following Sobolev spaces: A : W 2m,p (Ω) → Lp (Ω), Bj : W 2m,p (Ω) → W 2m−mj −1/p,p (∂Ω), 1 < p < ∞, (2.98) and the corresponding spaces for homogeneous boundary conditions X := W k+2m,p (Ω), X0 := {u ∈ X : Bu = 0}, Y := W k,p (Ω), Z := L1 (Ω) and for inhomogeneous boundary conditions: X , Y as in (2.99), Z := Lp (Ω), Yj := W k+2m−mj −1/p,p (∂Ω).

(2.99) (2.100)

We deﬁne solutions, for details see Subsections 2.4.4, 2.5.6 and Zeidler [676], Problems 6.8, p. 261, as u0 ∈ W 2m,p (Ω), Au0 = f in Ω, Bu0 = 0 or g = (gj )m j=1 on ∂Ω.

(2.101)

Theorem 2.33. Results for the Sobolev case: In Theorems 2.28–2.32 and the operators in (2.88), replace the H¨ older spaces (2.93), (2.95) by the Sobolev spaces (2.98), (2.99), (2.100). Correspondingly, replace the homogeneous and inhomogeneous boundary value problems (2.89), (2.96) by (2.101), require (2.92) and a normal boundary

62

2. Analysis for linear and nonlinear elliptic problems

¯ in value problem. Finally, in Theorems 2.30–2.32 replace the conditions for C j,γ (Ω) (2.93), (2.95) by W j,p (Ω) in (2.99), (2.100). Then Theorems 2.28–2.32 remain valid in this modiﬁcation and the index remains unchanged. This astonishing similarity of the results is due to the fact that H¨ older and Sobolev spaces are special cases of the so-called Besov–Triebel–Lizorkin spaces. For these spaces the above results are correct, see Runst and Sickel [561]. For our later applications to numerical methods we reformulate the original linear ¯ diﬀerential operator in (2.76) into the so-called divergence form. 10 The aη ∈ C ∞ (Ω) η ∞ ¯ and ∂ in (2.88) and (2.73) deﬁne appropriate aα,β ∈ C (Ω) with aη ∂ η u = (−1)|β| ∂ β (aα,β ∂ α u), (2.102) Au := |η|≤2m

|α|,|β|≤m

e.g. with aη = (−1)m aα,β for α + β = η and |α| = |β| = m. In Subsection 2.4.3 we discuss, for 2m = 2, general boundary operators, see (2.82). However here, to avoid too many technicalities, we restrict the discussion to special Dirichlet systems, see Theorem 2.50 and (2.159) below, satisfying bj,α (x)∂ α u = gj ⇐⇒ ∂ j−1 u/∂ν j−1 = g˜j on ∂Ω for j = 1, . . . , m : Bj u := |α|≤j−1

⇐⇒ (∂ α u)|α|≤m−1 = (˜ gα )|α|≤m−1 for ∂Ω ∈ C m−1 and g˜j ∈ C m−j .

(2.103)

For more general cases combine Oden and Reddy [518], pp. 184ﬀ. and Zeidler [677], pp. 331 ﬀ. By Lemma 2.26 we use the standard trick for modifying f := f˜ in (2.101) for obtaining homogeneous boundary conditions Au(x) = f (x) ∀ x ∈ Ω, (∂ j−1 u/∂ν j−1 )m j=1 = 0 ∀ x ∈ ∂Ω,

(2.104)

or abbreviated into u ∈ W0m,p (Ω). So (2.88), (2.89), (2.92) has the simpliﬁed form for the aη or aα,β , ¯ A in (2.102) or (2.102) satisﬁes aη or aα,β ∈ C ∞ (Ω)∀|η| ≤ 2m or |α| = |β| = m, A is strongly (so uniformly) elliptic and ∂Ω ∈ C ∞ .

(2.105)

With (2.79), (2.102), uniform ellipticity means, noting that ϑα ∈ R, ϑ ∈ Rn , |ϑ|n ∈ R, aα,β (x)ϑα ϑβ ≤ Γ|ϑ|2m (2.106) γ|ϑ|2m n ≤ n for 0 < γ, Γ < ∞ ∀ x ∈ D. |α|,|β|=m

The special form of f in (2.107) is proved by Adams [1], pp. 47 ﬀ. 10 As in the literature, we use diﬀerent notations for the standard and the divergence form as indicated, e.g. in (2.102) systematically with consequences for the diﬀerent ellipticity conditions; compare, e.g. (−1)m in (2.79) versus (2.106).

2.4. Linear elliptic diﬀerential operators

63

Proposition 2.34. Continuous forms, f ∈ Vb = W0−m,p (Ω) : For 1 < p < ∞, 1/p + 1/p = 1, we obtain a special unique representation: f, vV ×Vb := b

fβ ∂ β vdx ∀ v ∈ Vb := W0m,p (Ω) with

(2.107)

Ω |β|≤m

⎛

fβ ∈ Lp (Ω) and f V = ⎝

⎞1/p

fβ pLp (Ω) ⎠

.

(2.108)

|β|≤m

For p = 1 or p = ∞ (2.108) has to be replaced by f V := minfβ

|β|≤m

1/p ... .

For f ∈ V = W −m,p (Ω) we obtain more generally f, vV ×V := fβ ∂ β vdx + ϕβ ∂ β vds ∀ v ∈ V := W m,p (Ω) with Ω |β|≤m

∂Ω |β|<m

⎛

ϕβ ∈ Lp (∂Ω) and f V = ⎝

fβ pLp (Ω) +

|β|≤m

⎞1/p

ϕβ pLp (∂Ω) ⎠

.

(2.109)

|β|<m

Note that for f ∈ V = W −m,p (Ω) the f0 := f |W −m,p (Ω) is reduced to an f0 ∈ 0

W0−m,p (Ω)as in (2.107). This is the reason that the following a(u0 , v) = Au0 , vV ×V = f, vV ×V ∀v ∈ V = H0m (Ω) u0

tested ∀v ∈ H0m (Ω) makes sense for f ∈ W −m,p (Ω) in (2.109) as well. We combine the standard partial integration with Green’s formula: Theorem 2.35. Solutions for the divergence form: 1. Assume the form (2.102), the conditions (2.103)–(2.106) and deﬁne the bilinear form a(u, v), the principal part ap (u, v), and the linear f as in (2.107) a(u, v) := Au, vV ×V := aα,β ∂ α u∂ β vdx, (2.110) Ω |α|,|β|≤m

ap (u, v) := Ap u, vV ×V :=

aα,β ∂ α u∂ β vdx, ∀u, v ∈ V = H0m (Ω).

Ω |α|=|β|=m

These a(u, v), ap (u, v), f, vV ×V are continuous. 2. Any solution u0 ∈ C k+2m,γ (Ω) or u0 ∈ W k+2m,p (Ω) ⊂ H m (Ω), k + m ≥ n(1/p − 1/2) of (2.104) necessarily satisﬁes the standard equation u0 ∈ H0m (Ω) :

a(u0 , v) = f, vV ×V ∀v ∈ V = H0m (Ω).

(2.111)

64

2. Analysis for linear and nonlinear elliptic problems

Vice versa, under the conditions (2.92) in (2.93) and (2.99), (2.100), every such solution u0 ∈ H0m (Ω) even satisﬁes u0 ∈ C k+2m,γ (Ω) and u0 ∈ W k+2m,p (Ω), respectively. Theorem 2.36. Ellipticity, and Fredholm alternative for the divergence form: In addition to the conditions in Theorem 2.35 let the coeﬃcients aα,β with |α| = |β| = m ¯ Then for a uniformly elliptic A in (2.105) the in (2.102), (2.106), be continuous in Ω. ap (u, v) is H0m (Ω)-coercive. For an H0m (Ω)-coercive principal part ap (u, v), the above a(u, v) is an H0m (Ω)-elliptic bilinear form. So the Fredholm alternative in Theorem 2.21 applies, with index (A, B) = 0. The unique solution of the variational problem (2.61) for ap (u, v) satisﬁes (2.111) for ap (u, v). Proof. a(u, v) − ap (u, v) are lower order terms. So a combination of (2.63) and ¯ with |α| = |β| = m the ap (u, v) Proposition 2.24 shows that since for aα,β ∈ C(Ω) by (2.105) is H0m (Ω)-coercive, see Theorem 2.43 below, the a(u, v) is H0m (Ω)-elliptic and the claim is proved, compare Theorems 2.89, 2.104. 2.4.3

Linear operators of order 2 under C k conditions

In Subsection 2.4.2 we studied results under strong C ∞ -conditions for coeﬃcients and the boundary of the diﬀerential and boundary operators. For the most important special case of second order equations it is possible to relax this C ∞ -condition and still get very similar results. We list the essential regularity results and the Fredholm alternative in Gilbarg and Trudinger [346]. It is appropriate to arrange, see (2.76), the aα and ∂ α as ai,j , ∂ 0 u = u and ϑ0 := 1, ϑi , ϑj ∈ R, i, j = 0, . . . , n. With ν, the outer normal unit vector for ∂Ω, the diﬀerential and one of the three boundary operators (2.88) and (2.89), the boundary value problem has the form Au :=

n

aij ∂ i ∂ j u : C 2 (Ω) → C(Ω),

(2.112)

i,j=0

Bu = B1 u = u, or B2 u = ∂u(x)/∂ν, or B3 u = b0 u +

n

ci ∂ i u on ∂Ω, (2.113)

i=1

determine u0 ∈ C

2,γ

¯ such thatAu0 = f ∀x ∈ Ω, Bu0 = ϕ∀ x ∈ ∂Ω. (2.114) (Ω)

Then the λ(x) and Λ(x) in (2.79) are the minimal and the maximal (real) eigenvalues of the (symmetric) matrix Ap deﬁning the diﬀerent types of ellipticity: 11 Ap (x) := (ai,j (x))ni,j=1 ∀ x ∈ Ω and (2.79) has the form λ(x)|ϑ|2n ≤ −

n

aij (x)ϑi ϑj ≤ Λ(x)|ϑ|2n ∀ϑ ∈ Rn .

(2.115) (2.116)

i,j=1 11

Note the − sign in (2.106) compared to [346]; formula (6.2) yields a00 (x) ≥ 0 in (2.118).

2.4. Linear elliptic diﬀerential operators

65

For the following results we require A to be a uniformly elliptic diﬀerential operator with smooth coeﬃcients, boundary and inhomogeneities; more precisely, let ¯ b0 , cj , ϕ ∈ C 1,γ (Ω), ¯ 0 < γ < 1, A, B in (2.112), (2.113) satisfy f, aij ∈ C γ (Ω), ¯ for B3 and c = (c1 , . . . , cn ) let ∀i, j = 0, . . . , n, ∂Ω ∈ C 2,γ , for B1 let ϕ ∈ C 2,γ (Ω), |(c(x), ν(x))n | > κ > 0 ∀ x ∈ ∂Ω, A is strongly (so uniformly) elliptic,

(2.117)

cf. (2.26), (2.33). The following results are proved by Gilbarg and Trudinger [346], see Theorems 6.14, 6.15, 6.17, 6.19, 6.30, 6.31 and the last lines in Section 6.7. We summarize the existence, uniqueness and regularity results in the interior Ω, its closure ¯ and the Fredholm alternative: Ω Theorem 2.37. Existence, (non-)uniqueness and estimates: in (2.112), (2.113) satisfy (2.116), (2.117), and

Let the operators A, B

a00 (x) ≥ 0 ∀ x ∈ Ω for B1 u and for B3 u with b0 (x) > 0 ∀ x ∈ ∂Ω, a00 (x) > 0 ∀ x ∈ Ω for B2 u.

(2.118)

¯ → C γ (Ω) ¯ × C 1,γ (∂Ω) for B = B2 , B3 and (A, B) : 1. Then (A, B) : C 2,γ (Ω) ¯ → C γ (Ω) ¯ × C 2,γ (∂Ω) for B = B1 . C 2,γ (Ω) 2. If λ = 0 is not an eigenvalue, the (A, B)u = (f, ϕ) is boundedly invertible, hence ¯ for (2.114). there exists a unique solution u0 ∈ C 2,γ (Ω) 3. If λ = 0 is not an eigenvalue, e.g. for violated (2.118), (A, B) is not boundedly ¯ for (2.114) can be estimated as, compare invertible, any solution u0 ∈ C 2,γ (Ω) (2.94), uC 2,γ (Ω) ¯ ≤ C(uC(Ω) ¯ + f C γ (Ω) ¯ + ϕC 1,γ (Ω) ) for B = B2 , B3 , but (2.119) uC 2,γ (Ω) ¯ ≤ C(f C γ (Ω) ¯ + ϕC 1,γ (Ω) ) for a boundedly invertible (A, B). For B = B1 the ϕC 1,γ (Ω) in (2.119) has to be replaced by ϕC 2,γ (Ω) . Theorem 2.38. Regularity in Ω: For the A, B1 in (2.112), (2.113) modify (2.117) ¯ by Ω, C γ by C k,γ -conditions for the coeﬃcients, no conditions for by replacing Ω the boundary and ϕ, and assume that u0 ∈ C 2 (Ω) solves the Dirichlet boundary value problem (2.114). Then u0 ∈ C k+2,γ (Ω); note in Ω. Here k = ∞ is allowed. Corresponding results for weak solutions u0 ∈ H01 (Ω) are formulated in Subsection ¯ requires additional conditions for the boundary. 2.4.4. Regularity in Ω ¯ For the operator A, B1 in (2.112), (2.113) modify Theorem 2.39. Regularity in Ω: (2.117) by replacing only the C γ , C 2,γ -conditions by C k,γ , C k+2,γ , let ∂Ω be in C k+2,γ ¯ ∩ C 2 (Ω) solves the Dirichlet boundary value problem and assume that u0 ∈ C 0 (Ω) ¯ (2.114), see Theorem 2.37. Then u0 ∈ C k+2,γ (Ω). Note that (2.118) is not imposed in the last two theorems. Corresponding u0 ∈ W k+2,p=2 (Ω) = H k+2 (Ω) results are formulated in Subsection 2.4.4, cf. [346], Chapter 8.

66

2. Analysis for linear and nonlinear elliptic problems

Theorem 2.40. Fredholm alternative: Let the operators A, B in (2.112), (2.113) satisfy (2.117). Then A ∈ L(X0 , Y) with X0 := {u ∈ C 2,γ (Ω) : Bu = 0}, Y := C γ (Ω),

(2.120)

is a Fredholm operator of index i = 0, satisfying the Fredholm alternative in Theorem 2.21. As in the last subsection we want to relate the original boundary value problem (2.112), (2.113) to the corresponding strong divergence form. In particular, we aim for existence and uniqueness results and the Fredholm alternative for diﬀerent combinations of second order equations (2.112), and boundary conditions (2.113). With a ˜ij := aij in (2.112), and appropriate new aij , let, see (2.16), Au := As u :=

n i,j=0

aij ∈ C

1,γ

n

a ˜ij ∂ i ∂ j u =

(−1)j>0 ∂ j (aij ∂ i u) : C 2 (Ω) → C(Ω),

i,j=0

¯ ai0 ∈ C (Ω) ¯ i, j = 0, . . . , n, a (Ω), ˜ij = −aij , i, j ≥ 1. γ

(2.121)

¯ so u ∈ H 2 (Ω). We modify (2.117) In this Subsection we have studied u ∈ C k+2,γ (Ω), by (2.121) and get instead of (2.116) for the uniform ellipticity for (2.121)

λ

|ϑ|2n

≤

n

aij (x)ϑi ϑj ≤ Λ |ϑ|2n ∀ϑ ∈ Rn , x ∈ Ω for 0 < λ .

(2.122)

i,j=1

As in (2.24) the strong and weak bilinear forms are related, with A = As in (2.121): n i j a(u, v) = aij ∂ u∂ vdx = (As u)vdx + (Ba u)vds (2.123) Ω i,j=0

with Ba u :=

Ω j=1,...,n

∂Ω

νj aij ∂ i u and the principal parts

i=0,...,n

ap (u, v) =

n

i

j

aij ∂ u∂ vdx =

Ω i,j=1

with Ba,p u :=

(As,p u)vdx + Ω

n i,j=1

(Ba,p u)vds

(2.124)

∂Ω

νj aij ∂ i u so Ba − Ba,p = ha :=

n

νj a0j ,

j=1

where here Ba and Ba,p are the natural boundary operators induced by A and Ap , respectively The special Au = −Δu + cu induces Ba u = Ba,p u = ∂u/∂ν. We discuss the mutual dependency of coercivity properties and boundary conditions for the ﬁrst (Dirichlet), the second or natural (generalized Neumann) and the third (generalized Robin) boundary value problem: With a function h ≥ h0 on ∂Ω and Vi , f−i , i = 1, 2, 3, below let, cf. (2.123)–(2.124), Bu = B1 u = u, B2 u = Ba u, B3 u = Ba,p u + hu on ∂Ω and determine

(2.125)

u0 ∈ H (Ω) ∩ Vi such that Au0 = f−i a.e.∀x ∈ Ω, Bi u0 = ϕ∀ x ∈ ∂Ω, i = 1 or 2 or 3. 2

2.4. Linear elliptic diﬀerential operators

67

These generalized Neumann and Robin boundary operators, B2 and B3 , modify the standard forms ∂u/∂ν and au + b∂u/∂ν. For diﬀerent boundary conditions in (2.124), (2.125), compared to (2.113), we aim for compact perturbations of the original bilinear forms a(u, v). They will turn out to be coercive for the above boundary operators Bi and the chosen function spaces Vi , i = 1, 2, 3, cf. (2.126) ﬀ. For i = 1, the principal part has this property; for i = 2, 3, it has to be modiﬁed. These Vi -coercive aic (u, v) yield unique existence and are combined with the Fredholm alternative for the Vi -elliptic a(u, v), and the linear forms f−i , vV ×V , cf. Theorems 2.20 and 2.21. The Vi -coercive aic (u, v) induce the linear operators Aic as compact perturbations of A induced by the original a(u, v) in (2.121). We consider: BVPs: Aic u = f−i ∈ V in Ω, Bi u|∂Ω = ϕ, i = 1, 2, 3, and ∀v ∈ V := Vi . Let for i = 1 : a1c (u, v) := ap (u, v) = f−1 , vV ×V V := V1 := H01 (Ω), (2.126)

n f0 v + fj ∂ j v dx, B1 u = u = 0 for the ﬁrst BVP, with f−1 , vV ×V := j=1

Ω

:= ap (u, v) + c(u, v)L2 = f−2 , vV ×V , B2 u = Ba u, c > 0, (2.127) ϕvds, V := V2 := {v ∈ H 1 (Ω) : Ba v = 0} f−2 , vV ×V := f−1 , vV ×V +

i=2:

a2c (u, v)

i = 3 : a3c (u, v) := ap (u, v) + ∂Ω

∂Ω

huvds = f−2 , vV ×V , V3 := H 1 (Ω),

(2.128)

B3 u = Ba,p u + hu, f−3 := f−2 , with h(x)|∂Ω ≥ h0 > 0 for the third BVP. The case i = 2 and the two alternatives have to be discussed. For Ba 1 = 0 and f2 , 1 = 0 with ha in (2.124), ap (u, v) = f2 , v∀v ∈ H 1 (Ω), Ba u = 0 and its dual problem admit the nontrivial solution u0 ≡ 1, thus the Fredholm alternative applies. Enforcing unique existence for Ba 1 = 0 = f2 , 1 is possible by deﬁning V2 := {v ∈ H 1 (Ω) : 1, vV ×V = 0}. This excludes the nontrivial solution u0 ≡ 1. Then the equivalent Sobolev norms in Proposition 1.25 allow the following generalization of Zeidler [677], Sections 22.2 ﬀ. for (2.126)–(2.128). Theorem 2.41. Unique existence, and Fredholm alternative for (2.126)–(2.128): 1. Assume the form (2.121), the conditions (2.122), (2.125), for Ba , c, h, f−i , and deﬁne aic (u, v), V = V i , and f−i , vV ×V as in (2.126)–(2.128). Then these bilinear and linear forms are continuous ∀u, v ∈ V and the aic (u, v) are Vi -coercive for i = 1, 2, 3 with the conditions for Ba , ha , f−i listed in (2.126)–(2.128). Then unique solutions ui0 exist for ∃1 ui0 = u0 ∈ H 1 (Ω) : aic (u0 , v) = f−i , vV ×V , Bi u0 = ϕ, ∀v ∈ Vi = V. (2.129) The unique solution of the minimal problems (2.61) for these aic (u, v) satisﬁes the variational problem (2.129). 2. Deﬁne the corresponding ai (u, v) by replacing the principal part of a(u, v) by the above aic (u, v) and impose the previous conditions. Then the ai (u, v) are

68

2. Analysis for linear and nonlinear elliptic problems

Vi -elliptic bilinear forms. So the Fredholm alternative in Theorem 2.21 applies to (A, Bi ), even with index (A, Bi ) = 0, i = 1, 2, 3. 3. Any solution u0 ∈ C k+2,γ (Ω) ⊂ H 1 (Ω) of (2.125) with the original ai (u, v), and the previous f−i , vV ×V satisﬁes the standard equation u0 ∈ H 1 (Ω) :

ai (u0 , v) = f−i , vV ×V ∀v ∈ V = Vi , Bi u0 = ϕ, i = 1, 2, 3.

Vice versa, under the conditions (2.117), (2.120), every such solution u0 ∈ V even satisﬁes u0 ∈ C k+2,γ (Ω). Proof. As in Theorem 2.35 obviously the a(u, v), aic (u, v) are bounded, except for i = 3, and the f−i v, except for i = 2, 3, in H 1 (Ω). For the exceptional cases, Remark 2.1 and the Trace Theorem 1.38 for v ∈ H 1 (Ω) imply v|∂Ω ∈ L2 (∂Ω) 2 2 2 v ds ≤ CvH 1 (Ω) =⇒ | vϕds| ≤ CϕL2 (∂Ω) vH 1 (Ω) , vL2 (∂Ω) = ∂Ω

∂Ω

and hence the boundedness of all the f−i , vV ×V , i = 1, 2, 3. For i = 3 we estimate huvds| ≤ hC(∂Ω) |uv|ds (2.130) | ∂Ω

∂Ω

≤ hC(∂Ω) u

vL2 (∂Ω) ≤ C1 hC(∂Ω) uH 1 (Ω) vH 1 (Ω) . 1/2 deﬁne norms equivalent to uVi , i = Finally, by Proposition 1.25, the aic (u, u) 1, 2, 3. The following proof, that the ac (u, v) := aic (u, v) are Vi -coercive, is a straightforward generalization of Zeidler [677], pp. 331 ﬀ. Relations (2.125), (2.123), (2.124) imply As u0 vdx − f−1 , vV ×V + (Bi u0 − ϕ)vds (2.131) 0= L2 (∂Ω)

Ω

∂Ω

= a(u0 , v) − f−1 , vV ×V +

((Bi − Ba )u0 − ϕ)vds ∂Ω

= ac (u0 , v) − f−1 , vV ×V +

((Bi − Ba )u0 − ϕ)vds ∂Ω

+ (a(u0 , v) − ac (u0 , v)). Now we drop the lower order terms (a(u0 , v) − ac (u0 , v)) and are left with ac (u0 , v). Then (2.121), (2.122) imply, with ha (x) ≥ 0, n n aij ∂ i u∂ j u + ha u2 dx ≥ γ (∂ i u)2 dx + min{ha (x)}u2L2 (Ω) ac (u, u) = Ω i,j=1

Ω i=1

¯ x∈Ω

= γ∂u2L2 (Ω) + min{ha (x)}u2L2 (Ω) ≥ γi u2H 1 (Ω) , ∀u ∈ Vi , i = 1, 3. ¯ x∈Ω

(2.132) and uH 1 (Ω) are The last inequality is correct for i = 1, since the norms ac (u, u) equivalent on V1 = H01 (Ω). For the diﬀerent combinations of Vi , i = 2, the Vi -coercivity 1/2

2.4. Linear elliptic diﬀerential operators

69

of ac (u, u) is a consequence of the discussion preceding Theorem 2.41, based upon the deﬁnition of the Vi and by, Proposition 1.25, the equivalence of the norms uH 1 (Ω) and (ac (u, u) + | Ω udx|2 )1/2 . For i = 3 it is correct with γ3 ≥ min{γ, h0 }. Again ai (u, v) − aic (u, v) are only lower order terms. Since aic (u, v) is Vi -coercive, a combination of (2.63) and Proposition 2.24 implies a V-elliptic ai (u, v). That these solutions satisfy the diﬀerential equation and the boundary conditions is shown as indicated in Remarks 2.6 and 2.7. A combination of (2.63) and Proposition 2.24 implies a V-elliptic ai (u, v), and thus the claim. We discuss in more detail the implications of the Fredholm alternative for one of the above examples. We consider for i = 2 the special case, cf. Section 8.6, a(u, v) =

n Ω i=1

∇u∇vdx ∀v ∈ V2 , inducing Ba u = ∂u/∂ν,

∂ i u∂ i vdx = Ω

with a(1, v) = 0 ∀v ∈ V2 , Ba 1 = 0 implying N (A) = N (Ad ) = R. Consequently, a (nontrivial) solution u0 for f0 vdx + ϕvds ∀v ∈ V2 , a(u0 , v) = f−2 , vV ×V = Ω

∂Ω

exists, by the Fredholm alternative, iﬀ f−2 , 1V ×V = 0 and every u0 + c, c ∈ R solves this problem. Otherwise the trivial solution exists only for the homogeneous problem a(u0 , v) = 0, f−2 = 0. As we have seen, enforcing a uniquely existing solution for Ba 1 = 0 = f2 , 1 1 is possible by deﬁning, according to the Fredholm alternative, V2 := {v ∈ H (Ω) : vdx = 0}. This excludes the nontrivial solution u0 ≡ 1. Ω 2.4.4

Weak elliptic equation of order 2m in Hilbert spaces

In Subsection 2.4.2, we studied strong and weak forms of elliptic operators in general Sobolev spaces W m,p (Ω), 1 < p < ∞, under continuity conditions for the coeﬃcients of the diﬀerential and boundary operators. The results in this subsection remain correct, in particular with respect to H m (Ω)-coercivity with appropriate modiﬁcations, for W m,p (Ω), 2 ≤ p ≤ ∞ as well, see below. This is most important in our context for the later linearization of operators and their stability, cf. Section 2.7 and Corollary 2.44. Regularity results in textbooks are usually formulated in the H m+s (Ω) setting. Beyond the previous W m,p (Ω) results, we turn therefore in this subsection to weak forms in Hilbert spaces H m (Ω) = W m,2 (Ω). This modiﬁes the approach in Section 2.2 for studying diﬀerential operators of order 2m. We test the diﬀerential operator of order 2m with elements in H m (Ω), often in H0m (Ω), and A : V → V , V = H m (Ω) ⊂ W = L2 (Ω) = W ⊂ V = H −m (Ω),

(2.133)

with Ω satisfying (2.5) and the scalar product (u, v)L2 (Ω) . For Sobolev spaces, see Subsection 1.4.3, we use the scalar product and norm in (2.74). In this context it is

70

2. Analysis for linear and nonlinear elliptic problems

appropriate to reformulate Au in divergence 12 form and the corresponding ellipticity condition. Deﬁnition 2.42. Divergent linear diﬀerential operator: For aαβ ∈ W |β|,∞ (Ω) and aαβ ∈ L∞ (Ω) the strong and weak form A = As and A are, see Deﬁnition 2.25, As : H 2m (Ω) → L2 (Ω), As u := (−1)|β| ∂ β (aαβ ∂ α u) (2.134) |α|,|β|≤m

and A : H (Ω) → H m

−m

(Ω), deﬁned by a(u, v) := Au, vH −m (Ω)×H m (Ω) :=

aαβ ∂ α u∂ β vdx.

(2.135)

Ω |α|,|β|≤m

The symbol, S, principal part, Ap , and the diﬀerent types of ellipticity for A, are deﬁned as obvious modiﬁcations of (2.77)–(2.79), possibly in D ⊂ Ω, as n aα,β (x)ϑα ϑβ ≤ Λ(x)|ϑ|2m (2.136) λ(x)|ϑ|2m n ≤ n ∀ϑ ∈ R , a.e. x ∈ D. |α|=|β|=m

We repeat the three sets of m boundary conditions imposed in the trace sense, see Remark 2.1 and Theorem 1.37, ∀x ∈ ∂Ω, or diﬀerent conditions in complementary sets: Dirichlet conditions: ∂ i u/∂ν i = 0 ∀ 0 ≤ j ≤ m − 1,

(2.137)

Neumann conditions: ∂ u/∂ν = 0 ∀ m ≤ j ≤ 2m − 1, General conditions: Bj u := bj,α ∂ α u, order mj < 2m.

(2.138)

i

i

(2.139)

|α|≤mj

Natural conditions: for m = 1 : Ba u :=

j=1,...,n

νj aij ∂ i u.

(2.140)

i=0...,n

The latter are generalized to m > 1, in Lions and Magenes [478], p. 120, and Oden and Reddy [518], p. 290 ﬀ., see below. For smooth ∂Ω, the Dirichlet boundary conditions (2.137) are equivalent to, analogously for W0m,p (Ω), u ∈ V := H0m (Ω) = {u ∈ H m (Ω) : ∂ α u(x) = 0 ∀|α| ≤ m − 1, x ∈ ∂Ω} for Ω ∈ C m−1 . In (2.41) and (2.63) we had introduced the concepts of V-coercive and V-elliptic bilinear forms. We recall that, e.g. for V = H0m (Ω) the bounded bilinear form a(·, ·) : V × V → R is called V-coercive, and V-elliptic, respectively, if constants α ∈ R+ , Cc ∈ R exist such that, e.g. for V = H0m (Ω), W = L2 (Ω) a(u, u) > αu2V and ≥ αu2V − Cc u2W ∀u V.

(2.141)

The second inequality is a so-called G˚ arding inequality. 12 The (−1)|β| in (2.134) usually yields an opposite sign compared to (2.76) and (2.112). This (−1)|β| produces the standard sign for a(·, ·) in (2.135).

2.4. Linear elliptic diﬀerential operators

71

For the following coercivity results for the principal part Ap , we have to impose either the Dirichlet conditions on a subset ∂Ω1 of ∂Ω of positive measure, hence measn−1 (∂Ω1 ) := ∂Ω1 1ds > 0, such that ∂Ω = ∂Ω1 ∪ ∂Ω2 , otherwise for natural boundary conditions the coercivity results holds for Ap + c, c > 0. We impose conditions on the aαβ in (2.135) and for Ω as in (2.5) for guaranteeing the H0m (Ω)-ellipticity of the a(·, ·) in (2.135). We summarize Hackbusch’s [387] Theorems 7.2.2, 7.2.3, 7.2.11, 7.2.13, Lemma 7.2.12 and extend his Theorem 7.2.7. Some of the following results have already been formulated under diﬀerent assumptions: Theorem 2.43. G˚ arding: V-coercivity and ellipticity, Fredholm alternative for Dirichlet and natural boundary conditions: Let A in (2.134) have a strongly elliptic principal part, see (2.79), and satisfy aαβ ∈ L∞ (Ω). Then 1. For m ≥ 1, and aαβ ∈ L∞ (Ω), the a(·, ·) in (2.135) are bounded. 2. For m = 1, the principal part ap (·, ·) (or for a00 (x) ≥ 0 the ap (u, v) + Ω a00 uvdx) and a(·, ·) are H01 (Ω)-coercive and H01 (Ω)-elliptic, respectively ¯ for |α| = |β| = m. Then 3. For m > 1 let, additionally, the aαβ be continuous in Ω ap (·, ·) and a(·, ·) are H0m (Ω)-coercive and H0m (Ω)-elliptic, respectively 4. For boundedly invertible A : H01 (Ω) → H0−1 (Ω) or H01 (Ω)-coercive a(u, v) we obtain ∀f ∈ H0−1 (Ω) : ∃1 u0 ∈ H01 (Ω) : Au0 = f ⇐⇒ a(u0 , v) = f, vH −1 (Ω)→H 1 (Ω) ∀v ∈ H01 (Ω). 0

0

(2.142)

5. If in a(·, ·) = a (., .) + a (., .) the a (., .) is H0m (Ω)-coercive or -elliptic and a (u, v) =

|α|+|β|<2m

|α|,|β|≤m

aαβ ∂ α u∂ β vdx, Ω

contains only derivatives of orders ≤ 2m − 1, then a(·, ·) is H0m (Ω)-elliptic. m (Ω) ⊂ H m (Ω) 6. These results remain correct, if V = H0m (Ω) is replaced by H0,p m V := H0,p (Ω) = {u : ∂ α u(x) = 0 ∀|α| ≤ m − 1, x ∈ ∂Ω1 ⊂ ∂Ω} (2.143) for, e.g. a connected ∂Ω1 ∈ C m−1 with measn−1 (∂Ω1 ) = ∂Ω ds > 0. In ∂Ω2 := ∂Ω \ ∂Ω1 natural or other boundary conditions and possibly a modiﬁed functional, f , cf. (2.38), might be imposed. 7. For natural boundary conditions, explicitly formulated in Section 2.2 only for m by the H m (Ω)-coercive ap (u, v) + m = 1, the ap (·, ·) and V = H0 (Ω) are replaced m c uvdx with c0 ≥ > 0 and V by H (Ω). Then the previous results remain Ω 0 correct.

¯ for |α| = |β| = m, then the Vice versa, if aαβ ∈ L∞ (Ω) holds and aαβ ∈ C(Ω) H0m (Ω)-ellipticity of a(u, v) implies a strongly elliptic principal part, see (2.79).

72

2. Analysis for linear and nonlinear elliptic problems

Proof. For m = 1, we indicate the nearly trivial proof. The strong ellipticity condition (2.136) yields, for ∂ i u(x) = ϑi ∈ R, ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn , |∇u(x)|2n

=

|ϑ|2n

≤

n i,j=1

i j

aij (x)ϑ ϑ =

n

aij (x)∂ i u(x)∂ j u(x) ∀ x ∈ Ω. (2.144)

i,j=1

Integration over Ω yields a(u, u) ≥ Ω |∇u(x)|2 dx ≥ u(x)2H 1 (Ω) by the equivalence of the semi- or energy-norm and · H 1 (Ω) in H01 (Ω). For m > 1, Fourier transforms, cf. Hackbusch’s [387] proof of Theorem 7.2.7, are combined with the technique of frozen coeﬃcients. In Theorem 2.104 we will give the proof for systems for the case of constant (frozen) aαβ for |α| = |β| = m, and aαβ ≡ 0 for 0 ≤ |α| + |β| ≤ 2m − 1. We do not give all the details, cf. Wloka [667], p. 282 ﬀ. or ¯ H¨ormander [403], but present only the main idea. By the continuity of the aαβ in Ω we can choose, for an > 0, a ﬁnite number of points xi ∈ Ω, i = 1, . . . , N, and their neighborhoods Uxi such that ∀|α| + |β| = 2m c,i ¯ ⊂ ∪N Ux with ac,i := aαβ (xi ) : Ω − a < ∀i = 1, . . . , N. (2.145) a αβ i=1 i αβ αβ C(Uxi )

We choose a ﬁrst generation of the i such that, after an appropriate renumbering, Uxi ∩ ∂Ω = ∅, i = 1, . . . , N1 ≤ N. Then the standard technique allows us to prove, for the strongly elliptic principal part, see (2.79), and for u ∈ H0m (Ω), the coercivity of ap on each of the Uxi , i = 1, . . . , N1 . Now choose the next inner generation of Uxi intersecting the union of the ﬁrst generation, so 1 Uxi ∩ ∪N i=1 Uxi = ∅ ∀i = N1 + 1, . . . , N2 . The normal derivatives ∂ k u/∂ν k H m−k−1/2 (∂Uxi ) , i = 1, . . . , N1 , k = 0, . . . , m − 1 can be estimated by Remark 2.1 and the Trace Theorem 1.38, (1.67), by uH m (Uxi ) , i = 1, . . . , N1 . Each of these ∂ k u/∂ν k |H m−k−1/2 (∂Uxi ) , k = 0, . . . , m − 1, can be employed proving, for each Uxi , i = N1 + 1, . . . , N2 in the next generation, the coercivity of ap for the above nontrivial boundary values ∂ k u/∂ν k H m−k−1/2 (∂Uxi ) , k = 0, . . . , m − 1, etc. The other claims are proved in [387]. We only indicate the case of natural boundary conditions. Then the strong elliptic principal part and Ω c0 u2 dx induce norms equiv2 sum induces a norm equivalent alent to the seminorm in H m (Ω) and to L2 (Ω), their m to that in H (Ω), hence ap (u, u) + Ω c0 u dx is W m,p (Ω)-coercive. Corollary 2.44. V-coercive and V-elliptic a(·, ·) induce a boundedly invertible A and imply the Fredholm alternative, respectively, cf. Theorem 2.21: 1. For, V = W m,p (Ω) or V = W0m,p (Ω), and a V-coercive a(·, ·), cf. 3., 4 below, Theorem 2.15 guarantees the unique existence of a solution u0 for Au0 = f ∈ V , cf. (2.135),(2.107).

(2.146)

2.4. Linear elliptic diﬀerential operators

73

If a(·, ·) is even symmetric, it solves, by Theorem 2.16, the minimal problem (2.61) as well. 2. For V-elliptic a(·, ·), e.g. induced by an elliptic operator A with V-coercive principal part ap (·, ·), Theorem 2.21 implies the Fredholm alternative with index A = 0. So, for an eigenvalue 0 of A, inﬁnitely many solutions for Au0 = f exist under additional conditions for f , otherwise exactly one solution exists and A−1 V←V < ∞, hence ∀ f ∈ V ∃1 solution u0 ∈ V s.t. Au0 = f ∈ V and u0 V ≤ Cf V .

(2.147)

3. All H m (Ω)-coercivity and ellipticity based existence, uniqueness and Fredholm results, including the numerical stability of all our discretization methods, remain correct with the following modiﬁcations: As a consequence of the continuous embedding of the W m,p (Ω) → H m (Ω), 2 ≤ p ≤ ∞, cf. Theorem 1.26, elliptic m operators deﬁned on W m,p (Ω) have a H0 (Ω)-coercive principal part. For the H m (Ω)-coercivity we add μ Ω uvdx with suﬃciently large μ. 4. These W m,p (Ω), 2 ≤ p ≤ ∞, results are most important for the later linearization of operators and their stability, cf. Section 2.7. Consequently, we will get for our discretization methods stability and convergence with respect to the discrete H m (Ω) norms for 2 ≤ p ≤ ∞. This applies to each of the diﬀerent cases in Theorem 2.43. In fact, these last claims are a consequence of the continuous embeddings. So let the bilinear form a(u, v) be H m (Ω)-coercive. Then the embeddings W m,p (Ω) → H m (Ω) → W m,q (Ω), 2 ≤ p ≤ ∞, 1 ≤ q ≤ 2, imply

(2.148)

uW m,q (Ω) ≤ CuH m (Ω) ≤ C uW m,p (Ω) for C, C ∈ R+ . So for 1 ≤ p ≤ 2, the W m,p (Ω)-coercivity would be a consequence of the H m (Ω)coercivity. Unfortunately the bilinear forms, cf. (2.135), usually are not deﬁned for this case. For 2 ≤ p ≤ ∞ we still obtain H m (Ω)-coercivity, and well-deﬁned bilinear forms. Again regularity results are valid for smooth or convex Ω with corners and for the interior of Ω and under conditions relaxed compared to Subsections 2.4.2, 2.4.3. For our Theorems 2.45–2.49 see [386], Theorems 9.1.16, 9.1.21, 9.1.22, 9.1.26. For a(·, ·) and its coeﬃcients we impose the conditions: Let a(·, ·)in (2.135) be H0m (Ω) − elliptic, and ∀α, β with |α|, |β| ≤ m

(2.149)

∞

∂ aα,β ∈ L (Ω) ∀ γ with |γ| ≤ max{0, t + |β| − m} for t ∈ N, γ

¯ for |β| > m − t, else aα,β ∈ L∞ (Ω), or for t ∈ N let aα,β ∈ C t+|β|−m (Ω) ﬁnally, let Ω ∈ C t+m in (2.5) for some t ≥ 0. Theorem 2.45. Regular solutions and estimates: 1. Let a(·, ·) in (2.135) and its coeﬃcients satisfy (2.149). Let s ≥ 0 and for t ∈ N let s + 1/2 ∈ {1, 2, . . . , m} and 0 ≤ s ≤ t or for t ∈ N let 0 ≤ s < t. Every (not

74

2. Analysis for linear and nonlinear elliptic problems

necessarily unique) weak solution u0 ∈ H0m (Ω) of a(u0 , v) = f, vV ×V ∀ v ∈ V = H0m (Ω), for f ∈ H −m+s (Ω),

(2.150)

see Corollary 2.44, is in H m+s (Ω) ∩ H0m (Ω) and satisﬁes the estimate u0 H m+s (Ω) ≤ Cs f H −m+s (Ω) + u0 H m (Ω) .

(2.151)

2. If u0 ∈ H m (Ω) solves the inhomogeneous Dirichlet problem ∂ l u0 /∂ν l = ϕl ∈ H m+s−l−1/2 (∂Ω), l = 0, . . . , m − 1, then again u0 ∈ H m+s (Ω) and we have to add to (2.151) the last term m−1 ϕl H m+s−l−1/2 (∂Ω) . u0 H m+s (Ω) ≤ Cs f ··· + u0 ··· +

(2.152)

(2.153)

l=0

3. Speciﬁcally, for m = 1 and f ∈ H −1+s (Ω), let u0 ∈ V = H 1 (Ω) solve the inhomogeneous natural boundary value problem, cf. (2.150), a(u0 , v) = f, vV ×V ∀v ∈ V, f ∈ H −1+s (Ω), Ba u0 = ϕ ∈ H s−1/2 (∂Ω).

(2.154)

Then u0 ∈ H 1+s (Ω) satisﬁes, cf. (2.151), (2.153) u0 H 1+s (Ω) ≤ Cs f H −1+s (Ω) + u0 H 1 (Ω) + ϕH s−1/2 (∂Ω) . Theorem 2.46. Regular solutions and estimates under weaker conditions, Neˇcas [509, 510]: Let Ω in C 0,1 , and a(·, ·) in (2.135) and its coeﬃcients satisfy (2.149) for 0 < s < t ≤ 1/2 and f ∈ H −m+s (Ω). Then every (not necessarily unique) weak solution u0 ∈ H0m (Ω) of (2.150) is in H m+s (Ω) ∩ H0m (Ω) and satisﬁes the estimate (2.151). Theorem 2.47. Regular solutions for convex Ω: Let Ω in (2.5) be convex, hence Ω ∈ C 0,1 and a(·, ·) in (2.135) and its coeﬃcients satisfy (2.149) for the case |α| + |β| < 2m, 1 = t ∈ N, and ¯ for |α| = |β| = m. aα,β ∈ C 0,1 (Ω)

(2.155)

Then the above result (2.151) is valid for s = 1. The transition from the weak to the strong form, so from Au0 = f ∈ H −m (Ω) to As u0 = f0 ∈ L2 (Ω), is only possible for smooth enough coeﬃcients and right-hand side. So for m = 1 and the boundary condition Ba u0 = ϕb this requires instead of f ∈ H −1 (Ω) the special form, cf. (2.109), f0 vdx + ϕb vds ∀ v ∈ V := H 1 (Ω). f, vV ×V := Ω

∂Ω

For our later studies of so-called nonconforming methods we have to be able to restrict the normal derivatives or more general boundary operators of functions to the boundary of or to interior curves in Ω. Then Remark 2.1 and Theorem 1.39 consider

2.4. Linear elliptic diﬀerential operators

75

the continuous tracing γ∂ α u in the sense Ω ∈ C t , and |α| + 1/2 < s ≤ t ∈ N, or s < t ∈ N =⇒ γ∂ ∈ L(H (Ω), H α

s

s−|α|−1/2

(2.156)

(∂Ω)).

Theorem 2.48. Traces of u ∈ H s (Ω) to H s−|α|−1/2 (∂Ω): Let Ω0 ∈ C 0,1 , u ∈ H 3/2+ (Ω), and |α| ≤ 1. Then γ∂ α u ∈ H (∂Ω). Related results are found in, e.g. Taylor [618], Chapter 4, Proposition 4.5; Jonsson and Wallin [425]; Jerison and Kenig [419]; Schwab [575]; and Grisvard [373]. For the analysis of nonconforming methods we restrict the discussion to second order problems, so to m = 1. Then we usually assume either H 2 (Ω) or H 3/2+ (Ω), cf. Remark 2.1. In contrast to Theorem 2.45 the following result relates data and interior regularity, cf. [618], Chapter 5, Theorem 11.1. Theorem 2.49. Interior regularity: Let Ω0 ⊂⊂ Ω1 ⊂⊂ Ω in (2.5), let a(·, ·) in (2.135) be H0m (Ω)-elliptic. Let the coeﬃcients satisfy (2.149) with Ω replaced by Ω1 , s ≥ 0, and t ≥ s ∈ N or t > s. For f |Ω1 ∈ H −m+s (Ω1 ) let u0 be a weak solution of (2.150). Then u0 |Ω0 ∈ H m+s (Ω0 ) satisﬁes a modiﬁed (2.151), (2.157) u0 H m+s (Ω0 ) ≤ C(s, Ω0 , Ω1 , Ω) f H −m+s (Ω1 ) + u0 H m (Ω) . The following result, answering the question of existence and uniqueness for a large class of boundary value problems, is a simple consequence of the equivalence of norms in Sobolev spaces. We only indicate its proof and assume bj,α (x)∂ α u hence (2.158) A special Dirichlet system Bj u := |α|≤j−1

m1 = 0 < m2 = 1 < · · · < mm = m − 1, and let bj,α ∈ L∞ (Ω). Theorem 2.50. Modiﬁed boundary conditions: 1. Let a(·, ·) in (2.135) be H0m (Ω)-elliptic, cf. Theorem 2.43. Assume (2.158) such that the principal part of the boundary operator, the Bp,j , deﬁned by the highest derivatives ∂ α , |α| = j − 1, multiplied by the bj,α , in the boundary operator yields the rearranged 13 form (2.159). The normal and the complementary tangential derivatives ∂ j−1 u/∂ν j−1 and ∂ β u/∂ tβ , cf. (2.26), (2.33), are split as Bp,j u(x) :=

|α|=j−1

bj,α (x)∂ α u = bj (x)

∂ j−1 u ∂β u + cj,β (x) β j−1 ∂ν ∂ t |β|=j−1

0 = bj ∈ C(∂Ω), j = 1, · · · , m, bj unique.

(2.159)

13 The rearranged form is obtained by a transformation of the original coordinates into a system of ν and orthogonal elements t1 , . . . , tn−1 in the tangential plane of ∂Ω. The ∂ α u : |α| = j − 1 are β

n−1 βn 1 transformed into ∂ j−1 u/∂ν j−1 and the complementary ∂ β u/∂ tβ , and ∂ tβ = ∂ tβ 1 · · · ∂ tn−1 ∂ ν with β1 · · · + βn−1 > 0.

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2. Analysis for linear and nonlinear elliptic problems

Then for a given g = (Bj u = gj )m j=1 unique ϕj exist, such that, in the trace sense, j j g = (Bj u = gj )m j=1 ⇐⇒ (∂ u)/∂ν = ϕj , j = 0, · · · , m − 1. m 2. The a(·, ·) in (2.135) is simultaneously H0m (Ω)- and HB (Ω)-elliptic with m HB (Ω) := {u ∈ H m (Ω) : Bj (u)(x) = 0 ∀x ∈ ∂Ω, j = 1, · · · , m}.

3. A combination with the index, regularity and embedding results in Theorems 2.28 and 2.45, 2.47 and 1.26, respectively, shows: index A = 0 for a H0m (Ω)-elliptic a(·, ·) in (2.135) and for all the linear operators, A, induced by a(·, ·), studied in this section. Proof. By Theorem 2.43 it suﬃces to prove the claim for the principal part of the diﬀerential and boundary operators. Obviously the Dirichlet conditions in the trace sense, see Remark 2.1 and Theorem 1.37, (∂ j u/∂ν j ) = 0, j = 0, . . . , m − 1, determine ∂ α u = 0, |α| ≤ m − 1, and hence the conditions Bj (u) = 0. This remains correct for nonhomogeneous boundary conditions as well. Vice versa we prove by induction that Bj (u) = 0 or = g imply the homogeneous or inhomogeneous Dirichlet conditions (∂ j (u)/∂ν j ) = 0 or = ϕj , j = 0, · · · , m − 1 : in fact, B1 (u) = 0 or = g1 is equivalent to u = 0 or = (g1 /b1 ). This information allows us to compute all the tangential partial derivatives ∂ α u/∂ tα , |α| = 1 along ∂Ω, vanishing for homogeneous Dirichlet conditions. We combine these known ∂ α u/∂ tα , |α| = 1 with B2 (u) = 0 or = g2 determining the ∂u/∂ν, again vanishing along ∂Ω for homogeneous Dirichlet conditions. Next, we combine the known u, ∂u/∂ν, ∂ α u/∂ tα , |α| = 1, for determining in a ﬁrst step all the ∂ β u/∂ tβ with |β| = 2. Then B3 (u) = 0 or = g3 yields ∂ 2 u/∂ν 2 along ∂Ω, etc. Finally we use Lemma 2.26 to transform the inhomogeneous to homogeneous Dirichlet conditions. Then by Theorem 2.43, the principal part ap (·, ·) is H0m (Ω)-coercive. The next claims are an immediate consequence of this result. We ﬁnish by indicating the relation between As and A for the order 2m and general boundary conditions. Similarly to Section 2.2 we have extended A = As in (2.134) to A in (2.135). For ﬁnding this relation we would have to determine a complementary system Sj , j = 1, . . . , m, of boundary operators for the Bj , j = 1, . . . , m. For these Bj , Sj , j = 1, . . . , m, a second pair of Cj , Tj , j = 1, . . . , m, can be determined such that a generalized Green’s formula is valid, see e.g. Lions and Magenes [478] and Oden and Reddy [518], pp. 162 ﬀ. If we restrict the discussion to Dirichlet boundary conditions we can avoid all these technicalities, since these boundary operators are multiplied with the vanishing ∂ α u(x) = 0 ∀|α| ≤ m − 1, x ∈ ∂Ω anyway. Multiplying As u in (2.134) with v ∈ H0m (Ω), integrating over Ω and iteratively applying the Green’s formula we get, generalizing the special case m = 1 in (2.9), (2.16)–(2.18), the weak form A. Note that

2.5. Nonlinear elliptic equations

77

we only require boundary conditions for the v, but not for u in vAs udx = v (−1)|β| ∂ β (aαβ ∂ α u)dx ∀ u ∈ H 2m (Ω), v ∈ L2 (Ω), Ω

Ω

|α|,|β|≤m

= (As u, v)L2 (Ω) with aαβ ∈ W |β|,∞ (Ω) aαβ ∂ α u∂ β vdx = a(u, v) := Au, vV ×V =

(2.160)

Ω |α|,|β|≤m

∀ u ∈ V = H m (Ω), v ∈ Vb = H0m (Ω), aαβ ∈ L2 (Ω) with

|Au, vV ×V | ≤

max

|α|,|β|≤m

aαβ L∞ (Ω) uH m (Ω) vH m (Ω) .

By once more using the Green’s formula we obtain for v ∈ H0m (Ω) ∩ H 2m (Ω) (As u, v)L2 (Ω) = a(u, v) = Au, vV ×V = u, Ad vV×V = u, Ads v L2 (Ω) (2.161) ∀u, v ∈ H 2m (Ω) ∩ H0m (Ω), with Ads v = ((−1)|α| ∂ α (aαβ ∂ β v), |β|,|α|≤m

for aα,β ∈ W max{|α|,|β|},∞ (Ω). Summarizing these results we get, see (2.160), As , Ads : Vs := H 2m (Ω) ∩ H0m (Ω) → W := L2 (Ω) for aα,β ∈ W max{|α|,|β|},∞ (Ω) ⎫ ⎧ ⎬ ⎨ with As W←Vs ≤ sup |∂ β (aαβ (∂ α u))|dx (2.162) ⎭ uH 2m (Ω) =1 ⎩ Ω ≤C

|α|,|β|≤m

∂ γ aαβ L∞ (Ω) ≤ C

|γ|≤|β|,|α|≤m

and similarly Ads W←V ≤ s

aαβ W max{|α|,|β|},∞ (Ω)

|α|,|β|≤m

aαβ W |α|,∞ (Ω) ,

|α|,|β|≤m

A, Ad : V := H0m (Ω) → V := H −m (Ω) for aα,β ∈ L∞ (Ω) and Ad V ←V , AV ←V ≤ C aαβ L∞ (Ω) .

(2.163)

|α|,|β|≤m

2.5 2.5.1

Nonlinear elliptic equations Introduction

There is a vast literature on this extremely active topic, e.g. we mention that a search for regular solutions of nonlinear elliptic problems yields ≈ 1000 hits. 14 14 We list some of them: Alt and Luckhaus [23], Amann and Quittner [25], Bensoussan and Frehse [81], Boccuto and Mitidieri [99], Bochniak and S¨ andig [128], Chemin and Xu [169], Dachkovski [232], De Donno [271], Deng [283], Garello [337], Giaquinta, Necas, John and Star´ a [343], Girardi, Mastroeni

78

2. Analysis for linear and nonlinear elliptic problems

Already in Section 2.4 we have indicated the need for diﬀerent concepts. We want to demonstrate this for the the case of nonlinear elliptic operators: −Δu = f (u) ∈ Ω, u|∂Ω = 0 with A = −Δ.

(2.164)

This is a special case of the so-called semilinear elliptic operators. For f (u) = eu we obtain the famous Bratu equation, a standard example for bifurcation problems, e.g. Allgower and Jepson [22]. Either the f and Ω are smooth enough and A = −Δ is boundedly invertible, see Section 2.4. Then the preceding linear theory for A can be combined via u = A−1 f (u) with ﬁxed-point theorems to get results for speciﬁc semilinear equations. This is possible in H¨ older spaces A : C 2,γ (Ω) → C γ (Ω). No growth conditions for the f (u) for u → ∞ are necessary. This is well presented by Zeidler [676], e.g. Sections 6.5, 7.16, 8.12. We may relax the above smoothness conditions for A and impose growth conditions for the f (u). In some cases this still allows us to apply the strong form A : H 2 (Ω) → L2 (Ω). Or the f and Ω are not smooth. Then we have to generalize the weak theory in the last sections to (2.164). The very powerful tool of weak solutions u ∈ H 1 (Ω) for the corresponding weak linear operator A requires Au − f (u) ∈ H −1 (Ω). This is usually no longer correct for a nonlinear f . So we have to generalize A : H 1 (Ω) → H −1 (Ω) to A : W 1,p (Ω) → W −1,q (Ω), 1/p + 1/q = 1, and have to impose appropriate growth conditions on the nonlinear function f (u) for u → ∞. This is necessary since the W 1,p (Ω) is not a Banach algebra, so, e.g. u ∈ W 1,p (Ω) does not imply u2 ∈ W 1,p (Ω). A generalization of Sobolev spaces, e.g. towards H r (Ω), W r,p (Ω), r ∈ R, or A : W m,p (Ω) → W −m,q (Ω), or to Besov or Besov–Triebel–Lizorkin spaces allows a deeper analysis, see e.g. Runst and Sickel [561]. A major topic in this direction are general semilinear, quasilinear and fully nonlinear elliptic equations, more generally the so-called monotone and coercive and noncoercive operators. A thorough presentation in the Sobolev and H¨ older space setting, appropriate for our context, is given by Zeidler [676–678] and Taylor [618–620]. Second order equations are studied in the classical Gilbarg and Trudinger [346], and in Showalter [589] and Chen and Wu [170]. Here we essentially study homogeneous Dirichlet boundary conditions. In [589] extensions to more general quasilinear equations and special systems and to non-Dirichlet boundary conditions are discussed. Showalter [589] considers Neumann type conditions, and boundary and interior inequality constraints for the solutions. Here we combine, for special cases, the classical, strong or weak formulation for linear operators, e.g. A : C 2,γ (Ω) → C γ (Ω), with Au − f (u) = 0 for and Matzeu [347], Grunau and Sweers [377], Johnsen [424], Lakkis [468], Lassoued [470], Le Dung [308], Laurence and Stredulinsky [468], Luckhaus [481], M¨ uller [506], Naniewicz and Panagiotopoulos [507], Necas, John and Star´ a [511], Nedev [512], Nguyen [633], Pacard [522], Shahgholian [582], Guo and Li [379], Uchida [639], Zhang [682] Yang [672]; compare Kozlov, Maz’ya and Rossmann [452]. Additional papers and books consider solution techniques, e.g. Altman [24]. A rather important feature of this theory of nonlinear evolution equations in Banach spaces is the following: Nonlinear problems can be solved via linearized evolution equations with many additional tricks. Dong [299], lists solvability and regularity of solutions for many applications. In his Chapter 9, fully nonlinear parabolic equations are studied.

2.5. Nonlinear elliptic equations

79

interesting existence results. Estimates for the solutions, in particular with respect to higher norms, hence regularity, are highly complicated and often seem to be missing in textbooks. Sometimes a complete discussion in Hilbert spaces H 1 (Ω) is still possible due to the speciﬁc structure of the nonlinearity. A famous example is the Navier–Stokes operator in Section 2.8. We present in this section a short introduction to semilinear, quasilinear and fully nonlinear elliptic equations, strongly inspired by Zeidler [676–678]. Following the essential deﬁnitions in Subsection 2.5.2 we discuss special semilinear equations in Subsection 2.5.3, generalizing the above (2.164). Subsection 2.5.4 studies quasilinear equations of second order. We continue with the so-called Nemyckii operators in Subsection 2.5.5. They generalize the above f (u) and allow the discussion of semilinear and quasilinear elliptic operators in Subsection 2.5.6 of order 2m. We ﬁnish with fully nonlinear operators in Subsection 2.5.7. They are mainly studied here for speciﬁc cases, see e.g. Example 2.78 and Gilbarg and Trudinger [346]. Quasilinear problems have been studied very intensively. So their existence and uniqueness results, applied to semilinear problems, might be more general than those formulated for semilinear problems, see e.g. Theorem 2.61. Recently nonlinear elliptic problems have been considered with more than one solution for speciﬁc nonlinearities, e.g. with continuation methods, cf. Allgower, Cruceanu and Tavener [18, 19]. For the many gaps in our knowledge of solutions of nonlinear elliptic problems, the results of this book open up new numerical techniques. Continuation and discretization methods with the “stability implies invertibility of linear operators” in Section 3.6 allow at least indications for solutions and their properties under appropriate conditions for these problems. 2.5.2

Deﬁnitions for nonlinear elliptic operators

For orders 2 and 2m, we recall the notations, variables, and diﬀerential operators in (2.73) as, e.g. ϑ0 = 1, ϑα , Θ0 ∈ R, ϑ, Θ = Θ1 ∈ Rn , Θk ∈ Rnk and ∂ 0 u = ∇0 u = u, ∂ α u, ∇k = (∂ α )|α|=k .

(2.165)

In ∇2 the uxy and uyx are diﬀerent. So α = (1, 1) appears twice, similarly in ∇2m , thus deﬁning the n2m and N2m below. We extend the previous linear to fully nonlinear equations and introduce their ellipticity. For nonlinear operators we consider real valued functions of the form Gw : D(Gw ) → R, w = (x, Θ0 , . . . , Θ2m ) ∈ D(Gw ) ⊂ Ω × RN2m 2n

(2.166)

where Ω ⊂ Rn satisﬁes (2.5), RN2n = R × Rn × · · · × Rn , and N2m := (n2m+1 − 1)/(n − 1) for n > 1, and N2m := 2m + 1 for n = 1.

80

2. Analysis for linear and nonlinear elliptic problems

For the important case 2m = 2 this reads as Gw : D(Gw ) → R, w = (x, .., Θ2 ) ∈ D(Gw ) ⊂ Uo := Ω × R × Rn × Rn×n . 0

1

(2.167)

2

Sometimes we replace x, Θ , Θ , Θ by x, z, p, r, and usually we need only the n(n + 1)/2-dimensional subspace of symmetric real valued matrices in Rn×n . In all the following nonlinear problems only functions, u, are considered with u : Ω → R and w(x) := wu (x) := (x, u(x), ∇u(x), . . . , ∇2m u(x)) ∈ D(Gw ).

(2.168)

Preparing the following deﬁnitions, we explicitly formulate diﬀerent (strong) forms G of quasi- and semilinear operators of order 2. The generalization to order 2m is obvious. 15 G(·) : C 2 (Ω) → C(Ω), for u, ∇u, G(u) we simplify the notation, writing G(u) :=

n

aij (·, u, ∇u)∂ i ∂ j u + a(·, u, ∇u)

(2.169)

i,j=1

is the quasilinear operator, usually with (aij )ni,j=1 symmetric, G(u) := −

n

∂ j (bj (·, u, ∇u)) + b(·, u, ∇u)

(2.170)

j=1

is the quasilinear divergent operator, G(u) :=

n

aij (·, u)∂ i ∂ j u + b(·, u, ∇u), is the semilinear operator,

(2.171)

aij (·)∂ i ∂ j u − f (u), is a special semilinear operator,

(2.172)

i,j=1

G(u) :=

n i,j=0

G(u) := Gw (·, u, ∇u, ∇2 u), is fully nonlinear, if it is none of the above.

(2.173)

For G(·) : C 2 (Ω) → C(Ω) we need, for the quasilinear case, real valued aij , a, b ∈ C or bj ∈ C 1 . An extension G(·) to G(·) : W 2,p (Ω) → Lq (Ω) requires appropriate growth conditions for the aij (·, u, ∇u), bj (·, u, ∇u), a(·, u, ∇u), b(·, u, ∇u), Gw . We will come back to this point below. Deﬁnition 2.51. Nonlinear diﬀerential operators and ellipticity: 1. A nonlinear PDE of order 2m (analogously for order m) for a function u as in (2.168) and Gw as in (2.166) has the form and the solution u0 G(u) := Gw (wu ) = Gw (·, u, ∇u, . . . , ∇2m u) and G(u0 )(x) = 0 ∀ x ∈ Ω (2.174) or a.e. for x ∈ Ω. It is quasilinear if Gw is an aﬃne function of the variables Θ2m , see (2.169)–(2.170). It is semilinear if Θ2m and its coeﬃcients in Gw , the principal part, deﬁne a linear operator in Θ2m , hence, these coeﬃcients only 15

Note the − sign in (2.170) in contrast to the + sign in [346], p. 87.

2.5. Nonlinear elliptic equations

81

depend upon x, u, . . . , ∇2m−1 u, see (2.171)–(2.172). In all other cases it is fully nonlinear. 2. Let x ∈ D ⊂ Ω and w(x) = wu (x) ∈ D(Gw ). Then we require G ∈ C 1 (D(Gw )) (or e.g. ∈ W 1,∞ (D(Gw ))) and a principal part (or the symbol) of the linearized operator, compare (2.73), (2.79), with the corresponding property. So it satisﬁes, with the Euclidean norm, |ϑ| = |ϑ|n , and (2.79) m λ(w)|ϑ|2m n ≤ (−1)

n aα (w)ϑα ≤ Λ(w)|ϑ|2m n ∀ ϑ ∈ R ,

|α|=2m

(2.175)

∀x ∈ Ω or, a.e.∀x ∈ Ω, with w = wu (x) ∈ D(G ), aα (w) := w

(Gw ϑα ) (w(x));

hence, 0 ≤ λ(w) for elliptic, 0 < ≤ λ(w) for strongly elliptic and 0 < ≤ λ(w), Λ(w)/λ(w) bounded for a speciﬁc w = w(x) ∈ D(Gw ), or w ∈ D, or ∀ w ∈ D(Gw ) for uniformly elliptic operators, G. The Gw ϑα denotes the partial derivative of the real valued function Gw with respect to ϑα , |α| = 2m. For Gw ϑα (w) ∈ L∞ (D(G)) this condition is only required a.e. in Ω. 3. For 2m = 2 , Gw ∈ C 2 (D(Gw )) we get the symmetric matrix n

w w α i j −Gw Θ2 (w) := − (Gϑα (w))|α|=2 := − (Gϑi ϑj (w))i,j=1 , ϑ = ϑ ϑ , ∀x ∈ D. (2.176)

Then λ(w) and Λ(w) can be chosen as the minimal and the maximal (real) eigenvalues of Gw Θ2 (w(x)).

2.5.3

Special semilinear and quasilinear operators

The topic of this subsection is special equations of the form (2.169)–(2.172) and their generalization to quasilinear elliptic equations of order 2m. We study three types of results. The common structure of the ﬁrst two types is their deﬁnition in H¨ older spaces. This allows applying the Schauder ﬁxed point theorem and the Schauder principle. In the special semilinear case, the nonlinearity only occurs in the lowest order term as f (u) as in (2.172). For the quasilinear case, we restrict the presentation to 2m = 2 with special coeﬃcients aij , a, bj , b in (2.169), (2.170), and (2.193), (2.194). The third type presents semilinear equations, where again the nonlinearity only occurs in the lowest order term as f (u) as in (2.172), but only deﬁned on nonlinear spaces. We repeat: the existence and uniqueness results for quasilinear problems might be more general than those for semilinear problems, see e.g. Theorem 2.61. Special semilinear operators in Sobolev and H¨ older spaces ¯ and C k,γ (Ω) ¯ conditions, respectively, We study the orders 2m and 2 and impose C ∞ (Ω) with the norms uC k,γ (Ω) , see (2.87). So the operators, considered here, use the linear operator A as in Section 2.4, homogeneous boundary and smoothness conditions as,

82

2. Analysis for linear and nonlinear elliptic problems

see (2.88), (2.93), (2.117), G(u) := Au − cf (u) = 0 in Ω, Bu = (Bj u)m j=1 = 0 on ∂Ω with k ∈ N0

(2.177)

¯ → C k,γ (Ω), ¯ Bj : C 2m+k,γ (∂Ω) → C 2m+k−mj ,γ (∂Ω), mj < 2m, G : C 2m+k,γ (Ω) Au := aα ∂ α u, Bj u = bjα ∂ α u, j = 1, . . . , m, and |α|≤2m

|α|≤mj

B satisﬁes the complementary condition, either Dirichlet or as (2.183), with 2m+k,γ ¯ ¯ : Bu|∂Ω = 0}, k ∈ N0 , CB,0 (Ω) := {u ∈ C 2m+k,γ (Ω)

(2.178)

¯ bjα ∈ C ∞ (∂Ω), ∀|α| ≤ 2m, j = 1, . . . , m, ∂Ω ∈ C ∞ , aα ∈ C ∞ (Ω),

(2.179)

2m+k,γ ¯ ¯ is strongly (so uniformly) elliptic and of index 0. A : CB,0 (Ω) → C k,γ (Ω) 2m+k,γ ¯ We use the notation CB,0 (Ω) indicating the more general boundary conditions allowed in this context. The conditions (2.177)–(2.179) and the following (2.180)– (2.181) are formulated for k ∈ N0 . In both cases we start with k = 0. The general k ∈ N0 will be needed for higher regularity results, see Theorem 2.56 below. Examples for (2.178), (2.179) are (2.80) and Theorem 2.50. Similarly as above, it is possible to reduce the conditions for the coeﬃcients and domain for the special case of diﬀerential equations of order 2, cf. Theorem 2.37 ﬀ. and (2.83), (2.112). We assume for 2,γ ¯ ¯ with index A = 0, (Ω) → C γ (Ω) 2m = 2, m1 = 1 let A : CB,0

Au :=

n

aij (x)∂ i ∂ j u, Bu = b0 u +

i,j=0

n

ci ∂ i u,

i=1

¯ b0 , cj ∈ C 1+k,γ (Ω), ¯ 0 < γ < 1 ∀i, j = 0, · · · , n, aij ∈ C k,γ (Ω), ∂Ω ∈ C

(2.180)

2+k,γ

(2.181)

, k ∈ N0 , for c = (c1 , · · · , cn ) ≡ 0 let b0 (x) = 0, otherwise

|(c(x), ν(x))n | > κ > 0 ∀ x ∈ ∂Ω, A is strongly (so uniformly) elliptic. Leung [473] discusses diﬀerent examples of interacting population reaction diﬀusion systems. These are PDEs of this type coupled with ODEs. For solvability results for the semilinear equations Au − f (u) = 0 in (2.177) we have to extend some results for the linear equations Au − g = 0. We present the main idea for the proofs of the following theorems, but only indicate the necessary modiﬁcations of the proofs for the R2 results in [676], Proposition 6.4, to the Rn case: We start with 2m,γ ¯ ¯ see (2.178), and Proposition 2.54, with (Ω) → C γ (Ω), a boundedly invertible A : CB,0 −1 ¯ hence K|C γ (Ω) ¯ → C 1,β (Ω), . k = 0. Its inverse can be extended to a K : C(Ω) ¯ = (A) B,0 This yields the two equivalent formulations G(u)(x) = Au − cf (u) = 0 ⇔ u = Kcf (u),

(2.182)

2.5. Nonlinear elliptic equations

83

¯ → for c, f , see Theorem 2.55. Then Schauder type results for the linear K : C(Ω) 1,β ¯ CB,0 (Ω) are combined with ﬁxed-point theorems. This yields the existence and uniqueness of solutions for (2.182). We combine the results in Subsections 2.4.2 and 2.4.3 for linear equations of orders ¯ → C 1,β (Ω), ¯ 2m and 2 in Rn , respectively, to obtain the continuous extension K : C(Ω) B,0 similarly to Zeidler [676]. We start summarizing conditions from Subsections 2.4.2 and 2m,γ ¯ ¯ Theorem (Ω) → C γ (Ω). 2.4.3 which guarantee a boundedly invertible linear A : CB,0 2.28 shows that for order 2m we cannot directly guarantee a boundedly invertible A. For order 2 the situation is modiﬁed due to the maximum principle reﬂected in Theorem 2.37. For orders 2m and 2 and k = 0 we obtain Theorems 2.52 and 2.53 by combining Theorems 2.21, 2.28, 2.33, 2.39, 2.40, 2.43, Corollary 2.44 and Theorem 2.50. In Theorem 2.50 we found a special Dirichlet system satisfying the complementary condition with index A = 0. We had transformed it into the form (j = 1, . . . , m), Bj u = bj

∂ j−1 u ∂β u + cj,β β + j−1 ∂ν ∂t |β|=j−1

bj,α ∂ α u, with 0 = bj (x)∀x ∈ ∂Ω. (2.183)

|α|≤j−2

Theorem 2.52. Results for the boundary conditions in (2.177)–(2.179): 2m,γ ¯ ¯ of order 2m assume (2.177)–(2.179) and let index (Ω) → C γ (Ω) 1. For an A : CB,0 (A) = 0. If λ = 0 is not an eigenvalue of A, it is boundedly invertible. Otherwise (A + μ) with suﬃciently small or large enough μ ∈ R is boundedly invertible. The ¯ : If A is boundedly invertible, following estimates are valid for Au0 = g ∈ C γ (Ω) compare Theorem 2.37, the u0 C γ (Ω) ¯ is dropped in (2.184) u0 C 2m,γ (Ω) ¯ ≤ C u0 C γ (Ω) ¯ + gC γ (Ω) ¯ .

2. The preceding results are correct if the complementary or Dirichlet conditions are replaced by these modiﬁed conditions Bj in (2.183). For the particularly important case of second order equations, 2m = 2, the boundary conditions in (2.183) can be generalized, by Theorem 2.37, as one of the Bu = B1 u = u, or B2 u = ∂u(x)/∂ν, or B3 u = bu +

n

ci ∂ i u on ∂Ω

(2.185)

i=1 2,γ ¯ ¯ : Bi u|∂Ω = 0}, for one of the i = 1, 2, 3. (Ω) := {u ∈ C 2,γ (Ω) with CB,0

(2.186)

2,γ ¯ Theorem 2.53. Results for 2m = 2 and Bi in (2.185), (2.186): For A : CB,0 (Ω) → γ ¯ C (Ω) of order 2 assume (2.180)–(2.181) with k = 0, (2.185), (2.186). Then index (A) = 0 and the results in Theorem 2.52 are valid for 2m = 2. If additionally, λ = 0 is not an eigenvalue of (A, B), e.g. the aij satisfy aij ≡ 0, for ij = 0, i + j > 0, and

a00 (x) ≥ 0 ∀ x ∈ Ω for B1 u and for B3 u with b(x) > 0 ∀ x ∈ ∂Ω, or

(2.187)

a00 (x) > 0 ∀ x ∈ Ω for B2 u ∀ x ∈ ∂Ω, 2,γ ¯ ¯ is boundedly invertible and (2.184) holds for 2m = 2. (Ω) → C γ (Ω) then A : CB,0

84

2. Analysis for linear and nonlinear elliptic problems

We combine these results, in particular Theorems 2.32, 2.37, 2.52 and 2.53, with the results in Zeidler [676], pp. 233 ﬀ., see his Proposition 6.4. 2m,γ ¯ ¯ in (2.177), (2.186), with B in Proposition 2.54. Let A : CB,0 (Ω) → C γ (Ω) ¯ → C 2m,γ (Ω). ¯ This is (2.178), be boundedly invertible, and yield K = (A)−1 : C γ (Ω) B,0 correct under the conditions in Theorems 2.52 and 2.53, hence for orders 2m and 2, respectively Then K can be uniquely extended to a continuous linear operator

¯ for a ﬁxed β, 0 < β < 1. ¯ → C 1,β (Ω) K : C(Ω) B,0

(2.188)

Proof. There are only minor diﬀerences compared to the proof in Zeidler [676], pp. 238, for 2m = 2. We combine Theorems 2.30 and 2.33, in particular (2.94) and (2.98) to generalize Zeidler’s condition (A) as KgW 2m,p (Ω) ≤ CgLp (Ω) ∀g ∈ Lp (Ω), 1 < p.

(2.189)

By Theorem 1.26 the embeddings ¯ → Lp (Ω) and W 2m,p (Ω) → C 1,β (Ω) C(Ω) are continuous for p > n/(2m − 1 − β). So (2.189) is modiﬁed into γ ¯ KgC 1,β (Ω) ¯ ≤ CgC(Ω) ¯ ∀g ∈ C (Ω).

¯ is dense in C(Ω). ¯ So K can be By the Weierstrass approximation theorem, C γ (Ω) extended as claimed. Now we are ready for applying this result to special semilinear equations, see Zeidler [676], pp. 233 ﬀ., his Proposition 6.7, and Corollaries 6.8, 6.9: Theorem 2.55. Existence and uniqueness of solutions for semilinear equations: 2m,γ ¯ ¯ in (2.177), with B in (2.178), satisfy the conditions (Ω) → C γ (Ω) 1. Let A : CB,0 of Proposition 2.54 and assume:

for ﬁxed R > 0 let f : [−R, R] → R, f ∈ C β , 0 < β < 1.

(2.190)

Then for every suﬃciently small |c|, c ∈ R, the semilinear boundary value problem (Au − cf (u))(x) = 0∀x ∈ Ω, Bj u|∂Ω = 0, j = 1, . . . , m,

(2.191)

¯ ∩ C m (Ω) ¯ with 0 < γ < 1, see (2.186). has a unique solution u0 ∈ C 2m,γ (Ω) B,0 ¯ 2. For c ≥ 0 and f (u) ≥ 0 on [0, R], the solution is nonnegative, u0 ≥ 0 on Ω. 3. For a Lipschitz-continuous f there exists exactly one solution with |u0 (x)| ≤ R ¯ It can be determined by the convergent iteration process on Ω. 2m,γ ¯ m ¯ ¯ ∩ CB,0 (Ω) := C 2m,γ (Ω) (Ω), u0 := 0, Auk+1 = cf (uk ), k ∈ N0 . uk ∈ CB,0

Zeidler’s proof [676], pp. 240,241, remains unchanged, if only his Proposition 6.4 is replaced by our Proposition 2.54. Modiﬁcations for f are discussed by von Wahl [658], and Leung [473], pp. 82 ﬀ. Higher regularity for this and later theorems is obtainable via diﬀerent kinds of socalled bootstrap arguments or inherited regularity, cf. [676]. They are chosen according

2.5. Nonlinear elliptic equations

85

to the results available for the speciﬁc situation, usually based upon inductive arguments. Our approach here uses the regularity results for the corresponding linear elliptic equations, presented in Section 2.4, see Theorems 2.28, 2.30–2.33, 2.38, 2.39. We modify a technique of Hackbusch [387], Remark 9.1.2. We will refer to this result later on, hence, we formulate it as Theorem 2.56. Regularity, inherited to compact and semilinear perturbations: 1. Choose k ∈ N0 in (2.177)–(2.179) for A of order 2m or (2.180)–(2.181) for order 2. Then Theorems 2.52, 2.53 and the estimates (2.184) have the stronger form 2m+t,γ ¯ ¯ 0 ≤ t ≤ k, (Ω) → C t,γ (Ω), A : CB,0 (2.192) Au0 = g satisﬁes u0 C 2m+t,γ (Ω) ¯ ≤ C u0 C 2m,γ (Ω) ¯ + gC t,γ (Ω) ¯ . Note that for boundedly invertible A the term u0 C 2m,γ (Ω) ¯ is deleted. 2. Let δA be a linear or nonlinear diﬀerential operator of order ≤ 2m − 1, such that ¯ ∩ C 2m,γ (Ω) ¯ → C r+1,γ (Ω) ¯ is bounded for all r = 0, 1, . . . , k − 1. δA : C r+2m,γ (Ω) B,0 2m+t,γ ¯ ¯ and (2.192) for the solution u0 of (Ω) → C t,γ (Ω) This implies A + δA : CB,0 (A + δA)u0 = g. 3. Under the conditions of Theorem 2.55 let the f in (2.190) satisfy ¯ → f (r) (u) := f (u(·)) ∈ C r+1,γ (Ω), ¯ r = 0, 1, . . . , k − 1, f (r) : u ∈ C r+2m,γ (Ω) e.g. these f (r) are in C r+1,γ (R). Then the solutions u0 of (2.191) are u0 ∈ 2m+t,γ ¯ (Ω). The norms of u0 can be estimated as in (2.192). CB,0 Proof. By induction: For k = t = 0 the result and estimates (2.192) are a consequence of Theorems 2.45 and 2.53. Now we inductively assume (2.192) for the solution ¯ and for 0 ≤ t − 1 ≤ k − 1 for k ∈ N. Consequently, u0 ∈ (A + δA)u0 = g ∈ C t−1,γ (Ω) t−1+2m,γ ¯ (Ω) solves Au0 = g˜ := g − δAu0 as well. By our inductive assumption for C A + δA and ﬁnally the condition for A we estimate u0 C t−1+2m,γ (Ω) ¯ ≤ Ct−1 u0 C 2m,γ (Ω) ¯ + gC t−1,γ (Ω) ¯ , hence, ˜ g C t,γ (Ω) t−1+2m,γ u0 C t−1+2m,γ + gC t,γ (Ω) ¯ ≤ δAC t,γ (Ω)←C ¯ ¯ u0 C 2m,γ (Ω) and, by (2.192) for A, ≤ Ct−1 ¯ + gC t,γ (Ω) ¯ u0 C t+2m,γ (Ω) ¯ ≤ Ct u0 C 2m,γ (Ω) ¯ + gC t,γ (Ω) ¯ , hence, the claim. The ﬁnal remark about the solutions u0 of (2.192) is obvious.

Special quasilinear operators of order 2 in H¨ older spaces The previous results were, modulo the diﬀerences in Theorems 2.52 and 2.53, valid for orders 2m and 2. The following theorem is restricted to quasilinear uniformly elliptic equations of order 2 in R2 , a special form of (2.169). So we assume n = 2 : G(u) :=

2 i,j=1

aij (x, u, ∇u)∂ i ∂ j u, (aij )2i,j=1 symmetric.

(2.193)

86

2. Analysis for linear and nonlinear elliptic problems

Similarly to (2.180), (2.181) we assume for the domain, the deﬁning coeﬃcients and the inhomogeneous boundary function, g, for 0 < γ < 1, let Ω in (2.5), ∂Ω ∈ C 2,γ be a single curve with

(2.194)

¯ × R3 ), positive curvature ∀ x ∈ ∂Ω, and in (2.193), let aij ∈ C γ (Ω ¯ ∀i, j = 1, 2, G be strongly (hence uniformly) elliptic, and g ∈ C 2,γ (Ω). Theorem 2.57. Existence and uniqueness of solutions for (2.193), Zeidler [676], pp. 246 ﬀ., Gilbarg and Trudinger [346], pp. 312: Let the nonlinear G in (2.193) satisfy (2.194). Then the boundary value problem G(u)(x) = 0 ∀ x ∈ Ω, u(x) = g(x) ∀ x ∈ ∂Ω

(2.195)

¯ with supx∈Ω |∇u(x)| ≤ K. has a solution u0 ∈ C 2,γ (Ω) ∩ C 0 (Ω) For regularity results we refer to the discussion at the end of this subsection and of 2.5.4 and 2.5.6. Many other related results are listed, e.g. in Zeidler [676], Gilbarg and Trudinger [346], and Runst and Sickel [561]; or the linearization techniques in Section 2.7 can be combined with the Fredholm alternative and the inherited regularity in Theorem 2.56 extending these results. Special semilinear operators in nonlinear spaces For this last type of results we return to semilinear equations. For aαβ ∈ W |β|,∞ (Ω) we assume the linear operator A in strong divergence form, cf. (2.134) (−1)|β| ∂ β (aαβ ∂ α u). (2.196) A = As : H 2m (Ω) → L2 (Ω), Au = |α|,|β|≤m

We want to solve the Dirichlet boundary value problem G(u)(x) := (Au − f (u))(x) = g(x) ∀ x ∈ Ω ∂ u(x) = 0 ∀|α| ≤ m − 1, ∀ x ∈ ∂Ω. α

(2.197) (2.198)

As usual we multiply (2.197) with v ∈ H0m (Ω), integrate over Ω and iteratively use partial integration, see (2.9). We ﬁnd the following generalized form, related to Au0 − f (u0 ) = g, u0 ∈ V = H0m (Ω), (As u − f (u) − g, v)L2 (Ω) = a(u, v) − a1 (u, v) − (g, v)∀u ∈ H 2m (Ω), v ∈ V aαβ ∂ α u∂ β vdx with aαβ ∈ L∞ (Ω), (2.199) a(u, v) = Au, vV ×V = a1 (u, v) =

Ω |α|,|β|≤m

f (u(x))v(x)dx, (g, v) = Ω

g(x)v(x)dx.

(2.200)

Ω

This shows that we certainly have to impose conditions upon f, g and choose appropriate u, v such that a1 (u, v), (g, v) are well deﬁned. So we assume for G and a(·, ·) in

2.5. Nonlinear elliptic equations

87

(2.197) and (2.199), a(·, ·) is V − coercive, aαβ ∈ L∞ (Ω) ∀ |α|, |β| ≤ m, Ω satisﬁes (2.5), f : R → R continuous with a ∈ R ﬁxed, (f (u) + a)u ≤ 0 ∀u ∈ R,

(2.201)

V=

(2.202)

H0m (Ω), D(G)

:= {u ∈ V : h := (f (u) + a)u ∈ L (Ω)} = V and 1

W = H0m (Ω) ∩ H k (Ω), k > n/2, k := max{k, m} and vW = vH k (Ω) . We still require (2.198), but instead of (2.197), we solve the generalized problem determine u0 ∈ D(G) s.t. a(u0 , v) − a1 (u0 , v) − (g, v) = 0 ∀v ∈ W.

(2.203)

The a(u0 , v), a1 (u0 , v), (g, v) are well deﬁned for u0 ∈ D(G), v ∈ W but the domain D(G) is not a linear space. Then Zeidler’s proof of his Proposition 27.21, [678], p. 605, for Au = −Δu can be easily generalized to our situation: Theorem 2.58. Existence and uniqueness for solutions of (2.197), (2.198): Under the conditions (2.201), (2.202), e.g. for f (u) = −eu , this boundary value problem has for all g ∈ L2 (Ω) a generalized solution in the sense of (2.203). Remark 2.59. The condition (f (u) + a)u ≤ 0 ∀u ∈ R in (2.201) is satisﬁed if f : R → R is continuous and monotonically decreasing, e.g. for f (u) = −eu , a = f (0). For Au = −Δu this is a standard example in (numerical) bifurcation, the so-called Bratu equation, cf. e.g. Allgower and Jepson [22] and Govaerts [357], p. 21 ﬀ.: −Δe + eu = 0 in Ω and u = 0 on ∂Ω.

(2.204)

We do not go into details, but only indicate some properties: This equation has mainly be discussed on the basis of very rough diﬀerence methods. So it is not quite clear whether the bifurcations scenarios in the many papers go back to spurious solutions, which do not correspond to solutions of the exact equation, or reﬂect the exact (2.204) scenarios. In fact Peitgen, Saupe and Schmitt [525] have shown that diﬀerence methods applied to (2.204) always posess spurious solutions, moving to ∞ with the step size h → 0. For n = 1, 2 one ﬁnds the situation indicated in Govaerts [357], p. 21 ﬀ., with a turning and a bifurcation point. For n = 3 the solutions oscillate around a speciﬁc value of λ inﬁnitely often with decreasing amplitudes. For n ≥ 4 there are no longer any solutions. For analytical results, cf. Joseph and Lundgren [428]. Here the situation is very similar to Theorem 2.56. There we studied A : ¯ → C t,γ (Ω); ¯ here we consider the weak form A : H m+s (Ω) ∩ H m (Ω) → C02m+t,γ (Ω) 0 −m+s (Ω). We nevertheless formulate, for the reader’s convenience, the updated result H once more. We omit the proof, since it is nearly identical in both cases, except that ¯ have to be replaced by H m+s (Ω), etc. the C02m+t,γ (Ω)

88

2. Analysis for linear and nonlinear elliptic problems

Theorem 2.60. Regular solutions for linear and semilinear equations: 1. Choose t ∈ N0 such that the conditions (2.149) and in Theorem 2.45 are satisﬁed. Then A : H m+s (Ω) ∩ H0m (Ω) → H −m+s (Ω) for 0 ≤ s ≤ t (or even for 0 ≤ s < t ∈ N) and we obtain for the solution u0 of Au0 = g that u0 H m+s (Ω) ≤ C u0 H m (Ω) + gH −m+s (Ω) . (2.205) Note that for boundedly invertible A the term u0 H m (Ω) is deleted. 2. Let δA be a linear or nonlinear diﬀerential operator of order ≤ 2m − 1 such that δA : H r+m (Ω) ∩ H0m (Ω) → H r+1−m (Ω) is bounded for all r = 0, 1, . . . , s − 1. This implies the same A + δA : H m+s (Ω) ∩ H0m (Ω) → H s−m (Ω) and (2.205) for the solution u0 of (A + δA)u0 = g. In particular, the condition for δA is satisﬁed, if the linear δA is induced by the lower order terms |α| + |β| < 2m of the bilinear form (2.199) and the aαβ are smooth enough, aαβ ∈ C max{0,s−2m+|β|} (Ω). 3. Under the conditions of Theorem 2.58 let the f in (2.201) and its derivatives satisfy f (r) : u ∈ H r+m (Ω) → f (r) (u(·)) ∈ L∞ (Ω), r = 0, 1, . . . , s − 1, ¯ e.g. these f (r) are bounded. Then the solutions u0 of (2.191) are u0 ∈ C02m+t,γ (Ω). The norms of u0 can be estimated as in (2.205). 2.5.4

Quasilinear elliptic equations of order 2

Throughout this subsection γ always denotes a real 0 < γ < 1. Furthermore, we will use W m,q (Ω) instead of W m,p (Ω), since we need the p in the triple x, z, p below. For quasilinear equations of second order, two competing approaches have been worked out in the monographs consulted here, essentially Ladyˇzenskaja and Uralceva [464], Gilbarg and Trudinger [346], Zeidler [678], Taylor [620], Showalter [589], and Chen and Wu [170]. We present in this subsection results based upon the ellipticity of these equations in the sense of Deﬁnition 2.51. Then maximum principles, ﬁxed point theorems, H¨ older estimates for the gradient, and boundary gradient estimates are combined yielding global, interior and boundary estimates and corresponding existence results. These comparison results yield fewer uniqueness results than the second approach in Subsection 2.5.6. This uses Nemyckii and monotone operators. It replaces the ellipticity of the equations, strongly related to the coercivity in the sense of Deﬁnition 2.65, by a generalized coercivity in the sense of (2.286) and monotonicity in (2.289). Relation (2.289) implies the standard ellipticity condition only for q = 2, hence in H m (Ω), cf. Proposition 2.121. The theory is in many respects independent of the order 2 or 2m. So we will formulate it in Subsection 2.5.6 below, only for the order 2m. It allows us existence results for a large class of operators and uniqueness under natural additional conditions. This diﬀerence inﬂuences our numerical methods. The approach via ellipticity allows the proofs for stability and convergence via linearization. For the monotonic situation this remains possible for all cases as well, but modiﬁcations for stabiliy have to be imposed, see Corollary 2.44 and Section 2.7 below.

2.5. Nonlinear elliptic equations

89

We repeat for order 2 the diﬀerent quasilinear equations, cf. (2.170), (2.169), Gu =

n

aij (x, u, ∇u)∂ i ∂ j u + a00 (x, u, ∇u) = 0, u ∈ C 2,γ (Ω), general

(2.206)

(−1)i>0 ∂ i (Ai (x, u, ∇u)) = 0 on Ω, u ∈ W 2,q (Ω), divergent form,

(2.207)

i,j=1

Gu =

n i=0

Gu, vV ×V

=

n

Ai (x, u, ∇u)∂ i vdx = 0 ∀ v ∈ V = W01,q (Ω), weak

(2.208)

Ω i=0

form, with u ∈ V = W01,q (Ω), sometimes u ∈ W 1,q (Ω), u|∂Ω = ϕ|∂Ω . The following results are examples of existence and uniqueness results, serving as an illustration of the scope of the theory in Gilbarg and Trudinger [346]. They are obtained by appropriate combinations of tools formulated in the introductory remarks of this subsection. For special cases, [346], Theorems 10.1 and 10.7 immediately imply uniqueness. This is combined below with uniqueness results from Taylor [620]. Simplifying the notation we use, in this subsection, the standard (z, p) ∈ Rn+1 instead of (Θ0 , Θ) ∈ Rn+1 with N1 = n + 1, cf. (2.73). Denote the minimal and maximal real n eigenvalues of {aij (x, z, p)}i,j=1 , cf. (2.206), an assumed positive matrix, as λ(x, z, p) and Λ(x, z, p) and ϑ ∈ Rn . We called (2.206) elliptic, etc. in (x, z, p), iﬀ 0 < λ(x, z, p)|ϑ|2 <

n

aij (x, z, p)ϑi ϑj < Λ(x, z, p)|ϑ|2 ∀0 = ϑ ∈ Rn ,

(2.209)

i,j=1

with aij (x, z, p), Ai (x, z, p) ∈ R, (x, z, p) ∈ Uo ⊂ Ω × R × Rn , p = (p1 , · · · , pn ). Obviously (2.207) implies (2.206) for Ai (·, ·, ·) ∈ C 1 with aij (x, z, p) = −∂(Ai (x, z, p))/(∂pj ). Conversely, it is usually not possible to transform (2.206) into (2.207). If in (2.209) the (x, z, p) is replaced by (x, u(x), ∇u(x)) then this (uniform) ellipticity holds (locally) with respect to u. For the following growth conditions, we introduce the operators: For u ∈ C (Ω × R × R ) let δu := 1

n

−2

∂z + |p|

n i=1

pi ∂xi

¯ := u, δu

n

pi ∂pi u.

i=1

(2.210) Uniformly elliptic equations We distinguish uniformly and nonuniformly elliptic equations in general and in divergence form. For the diﬀerent forms we require growth and regularity conditions, assuming for (2.211)–(2.219), |p| → ∞, uniformly for x ∈ Ω, and bounded z, with

90

2. Analysis for linear and nonlinear elliptic problems

(x, z, p) ∈ Uo , cf. (2.209), (2.210). ¯ a00 , aij ∈ C 1 (Ω ¯ × R × Rn ), ∂Ω ∈ C 2,γ , For (2.206): G unif. ell. on Ω,

(2.211)

ϕ ∈ C 2,γ (∂Ω) and assume the estimates: ¯ ij = O(λ), δaij = o(λ), a00 , δa ¯ 00 = O(λ|p|2 ), δa00 = o(λ|p|2 ). aij , δa ¯ Ai ∈ C 1,γ (Ω ¯ × R × Rn ), A0 ∈ C γ (Ω ¯ . . .), For (2.207): G unif. ell. on Ω, ∂Ω ∈ C

2,γ

,ϕ ∈ C

|p|τ = O(λ),

n

2,γ

(2.212)

(∂Ω), and assume the estimates: ∃τ > −1 :

∂pi Ai = O(|p|τ ), |p|∂z Ai ,

i=1

n

∂xi Ai , A0 = O(|p|τ +2 )

i=1

Speciﬁc results are possible for the following special case of the general form (2.206), noting that i, j ≥ 1: Gu = Gu =

n

aij (x, u)∂ i ∂ j u + a00 (x, u, ∇u) = 0 on Ω, aij ∈ C 1 (Ω, R).

(2.213)

i,j=1

Then a transformation into divergence form (2.207) is possible by choosing Ai (x, z, p) := −

n

aij (x, z)pj , for i = 1, . . . , n,

j=1

A0 (x, z, p) = −a00 (x, z, p) +

n i,j=1

∂z aij (x, z)pi pj +

n

∂xi aij (x, z)pj .

i,j=1

¯ aij ∈ C 1 (Ω ¯ × R), a00 ∈ C γ (Ω ¯ × R × Rn ) For (2.213): G unif. ell. on Ω,

(2.214)

∂Ω ∈ C 2,γ , ϕ ∈ C 2,γ (∂Ω), and the above p, x, z, assume a00 = O(|p|2 ), ∃μ1 , μ2 > 0 :

n a00 (x, z, p)sign z 2 |p| ≤ E(x, z, p) := aij (x, z, p)pi pj . μ1 |p| + μ2 i,j=1

(2.215)

Note that for (2.213) these aij (x, z, p), i, j = 1, . . . , n, are independent of p. Nonuniformly elliptic equations ¯ and Λ/λ is no longer bounded. We no longer require G to be uniformly elliptic on Ω, This situation is important, e.g. for minimal surface operators. We assume the original aij (x, z, p) can be split, according to aij (x, z, p) = a∗ij (x, z, p) + (pi cj (x, z, p) + pj ci (x, z, p))/2,

i, j = 1, . . . , n, (2.216) ¯ × R × (Rn \ {0}), and a positive symmetric matrix a∗ n . with aij , a∗ij , ci ∈ C 1 (Ω ij ∗ i,j=1 n ∗ Furthermore, let λ indicate the lower bound in (2.209) deﬁned by the matrix aij i,j=1

2.5. Nonlinear elliptic equations

91

instead of the original (aij )ni,j=1 and use E in (2.215). We impose the structure and regularity conditions, modifying the above (2.211), (2.212) For (2.206): The ﬁrst two lines in (2.211) remain, except delete “unif.ell.on”, √ ¯ 00 = O(E), δa ¯ ∗ = O( λ∗ E/|p|), change the last line into |p|aij , a00 , δa ij √ δa∗ij = o( λ∗ E/|p|), δa00 = o(E), μ1 , μ2 > 0 exist, s.t. (2.217) n a00 (x, z, p)sign z 2 |p| ≤ E(x, z, p) = aij (x, z, p)pi pj ∀(x, z, p = 0) ∈ Uo . μ1 |p| + μ2 i,j=1

If we replace uniformly elliptic by elliptic, we impose ¯ For (2.206): The ﬁrst two lines in (2.211) remain, but require G elliptic on Ω, change the last line into Ω is uniformly convex (2.218) ∗ 2 % ∗& 2 ∗ a00 = o(|p|Λ), δa00 ≤ |O a00 /|p| T , T := trace aij . Finally we consider the divergence form (2.207). In the last two lines of (2.219) we use the general form (2.206) and introduce a family of equations, depending upon σ with 0 ≤ σ ≤ 1. We assume: ¯ For (2.207): The ﬁrst two lines in (2.212) remain, but require G elliptic on Ω, change the last line into Ω is uniformly convex ∃τ ∈ R s.t. |p|τ = O(λ), |p|∂z Ai ,

n

(2.219)

∂xi Ai , A0 = o(|p|τ +1 ), for

i=1

Gσ u :=

n

aij (x, u, ∇u)∂ i ∂ j u + σa00 (x, u, ∇u) = 0 in Ω, u = σϕ on ∂Ω,

i,j=1

let the set of solutions u0 = u0,σ ∈ C 2,γ (∂Ω) be uniformly bounded in Ω.

(2.220)

Existence for these quasilinear elliptic equations We summarize the existence results of the main cases in Gilbarg and Trudinger’s [346] Theorems 15.10–15.15. We omit his results for problems with boundary curvature conditions. Theorem 2.61. Existence for quasilinear operators: For (2.206), (2.207) or (2.213) on a bounded Ω in (2.5), let one of the corresponding conditions be satisﬁed: For uniformly elliptic equations impose one of (2.211), (2.212) or (2.214), for nonuniformly elliptic equations impose one of (2.217), (2.218), (2.219). Then there exists a solution ¯ of the Dirichlet problem Gu0 = 0 in Ω, and u0 |∂Ω = ϕ. u0 ∈ C 2,γ (Ω) For regularity results we refer to the end of this section and to subsection 2.5.6.

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2. Analysis for linear and nonlinear elliptic problems

Uniqueness results For the following results, we have to impose additional conditions: ¯ ∩ C 2 (Ω), For (2.206): G unif. ell. with respect to u, consider solutionsu ∈ V = C(Ω) aij independent of z, a00 nonincreasing in z for each (x, p) ∈ Ω × Rn , and a00 , aij ∈ C 1 with respect to the p variable in Uo ⊂ Ω × R × Rn .

(2.221)

o(λ|p|2 ) in A combination of (2.211) with n δaij = o(λ), δa00 = n (2.221) shows that 2 (2.211) would be implied by k=1 ∂xk aij = O(λ), i=1 ∂xi a00 = O(λ|p| ). ¯ For (2.207): G elliptic in Ω, consider solutions u ∈ V = C 1 (Ω),

(2.222)

¯ × R × Rn , A0 , Ai ∈ C 1 with respect to the z, p variables in Ω A0 is nonincreasing in z for each ﬁxed (x, p) ∈ Ω × Rn ﬁnally, one of the following conditions (1), (2), (3) is satisﬁed: (1) Ai , i = 1, . . . , n, is independent of z; or (2) A0 is independent of p; or (3) the following (n + 1) × (n + 1) matrix satisﬁes componentwise

[∂pj Ai (x, z, p)

−∂pj A0 (x, z, p)]ni,j=1

∂z Ai (x, z, p)

−∂z A0 (x, z, p)

≥ 0 in Ω × R × Rn .

By considering a special case of (2.206), Taylor [620] obtains uniqueness under, in some sense, less restrictive conditions, see (2.223)–(2.5.4). We deﬁne the Taylor case : Gu =

n

¯ aij (∇u)∂ i ∂ j u = 0 on Ω, u ∈ V = C ∞ (Ω),

(2.223)

i,j=1

cf. [620], Chapter 14, (10.1). It is combined with the following diﬀerent conditions (2.224), (2.225), (2.226) for (2.223), or a generalization with (2.5.4). So we assume For (2.223): ellipticity 0 < λ(p)|ϑ|2 <

n

aij (p)ϑi ϑj < Λ(p)|ϑ|2 ∀ϑ = 0,

i,j=1 ∞

aij ∈ C (Rn ), ∂Ω ∈ C ∞ , ϕ ∈ C ∞ (∂Ω), and let (2.224) the solutions uσ for

n

aij (∇u)∂ i ∂ j u = 0 on Ω, u|∂Ω = σϕ

i,j=1

satisfy |∂xi uσ (x)| ≤ K ∀1 ≤ i ≤ n, x ∈ ∂Ω, ∀0 ≤ σ ≤ 1.

2.5. Nonlinear elliptic equations

For (2.223): For strictly convex Ω, retain lines 1,2 in (2.224)

93

(2.225)

and cancel lines 3,4. For (2.223): For uniformly elliptic G, hence 0 < Λ(p)/λ(p) ≤ K ∀p

(2.226)

retain lines 1,2 in (2.224) and cancel lines 3,4. Allowing an additional zero order term a00 (u), cf. [620], Chapter 14, (10.49), we consider the n ¯ aij (∇u)∂ i ∂ j u + a00 (u) = 0 on Ω, u ∈ V = C ∞ (Ω), General Taylor case: Gu = i,j=1 2

unif. ellipt.: 0 < Λ(p)/λ(p) ≤ K ∀p ∈ Rn , let |a00 (z)| ≤ A1 + A2 |z|, A1 , A2 > 0, a00 (z) ≥ 0∀z ∈ R, aij ∈ C ∞ (Rn ), a00 ∈ C ∞ (R), ∂Ω ∈ C ∞ , ϕ ∈ C ∞ (∂Ω).

(2.227)

Here we summarize Gilbarg and Trudinger’s [346] uniqueness results from Theorem 10.7 and the remarks following Theorem 15.10, see (2.221), (2.222), and Taylor [620], cf. Chapter 14, Theorems 10.2, 10.6, 10.7, Proposition 10.9, see (2.224)–(2.5.4). Theorem 2.62. Uniqueness: For the quasilinear operators (2.206) or (2.207) or (2.208) or (2.223) on a bounded domain Ω ⊂ Rn , let one of the corresponding conditions (2.221), (2.222) or (2.224)–(2.5.4) be satisﬁed. Then the solution u0 ∈ V of the Dirichlet problem Gu0 = 0 in Ω, and u0 |∂Ω = ϕ exists, possibly except for (2.222), and is unique. Regularity results under Ladyˇ zenskaja–Uralceva conditions Similarly to Subsection 2.6.2 below, see Theorems 2.106, 2.107, regularity results distinguish estimates for the interior or for the entire Ω. To give a ﬂavor for the type of available results, for second order equations, we summarize Theorems 1.1, 2.1, 3.1., 5.1., 6.4., 6.5 in Chapter 4, from the great old ladies Ladyˇzenskaja and Uralceva [464]. We formulate the necessary modiﬁcations of the later growth and coerciveness conditions (2.284), (2.285), (2.286) in Subsection 2.5.6, here for the special case m = 1 instead of a general m. For avoiding misunderstandings with p ∈ Rn we again formulate the results for W 1,q (Ω) with 1 < q < ∞, cf. the beginning of this subsection. We consider the strong divergence form Gu0 =

n

(−1)i>0 ∂ i (Ai (x, u0 , ∇u0 )) = 0 on Ω, and its weak form

(2.228)

i=0

Gu0 , vV ×V =

n

Ai (x, u0 , ∇u0 )∂ i vdx = 0 ∀ v ∈ V = W01,q (Ω), (2.229)

Ω i=0

for u0 ∈ V = W 2,q (Ω) and u0 ∈ V = W 1,q (Ω), respectively The Ai : (x, z, p) → R are ¯ × R1+n . Boundary conditions u|∂Ω = ϕ|∂Ω are suﬃciently diﬀerentiable in x, z, p ∈ Ω imposed below for some of the results. We impose the following conditions for ¯ z ∈ R, and p = (p1 , . . . , pn ), p = (p , . . . , pn ) ∈ Rn : Let μ and ν be positive x ∈ Ω, 1

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2. Analysis for linear and nonlinear elliptic problems

nondecreasing and nonincreasing functions. 1. Growth cond.

n

|Ai (x, z, p)|(1 + |p|) + |A0 (x, z, p)| ≤ μ(|z|)(1 + |p|)q , (2.230)

i=1

Coerciveness:

n

Ai (x, z, p)pi ≥ ν(|z|)|p|q − μ(|z|), see (2.269),

(2.231)

i=1

Ellipticity:

ν(|z|) ≤ (1 + |p |)2−q |p|−2 '

n ∂Ai (x, z, p ) pi pj ∂pj i,j=1

≤ μ(|z|), (2.232)

n ∂Ai ∂A0 2. Growth cond . |Ai | + + ∂p (1 + |p|), ∂z i i=1 ( ∂A0 (x, z, p) ≤ μ(|z|)(1 + |p|)q , +|A0 | + ∂z ' n ∂Ai (1 + |p|) 3. Growth cond . |Ai | + ∂z i=1 ⎤ n ∂Ai + + |A0 |⎦ (x, z, p) ≤ μ(|z|)(1 + |p|)q , ∂pj

(2.233)

(2.234)

i,j=1

⎡⎛

⎞ n n ∂Ai ∂A0 ⎠ 4. Growth cond . ⎣⎝ ∂xj + ∂p (1 + |p|), i i,j=1 i=1 n ∂A0 ∂A0 q (2.235) ∂z + ∂xi (x, z, p) ≤ μ(|z|)(1 + |p|) , i=1 . / ∂A ∂A ∂A n i i i , Boundedness: max , |A0 | (x, z, p) ≤ μ1 < ∞, (2.236) , x∈Ω,i,j=1 ∂pj ∂z ∂xj Condition A for

∂Ω : ∃ 0 < a1 , a2 , a2 < 1 : ∀Bρ (x), x ∈ ∂Ω, ρ ≤ a1 ,

ˆ ⊂ Bρ (x) ∩ Ω : measure Ω ˆ ≤ (1 − a2 ) measure Bρ (x), (2.237) ∀ components Ω ⎛ ⎞ n ∂ 2 u 2 q−2 ⎝ q+2 ⎠ Integr. cond. (1 + |∇u|) ∂xi ∂xj dx, (1 + |∇u|) dx < ∞. Ω

i,j=1

Ω

(2.238) These (2.231)–(2.235) are not independent. For example, (2.232), (2.233) imply parts of (2.231), in particular, the standard coercivity, see Ladyˇzenskaja and Uralceva [464], p. 255. This fact is generalized in Lemma 2.77 below. The above (2.232), and (2.289) below, are in some sense competing. Obviously, the above (2.232) indicates the

2.5. Nonlinear elliptic equations

95

ellipticity for the linearized diﬀerential equation. By diﬀerent combinations of these conditions diﬀerent results are achieved, see Theorems 1.1, 2.1, 5.2, 6.4, 6.5 in Chapter 4 [464]. The existence of solutions u0 ∈ W 1,q (Ω) is discussed in Chapter 4, Section 8 [464], under highly technical additional conditions. There are some interesting diﬀerences between the conditions (2.230)–(2.232), imposed for m = 1, and for m ≥ 1, see (2.284)–(2.289) below. Theorem 2.63. Regular, bounded solutions and uniqueness on small balls: We assume a generalized bounded solution u0 ∈ W 1,q (Ω) in the sense of (2.229) with M0 := u0 L∞ (Ω) < ∞, 0 ≤ α ≤ 1. 1. Bounded generalized solutions: Under the conditions (2.230) and (2.231), we get u0 ∈ C 0,α (Ω) with α depending upon M0 , q, μ(M0 ), ν(M0 ). For an arbitrary Ω1 ⊂⊂ Ω the u0 C 0,α (Ω1 ) can be estimated from above by a constant depending upon the previous constants and dist(Ω1 , ∂Ω). If additionally ∂Ω satisﬁes (2.237) and if u|∂Ω = ϕ|∂Ω ∈ C 0,β (∂Ω) for α ≤ β, then u0 C 0,α (Ω) over the entire Ω can be estimated from above by a constant depending upon the previous constants and the a1 , a2 , in (2.237), β and ϕC 0,β (∂Ω) (Theorem 1.1). 2. Uniqueness in the small: Assume (2.232), (2.233) and let u1 , u2 ∈ W 1,q (Ω) be two bounded solutions of (2.229), coinciding on the surface of a ball Bρ ⊂ Ω1 ⊂⊂ Ω with radius ρ < ρ1 , and max{u1 L∞ (∂Ω) , u2 L∞ (∂Ω) } =: M1 < ∞. Then they coincide in Bρ as well. This ρ1 is determined by the M1 , q, μ(M1 ), ν(M1 ) and dist(Ω1 , ∂Ω) (Theorem 2.1). 3. A bound for ∇u0 L∞ (Ω1 ) over Ω1 ⊂⊂ Ω and u0 ∈ H 2 (Ω1 ): Assume differentiable Ai (x, z, p), i = 0, . . . , n, and (2.232), (2.234), (2.235), and for x ∈ ¯ |u| ≤ M , and arbitrary q the (2.238). Then for an arbitrary Ω1 ⊂⊂ Ω we get Ω, u0 ∈ H 2 (Ω1 ). It satisﬁes (2.228) almost everywhere in Ω and ∇u0 L∞ (Ω1 ) can be estimated by constants depending only upon M0 , q, μ(M0 ), ν(M0 ), ∂Ω, ϕC 2,0 (Ω) , and dist(Ω1 , ∂Ω) (Theorem 5.2). 4. Bounds for u0 C l,α (Ω) , l ≥ 2. Let M2 ≥ ∇u0 L∞ (Ω) and M := {(x, z, p) ∈ ¯ × R × Rn : |z| ≤ M0 , |p| ≤ M2 }. If, in addition, Ai (x, z, p) ∈ C l−1,α (M), i = Ω ¯ then u0 ∈ 1, . . . , n, A0 (x, z, p) ∈ C l−2,α (M), and ∂Ω ∈ C l,α , ϕ ∈ C l,α (Ω), ¯ and the u0 C l,α (Ω) are bounded by constants depending upon C l,α (Ω) ¯ M0 , M2 , q, μ(M0 ), ν(M0 ), ϕC l,α (Ω) ¯ , ∂Ω and Ai C l−1,α (M) , A0 C l−2,α (M) (Theorem 6.5). Finally, we summarize regularity results from Chen and Wu [170] for quasilinear equations (2.206). Essentially based upon the Leray–Schauder theorem, Chen and ¯ under modiﬁed conditions. We present their Theorem Wu get solutions u0 ∈ C 2,γ (Ω) 5.1 in Chapter 5, for divergent quasilinear second order equations, but only refer to Theorem 4.1 in Chapter 7 for nondivergent quasilinear second order equations. We use the remark on p. 117 in Chapter 7 for fully nonlinear second order equations ¯ k > 0, in the last part of formulating the higher regularity results u0 ∈ C 2+k,γ (Ω), the following Theorem 2.64, compare [170], p. 78. Their Theorem 7.4. in Chapter 7 for fully nonlinear second order equations will be discussed as Theorem 2.82 below.

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2. Analysis for linear and nonlinear elliptic problems

Additional results are available as special cases of Theorem 2.91 below. Chen and Wu study divergent equations of the form (2.207) and require ellipticity and other conditions in the form Ai (x, z, p) ∈ R, ∀ (x, z, p) ∈ Uo ⊂ Ω × R × Rn , 0 = p = (p1 , . . . , pn ) ∈ Rn , ∃Λ > λ > 0 : 0 < λ|p |2 <

n ∂Ai (x, z, p)pi pj < Λ|p |2 , ∂p j i,j=1

(2.239)

|Ai (x, z, 0)| ≤ g(x) ∀i = 1, · · · , n, g ∈ Lq (Ω), q > n, and ﬁnally 0 ∂Ai ∂Ai ∂Ai (·) + |Ai (·)| + (1 + |p|2 ) (·) + (1 + |p|)| (·) (2.240) ∂pj ∂z ∂xj ( + |A0 (·)| (x, z, p) ≤ μ(|z|)(1 + |p|2 ) ∀i, j = 1, · · · , n, ∀(x, z, p) ∈ Uo , ∗

−A0 (x, z, p) sign z ≤ Λ[|p| + f (x)] for an f ∈ Lq (Ω), q ∗ := nq/(n + q). The relation for the ellipticity conditions in (2.232) and (2.239) is discussed in Lemma 2.77 below for order 2m. Theorem 2.64. Existence and regularity of solutions: 1. Let the conditions (2.239)–(2.240) be satisﬁed and assume ∂Ω ∈ C 2,γ , Ai ∈ ¯ × R × Rn ), A0 ∈ C 0,γ (Ω ¯ × R × Rn ), ϕ ∈ C 2,γ (Ω) ¯ with 0 < γ < 1. Then C 1,γ (Ω 2,γ ¯ there exists a solution u0 ∈ C (Ω) for the Dirichlet problem (2.206). 2. If additionally Ai ∈ C k+1,α (Uo ), A0 ∈ C k,α (Uo ), k ≥ 1, and ∂Ω ∈ C 2+k,γ , ¯ ¯ ∩ C k+2,α (Ω), then the solution u0 ∈ C 2,γ (Ω) for ϕ ∈ C 2+k,γ (Ω), k+1,α ¯ k,α ¯ k+2,γ ¯ (Uo ), A0 ∈ C (Uo ), even u0 ∈ C (Ω). Ai ∈ C Further regularity results are formulated in Subsection 2.5.7 and are studied by Ladyˇzenskaja and Uralceva [463, 464, 466], Koshelev and Chelkak [449–451], Morrey [500], Neˇcas [510], and Giaquinta and Hildebrandt [341, 342]. These quasilinear equations are elliptic equations in the sense of Deﬁnition 2.51, since the linearized equation satisﬁes one of the ellipticity conditions (2.209), (2.214), (2.217), (2.224), (2.232), (2.239). In Lemma 2.77 below, we will discuss the relation of growth and coercivity conditions versus ellipticity for quasilinear equations of order 2m. 2.5.5

General nonlinear and Nemyckii operators

In the last Subsections 2.5.3, 2.5.4 we studied nonlinear operators. We had to evaluate, e.g. f (u) or aij (x, u, ∇u) usually for given continuous functions. Therefore the evaluation was obvious. The situation changes if u ∈ W m,p (Ω), 1 ≤ p ≤ ∞. This is necessary for studying monotone operators. They are, in some sense, evaluations of quasilinear elliptic operators and the main tool for analytical results for their solutions. So we deﬁne diﬀerent monotone and Nemyckii operators and summarize their essential properties, see Skrypnik [590, 591], Zeidler [676–678], and Runst and Sickel [561].

2.5. Nonlinear elliptic equations

97

For the next deﬁnition we need a function, see Zeidler [678], pp. 500 ﬀ. a : R+ → R+ used in (2.244), (2.253), continuous and strictly increasing with a(0) = 0, lim a(t) = ∞, e.g. a(t) = c|t|p−1 , 0 < c, 1 < p. t→∞

(2.241)

Deﬁnition 2.65. Let X be a Banach space, A : X → X an operator. The following conditions are imposed ∀u, v ∈ X . Then we call A 16 monotone ⇔ Au − Av, u − v ≥ 0, strictly monotone ⇔ Au − Av, u − v > 0 for u = v, uniformly monotone ⇔ Au − Av, u − v

(2.242) (2.243) (2.244)

≥ a(u − vX )u − vX , e.g., ≥ cu − vpX , strongly monotone ⇔ Au − Av, u − v ≥ cu − v2X , c ∈ R+ , (nonlinear) coercive ⇔ weakly coercive ⇔

(2.245)

lim

Au, u = ∞, uX

(2.246)

lim

AuX = ∞ for Hilbert spaces,

(2.247)

uX →∞ uX →∞

where we identify X = X , Lipschitz-continuous ⇔ ∃L > 0 : Au − AvX ≤ Lu − vX ,

(2.248)

continuous ⇔ ∀ > 0∃δ > 0 : Au − AvX < ∀u − vX < δ, (2.249) hemicontinuous ⇔ t → A(u + tv), w is continuous on [0, 1], strongly continuous ⇔ un u ⇒ Aun → Au for n → ∞, bounded ⇔ A maps bounded sets into bounded sets, stable ⇔ Au − AvX ≥ a(u − vX ).

(2.250) (2.251) (2.252) (2.253)

In (nonlinear) coercivity we often omit the “nonlinear”. Remark 2.66. Obviously strong monotonicity implies uniform implies strict and implies monotonicity, so (2.245) ⇒ (2.244) ⇒ (2.243) ⇒ (2.242). Lipschitz-continuity implies continuity and hemicontinuity, so (2.248) ⇒ (2.249) ⇒ (2.250). Obviously, a uniformly monotone A is coercive and stable. The preceding properties of A allow important consequences, see Zeidler [677], pp. 455 ﬀ: Let X be a reﬂexive Banach space. We replace the bilinear forms a(·, ·) and the induced linear diﬀerential operators A : X → X by nonlinear operators A : X → X , and the corresponding nonlinear forms. This is motivated by the results in 16

Note that for (2.251), un u ∈ X is deﬁned by un − u, v → 0∀v ∈ X .

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2. Analysis for linear and nonlinear elliptic problems

Subsection 2.5.3. For generalizing Proposition 2.10 we start, cf. [677], pp. 314 ﬀ., with a nonlinear form a(·, ·) : X × X → R, s.t. |a(u, v)| ≤ C(u)vX ∀u, v ∈ X .

(2.254)

Proposition 2.67. Under the condition (2.254) there exists a usually nonlinear but unique operator, such that, for all u ∈ X , Au, vX ×X := a(u, v) ∀ v ∈ X , denoted as A : X → X , Au ∈ X .

(2.255)

Then for any f ∈ X the following three problems are equivalent: Determine u0 ∈ X s.t. Au0 = f ∈ X ⇔ a(u0 , v) = f, vX ×X ∀v ∈ X

(2.256)

⇔ Au0 , vX ×X = f, vX ×X ∀v ∈ X . Obviously (2.254) and (2.255) and the following considerations can be generalized to a(·, ·) : D(a) ⊂ X × X → R, and A : D(A) ⊂ X → R(A) ⊂ X , respectively. Theorem 2.68. Properties of nonlinear operators, Zeidler [678], based upon Zarantonello, Browder, and Minty [150, 491, 674]: Let X be a real Banach space and A : X → X a (nonlinear) operator. Then 1. If A is linear and monotone, then A is continuous, see [678], Proposition 26.4. 2. If A is monotone, coercive, and hemicontinuous on a reﬂexive Banach space, then A is surjective and, for each f ∈ X , the set of solutions of Au = f is bounded, convex and closed, cf. [678], Theorem 26A(a) [150, 491]. 3. If A is additionally strictly monotone, the above solution is unique. Then A−1 : X → X exists and, cf. [678], Theorem 26A(c), (d) [150, 491], a uniformly monotone A implies a continuous A−1 ; a strongly monotone A implies a Lipschitz-continuous A−1 . 4. Let X be a real Hilbert space and A be strongly monotone and Lipschitzcontinuous with constant L. Then for each f the Au = f ∈ X has a unique solution, depending continuously upon f , more precisely, see [674], [678], Theorem 25B, u1 − u2 X ≤ L−1 f1 − f2 X for Au1 = f1 , Au2 = f2 , f1 ∈ X . Deﬁnition 2.69. For given functions f and appropriate u ∈ D(F ), deﬁne F (u) as f : D(f ) ⊂ Ω × Rk → R, u : Ω → Rk : F (u)(x) := f (x, u1 (x), · · · , uk (x)), (2.257) with (x, u1 (x), · · · , uk (x)) ∈ D(f ). This F is called a Nemyckii operator. Now we formulate two conditions guaranteeing that F (u)(x) makes sense. The Carath´eodory and the growth condition enforce measurability and appropriate growth

2.5. Nonlinear elliptic equations

99

of the composite function F ; more precisely Carath´eodory condition: For f in (2.257) let

(2.258)

Ω ⊂ R be nonempty, measurable and x → f (x, u) measurable on Ω ∀u ∈ Rk n

u → f (x, u) is continuous on Rk almost everywhere (a.e.) for x ∈ Ω. Growth condition: ∀(x, u) ∈ Ω × Rk let |f (x, u)| ≤ a(x) + b

k

|ui |pi /q ,

(2.259)

i=1

with ﬁxed reals 0 < b, 1 ≤ pi , q < ∞ and ﬁxed a ∈ Lq (Ω), 0 ≤ a(·)a.e. Theorem 2.70. Estimates for nonlinear operators, Zeidler [678], Proposition 26.6: If (2.258), (2.259) are satisﬁed, then F : Πki=1 Lpi (Ω) → Lq (Ω) is continuous and bounded (this is not equivalent for nonlinear operators!) such that F (u)Lq (Ω) ≤ C

aLq (Ω) +

k

p /q ui Lipi (Ω)

∀u ∈ Πki=1 Lpi (Ω).

(2.260)

i=1

Particularly important is the case k = 1. Then the Nemyckii operator has the form F (u)(x) := f (x, u(x)) and f : D(f ) ⊂ Ω × R → R, mainly F : X = Lp (Ω) → X = Lq (Ω), 1/p + 1/q = 1.

(2.261)

We summarize appropriate conditions for f . The Carath´eodory condition is only modiﬁed by replacing Rk by R. We specialize the growth condition as Growth condition: For ﬁxed reals 0 < b, 1 < p, q < ∞, 1/p + 1/q = 1 ⇔ p − 1 = p/q, a ∈ Lq (Ω), 0 ≤ a(·) a.e., let |f (x, u)| ≤ a(x) + b|u|

p−1

(2.262)

∀ (x, u) ∈ Ω × R.

We assume u, v ∈ R, and ∀x ∈ Ω, and denote the function f in (2.261) to be monotone ⇔ f (x, u) ≤ f (x, v) for u ≤ v,

(2.263)

strictly monot. ⇔ f (x, u) < f (x, v) for u < v,

(2.264)

coercive ⇔ ∃d > 0, g ∈ L1 (Ω) : f (x, u)u ≥ d|u|p + g(x),

(2.265)

positive ⇔ f (x, u)u ≥ 0,

(2.266)

asympt. positive ⇔ ∃R > 0 : f (x, u)u ≥ 0 for |u| ≥ R, meas Ω < ∞.

(2.267)

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2. Analysis for linear and nonlinear elliptic problems

Theorem 2.71. Properties of Nemyckii operators, Zeidler [678], Proposition 26.7: For the function f in (2.261) we assume (2.258) with Rk replaced by R, (2.262) and let F : X = Lp (Ω) → X be the Nemyckii operator deﬁned by f, cf. (2.261). 1. Then F is continuous and bounded with

and F (u)Lq (Ω) ≤ C aLq (Ω) + up−1 Lp (Ω) F (u), uX ×X = f (x, u(x))u(x)dx∀ u ∈ X .

(2.268)

Ω

2. A (strictly) monotone f , cf. (2.263), (2.264), implies a (strictly) monotone F. 3. A coercive f , see (2.265), implies a coercive F and F (u), uX ×X ≥ dupLp (Ω) + g(x)dx∀ u ∈ X . Ω

4. A positive f , see (2.266), implies a positive F, hence F (u), uX ×X ≥ 0 ∀ u ∈ X . 5. An asymptotic positive f , see (2.267), implies an estimate for F in the form F (u), uX ×X ≥ − c ∀ u ∈ X for a constant c > 0. These properties of general nonlinear and Nemyckii operators are essential tools in the study of divergent quasilinear elliptic equations. 2.5.6

Divergent quasilinear elliptic equations of order 2m

We start with the strong form of a quasilinear elliptic operator, G. The presentation is motivated by the properties of Nemyckii operators in the last Subsection 2.5.5. At the end of this subsection, we will show that, for the most important cases, under slightly stronger conditions, these quasilinear equations are elliptic equations in the sense of Deﬁnition 2.51. This allows us to prove convergence of our discrete solutions to the exact solutions by our general techniques with the correct rate of convergence for all the classes of space discretization methods, considered here, and the problems in W m,p (Ω), with 2 ≤ p ≤ ∞. A general convergence theory for monotone and quasilinear operators with its “pros and cons” is elaboreted in Section 4.5 for 1 < p < ∞. We only consider Dirichlet boundary conditions on bounded domains Ω, compare Theorem 2.50, and recall the standard notation in (2.73), (2.165), (2.166): ϑ ∈ Rn , Θ0 , ϑi , ϑα ∈ R, Θk = (tα ∈ R)|α|=k ∈ Rnk , often = (ϑα )|α|=k , Aα deﬁned as Aα : Ω × R

Nm

≤m

→ R, w = (x, Θ , . . . , Θ ) =: (x, Θ 0

m

) ∈ Ω × RNm ,

with RNm = R × Rn × · · · × Rnm , norms, e.g. |ϑ|, |Θm |, and Nm := (nm+1 − 1)/(n − 1) for n > 1, Nm := m + 1 for n = 1, partials ∂ α u :=

1 ∂xα 1

(2.269)

∂ |α| u, ∇0 u := ∂ α u := u for |α| = 0, ∇k u := (∂ α u)|α|=k . . . . ∂xn αn

2.5. Nonlinear elliptic equations

101

We discuss the general form of quasilinear elliptic equations with Dirichlet boundary conditions, see Skrypnik [590, 591] and Zeidler [678]: (−1)|α| ∂ α (Aα (x, u, · · · , ∇m u) − fα ) = 0 on Ω, (2.270) Gu − f = Gs u − f = |α|≤m

=

Aα,β (x, u, · · · , ∇m u)∂ α ∂ β u − (−1)|α| ∂ α fα = 0 on Ω,

with, e.g.

|α|,|β|≤m

∂ m Aα (x, u, · · · , ∇m u) ∀|α| = |β| = m ∂ϑβ (2.271) Ω a bounded nonempty measurable domain in Rn and Aα,β (x, u, · · · , ∇m u) = (−1)|α|

Dirichlet boundary conditions ∂ i u(x)/∂ν i = 0 ∀ 0 ≤ i ≤ m − 1, on ∂Ω ⇔ ∂ α u|∂Ω = 0 ∀|α| ≤ m − 1 in the trace sense for smooth enough ∂Ω. (2.272) A possible condition u ∈ W 2m,p (Ω) is still too restrictive. Below, we will list conditions allowing us to multiply (2.270) with v ∈ V, and to integrate over Ω, where V := W0m,p (Ω) := {v ∈ W m,p (Ω) : ∂ α v = 0 ∀|α| ≤ m − 1 on ∂Ω}, 1 < p < ∞. We iteratively use the fundamental Green’s formula (2.9). For smooth enough w1 , w2 and |α| ≤ m − 1 for w1 we get ∂(w1 w2 )/∂xi dx = (w1 w2 )νi ds. w1 = ∂ α v, w2 = ∂ α (Aα (x, u, · · · , ∇m u)) : Ω

∂Ω

For v, w ∈ H (Ω) we obtain by partial integration and with the outer normal ν = (ν1 , . . . , νn )T on a Lipschitz-continuous manifold ∂Ω, the fundamental Green’s formula, cf. Ciarlet [174], p. 14, w∂v/∂xi dx + v∂w/∂xi dx = ∂(vw)/∂xi dx = vwνi ds, (2.273) 1

Ω

Ω

Ω

∂Ω

∂Ω ∈ CL , ∀w, v ∈ H 1 (Ω). This yields the following weak form for G deﬁned for u, v ∈ V = W0m,p (Ω). as (u, v) = vGs udx = v(−1)|α| ∂ α (Aα (x, u, · · · , ∇m u))dx, u ∈ W 2m,p Ω

=

Ω |α|≤m

Ω |α|≤m

Aα (x, u, · · · , ∇m u)∂ α vdx =: a(u, v) =: Gu, vV ×V ∀ u, v ∈ V

Gu0 , vV ×V = f, vV ×V =

(2.274)

Ω |α|≤m

fα ∂ α vdx ∀ u0 ∈ V ∀ v ∈ V. (2.275)

The Aα (x, u, . . . , ∇ u) are Nemyckii operators, studied in the last subsection. m

102

2. Analysis for linear and nonlinear elliptic problems

These weak forms of quasilinear equations are often obtained as Euler equations for variational problems. For a smooth function F (x, u, . . . , ∇m u), a functional J(u) := F (x, u, . . . , ∇m u)dx (2.276) Ω

has to be (locally) minimized. The ﬁrst variation of this functional applied to v yields a form as in (2.274), for more details cf. Subsection 2.6.4. This is one of the reasons that for the class of quasilinear diﬀerential equations the most complete and ﬁnal results are available. Example 2.72. Important model problems of second order: 1. For this example it is important that [678], p. 592 ﬀ., H01 (Ω) is compactly embedded into L4 (Ω) for dimension n = 1, 2, 3: Gu = −Δu + α

n

sin(u)∂ i u = f ∈ L2 (Ω) on Ω,

(2.277)

i=1

n = 1, 2, 3, and the Dirichlet condition u(x) = 0 on ∂Ω.

(2.278)

We deﬁne linear, bilinear and nonlinear forms related to Gu = f as n f vdx, a1 (u, v) := ∂ i u∂ i vdx ∀u, v, w ∈ H01 (Ω) = V, b(v) := Ω Ω i=1 n (sinu)∂ i wvdx, a2 (u, v) := c(u, u, v), and (2.279) c(u, w, v) := Ω i=1

determine u0 ∈ H01 (Ω) s.t. a1 (u0 , v) + αa2 (u0 , v) = b(v) ∀v ∈ H01 (Ω). 2. We assume 2 ≤ p < ∞, 1/p + 1/q = 1, R t ≥ 0, and Dirichlet boundary conditions, see [678], p. 567 ﬀ., for Gu = Gs u = −

n

∂ i (|∂ i u|p−2 ∂ i u) + tu = f ∈ Lq (Ω) on Ω.

(2.280)

i=1

With the above b ∈ L2 (Ω) we obtain the weak form of G n i i p−2 i vGs udx = v − ∂ (|∂ u| ∂ u) + tu dx Ω

Ω

i=1

= a(u, v) := Gu, vW −1,q (Ω)×W 1,p (Ω) (2.281) n := |∂ i u|p−2 ∂ i u∂ i v + tuv dx and determine Ω

u0 ∈

W01,p (Ω)

i=1

s.t. a(u0 , v) = f, vV ×V ∀v ∈ W01,p (Ω).

(2.282)

2.5. Nonlinear elliptic equations

103

3. Minimal surface equation, see Gilbarg and Trudinger [346], p. 339: Let the minimal surface be represented by the function u : Ω → R. Then, with prescribed boundary conditions u = ϕ on ∂Ω, it has the form Gu = Gu = (1 + |∇u|2 )Δu +

n

∂ i u∂ j u∂ i ∂ j u = 0 on Ω.

(2.283)

i,j=1

This is an example where the Schauder techniques with boundary gradient estimates are applied, see [346]. Now we impose conditions which guarantee that G(u) in (2.270), (2.277), (2.280), (2.283), and its weak form (2.274), (2.283) are well deﬁned. The Carath´eo-dory and growth conditions enforce measurability and appropriate growth of the composite function Aα , see [678], p. 571 ﬀ. and (2.269) for the notation. More generally than the coercivity for bilinear forms we introduce a nonlinear coercivity, monotonicity and growth condition for these quasilinear operators, cf. (2.230)–(2.235), (2.242)–(2.247), (2.258), (2.259). This is again essential for the existence of solutions. Carath´eodory condition: Ω is bounded and measurable in Rn , Aα : (x, Θ≤m ) ∈ Ω × RNm → R, ∀ |α| ≤ m, Θ≤m → Aα (x, Θ≤m ) is continuous in RNm a.e. for x ∈ Ω,

(2.284)

x → Aα (x, Θ≤m ) is measurable in Ω ∀ Θ≤m ∈ RNm , e.g. for continuous Aα . For the following conditions, we assume 1 < p < ∞, 1/p + 1/q = 1, the special form = (ϑγ1 )|γ|≤m ∈ RNm and ∃ c, d, C ∈ R+ , such that Θ≤m = (ϑγ )|γ|≤m , Θ≤m 1 ∃ a ∈ Lq (Ω), h ∈ L1 (Ω), s.t. ∀ x ∈ Ω, Θ≤m , Θ≤m ∈ RNm , ϑ0 , ϑα , ϑα 1 ∈ R. 1 ≤m γ p−1 Growth condition: |Aα (x, Θ )| ≤ C(a(x) + |ϑ | ) ∀|α| ≤ m. (2.285)

(Nonlinear) coercivity: Monotonicity:

|γ|≤m

Aα (x, Θ≤m )ϑα ≥ c

|α|≤m

|ϑγ |p − h(x). (2.286)

|γ|=m

α α (Aα (x, Θ≤m ) − Aα (x, Θ≤m 1 )) (ϑ − ϑ1 ) ≥ 0.

(2.287)

|α|≤m

Strict monotonicity:

|ϑγ − ϑγ1 | > 0 implies

(2.288)

|γ|≤m α α (Aα (x, Θ≤m ) − Aα (x, Θ≤m 1 )) (ϑ − ϑ1 ) > 0.

|α|≤m

Uniform monotonicity: (2.289) p α α (Aα (x, Θ≤m ) − Aα (x, Θ≤m |ϑγ − ϑγ1 | . 1 )) (ϑ − ϑ1 ) ≥ C |α|≤m

|γ|=m

104

2. Analysis for linear and nonlinear elliptic problems

Other types of coercivity estimates towards (2.293) are discussed, e.g. by Showalter [589], p. 38 ﬀ. By [678], Proposition 26.12, (2.284)–(2.287) imply the conditions of his Theorem 26A. Theorem 2.73. Generalized solutions for (2.270), (2.275), [678], pp. 572 ﬀ.: Let (2.284)–(2.285) be satisﬁed and choose V = W0m,p . Let Fα : V = W0m,p (Ω) → Lq (Ω), (Fα u)(x) := Aα (x, u(x), · · · , ∇m u(x)). 1. Then for all |α| ≤ m, ∀x ∈ Ω, these Nemyckii operators Fα are well deﬁned, continuous and bounded, see [678], Proposition 26.12, Proof. 2. With the above Aα (x, u(x), . . . , ∇m u(x)) deﬁne the a(u, v) as in (2.274). Then there exists exactly one bounded operator G, cf. Proposition 2.67, such that, see [678], Proposition 26.12, (2.290) G : V → V : a(u, v) = Gu, vV ×V ∀u, v ∈ V = W0m,p (Ω)

p/q . and Fα uLq (Ω) , GuW −m,q (Ω) ≤ C aLq (Ω) + uW0m,p (Ω) Similarly to (2.275) and with f, vV ×V , the original problem (2.270) corresponds to the generalized problem: Determine for a(u, v) in (2.290) u0 ∈ V = W0m,p (Ω) s.t. a(u0 , v) = f, vV ×V ∀v ∈ V ⇐⇒ Gu0 = f.

(2.291)

3. Let, additionally, all the Aα be coercive and monotone, see (2.286) and (2.287). Then G is monotone, coercive, continuous and bounded, see [678], Proposition 26.12. So by Remark 2.66, parts 2.−4. of Theorem 2.68 are valid for for (2.291). 4. In particular, for any f ∈ W −m,q (Ω) = V the (2.291) has a solution u0 ∈ W0m,p (Ω) and the set of solutions is bounded, convex and closed. If all the Aα are additionally strictly monotone, see (2.288), Theorem 2.68, then G is strictly monotone as well and the above solution is unique. So the inverse G−1 : V → V exists and is strictly monotone and bounded, see [678], Theorem 26A. 5. If all the Aα are, and hence G is uniformly monotone, see (2.289), or if G is strongly monotone, see (2.245), this G−1 is continuous or even Lipschitzcontinuous. Remark 2.74. For our above model problems (2.277), (2.280) and (2.283) with homogeneous Dirichlet boundary conditions we ﬁnd: 1. There exists an α0 > 0, such that for all α ∈ R, |α| ≤ α0 , (2.279), corresponding to (2.277), has exactly one solution u0 ∈ H01 (Ω), see Proposition 27.11 [678]. 2. (2.281), corresponding to (2.280), has exactly one solution u0 ∈ W01,p (Ω). (2.281) is equivalent to Gu0 = b and G is uniformly monotone, coercive, continuous and bounded, see Proposition 26.10 [678]. 3. More complicated conditions for (2.283) are discussed in [346], Chapters 12–15. We modify the preceding results, see Zeidler [678], Problem 27.6, by requiring, for a quasilinear operator G, some conditions only for the principal part:

2.5. Nonlinear elliptic equations

105

Theorem 2.75. Modiﬁed coercivity and monotonicity only for the principal part: For (2.270), let Ω be bounded, and choose V = W0m,p (Ω). Assume the Carath´eodory, the growth and the coercivity conditions, (2.284)–(2.286). Additionally impose the following modiﬁed coercivity (2.293) and the special case (2.294) of the strict monotonicity (2.288), both only for the principal part: With, cf. (2.269), Θ≤m = (Θ0 , . . . , Θm−1 , Θm ) =: (d, Θm ), Θm = (ϑα )|α|=m , let a.e. ∀x ∈ Ω ∀Θ ∀ bounded sets H ⊂ R Aα (x, d, Θm )ϑα lim sup = ∞, |Θm | + |Θm |p−1 |Θm |→∞ d∈H m

Nm −nm

(2.292)

: (2.293)

|α|=m

α α m m nm (Aα (x, d, Θm ) − Aα (x, d, Θm . 1 )) (ϑ − ϑ1 ) > 0, ∀Θ = Θ1 ∈ R

(2.294)

|α|=m

Then ∀f ∈ Lq (Ω) the Gu = f has a generalized solution u0 in the sense of u0 ∈ W0m,p (Ω) : a(u0 , v) = b(v) ∀v ∈ W0m,p (Ω).

(2.295)

Uniqueness results are available via linearization in Section 2.7. The following result, again for special semilinear operators, is based upon the Carath´eodory and growth conditions for the involved Nemyckii operator. We combine the linear operator A in strong divergence form, see (2.196), (−1)|β| ∂ β (aαβ ∂ α u), u ∈ H 2m (Ω) (2.296) Au = |α|,|β|≤m

with Dirichlet boundary conditions. We replace in (2.197) the f (u) by f (x, u) and determine u0 such that G(u0 )(x) := (A + μ)u0 (x) + f (x, u0 (x)) = 0 a.e. ∀x ∈ Ω, u ∈ H0m ∩ H 2m (Ω). (2.297) As usual we ﬁnd the following generalized form, for (2.297) aαβ ∂ α u∂ β v + μuvdx determine with aμ (u, v) = Ω |α|,|β|≤m

(2.298)

u0 ∈ V := H0m (Ω) : aμ (u0 , v) −

f (x, u0 (x))v(x)dx = 0 ∀v ∈ V.

(2.299)

Ω

For A, aμ (·, ·), f in (2.297), (2.298), and the domain Ω we impose the conditions Ω bounded, aαβ : Ω → R measurable and bounded ∀|α|, |β| ≤ m,

(2.300)

a(·, ·) is V-elliptic, ⇐⇒ aμ (·, ·) is V-coercive, for suﬃciently large μ, f : Ω × R → R satisﬁes the Carath´eodory condition (2.258) for n = 1, (e.g. f continuous) and the growth condition (2.262) for p = 2, so ﬁxed b ∈ R+ , 0 ≤ a ∈ L2 (Ω) exist, s.t. |f (x, u)| ≤ a(x) + b|u| ∀(x, u) ∈ Ω × R.

106

2. Analysis for linear and nonlinear elliptic problems

Theorem 2.76. Existence and uniqueness for semilinear problems, [678], Proposition 28.9: Assume the conditions (2.300) for a V-coercive aμ (·, ·), and for the boundary value problem (2.297) or its generalized form (2.299). 1. Existence and uniqueness. Let the function u → f (x, u) be monotone increasing on R for all x ∈ Ω. Then (2.299) has exactly one solution. 2. Existence. Equation (2.299) has a solution, if the function (x, u) → f (x, u) satisﬁes ∃ R > 0 s.t. ∀u with |u| ≥ R : f (x, u)u ≥ 0 ∀ (x, u) ∈ Ω × R. We return to the question of whether quasilinear equations are elliptic equations in the sense of Deﬁnition 2.51, hence, the linearized equation is an elliptic equation. This has several important implications for our later discussion of the convergence of discretization methods and for the Fredholm alternative for quasilinear equations, see Section 2.7. For analytical results and our applications in discretizations, diﬀerent types of conditions are imposed. We want to show that the above monotonicity implies ellipticity and we relate two important concepts. We prove that an ellipticity condition for the linearization of the principal part essentially implies the nonlinear coercivity in the sense of (2.293) for the original quasilinear equation, cf. Theorem 2.75. Generalizing a remark by Ladyˇzenskaja and Uralceva [464], p. 255, we formulate the following lemma, cf. (2.285), (2.289), (2.292)– (2.294). Lemma 2.77. Quasilinear elliptic and nonlinear coercive equations: Let the equation (2.270), (2.274) satisfy the following simpliﬁed growth and nonstandard ellipticity condition: 17 There exist positive continuous functions γ, Γ : RNm−1 → R+ and μ : Rnm → R+ such that with |Aα (x, d, Θm = 0)| ≤ Γ(d) ∀|α| ≤ m, and μ(Θm ) := (1 + |Θm |)p−2 , p ≥ 2, (2.301) ∂Aα m β α m |ϑγ |2 ≤ (x, d, Θ )ϑ ϑ ≤ Γ(|d|)μ(Θ ) |ϑγ |2 . γ(|d|)μ(Θm ) ∂ϑβ |γ|=m

|α|=|β|=m

|γ|=m

1. If 0 ≤ γ ≤ γ(|d|) and Γ(|d|)/γ(|d|) is bounded, we obtain uniform ellipticity. Then the principal part of the linearization evaluated in u0 , hence in (x, d, Θm ) = (x, ∂ α u0 , |α| ≤ m), is uniformly elliptic, cf. Deﬁnition 2.51. 2. For p ≥ 2, the principal part is nonlinear coercive for m ≥ 1, cf. (2.293) and, for m = 1 in the sense of (2.231) only for bounded |Θm |. More precisely, Aα (x, d, Θm )ϑα ≥ C (γ(|d|)|Θm |p − Λ(d)|Θm |) . (2.302) |α|=m

Proof. The uniform ellipticity of the principal part of the linearization is an immediate consequence of (2.301) combined with the continuity of the γ, Γ, μ evaluated in a bounded subdomain deﬁned by (x, d, Θm ) = (x, ∂ α u0 , |α| ≤ m), x ∈ Ω. 17 We have called (2.301) a nonstandard ellipticity condition, since we have chosen the nonstandard form of γ(|d|)μ(Θm ) instead of the usual .

107

2.5. Nonlinear elliptic equations

For the coercivity, we modify the proof in [464], p. 255, and start with (1 + x)α ≥ 1 + xα , ∀x ≥ 0, 1 ≤ α ∈ R.

(2.303)

We rescale, for 1 < x, the (1 + x)α = xα (1 + 1/x)α and 1 + xα = xα (1 + (1/x)α ). So it suﬃces to apply the Taylor formula and ﬁnd, with 0 < θ < 1, for ∀0 ≤ x ≤ 1, 1 ≤ α ∈ R : (1 + x)α − 1 = α(1 + θx)α−1 x ≥ x ≥ xα . The last inequality α(1 + θx)α−1 x ≥ xα certainly does not hold ∀0 ≤ x, α ≤ 1. In fact for a ﬁxed x > 0 we get lim α(1 + θx)α−1 x = 0 < lim xα = 1.

α→0+

α→0+

Combining (2.303) with directional derivatives, integration and (2.301) we ﬁnd with μ(Θm ) = (1 + |Θm |)p−2 and the mean value theorem 1.43 for operators, with Θm = (ϑβ |β| = m), 1 ∂Aα ϑβ (x, d, τ Θm )dτ, hence, Aα (x, d, Θm ) = Aα (x, d, 0) + β 0 ∂ϑ

|β|=m

Aα (x, d, Θm )ϑα

|α|=m

=

Aα (x, d, 0) +

|α|=m

|β|=m

1

≥ −CΓ(|d|)|Θm | + γ(|d|)

1

0

∂Aα (x, d, τ Θm )dτ ϑβ ϑα β ∂ϑ

(1 + τ |Θm |)p−2 dτ |ϑγ |2

0

≥ −CΓ(|d|)|Θm | + γ(|d|)

|γ|=m

1 m p−1 1 (1 + τ |Θ |) | |ϑγ |2 0 |Θm |(p − 1) |γ|=m

≥ −CΓ(|d|)|Θm | + γ(|d|) (2.303)

≥

1 |Θm |(p

−CΓ(|d|)|Θm | + γ(|d|)

− 1)

((1 + |Θm |)p−1 − 1p−1 )

|ϑγ |2

|γ|=m

|Θm |p−1 |Θm |2 , for p ≥ 2, p ∈ R, |Θm |(p − 1)

≥ −CΓ(|d|)|Θm | + γ(|d|)|Θm |p . γ, Γ are continuous with respect to d, so they are bounded for all bounded sets H ⊂ RNm −nm , see (2.73), implying (2.293) since with p ≥ 2 lim

|Θm |→∞

sup C d∈H

γ(|d|)|Θm |p − Γ(|d|)|Θm | = ∞ ∀ bounded H ⊂ RNm −nm . |Θm | + |Θm |p−1

This proves the claim.

108

2. Analysis for linear and nonlinear elliptic problems

General regularity results of an as “simple” form as in Section 2.4 do not seem to exist or did not ﬁnd their way into the general monographs summarized here. In Subsection 2.5.4 we presented some examples for order 2, and in Subsection 2.6.6 and Theorem 2.116 we will present results for general systems of order 2m. For a lot of other results we refer to the literature. Morrey [500] and Giaquinta and Hildebrandt [341, 342] study quasilinear equations induced by variational problems. Giaquinta and Hildebrandt [341, 342] essentially consider second order problems. b Then the functional in (2.276) is often assumed in the form J(u) := a F (u, ∇u)dt where u = u(t), a ≤ t ≤ b, indicates a curve on a surface. Under appropriate conditions a minimizing curve is then u0 ∈ C 2 ([a, b]), cf. [341, 342], Chapter 1, Section 3, Proposition 3 and Chapter 8, Section 4, Proposition 2. Relations of minima of convex functionals or duality and conjugate functionals and quasilinear equations of orders 2 and 2m are discussed as well by Zeidler [675], Sections 42.7 and 51.7. Koshelev and Chelkak [449, 450] study regularity for systems, and Ladyˇzenskaja and Uralceva [463, 464, 466], for second order problems of elliptic and parabolic type. Ladyˇzenskaja [462] discusses boundary value problems of mathematical physics. Morrey in his many papers uses multiple integrals and the calculus of variations, and studies diﬀerentiability of solutions for nonlinear elliptic equations and systems, e.g. [500–504]. 2.5.7

Fully nonlinear elliptic equations of orders 2, m and 2m

We turn to the fully nonlinear case and start with the most important order 2, see Gilbarg and Trudinger [346] and Chen and Wu [170]. We require Ω ⊂ Rn satisfying (2.5) and a real valued function, Gw , cf. (2.165), (2.166), of the form 2

Gw : D(Gw ) → R, w = (x, z, p, r) ∈ D(Gw ) ⊂ Uo = Ω × R × Rn × Rn ; (2.304) we restrict the elements in Rn×n to the n(n + 1)/2-dimensional subspace of symmetric real valued n × n matrices. Choose functions u : Ω → R, such that D(G) := {u ∈ H 2 (Ω) : wu (x) := (x, u(x), ∇u(x), ∇2 u(x)) ∈ D(Gw )∀x ∈ Ω}. (2.305) Remark.

Fully nonlinear PDE of order 2 and 2m

1. A fully nonlinear PDE of order 2 is none of the above special cases (2.169)– (2.172). It has, for a Dirichlet boundary value problem, the strong classical form, hence G : D(G) ⊂ H 2 (Ω) → L2 (Ω), with u, u0 ∈ D(G) s.t.

(2.306)

L (Ω) G(u0 ) := G (·, u0 (·), ∇u0 (·), ∇ u0 (·)) = 0 on Ω, u0 = ϕ on ∂Ω. 2

w

2

In particular, the spaces of ansatz functions, u, u0 ∈ D(G) ⊂ H 2 (Ω), and of test functions, v ∈ L2 (Ω), are diﬀerent. 2. The same situation holds for the equations of order 2m and m at the end of this subsection and the systems in Subsection 2.6.8.

2.5. Nonlinear elliptic equations

109

In Deﬁnition 2.51 we have introduced (uniformly) elliptic G. For second order equations, m = 1, the −1 in (2.175), (2.176) is often omitted. Choose w = (x, z, p, r) ∈ D(Gw ) as in (2.304), a u1 with its w1 (x) := (x, u1 (x), ∇u1 (x), ∇2 u1 (x)) ∈ D(Gw ), and w ∈ Uw1 , a neighborhood of w1 . We assume for 0 = ϑ ∈ Rn n ∂Gw 0 < λ|ϑ| ≤ (w)ϑi ϑj ≤ Λ|ϑ|2 ∀w = (x, u, p, r) ∈ Uw1 ⊂ D(Gw ), ∂rij i,j=1 2

(2.307)

for its principal part. For a real symmetric matrix ∂Gw /∂rij (w) with its real eigenvalues the minimal and maximal eigenvalues satisfy λ ≤ λ(w) ≤ Λ(w) ≤ Λ. Example 2.78. The two of the most important examples: 1. Monge–Amp`ere equation This is, for u : Ω → R and n ≥ 2 0 = G(u) := det ∇2 u − f (x, u, ∇u) = uxx uyy −u2xy −f (x, u, ∇u) for n = 2. (2.308) We claim that this G is uniformly elliptic in u1 if ∇2 u1 is strictly positive deﬁnite and symmetric in Ω: standard textbooks show that, for a strictly convex u1 ∈ C 2 (Ω) on a convex Ω, this is correct. Furthermore, a matrix and its inverse are simultaneously strictly positive deﬁnite and have a positive determinant. We deﬁne the cofactor of the element ri,j = ∂ i ∂ j u = ∂ 2 u/∂xi ∂xj of the matrix ∇2 u = r = [rij = ∂ i ∂ j u]ni,j=1 , symmetric for u ∈ C 2 (Ω): cancel row i and column j in ∇2 u to get Rij and deﬁne, independently of ri,j = ∂ i ∂ j u, the cofactor of ∂ i ∂ j u as (−1)i+j det Rij . Then det ∇2 u =

n

∂ i ∂ j u cofactor ∂ i ∂ j u =

j=1

n

∂i∂j u

j=1

⇒ (det ∇2 ) (u1 )u =

n j=1

∂i∂j u

∂ det ∇2 u (u) ∂rij ∂ det ∇2 u (u1 ). ∂rij

(2.309)

The inverse of a nonsingular matrix is obtained by the matrix of its cofactors as: with B := [ cofactor ∂ i ∂ j u1 ]ni,j=1 we get (∇2 u1 )−1 = B T / det ∇2 u1 . This (2.309) implies with (2.175), (2.307), n ∂ det ∇2 u ϑT (det ∇2 ) (u1 )ϑ = ϑi ϑj (u1 ) = det ∇2 u1 ϑT (∇2 u)−1 (u1 ) ϑ, ∂r ij i,j=1 hence our claim by (2.307). So the G(u) in (2.308) is uniformly elliptic in u1 for a symmetric strictly positive matrix r = ∇2 u1 in Ω, hence for a function u1 ∈ C 2 (Ω), strictly convex ∀x ∈ Ω on a convex Ω. With det(∇2 u) > 0, uniformly convex solutions are only possible for positive f , [345], pp. 441 ﬀ. For partially nonelliptic generalizations, see [345], p. 467ﬀ. and Courant and Hilbert [214], p. 324. According to Haltiner and Kasahara the leading term in the balance equation in dynamical meteorology

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has the form (2.308) [389,437]. In Westcott [664] it is related to geometric optics. Among the many applications in geometry we mention the 2. Equation for a surface with prescribed Gauss curvature K = K(x) at x This is closely related to the Monge–Amp`ere Equation. Equation (2.308) has to be modiﬁed as G(u) := det ∇2 u − K(1 + |∇u|2 )(n+2)/2 = 0, K = K(x) > 0.

(2.310)

Again G(u) is elliptic only for uniformly convex functions u ∈ C 2 (Ω) in a convex Ω. Our FEMs for this class of problems are interesting for aspects of global geometry. The existence of surfaces with prescribed Gauss curvature is known for appropriate K. Speciﬁc properties would become more ﬂexibly available via our FEMs than by the available ﬁnite diﬀerence methods, see e.g. Crandall and Lions [227]. Our FEM allows us FEs in C 1 ∩ W on quasiuniform grids. It yields higher orders of convergence, cf. Theorem 5.18, than the diﬀerence methods of order 2, e.g. in Chapter 8 or in [227]. Both methods are only deﬁned on equidistant grids. For existence and uniqueness results we have to impose growth conditions. We w w w distinguish the cases n = 2 and n ≥ 2 and evaluate the derivatives Gw x , Gz , Gp , for Gr , compare (2.307). We assume nonnegative λ = λ(·) and Λ = Λ(·), μ = μ(·) in (2.307), (2.311)–(2.312) depending upon |z|, nonincreasing and nondecreasing in z ∈ [0, ∞), respectively, and require for w = (x, z, p, r), x ∈ ∂Ω, w for n = 2 : |Gw z (w)| , Gp (w) ≤ λμ |Gw x (w)| ≤ μλ(1 + |p| + |r|) and w w w w for n ≥ 2 : |Gw z (w)| , Gp (w) , |Grx (w)| , Gpx (w) , |Gzx (w)| ≤ λμ w |Gw x (w)| , |Gxx (w)| ≤ μλ(1 + |p| + |r|).

(2.311)

(2.312)

Theorem 2.79. Existence and uniqueness for fully nonlinear problems in R2 , [346], ¯ Let G satisfy the conditions Theorem 17.12: Assume n = 2, ∂Ω ∈ C 3 , ϕ ∈ C 3 (Ω). w (2.307), (2.311), and Gz ≤ 0 in Uo , see (2.304). Then the classical Dirichlet problem ¯ for all 0 < α < 1. (2.306) has a unique solution u0 ∈ C 2,α (Ω) Ω is called satisfying an exterior sphere condition for every ξ ∈ ∂Ω, if there exist a ¯ ∩Ω ¯ = {ξ}. ball B = BR (y) ⊂ Rn \ Ω such that B Theorem 2.80. Regular solutions for fully nonlinear problems in Rn , n ≥ 2, [346], Theorem 17.17: Assume n ≥ 2, ϕ ∈ C(∂Ω). Let Ω satisfy an exterior sphere condition for every ξ ∈ ∂Ω, Gw ∈ C 2 (Uo ) be concave (or convex) with respect to (z, p, r), nonincreasing with respect to z and let G satisfy the conditions (2.307), (2.312). Then ¯ the classical Dirichlet problem (2.306) has a unique solution u0 ∈ C 2 (Ω) ∩ C(Ω). More detailed results for the above special cases in Example 2.78 are presented in [346].

2.5. Nonlinear elliptic equations

111

Skrypnik [591], Theorem 3.2.3, considers the Monge–Amp`ere equation (2.308) on the two-dimensional ball Ω = BR (0) with center 0 and imposes the boundary condition (u0 − g)|∂BR (0) = 0. He assumes the existence of functions φ : BR (0) → R+ , f0 : R2 → R+ , s.t., with m := max g(x) x∈BR (0)

f (x, u, p) ≤ φ(x )f0 (p ) for x, p, s ∈ R , R u ≤ m and let φ(x2 )dx ≤ inf 1/f0 (s2 )dp, 2 2

BR (0)

2

2

(2.313)

R2 s−p <MH

where MH is the lower winding number of the curve deﬁned by (u0 − g)|∂BR (0) = 0, see, e.g. Bakelman [55]. Finally, let G := {(x, u, p) : x ∈ BR (0), u ∈ R, p ∈ R2 } and 4 H+ (Ω) := {u ∈ H 4 (Ω) : (∂ 20 u∂ 02 u − (∂ 11 u)2 )(x) > 0, ∂ 20 u(x) > 0∀x ∈ BR (0)}.

Theorem 2.81. Regular solutions for the Monge–Amp`ere equation, Skrypnik [591], Theorem 3.2.3: Let (2.313) be satisﬁed and f ∈ C 2,γ (G), g ∈ C 4,γ (∂Ω). Then (2.308) 4 (Ω). has at least one solution in H+ We continue with some existence and regularity results, elaborated by Chen and Wu [170], Chapter 7, Theorem 7.4 and Remark on p. 117, for general fully nonlinear equations (2.306) of order 2. They impose six conditions (F1 ) − (F6 ) for (2.306): Let

2

G(x, z, p, r) ∈ R, (x, z, p, r) ∈ D(G) ⊂ Uo = Ω × R × Rn × Rn , and (2.314)

assume a constant μ ¯ and nonnegative, nondecreasing μi : [0, ∞) → R+ , i = 1, 2, 3, s.t. (F1 ) : ∃0 < λ = λ(x, z, p), Λ = Λ(x, z, p) s.t., with the identity matrix I n ∂G ≤ Λ I, and Λ/λ ≤ μ1 (|z|) on Uo ; λ I≤ ∂rij i,j=1 (F2 ) : |G(x, z, p, 0)| ≤ λμ2 (|z|)(1 + |p|2 ); % & (F3 ) : (1 + |p|)−1 |Gx | + |Gz | + (1 + |p|)|Gp | (x, z, p, r) ≤ λμ3 (|z|)(1 + |p|2 + |r|)∀(x, z, p, r) ∈ Uo ; (F4 ) :

on Uo : G(x, z, p, r) is concave with respect to r;

μ(1 + |p|); (F5 ) : G(x, z, p, 0) sign z ≤ λ¯ (F6 ) : Gz (x, z, p, r) ≤ 0 ∀(x, z, p, r) ∈ Uo .

(2.315)

Theorem 2.82. Regular solution for fully nonlinear problems: 1. Let the conditions (2.314)–(2.315) be satisﬁed, Λ be bounded on Ω × R × Rn . Then there exists 0 < α < 1 with the following property: For any 0 < γ < α < 1 ¯ for the Dirichlet problem (2.306) there exists a with ∂Ω ∈ C 2,γ ϕ ∈ C 2,γ (Ω), 2,γ ¯ unique solution u0 ∈ C (Ω). This α depends only upon n, μ1 , μ2 , μ3 , ϕC 2,γ (Ω) ¯ and Ω.

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2. Analysis for linear and nonlinear elliptic problems

¯ then the solution u0 ∈ 2. If G ∈ C k,α (Uo ), k ≥ 1, and ∂Ω ∈ C 2+k,γ , ϕ ∈ C 2+k,γ (Ω), ¯ ∩ C k+2,α (Ω). For G ∈ C k,α (U¯o ), it is u0 ∈ C k+2,γ (Ω). ¯ C 2,γ (Ω) We turn to some existence and regularity results for fully nonlinear equations of order 2m. Skrypnik [591] seems to be the author of the only textbook studying these problems. He combines monotone operators and degree theory to get his results for quasilinear and fully nonlinear elliptic equations. He discusses some of these problems on narrow strips and in perforated domains. To give a ﬂavor of his existence and regularity results we cite one of his theorems, a “conditional” result. For the original problem, he formulates a family of parameter dependent problems (t ∈ [0, 1]) with t = 0 and t = 1 for the original problem and a problem with appropriate degree properties. He determines a solution u0 for the nonlinear Dirichlet problem. Let a function G be given such that, with N2m as in (2.285), cf. Remark concerning (2.306) N2m ¯ Gw ∈ C(D(Gw )), D(Gw ) := [0, 1] × D (Gw , 0 ) = [0, 1] × Γ ⊂ [0, 1] × Ω × R

with appropriate N2m , let Gw (0, x, −ΘN2m ) = −Gw (0, x, ΘN2m ), and ∀t ∈ [0, 1] : Gw (t, . . .) ∈ C l+1 (D (Gw 0 )) , l ≥ [n/2] + 1, where

(2.316)

u, u1 ∈ D(G) if, e.g. w1 (x) := (x, ∂ α u1 (x), |α| ≤ 2m) ∈ D (Gw 0 ) ∀x ∈ Ω, G(u) := G(t = 0, u) := Gw (t = 0, ·, u(·), . . . , ∇2m u(·)) G : D(G) ⊂ U := H0m (Ω) ∩ H 2m (Ω) → V := L2 (Ω), determine u0 from (2.317) G(u0 (·)) = Gw (t = 0, ·, u0 , . . . , ∇2m u0 )(·) = 0, u0 ∈ X0 := H0m (Ω) ∩ H 2m+l (Ω). He assumes the existence of constants 0 < k, 0 < γ < 1, 0 < Λ < ∞ for (2.316), and modiﬁes the ellipticity condition, compare (2.232), such that ∃k ∈ R+ , 0 < γ < 1 : ∀t ∈ [0, 1], v ∈ X0 : G(t, v) = 0 implies vC 2m,γ (Ω) ¯ ≤ k, and % & ∂ α Gw (t, ·, v, · · · , ∇2m v) (x) ϑα ≤ Λ|ϑ|2m ∀x ∈ Ω. (2.318) k −1 |ϑ|2m ≤ |α|=2m

Theorem 2.83. Regular solutions, Skrypnik [591], Theorem 3.2.2: Let G satisfy (2.316), (2.318) and let ∂Ω ∈ C ∞ for a bounded domain in Rn . Then (2.317) has, for t = 0, at least one solution u0 ∈ U. We ﬁnish with a nonlinear PDE of order m, see Deﬁnition 2.85, in the form G(u) := G(x, u, ∇u, . . . , ∇m u) and G(u)(x) = g(x) ∀ x ∈ Ω.

(2.319)

Theorem 2.84. Local solvability and regularity result – Taylor, [620], Chapter 14, Theorem 3.1: Let g ∈ C ∞ (Ω), and let u1 ∈ C ∞ (Ω) satisfy (2.319) at x = x0 , G(u1 )(x) = g(x) for x = x0 and let G be m-elliptic at u1 . Then for any there exists a u ∈ C (Ω) such that G(u)(x) = g(x) in a neighborhood of x0 .

2.6. Linear and nonlinear elliptic systems

2.6 2.6.1

113

Linear and nonlinear elliptic systems Introduction

In this section we discuss systems of q elliptic diﬀerential equations essentially in the Hilbert Sobolev space setting for two reasons: Practically all available results in the textbooks consulted are formulated for the Hilbert case. It is straightforward as in Subsection 2.4.4 to extend the following results, based upon the H m -coercivity, to the W m,p case for 2 ≤ p < ∞. The 1 ≤ p < 2 usually have to be excluded. As in Corollary 2.44, 3., 4., we modify them for 2 < p ≤ ∞: these W m,p (Ω) → H m (Ω) are densely and continuously embedded. Consequently, all H m -coercivity based existence, uniqueness and Fredholm results, including the numerical stability and convergence of all our discretization methods, remain correct for the W m,p case for 2 < p ≤ ∞ with respect to (discrete) H m norms. We will explicitly formulate this fact for the particularly ineresting Theorems 2.89 and 2.104. Comparing equations with systems shows an astonishing diﬀerence: the deﬁnition of ellipticity for linear and nonlinear equations coincides for nearly all aspects. For order 2m they are direct generalizations of those for second order. This is no longer correct for systems. In Subsection 2.6.2, Deﬁnition 2.85, we introduce the concept of systems of elliptic diﬀerential equations, a concept of ellipticity, pretty diﬀerent from those previously introduced. It is motivated by important examples, including the Navier–Stokes and von K´ arm´an equations. Starting with Subsection 2.6.3, and excluding Navier–Stokes equations, we turn back to the deﬁnitions of ellipticity, similar to or directly generalizing those for the case m = q = 1. However, depending upon the chosen goal, e.g. existence and uniqueness or regularity, diﬀerent types of ellipticity become relevant. Subsection 2.6.3 is devoted to linear second order systems, cf. (2.343), extended to quasilinear second order systems in Subsection 2.6.4, cf. (2.370), (2.371) for existence and (2.384) for regularity. The last three Subsections 2.6.5, 2.6.6 and 2.6.8 generalize some of these results to order 2m of linear, cf. (2.394), (2.401), quasilinear, cf. (2.437), and fully nonlinear elliptic systems, cf. (2.446). Not much seems to be known about uniﬁed conditions for the diﬀerent aspects and orders. Systems of elliptic diﬀerential equations are intensively studied in diﬀerent monographs, e.g. H¨ ormander [402], Garabedian [336], Lions and Magenes [478–480], Chen and Wu [170], Taylor [618–620], and Koshelev and Chelkak [449–451]. For systems the ellipticity is not uniformly deﬁned. In particular, existence and regularity are discussed by Ladyˇzenskaja and Uralceva [463–466], Koshelev and Chelkak [450, 451], Skrypnik [590, 591], Morrey [500], Agmon, Douglis and Nirenberg [3], Taylor [620], Zeidler [675–678], Giaquinta and Hildebrandt [341], Neˇcas [510], and Kozlov Maz’ya and Rossmann [452]. Here we include some papers as well: Wahl and Grunau [378,659], Luckhaus and Alt [23, 481], Nedev [512], and Boccuto and Mitidieri [99]. These books and papers show, e.g. [23,99,481,659], that usually generalizations of regularity results from the linear to the nonlinear case require new techniques and a careful analysis. Regularity for nonlinear elliptic systems with interesting applications to stochastic games, ergodic control, semiconductors, plasticity, and the stationary Navier–Stokes equations are studied by Bensoussan and Frehse [81]. The regularity results for the

114

2. Analysis for linear and nonlinear elliptic problems

stationary Navier–Stokes equations would be interesting for Section 2.8. However, in [81] the main goals are speciﬁc “maximum-like solutions” for higher dimensions n > 3, which might give insight into the nonstationary Navier–Stokes equations. We summarize some essential results of Koshelev and Chelkak [449–451] and Skrypnik [591], in Subsections 2.6.4 and 2.6.6, 2.6.8. 2.6.2

General systems of elliptic diﬀerential equations

In this subsection we give information for general systems of elliptic equations in the sense of Agmon, Douglis and Nirenberg [3]. The stationary Navier–Stokes equation is one of the most important of these systems of elliptic diﬀerential equations. It certainly motivated the new deﬁnition of ellipticity in this and the next subsections. Although we discuss the Navier–Stokes equation in a separate section, we memorize it here. We want to motivate the general systems, discussed here in contrast to the following subsections. As before we usually omit the index s for the strong form, possible for a regular enough situation. The context indicates the appropriate form. In this subsection essentially all operators are in strong form. Already in Chapter 1 we introduced the Navier–Stokes equation. For the stationary form, the solution, u0 , p0 satisﬁes ⎛ ⎞ n 0 0 i 0 −νΔ u + u ∂ u + ∇ p f 0⎠ 0 i ⎝ G( u , p0 ) := = in Ω, (2.320) i=1 0 − div u0

0

u = 0 on ∂Ω,

p0 dx = 0, Ω satisﬁes (2.5) and Ω

u = (u1 , . . . , un )T , v = (v1 , . . . , vn )T , f = (f1 , . . . , fn )T : Ω ⊂ Rn → Rn , div u := ∂u1 /∂x1 + · · · + ∂un /∂xn , p : Ω → R, u, p, . . . suﬃciently smooth, where u, p and f denote the velocity, pressureand forcing term of an incompressible medium and v a test function. The condition Ω p0 dx = 0 enforces a unique p0 . Note that (2.320) is a system of n + 1 diﬀerential equations for n + 1 unknown functions submitted to the boundary conditions and an integral constraint. For physical applications only n ≤ 3 is interesting, for higher n, cf. [81]. The derivative G of G, evaluated at ( u, p), u = 0, and applied to an increment (w, r), is the linearized Navier–Stokes operator, again an operator with n + 1 components, n + ∇r −νΔw + i=1 (wi ∂ i u + ui ∂ i w) . (2.321) r) = G ( u, p)(w, − div w The special case G ( u ≡ 0, p0 ) and ν = 1 applied to an increment ( u, p) yields the Stokes operator, S, and the corresponding Stokes equation, −Δ u + ∇p f = in Ω, (2.322) Gu (0, p0 )( u, p) = S( u, p) = 0 − div u u = 0 on ∂Ω, pdx = 0. Ω

2.6. Linear and nonlinear elliptic systems

115

The nonlinear system (2.320) and its linearizations (2.321) and (2.322) do not allow a direct generalization of the previous framework (in contrast to the situation in the next subsections). Again, the following notation and even some results for linear systems of q equations for q functions u = (u1 , . . . , uq ) are often similar to the concepts of linear algebra, cf. (2.73), (2.269), for ϑ ∈ Rn and ϑα , ∂ α uj . Deﬁnition 2.85. Elliptic systems – Agmon et al. [2, 3]: 1. A linear system of diﬀerential equations has the form, ⎛ ⎞q q q L u := ⎝ Lij uj ⎠ = f := f i i=1 , u = (u1 , . . . , uq )T deﬁned in Ω ⊂ Rn , j=1

Lij :=

i=1 α α aij α (x)∂ , i, j = 1, . . . , q, ∂ uj = uj for |α| = 0,

(2.323)

|α|≤kij 18 usually with aij We choose obviously not unique α ∈ C(Ω).

m1 , . . . , mq , m1 , . . . , mq s.t. kij ≤ mi + mj , i, j = 1, . . . , q,

(2.324)

with the 2m in (2.328). The principal part of Lij or its symbol are deﬁned by the components of its “highest order” tems α ij α aij aij (2.325) Lij p := α ∂ or Lp (x, ϑ) := α (x)ϑ , |α|=mi +mj

|α|=mi +mj

for i, j = 1, . . . , q. If kij < mi + mj , these aij α = 0, for |α| = mi + mj , hence Lij p = 0. 2. We deﬁne the symbol, Lp , and characteristic polynomial, det Lp , as q ij q Lp (x, ϑ) := Lij p (x, ϑ) i,j=1 ∀x ∈ Ω, det Lp (x, ϑ) = det Lp (x, ϑ) i,j=1 . (2.326)

3. A linear system (2.323) is called elliptic in x ∈ Ω, if Lp (x, ϑ) is invertible, the q × q matrix Lp (x, ϑ)is invertible ⇔ det Lp (x, ϑ) = 0 ∀ 0 = ϑ ∈ Rn . (2.327) Often, stronger than (2.327), > 0 exists, such that (only a.e. for aα ∈ L∞ (Ω)) | det Lp (x, ϑ)| ≥ |ϑ|2m ∀ × Ω, 0 = ϑ ∈ Rn with 2m :=

q

(mi + mi ) . (2.328)

i=1

4. We call 2m : =

max

i,j=1,...,q

(mi +mj )

the order of an elliptic system.

5. As in Deﬁnition 2.51 we call a nonlinear system elliptic if its linearization is elliptic. Obviously, we obtain the original ellipticity for 1 = q ≤ m = m1 = m1 . 18

Several types of notation are used in the literature, each with pros and cons. For linear systems q (2.323) the notation L u as a matrix of operators (Lij )qi,j=1 or coeﬃcients (aij α )i,j=1 applied to a T vector u = (u1 , . . . , uq ) of functions is advantageous. This does not work too well for quasilinear systems. In the Einstein convention the same index and exponent i indicatesthe sum over i. In (2.323) and (2.339) this could be used directly for the j and k, l, so qj=1 and n k,l=1 , respectively, would be omitted. This Einstein convention is not too common for numerical methods and can be applied only in parts of this section. So we decided to use the componentwise or vectorial notation.

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6. A modiﬁcation is discussed by Taylor [618], pp. 245 ﬀ. and pp. 379 ﬀ.: Replace in (2.323) the |α| ≤ kij and in (2.325) the |α| = mi + mj by |α| ≤ m and |α| = m. We denote this modiﬁed linear system (2.323) as m-elliptic, if Lp (x, ϑ) is invertible ∀x ∈ Ω, ϑ ∈ Rn . In the literature “strong ellipticity” is used in diﬀerent ways. We use “strong ellipticity” for (2.343) below, but not for (2.328). Example 2.86. We list several examples of elliptic systems, in particular two systems introduced in Chapter 1. 1. The Stokes operator L = S and the linearized Navier–Stokes operator: In (2.322) we choose q := n + 1, u = (u1 , . . . , un , p)T , f = (f1 , . . . , fn , 0)T ,

(2.329)

Lii := −Δ, Liq := −Lqi = ∂ i = ∂/∂xi for i = 1, . . . , n, otherwise Lij := 0. Then kii = 2, kiq = 1 = kqi for i = 1, . . . , n and kij = 0 otherwise. Hence the following mi , mi satisfy (2.324): mi = mi = 1 for i = 1, . . . , n = q − 1, mq = mq = 0. Furthermore Lij p are independent of x and we get 2 iq qi i ij Lii p (ϑ) = −|ϑ| , Lp (ϑ) = −Lp (ϑ) = ϑ , for i = 1, . . . , n, else Lp (ϑ) = 0.

This yields ﬁnally, with the 2m in (2.328) ⎞ ⎛ 0 ϑ1 −|ϑ|2 ⎜ .. ⎟ .. ⎜ . . ⎟ | det Lp | = det ⎜ 0 ⎟ = |ϑ|2m with 2m = 2n. 2 n ⎠ ⎝ −|ϑ| ϑ −ϑ1 . . . −ϑn 0

(2.330)

Note that this 2m = 2n and the order 2 for the Stokes operator introduced in Deﬁnition 2.85, 4. and Deﬁnition 2.102 are diﬀerent. If we linearize the Navier–Stokes operator in an arbitrary ( u, p), the additional terms are of lower order, and the corresponding Lij p (ϑ) are ν× those of the Stokes operator. Hence, (2.328) is satisﬁed with = 1 and = ν n for the Stokes and the linearized Navier–Stokes operator. So both are elliptic, with 2m = 2n. This = ν n indicates one of the most fascinating properties of the Navier– Stokes equations: for ν→ 0 the elliptic aspects are dominated by convection n phenomena, caused by i=1 ui ∂ i u terms in (2.320) and their linear counterparts in (2.321). For the time-dependent form of the equations this corresponds to a transition from a parabolic to a hyperbolic system. We will come back to (2.320) in Section 2.8. 2. The von K´ arm´ an equations, see Example 1.4: Berger and Five [82–84] have elaborated the appropriate spaces for these equations and proved the existence of solutions. Ciarlet [175] has identiﬁed these equations as the ﬁrst term in an

2.6. Linear and nonlinear elliptic systems

117

asymptotic expansion, describing the mechanical bending of a plate. He analyzes these equations and the corresponding FEMs. An external load λ compresses a plate located at Ω. The Airy stress function w(x, y) at the point (x, y) and the deviation or deﬂection u(x, y) of the plate from its ﬂat state u(x, y) ≡ 0, u, w : Ω → R, are described by ⎞ ⎛ ⎞ ⎛ 2 −[u, w] +λuxx Δ u f ⎠ , G(u0 , w0 ) = ⎝ ⎠ (2.331) G(u, w) := ⎝ 0 Δ2 w + 12 [u, u] in Ω. Here Δ2 = ΔΔ is the biharmonic operator in R2 , in some papers the λuxx is missing. The bracket operator [ ·, · ] is deﬁned by [u, v] = [v, u] = uxx vyy − 2uxy vxy + uyy vxx . The derivative G (u1 , w1 ) applied to (v, z) has the form ⎞ ⎛ 2 Δ v −([u1 , z] + [v, w1 ]) +λvxx ⎠. G (u1 , w1 )(v, z) := ⎝ Δ2 z +[u1 , v] Here we choose q := 2, Lii := Δ2 for i = 1, 2, Lij := 0 for i = j so kii := 4 for i = 1, 2, kij := 0 for i = j, mi = mi = 2, and

(2.332)

4 ij Lii p (ϑ) := |ϑ| for i = 1, 2, Lp (ϑ) := 0 for i = j, q = 2.

This yields ﬁnally

4 |ϑ| 0 = |ϑ|8 . | det Lp (ϑ)| = | det Lp | = det 0 |ϑ|4

(2.333)

Hence, the von K´arm´an equations are strongly ( = 1) elliptic and 2m = 8. We will come back to this in Subsection 2.6.6. 3. The system of Lam´e: This describes the deﬂection function u of an elastic plate by a linear system: with μ, λ > 0 we determine the solution u0 from u0 : Ω ⊂ R3 → R3 , μΔ u0 + (λ + μ)∇ div u0 = f in Ω, = ϕ on ∂Ω. Similarly as above one shows that | det Lp | = μ2 (2μ + λ)|ϑ|6 , so 2m = 6.

(2.334)

Agmon et al. [2, 3] strongly stimulated research on partial diﬀerential equations and systems. It is absolutely impossible to appropriately summarize these results in this chapter. Furthermore, some of these general systems require mixed discretization methods, e.g. the Navier–Stokes equations in Section 2.8, requiring special treatment. So, in the following subsections we formulate for special elliptic systems the results for existence and regularity, similar to those for elliptic equations. We give a ﬂavor for general systems, by summarizing a few examples on local existence and regularity from Taylor [618] for linear systems in Subsections 2.6.3 and 2.6.5.

118 2.6.3

2. Analysis for linear and nonlinear elliptic problems

Linear elliptic systems of order 2

In this and the following subsections we will study special elliptic systems of order 2 and 2m. The linear systems are obtained, very similarly to elliptic equations, as compact perturbations of the usually H0m -coercive principal parts of these operators. As in Corollary 2.44, 3., 4., this implies only the H0m -coercivity for 2 ≤ p ≤ ∞. Then existence, uniqueness and Fredholm alternatives are easily obtainable by Section 2.3, see e.g. Theorem 2.21. We elaborate the corresponding results for linear and divergent quasilinear elliptic systems. Our presentation generalizes Chen and Wu [170] with respect to existence and regularity. Their regularity results are only valid for equations deﬁned by the principal part, but will be extended to the general case by the inherited regularity technique in Theorem 2.56. We refer to Subsections 2.6.4 and 2.6.6 for more general types of regularity results, even for quasilinear equations, see Taylor [618–620], Koshelev and Chelkak [449–451] and Skrypnik, [591]. For these systems we can replace the |α| ≤ kij in (2.323) by |α| ≤ m. So we do not need the mi , mj in (2.324) any more. This allows a simpler matrix formulation for the equations. We start with the discussion of second order linear problems and extend it to quasilinear systems of second and then to higher order. Before we introduce the following setting for homogeneous Dirichlet boundary conditions, we want to recall the above generalizations. For a nonhomogeneous Dirichlet boundary function, extended to a function on Ω, Lemma 2.26, and the standard trick, reduces the nonhomogeneous to homogeneous Dirichlet boundary conditions. Then similarly to the cases following (2.124), generalized second (modiﬁed Neumann), third or natural boundary value problems, compare (2.125), can be introduced and treated for second order problems. Similarly to (2.38), diﬀerent boundary conditions on diﬀerent parts of ∂Ω are possible, but will not be formulated here. In this subsection, considering systems of order 2, we have to apply the partial derivatives ∂ l , l = 1, . . . , n, to q components uj , vi , i, j = 1, . . . , q. We modify the original notation in (2.73), replacing reals ∂ l u(x), ∂ α u(x), ϑl , ϑα ∈ R, and l , ϑ α ∈ Rq , and n-vectors ∂u(x), ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn by vectors ∂ l u(x), ∂ α u(x) ϑ ∈ Rn×q , using just one vector sign to keep the corresponding n × q matrices ∂ u(x), ϑ consistency with, but indicate the diﬀerence to (2.73). Evaluated in x (ev. at x) we get u := (u1 , · · · , uq ), ∂ l u := (∂ l u1 , · · · , ∂ l uq ), ∂ α u := (∂ α u1 , · · · , ∂ α uq ) ∈ Rq , ev. at x, ∇ u = (∇u1 , · · · , ∇uq ) = (∂u1 , · · · , ∂uq ) = ∂ u = (∂ 1 u, · · · , ∂ n u) ∈ Rn×q , ev. at x, ∇k u := (∂ α u)|α|=k ev. in x :∈ Rnk ×q , ∇≤k u := (∂ α u)|α|≤k ∈ RNk ×q , ev. at x, (2.335) with ∇k u = ∂ l u = ∂ α u = u for k = l = |α| = 0 and 1 l n n l l q = ϑ , . . . , ϑ ∈ R , l = 1, . . . , n, with j = ϑ , . . . , ϑ ∈ R , j = 1, . . . , q, ϑ ϑ j j 1 q n ) := (ϑ 1, · · · , ϑ q ) ∈ Rn×q , and |ϑ| = |ϑ| nq ∈ R, := (ϑ 1, · · · , ϑ ϑ α )|α|=k ∈ Rnk ×q , Θ α )|α|≤k ∈ RNk ×q . α := ϑα , · · · , ϑα ∈ Rq , Θ k := (ϑ ≤k := (ϑ ϑ 1 q

119

2.6. Linear and nonlinear elliptic systems

We recall and extend the usual Sobolev spaces and norms for Ω in (2.5). Sobolev (Hilbert) spaces and their inner products, norms, and seminorms are deﬁned as W m,p (Ω, Rq ) := (W m,p (Ω))q , W0m,p (Ω, Rq ), H m (Ω, Rq ) := W m,2 (Ω, Rq ), ( u, v )m := ( u, v )H m (Ω,Rq ) :=

q

(∂ α ui , ∂ α vi )L2 (Ω) and

(2.336) (2.337)

i=1 |α|≤m

uW m,p (Ω) := uW m,p (Ω,Rq )

⎛ ⎞1/p q := ⎝ ∂ α ui pLp (Ω) ⎠ in particular i=1 |α|≤m

W m,p (Ω, Rq ) → H m (Ω, Rq ) densely and continuously embedded for 2 ≤ p. We return to the second order equation with a functional, f, in general form: (2.338) f := (f0 , · · · , fn ) ∈ Lp (Ω, Rq(n+1) ), fk (x) = fk1 , · · · , fkq (x), ∂ k v (x) ∈ Rq , f , v V ×V :=

n

(fk , ∂ k v )q dx, ∀f ∈ V := W −1,p (Ω, Rq ), 1/p + 1/p = 1,

Ω k=0

cf. (2.107), (2.339), (2.340), noting that the fk (x), . . . , are only deﬁned a.e. Remark 2.87. As a consequence of (2.337), (2.338), the following integrals are deﬁned for u, v ∈ W01,p (Ω, Rq ), f ∈ W0−1,p (Ω, Rq ), p ≥ 2. Hence the following arguments remain valid for this case as well, possibly with a H01 (Ω, Rq )-coercivity. The strong form L is deﬁned for u, u0 ∈ H 2 (Ω, Rq ) and with f instead of f0 : 19 ⎞⎤q ⎛ ⎡ q n

% i &q l 0 ⎠⎦ ⎝ f0 = f0 i=1 = Ls u0 = L u0 := ⎣ (−1)k>0 ∂ k aij (2.339) kl ∂ uj j=1

k,l=0

i=1

⎤q ⎡ q n 0⎦ = (−1)k>0 ∂ k (Akl ∂ l u0 ) = ⎣ Lij s uj k,l=0

j=1

q , with Akl (x) = aij kl

,

i,j=1

i=1

for Akl ∈ W 1−δk,0 ,∞ (Ω, Rq×q ) and |∂ k aij kl (x)| ≤ Λ0 . For Dirichlet boundary conditions we take the standard scalar product of (2.339) with v = (v1 , . . . , vq ) ∈ H01 (Ω, Rq ) and integrate by parts. Then the weak solution u0 = (u01 , . . . , u0q ) ∈ H01 (Ω, Rq ) for (2.339) is determined, cf. (2.274), by equating the 19 Trying to keep the notation as for quasilinear equations, we have changed the order of diﬀerentiation in the strong form (2.339) compared to (2.134).

120

2. Analysis for linear and nonlinear elliptic problems

following linear and bilinear forms, f , v , and a( u0 , v ): -q 3' n 4 n 0 1 q i k fk ∂ , v = (fk , ∂ k v )q dx (2.340) u ∈ V := H0 (Ω, R ) s.t. f , v = k=0

= a( u0 , v ) := (L u0 , v )V :=

n

Ω k=0

i=1

(Akl ∂ l u0 , ∂ k v )q dx∀ v ∈ V, with

Ω k,l=0

Akl ∈ L∞ (Ω, Rq×q ), |aij kl (x)| ≤ Λ0

(a.e.) ∀x ∈ Ω, fki ∈ L2 (Ω) ∀i, j, k, l, or

(2.341)

i=1,...,q

=⇒ f ∈ H −1 (Ω, Rq(n+1) ), with f H −1 (Ω,Rq(n+1) ) ≤ max {fki L2 (Ω) }. k=0,...,n

Note the change in notation for f ∈ H0−1 (Ω, Rq(n+1) ) here to fki ∈ L2 (Ω) above and in Chen and Wu. The principal part and principal symbol Lp of L are deﬁned as n principal part: (Lp u, v )V := (Akl ∂ l u, ∂ k v )q dx, and symbol: (2.342) Ω k,l=1

:= Lp (x, ϑ)

⎛

q

Lij kl (x, ϑ)

=⎝

i,j=1

⎞q

n

l k⎠ aij kl (x)ϑj ϑi

k,l=1

i,j=1

=

n

l , ϑ k )q . (Akl (x)ϑ

k,l=1

Accordingly, for the Stokes operator the matrix in (2.330) would have to be modiﬁed by replacing the last row and column by zeros. For our special linear system of divergent diﬀerential equations we further update the above ellipticity in Deﬁnition 2.85, see (2.328), towards the Legendre, sometimes called the strong Legendre condition, and the Legendre–Hadamard condition. Deﬁnition 2.88. Ls or L in (2.339) or (2.340) are called strongly elliptic and uniformly elliptic, respectively, if they satisfy the strong and uniform Legendre condition. That is, there exist 0 < λ and 0 < λ < Λ < ∞, respectively, independent of x (sometimes only 0 < λ < λ(x) < Λ(x), Λ(x)/λ(x) < C < ∞), such that a.e. ∀x ∈ Ω, cf. (2.342), 2 ≤ 2 = λ|ϑ| λ|ϑ| nq

q i,j=1

Lij p (x, ϑ)

=

n

l , ϑ k )q ≤ Λ|ϑ| 2. (Akl (x)ϑ

(2.343)

k,l=1

∞ So the Stokes operator violates (2.343). For our aij kl ∈ L (Ω), the inequality in (2.343) indeed only makes sense a.e. ∀ x ∈ Ω. We will discuss the relation of the diﬀerent types of ellipticity in Remark 2.90. For proving existence and uniqueness results for (2.340) we reinterpret the bilinear form a( u, v ) as the principal part, see (2.342), and additional lower order terms n (Akl ∂ l u, ∂ k v )q dx. (2.344) a( u, v ) = ap ( u, v ) + b( u, v ), with b( u, v ) := Ω k,l=0,k×l=0

2.6. Linear and nonlinear elliptic systems

121

We repeat the concepts of V-coercive and V-elliptic bounded bilinear forms, see (2.41) and (2.63). For V = H0m (Ω, Rq ), m ≥ 1, a(·, ·) is called V = H0m (Ω, Rq )-coercive, and V-elliptic, respectively, if constants α ∈ R+ , Cc ∈ R exist such that a( u, u) ≥ α u2V and ≥ α u2V − Cc u2W ∀ u ∈ V = H0m (Ω, Rq ), W = L2 (Ω). (2.345) Applying Theorems 2.21 and 2.16 we ﬁnd Theorem 2.89. V-coercivity and ellipticity: 1. Let (2.340)–(2.341) be a uniformly elliptic system, hence let (2.343) with 0 < λ ≤ Λ < ∞ be satisﬁed. Then f , v , ap ( u, v ), a( u, v ) in (2.340), (2.344) are bounded with respect to V, and ap ( u, v ) and a( u, v ) are V-coercive and V-elliptic, respectively. 2. All these results remain correct, if the above strong Legendre condition (2.343) is replaced by the weaker Legendre–Hadamard condition (2.348). 3. For the principal part, Lp , and its ap ( u, v ), the unique solution u0 of (2.346) solves the variational problem (2.61). 4. Finally, all H 1 -coercivity based existence, uniqueness and Fredholm results remain correct for the W 1,p case for 2 ≤ p ≤ ∞, however with respect to the H 1 norm. Under these conditions Corollary 2.44 applies. It implies for a V-coercive a(·, ·) the existence of a unique generalized or weak solution u0 and a boundedly invertible L, and for a V-elliptic a(·, ·) the Fredholm alternative with its consequences. Hence, for a boundedly invertible L, with the induced a( u, v ), ∀f ∈ V = H0−1 (Ω, Rq ) ∃1 u0 ∈ V : a( u0 , v ) = f , v ∀ v ∈ V.

(2.346)

Proof. For completeness we include this nearly obvious proof: by (2.341) the f , v , ap ( u, v ), a( u, v ) are bounded with respect to V. Furthermore (2.343) implies q n ap ( u, u) ≥ λ |∂ u|2 dx with |∂ u|2 = |∂ k ui |2 ∀ u ∈ V. (2.347) Ω

i=1 k=1

The uV and Ω |∂ u|2 dx are equivalent norms in V, so ap ( u, v ) is V-coercive. Since a( u, v ) − ap ( u, v ) are lower order terms, a combination of Theorem 1.26 and Proposition 2.24 yields the V-ellipticity of a( u, v ). Theorems 2.15 and 2.21 thus verify the above claims. The ﬁnal remark concerning the strong Legendre–Hadamard condition is a special case of Theorem 2.104 below. The W 1,p -coercivity is a consequence of the initial discussion of Remark 2.87. Remark 2.90. We list some properties, relating the diﬀerent concepts of elliptic systems (q ≥ 1) to elliptic equations (q = 1) in (2.327)–(2.343). 1. We start by showing that (2.343) for 2m = 2 implies the Taylor modiﬁcation of (2.327): In fact let (2.327) be violated for our special case. Then a nontrivial

122

2. Analysis for linear and nonlinear elliptic problems

solution y = (y1 , · · · , yq ) for the linear homogeneous system exists: q

Lij p (x, ϑ)yj = 0, y = (y1 , · · · , yq ) = 0, hence,

j=1 q n

l k αi aij kl (x)ϑj ϑi yj = 0 ∀ αi ∈ R, a.e. x ∈ Ω.

i,j=1 k,l=1

By an appropriate rescaling of the αi ϑlj ϑki yj into new ϑlj := ϑlj yj , ϑki := αi ϑki we obtain the desired contradiction to (2.343). 2. The linearized von K´ arm´ an and Navier–Stokes operators satisfy and violate (2.343), respectively. We will see in Section 2.8 that the Stokes operator (and all of its compact perturbations) is not coercive. So the results of this and the next subsection are not applicable. Hence the Navier–Stokes operator and the related saddle point problems require a separate treatment. 3. For q = 1 the coercivity in (2.347) for the principal part and the two preceding ellipticity conditions (2.328) and (2.343) are essentially equivalent. This is no longer correct for q > 1. Chen and Wu [170], see Chapter 8, Theorems 1.1, and 1.2, show (compare the Taylor form of 2m-ellipticity in Deﬁnition 2.85, 6.):

under the condition (2.341) for the Akl = aij kl

q

the strong Legendre condii,j=1

tion implies the coercivity in (2.345) for the principal part. 4. Conversely, the coercivity of the principal part and (2.341) imply the so-called strong Legendre–Hadamard condition: with Lp (x, ϑ) in (2.326) ∃λ > 0 : ∀x ∈ Ω, ϑ ∈ Rn , η ∈ Rq , η T Lp (x, ϑ)η =

q n

l k 2 2 aij kl (x)ϑ ϑ ηj ηi ≥ λ|ϑ| |η| .

(2.348)

i,j=1 k,l=1

So instead of the above equivalence for q = 1 we ﬁnd for q > 1 the following implication F or q > 1 assume (2.341) for Lp . Then: (2.343) =⇒ (2.347) =⇒ (2.348)(2.349) q ∞ or verbatim, for Akl = (aij kl ∈ L (Ω))i,j=1 , the strong Legendre condition =⇒

coercivity for ap (·, ·) =⇒ strong Legendre–Hadamard condition for Akl . ¯ By the proof of Theorem 2.104 we get for aij kl ∈ C(Ω), k, l = 1, . . . , n, even the equivalence (2.347) ⇐⇒ (2.348). For our numerical applications again regularity results u0 ∈ H s (Ω, Rq ), s ≥ 1, for the general equation (2.340) are important. Similarly to the preceding sections, diﬀerent regularity results are known. In the remainder of this subsection we combine Chen and Wu [170], and the proof of Theorems 2.56 and 2.60. For linear L, Chen and Wu prove their results only for the solution u0 of the equation (2.350) deﬁned by the principal

2.6. Linear and nonlinear elliptic systems

123

part Lp and with the general f in (2.340) ap ( u0 , v ) = f , v ∀ v ∈ H01 (Ω, Rq ).

(2.350)

Then the proof of Theorem 2.56 allows us the extension to the general problem (2.340). We summarize the results, obtained by this combination with Chen and Wu [170]: Theorem 2.91 1. and 2. correspond to [170], Chapter 8, Theorems 2.3, 2.6, 2.7, and to Corollary 2.4. Theorem 2.91 1. yields u0 ∈ H s+2 (Ω, Rq ), s ≥ −1, Theorem 2.91 3. generalizes [170], Chapter 9, Theorems 2.7, 2.8. The resulting u0 ∈ C s,δ (Ω, Rq ), s ≥ 1 is needed in Subsection 2.6.4. ¯ Rn×q ): Theorem 2.91. Regular solution u0 ∈ H s+2 (Ω, Rn×q ) and ∂ u0 ∈ C s,δ (Ω, 1. Let a(·, ·) be strongly elliptic, see (2.343), and f of the form (2.340). For s ∈ Z, s ≥ −1, let s+1,∞ ∂Ω ∈ C s+2 , aij (Ω), fki ∈ H s+1 (Ω)∀i, j, k, l, kl ∈ W

(2.351)

equivalent to f ∈ H s+1 (Ω, Rn ). Then any weak solution u0 ∈ H01 (Ω, Rq ) of (2.340) satisﬁes u0 ∈ H s+2 (Ω, Rq ) and (2.352) u0 H s+2 (Ω,Rq ) ≤ C u0 H 1 (Ω,Rq ) + f H s+1 (Ω,Rq ) . 2. We even get u0 ∈ C0∞ (Ω, Rq ) if (2.351) is replaced by i ∞ ∞ aij kl , fk ∈ C (Ω) ∀i, j, k, l, ∂Ω ∈ C .

(2.353)

3. For a strongly elliptic a(·, ·) and the Dirichlet problem let i s,δ ¯ (Ω), ∂Ω ∈ C s+1,δ , 0 < δ < 1, 0 ≤ s, ∀i, j, k, l. aij kl , fk ∈ C

(2.354)

¯ Rn×q ). Then for any weak solution u0 ∈ H01 (Ω, Rq ) of (2.340) the ∂ u0 ∈ C s,δ (Ω, 2+k q 1 (Ω, R ) ∩ H0 (Ω, Rq ) → Finally, the operator L, induced by a(., .), with L : H k q H (Ω, R ) in (2.340) is Fredholm. Proof. We only prove Theorem 2.91, 1.; for 3. the proof is analogous. For s = 0 and ap (·, ·) the (2.352) is Chen and Wu’s Theorem 2.6, however with u0 H 1 (Ω,Rq ) replaced by u0 L2 (Ω ,Rq ) . For s > 0 and u0 ∈ H s+2 (Ω, Rq ) it is their Theorem 2.7 with (2.352) missing, for s = ∞ their Corollary 2.4. Applying the arguments in the modiﬁcation of the proof of Theorem 2.56 now for Theorem 2.60 veriﬁes these results for the general a(·, ·) in (2.340). In fact, (2.339) shows that the aij kl are derived at most once, hence any u ∈ H s+2 (Ω, Rq ) yields L u ∈ H s (Ω, Rq ) and thus (2.352). Since Hackbusch’s [387] proof for Remark 9.1.1 remains valid, we can improve Theorem 2.91 1. as Theorem 2.92. Let (2.340) with f ∈ V have a unique solution u0 ∈ V, let a(·, ·) induce L ∈ L(V, V ), and let the conditions of Theorem 2.91 be satisﬁed. Then L−1 ∈ L(H s (Ω, Rq ), H s+2 (Ω, Rq ) ∩ V) for s ∈ Z, s ≥ −1, and the following estimates are equivalent for f ∈ V−s := V ∩ H s (Ω, Rq ) : L−1 H s+2 (Ω,Rq )∩V←H s (Ω,Rq ) ≤ C ⇐⇒ u0 H s+2 (Ω,Rn ) ≤ Cf H s (Ω,Rn ) .

(2.355)

124

2. Analysis for linear and nonlinear elliptic problems

We ﬁnish with some Schauder type results. Again we generalize Chen and Wu’s results for the principal part to the full operator. They distinguish regularity in the interior and the closure of Ω. Theorem 2.93 1. combines the results [170], Chapter 9, Theorems 2.5, 2.6, Theorem 2.93 2. with [170], Chapter 10, Theorems 2.1, 2.2. ¯ 1 , Rn×q ) and ∈ Lp (Ω1 , Rn×q ): Theorem 2.93. Interior regularity for ∂ u0 ∈ C 0,δ (Ω 1. Assume ap (·, ·) is strongly elliptic, see (2.343), and ij k 0,δ , f ∈ C (Ω), 0 < δ < 1, a (x) ∂Ω ∈ C 1,δ , aij ≤ Λ ∀x ∈ Ω, ∀i, j, k, l. i kl kl

(2.356)

Then for any weak solution u0 ∈ H01 (Ω, Rq ) of the original equation (2.340) its derivative is ∂ u0 ∈ C 0,δ (Ω, Rn×q ). For any Ω1 ⊂⊂ Ω it is estimated as ∂ u0 C 0,δ (Ω¯1 ,Rn×q ) ≤ C ∂ u0 L2 (Ω ,Rn×q ) + f C 0,δ (Ω¯ ,Rn×q ) , (2.357) where Ω := {x ∈ Ω : dist(x, ∂Ω) > dist(Ω1 , ∂Ω)/2} and C depends on n, q, λ, Λ, δ, dist(Ω1 , ∂Ω), aij , and diam Ω. kl C 0,δ (Ω )

2. Assume ap (·, ·) is ¯ =⇒ aij ∈ C 0 (Ω), kl

ij i 0 strongly elliptic and replace in (2.351) akl , fk ∈ C (Ω) by ij i p akl (x) ≤ Λ ∀x ∈ Ω, fk ∈ L (Ω), p ≥ 2. Then for any weak

solution u0 ∈ H01 (Ω, Rq ) of (2.340) its derivative is ∂ u0 ∈ Lp (Ω, Rn×q ). For any Ω1 ⊂⊂ Ω it can be estimated as ∂ u0 Lp (Ω1 ,Rn×q ) ≤ C u0 H 1 (Ω,Rq ) + f Lp (Ω,Rn×q ) , % & with C = C n, q, p, λ, Λ, dist(Ω1 , ∂Ω), modulus of continuity of the aij kl . Similarly to Theorem 2.47 we obtain regularity results for convex domains, however only for systems of order 2 and only in Rn with n = 2, 3. For a(·, ·) and its coeﬃcients we impose the conditions, cf. Rossmann [558], and Kozlov, Maz’ya and Rossmann [452], Sections 8.6 and 11.4: Let a(·, ·) and its coeﬃcient matrices Akl ∈ Rq×q in (2.344) satisfy Akl =

A∗lk , Akl

∈W

1,∞

(2.358)

(Ω), Akl ∈ C(Uv ) ∀1 ≤ k, l ≤ n, and chosen

neighborhoods Uv ⊂ Rn ∀ vertices v ∈ ∂Ω, ﬁnally f ∈ W 1,∞ (Ω, Cq ), for f (j) (x) ∈ Cq , j ≤ n, let

n k,l=1

(Akl f (l) , f (k) )q ≥ c0

n

|f (j) |2q .

j=1

This last inequality obviously implies the coercivity of the principal part. Theorem 2.94. Regular solutions for convex Ω: Let Ω ⊂ Rn , in (2.5) be convex, n = 2 or n = 3, and Ω ∈ C 2 , except in the vertices v ∈ ∂Ω, and a(·, ·) in (2.344) and its coeﬃcients satisfy (2.358). Then the exact solution is u0 ∈ H 2 (Ω, Rq ). Proof. This result is not an immediate consequence of results in Subsections 8.6 and ˜ in 11.4 of [452]. So we indicate the missing arguments, by locally modifying Ω into Ω,

2.6. Linear and nonlinear elliptic systems

125

the neighborhoods, Uv , of the vertices, v. Similarly to [452], the system and coeﬃcients ˜ are modiﬁed such that Theorem 2.91 1. yields for s = 1 the regularity u0 ∈ H 2 (Ω). For the local analysis, the condition Akl ∈ C(Ue ) allows us the technique of frozen coeﬃcients near the vertex, v. Then the local H 2 (Ω)-regularity is correct, if the frozen system does not admit a singularity function rλ U (ϕ) with 0 < Reλ ≤ 1 and −1/2 < Reλ ≤ 1/2 for two and three dimensional problems. In [452], Sections 8.6 and 11.4, it is shown that 1 < Reλ and 1/2 < Reλ if the angle between edges is < π. That is exactly the situation in vertices of convex domains under the conditions of our theorem. Regularity estimates of a diﬀerent type, not used in our numerical context, are due to Morrey [500], Chen and Wu [170], pp. 137 ﬀ., and Koshelev [449], pp. 108 ﬀ. For example, for 1 ≤ p, 0 ≤ μ and Ω(x, ρ) := Ω ∩ B(x, ρ) we deﬁne a norm as ' -1/p uLp,μ (Ω) :=

sup x∈Ω,0<ρ
ρ−μ

| u(z)|p dz

.

(2.359)

Ω(x,ρ)

Then the space Lp,μ (Ω) := { u ∈ Lp (Ω) : uLp,μ (Ω) < ∞} is a normed linear space, the Morrey space, with respect to the norm uLp,μ (Ω) . Theorem 2.95. Regularity in Morrey spaces: Assume the aij kl in (2.342) satisfy (2.343). If 0 ¯ i 2,μ (Ω) ∀i, j, k, l, aij kl (x) ∈ C (Ω), fk ∈ L

(2.360)

(hence (2.341) is valid!), then for any weak solution u ∈ of (2.350) its derivative is in ∂ u0 ∈ L2,μ (Ω, Rn×q ). For Ω1 ⊂⊂ Ω it can be estimated as ∂ u0 L2,μ (Ω1 ,Rn×q ) ≤ C ∂ u0 L2 (Ω,Rn×q ) + f L2,μ (Ω,Rn×q ) , with C = C n, q, λ, μ, Λ, dist(Ω1 , ∂Ω), and the modulus of continuity of the aij kl . 0

2.6.4

H01 (Ω, Rq )

Quasilinear elliptic systems of order 2 and variational methods

We generalize some of the results from the last Subsection 2.6.3 to quasilinear systems and use the notation in (2.335). Our presentation again follows Chen and Wu [170], Chapter 11, and ﬁnally indicates some results from Leung [473] and Koshelev [449]. We consider the following divergent quasilinear system in strong form: ' n -q (−1)k>0 ∂ k aik (x, u0 (x), ∂ u0 (x)) (2.361) Gs u0 (x) := G u0 (x) = k=0

=

n

(−1)k>0 ∂ k ( ak (x, u0 (x), ∂ u0 (x))) = 0,

k=0

% &q with u0 , u ∈ H 2 (Ω, Rq ), ak = aik i=1 ,

i=1

126

2. Analysis for linear and nonlinear elliptic problems

and aik : Ω × Rq × Rn×q w = (x, z, p) → aik (x, z, p) ∈ R. Often this divergent quasilinear system characterizes a critical point of a variational problem as in (2.366). The corresponding strong form of the nondivergent quasilinear system, (2.361), originates, if (2.366) can be diﬀerentiated, see (2.369). In this sense, Taylor [620], pp. 198 ﬀ., presents his similar results on surfaces, and for smooth enough coeﬃcients. The standard multiplication of (2.361) with v ∈ V := H01 (Ω, Rq ), the scalar product (·, ·)q in Rq , and integration by parts yields the corresponding weak system, n q aik (x, u0 (x), ∂ u0 (x)), ∂ k vi (x) dx (2.362) u0 ∈ V : G u0 , v V ×V := Ω i=1

=

n

k=0

( ak (x, u0 (x), ∂ u0 (x)), ∂ k v (x))q dx =: a( u0 , v ) = 0 ∀ v ∈ V = H01 (Ω, Rq ).

Ω k=0

The most important examples for (2.361) or (2.362) are the so-called Euler systems of a corresponding variational problem: We start with a function F , a functional J( u), an admissible set A and formulate the following conditions essentially with respect to F (note that the following Fp etc. indicates the partial with respect to p etc.) F : Ω × Rq × Rn×q w = (x, z, p) → F (x, z, p) ∈ R F (x, u(x), ∂ u(x))dx, J : H 1 (Ω, Rq ) → R J( u) :=

(2.363) (2.364)

Ω

with boundary conditions in the form g ∈ H 1 (Ω, Rq ) s.t. J( g ) < ∞ e.g. choose A := { u ∈ H 1 (Ω, Rq ) : u − g ∈ V = H01 (Ω, Rq )} and

(2.365)

determine u ∈ A s.t. J( u ) ≤ J( v )

(2.366)

0

0

∀ v ∈ A.

If F is convex in p, hence Fp (x, z, pˆ)(p − pˆ) ≤ F (x, z, p) − F (x, z, pˆ), then (2.366) and J in (2.364) are called a regular variational problem and a regular integral functional. Obviously the admissible set A in (2.365) is a closed convex subset of H 1 (Ω, Rq ). The uniqueness of the solution, u0 , is an immediate consequence of the arguments on top of page 166 in [170]. Theorem 2.96. Variational problems, [170], Chapter 11, Theorem 2.1: 1. Let F, Fp be continuous in Ω × Rq × Rn×q , F convex in p, and F (x, z, p) ≥ λ|p|2 with λ > 0. Then the regular variational problem (2.366) has a solution u0 ∈ H 1 (Ω, Rq ). 2. If F is strictly convex in p, hence Fp (x, z, pˆ)(p − pˆ) < F (x, z, p) − F (x, z, pˆ) for p = pˆ, then u0 is unique. So Chen and Wu [170] have shown the existence and uniqueness of a solution u0 ∈ H 1 (Ω, Rq ) for the regular variational problem (2.366). In their Chapter 11, 2.2

2.6. Linear and nonlinear elliptic systems

127

they elaborate the relation between this variational problem and quasilinear equations. For the following considerations we need the derivatives of J( u). We start with formal derivatives and present conditions in Theorem 2.97, justifying these derivatives. We introduce, similarly to (2.336), the notation for the components of z ∈ Rq , p ∈ Rn×q , without completely indicating their vector structure, as u(x) = (u1 , . . . , uq )(x), z = (z1 , . . . , zq ) ∈ Rq , and corresponding p = (p1 , . . . , pq ) = (p1 , . . . , pn ), ∂ u(x) = (∂u1 , . . . , ∂uq )(x) ∈ Rn×q , where pi = p1i , . . . , pni , ∂ui (x) = ∂ 1 ui , . . . , ∂ n ui (x), x ∈ Rn , pki ∈ R, pk = pk1 , . . . , pkq , ∂ k u(x) = ∂ k u1 , . . . , ∂ k uq (x) ∈ Rq , and for w := (x, z, p) ∈ Ω × Rq × Rn×q , i = 1, . . . , q, k = (0), 1, . . . , n, Fz (w) v =

q

Fzi (w)vi , Fp (w)∂ v =

i=1

q n

Fpki (w)∂ k vi ,

(2.367)

i=1 k=1

F (w) v = Fz (w) v + Fp (w)∂ v , note that for w(x) = (x, u(x), ∂ u(x)), [∂Fp (w(x))] v (x) = [∂Fp (x, u(x), ∂ u(x))] v (x) =

q n

[∂ k Fpki (w(x))]vi (x)

i=1 k=1

=

q

n

5

Fpki plj (w(x))∂ k ∂ l uj (x)

i,j=1 k,l=1

6 +Fpki zj (w(x))∂ k uj (x) + Fpki xj (w(x)) vi (x).

Chen and Wu [170] derive two necessary conditions by the standard and a modiﬁed trick in the classical calculus of variations: Firstly, under appropriate growth condition, a solution u0 ∈ A ∩ C 1 (Ω, Rq ) of (2.366) satisﬁes the weak and strong form of the Euler system for J, see (2.368), (2.369) and compare Theorem 2.73. We formulate them componentwise and in a short form. With (2.367), (2.361) or (2.362) and ak : Ω × Rq × Rn×q → Rq , we get

q q , for k > 0, and a0 (w) = ai0 (w) := Fzi (w) i=1 . ak (w) = aik (w) := Fpki (w) i=1

With w0 (x) = (x, u0 (x), ∂ u0 (x)), we obtain the weak form a( u0 , v ) : = (J ( u0 ), v )H01 (Ω,Rq ) = Fp (w0 (x))∂ v + Fz (w0 (x)) v (x) dx = Ω

Ω

(F (w0 ) v )(x)dx

(2.368)

128

2. Analysis for linear and nonlinear elliptic problems

=

n

( ak (w0 (x)), ∂ k v (x))q dx = 0 ∀ v ∈ V = H01 (Ω, Rq ) u0 .

Ω k=0

By standard partial integration (2.368) is transformed into H01 (Ω, Rq ). This implies the system in its strong form as (· · · ) = −∂Fp (w0 (x)) + Fz (w0 (x)) =

n

Ω

(· · · ) v (x)dx = 0 ∀ v ∈

(−1)k>0 (∂ k ( ak (w0 (x))

k=0

⎡ = ⎣Fzi (w0 (x)) −

q n

Fpki plj (w0 (x))∂ k ∂ l uj

j=1 k,l=1

⎤q

+Fpki zj (w0 (x))∂ k uj + Fpki xj (w0 (x)) ⎦

= 0.

(2.369)

i=1

So the quasilinear equations (2.361), (2.369) and (2.362), (2.368) represent the strong and weak Euler system for J for the solution u0 . These Fpki (w0 (x)), and their gradients ∂ k (Fpki )(w0 (x))), k > 0, . . . , are Nemyckii operators of a type similar to those studied in Subsections 2.5.5 and 2.5.6. We will impose appropriate conditions to have them well deﬁned. Secondly, for F ∈ C 2 , a solution u0 of (2.366) necessarily satisﬁes, cf. [170], p. 169, the condition for local minima, the so-called Legendre–Hadamard condition for the corresponding principal part, see (2.348). q n

Fpki plj (x, u0 (x), ∂ u0 (x))ϑk ϑl ζ i ζ j ≥ 0 ∀ ϑ ∈ Rn , ζ ∈ Rq .

(2.370)

i,j=1 k,l=1

For our later linearization we assume, stronger than (2.370), cf. (2.349), that (2.368), (2.369) satisﬁes for u0 the so-called strong Legendre condition, compare (2.342), (2.343), so F is strictly convex in w(x), q n

2 ∀ϑ ∈ Rn×q . Fpki plj (w0 (x))ϑki ϑlj ≥ λ|ϑ| nq

(2.371)

i,j=1 k,l=1

This implies, for Fpki plj (w0 (x)) ∈ L∞ (Ω, Rq×q ) by Theorem 2.89, the coercivity of the linearized operator. Similarly to quasilinear equations, Chen and Wu [170], pp. 163 ﬀ., guarantee solutions by imposing Carath´eodory type and controllable and natural growth conditions for F, measurable in x, cf. [170], pp. 171 ﬀ. F : Ω × Rq × Rn×q → R measurable in x, let F ∈ C 1 with respect to( z, p) a.e.x ∈ Ω.

(2.372)

2.6. Linear and nonlinear elliptic systems

129

Controllable growth conditions: With constants λ, Λ, C ∈ R+ and functions 20 gi ≥ 0, gi ∈ L1 (Ω), fki , fi ≥ 0, fki ∈ L2 (Ω), fi ∈ Lr/(r−1) (Ω), deﬁne V := (1 + | z|2 + |p|2 ). We assume for F, Fxk , Fpki :

(2.373)

λV − g1 (x) ≤ F (x, z, p) ≤ ΛV + g2 (x), mind our V is V 2 in [170], Fx (x, z, p) ≤ C(|p|2(1−1/r) + | z|r−1 + fk (x)), k = 1, . . . , n, k

(2.374) Fpki (x, z, p) ≤ C |p| + | z|r/2 + fki (x) , i = 1, . . . , q. Natural growth conditions: For | z| ≤ M, and constants λ, μ, Λ = Λ(M ) ∈ R+ let λ|p|2 − μ ≤ F (x, z, p) ≤ Λ(1 + |p|2 ) |Fxk (x, z, p)| ≤ Λ(1 + |p|2 ), Fpki (x, z, p) ≤ Λ(1 + |p|).

(2.375) (2.376)

Theorem 2.97. J ( u)v is a weak Euler system, Chen and Wu, [170], Chapter 11, Theorems 2.3, 2.4.: Let (2.372)–(2.374) or (2.372), (2.375)–(2.376) be satisﬁed for u ∈ H 1 (Ω, Rq ) ∩ L∞ (Ω, Rq ). Then J( u) is diﬀerentiable in u and J ( u)v, a quasilinear operator, has the form of the weak Euler system (2.368). We call u0 ∈ A ⊂ L∞ (Ω, Rq ), cf. (2.365), a generalized solution of (2.362) if for controllable growth conditions, (2.368) is satisﬁed for u0 and for all v ∈ A. For natural growth conditions, we have to test (2.368) with all v ∈ L∞ (Ω, Rq ). It is well known that F is strictly convex in p iﬀ the strong Legendre condition (2.371) is satisﬁed. Theorem 2.98. Generalized solution of (2.362), [170], Chapter 11, Summary: Under the conditions of Theorems 2.96 and 2.97 and for F convex in p, there exists, and for F strictly convex there uniquely exists, a generalized solution u0 of (2.362), characterized as a weak solution of an elliptic quasilinear equation. Now, we formulate regularity results, see Chen and Wu [170], Chapter 12, pp. 173 ﬀ., directly for the quasilinear systems in (2.361) or (2.362). Accordingly we introduce modiﬁcations of the above strong Legendre and growth conditions, obviously stronger than (2.373), (2.374), and (2.375), (2.376), as

q n q , b := ai0 , e.g. ai0 = Fzi i=1 , A, b ∈ C 1 , (2.377) A := aik , e.g. aik = Fpki i=1,k=1

∃λ, Λ > 0 :

q

n

i,j=1 k,l=1

aik

plj

2 ∀ϑ ∈ Rn×q , with w = (x, z, p),(2.378) (w)ϑki ϑlj ≥ λ|ϑ|

|A(w)|, |Ax (w)|, | b(w)|, | bx (w)| ≤ Λ(1 + | z|2 + |p|2 )1/2 ,

(2.379)

|Az (w)|, |Ap (w)|, | bz (w)|, | bp (w)| ≤ Λ for the controllable growth conditions, and 20

r is the so-called Sobolev conjugate of 2, r = 2n/(n − 2) for n > 2, else r ∈ [2, ∞).

130

2. Analysis for linear and nonlinear elliptic problems

|A(w)|, |Ax (w)|, |Az (w)|, | bp (w)| ≤ Λ(1 + |p|), |Ap (w)| ≤ Λ,

(2.380)

| b(w)|, | bx (w)|, | bz (w)| ≤ Λ(1 + |p|2 )∀| z| ≤ M, for the natural growth conditions. The constants λ, Λ may depend upon M for natural growth conditions. We have to slightly modify the concept of generalized solutions: For the two cases of controllable and natural growth conditions (2.379) and (2.380), we test a possible generalized solution for (2.362), u0 ∈ H 1 (Ω, Rq ) and u0 ∈ H 1 (Ω, Rq ) ∩ C 0 (Ω, Rq ), by ∀ v ∈ V := V1 := H01 (Ω, Rq ) and v ∈ V := V1 ∩ L∞ (Ω, Rq ), respectively. Independently of boundary conditions we get: Theorem 2.99. Regularity of solutions, [170], pp 173 ﬀ., Theorems 1.1, 1.2: 1. Let the quasilinear equation (2.362) have smooth coeﬃcients, A, b ∈ C 1 , and satisfy the strong (linearized) Legendre condition for u0 , see (2.377), (2.378), and the controllable or natural growth condition (2.379) or (2.380). Then a generalized solution u0 ∈ H 1 (Ω, Rq ) is in H 2 (Ω, Rq ). 2. The derivatives ∂ u0 , = 1, . . . , n satisfy some kind of linearized form of (2.362):

q n 5

Ω i,j=1 k,l=1

aik

plj

w0 (x)∂ j ∂ u0l + aik z w0 (x) ∂ u0l + aik x w0 (x) l

6 −δi, a0k w0 (x) ∂ i vk dx = 0∀ v ∈ V with w0 (x) := (x, u0 (x), ∂ u0 (x)).

(2.381)

For the two cases of controllable and natural growth conditions the chosen V are dense in H01 (Ω, Rq ). Hence, the system (2.381) is, by (2.378), a linear elliptic system for each of the ∂ u0 , = 1, . . . , n. So we can achieve higher regularity by induction, cf. Theorems 2.38, 2.39, 2.45. We only discuss Theorem 2.91 1. Applications of 2. are only indicated. Theorem 2.100. Regular solution of (2.362) with ∂ u0 ∈ H s+2 (Ω, Rq ): 1. Under the conditions of Theorem 2.99 the relation (2.381) is strongly elliptic. We assume for a w1 (x) := (x, u1 (x), ∂ u1 (x)) with ∂ u1 (x)) ∈ H s+1 (Ω, Rn×q ) that, cf. (2.351), (2.382) ∂Ω ∈ C s+2 , aik pl (w1 (x)), aik z (w1 (x)) ∈ W s+1,∞ (Ω), ⎛ f := ⎝

j

j

⎞q, aik x (w1 (x)) − δk, a0i (w1 (x)) ∈ H s+1 (Ω)⎠

n

n

k,l=1

.

i=1,k=1

2. Then any weak solution u ∈ H (Ω, R ) and u ∈ H (Ω, R ) ∩ C 0 (Ω, Rq ) of (2.362) satisﬁes ∂ u0 ∈ H s+2 (Ω, Rn×q ), = 1, . . . , n, and 0 ∂ u s+2 (2.383) ≤ C ∂ u0 L2 (Ω ,Rn ) + f H s+1 (Ω,Rn ) . H (Ω,Rn ) 0

1

q

0

1

q

2.6. Linear and nonlinear elliptic systems

131

¯ Rn×q ), 3. Theorem 2.91 2., 3. implies u0 ∈ C0∞ (Ω, Rn×q ) and ∂ u0 ∈ C s,δ (Ω, = 1, . . . , n, if (2.382) is replaced by the C-conditions in (2.353) and (2.354), respectively Speciﬁc systems allow u0 ∈ C 0,δ (Ω1 , Rq ) for Ω1 = Ω, or Ω1 ⊂⊂ Ω, [170], pp. 178, 181 ﬀ. is extended to the general form by our Theorem 2.91. Indirect methods, reverse estimates for ∂ u0 and singular sets, Ω \ Ω1 , are studied. ¯ Rq ) for the solution of a Leung [473], Theorem 3.2-1, shows u0 ∈ C 2 (Ω, Rq ) ∩ C 1 (Ω, special case of our quasilinear systems, the semilinear systems of the form Δ u + f (x, u) = 0 ∀x ∈ Ω, ∂ u/∂ν + g(x, u) = b(x) ∀x ∈ ∂Ω. Koshelev and Chelkak [449,450] present many diﬀerent regularity results for elliptic and parabolic equations. For example, for special cases of the quasilinear equation (2.362), H¨ older continuity for the derivatives of solutions in weighted spaces, see Theorem 3.2.1, is proved. A combination of Liouville’s theorem, generalized to elliptic problems, with the method of elastic solutions describes appearing cracks. For the Navier–Stokes problem [450] estimates smoother norms for time dependent solutions. We give one example of [450] results. Again, we consider the above quasilinear boundary value problem (2.361), (2.365), see (2.16). In [449] he assumes a modiﬁed Carath´eodory condition (2.284) for all terms, so all aki (x, z, p), i = 1, . . . , q, k = 0, . . . , n, are measurable with respect to x for all ﬁnite ( z, p) ∈ Rn+nq and continuously differentiable with respect to ( z, p) a.e. for x ∈ Ω. As growth conditions he requires a sharpened strong Legendre condition for the linearized operator, compare (2.378). For ∈ Rn×q , and with w = (x, z, p), let all ϑ 2 λ(1 + |( z, p)|2 )(s−1)/2 |ϑ| ≤

q

n

(2.384)

2, ∂aik /∂pjl (w)ϑki ϑlj ≤ Λ(1 + |( z, p)|2 )(s−1)/2 |ϑ|

i,j=1 k,l=0

for 1 < s ≤ 2, λ, Λ = const . > 0 and

j ∂aik /∂pl (w) ≤ Λ(1 + |( z, p)|2 )(s−1)/2 ∀ i, j, k, l, and for some

(2.385)

1 < p ∀ u ∈ W 1,p (Ω ) and Ω ⊂ Ω let aik (x, u(x), ∂ u(x)) ∈ Lp /(s−1) (Ω ). ¯ and for 0 < γ < 1 let α := 2 − n − 2γ and K satisfy Choose g ∈ Hα1 (Ω) (0 (−1 0 α(α + n − 2) α(n − 2) α(α + n − 2) 2 > 0, K 1 − 1− < 1, 1− 2(m − 1) m−1 2(m − 1) . 1 ¯ Hα (Ω) := u ∈ H 1 (Ω) : uHα1 (Ω) := sup

¯ x0 ∈Ω

⎛ ⎝| u(x)| + 2

Ω

|α|=1

⎞ α⎠

|∂ u(x)| |x − x0 | α

2

(2.386) (2.387) /

dx < ∞ .

132

2. Analysis for linear and nonlinear elliptic problems

¯ Under the conditions (2.384)– Theorem 2.101. Solution of (2.362) in Hα1 (Ω): (2.386), the nonhomogeneous Dirichlet problem (2.362), (2.365) has a unique solution ¯ and u0 ∈ C (2−m−α)/2 (Ω). Stronger results are possible for weighted norms u0 ∈ Hα1 (Ω) ¯ setting. in Sobolev spaces, extending the Hα1 (Ω) 2.6.5

Linear elliptic systems of order 2m, m ≥ 1

We essentially study weak forms of systems of diﬀerential operators of order 2m. We start with coercivity and the related existence and Fredholm results. Again the following results are valid for W m,p (Ω, Rn ), p ≥ 2, in the sense of Corolary 2.44. But we essentially restrict the discussion here to the Hilbert space setting and extend it to Banach spaces in the next subsection. For the necessary higher order Fourier transforms we need Hilbert spaces anyway. For the case of second order systems, Theorem 2.89 shows that the coercivity is a consequence of the strong Legendre condition (2.343). This implies the strong Legendre–Hadamard condition, see (2.349). This simple situation does not seem to remain correct for higher order 2m. In fact, the strong Legendre–Hadamard condition is needed to prove the coercivity of the principal part. We combine V = H0m (Ω, Rq ) ⊂ W = L2 (Ω, Rq ) = W ⊂ V = H0−m (Ω, Rq ),

(2.388)

with the scalar products and norms in (2.337) and L u in divergence form. We recall and update the notation, cf. (2.73), (2.79), (2.102), (2.336), (2.335). in (2.335) the η, ϑ are here more appropriate: Instead of ϑ u(x) = (u1 , . . . , uq )(x), η = (η1 , . . . , ηq ) ∈ Rq , ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn , ϑα = (ϑ1 )α1 . . . (ϑn )αn ∈ R, ϑα = 1 for |α| = 0, ∂ α ui =

∂ |α| 1 αn ∂xα 1 . . . ∂xn q α

(2.389) ui ,

∂ α u = (∂ α u1 , . . . , ∂ α uq ), with ∂ α ui (x) ∈ R, ∂ α u(x) ∈ R , ∂ u = u for |α| = 0. Similarly to our Lemma 2.26 above, see (2.82), and the beginning of Subsection 2.6.3, generalizations of Dirichlet boundary conditions to Dirichlet system are possible. Several cases are studied by Taylor [620], Chapter 5, Section 11. For avoiding the many technicalities we have restricted the discussion to homogeneous Dirichlet boundary conditions and choose V = H0m (Ω, Rq ) instead of a possible H0m (Ω1 , Rq ), Ω1 ⊂ Ω. For smooth enough Ω we get the equivalence ∂ i u(x) = 0 ∀0 ≤ j ≤ m − 1 ⇔ ∂ α u(x) = 0 ∀|α| ≤ m − 1 ∀x ∈ ∂Ω, (2.390) ∂ν i in the trace sense; the usual weak form below and the strong form are well deﬁned (−1)|α| ∂ α (Aαβ ∂ β u) = f (2.391) L : H 2m ∩ V → W = L2 (Ω, Rq ), L u := |α|,|β|≤m

for smooth enough Aαβ . For the weak form we take the standard scalar product with v = (v1 , . . . , vq ) ∈ V and integrate by parts. A weak solution u0 = u01 , . . . , u0q ∈ V for

133

2.6. Linear and nonlinear elliptic systems

(2.391) is deﬁned by equating the corresponding linear and bilinear forms, f , v and a( u0 , v ), compare (2.146), (2.274), (2.275), ⎞q 3⎛ 4 0 m q i α u ∈ V = H0 (Ω, R ) : f , v = ⎝ fα ∂ ⎠ , v = (f α , ∂ α v )q dx |α|≤m

= a( u0 , v ) := L u0 , v V ×V :=

=

q

Ω

i,j=1 |α|,|β|≤m

Ω |α|≤m

i=1

(Aαβ ∂ β u0 , ∂ α v )q dx

(2.392)

Ω |α|,|β|≤m

β 0 α ∞ aij v ∈ V, Aαβ (x) ∈ Rq×q , aij αβ ∂ uj ∂ vi dx, ∀ αβ ∈ L (Ω),

q f = fαi i=1,|α|≤m = (f α )|α|≤m , fαi ∈ L2 (Ω)∀i, j, α, β, or f ∈ V .

(2.393)

We generalize the above strong Legendre–Hadamard condition for m = 1, cf. (2.348), to order 2m and will combine it with variable coeﬃcients aij αβ , see Theorem 2.43. Deﬁnition 2.102. A system of linear diﬀerential operators of order 2m is called elliptic, if its principal part, Lp , satisﬁes the strong Legendre–Hadamard condition Lp u v dx := (Aαβ ∂ β u, ∂ α v )q dx, and for (2.394) ap ( u, v ) := Ω

⎛

Ω |α|=|β|=m

⎞q

Lp (x, ϑ) := ⎝

β α⎠ aij αβ (x)ϑ ϑ

|α|=|β|=m T

η¯ Lp (x, ϑ)η =

q

=⎝

i,j=1

⎛

⎞

Aαβ (x)ϑβ ϑα ⎠ let ∃λ > 0 :

|α|=|β|=m

β α aij ¯i ≥ λ|ϑ|2m |η|2 ∀ϑ ∈ Rn , η ∈ Cq , x ∈ Ω. αβ (x)ϑ ηj ϑ η

i,j=1 |α|=|β|=m

For 1 = m < q the condition (2.343) was required, cf Remark 2.90. For proving existence and uniqueness results for (2.392) we reinterpret the bilinear form as a( u, v ) =: ap ( u, v ) + b( u, v ). As above we prove that a(·, ·), ap (·, ·) : V × V → R and f ∈ W −m,p (Ω, Rq ) = V → R are bounded bilinear and linear forms and ap (·, ·) is V-coercive. We summarize Theorem 2.103. Strong and weak solutions of (2.392): For a system of the form (2.392) deﬁne the bilinear and linear forms a( u, v ) and f , v V ×V , and the principal part ap ( u, v ) as in (2.392), (2.394). Under the conditions

q (2.393), are continuous in V = H m (Ω, Rq ). For ajk these a( u, v ), ap ( u, v ), f , v V ×V

|β|,∞

0

= Aαβ ∈ W (Ω, R ), any solution u ∈ the weak equation, cf. (2.392), q

0

H02m (Ω, Rq )

αβ

i,j=1

of (2.391) necessarily satisﬁes

u0 ∈ V : a( u0 , v ) = f , v V ×V ∀ v ∈ V.

(2.395)

134

2. Analysis for linear and nonlinear elliptic problems

Vice versa, every smooth solution u0 ∈ H 2m (Ω, Rq ) ∩ H0m (Ω, Rq ) of (2.392) satisﬁes (2.391). Our next goal are analogies to Theorems 2.43, 2.89 and Corollary 2.44. The key result towards this goal is the V-coercivity of the principal part. Theorem 2.104. Coercivity, unique solutions, Fredholm alternative: In (2.392) ∞ ¯ let the coeﬃcients ajk αβ ∈ L (Ω), but for |α| = |β| = m > 1 be continuous in Ω, see Theorem 2.43. Furthermore, let (2.394), the strong Legendre–Hadamard condition, be satisﬁed. Then the principal part ap ( u, v ) is V = H0m (Ω, Rq )-coercive. Under these conditions Corollary 2.44 implies for a V-coercive a(·, ·) the existence of a unique generalized or weak solution u0 and a boundedly invertible L. Hence, for a V-elliptic a(·, ·) the Fredholm alternative with its consequences is valid. For a Vcoercive, symmetric a(·, ·), the unique u0 solves the variational problem, the updated (2.61). Finally, all H m -coercivity based existence, uniqueness and Fredholm results, including numerical stability and index L = 0, remain correct for the W m,p case and for p ≥ 2, with the modiﬁcations in Corollary 2.44 3., 4.. Proof. We cannot repeat the argument in the proof of Theorem 2.89, see (2.347). i The reason is the inequality (∂uj /∂xi )αi = ∂ αi uj /∂xα i for αi > 1. Instead we use the standard Fourier transforms, see, e.g. Hackbusch [388], Theorem 7.2.7. We start the proof by assuming constant highest coeﬃcients. But the same ideas as in the proof of Theorem 2.43 can be used here as well to obtain the coercivity for continuous highest coeﬃcients. So for constant highest coeﬃcients we get ⎛ ⎞ q β α ⎠ ⎝ dx ∀ u, v ∈ V. (2.396) ajk ap ( u, u) = αβ ∂ uk ∂ uj Ω

j,k=1 |α|=|β|=m

¯ We extend the u, v ∈ V by u(x) = 0 ∀ x ∈ Rn \ Ω. This u has compact support Ω. 21 Thus the Fourier transform, see, e.g. [388], Subsection 6.2.3, is well deﬁned for this 21

It is deﬁned, with the imaginary unit ι with ι2 = −1, via the components u = uj ∈ C0∞ (Rn ) : F : C0∞ (Rn ) → L2 (Rn ), with (ξ, x)n = u ˆ(ξ) := (F u)(ξ) := (2π)−n/2 := lim (2π)−n/2 R→∞

ξk xk ∀ ξ, x ∈ Rn , as

k=1

Rn

|ξ|∞ ≤R

n

e−ι(ξ,x)n u(x)dx

e−ι(ξ,x)n u(x)dx.

(2.397)

Since C0∞ (Rn ) is dense in L2 (Rn ), this F can be extended to L2 (Rn ) → L2 (Rn ) and inverted on C0∞ (Rn ), dense in L2 (Rn ), and again extended to L2 (Rn ) → L2 (Rn ) as ˆ)(x) := (2π)−n/2 (F −1 u

Rn

eι(ξ,x)n u ˆ(ξ)dξ ∀ u ∈ C0∞ (Rn ), extended to L2 (Rn ).

Its characteristics are summarized in

2.6. Linear and nonlinear elliptic systems

135

u. For ap ( u, u) we obtain for constant coeﬃcients q β ∂ uk · ∂ α uj dx ∀ u ∈ V. ap ( u, u) = ajk αβ Rn

j,k=1 |α|=|β|=m

We apply the Fourier transform to each term ∂ β uk and ∂ α uj . By the invariance of the scalar products in (2.398), the strong Legendre–Hadamard condition (2.394), and ˆk (ξ) = F (u)(ξ) we get with ηk = u ap ( u, u) =

q

ajk αβ

Rn

j,k=1 |α|=|β|=m

q

=

Rn j,k=1 |α|=|β|=m

≥

λ Rn

n

m

2

(ξ )

[ι|β| ξ β u ˆk (ξ)][ι|α| ξ α u ˆj (ξ)] dξ

β |β| ajk ˆk (ξ)ξ α [ι|α| u ˆj (ξ)] dξ αβ ξ ι u q

(2.400)

|ˆ ui (ξ)|2 dξ

i=1

=1

n 2 m For a ﬁxed m > 1 there exists an > 0, such that > |α|=m (ξ α )2 , =1 (ξ ) see (2.75). So we ﬁnd with (2.399) ⎛ ⎞ q ⎝ ui (ξ)|2 dξ = λ| u|2H m (Ω,Rq ) . (ξ α )2 ⎠ |ˆ ap ( u, u) > λ i=1

Rn

|α|=m

Since | u|H m (Ω,Rq ) and uH m (Ω,Rq ) are equivalent norms on V = H0m (Ω, Rq ), the Vcoercivity of ap ( u, u) is proved. The extension to “continuous coeﬃcients” is obtained as in Theorem 2.43. Theorem 2.105. The Fourier transform and its inverse are linear isometries: F, F −1 : L2 (Rn ) → L2 (Rn ), with F L2 (Rn )←L2 (Rn ) = F −1 L2 (Rn )←L2 (Rn ) = 1, u, vˆ)L2 (Rn ) ∀ u, v ∈ L2 (Rn ), (u, v)L2 (Rn ) = (ˆ

(2.398)

ˆ(ξ), and ∀ u ∈ H k (Rn ), |α| ≤ k F (∂ α u)(ξ) = ι|α| ξ α u 7 u H k (Rn ) = |ξ α |2 u ˆ(ξ) and |α|≤k L2 (Rn )

the seminorm |u|H k (Rn )

7 = |ξ α |2 u ˆ(ξ) |α|=k

.

(2.399)

L2 (Rn )

By applying F componentwise, this F can be extended to F : L2 (Rn , Rq ) → L2 (Rn , Rq ), noting that ui , vˆj )L2 (Rn ) ∀ i, j = 1, . . . , q. then (ui , vj )L2 (Rn ) = (ˆ

136

2. Analysis for linear and nonlinear elliptic problems

For an elliptic linear system in the sense of Deﬁnition 2.85 6., we give a ﬂavor of some results for local existence and regularity, by summarizing a few examples from Taylor [618]. For regularity results we combine several Propositions in Taylor [618], Chapter 5, Section 11 with our previous results. We formulate an interior estimate, a near boundary “collar” estimate, and a local solvability and regularity result. We simplify Taylor [618], Chapter 5, Theorem 11.1 ﬀ., originally for elliptic systems on manifolds, to our Ω ⊂ Rn . Taylor [618] discusses regular boundary value problems, cf. Chapter 5, Propositions 11.9–11.10. For example, a system (2.392) with Dirichlet boundary conditions is regular, if it satisﬁes a modiﬁed strong Legendre condition, cf. Chapter 5, (11.79), ⎛ ∃λ > 0 : ∀x ∈ Ω, ∀ϑ ∈ Rn : ⎝

⎞q

β α⎠ aij αβ ϑ ϑ

|α|=|β|=m

≥ λ|ϑ|2m I, I ∈ Rq×q ,

(2.401)

i,j=1

with the idendity matrix, I. This implies the strong Legendre–Hadamard condition (2.394) and a G˚ arding inequality, cf. (2.141), [618], Chapter 5, Exercise 1. Other types of boundary conditions are discussed as well Taylor [618], Chapter 5, Proposition 11.10 and Theorem 11.11 apply to our situation: Theorem 2.106. Interior estimate: Let L, a system of linear diﬀerential operators of order m, be m-elliptic, cf. Deﬁnition 2.85, 6., and u ∈ H m (Ω, Rq ) (a distribution in [618], Theorem 11.1) and L u = f ∈ H s (Ω, Rq ), 0 ≤ s ∈ N0 . Then u ∈ H m+s (Ω, Rq ). Furthermore, for any Ω1 ⊂⊂ Ω2 ⊂⊂ Ω and σ < m + s the following estimate holds uH m+s (Ω1 ,Rq ) ≤ C(L uH s (Ω2 ,Rq ) + uH σ (Ω2 ,Rq ) ). For the next theorem we modify some properties related to the atlas of a domain Ω, see Proposition 1.32, (1.63). In a ﬁrst step we choose the V i , i = 1, . . . , N , with i i ˆ N ˆi 0,1 ˆi ¯ Proposition 1.32, implies ∂Ω ⊂ ∪N i=1 V . Let V := Ω ∩ V , V := ∪i=1 V . For Ω ∈ C that there exist Ω3 ⊂⊂ Ω4 ⊂⊂ Vˆ with Ω3 ∩ ∂Ω = Ω4 ∩ ∂Ω = ∂Ω, such that b3 : ∂Ω × [0, 2] → Ω3 , and b4 : ∂Ω × [0, 1] → Ω4 are bijections. Now boundary conditions and “collar” norms are relevant, see [618], Chapter 5, Propositions 11.14, 15, 16. With u(·, y)H s (∂Ω) , 0 ≤ y ≤ 2, and ∇jy , the jth partial derivative into the y-direction, let the norms 1 u2H0,s (Ω4 ,Rq ) := u(·, y)2H s (∂Ω) dy, s, k ∈ N0 , 0

u2Hk,s (Ω4 ,Rq ) :=

k 1 j=0

0

j ∇y u(·, y)2

H k−j+s (∂Ω)

dy,

(2.402)

deﬁne the corresponding Banach spaces Hk,s (Ω4 , Rq ), and similarly Hk,s (Ω3 , Rq ) with 1 2 replacing 0 . Additionally, let L, a system of linear diﬀerential operators now of 0 order 2m, be strongly 2m-elliptic, cf. Deﬁnition 2.85, 6. By Taylor [618], Chapter 5, Proposition 11.10. a strongly 2m-elliptic system (2.392) or (2.323) with Dirichlet boundary conditions is regular. For other types of boundary conditions see [618].

2.6. Linear and nonlinear elliptic systems

137

Theorem 2.107. “Collar” estimate, Taylor, [618], Chapter 5, Propositions 11.14: Under these conditions let u 0 ∈ H2m,σ (Ω3 ) solve the (nonhomogeneous) “collar” Dirichlet boundary problem 22 Lu 0 = f ∈ Hk,s (Ω3 , Rq ), ∂ j u 0 /∂ν j = ϕj ∈ H 2m+k−j+1/2+s (∂Ω, Rq ), j = 0, . . . , m − 1, for some 0 ≤ σ < s ∈ R. Then we even obtain u 0 ∈ H2m+k,s (Ω4 , Rq ) and it satisﬁes the estimate ⎛ ⎞ m−1 u 0 H2m+k,s (Ω4 ,Rq ) ≤ C ⎝Lu 0 Hk,s (Ω3 ,Rq ) + ∂ j u 0 /∂ν j H 2m+k−j+1/2+s (∂Ω,Rq ) ⎠. j=0

Proposition 11.16 of [618] applies to our situation and states, compare Zeidler [676], Section 4.17: Theorem 2.108. Regularity and Fredholm alternative for order 2m: Let the ∞ coeﬃcients aij αβ , ∂Ω in (2.392) be in C , and let Deﬁnition 2.85 6. be satisﬁed. 2m+k q m (Ω, R ) ∩ H0 (Ω, Rq ) → H k (Ω, Rq ), q ≥ 1, in (2.392) is Then the operator L : H k Fredholm, hence, for f ∈ H (Ω, Rq ), k ≥ 0, any solution for (2.392), if it exists, is u 0 ∈ H 2m+k (Ω). 2.6.6

Divergent quasilinear elliptic systems of order 2m

In this subsection we study strong and weak forms of systems of quasilinear diﬀerential operators of order 2m. The results for this case are not available in a similarly general form as before. We summarize parts of Koshelev and Chelkak [450] but start with one of the most important examples: von K´ arm´ an equations Diﬀerent types of existence and regularity results for this system are discussed by John and Necas [423], John and Naumann [422], Ciarlet [175], Giaquinta et al. [343], Kondrat’ev and Olejnik [447, 448], Horn [404], and Bock [100]. In John and Naumann [422], the strong form of the von K´ arm´an equations is studied: ⎞ ⎛ ⎞ ⎛ 2 Δ u −[u, w] f ⎠ = ⎝ ⎠ in Ω ⊂ R2 , (2.403) Gs (u, w) := ⎝ 0 Δ2 w + 12 [u, u] with Δ2 = ΔΔ and the bracket operator [u, v] := uxx vyy − 2uxy vxy + uyy vxx = [v, u]. Speciﬁc existence and uniqueness results for the von K´ arm´an equations are discussed in the papers of Hlav´ acek and Naumann [399, 508], John et al. [422, 423], even with diﬀerent boundary conditions on diﬀerent parts of ∂Ω, and Giaquinta et al., [343]. For a full formulation, many technical details would have to be discussed. So we just refer 22 Or let L with the boundary operators B be regular elliptic, see [620], pp. 379 ﬀ.; essentially L, B satisfy a generalized complementary condition.

138

2. Analysis for linear and nonlinear elliptic problems

to [343, 399, 422, 423, 508], and only combine John and Naumann [422] in (2.403) with some of our earlier results. Existence and uniqueness can also be elaborated numerically. The above (2.331) and (2.403) represent compact perturbations of its strong principal part T T Gp (u, w) := Δ2 u, Δ2 w = f, 0 in

Ω.

(2.404)

The weak form of (2.404) is obviously H02 (Ω, R2 )-coercive, so the linearized (2.403) is H02 (Ω, R2 )-elliptic. Furthermore, (2.404) satisﬁes the conditions in Subsection 2.6.5. Hence, existence and uniqueness for (2.331) and (2.403) can be guaranteed by combining the Fredholm alternative for the linearized form of (2.403) with discretizations and numerical continuation techniques, starting with (2.404) and ending with (2.331) or (2.403). Equation (2.403) is slightly simpliﬁed compared to (2.331). In the ﬁrst equation the +λuxx is missing. As a consequence of the special form of (2.331), and since the +λuxx term represents a compact perturbation, the arguments for Theorem 2.56 3. remain valid for this nonlinear case as well. Hence, we formulate John and Naumann [422] directly for the more general (2.331) case. They discuss two sets of non-Dirichlet boundary conditions. Using the unit outer normal vector ν = (νx , νy ), the boundary tangent vector σ = (−νy , νx ) and the Poisson ratio of the plate material, μ, they introduce the operators M u := μΔu + (1 − μ) uxx νx2 + 2uxy νx νy + uyy νy2 with constant μ ∈ (0, 1/2), or T u := −μ

∂ ∂ Δu + (1 − μ) (uxx νx νy − uxy νx2 − νy2 − uyy νx νy ). ∂ν ∂σ

(2.405)

This allows us to formulate the two sets of boundary conditions, considered in [422], ∂ w = 0 on ∂Ω and ∂ν ∂ M u = T u = 0, w = w = 0 on ∂Ω. ∂ν

u = M u = 0, w =

(2.406) (2.407)

We denote (2.403), (2.406) as Problem I, and (2.403), (2.407) as Problem II. For weak solutions of (2.403), John and Naumann [422] modify the standard approach. With a(u, v) := [uxx vxx + 2(1 − μ)uxy vxy + uyy vyy + μ(uxx vyy + uyy vxx )]dxdy Ω

and ∃c1 , c2 : c1 u2H 2 (Ω) ≤ a(u, u) ≤ c2 u2H 2 (Ω) , ∀u, v ∈ H 2 (Ω),

(2.408)

they get a modiﬁed Green’s formula, based upon M, T ∂u 2 a(u, v) = M vds ∀u, v ∈ H 4 (Ω), (2.409) uΔ vdxdy + uT vds + Ω ∂Ω ∂Ω ∂ν

2.6. Linear and nonlinear elliptic systems

or combined as (uΔ2 v − Δ2 uv)dxdy = + Ω

(vT u − uT v)ds +

∂Ω

∂Ω

139

∂u ∂v Mu − M v ds. ∂ν ∂ν

Correspondingly we have to determine the solution (u0 , w0 ) ∈ W := {u ∈ H 2 (Ω) : u|∂Ω = 0}) × H02 (Ω) ,

(2.410)

by testing (2.403) in the ﬁrst and second equation as a(u0 , ϕ) = [u0 , w0 ]ϕdxdy + f ϕdxdy∀ϕ ∈ W

(2.411)

Ω

(w0 , ψ)H02 (Ω) :=

|α|=2

Ω

∂ α u0 ∂ α vdxdy = −

Ω

[u0 , u0 ]ψdxdy ∀ψ ∈ H02 (Ω). (2.412) Ω

Equation (2.331) has to be tested by slightly changing (2.411) into (2.413) and leaving (2.412) unchanged: λu0,xx ϕdxdy + f ϕdxdy ∀ϕ ∈ W. (2.413) a(u0 , ϕ) = [u0 , w0 ]ϕdxdy − Ω

Ω

Ω

Obviously with a solution (u0 , w0 ) of Problem II, (2.403), (2.407), the (u0 + p1 , w0 ), ∀p1 ∈ Π1 , the polynomials of degree 1, is again a solution. We eliminate the nonuniqueness for u0 in (2.403), (2.407) by deﬁning K as H 2 (Ω) =: K ⊕ Π1 . So the ˜, such that u, v ∈ u ˜ iﬀ u − v ∈ Π1 . quotient space H 2 (Ω)/Π1 is the space of all classes u Each u ∈ u ˜ has a unique decomposition u = uK + p1 with uk ∈ K, p1 ∈ Π1 and uK uniquely determines u ˜. A (u0 , w0 ) ∈ W × H02 (Ω), cf. (2.410), is a weak solution of Problem I, (2.403), (2.406), if it satisﬁes (2.411) ∀ϕ ∈ W , and (2.412) ∀ψ ∈ H02 (Ω), and (2.406). Similarly, (u0 , w0 ) ∈ H 2 (Ω) × H02 (Ω) weakly solves Problem II, if it satisﬁes (2.411) ∀v ∈ H 2 (Ω), and (2.412) ∀ψ ∈ H02 (Ω), and (2.406). If we consider (2.331), we have to replace (2.411) by (2.413). Next, we present regularity results for the solutions of (2.331), (2.403). We start with the result in [422], for (2.403). Theorem 2.109. Regular K´ arm´an solutions: 1. For f ∈ Lp (Ω), 1 < p < ∞ let (mind [f ] = |f |)

[f ] := f Lp (Ω) 1 + f 2Lp (Ω) . Let (u0 , w0 ) be a solution of Problem I or II. Then u0 ∈ W 4,p (Ω), M u0 = 0 pointwise on ∂Ω, w0 ∈ W 6,p (Ω) ∩ H02 (Ω).

(2.414)

140

2. Analysis for linear and nonlinear elliptic problems

2. If (u0 , w0 ) solves Problem I, then additionally u0 ∈ W 4,p (Ω) ∩ W . For Problem II we get T u0 = 0 a.e. on ∂Ω in the trace sense. Moreover, for 1 < p < 2 : u0 W 4,p (Ω) ≤ C[f ], w0 W 6,p (Ω) ≤ C[f ]2 and for 2 ≤ p < ∞ : u0 W 4,p (Ω) ≤ C(f Lp (Ω) + [f ]3 ), w0 W 6,p (Ω) ≤ C(f Lp (Ω) + [f ]3 )2 . Finally, we combine Theorem 2.91 with Corollaries 3.1 and 3.2 in [422] to get Corollary 2.110. Enforce the conditions of Theorem 2.109 by requiring f ∈ W k,p (Ω). ¯ then we Then the solutions are u0 ∈ W 4+k,p (Ω), w0 ∈ W 6+k,p (Ω). If even f ∈ C ∞ (Ω) ∞ ¯ even get u0 , w0 ∈ C (Ω). In all these cases, T u0 = 0 pointwise on ∂Ω for Problem II. These results remain correct for (2.331). 2.6.7

Nemyckii operators and quasilinear divergent systems of order 2m

Generalized solutions of quasilinear divergent systems of order 2m Zeidler [678] does not study quasilinear divergent systems. But his results for quasilinear equations, cf. Zeidler [678], pp. 500 ﬀ., 554 ﬀ., Theorem 25B, Proposition 26.6 and pp. 572 ﬀ., are easily generalized to the present case. So we update the necessary deﬁnitions and results from Subsections 2.5.5 and 2.5.6. For the Banach space X we use here the product space X = W m,p (Ω, Rq ) = (W m,p (Ω))q . For 1 < p < ∞ this X is reﬂexive and for X = W m,p (Ω, Rq ) with 1 < p < ∞ let A u − A v , u − v = A u − A v , u − v X ×X . Then equations (2.242)–(2.253) remain unchanged, and we have called A monotone ⇔ A u − A v , u − v ≥ 0, A strongly monotone ⇔ A u − A v , u − v ≥ c u − v 2X , c ∈ R+ , A stable ⇔ A u − A v X ≥ a( u − v X ), with a : R+,0 → R+,0 a(0) = 0,

(2.415) (2.416) (2.417)

continuous and strictly increasing, cf. (2.418), with lim a(t) = ∞, e.g. a(t) = c|t|p−1 , 0 < c, 1 < p. (2.418)

t→∞

Then Proposition 2.67–Theorem 2.70 remain valid. We only reformulate the most important parts of Theorem 2.68: Theorem 2.111. Properties of nonlinear operators: Let X be a real Banach space and A : X → X a (nonlinear) operator. Then

2.6. Linear and nonlinear elliptic systems

141

1. If A is monotone, coercive, and hemicontinuous on a reﬂexive Banach space, then A is surjective and, for each f ∈ X , the set of solutions of Au = f is bounded, convex and closed, cf. [678], Theorems 25B and 26A(a). 2. If A is additionally strictly monotone, the above solution is unique. Then A−1 : X → X exists and, cf. [678], Theorem 26A(c), (d), a uniformly monotone A implies a continuous A−1 , a strongly monotone A implies a Lipschitz–continuous A−1 ∈ CL . We have to update the deﬁnition of Nemyckii operators, cf. Deﬁnition 2.69: Deﬁnition 2.112. For given functions f and appropriate u ∈ D(F ), deﬁne F ( u) as f = (fi )qi=1 : D(f ) ⊂ Ω × RNm ×q → Rq , u : Ω → Rq : F ( u)(x) := f (x, ∂ ≤m u(x)), (2.419) cf. (2.423), with (x, ∂ ≤m u(x)) ∈ D(f ). This F is called a Nemyckii operator. Now we formulate two conditions guaranteeing that F ( u)(x) makes sense. The Carath´eodory and the growth condition enforce measurability and appropriate growth of the composite function, F . More precisely, we assume Carath´eodory condition: Ω is bounded and measurable in Rn , ≤m ) ∈ D(f ⊂ Ω × RNm ×q → Rq , ∀ |α| ≤ m, cf. (2.423), f : (x, ϑ ≤m ) is continuous in RNm ×q a.e. for x ∈ Ω, ≤m → f (x, ϑ ϑ

(2.420)

≤m ∈ RNm ×q , e.g. for continuous f , ≤m ) is measurable in Ω ∀ ϑ x → f (x, ϑ

and with 1 < p < ∞, 1/p + 1/p = 1, p − 1 = p/p and ∃ a ∈ Lp (Ω), 0 ≤ a(·) a.e. ≤m ∈ RNm ×q , ϑ γ ∈ Rq : ∃ C ∈ R+ , s.t. ∀ x ∈ Ω, ϑ γ |p−1 ) ∀|α| ≤ m. ≤m )| ≤ C(a(x) + |ϑ Growth condition: |f (x, ϑ

(2.421)

|γ|≤m

Theorem 2.113. Estimates for nonlinear operators: If (2.420), (2.421) are satisﬁed, then F : W m,p (Ω, Rq ) → Lp (Ω, Rq ) in (2.419) is continuous and bounded (this is not equivalent for nonlinear operators!) such that F ( u)(Lp (Ω))q ≤ C

aLp (Ω) +

k

∂ γ up−1 (Lp (Ω))q

∀ u ∈ W m,p (Ω, Rq ). (2.422)

i=1

The following general quasilinear systems are studied by Koshelev and Chelkak [450]. They consider Sobolev spaces W m,p (Ω, Rq ) ⊂ L2 (Ω, Rq ) ⊂ W −m,p (Ω, Rq ),

142

2. Analysis for linear and nonlinear elliptic problems

1/p + 1/p = 1. We extend their existence results to the case 1 < p < ∞, by generalizing our results in Subsections 2.5.5 and 2.5.6, cf. Zeidler [678]. We recapitulate the notation for partials and vectors so, cf. (2.367), (2.389), (2.73), (2.79), (2.102), (2.336), (2.335): u ∈ W m,s (Ω, Rq ), u(x),

β = ϑβ1 , · · · , ϑβq , ∈ Rq ∂ β u(x), ϑ q 1

∂ ≤m u(x) = (∂ β u(x))0≤|β|≤m ,

(2.423)

≤m = (ϑ β )0≤|β|≤m , ∈ RNm ×q . ϑ

We discuss the general form of quasilinear elliptic equations on bounded domains Ω, compare Theorem 2.50, with Dirichlet boundary conditions, see Skrypnik [590,591] and Zeidler [678], and recall the standard notation in (2.165), (2.166), (2.269), (2.16), (2.361). The strong quasilinear system is G u0 = Gs u0 :=

⎡ α (·, ∂ ≤m u0 ) = ⎣ (−1)|α| ∂ α A

|α|≤m

= 0, u ∈ W 0

2m,p

⎤q (−1)|α| ∂ α Aiα (·, ∂ ≤m u0 )⎦

|α|≤m

(Ω, R ) ∩ q

W0m,p (Ω, Rq ),

i=1

% & α = Ai q : Ω × RNm ×q → Rq . A α i=1

(2.424) Here, we consider the weak solution u0 ∈ W0m,p (Ω, Rq ), a(u 0 , v ) := G u0 , v =

α (·, ∂ ≤m u0 ), ∂ α v )q dx = 0 ∀ v ∈ W m,p (Ω, Rq ). (A 0

Ω |α|≤m

(2.425) These integrals are well deﬁned under the following sharper Carath´eodory and growth conditions, compare (2.420), (2.384), (2.385). α (x, ∂ ≤m u) are Nemyckii operators, generalizing Subsection 2.5.5. Now we The A impose conditions which guarantee that the G(u) in (2.424) and its weak form (2.425) are well deﬁned. The Carath´eodory and growth conditions enforce measurability and α , cf. Theorems 2.73, 2.113, and [678], appropriate growth of the composite function A pp. 571 ﬀ. (2.426). The following nonlinear coercivity, monotonicity and growth conditions for these quasilinear operators generalize (2.230)–(2.235), (2.242)–(2.247). γ | in Rq , essential for the existence of They are, with the equivalence of all norms, |ϑ solutions. Carath´eodory condition: Ω is bounded and measurable in Rn , ≤m ) ∈ Ω × RNm ×q → Rq , α : (x, ϑ ∀ |α| ≤ m : A ≤m ) is continuous in RNm ×q a.e. for x ∈ Ω, ≤m → A α (x, ϑ ϑ

(2.426)

≤m ) is measurable in Ω ∀ ϑ ≤m ∈ RNm ×q , e.g. for continuous A α. α (x, ϑ x → A

2.6. Linear and nonlinear elliptic systems

143

For the following conditions, we assume 1 < p < ∞, 1/p + 1/p = 1, the special form γ )|γ|≤m , ϑ ≤m = (ϑ γ )|γ|≤m ∈ RNm ×q and ∃ c, d, C ∈ R+ , such that ≤m = (ϑ ϑ 1 1 ≤m , ϑ ≤m ∈ RNm ×q , ϑ 0, ϑ α , ϑ α ∈ Rq : ∃ a ∈ Lp (Ω), h ∈ L1 (Ω), s.t. a.e. ∀ x ∈ Ω, ϑ 1 1 ≤m )| ≤ C(a(x) + γ |p−1 ) ∀|α| ≤ m. α (x, ϑ Growth condition: |A |ϑ (2.427)

(Nonlinear) coercivity: Monotonicity:

|γ|≤m

≤m ), ϑ α )q ≥ c α (x, ϑ (A

|α|≤m

γ |p − h(x). |ϑ

≤m ) − A ≤m ), ϑ α − ϑ α α (x, ϑ α (x, ϑ A 1 1

|α|≤m

Strict monotonicity:

≤m ) − A ≤m ), ϑ α − ϑ α α (x, ϑ α (x, ϑ A 1 1

|α|≤m

q

≥ 0.

γ > 0 implies γ − ϑ ϑ 1

|γ|≤m

(2.429) (2.430)

> 0. q

Uniform monotonicity: p

≤m ) − A ≤m ), ϑ α − ϑ α ≥ C γ . γ − ϑ α (x, ϑ α (x, ϑ A ϑ 1 1 1 |α|≤m

(2.428)

|γ|=m

q

(2.431)

|γ|=m

For regularity results, p = 2 and additional conditions have to be imposed, cf. Theorem 2.116. By [678], Proposition 26.12, the conditions (2.426)–(2.429) imply the conditions of his Theorem 26A. So we refer in the next theorem to the corresponding results for equations in [678]. Theorem 2.114. Generalized solutions for (2.425): Let (2.426)–(2.427) be satisﬁed and choose V = W0m,p (Ω, Rq ). Let α (x, ∂ ≤m u(x)). F α : V = W0m,p (Ω, Rq ) → Lp (Ω, Rq ), (F α u)(x) := A

1. Then for all |α| ≤ m, a.e. ∀x ∈ Ω these Nemyckii operators F α are well deﬁned, continuous and bounded, see Theorem 2.113 and [678], Proposition 26.12, proof. α (x, ∂ ≤m u(x)) deﬁne the a( u, v ) as in (2.425). Then there exists 2. With the above A exactly one bounded operator G, cf. Proposition 2.67, [678], Proposition 26.12, (2.432) G : V → V : a( u, v ) = G u, v V ×V ∀ u, v ∈ V = W0m,p (Ω, Rq ) and

p/p m,p . Fα uLp (Ω,Rq ) , G uW −m,p (Ω,Rq ) ≤ C aLp (Ω,Rq ) + uW0 (Ω,Rq )

144

2. Analysis for linear and nonlinear elliptic problems

Similarly to (2.425) and with f , v V ×V replacing 0, the original problem (2.424) corresponds to the generalized problem: Determine for a( u, v ) in (2.432) u0 ∈ V = W0m,p (Ω, Rq ) s.t. a( u0 , v ) = f , v V ×V ∀ v ∈ V ⇐⇒ G u0 = f . (2.433) α be coercive and monotone, see (2.428) and (2.429). 3. Additionally let all the A Then G is monotone, coercive, continuous and bounded, see [678], Proposition 26.12. So by Remark 2.66, Theorem 2.68 2.–4. are valid for (2.425). 4. In particular, for any f ∈ W −m,p (Ω, Rq ) = V , (2.433) has a solution u0 ∈ α W0m,p (Ω, Rq ) and the set of solutions is bounded, convex and closed. If all the A are additionally strictly monotone, see (2.430), Theorem 2.68, then G is strictly monotone as well and the above solution is unique. So the inverse G−1 : V → V exists and is strictly monotone and bounded, see [678], Theorem 26A. If all the α are, and hence G is uniformly monotone, see (2.431), this G−1 is continuous. A 5. If G is strongly monotone, cf (2.416), Theorem 2.111, then G−1 is Lipschitz– continuous. Koshelev–Chelkak existence and regularity results for quasilinear systems As mentioned above, these problems are discussed by Koshelev and Chelkak [450]. Here the Sobolev spaces W m,s (Ω, Rq ) ⊂ L2 (Ω, Rq ) ⊂ W −m,s (Ω, Rq ), 1/s + 1/s = 1, in (2.388) have to be combined with the scalar products and norms in (2.337) and Gs u in divergence form. We introduce, similarly to the partials and vectors β , u(x), ∂ β u(x) ∈ Rq in (2.423), (2.367), (2.389), the notation for the components p β : ϑ

p := (p1 , · · · , pq ), p β = pβ1 1 , · · · , pβq q ∈ Rq , p ≤m := (pβ )0≤|β|≤m ∈ RNm ×q . (2.434) We consider and determine the solution, u0 ∈ W 2m,s (Ω, Rq ) ∩ W0m,s (Ω, Rq ), of the strong quasilinear system (2.424), see (2.16), (2.361). Koshelev and Chelkak [450] restrict (2.424) essentially to the case 1 < s ≤ 2. In their introduction they intensively discuss other possibilities and aspects of the problem (2.424) and give references. The weak solution u0 ∈ W0m,s (Ω, Rq ) is determined from (2.425). The integrals in (2.425) are well deﬁned under the following modiﬁed Carath´eodory and growth conditions, compare (2.426)–(2.431), and (2.284), (2.384), (2.385). We impose a strong Legendre condition for the linearizedoperator, compare (2.384). In fact, compared to (2.378), |α|,|β|=m is replaced by |α|,|β|≤m in (2.437). This appears again in the q growth condition (2.437): For w = x, p ≤m and with | p ≤m |2 = |β|≤m i=1 |pβi i |2 and λ, Λ = const. > 0, let, for almost all x ∈ Ω, Aiα (x, p

≤m

) are measurable and continuouslydiﬀerentiable with respect to p

∀|α| ≤ m in any domain of boundedvariation of p

≤m

= ∂ ≤m u of u,

≤m

(2.435)

2.6. Linear and nonlinear elliptic systems

λ(1 + | p ≤

≤m 2 (s−2)/2

| )

q

≤m |2 |ϑ

145

(2.436)

β ∂Aiα /∂pjβ (w)ϑα p i ϑj ≤ Λ(1 + |

≤m 2 (s−2)/2

| )

≤m |2 , |ϑ

i,j=1 |α|,|β|≤m

for 1 < s ≤ 2,

β p | ∂Aiα /∂pj j (w)| ≤ Λ(1 + |

≤m 2 (s−1)/2

| )

∀ i, j, α, β, and for any 1 < q , (2.437)

α (·, ∂ ≤m u(x)) ∈ Lq /(s−1) (Ω ). (2.438) domain Ω ⊂ Ω and u ∈ W 1,q (Ω , Rq ) : A

Koshelev and Chelkak call them systems with bounded nonlinearities. Similarly to our Lemma 2.26, see (2.82), generalizations of Dirichlet boundary conditions to Dirichlet systems are possible. To avoid the many technicalities we do not present the iteration process, which plays an important role in [450], for determining the solution. We restrict the discussion to Dirichlet boundary conditions ∂ j u(x) = gj ∀0 ≤ j ≤ m − 1 ⇔ ∂ α u(x) = g α ∀|α| ≤ m − 1 ∀x ∈ ∂Ω. ∂ν j

(2.439)

Theorem 2.115. Existence – Theorems 2.1.1 and 2.3.1 in [450]: 1. Assume s = 2, and the above modiﬁed Carath´eodory and growth conditions β (2.435)–(2.438), implying |(∂Aiα /∂pj j )(x, p ≤m )| ≤ C. Then the quasilinear system (2.425) with the Dirichlet boundary conditions (2.439) admits a solution u0 ∈ H m (Ω, Rq ). 2. If, more generally, for 1 < s ≤ 2, a weak solution u0 for (2.425), (2.439) exists and satisﬁes the estimate s |Aα (x, ∂ ≤m u0 (x))|s/(s−1) dx ≤ C1 u0 W m,s (Ω,Rq ) + C2 , (2.440) Ω |α|≤m

then this u0 is the unique solution of the problem. To get regularity results, again s = 2 and additional conditions have to be imposed. α /∂xk are required, the existence of these Whenever conditions for partials, e.g. ∂ A partials is silently assumed. For Ω ⊂⊂ Ω we introduce ⎛ ⎞ 2 ⎝ uH m (Ω ,Rq ) := sup |∂ α u(x)|2 + |∂ α u(x)|2 |x − x0 |δ ⎠dx (2.441) δ

Hδm (Ω , Rq )

x0 ∈Ω

Ω

|α|<m

|α|=m

:= { u ∈ H (Ω , R ) : uHδm (Ω ,Rq ) < ∞}, L2δ (Ω , Rq ) := Hδ0 (Ω , Rq ). m

q

We impose three sets of conditions for the three cases: (2.443) for u ∈ H m (Ω , Rq ), (2.444) for u ∈ Hδm (Ω , Rq ), and (2.445) for u ∈ W m+1,q (Ω , Rq ) : for all three cases let Ω ⊂⊂ Ω : ∃ C(R) independent of u, such that

(2.442)

146

2. Analysis for linear and nonlinear elliptic problems

∀ u ∈ H m (Ω , Rq ) with uH m (Ω ,Rq ) ≤ R,

(2.443)

α (·, ∂ ≤m u(·))L2 (Ω ,Rq ) ≤ C(R), and ∀ 0 ≤ |α| ≤ m : A ∂A α ≤m (·, ∂ u(·)) ≤ C(R) or ∀|α| = m, k = 1, . . . , n, let 2 q ∂xk L (Ω ,R )

∀ u ∈

Hδm (Ω , Rq )

with uHδm (Ω ,Rq ) ≤ R, δ ∈ [2 − m − 2γ, 0],

(2.444)

α (·, ∂ ≤m u(·))L2 (Ω ,Rq ) ≤ C(R) and 0 < γ < 1, ∀ 0 ≤ |α| ≤ m let A δ ∂A α (·, ∂ ≤m u(·)) ≤ C(R), ﬁnally ∀|α| = m, k = 1, . . . , n, let 2 q ∂xk Lδ (Ω ,R )

∀ u ∈ W m+1,q (Ω , Rq ), m < q ,

α ∂A (·, ∇≤m u(·)) ∈ Lq (Ω , Rq ). ∂xk

(2.445)

Theorem 2.116. Regularity, Theorem 4.3.1 in [450]: Assume the conditions (2.442)– (2.445), those in Theorem 2.115 (note s = 2), except (2.440), and some highly technical conditions, see (1.3.6), (1.3.7), (1.3.18), (1.3.20), (1.3.46), (4.1.11), (4.3.3) in [450]. Let the boundary values in (2.439) be deﬁned by one function g as g ∈ H m (Ω, Rq ) : gj =

∂ j g(x) ∀0 ≤ j ≤ m − 1. ∂ν j

Then the weak solution u0 , for (2.425), (2.439) belongs to u0 ∈ C m,γ (Ω , Rq ). The techniques for proving higher regularity as in Theorem 2.91 3. do not apply. 2.6.8

Fully nonlinear elliptic systems of orders 2 and 2m

In contrast to the earlier subsections, only few results for fully nonlinear systems have found their way into the monographs consulted here. Formally, they are obtained by the obvious generalizations of (2.174), (2.306), (2.319) as c.f. Remark around (2.306), G(·) : D(G) ⊂ U := H01 (Ω, Rq ) ∩ H 2 (Ω, Rq ) → V := L2 (Ω, Rq ), with G( u(·)) := Gw (·, u(·), ∂ u(·), ∇2 u(·)), G( u0 ) = 0

(2.446)

for systems of order 2. The corresponding equations are updated by replacing the original U, V, W . For systems of order 2m we obtain G(·) : D(G) ⊂ U := H0m (Ω, Rq ) ∩ H 2m (Ω, Rq ) → V := L2 (Ω, Rq ), with G( u) := Gw (·, u(·), . . . , ∇2m u(·)), G( u0 ) = Gw (·, ∂ γ u0 (·), |γ| ≤ 2m) = 0. (2.447) Often H 2m (Ω, Rq ) is replaced by C 2m,γ (Ω, Rq ) or C ∞ (Ω, Rq ), disregarding boundary conditions. In Taylor [620], pp. 108 ﬀ., some “local existence results” are proved: if at a point x0 ∈ Ω the equation Gw (·, ∂ γ u1 (·), |γ| ≤ 2m)(x0 ) = 0 is satisﬁed for a u1 , then

2.7. Linearization of nonlinear operators

147

there exists a solution u in a neighborhood of x0 . For second order some results can be found, e.g. in [79, 80, 311, 475, 671]. One of the few exceptional results is proved by Taylor [620], Chapter 14, Theorem 3.1, for nonlinear systems of PDEs of order m, see Deﬁnitions 2.51 and 2.85. It extends verbatim (2.319) and Theorem 2.84 to systems, by simply replacing u by u. Nonlinear systems of PDEs of order m have the form G( u) := G(x, u, ∇ u, . . . , ∇m u) and G( u)(x) = g(x) ∀ x ∈ Ω.

(2.448)

Theorem 2.117. Local solvability result for nonlinear elliptic systems: Let g ∈ C ∞ (Ω), and let u1 ∈ C ∞ (Ω) satisfy (2.448) at x = x0 , hence G( u1 )(x) = g(x) for x = x0 and let G be m-elliptic at u1 . Then for any there exists a u ∈ C (Ω) such that G( u)(x) = g(x) in a neighborhood of x0 . In this situation with only few known results, a combination of the linearized operators in Subsection 2.7.3, our FE or other discretization methods below and continuation techniques, allows us to study existence of solutions of these problems. So we give some details in Subsections 2.7.3 and 2.7.4 below.

2.7 2.7.1

Linearization of nonlinear operators Introduction

Linearization of operators including systems plays an essential role in our approach. We are mainly interested in numerical methods for these PDEs and their bifurcation scenarios. The stability of numerical methods can be proved via the stability of the method for the linearized operator. For the numerical approximation of their bifurcation scenarios we do need the Fredholm alternative, again formulated for the linearized operator. For this reason we have not included the many Fredholm type results, directly available for nonlinear diﬀerential equations, see, e.g. Zeidler [676–678]. In many contexts, nonlinear parameter-dependent operators G(u, λ) are studied. It is important to start with a parameter value λ0 such that the existence and (local) uniqueness of a solution u0 of G(u0 , λ0 ) = 0 is guaranteed. The results in Sections 2.4– 2.6 yield this information for solutions u0 for a ﬁxed parameter λ0 . Based upon this knowledge, the Implicit Function Theorem 1.46 deﬁnes a uniquely determined solution curve of the nonlinear problem G(u, λ) = 0 through (u0 , λ0 ). Note that, e.g. for an operator G : V × Rp → V the derivative G (u, λ) = (Gu (u, λ), Gλ (u, λ)) always maps V × Rp → V , whenever it exists as a bounded operator. The corresponding continuation process again is based upon the Fr´echet derivative G (u, λ) = (Gu (u, λ), Gλ (u, λ)) at a solution point (u, λ). The continuation breaks down and the solution curve usually bifurcates, whenever G (u, λ) is not surjective in a solution point (u, λ), cf. B¨ohmer [120]. Then usually Gu (u, λ) has a nontrivial kernel. So results from the Fredholm alternative for the linearized operator Gu (u, λ) apply in combination with the Liapunov–Schmidt methods studied in [120]. For both aspects we have to verify

148

2. Analysis for linear and nonlinear elliptic problems

that for the diﬀerent types of nonlinear operators G(u), studied in the last Sections 2.5, 2.6, the linearized operators G (u0 ) or the Gu (u0 , λ1 ) are well deﬁned and that Fredholm alternatives are available for them. By appropriate combinations of the corresponding Banach spaces, linearizations for a wide class of nonlinear elliptic equations and systems make sense. In this monograph we use discretization concepts diﬀerent from the Brezzi–Rappaz–Raviart [147–149] ideas. So, most of Santos’s [566] negative arguments concerning numerical methods as in [147–149], related to linearization, do not apply. Prima vista, linearization does not seem to be appropriate for some of the quasilinear elliptic equations and systems. In fact the results for quasilinear equations from Subsection 2.5.6 strongly depend upon monotony and nonlinear coercivity arguments. But we show in Subsection 2.7.3 that the linearization approach works for most quasilinear equations. Noncoercive equations, e.g. of the form Au0 + N u0 = b, with A ∈ L(U, U ) linear, N nonlinear

lim N u/u = 0,

u→∞

require speciﬁc handling. However, some type of Fredholm alternative still holds. This equation has a solution u0 ∀b ∈ U if Au = 0 has the unique solution u = 0, otherwise solutions u0 exist only for special b ∈ U , see Zeidler [678], Chapter 29. The situation for systems is diﬀerent, since for orders 2 and 2m the ellipticity of (the linearized) operator plays a dominant role compared to analogous monotony arguments. The situation changes if viscosity solutions are admitted instead, as studied in an impressive series of papers by Crandall et al. [215, 217–228], which we did not consider here. Above we have documented mainly two sets of results for nonlinear equations, either based upon (linear) coercivity arguments and the corresponding H m (Ω) or W m,p (Ω) setting, combined with Schauder results for second order equations. Or they are based upon monotony arguments in the W m,p (Ω) setting. In both cases the linearization usually yields operators as discussed in Section 2.4 and Subsections 2.6.3, 2.6.5, and satisﬁes the Fredholm alternative. An essential fact here is the strong ellipticity or the Legendre condition of the linearizations and, for systems of order 2m, m > 1, the strong Legendre–Hadamard condition. In this section we will show that the linearization of all nonlinear problems studied before in the H m (Ω) or W m,p (Ω) setting, 2 < p ≤ ∞, yields bounded linear operators, with the corresponding bounded bilinear forms a(u, v). They are compact perturbations of their principal parts. For 2 < p ≤ ∞, we use the compact embedding of the W m,p (Ω) into H m (Ω). Then the principal parts are H m (Ω)-coercive, and the convergence for numerical methods is measured with respect to a discrete H m (Ω) norm, cf. Corollary 2.44. For 1 ≤ p < 2 corresponding bilinear forms a(u, v) are not even deﬁned. So we have to exclude this case. Finally, for the fully nonlinear systems in Subsection 2.6.8 no existence results are available. However, linearizations may be combined with the Fredholm alternative and discretization and continuation techniques to obtain existence results. In Section 4.5 we will apply the general discretization theory from Chapter 3 to all our general quasilinear problems. This is formulated for FEMs, but is valid for all

2.7. Linearization of nonlinear operators

149

other methods as well. It is based upon monotony and related properties and allows the previously excluded cases 1 < p < ∞. For all the following cases we do not repeat, but only refer to, the conditions for the deﬁning coeﬃcients and the domain imposed for the original nonlinear problem. We formulate the stronger conditions where they are necessary for the linearization. To simplify the notation we always omit parameters λ and use the abbreviations ∂G ∂f ∂f ∂G (u) := (u, λ), (u) := (u, λ), etc., often for (u, λ) = (u0 , λ0 ). ∂u ∂u ∂u ∂u

(2.449)

We summarize the main result of this section as Theorem 2.118. For the diﬀerent types of nonlinear equations and systems G(u), studied in Sections 2.5, 2.6, the corresponding linearized operators are well deﬁned and bounded under the conditions in the following theorems, throughout assuming existing derivatives. Then Corollary 2.44 implies for 2 ≤ p ≤ ∞ and the H m -coercive essentially principal parts, the existence and uniqueness of solutions and a boundedly invertible A, for a H0m -elliptic a(·, ·), the Fredholm alternative and, in Chapter 3, the numerical stability, for the discertization methods studied in our books, including bifurcation and the related dynamics with respect to discrete H m norms. For natural limitations see the Introduction in Subsection 2.7.3. 2.7.2

Special semilinear and quasilinear equations

In contrast to the general quasilinear diﬀerential equations in Subsection 2.5.6 the linearization of the equations of the special cases of semilinear and quasilinear diﬀerential equations in Subsection 2.5.3 is straightforward. A: Special semilinear operators of order 2m For the equations of order 2m in (2.177), (2.191) with boundary conditions (2.178) and the conditions in (2.179) and (2.190) for f we get the strong form (the linear boundary operators are unchanged and satisfy the same complementary condition), ∂f ∂G (u0 )v := Av − (u0 )v : C 2m,γ (Ω) → C γ (Ω), ∂u ∂u ∂f ∈ C ∞ (R2 ). Av = aα ∂ α u, unif. elliptic of index 0, fu = ∂u

Gu = A − f (u),

(2.450)

|α|≤2m

For 23 order 2 we only have to replace 2m by 2 in (2.450), reduce the smoothness of fu , and choose Au, Bu and the conditions by the special form with Av :=

n

aij (x)∂ i ∂ j u, with Bu as in (2.185), and fu ∈ C γ (R2 ).

(2.451)

i,j=0

23

According to (2.449), fu ∈ C ∞ (R2 ) often means, here and below, fu (u, λ) ∈ C ∞ (R3 ).

150

2. Analysis for linear and nonlinear elliptic problems

These are exactly the linear operators in Theorems 2.37 ﬀ. with (2.83), (2.88), (2.93), (2.117), (2.181). They are studied in Subsections 2.4.2 and 2.4.3. In Theorems 2.32, 2.41, 2.52, and Proposition 2.54, the corresponding results and Fredholm alternatives are formulated. B: Special quasilinear operators of order 2 For the linearization of the special quasilinear operators in (2.193) and (2.452) we combine (2.194) and (2.452a), (2.453) of order 2m = 2 and n = 2. We recall the notation in (2.269), ∂v/∂ϑk , k ≥ 0 with = v for k = 0, and ∂aij /∂ϑ0 = ∂aij /∂u. The original form and its linearization are n = 2 : G(u) =

2

aij (x, u, ∇u)∂ i ∂ j u,

i,j=1

∂G ¯ → C γ (Ω), ¯ (2.452) (u0 ) : C 2,γ (Ω) ∂u

2 2 ∂aij ∂G (u0 )v = aij (x, u0 , ∇u0 )∂ i ∂ j v + (x, u0 , ∇u0 )∂ i ∂ j u0 ∂ k v , (2.452a) k ∂u ∂ϑ i,j=1 k=0

see Theorem 2.57, but with stronger condition for coeﬃcients, than (2.194), aij ,

∂aij ∂ϑk

¯ × R3 ), ∀i, j = 1, 2, k = 0, 1, 2. ∈ C γ (Ω

(2.453)

C: Special semilinear operators in nonlinear spaces For the last case in Subsection 2.5.3, (2.197), (2.198), we require (2.196), (2.199)– (2.202) and (2.455). Then we consider (−1)|β| ∂ β (aαβ ∂ α u) − f (u) − g (2.454) G(u) = Au − f (u) − g = |α|,|β|≤m

in (2.196) and (2.199) in nonlinear spaces. The linearized strong form is now deﬁned, with appropriate fu (u0 ), as (∂G/∂u)(u0 ) : H 2m (Ω) → L2 (Ω). Applied to v we get ∂G ∂f (u0 )v = A − (u0 ) v, with fu (u0 ) ∈ L∞ (Ω). (2.455) ∂u ∂u The conditions for the weak form are listed in (2.201), (2.202) and in Theorem 2.58. D: Semilinear nonautonomous operators For this (2.296), (2.297) we replace the previous f (u) + g(x) by f (x, u) and obtain, with A as in (2.454), the linearized strong form ∂G (u0 ) : H 2m (Ω) → L2 (Ω), with fu (·, u0 (·)) ∈ L∞ (Ω) (2.456) ∂u ∂f ∂G (u0 )v = A − (·, u0 (·)) v, fu ∈ L∞ (R2 ), G(u) = Au − f (x, u), ∂u ∂u G, A,

with Dirichlet boundary conditions. The conditions for the weak form are listed in (2.201), (2.202), (2.298)–(2.300) and Theorem 2.76. f (x, u) has to be monotone in

2.7. Linearization of nonlinear operators

151

u ∀x ∈ Ω for existence and uniqueness (or, only for existence, f (x, u)u ≥ 0 ∀x ∈ Ω, u ∈ R for large enough |u|) to get the linearization in (2.456). We summarize the conditions for linearization of the special cases of semilinear and quasilinear diﬀerential equations in Subsection 2.5.3 and the last example (2.297) in Subsection 2.5.6. Theorem 2.119. Linearization for diﬀerent problems: We consider the special semiand quasilinear equations in the above cases A, B, C, D and the necessary conditions for their elliptic and bounded linearizations listed above. Then the original problems and their linearizations have the form ∂f ∂G (u0 )v = Av − (u0 )v, for A, C, and (2.457) ∂u ∂u ∂G ∂f (u0 )v = Av − (·, u0 )v for D, and (2.452) for B. G(u) = Au − f (·, u) = g, ∂u ∂u G(u) = Au − f (u) = g,

Under the listed conditions all the linearizations in (2.457) are bounded linear operators, induced by compact perturbations of H0m -coercive bilinear forms, hence Corollary 2.44 holds, e.g. implying the Fredholm alternative.

2.7.3

Divergent quasilinear and fully nonlinear equations

We begin with the general divergent quasilinear equations studied in Subsections 2.5.4 and 2.5.6, see, e.g. (2.206)–(2.208), (2.228)–(2.229), (2.270)–(2.275). Here the linearization for these equations makes sense if we exclude some cases for the following reasons: Since many of the existence, uniqueness and regularity results are only known for the case H m (Ω), generalizations to the W m,p (Ω, Rq ) situation would be worthwhile. But for the appropriate W m,p (Ω) setting and for 1 ≤ p < 2, the bilinear forms, a(u, v), for the linearized operator, G (u0 ), usually are not even deﬁned, cf. (2.459). So we exclude 1 ≤ p < 2. For 2 ≤ p ≤ ∞, and a satisﬁed ellipticity condition (2.461), the bilinear form, ap (u, v), for the principal part of the linearized operators turns out to be only H0m -coercive. The compact embedding of W m,p (Ω) into H m (Ω) allows us to formulate numerical results for a discrete H m (Ω) norm, cf. Corollary 2.44. But, the above (linear) H0m -coercivity arguments and the corresponding C 2,γ (Ω) for a Schauder type approach do not always apply to m > 1. The original conditions for these quasilinear elliptic equations are related, but not identical with those garanteeing the H0m -coercivity of their linearized principal part, see Lemma 2.77 and the following discussion. So, for some cases, based upon monotony arguments, we impose modiﬁed conditions, cf. Proposition 2.121 and Theorem 2.122. We discuss the relations between these conditions. We had assumed Ω as a bounded nonempty measurable domain in Rn . The weak form, see (2.274), for general m is deﬁned for u, v ∈ V = W0m,p (Ω), as a(u, v) = Gu, vV ×V = Aα (x, u, . . . , ∇m u)∂ α vdx = f, vV ×V (2.458) Ω |α|≤m

152

2. Analysis for linear and nonlinear elliptic problems

with the exact solution u0 . We linearize analogously to (2.455) and get ∂G (u0 ) : V := W0m,p (Ω) → V = W0−m,q (Ω), 1 < p < ∞, 1/p + 1/q = 1, with (2.459) ∂u ⎛ ⎞ 8 9 ∂G(u0 ) ∂Aα ⎝ w, v = (x, u0 , . . . , ∇m u0 )∂ β w⎠ ∂ α vdx, u0 , v, β ∂u ∂ϑ Ω V ×V |α|≤m

|β|≤m

w ∈ V. Although the condition G, ∂G/∂u(u0 ) : V := W0m,p (Ω) → V = W0−m,q (Ω), makes sense, the previous (∂G(u0 )/∂u)w, vV ×V needs ∂Aα /∂ϑβ (x, u0 , . . . , ∇m u0 ) ∂ β w∂ α v ∈ L1 (Ω) ∀u0 , v, w ∈ W0m,p (Ω), requiring p ≥ 2. The usual ellipticity condition, and sometimes the not quite usual ellipticity condition (2.301) is imposed, with d ∈ RNm−1 , Θm ∈ Rnm λ(d), Λ(d), μ(Θm ), deﬁned there. This implies the usual ellipticity. Recalling (2.75), (2.79) with 2m 2m 2m 2m 2m ϑ2α and |ϑ|n /C ≤ |ϑ|n,α ≤ |ϑ|n ≤ C |ϑ|n,α we have (2.460) |ϑ|n,α := |α|=m 2m

2m

λ(d)μ(Θm ) |ϑ|n /C ≤ λ(d)μ(Θm ) |ϑ|n,α ≤

|α|=|β|=m

∂Aα 2m (x, d, Θm )ϑβ ϑα ≤ Λ(d)μ(Θm ) |ϑ|n,α . ∂ϑβ

We have to linearize in the exact solution u0 . The preceding regularity and boundedness results show that the special values for the d = (ϑβ = ∂ β u0 , |β| < m), Θm = (ϑβ = ∂ β u0 , |β| = m), and hence the λ(d), μ(Θm ), Λ(d) are bounded. Then (2.460) has the form of a standard ellipticity condition for the linearized operator, evaluated in u0 , formulated either with |ϑ|n or |ϑ|n,α , cf. (2.75) and the following Deﬁnition 2.25: ∂Aα 2m 2m (x, u0 (x), . . . , ∇m u0 (x))ϑβ ϑα ≤ Λ|ϑ|2m λ |ϑ|n /C ≤ λ |ϑ|n,α ≤ n,α . ∂ϑβ |α|=|β|=m

(2.461) For the above Example 2.72 the derivatives do exist. Similarly to the Monge–Amp`ere equation, the ellipticity of G (u0 ) depends upon u0 . Example 2.120. 1. For n = 1, 2, 3, and u ∈ H 2 (Ω) ∩ H01 (Ω) the linearization of Gu = −Δu + α

n i=1

sin(u)∂ i u = f ∈ L2 (Ω) on Ω,

2.7. Linearization of nonlinear operators

yields G (u0 )v = −Δv + α

n i=1

⎛ ⎝

n

153

⎞ cos(u0 )∂ i u0 ∂ j v + sin(u0 )∂ i v ⎠.

j=1

2. For 2 ≤ p < ∞, 1/p + 1/q = 1, s ∈ R, s ≥ 0, we obtain Gu = −

n

∂ i (|∂ i u|p−2 ∂ i u) + su = f ∈ Lq (Ω) on Ω,

i=1

G (u0 )v = −

n

∂ i |∂ i u0 |p−2 1 + (p − 2)(∂ i u0 )−1 ∂ i v

i=1

+lower order derivatives for v, with a singularity in ∂ i u0 (x) = 0. 3. The linearization of the minimal surface equation, Gu = (1 + |∇u|2 )Δu +

n

∂ i u∂ j u∂ i ∂ j u = 0 on Ω

i,j=1

yields G (u0 )v = (1 + |∇u0 |2 )Δv +

n

∂ i u0 ∂ j u0 ∂ i ∂ j v

i,j=1

+lower order derivatives for v. Some of the results for the general divergent quasilinear equations of order 2 in Subsection 2.5.4 are based upon the ellipticity conditions in the form of (2.460) with m = 1, see (2.209), (2.239), for the linearized operators. In Theorems 2.61 and 2.62 existence and uniqueness results are summarized under appropriate growth conditions listed there. Theorems 2.63 and 2.64 yield existence and regularity results in the C m,γ (Ω) setting, combining appropriate growth and H0m -coercivity conditions. General divergent quasilinear equations of order 2m are studied in Subsection 2.5.6, see, e.g. (2.270)–(2.275). Conditions (2.284)–(2.289) are chosen in diﬀerent combinations. Comparing condition (2.461), see Lemma 2.77, its evaluation in (2.461) is related to a modiﬁed H0m monotony (2.294) in Theorem 2.75. This is relevant for the existence of a solution u0 for (2.295), but not for the ﬁner results of Theorem 2.73 based upon the nonlinear H0m -coercivity in (2.286). Hence, if we impose the conditions of Theorem 2.75 and Lemma 2.77, in particular (2.301), we obtain an H0m (Ω)-coercive linearization of the principal part. In fact we ﬁnd Proposition 2.121. The monotonicity conditions (2.287), (2.288) and (2.289) for quasilinear equations imply, for p = 2, Θ≤m = (d, Θ≤m ), and Aα (x, d, Θm ) ∈ C 1 (Rnm )

154

2. Analysis for linear and nonlinear elliptic problems

with respect to Θm , in a neighborhood of Θm , classical and strong and uniform H0m (Ω)ellipticity. This implies, for uniform ellipticity, an H0m (Ω)-coercive linearization of the principal part. Proof. We only consider (2.289); the other cases are similar. In (2.289) we choose the , and obtain = d, Θ≤m above Θ≤m and Θ≤m 1 1 γ p m α α = (Aα (x, d, Θm |ϑ1 − ϑγ | . 1 ) − Aα (x, d, Θ )) (ϑ1 − ϑ ) ≥ C |α|≤m

|α|=m

|γ|=m

m This implies, by Theorem 1.43 and for suﬃciently small (ϑγ )|γ|=m := Θm 1 −Θ , ⎞p/2 ⎛ (∂Aα (x, d, Θm )/∂β)ϑα ϑβ ≥ C/2 |ϑγ |p ≥ C2,p C/2 ⎝ |ϑγ |2 ⎠ , |α|=|β|=m

|γ|=m

|γ|=m

γ 2 p/2 for the equivalent norms in Rnm . Skipping since |γ|=m |ϑγ |p ≥ C2,p |γ|=m |ϑ | γ 2 the middle term in this equation and dividing it by yields |γ|=m |ϑ | ⎞ ⎞p/2−1 ⎛ ⎛ (∂Aα (x, d, Θm )/∂β)ϑα ϑβ / ⎝ |ϑγ |2 ⎠ ≥ C2,p C/2 ⎝ |ϑγ |2 ⎠ . |α|=|β|=m

|γ|=m

|γ|=m

For p = 2 the right-hand side in this inequality can become arbitrarily small or large. So (2.289) implies uniform ellipticity only for p = 2. We have excluded the case W0m,p (Ω), 1 ≤ p < 2. The continuous embedding of W0m,p (Ω), 2 ≤ p ≤ ∞, into H0m (Ω), allows, under slightly enforced conditions, a H0m (Ω)-coercive linearization and full convergence results with respect to discrete H m (Ω) norms later on for the diﬀerent discretization methods. For quasilinear equations of order 2m, m ≥ 1, in Subsection 2.5.6 the results refer to the nonlinear H0m -coercivity in (2.286) and the monotonicity conditions (2.287)– (2.289), but not to ellipticity. In particular, linearization via the standard (2.461) is certainly not equivalent to (2.289). If the derivatives of the Aα are bounded and m − ΘNm is suﬃciently ΘN 1 small, thenp (2.461) implies with the Taylor formula, a modiﬁed (2.289) with |γ|≤m |ϑγ − ϑγ1 | on the right-hand side, instead of the usual γ 2 γ |γ|=m |ϑ − ϑ1 | , cf. the proof of Proposition 2.121. In this sense, (2.461) implies uniform monotonicity (2.294), and H0m -coercivity for the principal part of G (u0 ). With the standard techniques and Corollary 2.14 this shows that (2.59) is satisﬁed, hence G is boundedly invertible. We have seen that for quasilinear problems of order 2, ellipticity conditions as in (2.461) are required anyway. For orders 2m, m ≥ 1, they are pretty close to the standard conditions. We now prove that for 2 ≤ p ≤ ∞, the derivatives G (u0 ) of the nonlinear G(u) in W0m,p (Ω) are bounded and the principal parts are H0m (Ω)-coercive. Theorem 2.122. Bounded linearization in W0m,p (Ω), 2 ≤ p ≤ ∞, H0m (Ω)-coercive principal part, Fredholm alternative: In addition to the conditions (2.284), (2.285),

2.7. Linearization of nonlinear operators

155

for the original quasilinear operator in (2.290), we impose for its linearization (2.459) the Legendre condition (2.461). Either for the solution u0 ∈ W m,p (Ω) of (2.295) or close to it, 24 cf. (1.49), ∂Aα let for 2 ≤ p ≤ ∞ : h := (·, u0 , . . . , ∇m u0 ) ∈ L∞ (Ω), or h ∈ Lp/(p−2) (Ω), ∂ϑβ ∂Aα β (·, ϑ , |β | ≤ m) ≤ C ∀ϑβ ∈ R, |ϑβ − ∂ β u0 | ≤ ρ > 0. (2.462) or let ∞ ∂ϑβ L (Ω) Then the linearization in (2.459) is bounded. We use the compact embedding of W m,p (Ω), 2 ≤ p ≤ ∞, into H m (Ω), and obtain G (u0 ) and its principal part as H m (Ω)-elliptic and H0m (Ω)-coercive, respectively. Remark 2.123. 1. Any types of estimates based upon linearization for u ∈ W k,p (Ω), 2 ≤ p ≤ ∞, are therefore still possible, however with respect to the H k (Ω) norm. This is particularly important for discretization methods. Then stability, consistency, and convergence are valid with respect to the discrete, e.g. H k (Ωh ) norm. 2. The results of Corollary 2.44 imply, e.g. existence and uniqueness of solutions and a boundedly invertible A, for the H0m -coercive and the Fredholm alternative for the elliptic case. Regularity results as indicated in Subsections 2.5.4, 2.5.7, and 2.5.6, are valid under the corresponding conditions. Proof. With ∂ β w, ∂ α v ∈ Lp (Ω), ∂Aα /∂ϑβ (·, u0 , . . . , ∇m u0 ) ∈ L∞ (Ω) or, for p ≥2, in Lp/(p−2) (Ω) and the continuous embedding of Lp (Ω) into L2 (Ω), the following Ω are deﬁned. By (1.49), h=

∂Aα (x, u0 , .., ∇m u0 ) ∈ Lp/(p−2) (Ω), ∂ β w, ∂ α v ∈ Lp (Ω) ⇒ h∂ β w∂ α v ∈ L1 (Ω) ∂ϑβ

∂Aα m β α and (x, u0 , .., ∇ u0 )∂ w∂ vdx β Ω ∂ϑ ∂Aα m β α ≤ C β (x, u0 , .., ∇ u0 ) p/(p−2) ∂ wLp (Ω) ∂ vLp (Ω) hence, ∂ϑ L (Ω) |α|,|β|≤m

≤C

∂Aα m β α (x, u , .., ∇ u )∂ w∂ vdx 0 0 β Ω ∂ϑ

∂ β wLp (Ω) ∂ α vLp (Ω) ≤ CwW m,p (Ω) vW m,p (Ω) , (2.463)

|α|,|β|≤m

hence (∂G/∂u)(u0 ) = G (u0 ) is bounded for 2 ≤ p ≤ ∞, but not for 1 ≤ p < 2. 24 Since the W m,p (Ω) are continuously embedded into W j,s (Ω) for 1/p − 1/s ≤ (m − j)/n the conditions for ∂Aα /∂ϑβ can be relaxed for |α|, |β| ≤ m − 1.

156

2. Analysis for linear and nonlinear elliptic problems

Theorem 2.43 with (2.461) for m ≥ 1 imply the H0m (Ω)-coercivity of the principal part, Gp (u0 ), and hence the H0m (Ω)-ellipticity of the operator. The continuous embedding of W m,p (Ω) → H m (Ω), p ≥ 2, for bounded Ω allows considering u ∈ W m,p (Ω) as u ∈ H m (Ω), with the H m (Ω) norm. These H0m -coercivity results imply all the other claims. In other cases the Fredholm alternative can be directly veriﬁed for the quasilinear problem as formulated by Zeidler [677, 678]. For fully nonlinear equations in (2.306), the G(u) and G (u0 )u have the form G, G (u0 ) : U = W 2,p (Ω) ∩ W01,p (Ω) → V = Lq (Ω), u, u0 ∈ D(G) ⊂ U s.t.

(2.464)

w = (x, z, p, r), w0 (x) := (x, u0 (x), ∇u0 (x), ∇2 u0 (x)) ∈ D(Gw ) ∀x ∈ Ω G(u0 (·)) = Gw (·, u0 (·), ∇u0 (·), ∇2 u0 (·)) = 0, hence, n n ∂Gw ∂Gw ∂Gw i (w0 )u + (w0 )∂ u + (w0 )∂ i ∂ j u, G (u0 )u = ∂z ∂pi ∂rij i=1 i,j=1

∂Gw ∂Gw ∂Gw (w0 (·)), (w0 (·)), (w0 (·)) ∈ Lp/(p−2) (Ω) ⊂ L∞ (Ω). (2.465) ∂z ∂pi ∂rij n The term i,j=1 in (2.464), its principal part, is used in (2.307) deﬁning the uniform ellipticity of G: For w ∈ Uw0 a neighborhood of w0 , we required for 0 = ϑ ∈ Rn for

0 < λ|ϑ|2 ≤

n ∂Gw (w)ϑi ϑj ≤ Λ|ϑ|2 ∀w = (x, u, p, r) ∈ Uw0 ⊂ D(Gw ). ∂r ij i,j=1

(2.466)

We do not explicitly formulate the case of nondivergent quasilinear equations. This is a special case of (2.306) or (2.464) and, compared to (2.459), G (u0 ) requires only slight changes. For our later studies in discretization methods it is important that, in contrast to the original nonlinear problem, it is possible to transform the linearization in (2.464) and in (2.467) below into the standard weak form. For the Skrypnik [591] setting, the derivative G (u0 )u for the fully nonlinear operator G(u) = G(0, u) of order 2m is obtained. In a Hilbert space, evaluated at u0 applied to u, see (2.316) and the notation in (2.269), it has the form u, u0 ∈ D(G), u, u0 ∈ U with w0 (x) := (0, x, ∂ α u0 (x), |α| ≤ 2m)

(2.467)

hence, G (u0 ) : U = H 2m (Ω) ∩ H0m (Ω) → V = L2 (Ω), G (u0 )u =

∂ α Gw (w0 ) ∂ α u

|α|≤2m

and we need, for the weak form, ∂ α Gw (w0 (·)) ∈ L∞ (Ω) ∀α : |α| ≤ 2m.

(2.468)

2.7. Linearization of nonlinear operators

157

We recapitulate the modiﬁed ellipticity condition (2.318): There exist constants 0 < k, 0 < μ < 1, 0 < Λ < ∞ such that (2.469) ∀t ∈ [0, 1], v ∈ U : G(t, v) = 0 implies vC 2m,γ (Ω) ¯ ≤ k, and in Ω the ∂ α Gw (t, ·, v, . . . , ∇2m v)(x) ϑα ≤ C|ϑ|2m . ellipticity condition k −1 |ϑ|2m ≤ |α|=2m

A combination of the last proof with these results and Theorem 2.33 yields Theorem 2.124. Elliptic bilinear form, Fredholm alternative: 1. For the linearization (2.464) of the fully nonlinear diﬀerential operator of order 2m = 2 in (2.306) we require: In addition to the conditions for the data of the original problem, listed in Theorems 2.79 and 2.80, e.g. (2.469), assume 2 ≤ p ≤ ∞, and (2.465), for any u0 ∈ U in (2.464) with w0 (x) ∈ D(Gw )∀x ∈ Ω. Then the linearization in (2.464) is bounded in W0m,p (Ω, Rq ) with an H0m (Ω, Rq )-coercive principal part for 2 ≤ p ≤ ∞. 2. This remains correct for order 2m, m > 1, p ≥ 2 in (2.467) with the correspond¯ for |α| = 2m. In both cases, ing (2.468), (2.469) for ∂ α Gw (w0 (·)) ∈ C(Ω) Corollary 2.44 is valid. 2.7.4

Quasilinear elliptic systems of orders 2 and 2m

Here we are in a situation similar to Subsection 2.7.3, now starting in the H0m (Ω, Rq ) setting. Most of the existence, uniqueness and regularity results are known only for this case. Generalizations to the W0m,p (Ω, Rq ) situation are possible and worthwhile. In fact results as in Theorem 2.122 form the basis for numerical methods for the W0m,p (Ω, Rq ) setting, opening the possibility for research into new problems beyond the previous scope. The same extensions are possible for fully nonlinear equations and systems. We have studied quasilinear elliptic systems as Euler equations of a variational problem and the general case. We start with the Euler equations and formally linearize 0 (x) := (x, u0 (x), ∇ u0 (x)) or this system (2.362) in a generalized solution u0 using w w(x) to get, compare (2.377) ﬀ., % & Fp (w(x))∇ v + Fz (w(x)) v dx ∀ v ∈ H01 (Ω, Rq ) G( u), v = Ω

G ( u0 ) u, v =

% & Fpz (w 0 (x)) u + Fpp (w 0 (x))∇ u ∇ v

Ω

% & + Fzz (w 0 (x)) u + Fzp (w 0 (x))∇ u v dx (2.470) ' q n n i aik pl (w ak z (w 0 (x))uj + 0 (x))∂ l uj ∂ k vi = Ω i,j=1 k=1

+

j

l=1

j

n i & % i a0 pl (w a0 zj (w 0 (x))uj + 0 (x))∂ l uj vi dx = 0. l=1

j

158

2. Analysis for linear and nonlinear elliptic problems

We have to impose, similarly to (2.462), slightly stronger conditions than the original (2.372)–(2.374) or (2.372), (2.375)–(2.376), and (2.377)–(2.379) or (2.377), (2.378), (2.380) for the diﬀerentiability of J and the regularity of solutions of (2.362). For 1 (x), w 0 (x) we require for p = q = 2 and general u1 or our solution u0 and the above w 0 β l nq ∈ R : Assume there exists, similarly to (2.462) and the usual ∂ u = u, ϑ , ϑj ∈ R, ϑ 0 , for the solution, such that (2.371), a neighborhood Uw0 of w 0 (x)) ∈ L∞ (Ω), and for w 1 ∈ Uw0 ∃λ > 0 u0 ∈ H 1 (Ω, Rq ) : Fpp , Fpz , Fzz (w s.t.

q

n

k l nq 2 Fpki plj (w(x))ϑ (ellipticity condition). i ϑj ≥ λ|ϑ|nq ∀ ϑ ∈ R

(2.471) (2.472)

i,j=1 k,l=1

Theorem 2.125. Quasilinear operators of order 2 with H01 -coercive linearization: For the quasilinear operator G( u) and its linearization, G ( u0 ), cf. (2.470), assume (2.471). Then G ( u0 ) : H01 → H0−1 is a bounded linear operator. Its principal part is uniformly H01 (Ω, Rq )-coercive if (2.472) is satisﬁed, hence G ( u0 ) is uniformly elliptic. As in Theorem 2.122 this remains correct for W02,p (Ω, Rq ), 2 ≤ p ≤ ∞. The full machinery of Corollary 2.44 applies, e.g. with respect to existence and uniqueness. Similarly as before we now diﬀerentiate the weak form (2.425) of a general quasilinear elliptic system. This yields ∂ |β| aα 0 ≤m 0 β α (·, ∂ u (x))∂ u(x), ∂ v (x) dx. (2.473) G ( u ) u, v = ∂ϑβ Ω q |α|,|β|≤m

This is the linear form (2.392) of the system of order 2m, now restricted to u, u0 ∈ H m (Ω) instead of the above u ∈ W m,p (Ω) for m = 1. The diﬀerentiation is correct if for the solution ∂ aα ≤m 0 u0 ∈ H m (Ω) : Aα,β (·) := (·; ∂ u (·)) ∈ L∞ (Ω), e.g. |Aα,β (·)| < Λ. ∂ϑβ Theorem 2.126. Quasilinear operators of order 2m with H0m -coercive linearization: Under these conditions, the operator G of the quasilinear system (2.425) of order 2m is diﬀerentiable in u0 ∈ H m (Ω). The weak linearized operator G ( u0 ) : H0m (Ω) → H0−m (Ω) has the form (2.473). Let the strong Legendre–Hadamard condition (2.394) ¯ for |α| = |β| = m. Then the be satisﬁed for the above (∂ aα /∂ϑβ )(·, ∂ ≤m u0 (x)) ∈ C(Ω) m principal part of G ( u0 ) is H0 (Ω)-coercive and G ( u0 ) is elliptic. As in Theorem 2.125 this remains correct for W0m,p (Ω, Rq ), 2 ≤ p ≤ ∞, and allows us the full machinery of Corollary 2.44. 2.7.5

Linearizing general divergent quasilinear and fully nonlinear systems

In Subsection 2.6.8 we have already indicated that for systems of q fully nonlinear equations no results seem to be listed in the consulted monographs. We had only hinted at Yang [671], Fanciullo [311], Belopolskaya [79, 80], Chen and Wu [170] and Li and Liu [475].

2.7. Linearization of nonlinear operators

159

In this situation, linearization cannot be used for theoretical results. However, it might be the only possibility, for obtaining existence and bifurcation results by combining the FEMs later on for these problems with numerical continuation techniques. In fact, the linearized operators do ﬁt the linear elliptic systems studied in Subsection 2.6.5 For convenience, we repeat the necessary generalizations, compare (2.446), (2.447). For v , u = (u1 , . . . , uq ) ∈ W = H0m (Ω, Rq ) we have to apply the ∂ l , l = 0, . . . , n, ∂ α , |α| ≤ 2m, to the u, uj , vi , i, j = 1, . . . , q and use the notation in (2.389). Accordingly, (2.306) and (2.317) have to be reinterpreted as q equations in strong form for the q components of u, deﬁned in U := H0m (Ω, Rq ) ∩ H 2m (Ω, Rq ), W := H0m (Ω, Rq ), V := L2 (Ω, Rq )

(2.474)

with the obvious norms. For systems of order 2 with m = 1 in (2.474), we obtain G : D(G) ⊂ U → V, G( u(·)) := Gw (·, u(·), ∇ u, ∇2 u(·)), G( u0 ) = 0.

(2.475)

0 (x) := The linearization G ( u0 ) : U → V in u0 applied to u has the form, with w 0 0 2 0 w q n q n (x, u (x), ∇ u (x), ∇ u (x)) ∈ D(G ) ∀x ∈ Ω, p = (pk ∈ R )k=1 , r = (rkl ∈ R )k,l=1 , n n ∂Gw 0 ∂Gw 0 k ∂Gw 0 k l (w ) u + (w )∂ u + (w )∂ ∂ u, G ( u ) u = ∂z ∂pk ∂rkl

0

k=1

(2.476)

k,l=1

and we require ∂Gw 0 ∂Gw 0 ∂Gw 0 (w (·)), (w (·)), (w (·)) ∈ L∞ (Ω, . . .). ∂z ∂pk ∂rkl Again, in contrast to the original nonlinear problem, it is possible to transform the linearization in (2.476), with the usual partial integration, into the standard weak form. For the strong operator G ( u0 ) : U → V we introduce new coeﬃcients Akl . This yields the strong bilinear form as (·, ·), and conditions for the partials of Gw : G ( u0 ) : U → V = V , as (·, ·) : U × V → R, and with δ0k , (−1)k>0 we ﬁnd G ( u0 ) u :=

n

(−1)k>0 ∂ k (Akl ∂ l u)

k,l=0

withAkl = Akl (w 0 (·)) ∈ W 1−δk,0 ,∞ (Ω, Rq×q ), Alk = Akl = −

∂Gw 0 (w ) for k, l = 1, . . . , n, ∂rkl

∂Gw ∂Gw 0 (w )+ ∂k (w 0 ), ∂pl ∂rkl n

A0l =

k=1

Al0 = 0 for l = 1, . . . , n, A00 =

∂Gw 0 (w ), then the ∂z

(2.477)

G ( u0 ) : U → V = V and as ( u, v ) := (G ( u0 ) u, v )V ∀ u ∈ U, v ∈ V are bounded.

160

2. Analysis for linear and nonlinear elliptic problems

With (·, ·) := (·, ·)V and ·, · := ·, ·W×W the strong and weak solutions u0 ∈ U and u0 ∈ W, respectively, of the second order linearized systems are determined by, cf. (2.476), (G ( u0 ) u, v ) =

n

(−1)k>0 ∂ k (Akl ∂ l u), v

k,l=0

a( u, v ) := G ( u ) u, v := 0

n

q

= (f , v ) ∀ v ∈ V,

(2.478)

(Akl ∂ l u, ∂ k v )q dx = f , v , ∀ v ∈ W

Ω k,l=0

and with Akl = Akl (w0 ) ∈ L∞ (Ω, Rq×q ) for the weak form.

(2.479)

The uniform Legendre condition deﬁnes a uniformly elliptic system: n

2≤ λ|ϑ|

2 ∀ϑ ∈ Rnq , |ϑ| 2= (Akl (x)ϑl )ϑk ≤ Λ|ϑ|

q n

ϑki

2

.

(2.480)

i=1 k=1

k,l=1

This implies the H01 (Ω, Rq×q )-coercivity of the principal part ap ( u, v ). For order 2m, m > 1, we again reinterpret, now (2.317), as a nonlinear system of q equations in U := H0m (Ω, Rq ) ∩ H 2m (Ω, Rq ). We update the above notations in (2.335) as: w := (x, ϑγ , |γ| ≤ 2m) ∈ D(Gw ), u, u0 ∈ D(G), w 0 (x) := (x, ∂ γ u0 (x), |γ| ≤ 2m). Then a systems of order 2m has the form G : U → V G( u(·)) := Gw (·, u(·), · · · , ∇2m u(·)) = Gw (·, ∂ γ u(·), |γ| ≤ 2m), G( u0 ) = 0. (2.481) The linearization G ( u0 ) : U → V = V in u0 applied to u has the form, G ( u0 ) u =

∂Gw ∂Gw 0 0 γ ( w )∂ u , with (w (·)) ∈ L∞ (Ω, Rq ) ∀|γ| ≤ 2m. (2.482) ∂ϑγ ∂ϑγ

|γ|≤2m

Introducing new coeﬃcients Aαβ yields the strong and weak bounded linear operators and bilinear forms Gs ( u0 ), G ( u0 ), and as (·, ·), a(·, ·), and conditions for the partials of Gw : as (·, ·) : U × V → R, a(·, ·) : W × W → R, G ( u0 ) : W → W , ∂Gw (−1)|α| ∂ α (Aαβ ∂ β u) := (w 0 )∂ γ u, with Gs ( u0 ) u := ∂ϑγ |α|,|β|≤m

|γ|≤2m

0 (·)) ∈ W |α|,∞ (Ω, Rq×q ), ∀|α|, |β| ≤ m, e.g., Aαβ = Aαβ (w Aαβ = (−1)m

(2.483)

w

∂G (w 0 ) = Aβα for |α| = |β| = m, and with ∂ϑαβ

(·, ·) := (·, ·)V , ·, · := ·, ·W×W we get the strong solution u1 ∈ U from

2.7. Linearization of nonlinear operators

161

(f , v ) = as ( u1 , v ) := (Gs ( u0 ) u1 , v ) ∀ v ∈ V or the weak solution u1 ∈ W, from 1 0 1 (Aαβ ∂ β u1 , ∂ α v )dx = f , v , a( u , v ) := G ( u ) u , v = Ω |α|,|β|≤m

∀ v ∈ H0m (Ω, Rq ) and with Aαβ ∈ L∞ (Ω, Rq×q ) for this weak form.

(2.484)

Here we have to impose the uniform Legendre–Hadamard condition: ∃Λ > λ > 0 : λ|η| |ϑ| 2

2m

≤

n

η¯Aαβ (w 0 (x))η ϑβ ϑα ≤ Λ|η|2 |ϑ|2m ∀ϑ ∈ Rn , η ∈ Cq .

(2.485)

|α|,|β|=m

To enforce the H0m -coercivity of the principal part, see Theorem 2.104, we need, in addition to the above Aαβ ∈ W |α|,∞ (Ω, Rq×q ), ∀|α|, |β| ≤ m, ¯ Rq×q ) ∀|α| = |β| = m > 1. 0 (·)) ∈ C(Ω, Aαβ (w

(2.486)

A combination of Theorems 2.104 and 2.33 yields the following Theorem 2.127. Bounded linearization, Fredholm alternative for fully nonlinear systems in W 2m,p (Ω, Rq×q ), 2 ≤ p ≤ ∞: 1. For the linearizations in (2.478), and (2.483), (2.484) of the fully nonlinear systems in (2.475), and (2.481), respectively, we require, for the solution u0 in (2.478) and (2.481), (2.483), the conditions in (2.477),(2.483). Furthermore, let Aij ∈ L∞ (Ω, Rq×q ) for m = 1, and Aαβ ∈ L∞ (Ω, Rq×q ) for |α| = |β| ≤ m, (2.487) and assume the conditions for the data of the original problem in Subsection 2.6.8. Then the above linearizations are bounded. 2. Additionally, we assume the strong Legendre condition (2.480) for m ≥ 1 = q and q ≥ 1 = m with (2.486) for m > 1 and the strong Legendre–Hadamard condition, (2.485), and (2.486) for m, q > 1. Then the above linearizations are even elliptic and the principal parts are H0m (Ω, Rq )-coercive. Thus Corollary 2.44 is valid for p ≥ 2. The strong and weak bilinear forms coincide in u ∈ H 2m (Ω, Rq ) ∩ H0m (Ω, Rq ), m ≥ 1, and v ∈ H0m (Ω, Rq ) if furthermore Akl ∈ W 1−δk0 ,∞ (Ω, Rq×q ) for m = 1 and Aαβ ∈ W |α|,∞ (Ω, Rq×q ) for m > 1. We discuss the problem of how nonexactly satisﬁed diﬀerential and boundary equations perturb the exact solution. This is an important problem for discretization methods. In particular, the mesh-free methods in [120] usually aim for a discrete solution of an approximate original problem. Similarly to Section 5.2, we introduce an operator F := (G, B) : D(G) ∩ D(B) ⊂ U → V × Z with diﬀerential and boundary operators, G amd B as components. We restrict the discussion here to, cf. (2.158), linear special Dirichlet systems. In Theorems 1.29, 2.50 we have extended functions deﬁned on the boundary to the whole domain, Ω. In Lemmas 2.26, 2.27, this was used for reducing the inhomogeneous to homogeneous boundary conditions. Similar strategies are possible for some more general, in particular nonlinear boundary conditions as well. We leave that to the interested reader. Note that for fully nonlinear problems (in strong

162

2. Analysis for linear and nonlinear elliptic problems

m−j+1−1/2 form) we have U = H 2m (Ω, Rq ) and V = L2 (Ω, Rq ), Z = Πm (∂Ω, Rq ). j=1 H The solutions are denoted for all cases q ≥ 1, as u0 . For all the other problems (in weak form) we have U = W m,p (Ω, Rq ) and V = W −m,p (Ω, Rq ), 1/p + 1/p = 1, m m−j+1−1/p,p q (∂Ω, R ). We combine Theorems 1.48 and 1.43 with our Z = Πj=1 W results for boundedly invertible linearizations. Therefore we present, in the next theorem, estimates for the norm of the diﬀerence ˜0 U , with the exact and the perturbed solutions, hence for u0 − u

u0 ) = (G(˜ u0 ), B(˜ u0 )) = (g, ϕ). (2.488) F (u0 ) = (G(u0 ), B(u0 )) = (0, 0) and F (˜ For many applications, e.g. for discretization methods, results for boundedly invertible G (u0 ), and bounds for its norm, are more directly available than for F (u0 ). ˜0 U , in terms of Therefore we present in the next theorem estimates for u0 − u (F (u0 ))−1 U ←V×Z and of (G (u0 ))−1 U ←V . Theorem 2.128. Error estimates for perturbed elliptic problems: With the diﬀerential and boundary operators, G and B, let the operator F := (G, B) : D(G) ∩ D(B) ⊂ U → V × Z have a locally unique solution u0 for F (u0 ) = 0, and let F ∈ C 1 in a neighborhood of u0 , hence F (u0 ) = (G (u0 ), B (u0 )) : U → V × Z exists and is boundedly ˜0 be the solutions of the exact and the perturbed problems in invertible. Let u0 and u (2.488) with small enough (g, ϕ) ∈ V × Z. Then ˜0 U ≤ 2(F (u0 ))−1 U ←V×Z (g, ϕ)V×Z . u0 − u

(2.489)

G(u) = 0, B(u) = 0 has the same locally unique solution u0 , and G (u0 ) : U ∩ N (B) → V exists in a neighborhood of u0 , and is also boundedly invertible. Then (2.490) ˜ u0 − u0 U ≤ 2(G (u0 ))−1 U ←V gV + C|ϕZ (1 + 2G (u0 )V←U ) . Proof. F is continuous in a neighborhood of u0 , hence equi continuous in a smaller closed neighborhood of u0 . Therefore, (2.489) is an immediate consequence of Theorems 1.48 and 1.43. We employ the lemmas and theorems mentioned above. In the ﬁrst step, we extend ϕ ∈ Z, deﬁned along ∂Ω, to a u ˜, deﬁned on Ω. By lemma 2.26 this yields Bu ˜ = ϕ on ∂Ω =⇒ ˜ uU ≤ CϕZ .

(2.491)

Next we estimate for the linear B, and with (2.491) and O(???) as deﬁned by (2.493), ˜) − F (u0 ) = G(˜ u0 − u ˜) − G(u0 ), B(˜ u0 − u ˜) − B(u0 ) (2.492) F (˜ u0 − u = (G(˜ u0 − u ˜) − 0, 0) = (G (u0 )(˜ u0 − u ˜ − u0 ) + O(???), 0) with ˜)V = G(˜ u0 ) − G (u0 )˜ u + O(???)V ≤ (gV + 2G (u0 )V←U CϕZ ). G(˜ u0 − u ˜0 , for the last inequality as We have modiﬁed (1.78), e.g. with x0 = u0 near u0 u u0 − u0 ) ≤ ˜ u0 − u0 sup G (x) − G (x0 ) =: O(???)V . G(˜ u0 ) − G(u0 ) − G (x0 )(˜ x∈u0 u ˜0

(2.493)

2.8. The Navier–Stokes equation

163

The right-hand term (2.493) is indicated in (2.492) as O(???). It gets very small and implies the previous factors 2 in the last two lines in (2.492) and in (2.494). The combination of the last two lines in (2.492) yields ˜ − u0 U ≤ 2(G (u0 ))−1 U ←V gV + 2G (u0 )V←U C|ϕZ . (2.494) ˜ u0 − u The triangle inequality ˜ u0 − u0 U ≤ ˜ u0 − u ˜ − u0 U + ˜ uU

ﬁnally yields the claim.

2.8 2.8.1

The Navier–Stokes equation Introduction

This is one of the most stimulating equations for research in mathematics, physics, and engineering. It models ﬂuid mechanics, e.g. the ﬂow of water around a vessel, and the air around a wing of an airplane. In this context (2.495) with small values of 0 < ν are interesting. Then boundary layers and turbulence develop, modeling the up-wind properties carrying an airplane. Mathematically, it was one of the ﬁrst extremely important systems of nonlinear diﬀerential equations. It required an update of the deﬁnition of elliptic or parabolic equations, and for small 0 < ν the borders between parabolic and hyperbolic problems are no longer well deﬁned. Furthermore, by varying 0 < ν and other parameters, extremely interesting regimes of bifurcation and the corresponding dynamics and chaotic phenomena originate. Still, for a lot of these problems satisfactory answers are missing. Some answers can be found in [120]. We only consider incompressible Navier-Stokes equations. From the huge number of publications we have to choose a very narrow selection of results. Further results can be found in, e.g. Kreiss and Lorenz [454], von Wahl and Grunau [378, 658, 659], K¨ otter [453], Bensoussan and Frehse [81], and in Ladyˇzenskaja and Uralceva [463–466]. Further regularity results for Navier-Stokes equations are due, e.g. to Cannone and Karch [157], Ladyzhenskaya [465], and Zubelevich [683]. Models of Navier-Stokes equations as Newtonian compressible ﬂuids in the steady and unsteady regime are studied, e.g. by Feistauer et al. [313, 315], and Novotn´ y and Straˇskraba [515]. The steady regime is discussed in Subsection 7.15.2 as well. 2.8.2

The Stokes operator and saddle point problems

We start with the Stokes operator and its properties, see Hackbusch [386, 387]. It is closely related to the stationary Navier-Stokes equation, cf. (2.320), ⎞ ⎛ n i u ∂ u + ∇ p −νΔ u + f i ⎠ ⎝ = in Ω, (2.495) G( u, p) := i=1 0 − div u

pdx = 0, Ω satisﬁes (2.5)

u = 0 on Γ = ∂Ω, Ω

164

2. Analysis for linear and nonlinear elliptic problems

and u, p and f denote the velocity, pressure n and forcing terms of an incompressible medium, v a test function with div u = i=1 ∂ui /∂xi , and ν = 1/R with the Reynolds number, R. The Stokes operator is obtained from (2.495) for ν = 1 and by deleting the nonlinear term as −Δ u + ∇ p f pdx = 0. (2.496) S( u, p) := = in Ω, u|∂Ω = 0, − div u 0 Ω We introduce Vb , W, see Subsection 2.6.2, as Vb := H01 (Ω, Rn ), V := H 1 , W := L2∗ (Ω) :=

⎧ ⎨

p ∈ L2 (Ω) :

⎩

Ω

⎫ ⎬ p(x)dx = 0 . (2.497) ⎭

Now we take the scalar product of the Stokes operator with v ∈ Vb and apply the Green’s formula for smooth enough u, v , p. With theEuclideannorm | · |n and inner n product (·, ·)n in Rn and the product norm v V := | vi H 1 (Ω) i=1 |n the ﬁrst line in (2.496) yields (−Δ u + ∇p, v )n dx = (∇ u, ∇ v )L2 − (p, div v )L2 (Ω) .

(−Δ u + ∇p, v )L2 (Ω,Rn ) := Ω

We deﬁne the bilinear and linear forms a(·, ·), b(·, ·) and (f , ·) as (∇ u(x), ∇ v (x))n dx for u, v ∈ Vb ,

a( u, v ) := Ω

b(p, v ) := − (f , v ) :=

Ω

p(x) div v (x)dx for p ∈ L2∗ (Ω),

(2.498)

(f, v )n dx for f ∈ L2 (Ω, Rn ) or ∈ V = H −1 (Ω, Rn ), cf. (2.110).

Ω

This yields for the two equations in (2.496) and for u ∈ H 2 (Ω, Rn ) ∩ Vb (−Δ u + ∇p, v )L2 (Ω,Rn ) = a( u, v ) + b(p, v ) = (f , v ), ∀ u, v ∈ Vb , p ∈ W and − q(x) div u(x)dx = b(q, u) = 0 ∀ q ∈ W, u ∈ Vb . Ω

This requires testing the ﬁrst equation by v and div u = 0 by q ∈ W. For our bounded Ω in (2.5), the a(·, ·), b(·, ·), and (f , ·) are continuous, cf. Theorem 2.43 and Proposition 2.34: |a( u, v )| ≤ Ca uV · v V , |b(p, v )| ≤ Cb v V · pW , |(f , v )| ≤ f L2 (Ω,Rn ) · v V , with f L2 (Ω,Rn ) or f V .

(2.499)

2.8. The Navier–Stokes equation

165

In the weak formulation of (2.496), we replace f ∈ L2 (Ω, Rn ) and 0 by f 1 ∈ V and f 2 ∈ W , respectively, and obtain for given f = (f 1 , f 2 ) ∈ Vb × W determine u0 ∈ Vb , p0 ∈ W such that a( u0 , v ) + b(p0 , v ) = (f 1 , v )

∀ v ∈ Vb ,

(2.500)

b(q, u0 ) = (f 2 , q) ∀ q ∈ W. Saddle point problems To obtain a situation similar to Section 2.3, we reinterpret (2.500): We replace the Vb , W, a(·, ·), b(·, ·), (f , ·), 0 in (2.497)–(2.500) by general Hilbert spaces Vb , W, continuous bilinear forms a, b and linear forms f 1 ∈ Vb , f 2 ∈ W , respectively. We call the Stokes problem as in (2.498), (2.500) and this generalized version a saddle point problem. Furthermore, we introduce X := Vb × W and xT := ( u, p), y T := ( v , q) ∈ X to obtain v u := a( u, v ) + b(p, v ) + b(q, u), , c( x, y ) := c q p

(2.501)

(f , y ) := (f 1 , v ) + (f 2 , q), f = (f1 , f2 ) ∈ X = Vb × W . Then c(·, ·) and (f , ·) are continuous bilinear and linear forms on X × X and X . Let A ∈ L (Vb , Vb ) , B ∈ L (W, Vb ) and C ∈ L(X , X ) be the linear operators induced by a(·, ·), b(·, ·) and c(·, ·), respectively. Then A B A u + Bp A B u , ), C x = ∈ L(X , X = C := p Bd 0 B d u Bd 0 9 8 A u + Bp v c( x, y ) = C x, y X ×X = , (2.502) q B d u X ×X = A u, v Vb ×Vb + Bp, v Vb ×Vb + B d u, qW ×W = a( u, v ) + b(p, v ) + b(q, u). Obviously, c(·, ·) is bounded, but it is not coercive, since c( x, x) = 0 for all x = (0, p). We formulate this saddle point problem in three equivalent forms as for givenf , = (f 1 , f 2 ) ∈ X determine x0 = ( u0 , p0 ) ∈ X such that c( x0 , y ) = f , y X ×X for all y ∈ X ⇔ C x0 = f ∈ X

(2.503)

⇔ a( u0 , v ) + b(p0 , v ) = (f 1 , v ) ∀ v ∈ Vb , b(q, u0 ) = (f 2 , q) ∀ q ∈ W, compare (2.55). The following theorem explains why we call this a saddle point problem. Let J( x) := J( u, p) = a( u, u) + 2b(p, u) − 2(f 1 , u) − 2(f 2 , p) = c( x, x) − 2(f , x) for x = ( u, p) ∈ Vb × W.

(2.504)

166

2. Analysis for linear and nonlinear elliptic problems

This J( u, p) is neither bounded from above nor from below. So a solution x0 = ( u0 , p0 ) ∈ X of (2.503) cannot be characterized by a minimum as in Theorem 2.16, but only as a saddle point, Hackbusch [387], Theorem 12.2.4. Theorem 2.129. Saddle point problem: Let a(·, ·) be symmetric and Vb -coercive. Then x0 = ( u0 , p0 ) ∈ Vb × W solves one of the equivalent problems in (2.503), if and only if J( u0 , p) ≤ J( u0 , p0 ) ≤ J( u, p0 ) for x = ( u, p) ∈ X = Vb × W ⇔ J( u0 , p0 ) = min J( u, p0 ) = max min J( u, p). u∈Vb

p∈W u∈Vb

(2.505)

Existence and uniqueness of solutions x0 = ( u0 , p0 ) ∈ X for (2.503) cannot be obtained with our previous methods. So we introduce a closed V0 and its orthogonal complement V ⊥ such that V0 = ker B d = { v ∈ Vb : B d v = 0} = { v ∈ Vb : b(q, v ) = 0∀q ∈ W}, Vb = V0 ⊕ V ⊥ . Conditions for the unique solvability of (2.503) require a slight reinterpretation. Let A00 A0⊥ with (2.506) A := A⊥0 A⊥⊥ ) , A⊥⊥ ∈ L (V⊥ , V⊥ ), A00 ∈ L (V0 , V0 ) , A0⊥ ∈ L (V⊥ , V0 ) , A⊥0 ∈ L (V0 , V⊥ ∗ ) = (0, B ∗ ), since B0∗ = 0. and an analogous B ∗ = (B0∗ , B⊥

Theorem 2.130. Unique solution for a saddle point problem: Let a(·, ·) : V × V → R, b(·, ·) : W × V → R,, and f 1 ∈ V , f 2 ∈ W be bounded and V0 , V ⊥ be deﬁned as above. Then (2.503) or equivalently (2.505) is uniquely solvable if and only if the inverses of A00 and B exist, hence V0 = ker B d = { v ∈ Vb : B d v = 0} = { v ∈ Vb : b(q, v ) = 0∀q ∈ W}, Vb = V0 ⊕ V ⊥ , V0 , V ⊥ ⊂ Vb .

(2.507)

Now we are able to modify Theorem 2.12 to this new situation. With the above A, B, C the C is boundedly invertible, if and only if the following modiﬁed Brezzi– Babuska condition or Ladyˇzenskaja–Brezzi–Babuska condition is fulﬁlled: C ∈ L(X , X ) is boundedly invertible ⇔ ∃ α, β > 0 s.t.

sup 0 = v0 ∈V0

sup 0 = v1 ∈V0

|a( v0 , v1 )|/ v0 Vb > 0 ∀0 = v1 ∈ V0 ,

(2.508)

|a( v0 , v1 )|/ v1 Vb ≥ α v0 Vb ∀ v0 ∈ V0 ,

(2.509)

sup |b(p, v )|/ v Vb ≥ βpW ∀p ∈ W.

(2.510)

0 = v ∈Vb

We do not always need all these conditions. If a(·, ·) is symmetric or Proposition 2.19 holds, then (2.508) can be skipped. The second line (2.509) implies the V0 -coercivity of a(·, ·). For other combinations, see Hackbusch [387], Lemma 12.2.9 ﬀ.

2.8. The Navier–Stokes equation

167

Solvability and regularity of the Stokes problem Hackbusch’s Lemmas 12.2.12 and 13 [387], yield (2.508), (2.509), (2.510). He refers to Neˇcas [509] for n ≥ 2, Ladyˇzenskaja [461], and Kellog and Osborn [443], see [387], Theorems 12.2.14, 18 and 19, for the following Theorem 2.131. Unique and regular solution for the Stokes problem: 1. Let Ω ∈ C 0,1 be bounded, see (2.5), C be deﬁned by (2.502) for the Stokes case with A, B as induced by (2.498). Then C is boundedly invertible, C −1 ∈ L(X , X ). So the Stokes problem −Δ u + ∇p = f 1 ∈ H −1 (Ω), − div u = f 2 ∈ L2∗ (Ω) on Ω, u = 0 on ∂Ω has a unique weak solution ( u0 , p0 ) ∈ Vb × W = H01 (Ω, R3 ) × L2∗ (Ω) for (2.500) satisfying u0 H 1 (Ω) + p0 L2 (Ω) ≤ CΩ (f 1 H −1 (Ω) + f 2 L2 (Ω) ).

(2.511)

2. Smoother data imply smoother solutions: If, in the above Stokes problem, Ω is smooth enough and f 1 ∈ H k (Ω), f 2 ∈ H k+1 (Ω) ∩ L2∗ (Ω), k ∈ N0 , we get for the solution u0 ∈ H k+2 (Ω) ∩ H01 (Ω), p0 ∈ H k+1 (Ω) ∩ L2∗ (Ω). There exists a constant C > 0, only depending upon Ω, such that u0 H k+2 (Ω) + p0 H k+1 (Ω) ≤ CΩ (f 1 H k (Ω) + f 2 H k+1 (Ω) ). For n = 2, k = 0 a bounded and convex Ω ⊂ R2 suﬃces. 2.8.3

The Navier–Stokes operator and its linearization

We want to study the linearization of the stationary Navier–Stokes operator in (2.495). Its derivative, the linearized Navier-Stokes operator, is ⎛ ⎞ n i i −νΔ w + (w ∂ u + u ∂ w) + ∇r i i ⎠. G ( u, p)(w, r) = ⎝ (2.512) i=1 − div w The special cases u ≡ 0, ν = 1, yields the above Stokes operator, S, see (2.496), and u ≡ 0, ν = 1, Sν := G (0, p) we call the ν-Stokes operator.

(2.513)

The results of the last subsection remain correct for this ν-Stokes operator, if throughout, the a( u, v ) are replaced by νa( u, v ) and ν is not too small. We present the stationary Navier-Stokes operator following Temam et al. [349, 623, 624]. They discuss the local existence of solutions for the nonstationary Navier-Stokes equations and many open problems, e.g. related to nonlocal existence. As in the transition from (2.497) to (2.498), we multiply the ﬁrst equation in (2.495) with the test function v = (v1 , . . . , vn ) and the second with q and integrate. We use Green’s formula to obtain, with a(·, ·), b(·, ·) in (2.498),

168

2. Analysis for linear and nonlinear elliptic problems

for given (f 1 , f 2 ) ∈ Vb × W determine u0 ∈ Vb , p0 ∈ W such that n −νΔ u0 + ( u0 )i ∂ i u0 + ∇ p0 , v dx Ω

i=1

(2.514)

n

= νa( u0 , v ) + d( u0 , u0 , v ) + b(p0 , v ) = (f 1 , v ) ∀ v ∈ Vb for the ﬁrst, and b(q, u0 ) = (f 2 , q)

∀ q ∈ W for the second equation;

here we use the previous a( u, v ), b(q, v ), (f , v ) in (2.498) and introduce the trilinear n n i d( u, w, v ) := (ui (∂ w), v )n dx = ui (∂ i wj )vj dx. (2.515) i=1

Ω

i,j=1

Ω

For bounded Ω, and n ≤ 4, see Temam [624] Lemma 1.2, Chapter II, Section 1. d( u, v , w) is a bounded trilinear form on Vb × Vb × Vb

(2.516)

with Vb → W ∗ := L2 (Ω, R3 ) compactly embedded. To linearize (2.514) with respect to u, we consider, for ﬁxed u, v and small w,

2 d( u + w, u + w, v ) − d( u, u, v ) = d( u, w, v ) + d(w, u, v ) + O w V v V =⇒

(G ( u, p)(w, r)), ( v , q) n+1 2 (2.517)

=

L (Ω)

νa(w, v ) + d( u, w, v ) + d(w, u, v ) + b(r, v ) b(q, w)

∀( v , q) ∈ Vb × W = X .

Now we consider the d( u, w, v ), d(w, u, v ) in (2.515). For ﬁxed u, both terms d( u, w, v ), d(w, u, v ) are continuous bilinear forms in w and v , corresponding to the operators n n i + i,j=1 wi ∂ i u dx in (2.515). Thus, the d( u, w, v ) and d(w, u, v ) ∈ i,j=1 ui ∂ w Ω

R deﬁne elements in Vb :

∈ Vb and for ﬁxed u ∈ Vb . d( u, w, ·), d(w, u, ·) ∈ Vb for variable w Hence, by Section 2.3 they induce linear continuous operators, see (2.514), := d( u, w, ·), D2 w := d(w, u, ·), u ﬁxed D1 , D2 ∈ L (Vb , Vb ) as D1 w

(2.518)

v Vb ×Vb = d( u, w, v ), D2 w, v Vb ×Vb = d(w, u, v )∀ v ∈ Vb . with D1 w,

(2.519)

The embedding I : H01 (Ω, Rn ) → L2 (Ω, Rn ) is continuous and compact, see Theorem 1.26. So (2.519) shows that = D1 I w, D2 w = D2 I w ∀ w ∈ H01 (Ω, Rn ). D1 w Hence, as a product of a compact and a continuous operator, D1 = D1 I and D2 = D2 I are compact operators.

2.8. The Navier–Stokes equation

169

Similarly to the transformation from (2.500) to (2.503) we use here, with the factor ν and the same A, B as in (2.502), νA B D1 + D2 0 . (2.520) , D := Cν := 0 0 Bd 0 Consequently, D1 + D2 is a compact perturbation of νA and D of Cν . As indicated above, this observation is only appropriate if ν is not too small, we call it moderate ν. Then (Cν )−1 (D1 + D2 ) is compact and allows us to apply Theorem 2.20. This shows that for the linearized Navier-Stokes operator Corollary 2.44, implying the Fredholm alternative, is valid. For 0 < ν ≈ 0 the convection terms D1 + D2 dominate the behavior of the equation. We apply Cν + D to the above x = ( u, p)T , multiply by y = ( v , q)T and integrate: νA u + Bp + D1 u + D2 u yields so (Cν + D) x = B d u (Cν + D) x, y X ×X = νA u, v Vb ×Vb + Bp, v Vb ×Vb

(2.521)

+ D1 u + D2 u, v Vb ×Vb + B d u, qW ×W . The exact solution x0 = ( u0 , p0 ) has to be determined from for given f ∈ X determine x0 = ( u0 , p0 ) such that (Cν + D) x0 , y X ×X = f , y X ×X ∀ y ∈ X ⇔ (Cν + D) x0 = f ∈ X

(2.522)

⇔ Cν (I + (Cν )−1 D) x0 = f ∈ X ⇔ (I + (Cν )−1 D) x0 = (Cν )−1 f ∈ X . Applying Theorems 2.20 and 2.131, we can guarantee the Fredholm alternative for Cν + D. Theorem 2.132. Unique existence, Fredholm alternative: 1. For ν > 0 not too close to 0 the linearized Navier-Stokes operator Cν + D in (2.512) is a compact perturbation of the ν-Stokes operator Cν in (2.513). This implies the spectrum of results of Corollary 2.44. 2. These statements remain correct if the Navier-Stokes operator is replaced by a generalized nonlinear saddle point problem, according to (2.521), where the nonlinearity satisﬁes (2.516) with the special Vb = H01 (Ω, Rn ), W ∗ := L2 (Ω) replaced by general compactly embedded Vb → W ∗ . So whenever −1 is not an eigenvalue of the compact operator Cν−1 D, or equivalently N (Cν + D) = {0}, the linearized Navier-Stokes operator Cν + D is boundedly invertible, and thus excludes bifurcation phenomena. On the other hand, this allows us to use continuation techniques for determining solutions of the nonlinear Navier-Stokes equations (2.495) and localizing bifurcation.

170

2. Analysis for linear and nonlinear elliptic problems

Finally we inherit the regularity from the Stokes to the Navier-Stokes operator by combining Theorem 2.56 with the second part of Theorem 2.131 to yield: Theorem 2.133. Smooth data imply smooth solutions: If, in the above NavierStokes problem, Ω is smooth enough and f 1 ∈ H k (Ω), f 2 ∈ H k+1 (Ω) ∩ L2∗ (Ω), k ∈ N0 , we get for the solution u0 ∈ H k+2 (Ω) ∩ H01 (Ω), p0 ∈ H k+1 (Ω) ∩ L2∗ (Ω). There exists a constant C > 0, only depending upon Ω, such that u0 H k+2 (Ω) + p0 H k+1 (Ω) ≤ CΩ (f 1 H k (Ω) + f 2 H k+1 (Ω) ).

Part II Numerical Methods

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3 A general discretization theory 3.1

Introduction

In this chapter we combine general Petrov–Galerkin methods, the standard approach for many elliptic discretizations, with a modiﬁcation of one of the classical general discretization methods, cf. Stetter [596]. This is a new way of proving stability for the general class of elliptic equations discussed in Chapter 2, cf. B¨ ohmer and Sassmannshausen [128] and B¨ ohmer [116]. It probably applies to all up-to-date discretization methods. As examples we demonstrate this for six diﬀerent types of discretization methods: we choose the still very alive grandfather, the diﬀerence methods, the most popular, the FEM including variational crimes, and their adaptive version, the DCGMs, then two methods still strongly in the process of development, the wavelets and the mesh–free methods, and ﬁnally, the spectral methods. The two last methods are particularly useful for bifurcation problems with their underlying symmetries. So they are described by B¨ ohmer [120]. The proof of the convergence of mesh–free methods applied to general nonlinear problems was an open problem for two decades and is solved for the ﬁrst time in [120]. Proofs of diﬀerent versions of linear problems are given by Schaback and his group, e.g. [327, 328, 401, 568, 569]. Speciﬁc properties, e.g. the adaptivity for wavelets, have to be studied additionally. We combine compact perturbations of coercive linear, hence boundedly invertible operators with additional approximation properties, see B¨ ohmer [116] and B¨ ohmer Sassmannshausen [128]. This technique applies to Navier–Stokes problems and to the bordered systems, required for numerical bifurcation. Or we directly apply this theory to nonlinear via monotone operators. This mix of diﬀerent concepts is strongly inﬂuenced by many earlier papers and books. Discrete convergence has been studied by Stummel [607–610], Reinhardt [547] and Vainikko [644]. Stetter [596] studies admissible discretization methods; the (inner and outer) admissible approximation schemes due to Petryshyn [528–530, 532] are nicely collected, presented and extended by Zeidler [677]. Keller [441] has combined, for special problems, linearization with stability and consistency for proving convergence. Anselone and Ansorge [28–31] strongly elaborate the functional analytic side of general discretizations, based in the concept of compact perturbations. In Ansorge’s iterated discretization [32] additional iteration steps in improved projection methods for more general discretization procedures, not only projection methods, and more general classes of problems, sometimes even yield superconvergence.

174

3. A general discretization theory

Brezzi, Rappaz and Raviart have applied a related version to ﬁnite dimensional approximations of nonlinear problems, including limit and simple bifurcation points, however excluding many operator equations and, necessarily, missing the new discretization methods and higher singularities [147–149]. Rappaz and Raugel extended this approach to special cases of ﬁnite dimensional approximation of bifurcation problems at a multiple eigenvalue [542–544], and to symmetric problems. Crouzeix and Rappaz gave a short survey book on numerical approximation in bifurcation theory [229]. Griewank and Reddien [366–370] prove convergence for general bifurcation scenarios under the conditions of [147–149]. Jepson and Spence [418] study bifurcation for perturbed operators, not covering discretization methods. Calos and Rappaz presented numerical approximations in nonlinear and bifurcation problems in [156]. Here, in contrast to the earlier [147–149, 542, 543], two projectors for the analysis of FEMs for nonlinear problems are appropriately introduced, but are not employed for the bifurcation analysis. Finally, Cliﬀe, Spence and Tavener [180] and Calos and Rappaz [156] give a well–written introduction to the numerical analysis of bifurcation problems with application to ﬂuid mechanics. All these papers, starting with [147], do not prove convergence for higher bifurcations and sometimes of more complicated problems such as porous media nor the new methods. Often the Navier–Stokes equation is studied. Extensions to higher bifurcation scenarios and the previously omitted other methods are discussed by B¨ ohmer et al. [122, 124, 128]. Approaches to these more complicated cases use the concepts of consistent diﬀerentiability and modiﬁed or bordered stability, introduced in [111], cf. [8, 41, 42, 113, 114, 114, 115, 127–129]. We have discussed many diﬀerent elliptic equations and systems. We consider linear and nonlinear, often parameter–dependent, problems of the form G : D(G) ⊂ U → V , G(u0 ) = 0, U, V Banach spaces, with duals U , V .

(3.1)

For some problems solutions with special features can be obtained by analytical methods. For general equations or interesting parameters in the problem this is usually impossible. So we introduce discretization methods. Let

Gh : D(Gh ) ⊂ U h → V h , Gh (uh ) = 0, U h , V h discrete Banach spaces,

(3.2)

represent a corresponding discrete problem. The classical convergence results for (nonlinear) problems (3.1), (3.2) are obtained by combining the well-known concepts of stability and consistency. Consistency requires small errors in 0 = G(u0 ) ≈ Gh uh0 + errors, with uh0 an “approximation” for u0 . The following Petrov–Galerkin and diﬀerence methods have to be included in the general discretization methods. Stability and consistency, cf. Subsections 3.3 and 3.4, are the basic concepts. They guarantee, with some technical conditions, the existence and uniqueness of the discrete solutions for uniquely existing original solutions and convergence for the discrete compared to the exact solution. Variational consistency for linear operators is more familiar in the FE and Petrov–Galerkin community, but classical consistency is more appropriate in combination with stability and nonlinear problems. It is unavoidable for fully nonlinear problems.

3.2. Petrov–Galerkin and general discretization methods

175

For the many diﬀerent discretization methods in the following chapters we choose the neutral notation of U, V and U h , V h for the Banach spaces and their discrete approximations. Ub , Vb and Ubh , Vbh is used for contrasting the full space to the subspace of functions satisfying the boundary conditions, e.g. U = H 1 (Ω), Ub = H01 (Ω). Often we work with the weak form. The strong form is important for all kinds of variational crimes, e.g. in FEMs or DCGMs or diﬀerence or mesh–free methods, and for fully nonlinear elliptic problems. Boundary conditions are imposed for unique existence. In contrast to many other books, we maintain this U versus Ub form, since nonconforming FE, DCG, and mesh–free methods violate boundary conditions and, e.g. continuity, hence Ubh ⊂ Ub . The linear and nonlinear operators and their discretizations are denoted as A, G and Ah , Gh . In the previous chapter we indicated, for most classes of problems, appropriate conditions for higher regularity of the solutions, which will allow better convergence. We will formulate the following Examples 3.1–3.4 as FEMs in Petrov–Galerkin form or as diﬀerence methods in classical form. We are aiming for a deﬁnition of a general discretization theory, applicable to all intended analytical problems. It will include these examples and all the discretization methods, considered in both books, and very probably most of the future methods as well. In addition, we will be able to avoid the dichotomy between interior and exterior approximation methods.

3.2

Petrov–Galerkin and general discretization methods

Our (stationary) nonlinear problems are usually formulated in weak, and sometimes in strong form. For unique existence, we have to impose boundary conditions, indicated by u ∈ Ub ⊂ U. So we determine u0 ∈ Ub by testing with v ∈ Vb : G : Ub → Vb ,

determine u0 ∈ Ub s.t. G(u0 ) = 0 ∈ Vb ⇐⇒ G(u0 ) ⊥ Vb e.g. Vb = H01 (Ω) ⇐⇒ < G(u0 ), v >V ×V = 0 ∀ v ∈ Vb .

(3.3)

In Remark 2.3, we have summarized the fact that for the problems in weak form in Chapter 2, V = Vb . However, here we have to use both duals for the discrete Vbh = V h , and sometimes use V , Vb as well, thus maintaining U versus Ub . Reducing the technical diﬃculties we start with a bounded linear operator, e.g. the derivative A := Gu (u1 ), in either weak or strong form. Nonlinear operators will be discussed below. For a second order linear A and f, A ∈ L(U, V ),

determine u0 ∈ Ub s.t. Au0 − f = 0 ∈ Vb n f j ∂ j vdx ∀ v ∈ Vb , ⇔ Au0 − f ⊥ Vb ⇐⇒ a(u0 , v) = f, vV ×V =

(3.4)

Ω j=0

with the standard a(u, v) = Au, vV ×V ∀ v ∈ V, ∀u ∈ U. The boundary conditions Ub and Vb , necessary for unique solvability of (3.3) and (3.4) are realized by continuous

176

3. A general discretization theory

trace operators, e.g. B(u) = 0. u ∈ Ub := {u ∈ U : B(u) = 0} ⊂ U, v ∈ Vb := {v ∈ V : B2 (v) = 0} ⊂ V, with closed subspaces Ub , Vb , and usually Vb = V , often B(u) = B2 (u). So, A ∈ L(U, V ) is a bounded linear operator, deﬁned, but usually only invertible as A ∈ L (Ub , Vb ). Before we give some general deﬁnitions and properties for the intended discretization methods, we formulate as motivation the simplest cases of the ﬁnite element and ﬁnite diﬀerence method for the Laplacian with Dirichlet boundary conditions. We choose, for the weak bilinear form, Ub , Vb = H01 (Ω), U = H 1 (Ω), and for the strong form, Ub = H 2 (Ω) ∩ H01 (Ω), V = L2 (Ω), similarly for natural boundary conditions. Example 3.1. Finite element method for the Laplacian Its weak and strong form, A and As , respectively, with the same boundary conditions for Ub = Vb , are u0 ∈ Ub = Vb = H01 (Ω) : Au0 , vV ×V = a(u0 , v) = (∇u0 , ∇v)n dx (3.5) Ω

= f, v

V ×V

∀v ∈ Vb , and for f ∈ L (Ω), u0 ∈ H (Ω) ∩ Ub : As u0 = −Δu0 = f. 2

2

Our goal is the reduction of the original problems, deﬁned and tested on inﬁnite dimensional spaces Ub , Vb , to related discrete problems, deﬁned and tested on spaces Ubh , Vbh of the same ﬁnite dimension. We choose two grids Ωh and Ωh0 in Ω = [0, 1]2 : Ωh0 := {Pi,j := (xi , yj ), xi := i/N, yj := j/N, 0 ≤ i, j ≤ N } Ω := {Pi,j , 1 ≤ i, j ≤ N − 1}, ∂Ω := h

h

Ωh0

and

(3.6)

\ Ω , cf. F igure 3.1. h

The following approximating spaces, Ubh , are deﬁned as the set of piecewise linear hat functions, φi,j , with 1 and 0 in the grid points, hence Ubh

:=

h

u :=

N −1

Ci,j φi,j : Ci,j ∈ R, φi,j (Pk,l ) = 1

for (i, j) = (k, l),

(3.7)

i,j=1

or Pi,j ∈ Ωh , and = 0 else, so ∀0 ≤ k, l ≤ N, (i, j) = (k, l) ⊂ H01 (Ω). Since Ubh ⊂ H01 (Ω), we call the uh conforming FEs. Then the weak form in (3.5) is reduced to determining the Ci,j for uh0 ∈ Ubh in (3.7), such that : h h; h h h Au0 , v V ×V = a u0 , v = ∇u0 , ∇v h n dx = f, v h V ×V ∀v h ∈ Vbh ,

Ω

∇uh0 , ∇φk,l

⇐⇒ a uh0 , φk,l = Ω

n

dx = f, φk,l V ×V ∀1 ≤ k, l ≤ N − 1.

We expect this uh0 to be a good approximation for u0 .

(3.8)

3.2. Petrov–Galerkin and general discretization methods

177

: points in Ωh : points in ∂Ωh Figure 3.1 Two-dimensional grid for Dirichlet boundary conditions.

Example 3.2. Nonconforming ﬁnite element methods are characterized by Ubh ⊂ H01 (Ω). In modiﬁed forms this ansatz is used in DCGMs as well. The basic case are the discontinuous Crouzeix–Raviart elements, cf. Figure 4.10. They are deﬁned as follows: For all Pi,j ∈ Ωh0 let Ωhm := {Qk,l are the midpoints between all neighboring Pi,j ∈ Ωh0 }

(3.9)

Ubh := span ψi,j : ψi,j (Qk,l ) = 1 for (i, j) = (k, l), and Qi,j ∈ Ω, and = 0 else and, in particular, ∀Qk,l ∈ ∂Ω ⊂ H01 (Ω). Then obviously the a uh0 , φk,l in (3.8) are no longer deﬁned. So we deﬁne a triangulation, T h , of Ω: through all points Pi,j ∈ Ωh0 the parallels to the x- and y-axes and to the diagonal x = y are drawn. Then let T h := {all these constructed open rectangular triangles T }. We introduce broken Sobolev spaces of functions and functionals on T h by UT h : = VT h := H 1 (T h ) := {u : Ω → R : ∀T ∈ T h : u|T ∈ H 1 (T )} ⊃ U,

(3.10)

UT h : = VT h := H −1 (T h ) := {fh ∈ L(VT h , R) : ∀T ∈ T h : fh |T ∈ H −1 (T )} ⊂ U .

178

3. A general discretization theory

Then we generalize A, a(uh , v h ), and

Ω

f v h dx in (3.8) by deﬁning

Ah : UT h → UT h , ah (·, ·) : UT h × UT h → R, fh (·) : UT h → R,

(3.11)

A = Ah |U , a(·, ·) = ah (·, ·)|U ×U , ah (·, ·) := ah (·, ·)|U h ×U h , h h h h (∇uh , ∇v h )n dx, Ah u , v U h ×UT h := ah (u , v ) := T

T ∈T h

fh ∈ UT h : fh , v h := fh , v h U h ×UT h := T

T

n T ∈T h

T j=0

fhj ∂ j vdx

with f (·) = fh (·)|U , f h (·) := fh (·)|U h . Finally, determine uh0 ∈ Ubh = Vbh = span {ψk,l , Qk,l ∈ Ω} ⊂ UT h , cf. (3.9), from h h ∇uh0 , ∇ψk,l n dx = f h , ψk,l U h ×UT h ∀ψk,l ∈ Ubh . (3.12) a u0 , ψk,l = T ∈T h

T

T

Note that for conforming FEMs all these new forms coincide, hence, for conforming FEMs A = Ah , a(·, ·) = ah (·, ·) = ah (·, ·), f (·) = fh (·) = f h (·). (3.13)

Again this uh0 in (3.12) should be a good approximation for u0 .

Example 3.3. Finite diﬀerence method for the Laplacian in its strong form, As u = Au = Δu, on Ω = [0, 1]2 We consider the strong form as A : Ub = C 2 (Ω) ∩ H01 → V = C(Ω). This unusual notation V = C(Ω) is a consequence of our aim, keeping the general discretization approach as close as possible to the usual Petrov–Galerkin methods, cf. below. 25 We evaluate Au in (3.5) in Pi,j ∈ Ωh , cf. (3.6), and approximate −(uxx + uyy )(Pi,j ) by the usual ﬁve-point star. This yields the diﬀerence approximation Ah , deﬁning an approximate solution uh0 ∈ Ubh with uhi,j := uh0 (Pi,j ) as indicated in Table 3.1. Let U h : = {uh : Ωh → R} ⊃ Ubh := {uh ∈ U h : uh |∂Ωh = 0}, V h = V

h

:= {uh : Ωh0 → R}

and uh Ubh := max {|uh (Pi,j )|}, f h V h := max {|(f h )(Pi,j )|}, Pi,j ∈Ωh 0

uh0 ∈ Ubh :

b

Pi,j ∈Ωh 0

(3.14)

1 Ah uh0 (Pi,j ) := 2 4uhi,j − uhi−1,j − uhi+1,j − uhi,j−1 − uhi,j+1 N

= f (Pi,j ), 1 ≤ i, j ≤ N − 1, h := 1/N, uh (Pi,j ) = 0 for i = 0, N or j = 0, N. 25 Then V would be {f : Ω → R, f of bounded variation}. At the same time, this example provides the possibility of indicating Stetter’s [596] approach in the original form: He avoids testing by v ∈ Vb and studies directly A : Ub → V = C(Ω), with the norms in (3.14).

3.2. Petrov–Galerkin and general discretization methods

179

Table 3.1: Stencil for the ﬁve-point star around Pi,j ∈ Ωh ⊂ R2 for −h2 Δh .

0

−uhi,j+1

0

−uhi−1,j

4uhi,j

−uhi+1,j

0

−uhi,j−1

0

abbreviated as

0

−1

0

−1

4

−1

0

−1

0

Note that (3.14) directly deﬁnes an Ah : Ubh → V h without any detour as for nonconforming FEMs. It will turn out to be obtainable as the strong form of a weak analogue by a less familiar testing process in Chapter 8. To this end, we will discuss discrete Sobolev spaces of grid functions. This will yield the more or less standard approach. A similar construction as for FEMs is possible: In the ﬁrst step, we reduce Ω to Ωr := [h, 1 − h]2 . Next, we approximate the previous A for functions u ∈ Ub = C 2 (Ω) ∩ H01 by applying the ﬁve-point star in (3.14), thus deﬁning (A˜h u)(P ) := ﬁve point star for u (P )∀P ∈ Ωr ⇒ Ah u(Pi,j ) = (A˜h u)(Pi,j ), (3.15) again well deﬁned for uh ∈ Ubh . This will be modiﬁed in Example 3.8 and in Chapter 8. Example 3.4. Summary of these discretizations For all three cases the original A : Ub → Vb , tested by elements of the space Vb , is transformed into Ah : Ubh → Vbh . All equations are tested with elements of Vbh and h h spaces Ub , Vb with the same ﬁnite dimension. For the above diﬀerence method this is realized by evaluation in Pi,j ; a more natural setting is presented in Chapter 8. There are still essential diﬀerences between Examples 3.1–3.3. As in (2.20), (2.22), the a(u, v) in Example 3.1, (3.5) is Ub -coercive, and, by Ubh ⊂ Ub , the a(u, v) in (3.8) is Ubh -coercive as well. This will imply the stability of the discrete Ah with respect to the · H 1 (Ω) . For conforming FEMs the variational consistency error will be shown to vanish, and the classical consistency error tends to 0. This will imply the convergence in the sense of u0 − uh0 H 1 (Ω) → 0 for h → 0. For the nonconforming FEMs in Example 3.2, and with H01 (T h ) := {u ∈ H 1 (T h ) : u(Qi,j ) = 0∀Qi,j ∈ ∂Ω} we show similarly as above that the a(u, v) in (3.11) is now Ubh -coercive. However the proof for variational or classical consistency is much more complicated, cf. Subsections 5.5.2, 5.5.6, and 5.5.7. Still, we get the same convergence, now with respect to the discrete · H 1 (T h ) . For the diﬀerence methods in Example 3.3, (3.14), the matrix for Ah is a weakly diagonal dominant matrix, a special case of an M -matrix. This implies the stability, cf. Deﬁnition 3.19, of Ah with respect to the discrete · L∞ (Ωh ) . The classical consistency is an immediate consequence of the local errors of the diﬀerence approximations, cf. Lemma 8.2. This allows convergence results with respect to the max norm, so

180

3. A general discretization theory

maxPi,j ∈Ωh u0 (Pi,j ) − uh0 (Pi,j ) → 0 for h → 0, cf. Deﬁnition 3.18. In contrast to Examples 3.1 and 3.2, with the optimal discrete · H 1 (T h ) norm, we use here, as in [596], the discrete · L∞ (T h ) norm. However, we will prove in Chapter 8 convergence for diﬀerencemethods even in a discrete H 1 Sobolev norm, · H 1 (Ωh ) , obtaining a much better u0 − uh0 H 1 (Ωh ) → 0. This is possible by transforming the h strong form (3.14) into a corresponding weak form, similar to (3.8), combined with general discretization methods. Examples 3.1–3.4 motivate the deﬁnition of a general discretization theory, applicable to all intended analytical problems and their discretization methods. The previous forms of FEMs are special cases of the so-called Petrov–Galerkin methods. Deﬁnition 3.5. Approximating spaces: Let U be a Banach space, and Ub ⊂ U, its closed subspace. Let the sequence {U h }h∈H assign to h, 0 < h ∈ H ⊂ R, inf h∈H h = 0 the 26 spaces U h . Often we use the short notation of U h . Then we call U h (a sequence of) approximating spaces for U if for well deﬁned ||u||U h , e.g. for conforming and nonconforming FEMs, and for DCGMs, dist(u, U h ) := inf ||u − uh ||U h → 0 ∀u ∈ U for h → 0. uh ∈U h

(3.16)

This is sometimes only required for u ∈ Ub and hence uh ∈ Ubh . The (sequence of ) approximating spaces Ubh for Ub , with b indicating the boundary conditions, are called conforming spaces, if Ubh ⊂ Ub for all h ∈ H. Otherwise they are called nonconforming spaces. In the latter case, either the Ubh violate the boundary conditions, or U h ⊂ U. The modiﬁcations for diﬀerence methods are discussed in Proposition 8.15 and Remark 8.16. For diﬀerence methods ||u||U h is not deﬁned, so inf uh ∈U h ||u − uh ||U h → 0 does not make sense. Nevertheless, the grid functions there do approximate Sobolev spaces in an appropriate sense as well, cf. Subsection 8.5.2. This problem motivates introducing the concept of external in contrast to internal approximation schemes, cf. Petryshyn [528–530, 532], Temam [621] and Zeidler [677]. These concepts only play a minor role for the approach chosen here. Deﬁnition 3.6. Admissible, conforming, nonconforming Petrov–Galerkin approximating spaces and general Petrov–Galerkin methods: 1. A pair of admissible approximating spaces Ubh ⊂ U h and Vbh ⊂ V h for Ub and Vb , respectively, has to have the same ﬁnite dimensions dim Ubh = dim Vbh < ∞ ∀ h.

(3.17)

2. For admissible approximating spaces, the terms conforming and Galerkin and Petrov–Galerkin approximating spaces are used if Ubh ⊂ Ub , Vbh ⊂ Vb and Ubh = Vbh and Ubh = Vbh , respectively. We denote Ubh ⊂ Ub , Vbh ⊂ Vb or Ubh ⊂ U, Vbh ⊂ V as nonconforming (Petrov–Galerkin) approximating spaces. Sometimes, all these 26 We do not want to over-formalize the notation and have chosen h ∈ H, instead of the possible hn = (h1 , · · · , hn ) ∈ H n .

181

3.2. Petrov–Galerkin and general discretization methods

cases are denoted as generalized Petrov–Galerkin approximating spaces. For a discretization method, see Deﬁnition 3.12, we use the same name as for its pair of approximating spaces. 3. As in Examples 3.1, 3.2, let the weak form of a linear, or more general quasilinear problem be given as in (3.5). Discretize it by replacing the Ub , Vb in (3.5) by its approximating spaces Ubh , Vbh as in (3.8) or (3.12). Then we call this transformation a Petrov–Galerkin (discretization) method. 4. If, in addition to (3.16) for Ub , the V h satisfy (3.16) ∀ v ∈ Vb and ∀ v ∈ V , the bi–dual space of V, then we choose V h = V h and call U h , V h a bi–dual pair of conforming or nonconforming (Petrov–Galerkin) approximating spaces. Remark 3.7. We will need the bi–duality only in Section 3.6. There we prove that for a bi–dual pair of approximating spaces, deﬁning a stable Ah , this stability and technical conditions imply the invertibility of A. This bi–duality condition is essentially satisﬁed in separable reﬂexive spaces. 27 The most important reﬂexive examples are Sobolev spaces, Wpm (Ω), with 1 < p < ∞. In these separable spaces the bi–duality is correct for all discretization methods studied here. Example 3.8. Structure of generalized Petrov–Galerkin methods 1. The form of Petrov–Galerkin methods in Examples 3.1, 3.2 are not directly applicable to ﬁnite diﬀerence methods, e.g. in Example 3.3. However, we observe that the arguments and images of the original A and their discrete counterparts Ah are related by two types of projectors: P h : Ub → Ubh map functions onto their FE approximations or restrict them to their grid values on Ωh . Similarly, we reinterpret the right-hand sides of the conforming and nonconforming FE equations (3.8) and (3.12) as an orthogonal projection of Au or f onto, e.g.

Q h : Vb → Vbh :

Q h f − f, v h V h ×V h = 0∀v h ∈ Vbh , generally,

Q h : Vb ∪ VT h → Vbh :

(3.18)

Q h f − f, v h V h ×VT h = 0∀v h ∈ Vbh . T

2. For the diﬀerence methods in ( 3.14), we use restriction operators, cf. Deﬁnition 8.13, P h : U → Uh :

Q h : V → Vbh :

P h u := {u(Pi,j ) : 0 ≤ i, j ≤ N, h := 1/N } ∈ U h , (3.19)

Q h f := {f (Pi,j ) : 1 ≤ i, j ≤ N − 1} ∈ Vbh .

3. Then the discrete operators Ah for Examples 3.1–3.3, can be formulated as

Ah = Q h Ah |Ubh cf. (3.8),(3.12), and Ah = Q h A˜h |Ubh

for (3.15).

(3.20)

This A˜h could, as well, be obtained by quadrature formulas for A. 27 Certainly, separability is not enough. This is shown by the well-known example of spaces of inﬁnite sequences: c0 ⊂ l1 ⊂ l∞ = (c0 ) , where a countable basis for c0 is not a basis for l∞ . However, a statement for Wpm (Ω), analogous to the c0 ⊂ l1 ⊂ l∞ = (c0 ) situation does not seem to be known.

182

3. A general discretization theory

4. In a similar way, we deﬁne the nonlinear discrete operators, Gh , replacing A by G, as

˜ h |U h Gh = Q h Gh |Ubh cf. (3.8),(3.12), and Gh = Q h G b

for (3.15).

(3.21)

Here Ah : U h ∪ UT h → VT h is either an extension, or A˜h : U h ∪ U → V h ∪ V is an approximation for A : U → V . We will denote Φh A := Ah as the discrete operators generated by Petrov–Galerkin methods as general discretization methods. Similarly, ˜ h : U h ∪ U → V h ∪ V is an approximation Gh : U h ∪ UT h → VT h is an extension, or G for G : U → V , or both.

The above Q h : Vb → Vbh or V → Vbh will be shown to satisfy the necessary conditions for a general discretization theory, cf. Deﬁnition 3.12, (3.24), even for nonconforming FEMs. However, for discussing their variational consistency and with Ah : UT h → VT h we have extended these Q h into Q h : Vb ∪ VT h → Vbh . For Petrov–Galerkin methods we sometimes obtain useful properties for the projectors and approximating spaces.

Lemma 3.9. Let V = V and Qhinf : V → V h be uniquely deﬁned by v − Qhinf v = inf v − v h V = min v − v h V . V v h ∈V h

v h ∈V h

(3.22)

Then Q h = Qhinf . Proof. For V = V

we have v = sup v, v V h ×V . Now, we choose in addition to v ∈V

V , annihilating an additional nontrivial vv ∈ V annihilating span{v, V h } and hV , let Vv := span V , vv . Then Vv annihilates Vvh := span{V h , v}. This implies, with V = Vv ⊕ Vcomp where v − v , v = 0 ∀v ∈ Vcomp , v − Qhinf v = min sup |v − v h , v |/v V V h

h

v h ∈V h v ∈V

= min sup |v − v h , v |/v V v h ∈V h v ∈Vv

= min

sup |v − v h , v |/v V = v − Q h vV ,

v h ∈V h v ∈V h

hence, the claim.

Lemma 3.10. Let U be a separable Banach space, e.g. our Sobolev spaces. Let U h be approximating spaces for a dense subspace Ud of U with respect to the norm · U . Then U h are approximating spaces for U as well. Proof. Let u ∈ U be given. Then ∀ > 0 we can ﬁnd ud ∈ Ud with u − ud U < /2. For this ud there exists a uh ∈ U h such that uh − ud U < /2. We can extend this to the dual spaces and deﬁne the corresponding projectors: usually, the dual space V of V consists of functions which admit approximating subspaces, V h . This is the background for

183

3.2. Petrov–Galerkin and general discretization methods

Lemma 3.11. Let V be a separable Banach space and V for V . Finally, deﬁne Pˆ h : V → V h := (V h ) by

for f ∈ V let Pˆ h f − f, v h V×V = 0 ∀ v

h

be approximating spaces

∈ V h.

h

(3.23)

Then V h are approximating spaces for V, Pˆ h is a projector and lim Pˆ h f − f V = 0

h→0

∀f ∈ V.

Proof. With V the V is separable as well. As a consequence of the separable V , we can and do choose a basis ϕi ∈ V such that V h = span ϕi , . . . , ϕi(h) . Furthermore, deﬁne a dual basis ϕi for V, hence ; : ϕi ∈ V , ϕj ∈ V and ϕi , ϕj V×V = δi,j . Obviously, V h := span{ϕi , . . . , ϕi(h) }. Then : ; f, ϕi ϕi , i.e. f ∈V ⇒f = i : ; f, ϕi ϕi V ∀ > 0 ∃ j = j() and a h() : > f − i≤j( )

: : ; ; f, ϕi ϕi V ≥ f, ϕi > ϕi : ϕi ∈ V h( ) V . = i>j( )

i>j( )

This V h( ) includes at least the ϕi , i ≤ j(), but usually some more. The above (3.23) implies ; : 0 = Pˆ h f − f, ϕi ∀ ϕi ∈ V h ⇔ ϕi ∈ V h , hence we obtain : ; Pˆ h f − f V ≤ f, ϕi ϕi : ϕi ∈ V h V → 0 for h → 0. Summarizing our examples, we have deﬁned an operator Φh such that the original A is discretized into Ah = Φh A. This process has to be generalized to linear and nonlinear operators A and G, respectively. Formalizing, we introduce general discretization methods. We modify Stetter’s [596] approach to ﬁt the testing as in Petrov–Galerkin methods and to the ﬁnite diﬀerence methods. Deﬁnition 3.12. A discretization method M, applicable to a problem G: D(G) ⊂ h : Ub → Vb is deﬁned by a sequence of quintuples, M := Ub , Vbh , P h , Q h , Φh h∈H

1. Let U, V be Banach spaces, Ub ⊂ U, Vb ⊂ V closed subspaces. To h, 0 < h ∈ H ⊂ R, inf h∈H h = 0 assign sequences {U h , V h }h∈H , of ﬁnite dimensional 28 spaces. Often we use the short notation of U h . The relation between the spaces is realized by the bounded approximation and projection operators, P h and Q h , usually 28

We do not want to over-formalize the notation and have chosen h ∈ H.

184

3. A general discretization theory U⊃

A,G

Ub

Vb ⊂ V

Q′h

Ph

Φh

Uhb

A ,G h

tested by

V ′b

h

V ′hb

(3.27) (approximately) tested by

Vbh

Figure 3.2 General discretization methods.

linear and generalized approximation and projection operators, respectively, such that, cf. Proposition 4.50, We use the notation V h = V h , Q h = Qh equivalently throughout the book.

P h : Ub → Ubh , and Q h : Vb → Vbh , s.t. P h 0 = 0, Q h 0 = 0, and

lim P uU h = uU ∀u ∈ U, lim Q h

h→0

h→0

h

f hV h b

= f V ∀f ∈ b

(3.24)

Vb ,

h 2. We need the mappings

Φ deﬁned for nonlinear mappings denoted as N L(Ub , Vb ) = Ub → Vb . They might be deﬁned on subsets of Ub .

Φh : (Ub → Vb ) → Ubh → Vbh .

(3.25)

If G ∈ D(Φh ) ∀ h ∈ H, h < h0 , then Φh or M is called a discretization applicable to G, and Gh := Φh G the discretization or discrete operator for G. 3. Let the restriction of Φh to linear operators L be linear and satisfy

Φh : N L(Ub , Vb ) → N L Ubh , Vbh , Φh |L : L (Ub , Vb ) → L Ubh , Vbh with (3.26) Φh (cA) = cΦh A, Φh (A + A1 ) = Φh A + Φh A1 , Φh (A + f ) = Φh A + Φh f and Φh (G + G1 ) = Φh G + Φh G1 ∀c ∈ R, A, A1 ∈ L (Ub , Vb ) , f ∈ V , G, G1 ∈ N L (Ub , Vb ) , all in D(Φh ) Then we call Φh a linear discretization method. 4. Often we use the short notation (for the inﬁnite sequence of ) Φh for indicating the discretization method M. 5. Whenever in the next chapters a problem is transformed and solved only in its weak form we will choose U = V, Ub = Vb . This is no longer correct, e.g. for fully nonlinear problem, cf. Subsections 2.5.7, 2.6.8 and 5.2.3 ﬀ.

Summarizing, we obtain the diagram in Figure 3.2 with uniformly bounded P h , Q h , cf. e.g. Proposition 4.50. Remark 3.13.

1. It has to be pointed out that the choice of P h , Q h for the given Ubh , Vbh and problem (3.3) or (3.4) is certainly not unique. However the combination Ubh , Vbh , P h , Q h , Φh should be chosen appropriately to yield consistency, see

3.3. Variational and classical consistency

185

below, with the highest possible order p. Otherwise the results are no longer optimal, see, e.g. (3.32). 2. In Example 3.8, we have formulated the discrete operators Ah for Examples 3.1– 3.3 with appropriate P h , Q h , Φh , Ah , A˜h , as

Φh A = Ah = Q h Ah |Ubh or Φh A = Ah = Q h A˜h |Ubh .

(3.28)

3. We have to show that the P h , Q h satisfy (3.24) for the case of generalized Petrov–Galerkin methods. Then these Φh A = Ah , Φh G = Gh deﬁne general discretization methods, in the sense of Deﬁnition 3.12. Obviously, the generalized discretization methods and some of the following results in this chapter include nonlinear methods. The later applications to bifurcation numerics, however, require linear methods. All the generalized discretization methods, studied here, applicable to linear and nonlinear elliptic problems, are linear discretization methods. It is important that Φh acts on the diﬀerent u dependent terms in A or G by applying linear operators. This holds even for more complicated cases, e.g. the Fourier collocation derivative in spectral methods, cf. [114, 115, 120, 123, 124, 158, 631] for details. The above linearity condition (3.26) is violated for the (nonlinear) Runge– Kutta methods. They are general discretization methods in the above sense, but are not relevant for our space discretizations in bifurcation and the small dimensional center manifolds. Therefor we avoid all the diﬃculties with parabolic problems. Our deﬁnition allows all forms of ﬁnite diﬀerence, ﬁnite element, discontinuous Galerkin, spectral, wavelet, and mesh–free methods, and approximating the discrete operators by quadrature formulas or diﬀerence approximations. These methods are necessary for studying eﬀects in space discretization for numerical bifurcation and center manifolds in partial diﬀerential equations. Deﬁnition 3.12 is a modiﬁcation of [596] and is related to [607–609, 677]. Diﬀerently from these, we study operators G : Ub → Vb and require, in some form, approximating spaces U h , V h for U, V. This emphasizes the testing with the Vb , Vbh and avoids, appropriately for all our methods, a direct consideration of the dual space V . So we do not have to distinguish the inner and outer discretization schemes, important in [528, 531, 532, 677]. Next, we have to study the relations between operators and their discretizations.

3.3

Variational and classical consistency

We want to know how well the discrete solution, uh0 , approximates the exact solution, u0 . One tool is an estimate of how well uh0 satisﬁes the original equation. We motivate by our previous examples, two diﬀerent, however, closely related concepts, the so-called variational and the classical consistency (errors) for linear and nonlinear operators. For conforming FEMs we have Ubh ⊂ Ub , Vbh ⊂ Vb ; for nonconforming FEMs and for diﬀerence methods the Ubh , Vbh have to be modiﬁed as in Example 3.8. Accordingly,

186

3. A general discretization theory

we have deﬁned the exact and discrete solutions u0 and uh0 , respectively, (3.29) u0 ∈ Ub : Au0 , vV ×V = a(u0 , v) = f, vV ×V ∀ v ∈ Vb and ; : ; : uh0 ∈ Ubh : Auh0 , v h V ×V = a uh0 , v h = f, v h V×V ∀v h ∈ Vbh , conform. FE ; : ; : uh0 ∈ Ubh : Ah uh0 , v h V h ×V h = ah uh0 , v h = f h , v h ··· ∀v h ∈ Vbh , nonc. FE, so uh0 ∈ Ubh : Q h Auh0 − f = 0 for conf. FEs or Q h Ah uh0 − f = 0 for nonconf. FEs

uh0 ∈ Ubh : Q h A˜h uh0 − f = 0, for diﬀ. meth., with the P h , Q h in (3.18), (3.19). Hence, the second equation represents a subset of the conditions in the ﬁrst equation for conforming FEs, the third and fourth equations for nonconforming Petrov–Galerkin and diﬀerence methods don’t. For conforming FEs with a(u, v) = ah (u, v)∀u ∈ U, v ∈ V, we obtain (3.30) a u0 − uh0 , v h = 0 ∀ v h ∈ Vbh , Q h Au0 − Auh0 = 0, hence, these variational consistency errors vanish, see below. h h For nonconforming FEs a direct comparison of Au0 and A u0h is impossible. So h h we have to employ the extended ah u0 − u0 , v or Ah u0 − Ah u0 . This variational consistency error is nontrivial for nonconforming methods, and with Ah = Q h Ah |Ubh |ah u0 − uh0 , v h |/v h V h = Q h Ah u0 − Ah uh0 h . sup (3.31) Vb

0 =v h ∈Vbh

This variational consistency error is the usual concept in the FE community. For the linear case the results in Lemma 5.52 will allow estimates of the form h h h u0 − u0 h ≤ C dist u0 , Ubh + sup |ah u0 − u0 , v | . (3.32) U v h V h 0 =v h ∈V h b

However, this variational consistency error diﬀers from the classical consistency error, Q h Au0 − Ah P h u0 , cf. Deﬁnition 3.14, (3.33), (3.35), and [596]. The latter is more appropriate for using the whole nonlinear machinery. We aim for the relation of the variational consistency to the classical consistency error. Again, we use the general notation of Ah for the diﬀerent cases, unless the diﬀerence has to be emphasized. To compare both “consistencies”, we modify (3.31) in operator form as Q h Ah u0 −Ah uh0 . This can be generalized to arbitrary (smooth enough) u, if we introduce fu := Au and the discrete solution uh of Ah uh = Q h fu := Q h Au:

Q h Ah u − Ah uh = Ah (P h u − uh ) + Q h Ah u − Q h Ah P h u < => ? var.cons.error

= Ah P h u − Q h fu + Q h Ah u − Q h Ah P h u

= Ah P h u − Q h Au + Q h Ah u − Q h Ah P h u . < => ? < => ? class.cons.error

related to interpol.error

(3.33)

187

3.3. Variational and classical consistency

As indicated, the term Q h Ah u − Ah uh left in the ﬁrst line in (3.33) and the ﬁrst term in the last line, Ah P h u − Q h Au, are called the variational and classical consis tency error for the linear operator A. For linear Q h we ﬁnd Q h Ah u − Q h Ah P h u = Q h Ah (u − P h u), hence Q h Ah (u − P h u) ≤ Cu − P h u is bounded by the interpolation error. For diﬀerence methods we simply replace the Ah in (3.33) by A˜h , however, the variational consistency error, Q h Ah u − Ah uh , is only deﬁned for smooth enough u. There is no problem with the classical consistency error, Ah P h u − Q h Au = Q h A˜h P h u − Q h Au. This (3.33) has to be slightly modiﬁed for nonlinear operators. Again, we introduce fu := Gu and assume uh , a discrete solution of Gh uh = Q h fu = Q h Gu. From the h possibly many solutions u for the nonlinear G we choose that with small uh − u for an appropriate norm. In general notation we ﬁnd with Gh = Q h Gh |Ubh in (3.21),

Q h Gh u − Gh uh = (Gh P h u − Q h Gu) + (Q h Gh u − Q h Gh P h u) < => ? < => ? < => ? var.cons.error

class.cons.error

(3.34)

related to interpol.error

˜ h u only makes sense for smooth enough u. Again, for diﬀerence methods the Q h G Similarly, for the general discretization methods in Deﬁnition 3.12, with Gh = Φh G, the classical discretization error, Gh P h u − Q h Gu = (Φh G)P h u − Q h Gu, is always deﬁned, but the other terms are usually not deﬁned. So we introduce in the following deﬁnition the classical discretization error for general discretization methods. Next, we discuss, for generalized Petrov–Galerkin methods, the variational and classical discretization errors in Example 3.15. Deﬁnition 3.14. Classical consistency, local discretization error: 1. Under the conditions of Deﬁnition 3.12 for a general discretization method

(Φh A)P h u − Q h Au = Ah P h u − Q h Au and (Φh G)P h u − Q h Gu

(3.35)

or its norm Ah P h u − Q h AuV h and Gh P h u − Q h GuV h , b

b

for A and G, respectively, are called the (classical) consistency error or local discretization error in u ∈ D(G) ∩ Ub . Note that for the exact solution u0 of G(u0 ) = 0 the classical consistency error reduces to Gh P h u0 V h . b 2. A method is called classically consistent for A or G in u, if

Ah P h u − Q h AuV h → 0 or Gh P h u − Q h GuV → 0 for h → 0. b

3. If even

Ah P h u − Q h AuV h or Gh P h u − Q h Gh uV = O(hp ), b

we call it consistent of order p in u. These O(hp ) are usually estimated by Chp uW k,q (Ω) with appropriate p, k, q. Example 3.15. Consistency errors

1. As in Example 3.8 we choose ah (·, ·) and the discrete operators Ah = Q h Ah |Ubh

and Gh = Q h Gh |Ubh by applying a generalized Petrov–Galerkin method to A and

188

3. A general discretization theory

G. Let u and uh be related as

Ah uh = Q h fu := Q h Au and Gh uh = Q h fu := Q h Gu for the linear and nonlinear operator A and G, respectively. Then

sup {|ah (u − uh , v h )|/v h } = Q h Ah u − Ah uh V h and

(3.36)

b

0 =v h ∈Vbh

Q h Gh u − Gh uh V h , for A and G, respectively, b

is called the variational consistency error in u. 29 A method is called variationally consistent for A in u, if

sup {|ah (u − uh , v h )|/v h } = Q h Ah u − Ah uh V h → 0,

(3.37)

b

0 =v h ∈Vbh

similarly for the nonlinear G, e.g. Gh P h u − Q h GuV h → 0 for h → 0. 2. Note that for all these cases the variational errors vanish for conforming FEMs. The variational consistency error usually is only considered and estimated for the exact solution, u0 , or if necessary for an approximant P h u0 , see Section 5.5 for estimates. A generalization to arbitrary u is formulated in (3.33). The classical consistency errors are often studied for general u. There is the simple relation for the diﬀerence of variational and classical consistency errors in (3.33) and (3.34). In fact, see Proposition 5.47 and Theorem 3.17, this diﬀerence is, under appropriate conditions, bounded as O(u − P h uU h ) → 0 ∀u ∈ Ub ∀h → 0. Hence, they even have the same size if both dominate or have the same size as the interpolation error u0 − P h u0 U . Therefore it suﬃces to estimate the more convenient of the two consistency errors. The general case needs the continuity or even Lipschitz-continuity of the Gh , a condition satisﬁed for all the examples studied here, simultaneously with G. We assume equicontinuous and equibounded Q h , cf. Theorem 4.54. Theorem 3.16. Conforming classical consistency, and interpolation errors:

1. Let Φh G = Gh = Q h G|Ubh : D(Gh ) = Ubh ∩ D(G) be a conforming FEM and let

uh ∈ D(Gh ) be given. Assume G is Lipschitz-continuous in Br (u) and Q h equicontinuous and equibounded. Then the classical consistency error satisﬁes

Gh P h u − Q h GuV h = Q h GP h u − Q h GuV h ≤ L(P h u − u)U h . (3.38) b

b

is chosen as the interpolation or approximation operator, then 2. If P : Ub → the classical consistency error vanishes as, e.g. the interpolation error. h

29

Ubh

We sometimes use the notation Q h Ah u − Ah uh V h instead of Q h Ah u − Ah uh V h to

b

b

emphasize that this Q h Ah u − Ah uh is tested by the v h ∈ Vbh . This notation in (3.36) is used similarly for the isoparametric variants.

3.4. Stability and consistency yield convergence

189

Nonconforming variational, classical consistency, interpolation

Theorem 3.17. errors:

1. For the case of nonconforming FEs we extend the original linear A : U → V and the nonlinear G : D(G) ⊂ U → V as Ah : UT h → VT h and Gh : D(Gh ) ⊂ UT h → VT h . Assume Ah and Q h to be equibounded and equicontinuous and (G ∈ CL (D(G)) usually implying) Gh ∈ CL (D(Gh )), with Lipschitz constant L. Furthermore, let Ah = Q h Ah |UT h , Gh = Q h Gh |UT h and u ∈ D(G), uh , P h u ∈ D(Gh ), and uh be the discrete approximation for u from Gh uh = Q h Gh uh = Q h fu = Q h Gu or its analogue for the linear A, Ah . 2. Then the diﬀerence between the variational and classical discretization errors

(Q h Gh u − Gh uh ) − (Gh P h u − Q h Gu) = (Q h Gh u − Q h Gh P h u)

(3.39)

∀ u ∈ D(G), uh , P h u ∈ D(Gh ) can be estimated ∀ u ∈ U as

Q h Gh u − Q h Gh P h uV h , Q h Ah u − Q h Ah P h uV h ≤ O(P h u − uUT h ). b

b

3. If only G ∈ C(D(G)), Gh ∈ C(D(Gh )), then

Q h Gh u − Q h Gh P h uV h ≤ ωGh (P h u − uUT h ), b

with the modulus of continuity of Gh , ωGh (t) :=

sup x−yU

Th

≤t,

x,y∈D(Gh )

Gh x − Gh yV h . b

4. For nonconforming FEs and DCGMs the estimates for variational and classical discretization errors are rather technically involved, cf. Section 5.5 and Chapter 7. Proof. With the equibounded and equicontinuous Q h and Gh ∈ CL Ub,c + Ubh , u ∈ D(G), uh , P h u ∈ D(Gh ), the ﬁrst estimate is obvious. The ωGh (P h u − uUc ) modiﬁcation is straightforward.

3.4

Stability and consistency yield convergence

We want to unfold this well-known fact for the general discretization methods introduced in Deﬁnition 3.12. We combine consistency introduced in Section 3.3 with the missing convergence and stability to be deﬁned now. An appropriate interplay of the diﬀerent concepts of consistency, convergence, stability, the operators P h , Q h , the norms, the diﬀerent approximations by spaces and operators is the basic condition for the success of this approach. These choices are by no means unique. Stetter [596] gives some examples for failing combinations. In Theorem 3.21 we prove the existence of a discrete solution and its convergence towards the exact solution essentially for a stable and consistent discretization as in Deﬁnition 3.12. This includes our general Petrov–Galerkin methods, cf. Remark 3.13. Except in Section 3.7, we skip, in the

190

3. A general discretization theory

remainder of this chapter, the condition of a “linear” Φh , see (3.25), and study the general discretization methods. Convergence is expected to describe a discrete solution, uh0 , well approximating the we can directly compare uh0 and u0 since exact solution, u0 . In for both some hmethods · U h , and hence u0 − u0 U h is well deﬁned with hopefully u0 − uh0 U h → 0 for h 0. Theoretically, two other measures are possible as well. We might consider → P h u0 − uh0 h with our known P h : U → U h , e.g. for diﬀerence methods. Or we U might study u0 − Ph uh0 U and need an additional Ph : U h → U. We apply this Ph , essentially deﬁned by an anticrime transformation, cf. Lemma 5.77, to nonconforming FEs for proving stability of linearized elliptic operators. Deﬁnition 3.18. Global discretization error, convergence:

1. Let G : D(G) ⊂ Ub → Vb and its discretization Gh = Φh G : D(Gh ) ⊂ Ubh → Vbh have the (locally) unique exact and approximate solutions u0 and uh0 , respectively. Then (the sequence) uh0 − u0 or uh0 − P h u0

(3.40)

is called the (global) discretization error of Φh for G. 2. The uh0 are called convergent to u0 and convergent of order p (sometimes, Gh = Φh G is called convergent to G in u0 ) if for h → 0, h ∈ H, respectively u0 − uh0 h → 0 or P h u0 − uh0 h → 0 and (3.41) U

u0 − uh0

Uh

U

= O(hp ) or P h u0 − uh0 U h = O(hp ).

3. As in Deﬁnition 3.5, modiﬁcations are necessary for diﬀerence methods. We want to prove convergence. This will not be possible directly. So we try to derive it from the local discretization error. Consistency alone will not do it,cf. [596] for counterexamples. Let us motivate what is missing: G(u0 ) = 0 and Gh uh0 = 0 have to be solved and P h u0 and uh0 have to be compared. The classical consistency, see (3.35), in u0 with Gu0 = 0 implies

Gh P h u0 − Q h Gu0 = Gh P h u0 − Q h 0 = Gh P h u0 → 0 ∀ h → 0, h ∈ H. (3.42) If (Gh )−1 is deﬁned in and close to Gh P h u0 , then with Gh uh0 = 0, P h u0 − uh0 = (Gh )−1 Gh P h u0 − (Gh )−1 0. For a vanishing consistency error in u0 and a continuous or even Lipschitz-continuous (Gh )−1 this implies h P u0 − uh0 h → 0 or ≤ LGh (P h u0 ) h → 0 ∀ h → 0, h ∈ H. (3.43) V U b

This kind of perturbation insensitivity is called stability. Deﬁnition 3.19. Stability: Under the conditions of Deﬁnition 3.12 we assume a non linear and linear operator Gh : D(Gh ) ⊆ Ubh → Vbh , and B h : Ubh → Vbh , respectively,

3.4. Stability and consistency yield convergence

191

to be assigned to every h ∈ H (usually deﬁned by a discretization method M). For uh and h0 > 0 let Br (uh ) = {v h ∈ U h : v h − uh U h < r} ⊂ D(Gh ) ∀h < h0 , h ∈ H. Furthermore, let h0 , r, S ∈ R+ be ﬁxed constants, such that, uniformly in h ∈ H, h < h0 : uhi ∈ Br (uh ), i = 1, 2, =⇒ uh1 − uh2 U h ≤ S Gh uh1 − Gh uh2 V h . (3.44) b

For B or B · +f this reduces to (for any r and u , but for h < h0 ) (B h )−1 ∈ L Vbh , Ubh exists and ||(B h )−1 ||U h ←V h ≤ S. h

h

h

h

b

h

h

b

h

Then G and B are called stable in u and stable, respectively. The S and r are called stability bound and stability threshold, respectively. The proof of stability is simple for some cases; for general problems it is usually rather complicated. For example, for the conforming FEs in Example 3.1 it is an immediate consequence of the coercivity and the Brezzi–Babuska condition in Theorem 2.12. For more general cases we will discuss in Theorems 3.23–3.29, conditions which guarantee stability. For example, it allows reducing stability for the nonlinear problem to that of the linearized problem, and from a linear coercive operator to its compact perturbations. In Theorem 2.122 we discuss the implication of H0m (Ω) ⇒ W0m,p (Ω)coercivity for p < 2 and ⇒ W0m,p (Ω)-coercivity for 2 < p. In modiﬁed form these arguments are applicable to the (linearized) Navier–Stokes operator as well. In [596] the ﬁrst inequality in (3.44), uhi ∈ Br (uh ), is replaced by the weaker Gh uhi − Gh (uh )V h < R, i = 1, 2. (3.45) b Since for nonlinear G (and hence Gh ) this (3.45) may be satisﬁed for large uh − uhi U h we have chosen the stronger assumption in (3.44), cf. [596], Corollary 1.2.2.

Proposition 3.20. In the discretization Gh : D(Gh ) ⊂ Ubh → Vbh let the Gh be continuous in Br (uh ) ⊂ D(Gh ) and satisfy (3.44). Then these uhi satisfy (3.45) with R = r/S, and thus are stable in this weaker sense. Assuming continuity, consistency and stability and some technicalities, we will now prove the existence of a discrete solution uh0 for Gh uh0 = 0, deﬁned by the general discretization methods in Deﬁnition 3.12, cf. [596]. Theorem 3.21. Let the original problem (3.3) have the exact solution u0 . Let Gh : D(Gh ) ⊂ Ubh → Vbh , h ∈ H, be its discretization, see Deﬁnition 3.12 (here nonlinear Φh are allowed) and satisfy the following conditions:

1. Gh = Φh G : D(Gh ) ⊂ Ubh → Vbh is deﬁned and continuous in Br (P h u0 ) ⊂ D(Gh ) with r > 0 independent of h; 2. Gh is (classically) consistent with G in P h u0 ; 3. Gh is stable for P h u0 .

192

3. A general discretization theory

Then the discrete problem Gh (uh ) = 0 possesses the unique solution uh0 ∈ D(Gh ) ⊂ Ubh near u0 for all suﬃciently small h and uh0 converges to u0 . If Gh is consistent and consistent of order p, then uh0 converges and converges of order p, hence h u0 − u0 h ≤ SGh (P h u0 ) h and ≤ O(hp ) for h → 0, h ∈ H, respectively (3.46) V U As essential tool for the proof is the following. Proposition 3.22. For a ﬁxed r > 0 and a sequence u ¯h , let the discrete

uh ) → Vbh , Gh ∈ C(Br (¯ uh )) be continuous. Gh : Br := Br (¯ Furthermore for S ∈ R+ let

uhi ∈ Br , i = 1, 2, imply uh1 − uh2 U h ≤ S Gh uh1 − Gh uh2 V h .

(3.47)

b

Then Gh is invertible on = v h ∈ Vbh : v h − Gh u ¯h V h < R , with R := r/S. BR

(3.48)

b

This (Gh )−1 : BR → Ubh is Lipschitz-continuous with Lipschitz-bound S.

Proof. By (3.47) the Gh are invertible on every set M ⊂ Gh (Br ). In fact, if uh1 , uh2 ∈ Br h h h have the same images G ui = v ∈ M, i = 1, 2, then (3.47) implies h u2 − uh1 h ≤ S Gh uh2 − Gh uh1 h = S · 0, such that uh2 = uh1 . (3.49) U V b

Hence, it suﬃces to prove BR ⊂ Gh (Br ).

(3.50)

\Gh (Br ) would exist. We deﬁne If this were incorrect, a v1h ∈ BR

¯h + λv1h , λ ≥ 0 for v1h ∈ BR \Gh (Br ). v h (λ) := (1 − λ)Gh u

(3.51)

¯h ∈ Gh (Br ), v h (1) = v1h ∈ Gh (Br ). With This implies v h (0) = Gh u

Λ := {λ > 0 : v h (λ) ∈ Gh (Br ) we deﬁne

. ¯ := λ

for λ ∈ [0, λ )}

supλ∈Λ λ for Λ = ∅, 0 for Λ = ∅.

(3.52)

¯ > 1. We construct a The inclusion (3.50) would be proved if we could show that λ ¯ ∈ Gh (Br ). This allows ¯ ≤ 1 in two steps: ﬁrst we show v¯h := v h (λ) contradiction to λ constructing the contradiction in step 2. ¯ = 0 we automatically have v¯h = v h (0) = Gh u ¯h ∈ Gh (Br ). For the compleFor λ ¯ − ) ∈ Gh (Br ), see (3.52) for any ¯ ≤ 1 we have by deﬁnition v h (λ mentary case 0 < λ ¯ and, see (3.51), 0<<λ ¯ − ) ∈ B , since v h (λ ¯ − ) − Gh u ¯h V h < v1h − Gh (¯ uh ) h < R. v h (λ R b

Vb

3.4. Stability and consistency yield convergence

193

¯ − ), see (3.52). A combination Therefore an uh ∈ Br exists such that Gh uh = v h (λ with (3.47) and (3.51) shows the property of a Cauchy sequence h ¯ − ) − v h (λ ¯ − ¯) h → 0 u − uh ¯ h ≤ S Gh uh − Gh uh ¯ h = Sv h (λ V U V b

b

for , ¯ > 0 and → ¯.

¯ − )) exist and we have convergence for the uh , v h (λ ¯ − ), Hence the (Gh )−1 (v h (λ

¯ − )) as well for → +0. This yields and, by (3.47), for the (Gh )−1 (v h (λ

¯ − )). uh0 := lim uh = lim (Gh )−1 (v h (λ

→+0

→+0

This uh0 is ∈ Br : in fact,

¯ − )) − u ¯ − ) − Gh u (Gh )−1 (v h (λ ¯h U h ≤ Sv h (λ ¯h V h b

¯ − ) v h − Gh u = S(λ ¯h V h . 1 b

¯ − < 1, we obtain and, see (3.51), v1h − Gh u ¯h V h < R, independent of . With 0 < λ b the following inequality ¯ − )) − u ¯ − )R ≤ (λ ¯ − )r ≤ r, hence uh − u ¯h U h < S(λ ¯h U h < r. (Gh )−1 (v h (λ 0 Finally Gh is continuous in Br and we have reached our ﬁrst goal

¯ = Gh uh ∈ Gh (Br ). v¯h := v h (λ) 0 This allows the contradiction: since uh0 ∈ Br and Gh is continuous in Br , there exists ¯ = Bδ uh ⊂ Br with Gh (B) ¯ ⊂ B . For this B ¯ the Gh : B ¯ → Gh (B) ¯ ⊂ a closed ball B 0 R BR is injective, as we have seen already in (3.49). ¯ is compact. Now Gh |B¯ is an injective continuous Since dim U h < ∞ then B ¯ →B ¯ is deﬁned and con¯ Then (Gh )−1 : Gh (B) mapping with compact domain B. tinuous as well, see e.g. [560], [74], p. 56. In particular, with compact S¯ := h u ∈ Ubh : uh − uh0 U h = δ the injectivity implies ρ := min Gh uh0 − Gh uh > 0, ¯ uh ∈S

and hence ¯ ⊂ Gh (Br ) with open Bρ0 . Gh uh0 = v¯h ∈ Bρ0 = Bρ0 (¯ v h ) ⊂ Gh (B) ¯ cannot be the supremum deﬁned in (3.52). This is our contradiction. Then the above λ ¯ > 1 or v h ∈ Gh (Br ). So either λ 1 The Lipschitz-continuity for (Gh )−1 and the Lipschitz constant S are immediate consequences from (3.47). Proof of Theorem 3.21. The conditions 1, 3 imply by Proposition 3.22 the exis¯h = P h u0 and tence of a unique discrete solution uh0 ∈ Br (P h u0 ): in fact we use u h h h h h h ¯ = G P u0 in (3.48), the local discretization error G P u0 , see Deﬁnition 3.14. G u

194

3. A general discretization theory

Furthermore, R is independent of h since r, S are independent of h. The consistency of the method implies Gh P h u0 V h → 0 for h → 0. b

for h < h1 suﬃciently small and uh0 = (Gh )−1 0 is uniquely Therefore 0 ∈ determined for h < h1 . In the next step we show convergence and convergence of order p for this discrete solution. For h < h1 let uh0 be the unique solution of Gh uh0 = 0. Now we use the global discretization error h = uh0 − P h u0 , see Deﬁnition 3.18, obtaining 0 (Gh P h u0 ) BR

Gh uh0 −Gh P h u0 = Gh (P h u0 + h ) −Gh P h u0 = −Gh P h u0 . < => ? < => ? =0

=0

Gh is stable for P h u0 hence h U h = uh0 − P h u0 U h ≤ SGh P h u0 V h for Gh P h u0 V h < R. b

b

Since G is consistent with G in P u0 , then G P u0 V h → 0. This implies, for h

h

h

h

b

h < h2 , that Gh P h u0 V h < R and hence b

h U h ≤ SGh P h u0 V h → 0 for h → 0. b

This shows the convergence and convergence of order p for this method.

3.5

Techniques for proving stability

The following theorems formulate known and well applicable criteria for stability. We start discussing the relation between stability of a nonlinear operator and its linearized form and with perturbed nonlinear problems. Most importantly, we combine a coercive principal part with its compact perturbations, cf. Theorem 3.29. This allows, for all problems and methods treated in both books, the proof of stability for (nonlinear) operators. Their linearizations have to be boundedly invertible in the exact solution, implying locally unique solutions. Natural extensions to the bifurcation situation are valid as well. Here we prove Theorem 3.29 only for the case of conforming FEMs. Generalizations to nonconforming methods require speciﬁc additional information, available in the corresponding chapters. Theorem 3.23. Under the conditions of Deﬁnition 3.12 let a discretization Φh , Gh = Φh G : D(Gh ) ⊂ Ubh → Vbh and uh ∈ D(Gh ) ⊂ U h be given. Assume Gh ∈ C 1 (Br (uh )), Br (uh ) ⊂ D(Gh ), with ﬁxed r > 0. Furthermore, for all v h ∈ Br (uh ) let the inverse of the derivative, ((Gh ) (v h ))−1 , exist for h < h0 and ((Gh ) (v h ))−1 U h ←V h ≤ S ∀v h ∈ Br (uh ) uniformly for h0 > h ∈ H. b

(3.53)

b

1. Then the discretization Gh is stable at uh with stability bound S and stability threshold r0 = r/S.

3.5. Techniques for proving stability

195

2. The above ((Gh ) (v h ))−1 in Br (uh ) can be replaced by ((Gh ) (uh ))−1 in Bs (uh ) with s < r and, for bounded Φh , the Gh ∈ C 1 (Br (uh )) by G ∈ C 1 (Br (u)), Br (u) ⊂ D(G) and S by S/2 in (3.53). Proof. The existence of an inverse operator (Gh )−1 : BR → Ubh in the balls BR = h h h BR (G u ), R = r/S, is shown as for Proposition 3.22. BR ⊂ Vb is an open neighborhood of Gh uh with respect to Vbh . Then Theorem 1.48 guarantees the diﬀerentiability of this inverse operator (Gh )−1 . Moreover it yields, uniformly for h0 > h ∈ H,

((Gh )−1 ) (Gh v h )U h ←V h = ((Gh ) (v h ))−1 U h ←V h ≤ S b

b

b

(3.54)

b

implying (Gh uh ), h0 > h ∈ H. (3.55) ((Gh ) )−1 (δ)U h ←V h ≤ S ∀ δ ∈ BR b b Now BR (Gh uh ) is convex, hence, Gh uh2 + λ Gh uh1 − Gh uh2 ∈ BR (Gh uh ) for 0 ≤ λ h h h h h h h < 1 ∀ G u1 , G u2 ∈ BR (G u ). For these ui and with the existence of ((Gh )−1 ) Theorem 1.43 yields h u1 − uh2 h = (Gh )−1 Gh uh1 − (Gh )−1 Gh uh2 |U h Ub b h h h −1 ≤ max ((G ) ) (δ)U h ←V h G u1 − Gh uh2 |V h δ∈BR (Gh uh )

b

b

b

≤ S Gh uh1 − Gh uh2 V h by (3.55), b

hence the claim. Theorem 1.46 and bounded Φh imply the modiﬁcations.

Theorem 3.24. Stability of a perturbed nonlinear operator: Under the conditions of ¯ h := Gh + F h : D(Gh ) ∩ Deﬁnition 3.12 let G, F, G + F ∈ D(Φh ), the discretization G D(F h ) ⊂ Ubh → Vbh and a sequence uh ∈ Ubh be given. Assume S, r, L > 0 exist, independent of h, such that Br (uh ) ⊂ D(Gh ) ∩ D(F h ) and 1. Gh are continuous and stable in Br (uh ) with the stability bound S; 2. F h are Lipschitz-continuous in Br (uh ), with h h F u1 − F h uh2 h ≤ L uh1 − uh2 h and L · S < 1. V U b

¯ h is stable in uh , with bound S/(1 − LS) and threshold (1 − LS)r/S. Then G Proof. For all h ∈ H and uhi ∈ Br (uh ) we have h h ¯ u −G ¯ h uh h = Gh uh − Gh uh h − F h uh − F h uh h G 1 2 V 1 2 1 2 V V b b b h h ≥ (1/S − L) u1 − u2 U h , hence the claim. Stetter [596] proves the following modiﬁcation:

196

3. A general discretization theory

Theorem 3.25. Under the conditions in Theorem 3.24 replace 1., 2. there by 1. Gh : Br (uh ) → Gh (Br (uh )) are diﬀerentiable, one-to-one and Gh (Br (uh )) are convex with ((Gh ) (v h ))−1 U h ←V h ≤ S ∀ v h ∈ Br (uh ). b

(3.56)

b

2. For all v h , uh1 , uh2 ∈ Br (uh ) and with L S < 1 let ((Gh ) (v h ))−1 F h uh1 − F h uh2 U h ≤ L uh1 − uh2 U h .

(3.57)

¯ h is stable in uh , with bound S/(1 − L ) and threshold (1 − L )r/S. Then again G In the previous sections two related conditions compete with each other for nonlinear operators, G. We required well deﬁned continuous G for consistency results and Gh for convergence. The continuity of G : D(G) ⊂ U → V can often be avoided, since consistency requires smoother functions u ∈ D(G) ∩ Us , where G is in fact well deﬁned and continuous, cf. Theorems 4.53 and 5.4 We will discuss some possibilities in Example 3.26 below. The continuity of Gh : D(Gh ) ⊂ U h → V h is h h much less restrictive than that of G, since the u ∈ U have a very special structure, e.g. piecewise polynomials for FE, DCG and wavelet methods on appropriate triangulations, or algebraic or trigonometric polynomials of a ﬁnite degree on the whole or domain decomposed Ω. For these concepts we refer to the corresponding chapters below. The condition of continuous Gh : D(Gh ) ⊂ U h → V h is unavoidable in the context of piecewise polynomial methods for fully nonlinear elliptic operators G on Ω. But it is applicable as well for other nonlinear operators G. E.g. a combination with the stability results in Section 3.6, implying boundedly invertible derivatives G (u0 ), allows extended search for discrete and related exact solutions of quasilinear problems. These results might relax the conditions imposed, e.g. in Chapters 4 and 5. Example 3.26. The G, Gh dilemma for the Monge-Amp`ere operator. G, Gh have to be well deﬁned and continuous, cf. Example 2.78: We recall G(u) := det ∇2 u − f (x, u, ∇u) and = uxx uyy − u2xy − f (x, u, ∇u) for n = 2.

(3.58)

Let Df := {u : f (x, u(x), ∇(x)) is deﬁned a.e. in Ω}. Obviously, this G(u) is well deﬁned for u ∈ D(G) := Df ∩ C 2 (Ω), but certainly is ∈ L2 (Ω) for u ∈ Df ∩ H 2 (Ω). So 4 (Ω) for (3.59), we choose the domain, D(G) ⊂ Df , as dense subset, here as C 2 (Ω) or H+ 2 2 of H (Ω) and the range, R(G) ⊂ L (Ω), cf. Theorems 2.79 ﬀ. Another possibility is D(G) = Df ∩ {u ∈ H 2 (Ω) ∩ W 2,∞ (Ω)}. Then the exact solution, u0 , satisﬁes the diﬀerential equation, and the boundary condition in the trace sense, cf. Subsections 5.2.4, 5.2.5. We assume D(G) to be deﬁned s.t. G : D(G) ⊂ U := H 2 (Ω) → R(G) ⊂ V := L2 (Ω) = V : G(u) := Gw (·, u, ∇u, ∇2 u), s.t. ∀u ∈ D(G) : G(u) ∈ L2 (Ω), G(u0 )(x) = 0∀ a.e. x ∈ Ω, ⇔ (G(u0 ), v)L2 (Ω) = 0∀v ∈ V, and u0 |∂Ω = 0 ⇐⇒ (u0 |∂Ω , vb )L2 (∂Ω) = 0 ∀vb ∈ L2 (∂Ω),

(3.59)

3.5. Techniques for proving stability

197

2

with Gw : Ω × R × Rn × Rn → R, (see (2.304)). For this second order fully nonlinear problem (3.59) on a polyhedral Ω, we will choose U h ⊂ C 1 (Ω), with piecewise polynomials, Pdn of degree d in n variables, and a triangulation, T h e.g. with simplices T , cf. Deﬁnition 4.2. So for a polyhedral Ω and a triangulation for Ω = ∪T ∈T h T ⊂ Rn let U h = {uh : uh |T ∈ Pdn } ∩ C 1 (Ω), Ubh = {uh ∈ U h : uh |∂Ω = 0}. Ubh

(3.60)

∈ ⊂ H (Ω) ∩ the FE solution, is determined by For this Ω, and testing only the diﬀerential equation with respect to v h ∈ V h , cf. Section 5.2. (3.61) uh0 ∈ (Ubh = V h ) ∩ D(G) s.t. G uh0 , v h L2 (Ω) = 0 ∀v h ∈ V h . uh0

2

H01 (Ω),

uh0 ,

Proposition 3.27. For the three cases of u ∈ D(G) ⊂ Df in the previous Example the G(u) ∈ L2 (Ω) and for uh ∈ U h ∩ Df in (3.60) the G(uh ) ∈ L2 (Ω) are well deﬁned a.e. for the Monge-Amp`ere equation in (3.58) and both G and Gh are continuous w.r.t. the G : H 2 (Ω) → L2 (Ω) norms. Generalized to curved Ω, and to the order 2m with U h ⊂ C 2m−1 (Ω), this remains correct and is proved exactly the same way. Proof. We formulate the proof only for G(uh ) and n = 2. It is well known that for u ∈ H 2 (Ω) the uxx uyy ∈ L2 (Ω). But for the piecewise polynomials uh in (3.60) we obtain uh , uhx , uhy , uhxx , uhyy , uhxy , uhxx uhyy , (uhxy )2 , . . . ∈ L2 (Ω) ∩ L∞ (Ω). For uh1 , uh2 ∈ U h , we apply the Cauchy-Schwarz inequality (1.48) to any T ∈ T h :

T

2 (uh1 T uh2 T )dx ≤

uh1 uh2 4L2 (T )

=

T

T

(uh1 T )2 dx

T

(uh2 T )2 dx implying

2 (uh1 uh2 T )2 dx =

h 2 h 2 2 u1 u2 dx ≤ T

T

T

(3.62)

T

(uh1 T )4 dx

T

(uh2 T )4 dx.

These integrals are deﬁned as limits of Riemann sums with appropriately distributed N knotes xi ∈ T . The maximal distance between them vanishes and the covering with N M i. open volume elements Mi xi and measure μ(Mi ) satisﬁes T = Ui=1 (uh |T )4 dx = lim T

N →∝

N

(uh (xi ))4 μ(Mi )

(3.63)

i=1

N N ≤ lim ( (uh (xi ))2 μ(Mi )) max(uh (xi ))2 = uh 2L2 (T ) uh 2L∝(T ) , N →∝

i=1

i=1

and similarly for all other up to second order derivatives of the uh , hence continuity. We prove stability for linear coercive, hence boundedly invertible operators and their compact perturbations. These linear coercive operators are usually the principal parts of elliptic operators, sometimes, e.g. for natural boundary conditions a term such

198

3. A general discretization theory

as c(u, v)L2 (Ω) , c > 0 has to be added. It is important to realize that for all elliptic problems discussed in this book, the weak principal parts always have the form ap (·, ·) : Vb × Vb → R or V × V → R. In a few exceptional cases, e.g. for mesh–free methods, in one of the V × V a dense subspace has to be chosen. For our generalized Petrov–Galerkin methods, always the Ubh , Vbh , even for Ubh = Vbh , are approximating subspaces for the Ub = Vb , with the same norms v h V h = v h U h . So we additionally require for generalized Petrov–Galerkin methods: ∀ > 0 ∃δ < ∈ R+ : (3.64) ∀v h ∈ Vbh ∃uh ∈ Ubh : uh − v h U h ≤ δv h U h , always correct for Ubh = Vbh . Fully nonlinear elliptic problems cannot be solved by generalized Petrov–Galerkin methods. In that case, (3.64) only plays the role for the weak linearized form of these problems. However, it has to be modiﬁed there for handling the violated boundary conditions. For nonconforming FE, the DCG, the diﬀerence, and the mesh–free methods, the following proof has to be generalized by employing speciﬁc techniques. So we will formulate the corresponding stability results in those chapters. So we restrict the discussion to conforming methods, cf. Examples 3.4, 3.8. These are, in our two books, the conforming FE and the spectral methods. Then the orthogonal projection Q h in (3.18) and the Ah uh and f h in (3.20) are strongly simpliﬁed. For ; : (3.65) Ubh ⊂ Ub , Vbh ⊂ Vb : Q h : Vb → Vbh : Q h f − f, v h V ×V = 0∀v h ∈ Vbh ,

Ah = Q h A|Ubh , with Q h A|Ubh

−1

b

b

: Vbh → Ubh , f h = Q h f |Vbh .

For Ub -coercive A ∈ L (Ub , Ub ), the stability of conforming generalized Petrov– Galerkin methods can be veriﬁed directly. Often this A ∈ L (Ub , Ub ) = A|Ub of an A ∈ L(U, U ). Theorem 3.28. Ub -coercive operators induce stable Petrov–Galerkin methods: 1. Let A ∈ L (Ub , Ub ) be a linear Ub -coercive operator, i.e. ∃M > 0 : a(u, u) = Au, uU ×U ≥ M ||u||2U ,

for all u ∈ Ub .

(3.66)

Then A−1 : Ub → Ub exists and A−1 Ub ←Ub ≤ 1/M . 2. For any conforming generalized Petrov–Galerkin method, satisfying (3.64) and with Q h in (3.65), the Ah = Q h A|Ubh is stable and

−1 h Q A|U h ≤ 2/M for suﬃciently small h. (3.67) b Ubh ←Vbh

Proof. Relations (3.66) and (3.67) follow from Theorem 2.12 and (3.64).

In general the existence of a bounded inverse of A is not suﬃcient to ensure that Ah = Q h A|Ubh is stable. However, the next result allows all types of elliptic

199

3.5. Techniques for proving stability

equation, bordered systems, hence numerical methods for bifurcation, and NavierStokes equations, see the following chapters. Theorem 3.29. For conforming generalized Petrov–Galerkin methods invertible A imply stable Ah : For the approximating spaces Ubh , Vbh h∈H for Ub , Vb , satisfying (3.64), let B ∈ L (Ub , Vb ) be boundedly invertible and the Q h B|Ubh be stable. Let A := B + C, with C ∈ C (Ub , Vb ) , the set of compact operators from Ub → Vb . Then:

A−1 ∈ L (Vb , Ub ) ⇒ Q h A|Ubh is stable. Remark 3.30. The existence of A−1 ∈ L (Vb , Ub ) has to be discussed in a little bit more detail. Since B −1 ∈ L (Vb , Ub ) exists, this implies B −1 A = I + B −1 C ∈ L(Ub , Ub ) and, as a product of a continuous and a compact operator, B −1 C ∈ C(Ub , Ub ) is a compact perturbation of the identity. Therefore the Riesz–Schauder theory applies, cf. Theorem 2.20. Hence for λ ∈ C \ {0} either (B −1 C − λI)−1 ∈ L(Ub , Ub ) or λ is one of the at most countably many eigenvalues of B −1 C. This implies that A−1 exists if and only if λ = −1 is no eigenvalue of B −1 C.

Proof. We have to show that A−1 ∈ L (Vb , Ub ) implies the stability of Q h A|Ubh . We determine for an arbitrary u ∈ Ub and v := Cu the, by assumption, unique exact and ˆ = v and Q h B|Ubh u ˆh = Q h B u ˆh = discrete solutions, u ˆ and u ˆh , of the equations B u

Q h v ∈ Vbh , cf. (3.4). With the notation B h := Q h B|Ubh , T := B −1 ∈ L (Vb , Ub ) and −1 T h := B h Q h ∈ L Vb , Ubh , Theorems 3.16 and 3.21 imply that lim ||ˆ u−u ˆh ||Ub = ||(T − T h )Cu||Ub = 0 and u ∈ Ub or

h→0

(3.68)

for any v ∈ Vb and B u ˜ = v, B h u ˜h = Q h v : lim ||˜ u−u ˜h ||Ub = ||(T − T h )v||Ub = 0. h→0

By Lemma 3.31 this second limit implies ||(T − T h )C||Ub ←Ub → 0 Now let u ∈ h

Ubh

for h → 0.

(3.69)

⊂ Ub . Because A is boundedly invertible, we can estimate

||uh ||Ub ≤ ||A−1 ||Ub ←Vb ||Auh ||Vb = ||A−1 ||Ub ←Vb ||B(I + T C)uh ||Vb

||(I + T C)uh ||Ub

≤ ||A−1 ||Ub ←Vb ||B||Vb ←Ub ||(I + T C)uh ||Ub , hence,

−1 ≥ ||A−1 ||Ub ←Vb ||B||Vb ←Ub ||uh ||Ub .

(3.70)

For the following we recall that Q h Bu is deﬁned for every u ∈ Ub . We apply the stability of B h to wh := B h uh to ﬁnd uh Ub ≤ (B h )−1 U h ←V h · wh Vb . Furthermore b

b

(I + T h C)uh Ub = (B h )−1 B h (I + T h C)uh Ub ≤ (B h )−1 U h ←V h B h (I + T h C)uh Vb b

implies

b

200

3. A general discretization theory

B h (I + T h C)uh Vb ≥

1 · (I + T h C)uh Ub . (B h )−1 U h ←V h

(3.71)

b

b

We combine (3.70), (3.71) for estimating h h Q Au = ||B h (I + T C)uh ||Vb ≥ B h (I + T h C)uh Vb − Q h B(T − T h )Cuh Vb Vb

(3.71)

≥ ||(I + T h C)uh ||Ub /||(B h )−1 ||U h ←V h − Q h B(T − T h )Cuh Vb b b h −1 h ≥ 1/||(B ) ||U h ←V h ||(I + T C)u ||Ub − ||(T − T h )Cuh ||Ub b

b

−Q B(T − T )CVb ←Ub uh Ub ' (3.70) 1 1

≥ −1 h −1 ||B ||U h ←V h ||B||V ←U ||A || U ←V b b b b b b

−O ||(T − T h )C||Vb ←Ub uh Ub . h

h

Because of (3.69) and the stability of (B h )h∈H there exists a positive constant K, independent of h, such that for all h ≤ h0 the following holds:

||Q h Auh ||Vb ≥ K||uh ||Ub

for all uh ∈ Ubh .

This and dim Ubh = dim Vbh < ∞ shows that Q h A|Ubh is invertible for h ≤ h0 . More

−1 over, we obtain || Q h A|Ubh ||U h ←V h ≤ 1/K, i.e. the stability. b

b

Lemma 3.31. Raasch [539]: With the notation and conditions introduced in the previous proof, the second line of (3.68) implies (3.69). Proof. We have to prove the implication that for any

∀v ∈ Vb and B u ˜ = v, B h u ˜h = Q h v ∀v ∈ Vb : lim ||˜ u−u ˜h ||Ub = ||(T − T h )v||Ub = 0 h→0

=⇒ lim ||(T − T h )C||Ub ←Ub = 0. h→0

(3.72) (3.73)

This limit in (3.72) is a consequence of the stability and consistency of B h with B, hence the convergence of the method. For the compact operator C : Ub → Vb , and with the previous v := Cu, we claim that M := {u ∈ Ub : uUb = 1} u implies ||(T − T h )Cu||Ub ≤ C = C uUb

(3.74)

implying (3.73). With C, the closed image CM is compact. Therefore by Deﬁnition 1.8 ﬀ. the following ﬁnite open coverings exist, such that

3.5. Techniques for proving stability

∀ > 0∃N = N () : v1 , . . . , vN ∈ Vb s.t. @N CM ⊂ i=1 v ∈ Vb : v − vi Vb ≤ and by (3.72)

201 (3.75)

=⇒ ∃h0 : ||(T − T h )vi ||Ub ≤ ∀h < h0 , i = 1, . . . , N. As a consequence of (3.73) and the equiboundedness of T, T h , hence ||(T − T h )||Ub ←Vb ≤ C ∗ , we ﬁnd ∀u ∈ M∃i = iu ∈ {1, . . . , N } with Cu − vi Vb ≤ hence

(3.76)

||(T − T h )Cu||Ub ≤ ||(T − T h )(Cu − vi )||Ub + ||(T − T h )vi ||Ub ≤ C ∗ + ≤ (1 + C ∗ ) ≤ C uUb , hence the claim by (3.73).

An important example for operators, which are compact perturbations of coercive arding inequality, cf. (2.141). operators, are A ∈ L (Ub , Ub ) satisfying a G˚ Theorem 3.32. G˚ arding inequality and compact perturbation: In the Gelfand triple of Banach spaces Ub ⊂ W = W ⊂ U , cf. Deﬁnition 2.17, let Ub be (continuously, densely) and compactly embedded into the Hilbert space W. Assume that A ∈ L (Ub , Ub ) fulﬁlls a G˚ arding inequality, i.e. ∃M > 0 and m ∈ R, such that < Au, u >Ub ×Ub ≥ M ||u||2Ub − m||u||2W ∀ u ∈ Ub .

(3.77)

Then A is a compact perturbation of a coercive operator. Proof. With the identiﬁed W with W , the above assumptions yield Ub ⊂ W ⊂ Ub . Additionally the embedding IUb →Ub is compact, see Zeidler [677]. We obtain the splitting

A = A + mIUb →Ub − mIUb →Ub , with coercive A + mIUb →Ub , hence the claim.

Finally, we discuss variational crimes of the easiest type. We allow inexact evalua tions of the operators Q h A|Ubh , Q h G|Ubh , and Q h Ah |Ubh , Q h Gh |Ubh , and of Q h f , cf. Example 3.8. This is necessary, since the exact values often are not available or are too expansive. So they are approximated, e.g. by numerical integration. The inﬂuence of this type of perturbation is analyzed in the following theorem. We only present the basic result for conforming Petrov–Galerkin methods. More detailed results including nonconforming FE and spectral methods will again be discussed in Chapters 4 and in [120]. Theorem 3.33. Let Ubh , Vbh be approximating spaces for Ub , Vb in a conforming Petrov–Galerkin method, and f ∈ Vb . Assume A˜h ∈ L Ubh , Vbh and f˜h ∈ Vbh are

202

3. A general discretization theory

“proper” perturbations of Ah = Q h A|Ubh and Q h f , respectively, i.e.

||A˜h − Q h A|Ubh ||V h ←U h → 0 and ||f˜h − Q h f ||Vbh h → 0 for h → 0. b

(3.78)

b

1. Then, for suﬃciently small h,

A˜h is stable ⇔ Q h A|Ubh is stable.

(3.79)

2. Furthermore, if Ah = Q h A|Ubh is stable, the uniquely determined solution u ˜h0 of the perturbed discrete equation ˜h0 = f˜h , A˜h u

u ˜h0 ∈ Ubh

converges to the uniquely determined solution uh0 of Ah uh0 = f h , more precisely, for suﬃciently small h, we obtain ﬁnally, with (Ah )−1 U h ←V h ≤ S, b

b

˜ u − u Ubh h

h

≤ 2S A˜h − Ah V h ←U h f V h + Sf˜ − f V h . 2

b

b

b

(3.80)

b

Proof. The condition (3.78) immediately shows that for small enough h the relation (3.79) is correct. For proving the estimate we consider uh = (Ah )−1 f

Ah uh = f,

˜h = (Ah + (A˜h − Ah ))˜ uh = Ah (I + (Ah )−1 (A˜h − Ah ))˜ uh = f˜ A˜h u u ˜h = (I + (Ah )−1 (A˜h − Ah ))−1 (Ah )−1 f˜ u ˜h − uh = (I + (Ah )−1 (A˜h − Ah ))−1 (Ah )−1 f˜ − (Ah )−1 f˜ + (Ah )−1 (f˜ − f ). We assume, e.g. for suﬃciently small h, that (Ah )−1 (A˜h − Ah )Ubh ←Ubh < 1/n. With the Neumann series we obtain ∞ h −1 ˜h h −1 (−(Ah )−1 (A˜h − Ah ))j =⇒ (I + (A ) (A − A )) = j=0 h −1

(I + (A )

˜h

−1

(A − A ))

This implies u ˜ h − uh =

h

= (I + (Ah )−1 (A˜h − Ah )) + O((Ah )−1 (A˜h − Ah )2U h ←U h ). b

b

(I + (Ah )−1 (A˜h − Ah )) + O((Ah )−1 (A˜h − Ah )2U h ←U h ) (Ah )−1 f˜ b

h −1

f˜ + (Ah )−1 (f˜ − f )

h −1

f˜ + (Ah )−1 (f˜ − f ).

b

− (A )

= (Ah )−1 (A˜h − Ah ) + O((Ah )−1 (A˜h − Ah )2U h ←U h ) b

(A )

b

So we obtain ﬁnally with (Ah )−1 U h ←V h ≤ S b

˜ u − u Ubh h

hence the claim.

h

b

≤ 2S A˜h − Ah V h ←U h f˜V h + Sf˜ − f V h , 2

b

b

b

b

3.6. Stability implies invertibility V ″b

Vb ⊂

Ad

Ub′

V hb

Ub

tested by Qd′ h

P hd =

V ″bh

(Ad )h

Ub′h

203

P hd , Qd′ h uniformly bounded, and (3.78)

tested by

Ub′h .

(Ad )h = Qb′ h Ad ⏐V ″bh

Figure 3.3 General discretization methods for dual problems.

3.6

Stability implies invertibility

In this section we study the opposite relation: under which conditions is the invertibility of an operator A a consequence of the stability of its discretization Ah . In fact we show that stability is, for some cases, the stronger condition, but both properties are equivalent under rather general conditions. Again, we restrict the discussion to conforming Petrov–Galerkin methods, with Ah = Q h A. Extensions to the other methods would be possible as well. We have to simultaneously study Au0 = f , cf. (3.4), and its dual problem. So the original equation and its dual form A : Ub → Vb , Au = f and Ad : Vb → Ub , Ad v0 = f

(3.75)

induce a pair of corresponding discrete equations of the form

Ah : Ubh ⊂ Ub → Vbh ,

Ah uh0 = Q h Auh0 = Q h f

and and

(Ad )h : Vbh = Vb h ⊂ Vb → Ubh ,

(3.76)

(Ad )h v0h = Qdh Ad v0h = Qdh f .

This Vbh = Vb h ⊂ Vb is a consequence of the assumed conforming Petrov–Galerkin method with the bi-dual pair of approximating spaces. For Pdh we choose an inter polation or approximation operator, similar to the previous P h for Ubh . The Qdh are deﬁned as special case of (3.18), as B A (3.77) Qdh : Ub → Ubh : Qdh g − g , uh h h = 0∀uh ∈ Ub h . Ub ×Ub

Therefore the situation in Figure 3.2 implies a corresponding diagram for Ad ∈ L (Vb , Ub ) in Figure 3.3. For the proof of Theorem 3.37 we need the following proposition for the bounded invertibility of a linear operator between Banach spaces. It is a consequence of Theorem 2.12, however it can be proved directly via Theorem 1.19. Proposition 3.34. Let U, V be Banach spaces and A ∈ L(U, V ). Then the following conditions are equivalent: (1) A−1 ∈ L(V , U); (2) (Ad )−1 ∈ L(U , V ); (3) there exist positive constants C1 , C2 such that (a) ||Au||V ≥ C1 ||u||U for all u ∈ U and (b) ||Ad v ||U ≥ C2 ||v ||V for all v ∈ V .

204

3. A general discretization theory

If one of the above conditions is satisﬁed, we obtain ||A−1 ||U ←V = ||(Ad )−1 ||V ←U ≤ min(1/C1 , 1/C2 ). Remark 3.35. The three conditions (1), (2), (3) are not symmetric: with D(A), N (A), R(A) denoting the domain, null-space and range of A, respectively, note that A ∈ L(U, V ) implies D(A) = U. Therefore Ad ∈ L(V , U ) is deﬁned and A−1 ∈ L(V , U) exists if and only if (Ad )−1 ∈ L(U , V ) exists. Hence, we only need one of the equivalent conditions (1) or (2). However, in (3) we indeed need booth conditions, since (3)(a) still admits AU = V . Proof. The equivalence “(1) ⇔ (2)” is well known. To show “(1) ⇒ (3)” let A−1 ∈ L(V , U) (and therefore (Ad )−1 ∈ L(U , V )). This yields immediately condition (3) with C1 = 1/||A−1 || (and C2 = 1/||((Ad )−1 || = C1 ). “(1) ⇐ (3)” 3(a) and 3(b) imply that A and Ad are injective. 3(a) yields, for a convergent sequence {Auk }, uk ∈ U, that {uk } is a Cauchy sequence and consequently converges to an u ∈ U. A is continuous by assumption, so we obtain Auk → Au and can conclude that R(A) is closed in V . The closed range theorem and the injectivity of Ad yield that A is surjective. Finally, the inverse mapping theorem implies the ﬁrst condition. Remark 3.36. Applying Proposition 3.34 to the bi-dual Petrov–Galerkin approxi mation case Ah : Ubh → Vbh , and (Ad )h : Vbh = Vb h → Ubh , shows that Ah is stable if d h and only if (A ) is stable, and for both sequences the same stability constants can be chosen. For applying Theorem 3.21 we need a solution, u0 , for Au0 = f, and the properties 1., 2., 3. Theorem 3.37. Stability implies the existence of A−1 :

1. For A ∈ L(Ub , Vb ) let the pair of bi-dual approximating spaces (Ubh , Vbh = Vb h ), deﬁne a generalized Petrov–Galerkin method. Then the stability of Ah = Q h A|Ubh implies the existence of A−1 . 2. If Ah = Q h A|Ubh is replaced, according to Theorem 3.33, by an approximation Aˆh the corresponding discrete operator (Aˆd )h for the dual of A has to satisfy (3.78) as well. In this sense the stability of Aˆh again implies the existence of A−1 ∈ L(Vb , Ub ). Proof. We have to show that the existence of a bounded inverse of A follows from the stability of Q h A|Ubh . Let u0 ∈ Ub and v0 ∈ Vb be given. Obviously, they solve f := Au0 ∈ Vb , g := Ad v0 ∈ Ub , respectively. The corresponding discrete solutions uh0 , v0 h , see (3.75), (3.76), uniquely exist, since Remark 3.36 and the conforming bi-dual Petrov–Galerkin method imply Q h = Qdh = 1, stability and consistency for Ah and (Ah )d = (Ad )h , hence uh0 − u0 Ub → 0, v0 h − v0 Vb → 0 for h → 0. One

205

3.7. Solving nonlinear systems

obtains the following estimates: h u0 = Q h A|U h −1 Q h f = Q h A|U h −1 Q h Au0 Ub U Ub b b b h h −1 h h Q h · Au0 ≤ C Au0 and ≤ Q A|Ubh U ←V V ←V V V h v0

Vb

b

b

b

b

b

−1 h −1 h d = Qdh Ad |V h Qd g V = Qdh Ad |V h Qd A v0 V b

b

b

b

−1 h h Ad v0 ≤ C Ad v0 . ≤ Q h A|Ubh U ←V U U b

b

b

b

b

Theorem 3.21 applied to the bi-dual Petrov–Galerkin method yields the convergence of uh0 and v0 h to the chosen solutions u0 and v0 , respectively. Consequently, we obtain with the above estimates and the continuity of the norms ||Au0 ||Vb ≥ K||u0 ||Ub and Ad v0 U ≥ K v0 V . b

−1

b

L (Vb , Ub )

does exist. If, instead of Therefore Proposition 3.34 implies that A ∈ h h h ˆ A = Q A|Ubh a sequence A is chosen, the corresponding (Aˆd )h has to satisfy (3.78) as well.

3.7 3.7.1

Solving nonlinear systems: Continuation and Newton’s method based upon the mesh independence principle (MIP) Continuation methods

For the many discretization methods discussed in our two books much more speciﬁc solution techniques, in particular for linear problems, have been studied. The joint structure of these methods is less strong than for general discretization methods. So we do not aim at a synopsis of these methods. Rather we indicate two principles, allowing for all presented cases a simultaneous solution technique. We determine the discrete solutions, u0 , by combing the well-known continuation methods with the mesh independence principle (MIP). Proving convergence results for Newton’s method for many discrete problems avoiding the MIP often require highly technical eﬀorts. In continuation methods we embed the discrete problem Gh (uh0 ) = 0 into a oneparameter problem. We identify uh0 and x1 ∈ Rm and deﬁne F h : Rm+1 → Rm with F h (x, λ) := (1 − λ)Gh0 (x) + λGh1 (x) where

(3.79)

Gh1 := Gh and a Gh0 : Rm → Rm with a known approximate solution x0 ∈ Rm . We start we the continuation process with λ = 0 and the known (approximate) solution x0 aiming for λ = 1: we only consider the easiest case and refer for other strategies to, e.g. Allgower and Georg [20, 21], Decker and Keller [278, 439, 440] Rheinboldt and Burkardt [550, 551], and Seydel [580, 581].

206

3. A general discretization theory

This allows formulating the following algorithm: Algorithm 3.1. Continuation method by parametrization h h h INPUT: (x0 , λ0 = 0) ∈ Rm × R, Gh 0 , G1 , G0 (x0 ) = G (x0 , 0) ≈ 0); starting point with h > 0 (Initial step size); OUTPUT: (xi , λi ), i = 1, 2, . . . , (Stop at λie = 1 with xie ≈ uh 0 , λie = 1 );

ITERATION: FOR (i = 1; Stopping criterion) (i) λi = min{λi−1 + h, 1}; xi = xi−1 ; (ii) With Newton-like methods (MIP) improve this xi into xi ∈ Rm , such that F h (xi , λi ) ≈ 0; (iii) Stop for λi = 1, else goto (iv); (iv) Adapt the stepsize h > 0, goto (i).

Figure 3.4 shows the predictor and the trace of a few corrector steps. 3.7.2

MIP for nonlinear systems

As a consequence of the general discretization approach, the number of iterations in the Newton methods for the original G in (3.81) and its discrete counterpart Gh in (3.84) are strongly related. The number of iterations for a given tolerance, uhi − uh0 2U h ≤ , for G and Gh , is nearly the same, independent of h. This is achieved by the MIP, cf. Theorem 3.40, Allgower and B¨ ohmer [9], and Allgower et al. [17]. We only present this main form of the MIP. There are many other papers on this subject, dealing with diﬀerent aspects of discrete Newton methods: Deuﬂhard and Potra [285] discuss the “aﬃne” Newton method, and B¨ ohmer [112] combines it with the defect correction method. In particular, Argyros [36, 37] discusses Newton methods, where the derivative of the operator is H¨ older instead of the original

( xi , λ i )

h (xi − 1, λi − 1) Figure 3.4 Natural parametrization.

3.7. Solving nonlinear systems

207

Lipschitz-continuous, or the discretized secant iterations. In both cases the quadratic convergence is reduced. Instead of discretizations he allows perturbed or inexact Newton methods or directly works in Banach spaces, sometimes based upon majorant principles. Eﬃcient strategies combining the MIP with sequences of increasingly ﬁner meshes have been studied by Allgower and B¨ ohmer [9, 10]. They can be combined with the nested sequences of, e.g. FE spaces Sd1 Tch,1 Sd1 Tch,2 · · · yielding multiscaling or multiresolution techniques. For formulating the MIP we recall the main condition for the existence of a unique solution u0 for the original G(u) = 0 and the convergence for a general discretization: it is the bounded invertibility of the linearized G (u0 ). For a G ∈ CL1 , this is simultaneously the main condition for the (quadratic) convergence of the Newton–Kantorovich method applied to G. We do not want to formulate the sharpest possible conditions for the following theorems, but rather give their ﬂavor. We compare Newton’s method applied to G and to Gh . We additionally assume a Lipschitz-continuous derivative, so Let u0 be an isolated solution of G(u0 ) = 0, s.t. G (u0 ) : U → V

(3.80)

is boundedly invertible and G (u) − G (v)V ←U ≤ Lu − vU

∀u, v ∈ BR (u0 ).

Theorem 3.38. Newton–Kantorovich method for G, Ortega and Rheinboldt [520]: Under the condition (3.80) and for small enough u0 − u1 U the method in (3.81) is well deﬁned and converges quadratically: ui+1 := ui − G (ui )−1 G(ui ), i = 1, . . . , and u0 − ui+1 U ≤ Cui − u0 2U ,

(3.81)

or ≤ C ui − ui−1 2U . These C, C depend upon u0 − u1 U , L, (G (u0 ))−1 . For maintaining the consistency of Gh of order p > 0 for u ∈ Us a smooth subspace of U, during the Newton process we have to require G : D(G) ∩ Us → Vs , and G (u0 ) : Us → Vs is boundedly invertible.

(3.82)

Furthermore, we have to extend the concept of consistency in Deﬁnition 3.14 to the ﬁrst derivative. Deﬁnition 3.39. Diﬀerentiable consistency: Under the conditions of Deﬁnition 3.14 a general discretization method is called diﬀerentiably consistent and of order p, if it is classically consistent and if for a Frechet diﬀerentiable nonlinear operator G,

(Gh ) (P h u)P h v − Q h (G (u)v)V h → 0 and

(3.83)

b

(Gh ) (P h u)P h v − Q h (G (u)v)V h ≤ Chp (1 + uUs )vUs for h → 0. b

These vUs are usually vW k,q (Ω) and p, k, q appropriately chosen. Theorem 3.40. Newton method for the discrete Gh , cf. [9, 10, 17]: Under the conditions (3.80), (3.82), and for a linear discretization method M assume a stable Gh , consistent and diﬀerentiably consistent of order p > 0 for u0 ∈ Us ⊂ U, a smooth

208

3. A general discretization theory

subspace, still with Lipschitz-continuous (Gh ) . Start the discrete Newton process for i = 1, . . . , with uh1 := P h u1 , u1 ∈ Us , as −1 h h (3.84) G ui . uhi+1 := uhi − Gh uhi Then, for small enough u0 − u1 U and h, the uhi+1 in (3.84) uniquely exist and converge quadratically to uh0 , such that h ui+1 − uhi h ≤ C uhi − uhi−1 2 h and U U h ui − P h ui h ≤ Chp ui U , i = 1, . . . , s U Gh uhi − Q h G(ui )V h ≤ Chp ui Us , i = 1, . . . , (3.85) h ui − uh0 − (P h (ui − u0 ))U h ≤ Chp ui Us , i = 1, . . . . Corollary 3.41. MIP, cf. [9, 17]: Under the conditions of the previous theorem there exists h such that for all h ∈ (0, h] min{i ≥ 1 : uhi − P h u0 h < } − min{i ≥ 1 : ui − u0 U < } ≤ 1. (3.86) U This result is the reason for the MIP notation and our remark concerning the essential h independence of the convergence. Now we are able to formulate the discrete Newton algorithm, based upon the MIP. We start with a moderately good approximation for uh0 obtained by Algorithm 3.2 Algorithm 3.2. Discrete Newton method and MIP

m INPUT: Starting approximation uh with Gh uh 1 ∈ R 1 ≈ 0 and tolerance > 0; h > 0;

h h h OUTPUT: Approximation uh n for u0 s.t. un − u0 U h < . ITERATION: FOR (i = 1; Stopping criterion)

h −1 h h h h ui G ui , see (3.84); (i) Calculate uh i+1 := ui − G (ii) Update i := i + 1; h (iii) Estimate uh i − u0 U h (with standard Newton techniques); h (iv) Stop if u − uh h < , else goto (i). i

0

U

4 Conforming ﬁnite element methods (FEMs) 4.1

Introduction

Finite element methods (FEMs) are nowadays the most advanced methods, applicable to nearly all types of PDEs. As throughout this book, we restrict the discussion to elliptic problems, from linear, to semilinear, to quasilinear to fully nonlinear equations and systems of orders 2 and 2m. The goal in this chapter is the proof of stability and convergence for diﬀerent types of conforming FEMs and to generalize many of the known the results. In Chapter 3, we developed appropriate tools for proving results for these operator equations and a relatively general class of discretization methods, based upon consistency and stability. Conforming FEMs are consistent for continuous problems. We proceed in two steps to stability for nonlinear a consequence of stability for the linearized operator. Let a boundedly invertible linear operator yield a stable discretization, e.g. for an operator with coercive bilinear form. Under some technical conditions, this stability is inherited to its compactly perturbed linear operators, if these are still boundedly invertible and consistent. These results are applicable essentially to all the elliptic problems in Chapter 2, if, for nonlinear problems, the linearization is bounded. As mentioned, a stable discretization of a boundedly invertible operator is a consequence of a coercive principal part or a G˚ arding inequality, cf. Theorem 3.32. This is correct, except for (nonlinear) elliptic systems in the sense of Agmon et al. [3], with diﬀerent highest orders of the derivatives in diﬀerent equations, cf. Deﬁnition 2.85. The most famous example is the Navier–Stokes operator. The stability and bounded invertibility of the Stokes operator has been proved by other techniques. Then the above arguments show the stability of the (linearized) Navier–Stokes operator for moderate Reynolds numbers. This is applicable to other examples as well. Convergence for monotone operator is available without linearization. Excellent books are available on FEMs. We list only some of them, best suited to support speciﬁc points in the text. We need discretizations of general problems and their convergence properties. Hackbusch’s [386, 387] studies of elliptic, including higher order PDEs, with ﬁnite diﬀerence and FEMs, and Ciarlet’s [174] Brenner and Scott’s [141] books on FEMs have very strongly inﬂuenced this book and this and the next chapters. Diﬀerent numerical methods and the corresponding analysis are studied by Grossmann et al. [374–376], for second order PDEs. FEMs, again combined with many interesting applications, are discussed by Babuˇska and Strouboulis [51]

210

4. Conforming ﬁnite element methods

and Oden and Reddy [518]. Other important applications with more or less theory for linear and nonlinear equations and systems are studied, e.g. by Braess [135], Brenner and Scott [141], Debnath et al. [277, 638], Leung [473], DuChateau and Zachmann [305–307], Larsson and Thomee [469], Quarteroni and Vali [537], and Thomas [626]. In all these books, nonlinear problems essentially are discussed via some practically important examples. Zen´ısek [681] approaches nonlinear problems with monotony methods. Haslinger et al. [393] study unilateral linear boundary value, mainly mechanical contact, problems via inequalities. The nonlinear problems in these books are so-called semilinear problems. The analytic theory, special Galerkin methods and a general discretization theory for nonlinear, here semilinear and quasilinear, problems are presented by Zeidler [675–678]. FEMs for fully nonlinear problems have not been available before. So they are not included in previous books, c.f. Section 5.2. In this chapter we only consider and approximately solve elliptic and monotone problems formulated as weak forms. Therefore we assume throughout this chapter that our operator and approximating subspaces satisfy

A : Ub → Ub , G : D(G) ⊂ Ub → Ub , Ubh = Vbh ∀ h usually A and G are deﬁned on U as well. The basic idea of FEMs is simple. We demonstrate it for a linear operator A : U = V := H 1 (Ω) → V := H −1 (Ω), s.t. A : Ub = Vb := H01 (Ω) → Vb bijective.(4.1) The index and exponent, b and h , e.g. in Ub and Ubh , indicate boundary conditions and ﬁnite dimensional subspaces. The exact solution, u0 , is determined by testing u0 ∈ Ub ⊂ U : a(u0 , v) = Au0 , vV ×V = f, vV ×V

(4.2)

∀ v ∈ Vb ⊂ V often with Ub = Vb ⊂ U = V. In FEMs the Ub , Vb in (4.2) are replaced by ﬁnite dimensional approximating subspaces Ubh , Vbh of piecewise, e.g. linear polynomials, deﬁned on a triangulation of Ω, cf. Deﬁnition 4.2, with Ubh = Vbh for conforming methods. We determine the discrete or approximate solution, uh0 , such that Ubh ⊂ Ub , Vbh ⊂ Vb , : ; (4.3) uh0 ∈ Ubh ⊂ Ub ⊂ U : a uh0 , v h = Auh0 , v V ×V = f, vV ×V ∀v h ∈ Vbh ⊂ Vb . The Ub , Ubh and Vb , Vbh , represent ansatz and test functions, respectively. If the functions uh ∈ Ub , Ubh are continuous and exactly satisfy the boundary conditions, we call (4.3) a conforming FEM. For a bounded and coercive a(u, u), so ∃α > 0 ∀u ∈ Ub : a(u, u) ≥ αu2U , the convergence of the uh0 to u0 via the Cea Lemma 4.48 is surprisingly simply proved and yields for piecewise polynomial FEs of degree d − 1 u0 − uh0 ≤ (C/α)dist(u0 , U h ) = C ∗ hd−1 u0 H d (Ω) . (4.4) U We do not study super convergence results, c.f. e.g. Bank and Xu, Griewank and Reddien, Jiang et al. [66, 368, 417] The conditions of continuous FEs, exactly satisfying the boundary conditions, is too restrictive for many applications. If we relax them we obtain the so-called nonconforming FEMs, cf. Chapter 5.

4.1. Introduction

211

Our approach to FEMs diﬀers from the usual versions in several respects. We generalize known results and modify proofs in many directions. The main tool is the systematic use of the general theory of discretization methods as presented in Chapter 3. This contrasts with the standard coercivity and ellipticity arguments for mainly linear problems with their bilinear forms and trilinear modiﬁcations as, e.g. for the Navier–Stokes equations, cf. Section 4.6, and to the monotony arguments for higher nonlinearities. In fact, conforming and nonconforming FEMs, as usual with one reference element and here with ﬁxed polynomial degree, are special cases of generalized Petrov–Galerkin methods. We have to ensure stability and convergence for FEMs for elliptic problems. Stability is a consequence of the fact that for a linear A the (modiﬁed) principal part Ap induces a stable Ahp . For conforming FEMs the (linear) discrete Ah are (classically) consistent with the original A. The goal is an extension to nonlinear problems and to higher order conforming and nonconforming FEMs for the large class of linear, semilinear, quasilinear and fully nonlinear elliptic equations mentioned above. This is achieved, including quasilinear problems, for FEMs based upon the classical weak form of the problems. For fully nonlinear problems we have to use nonconforming FEMs for their strong form. Applying FEMs in the context of path-following of parameter-dependent solutions of nonlinear problems and avoiding spurious solutions for turning and bifurcation points requires high order methods. They can conveniently and with a wide range of applications be deﬁned as conforming and nonconforming FEMs. We organize this chapter on conforming FEMs as follows. In Section 4.2 we start with the standard approximation theory for FEs. We only prove results not included in standard textbooks, e.g. [141], and in easily accessible papers, e.g. [362]. As preparation for the later nonconforming FEMs, we present FEs on curved domains Ω and smooth FEs ∈ C r (Ω), r ≥ 1 in Subsections 4.2.7, and with Davydov, in 4.2.6. In Section 4.3 a simple example, the Helmholz equation, motivates the deﬁnition of conforming FEMs. They are extended to general linear and in Section 4.4 to quasilinear elliptic diﬀerential equations and systems of orders 2 and 2m. Another type of generalization represents the very active area of mixed FEMs, presented here only for the Navier–Stokes equation in Section 4.6. For the case of moderate Reynolds numbers, it is a compact perturbation of the well studied boundedly invertible Stokes operator with stable discretization. Consequently a boundedly invertible linearized Navier–Stokes operator implies stability for the linearized and the nonlinear operator. This stability result for the Navier–Stokes operator seems to be the state of the art in discretizations via FEM, spectral, diﬀerence and wavelet methods anyway. Extensions to nonconforming and adaptive forms are not considered here. In mixed FEMs the diﬀerent components of the solution, here the velocity, u, and the pressure, p, live in diﬀerent spaces and satisfy side conditions, div u = 0. Correspondingly, for uh , ph , diﬀerent and appropriate spaces, U h , W h , mixed FEMs have to be chosen. In particular, for the later studies of bifurcation, FEMs for eigenvalue problems are important. We discuss them in Section 4.7. The extension of the results to nonlinear scenarios is not interesting here, since the eigenvalues that decide bifurcation are always deﬁned by the linearized operators.

212

4. Conforming ﬁnite element methods

Adaptive FEMs are very well advanced these days. So the main tools and results are presented in Chapter 6 by W. Doerﬂer. We have already mentioned that we delay the study of FEMs with crimes, or nonconforming FEMs to the next chapter. Another possibility for violated continuity and boundary conditions, the discontinuous Galerkin methods (DCGMs) follows. We exclude the following important areas. The hp-methods and their discontinuous versions are found, e.g. in Babuska et al. [52, 53, 517]. Even some of the recent results towards superconvergence in Babuska and Strouboulis [51], and exponential accuracy as in Schwab and Melenk [485–487], would probably ﬁt our approach. Multigrid methods, multigrid solutions of bifurcation problems, hierarchical bases and all kinds of a posteriori and reliability estimates are studied by, e.g. Hackbusch [385], Babuska and Strouboulis [51], Brenner and Scott [142], Mittelmann and Weber [492–494, 660, 661], and Babuska et al. [51, 59–61, 63–66]. Special high order FEMs are studied, e.g. by ˇ ın, Segeth and Doleˇzel [593]. Sol´ Throughout the chapter, all convergence results require a suﬃciently small discretization parameter, h. In fact, we characterize, strongly simplifying, an FE discretization method by a parameter h. In the following we indicate the essential constants depending upon essential parameters, by indices listing these constants, e.g. Ch,μ depends upon h, μ. Sometimes, we omit parameters to show the independence of constants, e.g. we would write C instead of Ch,μ to indicate the independence of C from h, μ.

4.2

Approximation theory for ﬁnite elements

The following introduction to the approximation theoretic properties of ﬁnite elements (FEs) and many other topics is strongly inﬂuenced by the classical Ciarlet [174], and by Hackbusch, Braess, Brenner and Scott [136, 141, 386, 387]. The basis for the success of FEMs are the good approximation properties and the high ﬂexibility of FEs. We present the basic results. After introducing subdivisions and general FEs, we consider the most interesting example of piecewise polynomials. We formulate general FE spaces and their approximation properties for functions. For vectors of functions, e.g. solutions of systems of elliptic equations, they remain unchanged if each component shares the same structural properties and has the same smoothness. We summarize interpolation errors, inverse estimates and extensions to curved boundaries. The results in this section are strictly local, except Subsection 4.2.6 with O. Davydov. Only here do we present and prove the essentially new results for smooth FEs on curved domains with the necessary splitting properties. We assume throughout the whole chapter that the (open) domain Ω satisﬁes Ω ⊂ Rn bounded domain, piecewise smooth Lipschitz-continuous ∂Ω. 4.2.1

(4.5)

Subdivisions and ﬁnite elements

The power of FEs is based upon approximating functions, uh : Ω ⊂ R, by splitting the domain into subsets Ti ⊂ Ω. For simpliﬁed computations we assume each of these Ti

4.2. Approximation theory for ﬁnite elements

213

to be aﬃnely (or isoparametrically, see Section 4.2.7), mapped onto the same reference domain, K. In the simplest case an approximating function uh , restricted to a triangle T = Ti , is uh ∈ P1 , hence a linear polynomial, deﬁned by its function values in the three vertices of T . This is generalized as, see [174]: Deﬁnition 4.1. We assume that 1. the element or reference domain K ⊆ Rn is an open domain with piecewise smooth boundary; 2. the space of shape functions P is a ﬁnite-dimensional space of functions deﬁned on K; and 3. the set, N = {N1 , N2 , . . . , Nd }, of nodal variables are linear, linearly independent, usually interpolation functionals deﬁned on P. They represent a basis for P , hence, P = span N . We need the same dimension dim P = dim P = d . Then (K, P, N ) is called a reference element or a ﬁnite element. There are many possible combinations of P, N . If P = span {φ1 , φ2 , . . . , φd } is a basis for P dual to N , hence Ni (φj ) = δij , then φ1 , . . . , φd is called the nodal basis for P. A set of nodal variables, N , is called unisolvent for P, or K, P, N a unisolvent FE, compare 3., if

∀(βi )di=1 ∈ Rd

∃1 φ ∈ P : Ni (φ) = βi , i = 1, . . . , d .

This choice of a reference element (K, P, N ) has the important consequence that much of the following analysis and computations can be performed in the reference domain K. In the previous simplest case, motivating Deﬁnition 4.1, K is a (unit) triangle with vertices P1 , P2 , P3 , the P = P1 and N are the point evaluations in the P1 , P2 , P3 . It is implicitly assumed that the nodal variables, Ni , lie in the dual space of some larger function space, U, allowing Ni (φ)∀φ ∈ U. The Ni usually denote evaluation of functions or derivatives in given interior and boundary points of K. Obviously N is unisolvent for P if and only if dim P = dim span N = d and φ ∈ P, Ni (φ) = 0, i = 1, . . . , d implies φ ≡ 0. For a given u ∈ U, a nodal basis of shape functions P = span{φ1 , . . . , φd } and a unisolvent N = {N1 , N2 , . . . , Nd }, applicable to functions in a space U : K → R, a local interpolation operator is uniquely deﬁned, as

IK := I : U → P, Iu :=

d

Ni (u)φi , Ni (u) = Ni (Iu), i = 1, . . . , d .

(4.6)

i=1

We use the standard notation of the Dirac delta function δ(Pi ) and δ(Pi ) ∈ N if Ni (u) = u(Pi ) = δ(Pi )u. We mainly consider these or more general interpolations. Usually the K is chosen as a triangle, tetrahedron, an n-dimensional simplex, rectangle, parallelogram or an n-dimensional aﬃne cuboid. Usually the following Ti are aﬃnely (or isoparametrically), see below, mapped onto this same K. Then properties 2.–4. in Deﬁnition 4.2 make sense. We introduce subdivisions for general ﬁnite elements and will combine it below with the above local interpolation operator on K.

214

4. Conforming ﬁnite element methods

Figure 4.1 Admissible triangulation for Ω with curved boundary.

Deﬁnition 4.2. Subdivision: A set T h = {T1 , . . . , TM } of open subsets Ti ⊂ Ω ⊂ Rn is called a subdivision and an admissible subdivision for Ω if it satisﬁes 1. and 1.–4., respectively. M

1. Ti ∩ Tj = ∅ for i = j and Ω = ∪ T i ; i=1

2. each T ∈ T h has at least n + 1 and at most n ≥ n + 1 vertices, e.g. triangles and quadrangles with straight or curved edges; 3. T i ∩ T j for i = j is either empty or has one vertex or a common edge or one, in general (n − 1) -dimensional surface spanned by joint vertices of Ti and Tj , see Figure 4.1; 4. for paralellotopes or cuboids, hence Qi = Πnj=1 [aj , bj ], generally Ti , 3. can be relaxed, e.g. for rectangles, T¯i ∩ T¯j may be one half of the larger edge of the larger Tj , see Figure 4.15. T h is called a triangulation when the T ∈ T h are triangles, tetrahedrons or ndimensional simplices, possibly with curved edges, cf. Figure 4.1. 4.2.2

Polynomial ﬁnite elements, triangular and rectangular K

Chosing piecewise polynomials as P are certainly the most important FEs. We start choosing K as triangles and quadrangles for n = 2 and their generalizations to n > 2. To avoid diﬃculties near the boundary, we start with a polyhedral domain Ω, for n = 2 called a polygon.

(4.7)

Curved boundaries Ω are treated by interpolation, approximation or isoparametric FEs in Subsections 4.2.7 and 4.2.6. We choose the reference domain K as√a unit rectangular n-triangle or a unit n-cube, hence diam K = 1 or diam K = n. We give some examples of FE triples (K, P, N ). The most important choice for P are polynomials with degree d: ⎧ ⎫ ⎨ ⎬ αi xi : x = (x1 , . . . , xn ) ∈ Rn , Pd := Pd2 , P d := Pd1 , (4.8) P = Pdn := ⎩ ⎭ |i|≤d

with i = (i1 , . . . , in ) ≥ 0, |i| = i1 + · · · + in , xi = xi11 · · · xinn , αi ∈ R.

4.2. Approximation theory for ﬁnite elements

215

Sometimes, we allow P ⊂ Pdn . The dimensions of these Pdn are dim Pd1 = dim P d = (d + 1), dim Pd2 = dim Pd = and in general

dim Pdn

=

n+d n

(d + 1)(d + 2) , 2

(4.9)

n+d . = d

A convenient criterion for checking the unisolvence property for n-triangles combines the edges L ∈ {L1 , L2 , . . . , Ln+1 } of the triangle K with Proposition 4.3, see [141]. Let a nondegenerate hyperplane L be deﬁned as αi xi − β = 0 with αi ∈ R, |αi | > 0. (4.10) L := x ∈ Rn : L(x) = |i|=1

Proposition 4.3. Let a polynomial P ∈ Pdn vanish on L. Then there exists a Q ∈ n , such that P (x) = L(x)Q(x). Otherwise apply computer algebraic tools. Pd−1 Triangular ﬁnite elements In Figures 4.2–4.11 we show combinations of K with Pd for d = 1, . . . , 5, and appropriate N , unisolvent for Pd . The diﬀerent cases are indicated by points zi in K or on the boundary of K according to zi is marked by • if Ni (φ) = φ(zi ), Ni ∈ N , for φ ∈ Pd , and by # if Ni (φ) = φ(zi ) and Nig (φ) = grad φ(zi ), Ni , Nig ∈ N . If only function values, •, are required, we have a Langrange ﬁnite element. If function values and ﬁrst (#) or, even additionally, second derivatives, indicated by two circles are required, e.g. the Argyris element, see below, we have Hermite ﬁnite elements. Proposition 4.3 is visualized in Figures 4.2–4.5. In some cases, e.g. for the Argyris P5 -element, see Figure 4.9, in addition to the values of the function and ﬁrst and second derivative, normal derivatives are z3

z1

z2

Figure 4.2 P1 : Linear Lagrange FEs.

216

4. Conforming ﬁnite element methods z3

z5

z4

z1

z6

z2

Figure 4.3 P2 : Quadratic Lagrange FEs.

z 10

z9

z8

z7

z1

z5

z2

z4

Figure 4.4 P3 : Cubic Lagrange FEs. z 15

z 14

z 13

z 12

z 10

z9

z1

z3

z5

Figure 4.5 P4 : Quartic Lagrange FEs.

4.2. Approximation theory for ﬁnite elements z3

z1

z2

Figure 4.6 P3 : Cubic Hermite FEs.

z3

z9 z1

z6

z2

Figure 4.7 P4 : Quartic Hermite FEs.

Figure 4.8 Inequivalent cubic Hermite FEs: Incompatible normal derivatives.

217

218

4. Conforming ﬁnite element methods

z3

z1

z2

Figure 4.9 Quintic Argyris FEs.

z3

z2

z1 Figure 4.10 P1 : Nonconforming linear Lagrange FEs (Crouzeix–Raviart).

prescribed. So in the sense of Deﬁnition 4.6 below, the Argyris elements are not aﬃne equivalent, but only nearly aﬃne equivalent [135, 174]. On the other hand, see Subsection 4.2.6, there are C 1 FEs as modiﬁed Argyris elements, important for the later FEMs. The Bell element corresponds to an 18-dimensional subset of P5 . For unisolvence we need the same number of values of the function (and/or its derivative) as dim P, e.g. dim Pd = (d + 1)(d + 2)/2. All ﬁgures show unisolvent, hence dim Pd = (n + 1)(n + 2)/2 = dim N , and conforming ﬁnite elements. Figures 4.10 and 4.11 show nonconforming elements, see Section 5.5. The case P1 for the linear conforming and nonconforming Lagrange elements shows that the N is certainly not uniquely determined by P. Remark 4.4. All the following ﬁgures show a high symmetry with respect to the boundary (and interior) points. For general nondegenerate subdivisions, see Deﬁnition 4.12 below, we have to impose Ni ∈ N for boundary points symmetric and of the same kind on all edges, corresponding to those in neighboring subtriangles. For highly

219

4.2. Approximation theory for ﬁnite elements

z5

z6

z3

z4

z1

z2

Figure 4.11 P2 : Nonconforming quadratic Lagrange FEs.

symmetric subdivisions, e.g. rectangular triangles in a rectangle, relaxations of this symmetry of the boundary points are possible. Remark 4.5. Simple inductive computation of the FE-interpolant: Let the knots be chosen in a triangle K on d + 1 parallel lines with altogether s = 1 + 2 + · · · (d + 1) points z1 , z2 , . . . , zs , see [135]. We determine (d + 1)(d + 2) = dim Pd . (4.11) 2 Using the aﬃne equivalence, see Deﬁnition 4.6, we can choose the parallel lines as y = constant. The φ for the d + 1 = 1 line is obvious. So we apply the following (4.12) 2 in (4.11). Let the last line, with inductively. We assume the ﬁrst d lines for φ ∈ Pd−1 the ﬁrst d + 1 knots, z1 , · · · ., zd+1 , be located at y = 0. φ ∈ Pd s.t. φ(zi ) = fi , i = 1, . . . , s =

Determine p0 ∈ Pd1 : p0 (zi ) = fi , i = 1, . . . , d + 1, and, by induction, 2 determine the unique q ∈ Pd−1 : q(zi ) =

(4.12)

1 (fi − p0 (zi )), i = d + 2, . . . , s. yi

Then p ∈ Pd2 = Pd , with p(x, y) := p0 (x) + y q(x, y) solves (4.11). A similar approach is possible for higher dimensions and some rectangular elements. Bilinear, biquadratic, . . . ﬁnite elements For rectangular K we often use instead of Pd the tensor products of bilinear, biquadratic, . . . polynomials, see Figures 4.12–4.15 and [141, 572] ⎧ ⎫ ⎞⎛ ⎞ ⎛ ⎨ ⎬ 2 ci xi ⎠ ⎝ dj y j ⎠ , i , dj , x, y ∈ R ⊂ P2d Q2d = Pd1 ⊗ Pd1 = u = ⎝ ⎩ ⎭ 0≤i≤d

Pdn ⊂ Qnd =

n i=1

n Pd1 ⊂ Pnd , Qd = Q2d ,

0≤j≤d

n dim Qnd = dim Pd1 = (d + 1)n .

(4.13)

z3

z4

z1

z2

Figure 4.12 Q1 : Bilinear Lagrange FEs.

z9

z7

z6

z1

z2

z3

Figure 4.13 Q2 : Biquadratic Lagrange FEs.

z 13

z 16

z 12

z8

z1

z2

z3

z4

Figure 4.14 Q3 : Bicubic Lagrange FEs.

4.2. Approximation theory for ﬁnite elements

221

quadratic interpolation

Figure 4.15 Quadratic subdivision (mesh-reﬁnement), violating Deﬁnition 4.2, 3., see 4..

4.2.3

Interpolation in ﬁnite element spaces, an example

We want to interpolate on general subdivisions of Ω. This requires the extension from ˆ = T ∈ T h , denoted as (K, ˆ P, ˆ N ˆ ). the (K, P, N ) situation to the collection of the K For eﬃcient computations they have to be closely related to the original (K, P, N ). Therefore we introduce ˆ P, ˆ N ˆ ) be two ﬁnite Deﬁnition 4.6. Aﬃne equivalent FEs: Let (K, P, N ) and (K, ˆ F (x) = Ax + b, A ∈ Rn×n nonsingular, b ∈ Rn , be an aﬃne elements, let F : K → K, (or isoparametric) map such that, see Figure 4.16, 1. 2. 3.

ˆ F (K) = K, ˆ P ◦ F = P or Pˆ = P ◦ F −1 and ˆ (fˆ) = N (fˆ ◦ F ) = N (f ) for fˆ : K ˆ → R and f := fˆ ◦ F. N

ˆ ˆ ˆ ˆ ˆ ˆ Then (K, P, N ) F (K, P, N ) and (K, P, N ) is called (aﬃne) equivalent to (K, P, N ). Obviously aﬃne equivalence is an equivalence relation. If P = span{φ1 , φ2 , . . . , φd }, cf. Deﬁnition 4.1, is a dual basis for N with Ni (φj ) = δij , then Pˆ = span{φˆ1 = φ1 ◦ ˆ and N ˆi vˆ = Ni (ˆ F −1 , . . . , φˆd = φd ◦ F −1 } is a dual basis for N v ◦ F ). The elements in Figures 4.8 and 4.17 are inequivalent. Now we can handle the local interpolation in (K, P, N ) and its aﬃne equivalent elements, e.g. the (T, PT , NT ). If (4.7) is satisﬁed, we assume for simplicity the same (K, P, N ) and appropriate FT and PT ◦ FT = P. A generalization to diﬀerent K is straightforward, but we need additional technical tools, e.g. combining triangular

222

4. Conforming ﬁnite element methods zˆ3

z3

F

Kˆ

K zˆ1 z1

z2

zˆ2

Figure 4.16 Aﬃne transformation.

Figure 4.17 Inequivalent quadratic Lagrange FEs: Aﬃne mappings impossible.

and rectangular elements. For introducing the space of FEs, we need K, P, N , the subdivision T h and the aﬃne (or isoparametric) mappings F = FT : K → T, F x = Ax + b, A = AT ∈ Rn×n , b = bT ∈ Rn ∀ T ∈ T h .

(4.14)

Deﬁnition 4.7. Let T h be a subdivision for Ω and let FT : K → T be chosen as in (4.14). Then U h := {uh ∈ L∞ (Ω) : uh |T ∈ PT := P ◦ (FT )−1 }

(4.15)

is denoted as (approximating) space of FEs, uh as FEs. Since the T ∈ T h are open, (4.15) does not imply transition properties for uh |T and uh |T1 if T¯ ∩ T¯1 = ∅. For nonconforming FEs, uh , see below, and an edge e ⊂ T¯ ∩ T¯1 , usually uh |T¯|e = uh |T¯1 |e . Therefore, in many cases additional properties are imposed such as U h ∩ L2 (Ω), U h ∩ C 0 (Ω), U h ∩ H01 (Ω). For the following local and global interpolation operators we combine K, P = {φ1 , . . . , φd }, and N = {N1 , . . . , Nd }, with the bijection FT , see Deﬁnitions 4.2 ﬀ. Choose U v : Ω → R such that ∀T ∈ T h the NiT (v) := Ni (v ◦ FT ) are deﬁned.

4.2. Approximation theory for ﬁnite elements

223

Deﬁnition 4.8. Choose the FE space U h : Ω → R with the underlying (K, P, N ), P a nodal basis for N , the subdivision T h and a function v : Ω → R: 1. Let (T, PT , NT ) and the ﬁxed (K, P, N ) be aﬃne equivalent ∀T ∈ T h such that FT x = AT x + bT satisﬁes FT (K) = T. cf. Figure 4.18. 2. Deﬁne the interpolation operator I h = ITh h : U → U h with NiT (v) := Ni (v ◦ FT ) = NiT (I h v ◦ FT ) and φTi := φi ◦ FT−1 , as, see (4.6),

h

I v|T =

ITh h v|T

= IT v =

d

NiT (v)φTi : T → R ∀ T ∈ T h .

(4.16)

i=1

3. The φTi = φi ◦ FT−1 , i = 1, . . . d , ∀ T ∈ T h are called an interpolation basis for PT for a ﬁxed T ∈ T h (for ∀T ∈ T h not a basis for U h , see 4. 4. Assume P ∈ T¯ ∩ T¯1 to be a joint vertex or a point on an edge e or an (n − 1)dimensional boundary surface of T¯ with, e.g. δ(P ) ∈ NT . Then ∀ Tj ∈ T h with P ∈ T¯j the δ(P ) ∈ NTj and identical function values in this point P have to be used for all these T¯j P deﬁning, e.g. an N (v ◦ FTj ) = δ(P )v. (This implies continuity of the interpolation function in P from T to Tj , but not on e and in Ω and δ(P ) yields exactly one basis element for U h .) Analogous restrictions are to be imposed for derivatives. Exceptions are the DCGMs in Chapter 7. 5. Let l be the highest derivative required in N . A reference element (K, P, N ), K ⊂ Ω is said to be a C r element if r is the largest nonnegative integer such that r+1 ¯ ⊆ C r (Ω) ∩ W∞ U h = I h (C l (Ω)) (Ω). n d 6. Let Pd−1 = P ⊂ W∞ (K). Then K or the FEs are called polynomial FEs of (total) n n n degree d − 1 or order d and denoted as Pd−1 (T h ). For Pd−1 ⊆ P ⊆ Pd+τ ,P⊂ d W∞ (K), with the smallest possible τ ≥ −1, e.g. the bilinear FEs, we call them FEs of at least degree d − 1.

Proposition 4.9. Under the conditions of Deﬁnition 4.8 we get, cf. (4.15), IK v = v ∀ v ∈ P and I h v h = v h ∀ v h ∈ U h implying

(4.17)

(IK )2 = IK on P and (I h )2 = I h on U h , hence, IK , I h are projectors. Remark 4.10. There are diﬀerent reasons to use the reference elements (K, P, N ). Most theoretical results can be proved by combining the speciﬁc situation (K, P, N ) with the aﬃne equivalence. For the numerical praxis another point is even more important: The Ni (v) and φi , i = 1, . . . , d can be computed once. Then the NiT (v) := Ni (v ◦ FT ) and φTi = φi ◦ FT−1 are obtained by simple transformation formulas via the FT . This applies to the terms needed in the FEMs below, e.g. elements of the mass and stiﬀness matrices as well, cf. (4.121).

224

4. Conforming ﬁnite element methods zˆ 3

z3

F = FT

K

T=K zˆ 1

z1

zˆ 2

z2

Figure 4.18 Aﬃne transformations here applied to linear FEs.

Example 4.11. We want to demonstrate this procedure for a triangulation applied to two cases of linear and cubic two-dimensional polynomials deﬁned by the values indicated in Figures 4.19 and 4.20, see [576], p. 68 ﬀ. We start with the transformation from the reference triangle K with the vertices z1 := (0, 0), z2 := (1, 0), z3 := (0, 1), and the coordinates (ξ , η) ∈ K onto the triangle T . We denote edges of T as zˆi := (xi , yi ), i = 1, 2, 3 and its coordinates as (x , y) ∈ T , thus deﬁning FT = ((FT )1 , (FT )2 )T as x = (FT )1 (ξ , η) := x1 + (x2 − x1 ) ξ + (x3 − x1 ) η y = (FT )2 (ξ , η) := y1 + (y2 − y1 ) ξ + (y3 − y1 ) η. Since FT is linear its derivative is constant and (x2 − x1 ) (x3 − x1 ) . FT = (y2 − y1 ) (y3 − y1 ) z3

z1

z2

Figure 4.19 P1 : Again linear Lagrange FEs.

(4.18)

4.2. Approximation theory for ﬁnite elements

225

z3

z1

z2

Figure 4.20 P3 : Again cubic Hermite FEs.

This FT is bijective if and only if the Jacobian, J, of FT is (x − x ) (x − x ) 1 3 1 2 J = det (FT ) = det (y2 − y1 ) (y3 − y1 ) = (x2 − x1 )(y3 − y1 ) − (y2 − y1 )(x3 − x1 ) = 0.

(4.19)

Then FT is invertible. (FT )−1 has the form ξ = (FT )−1 1 (x , y) := a1 + a2 x + a3 y η = (FT )−1 2 (x , y) := b1 + b2 x + b3 y.

(4.20)

By equating the coeﬃcients in FT ◦ (FT )−1 = I we determine the a1 , . . . , b3 as a1 = −(x1 (y3 − y1 ) − y1 (x3 − x1 ))/J, a2 = (y3 − y1 )/J, a3 = −(x3 − x1 )/J, b1 = (x1 (y2 − y1 ) − y1 (x2 − x1 ))/J, b2 = −(y2 − y1 )/J, b3 = (x2 − x1 )/J.

(4.21)

We use the notation u : (ξ , η) ∈ K → R and v : (x , y) ∈ T → R. For the later computations of the FE terms in mass and stiﬀness matrices, see Example 4.51 below, we need the corresponding transformations of the partials via the chain rule: For v := u ◦ FT−1 we ﬁnd vx = uξ ξx + uη ηx vy = uξ ξy + uη ηy .

226

4. Conforming ﬁnite element methods

For computing the partials of the ξ, η we diﬀerentiate (4.18) with respect to x and ﬁnd 1 = (x2 − x1 ) ξx + (x3 − x1 ) ηx 0 = (y2 − y1 ) ξx + (y3 − y1 ) ηx . Analogously, we diﬀerentiate with respect to y. Both systems yield the solutions ξx = (y3 − y1 )/J, ξy = − (x3 − x1 )/J ηx = −(y2 − y1 )/J, ηy = (x2 − x1 )/J.

(4.22) To give a feeling for the interplay of the transformations in Ni (v ◦ FT ) · φi ◦ FT−1 we will study below a special case of (4.18), the scaling

F (ξ, η) := FT (ξ, η) := (x = hξ, y = hη) with ξx = 1/h, ξy = 0, ηx = 0, ηy = 1/h

(4.23)

implying vx = uξ /h, vy = uη /h.

(4.24)

We apply these results to two special cases: Linear Lagrange FEs:

We choose P = PT = P1 and v and u have the form v = v(x, y) = c1 + c2 x + c3 y, u = u(ξ, η) = α1 + α2 ξ + α3 η.

For the z1 = (0, 0), z2 = (1, 0), z3 = (0, 1), and N = {δ(z1 ), δ(z2 ), δ(z3 )} we want to determine a nodal basis P = span {φ1 , φ2 , φ3 }. More generally, to determine a φ interpolating in the zi or the φi in (4.6), we compute in a ﬁrst step the αj for given ui := u(zi ) in Figure 4.19. We introduce the vectors α := (α1 , α2 , α3 )T and u := (u1 , u2 , u3 )T and obtain the system α1 = u1 α1 +α2 = u2 α1 +α3 = u3 This (4.25) has the form ⎛

1 u = Bα with B := ⎝ 1 1

0 1 0

implying

⎞ 0 0 ⎠, 1

u1 = α1 −u1 +u2 = α2 −u1 +u3 = α3 . ⎛

1 or α = Au with A := ⎝ −1 −1

(4.25)

0 1 0

⎞ 0 0 ⎠, (4.26) 1

where AB = I. We employ the unit vectors ei , i = 1, 2, 3, e.g. e1 := (1, 0, 0)T , for determining the T for the φi as nodal basis φi . We obtain the coeﬃcients αi = α1i , α2i , α3i αi = Aei , e.g. α1 = (1, −1, −1)T , φ1 (ξ, η) = 1 − ξ − η.

4.2. Approximation theory for ﬁnite elements

227

For determining the φi (FT−1 ) we insert in φi (ξ, η) for the ξ, η the terms in (4.20) 3 with the a1 , . . . , b3 determined in (4.21). Finally IK u = j=1 φj u(Pj ) and IT v = 3 −1 v(Pˆj ). j=1 φj FT In cases of simple T, P, N we can often determine the φj and φj FT−1 and the I h v|T directly, e.g. for our linear Lagrange FEs and the scaling in (4.23). With T = hK, we get obviously φ1 = 1 − ξ − η, φ2 = ξ, φ3 = η from (4.26) and with FT−1 in (4.23) we ﬁnd, e.g. φT1 = φ1 ◦ FT−1 = (h − x − y)/h. The zi = Pi and zˆi = Pˆi show NiT v = Ni (v ◦ FT ) = v(FT (Pi )) = v(Pˆi ). Then (4.16) yields I h v|T = [v(Pˆ1 )(h − x − y) + v(Pˆ2 )x + v(Pˆi )y]/h (I h v|T ) = ((v(Pˆ2 ) − v(Pˆ1 )), (v(Pˆ3 ) − v(Pˆ1 )))/h with I h v|T ∞ ≤ max{|v(Pˆi )| : i = 1, 2, 3} ≤ v|T¯ ∞ , (I h v|T ) ∞ ≤ max{|(v(Pˆ2 ) − v(Pˆ1 ))|, |(v(Pˆ3 ) − v(Pˆ1 ))|}/h ≤ v |T ∞ . Cubic Hermite FEs: The ansatz for v and u is now v = v(x, y) = c1 + c2 x + c3 y + c4 x2 + c5 xy + c6 y 2 + c7 x3 + c8 x2 y + c9 xy 2 + c10 y 3 , u = u(ξ, η) = α1 + α2 ξ + α3 η + α4 ξ 2 + α5 ξ η + α6 η 2 +α7 ξ 3

+ α8 ξ 2 η + α9 ξ η 2 + α10 η 3 .

(4.27)

In addition we need the partials of u +2α4 ξ +α5 η +3α7 ξ 2 +2α8 ξη +α9 η 2 u ξ = α2 uη = α3 +α5 ξ +2α6 η +α8 ξ 2 +2α9 ξη +3α10 η 2 . With zi = Pi , zˆi = Pˆi , z4 := (z1 + z2 + z3 )/3, we determine the FEs in (4.6), see Figure 4.20, by the following function values and derivatives: ui := u(zi ), i = 1, . . . , 4, pi := uξ (zi ), qi := uη (zi ), i = 1, 2, 3.

(4.28)

We determine any φ in (4.6), by evaluating in the ﬁrst step, the u and its derivatives in (4.27), then compute, in the next step, the α := (α1 , . . . , α10 )T from the system u := (u1 , p1 , q1 , u2 , p2 , q2 , u3 , p3 , q3 , u4 )T = Bα, with a matrix B obtained as follows. Every row of B, e.g. numbers 1 and 5, yield the values of the corresponding components of u. For numbers 1 and 5 these are the u1

228

4. Conforming ﬁnite element methods

and p2 . Then (4.28) tells us u1 = u(z1 = (0, 0)), p2 = uξ (z2 = (1, 0)). The evaluation according to (4.27) and the following uξ then yields rows 1 and 5 of B, and so on. ⎛ ⎞ 1 0 0| 0 0 0| 0 0 0 0 ⎜ 0 1 0| 0 0 0| 0 0 0 0⎟ ⎜ ⎟ ⎜ 0 0 1| 0 0 0| 0 0 0 0⎟ ⎜ ⎟ ⎜− − − | − − − | − − − −⎟ ⎜ ⎟ ⎜ 1 1 0| 1 0 0| 1 0 0 0⎟ ⎜ ⎟ ⎜ 0 1 0| 2 0 0| 3 0 0 0⎟ ⎜ ⎟ 0 1 0 0⎟ B := ⎜ ⎜ 0 0 1| 0 1 0| ⎟. ⎜− − − | − − − | ⎟ − − − − ⎜ ⎟ ⎜ 1 0 1| 0 0 1| 0 0 0 1⎟ ⎜ ⎟ ⎜ 0 1 0| 0 1 0| 0 0 1 0⎟ ⎜ ⎟ ⎜ 0 0 1| 0 0 2| 0 0 0 3⎟ ⎜ ⎟ ⎝− − − | − − − | − − − −⎠ 1 1/3 1/3 | 1/9 1/9 1/9 | 1/27 1/27 1/27 1/27 For computing the φi we recall Ni (φj ) = δi,j , hence, we determine αi from Bαi = ei , with the unit vectors ei , i = 1, . . . , 10, or we invert B as A := B −1 . Then the coeﬃcient vectors α for a FE and the αi for the φi are obtained from the systems α = Au and αi = Aei , e.g. for i = 8 as column number 8 of A, as α = Au with αi = Aei , e.g., φ8 = −ξη + ξ 2 η + 2ξη 2 and ⎛ ⎞ 1 0 0| 0 0 0| 0 0 0 0 ⎜ 0 1 0| 0 0 0| 0 0 0 0⎟ ⎜ ⎟ ⎜ 0 0 1| 0 0 0| 0 0 0 0⎟ ⎜ ⎟ ⎜ − − − | − − − | − − − −⎟ ⎜ ⎟ ⎜ −3 −2 0 | 3 −1 0 | 0 0 0 0⎟ ⎜ ⎟ ⎜ −13 −3 −3 | −7 2 −1 | −7 −1 2 27 ⎟ ⎟. A := ⎜ ⎜ −3 0 −2 | 0 0 0 | 3 0 −1 0⎟ ⎜ ⎟ ⎜ − − − | − − − | − − − −⎟ ⎜ ⎟ ⎜ 2 1 0 | −2 1 0 | 0 0 0 0⎟ ⎜ ⎟ ⎜ 13 3 2 | 7 −2 2 | 7 1 −2 −27 ⎟ ⎜ ⎟ ⎝ 13 2 3 | 7 −2 1 | 7 2 −2 −27 ⎠ 2 0 1 | 0 0 0 | −2 0 1 0 Then we proceed as for the Lagrange case, however we do not directly compute the φTi . In addition we do need the transformation of the partials in the zˆi to the zi . We explicitly formulate this for the above special case of scaling in (4.23). With (4.24) and, e.g. N8 (v ◦ FT ) · φ8 ◦ FT−1 , we obtain φ8 ◦ FT−1 (x, y) = −xy/h2 + (x2 y + 2xy 2 )/h3 N8 (v ◦ FT ) = (v ◦ FT )ξ (0, 1) = v (FT (0, 1)) · (FT )ξ (0, 1) = ((vx , vy )(0, h))(h, 0)T = hvx (0, h)

4.2. Approximation theory for ﬁnite elements

229

hence N8 (v ◦ FT ) · φ8 ◦ FT−1 (x, y) = vx (0, h)(−xy/h + (x2 y + 2xy 2 )/h2 ) for 0 ≤ x, y, x + y ≤ h. This shows, in particular, that the contribution of N8 (v ◦ FT ) · φ8 ◦ FT−1 (x, y) in T is O(h). 30 Similarly the other terms Ni (v ◦ FT ) · φi ◦ FT−1 can be computed. We do not intend formulating the sharpest results under the weakest possible conditions for the diﬀerent cases. Rather we formulate Condition 4.16 below, which allows the following implications: The interpolation operator I h is a bounded operator, a good approximation and the FEs allow inverse estimates. We introduce Deﬁnition 4.12. Let {T h }, 0 < h ≤ 1, be a family of subdivisions for Ω such that max{diam T : T ∈ T h } ≤ h diam Ω. Then h is denoted as the maximal step size. The family is said to be quasiuniform if there exists χ > 0 such that min{diam BT : BT ⊂ T ∈ T h } ≥ χ h

(4.29)

for all h ∈ (0, 1]; here BT is the largest ball such that T is star-shaped with respect to BT , cf. Deﬁnition 4.14. The family is said to be nondegenerate if there exists χ > 0 such that diam BT ≥ χ diam T

∀ T ∈ T h , hT := diamT, h ∈ (0, 1].

(4.30)

Remark 4.13. 1. The sequence of {T h } is nondegenerate if and only if the chunkiness parameter γ = γT , see (4.32), is uniformly bounded for all T ∈ T h and for all h ∈ (0, 1], hence γT = 1/χ. Let all T ∈ T h be triangles. Then T h is nondegenerate if and only if for every T ∈ T h the interior angles α ≥ α0 > 0. This is used as the deﬁnition for quasiuniform T h e.g. in [135, 387]. A quasiuniform family is nondegenerate, but not conversely. 2. If we start with an arbitrary triangulation in two dimensions and repeatedly subdivide by connecting midpoints of edges, we obtain a nondegenerate family of subdivisions. For a well-deﬁned alternative, see [58]. 3. For quasiuniform T h each T is uniformly star-shaped. 4.2.4

Interpolation errors and inverse estimates

We have already formulated the general interpolating uh = I h u on Ω in (4.16). It reproduces function values and/or derivatives of the original u according to N . We want to study the question of how well these I h u approximate the original u. We collect 30

f (h) = O(hp ) and f (h) = o(hp ) iﬀ |f (h)|/hp ≤ C and |f (h)|/hp → 0 for h → 0.

230

4. Conforming ﬁnite element methods

the appropriate assumptions in Condition 4.16. Mainly for the study of variational crimes we need estimates of higher Sobolev norms of FEs by lower Sobolev norms, the so-called inverse estimates. We need the standard and broken Sobolev norms, cf. (4.36), and some lemmas. We use the notation for multi-indices, α, and partials, ∂ α , see (2.73), vW d,p (Ω) :=

d j=0

(|v|W j,p (Ω) )p

1/p

, |v|W j,p (Ω) :=

∂ α vpLp (Ω)

1/p .

(4.31)

|α|=j

For w ∈ W m,p (Ω) the classical derivatives and hence their point evaluation and Taylor polynomials Tym w(x) = |α|<m ∂ α w(y)(x − y)α /α! are not deﬁned. So we use the ∂ α w ∈ L2 (Ω), |α| < m, directly and introduce the averaged Taylor polynomials Qm w for w ∈ W m,p (Ω), see [141], in Deﬁnition 4.14. Let G ⊂ Rn , e.g. G = Ω, have ﬁnite diam G and let B := Bρ (x0 ) ¯ ⊂ G such that for all x ∈ G the closed convex hull be a ball of radius ρ. We assume B ¯ ⊂ G. Then G is called star-shaped with respect to B. With of {x} ∪ B ρmax := sup{ρ : G star-shaped with respect to B}, and γ = diam G/ρmax , (4.32) this γ is called the chunkiness parameter of G. Let G be star-shaped with respect to B ¯ and (ii) n φ(x)dx = 1. and let φ be a cut-oﬀ function for B, that is (i) supp φ = B R Then the following averaged Taylor polynomials Qd w of degree d − 1, and the Tyd w(x) are well deﬁned, ∀x ∈ B and ∀y ∈ B \ M , M a set of Lebesgue measure 0, as d Tyd w(x)φ(y)dy with Tyd w(x) = ∂ α w(y)(x − y)α /α!. (4.33) Q w(x) := B

|α|
Obviously, triangles are star-shaped. Note that only ∂ α u with |α| < d are employed in (4.33). A standard example for a cut-oﬀ function on B = Bρ (x0 ) is φ(x) = c exp (−(1 − (|x − x0 |/ρ)2 )−1 ) for x ∈ B and φ(x) ≡ 0 outside B. We only consider star-shaped G (or T ∈ T h , see below) with bounded chunkiness parameter, γ, see (4.32). We obtain, see [141], Lemma 4.15. Bramble–Hilbert lemma: Let G be star-shaped with respect to B of radius ρ such that ρ > (1/2)ρmax , G has the chunkiness parameter γ, and Qd w be the averaged Taylor polynomial. Then for all w ∈ W d,p (G), = 0, 1, . . . , d, 1 ≤ p ≤ ∞, there exist constants Cd,n,γ such that |w − Qd w|W ,p (G) ≤ w − Qd wW ,p (G) ≤ Cd,n,γ (diam G)d− |w|W d,p (G) .

(4.34)

Since the uh ∈ U h might not be continuous across common edges e ⊂ T ∩ T 1 , and hence uh ∈ H 1 (Ω) [174], pp. 207–208, the usual Sobolev norms · W d,p (Ω) might not be deﬁned. However, for all practically important cases the uh |T W d,p (T ) and

231

4.2. Approximation theory for ﬁnite elements

u|T¯ C d (T¯) . So the broken Sobolev spaces and norms uh W d,p (T h ) make sense: UT h := W k,p (T h ) := {uh : Ω → R : ∀ T ∈ T uh |T ∈ W k,p (T )}, with

1/q ∂ α uh |T qLq (T ) , uh hk,q := uh W k,q (T h ) =

(4.35)

T ∈T h |α|≤k

for 1 ≤ q < ∞,

(4.36)

α h uh W∞ ∈ T h }. k (T h ) := ess sup{|∂ u (x)| : ∀α |α| ≤ k∀x ∈ T ∀ T

(4.37)

These norms satisfy H¨ older inequalities as the standard norms: uh v h W k,1 (T h ) ≤ uh W k,p (T h ) v h W k,q (T h ) for 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1 The corresponding seminorms with |α|=k in (4.36) are |uh |W k,q (T h ) =

T ∈T

h

|α|=k

∂ α uh |T qLq (T )

1/q =

T ∈T

|uh |qW k,q (T )

(4.38)

1/q .

(4.39)

h

These norms coincide with the original norm uh W k,q (T h ) for uh ∈ W d,p (Ω). Analogous deﬁnitions and equalities may have to be used for scalar products and bilinear forms as well. The decision about which of the following choices of d, p in the spaces h h ∈ U h : uh W d,p (T h ) < ∞} for h → 0 UW d,p (Ω) := {u

(4.40)

is appropriate, depends upon the interplay of ansatz- and test-functions and the (non)linear elliptic problem. They are related by the corresponding bilinear or other forms. Certainly, for each ﬁxed h we want the uh W d,p (T h ) < ∞. We start with a polyhedral domain Ω ⊂ Rn , for curved ∂Ω see Subsections 4.2.7, 4.2.6. Condition 4.16. For the reference FE (K, P, N ) and the subdivision T h assume: 1. K is star-shaped with respect to some ball. n n d ⊆ P ⊆ Pd+τ , P ⊂ W∞ (K), usually 2. Choose the smallest τ ≥ −1 such that Pd−1 n n , hence, there exist v ∈ P ∩ Pd+τ . τ = −1. For τ ≥ 0, let P Pd+τ l ¯ 3. N ⊂ (C (K)) , hence the elements of N are evaluations of u and derivatives ¯ and additionally let of u up to the order ≤ l in (diﬀerent) points P for P ∈ K 1 ≤ p ≤ ∞ and d − l − n/p > 0 for p > 1 or d−l−n≥0 for p = 1. 4. Alternatively to 3. we assume N ⊂ (W d,p (K)) . 5. {T h }, 0 < h ≤ 1 , is a nondegenerate family of admissible subdivisions of a bounded polyhedral domain Ω ⊂ Rn with the parameter χ, see (4.30). 6. For all T ∈ T h , 0 < h ≤ 1, the (T, PT , NT ) is aﬃne equivalent to (K, P, N ). 7. The combination of 1. and 5. shows that T is uniformly star-shaped ∀ T ∈ T h and that the chunkiness parameter γ of all T ∈ T h is bounded, see e.g. [141], Theorem 4.4.20.

232

4. Conforming ﬁnite element methods

n n The unusual condition τ ≥ −1 such that Pd−1 ⊆ P Pd+τ allows us to study, e.g the families of tensor product FEs as in (4.13), the serendipity FEs, see [141], pp. 88 ﬀ. and pp. 112 ﬀ. For all these families and for Davydov’s C 1 splines, see Subsection 4.2.6 and [263], speciﬁc error estimates are possible. For linear and quadratic tensor product FEs we get τ = 1 and τ = 3 for n = 2 or τ = 4 and τ = 17 for n = 3. The following results are proved in [141], Theorems 4.4.4, 4.4.20, 4.6.11 and 4.6.14, compare [135, 387] as well. Note that the conditions, relating d, l, n, p in [141], Theorems 4.4.4 and 4.4.20 are imposed to guarantee Condition 4.16, 3. and to apply the Sobolev embedding lemma. They are not necessary for 4. Furthermore, these results extend to a combination of v ∈ W d,p (Ω) and FEs of higher order.

Theorem 4.17. Interpolation errors: Let the {T h }, 0 < h ≤ 1, and the reference element, (K, P, N ), satisfy Condition 4.16 for some l, d, p, 1 ≤ p ≤ ∞, cf. (4.36), (4.37). Let I h be the (local) interpolation operator in (4.16) deﬁned by (K, P, N ). 1. Then there exists a constant C > 0, such that ∀ 0 ≤ s ≤ d, the following local and global interpolation error estimates hold ∀v ∈ W d,p (Ω): v − I h vW s,p (T ) ≤ C (diam T )d−s |v|W d,p (T ) , I h vW s,p (T ) ≤ vW s,p (T ) + C(diam T )d−s |v|W d,p (T ) . v − I vW s,p (T h ) ≤ C h h

d−s

(4.41)

|v|W d,p (Ω) ,

I vW s,p (T h ) ≤ vW s,p (Ω) + C hd−s |v|W d,p (Ω) ≤ CvW d,p (Ω) . h

(4.42)

C depends on (K, P, N ), n, d, p and the number χ in (4.30). 2. For 0 ≤ s ≤ l < d, see 3. in Condition 4.16, we have v − I h vW s,∞ (T h ) ≤ C hd−s−n/p |v|W d,p (Ω)

∀ v ∈ W d,p (Ω).

(4.43)

3. For the case of tensor product FEs the estimates for the interpolation errors in (4.41)–(4.43) can be modiﬁed by replacing the seminorms |v|W d,p (T ) , |v|W d,p (Ω) 1/p n d d p by |v|tW d,p (T ) , |v|tW d,p (Ω) := . i=1 ∂ v/∂xi Lp (Ω) Proposition 4.18. Our FE spaces U h are approximating spaces for U = W s,p (Ω) and d > s ≥ 0, in the sense of Deﬁnition 3.5, hence, lim dist (u, U h ) = inf ||u − uh ||U h = 0 ∀ u ∈ U ⇒ lim ||I h u||U h = ||u||U . (4.44)

h→0

uh ∈U h

h→0

Proof. For ﬁxed u ∈ W s,p (Ω) and arbitrary > 0 we choose a v ∈ W d,p (Ω) such that v − uW s,p (T h ) < /2. Now by v − I h vW s,p (T h ) ≤ C hd−s |v|W d,p (Ω) we can choose h0 such that v − I h vW s,p (T h ) < /2 for h < h0 , so ﬁnally u − I h vW s,p (T h ) < for h < h0 , Hence, our FE spaces for d > s are approximating spaces. For the last claim we combine (4.42) with u − I h u < u − v + v − I h v + I h v − I h u. Inverse estimates, presented in the sequel, relate various norms v h W s,p (T h ) for ﬁnite element spaces for s ≥ d. This is necessary to handle variational crimes. We again

4.2. Approximation theory for ﬁnite elements

233

present local and global estimates. Note that for the next two results, see [141], pp. 110 ﬀ., we need T h and the aﬃne (isoparametric) maps FT : K → T . However, no reference to N , NT nor the unisolvence is made. Since P always satisﬁes P ⊆ W j,p (T ) ∩ Wql (T ), 1 ≤ q ≤ p, 0 ≤ l ≤ j, we get Theorem 4.19. Inverse estimates: Let {T h }, 0 < h ≤ 1, (K, P, N ), PT = P satisfy Condition 4.16, let j ≤ d + τ , 1 ≤ q ≤ p, 0 ≤ l ≤ j, and U h = {v h : Ω → R, v h |T ∈ PT ∀ T ∈ T h }.

(4.45)

Then there exists C = C(j, p, q, χ), with χ in (4.30), such that v h W j,p (T ) ≤ C (diam T )l−j+n/p−n/q v h Wql (T ) , ∀v h ∈ U h

(4.46)

v h W j,p (T h ) ≤ C hl−j+min(0,n/p−n/q) v h Wql (T h ) ,

(4.47)

and

where {T h }, 0 < h ≤ 1, is quasiuniform for (4.47), and for all v h ∈ U h . These inequalities only make sense for j ≤ d + τ , otherwise v h W j,p (T h ) = v h W d+τ,p (T h ) . 4.2.5

Inverse estimates on nonquasiuniform triangulations

This condition of quasiuniformity for (4.47) can be relaxed by applying the results for boundary and ﬁnite element methods in Graham, Hackbusch and Sauter [360–364], and with Dahmen, Faermann and Grasedyck in [244, 359]. We present some of their wide range of inverse-type inequalities, here applications to ﬁnite element functions on general classes of meshes, including degenerate meshes obtained by an isotropic reﬁnement. These are obtained for Sobolev norms of positive, zero and negative order. In contrast to classical inverse estimates, negative powers of the minimum mesh diameter are avoided. In the whole subsection the functions u = uh are always FEs. For n = 2 or 3, let Ω ⊂ Rn denote a bounded domain. Suppose that Ω is decomposed into a mesh, T = T h , of tetrahedra/bricks (n = 3) or curvilinear triangles/ quadrilaterals (n = 2). Then classical inverse estimates give, cf. (4.47) for the left inequality, for ∼, compare the index, −2s uH s (T h ) h−s min uL2 (T h ) hmin uH −s (T h ) ,

(4.48)

for a suitable range of positive s and for all functions u ∈ H s (T h ) which are piecewise polynomials of degree ≤ d − 1 with d − 1 ≥ 0 with respect to this mesh. The quantity hmin is the minimum diameter of all the elements of the mesh and (4.48) holds under the assumption of shape regularity, i.e. ρτ hτ for each τ ∈ T h where hτ is the diameter of τ and ρτ = ρ is the diameter of the largest inscribed sphere, cf. Deﬁnition 4.14. We will use such estimates regularly in the ﬁnite element analysis, particularly for nonconforming FEMs, cf. Chapter 5. When the mesh is quasiuniform, cf. Deﬁnition 4.12, so h hmin , with h the maximum diameter of all the elements, they can be used to obtain convergence rates in powers of h for various quantities in various norms.

234

4. Conforming ﬁnite element methods

However, practical meshes are often nonquasiuniform and then the negative powers of hmin in (4.48) may give rise to overly pessimistic convergence rates. In [361], less pessimistic replacements for (4.48) have been derived, a particular case being uH s (T h ) h−s uL2 (T h ) h−2s uH −s (T h ) ,

(4.49)

where h : Ω −→ R is now a continuous piecewise linear mesh function whose value on each element τ reﬂects the diameter of that element, i.e. hτ h|τ hτ . Estimates (4.49) have several applications, e.g. to the analysis of the Mortar element method [361]. In fact [361] contains more general versions of (4.49), e.g. in the Sobolev space W s,p (T h ) and in related Besov spaces. While the left-hand inequality in (4.49) is well-known, at least in the Sobolev space case, the right-hand inequality requires rather delicate analysis. In this subsection we obtain more general versions of (4.49) which do not require the mesh sequence to be shape-regular. A typical estimate is uH s (T h ) ρ−s uL2 (T h ) ρ−2s uH −s (T h ) ,

(4.50)

where the mesh function ρ : Ω −→ R is now a continuous piecewise linear function whose value on each element τ reﬂects the diameter of the largest inscribed sphere, introduced in Deﬁnition 4.12. Estimates (4.50) hold under the rather weak assumptions that (i) the quantities hτ and ρτ are locally quasiuniform (i.e. hτ /hτ 1 and ρτ /ρτ 1 for all neighboring elements τ, τ ) and (ii) the number of elements which touch any element remains bounded as the mesh is reﬁned (see Assumption 4.26). These assumptions admit degenerate meshes, containing long thin “stretched” elements, which are typically used for approximating edge singularities or boundary layers in solutions of PDEs. Our estimates (4.50) hold true when all the elements τ of a mesh are obtained by suitable maps from a single unit element, as for the above ﬁnite element spaces. For the purpose of a readable introduction we have here written our estimates (4.50) in a very compact form. In fact the range of s for which the right-hand inequality in (4.50) holds may be greater then that for which the left-hand inequality holds and we shall give precise ranges in the last subsubsection. Meshes and ﬁnite elements Throughout, Ω will denote a bounded n-dimensional subset of Rn , for n = 2 or 3. We deﬁne the Sobolev spaces H s (Ω), and H s (T h ), s ≥ 0, as in Subsection 1.4.3. Throughout, we let [−k, k] denote the range of Sobolev indices for which we are going to prove the inverse estimates (where k is a positive integer), and we assume that H s (Ω) is deﬁned for all s ∈ [−k, k], and that H −s (Ω) is the dual of H s (Ω), for s > 0. We assume that Ω is decomposed as above into a mesh T = T h of relatively open τ : τ ∈ T }. pairwise-disjoint ﬁnite elements τ ⊂ T h with the property T h = ∪{¯ Deﬁnition 4.20. Mesh parameters: For each τ ∈ T , |τ | denotes its n-dimensional measure, hr denotes its diameter and ρτ is the diameter of the largest sphere centered at a point in τ whose intersection with T h lies entirely inside τ¯. For any other simplex or cube t ∈ Rn (not necessarily an element of T ) we deﬁne ht and ρt in the same way.

235

4.2. Approximation theory for ﬁnite elements

In order to impose a simple geometric character on the mesh τ , we assume that each τ ∈ T is diﬀeomorphic to a simple unit element. More precisely, let σ ˆ 3 in R3 denote the unit simplex with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), and let κ ˆ 3 denote the unit 2 2 ˆ to be the unit simplex cube with vertices {(x, y, z) : x, y, z ∈ {0, 1}}. In R , deﬁne σ with vertices (0, 0), (1, 0) and (0, 1) and deﬁne κ ˆ 2 to be the unit square with vertices {(x, y) : x, y ∈ {0, 1}}. Then we assume, as above, that for each τ ∈ T , there exists a unit element τˆ = σ ˆn n −1 ∞ or κ ˆ and a bijective map χτ : τˆ → τ , with χτ and χτ both smooth, here ∈ C . Since χτ is smooth, each element τ ∈ T is either a curvilinear tetrahedron/brick (n = 3) or a curvilinear triangle/rectangle (n = 2). The mesh T is allowed to contain both types of elements. Each element has vertices and edges. In the R3 case the element also has faces, comprising the images of the faces of the unit element. For a suitable index set Q, we let {xp : p ∈ Q} denote the set of all vertices T . We assume the mesh is conforming, cf. Deﬁnition 4.2, i.e. for each τ, τ ∈ T with τ = τ , τ¯ ∩ τ is allowed to be either empty, a node, an edge or (when n = 3) a face of both τ and τ . The requirement that χτ is smooth ensures that edges of T h (n = 2) and edges ∂T h (n = 3) are conﬁned to edges of elements τ ∈ T . Let Jτ denote the n × n Jacobian of χτ . Then 1/2 gτ := det JτT Jτ is the Gram of the map χτ , which appears in the change of variable determinant x))gτ (ˆ x)dˆ x. To ensure that the map χτ is suﬃciently formula: τ f (x)dx = τˆ f (χτ (ˆ regular we shall make the following assumptions on Jτ : Assumption 4.21. Mapping properties: x)T Jτ (x)} ≤ D|τ |2 , D−1 |τ |2 ≤ det{Jτ (ˆ Eρ2r

(4.51)

≤ λmin {Jτ (ˆ x) Jτ (x)} T

ˆ ∈ τˆ, for all τ ∈ T , with positive constants D, E independent of τ . uniformly in x (Throughout this section, for a symmetric matrix A, λmin (A) and λmax (A) denote respectively the minimum and maximum eigenvalues of A.) Assumption 4.21 is satisﬁed in a number of standard cases. Example 4.22. Suppose τ ⊂ Ω, where τ is for n = 2 and n = 3 a polygon and a polyeder, respectively. Then χτ can be chosen as an aﬃne map. Then the Jacobian Jτ is identical to an n × n τ | and that Jτ−1 2 ≤ hτˆ ρ−1 constant matrix and it is well known that det Jτ = |τ |/|ˆ τ , from which the estimates (4.51) follow. Example 4.23. In many applications quadrilateral or hexahedral elements are important. Consider, for example, quadrilateral elements τ obtained by mapping from the unit element κ ˆ 2 = (0, 1)2 . If the map is aﬃne, then the estimates for (4.51) obtained in Example 4.22 carry over verbatim. However only parallelograms can be obtained by applying

236

4. Conforming ﬁnite element methods

aﬃne maps to κ ˆ 2 . More general quadrilaterals can be obtained using bilinear maps and it turns out that, under quite moderate assumptions, (4.51) still hold. This is a consequence of the following considerations for quadrilateral elements τ . A parallelogram π (considered as an open subset of R2 ) will be called an inscribed parallelogram for τ if π ⊂ τ and if two adjacent edges of π are identical with two adjacent edges of τ . It is easy to show that any convex quadrilateral τ has at least one inscribed parallelogram. Proposition 4.24. cf. [364]: Let τ be a convex planar quadrilateral obtained by applying a bilinear mapping to κ ˆ 2 . Let π be any inscribed parallelogram for τ . Then ˆ ∈ τˆ with D, E depending continuously on the the estimates (4.51) hold uniformly in x ratio r := |π|/|τ | ∈ (0, 1]. Remark 4.25. If follows that if the ratio |π|/|τ | is bounded below by some constant γ > 0 say, for all elements τ as the mesh is reﬁned, then (4.51) hold (with D and E dependent on γ). There are obvious degenerate elements which satisfy this, for example any parallelograms (no matter how small the smallest interior angles are) satisfy it. Similarly “moderately” distorted parallelograms also satisfy it. Assumption 4.21 describes the quality of the maps which take the unit element τˆ to each τ . We also need assumptions on how the size and shape of neighboring elements in our mesh may vary. Here we impose only very weak local conditions which require the meshes to be neither quasiuniform nor shape-regular. Throughout the rest of this subsubsection we make the following assumption. Assumption 4.26. Mesh properties: For some K, L ∈ R+ and M ∈ N, we assume that, for all τ, τ ∈ T with τ¯ ∩ τ = ∅, hτ ≤ Khτ ,

ρτ ≤ Lρτ ,

max #{τ ∈ T : xp ∈ τ¯} ≤ M. p∈Q

(4.52) (4.53)

Note that condition (4.52) requires that hτ and πτ do not vary too rapidly between neighboring elements. This allows elements with large aspect ratio, provided their immediate neighbors have a comparable aspect ratio. Example 4.27. Shape-regular meshes are easily shown to satisfy Assumption 4.26 with moderate K, L, M . Also, meshes which are anisotropically graded towards an edge typically lie in this class. Near an edge, but away from the corners, the solution typically is badly behaved only in the direction orthogonal to the edge and eﬃcient approximations require meshes which are anisotropically graded. For example, for the square screen [0, 1] × [0, 1], a typical tensor product anisotropic mesh would be: xi,j = (ti , tj ), where ti = (i/n)g /2 and t2n−i = 1 − (i/n)g /2 for i = 0, . . . , n, for some grading exponent g ≥ 1. In this case the elements become very long and thin near smooth parts of edges. In the hp version of the ﬁnite element method

4.2. Approximation theory for ﬁnite elements

237

similar meshes but with more extreme grading may be used and these also satisfy Assumption 4.26. We denote the class of meshes which satisfy Assumptions 4.21 and 4.26 as MD,E,K,L,M . From now on, if A(T ) and B(T ) are two mesh-dependent quantities, then the inequality A(T ) B(T ) means that A(T ) ≤ CB(T ) with the constant C independent of T ∈ MD,E,K,L,M , but not necessarily independent of D, E, K, L, M , similarly for A(T ) ∼ B(T ) and B(T ) A(T ). Now we generalize the above deﬁnition of ﬁnite element spaces on the mesh T . ˆ n }, we deﬁne Deﬁnition 4.28. Finite element spaces: For d − 1 ≥ 0 and τˆ ∈ {ˆ σn , κ polynomials of total degree ≤ d − 1 on τˆ if τˆ = σ ˆn, τ) = Pd−1 (ˆ polynomials of coordinate degree ≤ d − 1 on τˆ if τˆ = κ ˆn , ˆ n on top of p. 234. Then we deﬁne with the σ ˆn, κ S0d−1 (T ) = {u ∈ L∞ (T h ) : u ◦ χτ ∈ Pd−1 (ˆ τ ), τ ∈ T

for d − 1 ≥ 0.

S1d−1 (T

for d − 1 ≥ 1.

) = {u ∈ C 0 (T h ) : u ◦ χτ ∈ Pd−1 (ˆ τ ), τ ∈ T

Inverse estimates In this subsubsection we summarize the inverse estimates from [364], which were motivated above (see (4.50)). To deﬁne the scaling function ρ, recall the parameters ρτ introduced in Deﬁnition 4.20. From these we construct a continuous mesh function ρ ∈ S11 on T h as follows. Deﬁnition 4.29. Mesh function: For each p ∈ Q, set ρp = max{ρτ : xp ∈ τ¯}. The mesh function ρ is the unique function in S11 (T ) such that ρ(xp ) = ρp , for each p ∈ Q. Clearly ρ is a positive, continuous function on Ω and, by Assumption 4.26, it follows that ρ(x) ∼ ρτ for x ∈ τ , and all τ ∈ T . The main results of this section are Theorems 4.30, 4.32 and 4.33. The ﬁrst two of these provide inverse estimates in positive Sobolev norms for functions u ∈ Sid−1 (T ) with continuity index i = 1, 0, respectively. The third theorem provides inverse estimates in negative norms. ¯ < ∞. Then Theorem 4.30. Inverse estimates: Let 0 ≤ s ≤ 1 and −∞ < α < α ρα uH s (T h ) ρα−s uL2 (T h ) , ¯ u ∈ S1d−1 (T ). uniformly in α ∈ [α, α], Remark 4.31. Since S1d−1 (T ) ⊂ H s (T h ) for all s < 3/2, it may be expected that the range of Sobolev indices for which Theorem 4.30 holds may be extended. Such an extension has been obtained in [361] for shape-regular meshes at the expense of working in Besov norms. We have avoided such extensions here in order to simplify the present subsubsection. ¯ < ∞. Then Theorem 4.32. Inverse estimates: Let 0 ≤ s < 1/2 and −∞ < α < α ρα uH s (T h ) ρα−s uL2 (T h ) ,

238

4. Conforming ﬁnite element methods

uniformly in α ∈ [α, α], ¯ u ∈ S0d−1 (T ). Theorem 4.33. Inverse estimates, negative norms: Let i ∈ {0, 1}, d − 1 ≥ i, 0 ≤ s ≤ 1 ¯ < ∞. Then the inequality and −∞ < α < α ρs+α uL2 (T h ) ρα uH −s (T h ) ,

(4.54)

¯ ]. If χτ is aﬃne for all τ then (4.54) holds uniformly in u ∈ Sid−1 (T ) and α ∈ [α, α holds for all 0 ≤ s ≤ k, where k is as described in the ﬁrst paragraph of the last subsubsection. Remark 4.34. Note that the test function w constructed in the proof of Theorem 4.33 in Graham et al. [364] vanishes at the boundaries of elements. Hence w belongs to the closure of the space C0∞ (Ω) with respect to the H s (Ω) norm (this space is usually denoted H0s (Ω)). Thus the result of Theorem 4.33 also holds if H −s (Ω) was deﬁned as the dual of H0s (Ω), although we have not so deﬁned it here. 4.2.6

Smooth FEs on polyhedral domains, with O. Davydov

This subsection is an adapted version of Davydov [264]. In the context of this book, we need smooth FEs for two kinds of problems. Conforming FEMs for weak 2m-th order equations require FEs (piecewise polynomial splines) in C m−1 (Ω). Clearly such piecewise polynomial functions belong to H m (Ω). The new FEM for fully nonlinear strong 2m-th order elliptic problems can only be used on (curved) domains with C 2m boundary, and require FEs in C 2m−1 (Ω) on such domains in the case m > 1. For m = 1 we still need C 1 (Ω) splines, but can admit convex domains, which allows convex polyhedral domains in particular. Moreover, applications to fully nonlinear equations in Section 5.2.5 require certain stable splitting of the spline space. Consequently, we consider smooth FEs in three diﬀerent settings: (1) Approximation theory of C r (Ω) splines of local degree d, for r ≥ 1 on polyhedral domains in Rn with d ≥ r2n + 1. For this setting we prove inverse estimates (Theorem 4.35) and error bounds for the quasi-interpolation operators (Theorem 4.36). (2) C 1 (Ω) splines on polygonal domains in R2 . A modiﬁcation of the Argyris FE presented below is shown to admit a stable splitting, see Theorem 4.39. (3) C 2m−1 (Ω) splines, for m ≥ 1, on curved C 2m domains in Rn as needed in FEMs for fully nonlinear problems. Although the question of constructing such splines remains widely open, we formulate Conjecture 4.41 as a basis for these methods. In this and the next chapter we, in particular, prove the full convergence theory for elliptic equations and systems of order 2m with FEs in H m (Ω). Smooth FEs in two variables have been studied, e.g. in the classical book by Ciarlet [174]. For n > 2, see Zen´ısek [679] and Le M´ehaut´e [484]. For most FEMs, smooth FEs or splines are successfully avoided nowadays by using the standard FEMs in C(Ω) and in H 1 (Ω). Then equations or systems of order 2m are transformed into a system of m second order

4.2. Approximation theory for ﬁnite elements

239

equations, analogously for systems, sometimes losing information. The corresponding variational crimes, see e.g. Sections 5.5, allow further generalizations. This is not yet possible for fully nonlinear elliptic problems. In the literature, several diﬀerent approaches for smooth FEs are discussed. Either FEs on a polyhedral or on approximate curved domains approximate the functions on the original polyhedral or curved domain. C 1 (Ω) FEs on polyhedral or C 2 domains, and C m−1 (Ω) FEs for m ≥ 1, are studied in many papers. arm´an equations by C 1 FEs on curved domains are studied and applied to the K´ Bernadou, e.g. [85,86] for the “curved” Argyris and Bell FEs, hence with degree d = 5. The results in [85, 86, 680, 681] and the compilations only apply to R2 and provide partial answers in the above setting (3). A totally diﬀerent approach is based on the weighted extended B-splines, see e.g. H¨ollig, Reif and Wipper [400] They allow the incorporation of curved boundaries into splines on uniform grids with a new technique. The results about stable splitting mentioned in (2) above are new (see Davydov [264]). On polyhedral domains many results are available for bivariate polynomial splines, see e.g. the recent book [467]. However, certain important questions have not been addressed, in particular the error bounds for multivariate smooth piecewise polynomials on general triangulations in Rn . The error bounds below are formulated for a triangulation T h of a polyhedral domain Ω. Thus, we deal with a family of triangulations {T h } and spaces {S h } parametrized by h. It is assumed that the other parameters, e.g. of a basis for S h characterizing its stability and locality, or n, d, p, ωT h , remain independent of h. In this sense, one obtains respective asymptotic estimates and rates of convergence. Splines on polyhedral domains in Rn The triangulation T h for polyhedral domains has been introduced in Deﬁnition 4.2. T h has to satisfy the standard conditions, see Deﬁnition 4.2. In Deﬁnition 4.12 we introduced the maximal and local step sizes h and hT = diam T. For a quasiuniform and nondegenerate T h , see (4.30), we require ∃χ > 0 :

minT ∈T h diam BT ≥ χ and h

1 diam BT := min ≥ χ, h ωT h hT T ∈T

(4.55)

respectively, and ωT h is called the shape regularity constant of T h . For any d ≥ 0, we consider the spaces of all (including discontinuous) piecewise polynomials Sd (T h ) and the space Sdr (T h ) of splines of smoothness r ≥ 0, deﬁned by s ∈ Sd (T h ) ⇔ s|T ∈ Pdn ∀T ∈ T h

and

Sdr (T h ) = Sd (T h ) ∩ C r (Ωh ), 0 ≤ r < d,

(4.56)

denotes the space of n-variate polynomials of total degree ≤ d. In particular, where Sd0 (T h ) is the classical space of continuous FEs. The spaces Sdr (T h ) are subspaces of the W r+1,p (T h ) with norms and subnorms in (4.35)–(4.39) and corresponding scalar products for p = 2. As a consequence of the diﬀerentiability of the Sdr (T h ) we have to change the assumptions here and in Section 5.2 compared with the earlier sections. The differentiability imposes a semilocal condition, cf. (4.62), not directly comparable with Pdn

240

4. Conforming ﬁnite element methods

the preceding Pd−1 ⊂ P ⊂ Pd+τ , cf. Deﬁnition 4.8, 6. Condition (4.62) allows us for example to include reduced FEs, e.g. the Bell element into consideration. h h Note that h for nested h sequences of triangulations T1 T2 · · · , nested spline r r spaces Sd T1 ⊂ Sd T2 ⊂ · · · are available, allowing multiresolution techniques, see Davydov and Petrushev [265] and Davydov and Stevenson [268]. Inverse estimates We start by establishing the inverse estimates needed in the proof of the interpolation error bound and of independent interest in the FEM. The multivariate Markov inequality for a simplex T ⊂ Rn , see Coatm´elec [181], Dα sL∞ (T ) ≤ c

nd2 sL∞ (T ) , diam BT

s ∈ Pdn ,

|α| = 1,

implies the following inverse estimates. Theorem 4.35. Inverse estimates: For any 1 ≤ p ≤ ∞, 0 ≤ k < μ ≤ d, we have |s|W μ,p (T ) ≤

K hμ−k T

|s|W k,p (T ) , s ∈ Sd (T h )

for simplices T ∈ T h ,

(4.57)

where the constant K depends only on n, d, p, ωT h , see (4.55). For a quasiuniform T h , hT = diam T can be replaced by h. Proof. Let |α| = μ and g = s|T ∈ Pdn . Since Dα g = Dβ Dγ p for some β, γ with |β| = n , a repeated application of the Markov inequality μ − k, |γ| = k, and since Dγ g ∈ Pd−k implies a1 a2 Dγ gL∞ (T ) ≤ μ−k Dγ gL∞ (T ) , Dα gL∞ (T ) ≤ μ−k diamBT hT where a1 depends only on n, d, and a2 = a1 ωTμ−k h . This already proves (4.57) in the case p = ∞. For 1 ≤ p < ∞, a simple scaling argument shows that 1/p (T )qL∞ (T ) ≤ qLp (T ) ≤ a4 vol1/p (T )qL∞ (T ) , a−1 3 vol

q ∈ Pdn ,

(4.58)

where vol (T ) is the n-dimensional volume of the simplex T , and the constants a3 , a4 depend only on n, d, p. Therefore, we get a2 a3 a4 Dα gLp (T ) ≤ μ−k Dγ gLp (T ) , hT and (4.57) follows. The statement for the quasiuniform case is obvious.

Error bounds For smooth splines on a polyhedral domain Ω in Rn we now prove error bounds of quasi-interpolation. For FE subspaces these bounds can be found e.g. in [141], and for bivariate spline spaces in [467], Theorem 5.18. Anisotropic triangulations have been considered in [265]. We introduce the star of a vertex v of T h , denoted by star(v), as the closure of the union of all n-simplices T ∈ T h with a vertex at v. We set star1 (v) := star(v),

4.2. Approximation theory for ﬁnite elements

241

and deﬁne starγ (v), γ ≥ 2, recursively as the union of the stars of all vertices of T h contained in starγ−1 (v). Assuming that S h is a subspace of Sd (T h ), we choose a basis for S h and its dual (S h ) , such that S h = span [s1 , . . . , sN ], (S h ) = span [λ1 , . . . , λN ], λi sj = δij , i, j = 1, . . . , N. (4.59) The basis [s1 , . . . , sN ] is said to be stable and local if for each k = 1, . . . , N , a set Ek ⊂ Ω exists, such that supp sk ⊂ Ek ⊂ starγ (vk ) sk L∞ (Ω) ≤ C1

and

for an appropriate vertex vk of T h , |λk s| ≤ C2 sL∞ (Ek ) , ∀s ∈ S h ,

(4.60) (4.61)

for some C1 , C2 and γ. Indeed, (4.60) implies that the basis functions sk have local support, and it follows from (4.60), (4.61) that they are stable in L∞ in the sense that for any real α1 , . . . , αN , N K1 max |αk | ≤ αk sk 1≤k≤N

≤ K2 max |αk |, 1≤k≤N

L∞ (Ω)

k=1

with K1 , K2 depending only on n, d, ωT h , γ, C1 , C2 . It can also be shown [263], Lemma 6.2, that, after renorming, the new basis s˜1 , . . . , s˜N with s˜k := vol−1/p (Ek ) sk , is stable in Lp , 1 ≤ p < ∞, that is K1

N

k=1

1/p |αk |

p

N k ≤ αk s˜ k=1

≤ K2

N

1/p |αk |

p

.

k=1

Lp (Ω)

In addition, we assume that S h contains the space of polynomials of certain degree, such that n ⊂ S h ⊂ Sd (T h ) for some 0 ≤ − 1 ≤ d. P−1

(4.62)

The following theorem provides error bounds for certain quasi-interpolation operator r n r (T h ) in (4.56), we have Pd−1 ⊂ Sd−1 (T h ). Therefore the I h : L1 (Ω) → S h . For Sd−1 role of the earlier d in the context of interpolation errors is taken over by − 1 here. Note that in applications to FEMs usually only the existence of an operator I h with desired approximation properties is important. Hence it is acceptable to use the Hahn– Banach Theorem 1.14 as a tool in the deﬁnition of I h . Theorem 4.36. Quasi-interpolation error on polyhedral domains Ω in Rn : 1. Under conditions (4.59)–(4.62) there exists a linear operator I h : L1 (Ω) → S h , such that for any T ∈ T h , 1 ≤ p ≤ ∞, 0 ≤ |α| ≤ − 1, u ∈ L1 (Ω) with |u|W ,p (Ωγ ) < ∞, T

−|α|

Dα (u − I h u)Lp (T ) ≤ KhT

|u|W ,p (Ωγ ) , T

(4.63)

242

4. Conforming ﬁnite element methods

@ where ΩγT := v∈T star2γ−1 (v), K = K(n, d, ωT h , γ, C1 C2 , L∂Ω ), and L∂Ω is the Lipschitz constant of the boundary ∂Ω of Ω. Moreover, if u ∈ W ,p (Ω), then for all 1 ≤ p ≤ ∞, 0 ≤ |α| ≤ − 1, Dα (u − I h u)Lp (T h ) ≤ Kh−|α| |u|W ,p (Ω) .

(4.64)

2. Rewriting (4.64) for Sobolev norms, we obtain for all 0 ≤ k ≤ − 1, u − I h uW k,p (T h ) ≤ Kh−k uW ,p (Ω) and

(4.65)

lim I h uW k,p (T h ) = uW k,p (Ω) .

h→0

3. Moreover, it follows that for all 0 ≤ k ≤ − 1, u − I h uW k,p (∂Ω) ≤ Kh−k−1/p uW ,p (Ω) , and

(4.66)

lim I h uW k,p (∂Ω) = uW k,p (∂Ω) ,

h→0

where p > 1 if k = − 1. Proof. (1) Let us deﬁne the operator I h . In view of (4.61), each functional λk , k = 1, . . . , N , is well deﬁned on S h |Ek . By the Hahn–Banach Theorem 1.14, we extend λk from S h |Ek to Sd (T h )|Ek , such that |λk s| ≤ C2 sL∞ (Ek ) ,

for all s ∈ Sd (T h )|Ek ,

and deﬁne Iˆh : Sd (T h ) → S h by Iˆh s =

N

λk (s|Ek )sk ,

s ∈ Sd (T h ).

(4.67)

k=1

Let T be a simplex in T h , and let T ⊂ Ek . In view of (4.60), diam (Ek ) ≤ c1 hT and vol (Ek ) ≤ c2 vol (T ), with some constants c1 , c2 depending only on n, γ and ωT h . 31 Hence, for any s ∈ Sd (T h ) we have in view of (4.58), 1/p (T )sL∞ (Ek ) ≤ sLp (Ek ) ≤ c4 vol1/p (T )sL∞ (Ek ) , c−1 3 vol

where c3 , c4 depend only on n, d, p, γ, ωT h . Since the basis splines sk are also piecewise polynomials, we have by (4.61), sk Lp (T ) ≤ c5 vol1/p (T )sk L∞ (T ) ≤ c5 vol1/p (T )C1 , 31 This is easy to show considering that the number of simplices in E is bounded by a constant k depending only on n, γ and ωT h , and that any two simplices T , T with a common facet satisfy hT ≤ c˜1 hT and vol (T ) ≤ c˜2 vol (T ), for some c˜1 , c˜2 depending only on n and ωT h .

4.2. Approximation theory for ﬁnite elements

243

where c5 depends only on n, d and p. Hence, assuming s ∈ Sd (T h ) and applying (4.60) and (4.61), we obtain for any T ∈ T h , N N h k k ˆ I sLp (T ) = λ (s|Ek )s ≤ |λk (s|Ek )|sk Lp (T ) k=1 k=1 T ⊂E T ⊂E k

k

Lp (T )

≤ c5 C1 C2

N

sL∞ (Ek ) vol1/p (T )

k=1 T ⊂Ek

N

≤ c3 c5 C1 C2

sLp (Ek ) ≤ c6 c3 c5 C1 C2 sLp (Ωγ ) , T

k=1 T ⊂Ek

where the last inequality follows from N C

Ek ⊂ ΩγT ,

T ∈ T h,

k=1 T ⊂Ek

and

n+d γ h n # T ∈ T : T ⊂ ΩT

#{k : T ⊂ Ek } ≤ dim Sd (T h )|ΩγT = ≤ n+d n c6 ,

where c6 depends only on γ and ωT h . Thus, we have shown that Iˆh sLp (T ) ≤ csLp (Ωγ ) ,

s ∈ Sd (T h ),

T

T ∈ T h,

(4.68)

with c depending only on n, p, d, ωT h , C, and γ. (2) In order to extend I h = Iˆh from Sd (T h ) to L1 (Ω), we consider, for any u ∈ L1 (Ω) and any T ∈ T h , the averaged Taylor polynomial pT (u) of degree − 1 with respect to the inscribed ball of T (see Deﬁnition 4.14). By Lemma 4.15, |u − pT (u)|W k,p (T ) ≤ Bh−k T |u|W ,p (T ) ,

0 ≤ k ≤ − 1,

(4.69)

where B depends only on , n, ωT h . We deﬁne sˆ(u) ∈ Sd (T h ) by sˆ(u)|T = pT (u),

T ∈ T h,

and set I h u := Iˆh sˆ(u),

u ∈ L1 (Ω).

Clearly, I h is a projector onto S h , and, in particular, in view of (4.62), I h p = p,

n for any p ∈ P−1 .

(4.70)

(3) Let us prove (4.63). Suppose that T ∈ T , 0 ≤ |α| ≤ − 1, and |u|W ,p (Ωγ ) < h

T

∞. By using the Stein extension theorem (see e.g. [141], Theorem 1.4.5), we

244

4. Conforming ﬁnite element methods

extend u|ΩγT to a function u ˜ deﬁned on the convex hull U of ΩγT such that |˜ u|W ,p (U ) ≤ c7 |u|W ,p (Ωγ ) , T

where c7 depends only on n, p, , ωTch and, possibly, on the Lipschitz constant L∂Ω if Ω is nonconvex, and the boundary of ΩγT contains a part of ∂Ω, compare n be the average Taylor polynomial for [467], Theorem 1.9. Now, let pT ∈ P−1 u ˜ with respect to a ball in U of the greatest diameter. Again by the Bramble– Hilbert Lemma 4.15, we have 32 |u − pT (u)|W k,p (Ωγ ) ≤ ch−k T |u|W ,p (Ωγ ) , T

0 ≤ k ≤ − 1,

T

(4.71)

which implies, in particular, −|β|

Dβ (u − pT (u))Lp (Ωγ ) ≤ c8 hT T

0 ≤ |β| ≤ − 1,

|u|W ,p (Ωγ ) , T

(4.72)

with c8 depending only on n, p, , ωT h , L∂Ω . Therefore, Dα (u − I h u)Lp (T ) ≤ Dα (u − pT (u))Lp (T ) + Dα (pT (u) − I h u)Lp (T ) −|α|

≤ c8 hT

|u|W ,p (Ωγ ) + Dα (pT (u) − I h u)Lp (T ) . T

By (4.70), (4.57), (4.68), and (4.72), Dα (pT (u) − I h u)Lp (T ) = Dα Iˆh (pT (u) − sˆ(u))Lp (T ) ≤ ≤ ≤

A |α| hT

c |α| hT

c |α| hT

Iˆh (pT (u) − sˆ(u))Lp (T ) pT (u) − sˆ(u)Lp (Ωγ ) T

u − pT (u)Lp (Ωγ ) +

−|α|

≤ chT

T

|u|W ,p (Ωγ ) + T

c |α| hT

c |α| hT

u − sˆ(u)Lp (Ωγ ) T

u − sˆ(u)Lp (Ωγ ) . T

Now, since hT ≤ c9 hT , for all T ∈ T h such that T ⊂ ΩγT , where c9 depends only on n, γ and ωT h , we have by (4.69) u − sˆ(u)L∞ (Ωγ ) = max u − pT (u)L∞ (T ) T

T ∈T h γ T ⊂ΩT

≤ B max hT |u|W ,∞ (T ) ≤ Bc9 hT |u|W ,∞ (Ωγ ) T ∈T h γ T ⊂ΩT

T

32 For a convex Ω it is possible to show (4.71) by arguments similar to the proof of Lemma 2.5 in [436], without any resort to the Stein extension theorem.

4.2. Approximation theory for ﬁnite elements

245

in the case p = ∞, and u − sˆ(u)pL

γ p (ΩT )

=

T ∈T h γ T ⊂ΩT

u − pT (u)pLp (T )

≤ Bp

T ∈T h γ T ⊂ΩT

p p p p p hp T |u|W ,p (T ) ≤ B c9 hT |u|W ,p

(ΩγT )

in the case 1 ≤ p < ∞, which completes the proof of (4.63). (4) To show (4.64), we ﬁrst consider the case p = ∞. For some T ∗ ∈ T h , we have by (4.63), Dα (u − I h u)TL∞ (Ω) = Dα (u − I h u)L∞ (T ∗ ) ≤ KhT ∗ h

−|α|

≤ K|T |

h −|α|

|u|W ,∞ (Ωγ ∗ ) T

|u|W ,∞ (Ω) .

Assume now that 1 ≤ p < ∞. Then by (4.63),

Dα (u − I h u)TLp (Ω) h

p =

T ∈T

≤ Kp

Dα (u − I h u)pLp (T )

h

(−|α|)p

hT

T ∈T h

≤ K p |T h |(−|α|)p

|u|pW ,p

(ΩγT )

T ∈T h

T ∈T h γ T ⊂Ω T

|u|pW ,p (T ) .

Now

T ∈T h

T ∈T h γ T ⊂ΩT

|u|pW ,p (T ) =

# T ∈ T h : T ⊂ ΩγT |u|pW ,p (T ) ,

T ∈T h

and since T ⊂ ΩγT ⇔ T ⊂ ΩγT , we have # T ∈ T h : T ⊂ ΩγT = # T ∈ T h : T ⊂ ΩγT ≤ c6 , and, hence,

Dα (u − I h u)TLp (Ω) h

p

≤ c6 K p |T h |(−|α|)p

|u|pW ,p (T )

T ∈T h

= c6 K p |T h |(−|α|)p |u|pW ,p (Ω) , which completes the proof of (4.64).

246

4. Conforming ﬁnite element methods

(5) The last statement of the theorem is a direct consequence of (4.64) and Theorem 1.38. Indeed, by (1.68), 1−1/p 1/p u − I h uW k+1,p (T h ) u − I h uW k,p (∂Ω) ≤ u − I h uW k,p (Ω) ≤ Kh−k−1/p uW ,p (Ω) , and (4.66) follows.

From the diﬀerent types of extension operators mentioned above we refer to ¯ and extends them Treves [632]. He proves Theorem 26.A.3 for functions in u ∈ C k (Ω) continuously to u ∈ W k,p (Ω), 1 ≤ p < ∞, in Corollary 26.A.1, yielding (4.73), (4.74). This technique can obviously be extended the following situation to show (4.75). For 2 the FEMs for fully nonlinear elliptic problems and Ω ∈ C we will need approximate h h h polyhedral Ω or curved Ωc with dist Ω, Ωc ≤ Ch. Theorem 4.37. Modiﬁed Sobolev–Stein extension operator: Let Ω ∈ C k , Ωhc , Ω0 ⊃ (Ω ∪ Ωhc ), and k ∈ N be given. Then there exist a linear Ec : bounded extension H k (Ω) → H k Ω ∪ Ωhc → H0k (Ω0 ) and a constant, C = C k, Ω, Ωhc , Ω0 , such that Ec u|Ω = u on Ω and Ec uH k (Ω∪Ωhc ) ≤ CuH k (Ω) ∀u ∈ H k (Ω). (4.73) The traces of the partials of u and Ec u ∈ H k Ω ∪ Ωhc to the boundary ∂Ω coincide and for dist Ω, Ωhc ≤ Ch we obtain (4.74) H k−|β|−1/2 (∂Ω) ∂ β (u − Ec u)|∂Ω = 0 ∀|β| ≤ k, ∀u ∈ H k (Ω) β β k =⇒ ∂ Ec u(x) − Ec ∂ u(x) L2 (Ω∪Ωhc ) ≤ ChuH |β| (Ω) = 0∀u ∈ H0 (Ω). (4.75) Based upon this theorem we immediately obtain Theorem 4.38. Interpolation errors: Under the conditions (4.59)–(4.62), those of p Theorems 4.36 and 4.37, and for Ω piecewise in C , 2 ≤ p ≤ 5, a positive constant K = K n, d, ωTch , γ, C := C1 C2 , L∂Ωh exists, such that for u ∈ H (Ω), and 0 ≤ s ≤ − 1, cf. (4.62), we obtain for the operator P h = I h Ec : Ec u − P h uH s (Tch ) ≤ C h−s |u|H (Ω) , P h uH s (Tch ) ≤ CuH (Ω) Ec u − P h uH s (∂Ωhc ) ≤ C h−s−1/2 |u|H (Ω) , P h uH s (∂Ωhc ) ≤ CuH (Ω) .

(4.76)

The U h : = {P h u : u ∈ H (Ω)} approximate the H s (Ω), ≥ s, and H s (∂Ω), ≥ s + 1/2, e.g. in the sense that dist(Ec u, U h ) := inf ||Ec u − uh ||H s (Tch ) → 0 uh ∈U h

∀u ∈ H s (Ω)

for h → 0.

(4.77)

Spaces of smooth piecewise polynomials with stable local bases Bases with properties (4.60), (4.61), with γ = 1 and C1 , C2 depending only on d, are available for Sdr (T h ) and certain subspaces of it on arbitrary T h if d ≥ r2n + 1, see [263] and references therein. In the case of two variables, [267] provides a construction

4.2. Approximation theory for ﬁnite elements

247

of bases for Sdr (T h ) satisfying (4.60)–(4.62), with γ = 3 and C1 , C2 depending only on d, as soon as d ≥ 3r + 2. There are many more results on stable local spline bases in the recent literature, especially in the contexts of high order macro-element and Lagrange interpolation methods, see e.g. [467, 516] and references therein. Some classical spaces of smooth ﬁnite elements, for example those based on the Argyris element, see Figure 4.9, and [141] Example 3.2.10, belong to the class of so-called superspline subspaces of Sdr (T h ) [573], and their associated bases satisfy the requirements of Theorem 4.36. In particular, for the Argyris FE we have in the notation of Theorem 4.36, (4.78) S h = s ∈ S51 (T h ) : s is C 2 smooth at any vertex v of T h . Clearly, (4.62) is satisﬁed with n = 2 and = 6. The functionals λk : S h → R are function evaluations, weighted ﬁrst and second derivatives at the vertices, s(v), hT

∂s (v), ∂x1

hT

∂s (v), ∂x2

h2T

∂2s (v), ∂x21

h2T

∂2s (v), ∂x22

h2T

∂2s (v), ∂x1 ∂x2

(4.79)

and weighted ﬁrst order normal derivatives at the midpoints of the edges of T h , ∂s (v1 + v2 )/2 , (4.80) hT ∂n where hT is the diameter of a triangle in T h containing the corresponding evaluation point v or (v1 + v2 )/2 from (4.79), (4.80). The sets Ek = supp sk are either star (v) for the functionals of type (4.79), or the unions of two triangles sharing the edge [v1 , v2 ] in case (4.80). Hence, (4.60) is satisﬁed with γ = 1. Furthermore, by estimating the norms of local Hermite interpolation operators, it can be shown that sk L∞ (Ω) ≤ C1 , where C1 is an absolute constant, see [263, Lemma 3.3]. Similarly, in view of the inverse estimates (4.57), |λk s| ≤ C2 sL∞ (Ek ) , for any s ∈ S h , where C2 is again an absolute constant, see [263, p. 292]. Thus, (4.61) is correct with C = C1 C2 being an absolute constant. Stable splitting S h = S0h ⊕ Sbh In the ﬁnite element method an important role is played by the spaces of ﬁnite elements vanishing on (parts of) the boundary. We set S0h = {s ∈ S h : s|∂Ω = 0}. For the sake of applications to the numerical solution of fully nonlinear problems, we construct a splitting of S h into a direct sum S h = S0h ⊕ Sbh such that there exists a basis {s1 , . . . , sN } for S h satisfying the hypotheses (4.60), (4.61) of Theorem 4.36 (i.e. a stable local basis), where {s1 , . . . , sN0 } and {sN0 +1 , . . . , sN } are bases for S0h and Sbh , respectively. We provide a construction for n = 2 and S h ⊂ S51 (Ω) based on a modiﬁed Argyris ﬁnite element. We replace (4.78) by (4.81) S h = s ∈ S51 (T h ) : s is C 2 smooth at any interior vertex v of T h .

248

4. Conforming ﬁnite element methods

Thus, in contrast to the Argyris element, the functions in S h are not necessarily C 2 at the boundary vertices. We now describe a set of functionals {λ1 , . . . , λN } ⊂ (S h ) , such that the desired basis {s1 , . . . , sN } for S h will be uniquely deﬁned by duality λi sj = δi,j . The set {λ1 , . . . , λN } includes (a) the functionals (4.79) for all interior vertices v of T h ; (b) the functionals (4.80) for all edges of T h ; and (c) the following functionals for each boundary vertex v of T h : s(v), hT

∂s (v), ∂e0

hT

∂s (v), ∂e⊥ 0

h2T

∂2s (v), . . . , ∂e20

h2T

∂2s (v), ∂e2n

h2T

∂2s (v), ∂e0 ∂e1

where e0 , . . . , en are all edges of T h emanating from v, in counterclockwise order, with e0 and en being the boundary edges. Here the symbol ∂/∂e denotes the usual directional derivative in the direction of edge e, and ∂/∂e⊥ in the orthogonal direction. The above second order edge derivatives ∂2s (v), . . . , ∂e20

∂2s (v), ∂e2n

∂2s (v) ∂e0 ∂e1

are well deﬁned despite s being only C 1 at the boundary vertices, see [266]. Note that these second order derivatives are independent of each other even if some edges are collinear. This choice of the degrees of freedom at boundary vertices in (c) is motivated by the construction of the Morgan–Scott basis [497] and is shown to be stable in [266]. Following the argument in [266], one can see that the basis {s1 , . . . , sN } for S h , deﬁned by duality, satisﬁes (4.60), (4.61) with γ = 1 and bounded C1 , C2 . Hence it is a stable local basis. Moreover, (4.62) is obviously true for S h with n = 2 and = 6. Therefore, Theorem 4.36 applies to this basis. To determine the subsets of {s1 , . . . , sN } which generate S0h and Sbh , respectively, we now split the functionals in (c) into two groups (c1) and (c2) as follows. (c1) The ﬁrst group includes h2T

∂2s (v), . . . , ∂e21

h2T

∂2s (v), ∂e2n−1

h2T

for all boundary vertices, and, in addition, hT vertices, where e0 and en are collinear. (c2) The second group includes s(v), hT

∂s (v), ∂e0

h2T

∂2s (v), ∂e20

∂2s (v), ∂e0 ∂e1

∂s (v) for those boundary ∂e⊥ 0

h2T

∂2s (v), ∂e2n

(4.82)

4.2. Approximation theory for ﬁnite elements

for all boundary vertices, and, in addition, hT vertices, where e0 and en are not collinear.

249

∂s (v) for those boundary ∂e⊥ 0

Now let {λ1 , . . . , λN0 } list all functionals λi in (a), (b), and (c1), and let {λN0 +1 , . . . , λN } be those in (c2). It is easy to see that S0h = {s ∈ S h : λN0 +1 s = · · · = λN s = 0}.

(4.83)

Therefore S0h = span {s1 , . . . , sN0 }, and Sbh := span {sN0 +1 , . . . , sN } is its complement in S h as required. Clearly, both {s1 , . . . , sN0 } and {sN0 +1 , . . . , sN } are stable local bases as subsets of the stable local basis {s1 , . . . , sN }. ∂s Note that hT ⊥ (v) belongs to (c1) or (c2) depending on whether e0 and en are ∂e0 exactly collinear or not. In particular it is in (c2) if e0 and en are near-collinear, but not collinear, a situation which may appear quite often when a polygonal domain is an approximation of a smooth domain. As e0 and en become exactly collinear, ∂s hT ⊥ (v) is moved into (c1). Thus, the dimensions of S0h and Sbh jump if a vertex ∂e0 shared by collinear boundary edges is slightly perturbed. This ‘dimension instability’ is not a new phenomenon in the theory of multivariate splines. Indeed, it is related to the well-known fact that the dimension formulas for the spline spaces may depend on some geometric information about the placement of the vertices, see [467]. This behavior is compatible with the availability of stable bases, see for example the discussion in [267], Remark 13.1. Since λi are function evaluations or derivatives of at most second order, they may be applied to any suﬃciently smooth functions, thus leading to the interpolation operator I h : C 2 (Ω) → S h , deﬁned by I h (v) =

N

u ∈ C 2 (Ω).

λi (u)si ,

(4.84)

i=1

The following is an obvious property of this operator: v|∂Ω = 0

=⇒

I h (v) ∈ S0h .

(4.85)

It is easy to see that the operator I h can be used in the proof of Theorem 4.36 in place of the quasi-interpolation operator assuming that u is suﬃciently smooth. Finally, consider the operator Ibh : C 2 (Ω) → Sbh , deﬁned by Ibh (u)

=

N

λi (u)si .

(4.86)

i=N0 +1

Clearly, Ibh (u)|∂Ω = I h (u)|∂Ω .

(4.87)

250

4. Conforming ﬁnite element methods

Hence, by Trace Theorem 1.38 we obtain for any 1 ≤ p ≤ ∞, 1−1/p 1/p u − Ibh (u) ≤ Cu − I h (u)Lp (Ω) u − I h (u)W 1,p (Ω) , L (∂Ω) p

where C is a constant depending only onp and the Lipschitz constant L∂Ω of ∂Ω. This allows us to ﬁnd a bound for u − Ibh (u)L (∂Ω) , see (4.90) below, using the estimates p

available for u − I h (u)Lp (Ω) and Dα (v − I h (u))Lp (Ω) , |α| = 1. We summarize the results for the stable splitting in the following theorem. Theorem 4.39. Stable splitting: For the space S h deﬁned in (4.81), there exists a basis {s1 , . . . , sN } satisfying (4.60)–(4.62), such that {s1 , . . . , sN0 } is a basis for S0h = {s ∈ S h : s|∂Ω = 0}. The interpolation operators I h : C 2 (Ω) → S h and Ibh : C 2 (Ω) → Sbh deﬁned by (4.84) and (4.86), respectively, have the following properties, cf. (4.63), (4.64). 1. If u|∂Ω = 0, then I h (u) ∈ S0h . 2. Let 1 ≤ p ≤ ∞, ≤ 6, be such that W ,p (Ω) ⊂ C 2 (Ω). Then for any 0 ≤ |α| ≤ − 1, −|α|

Dα (u − I h (u))Lp (T ) ≤ KhT

|u|W ,p (Ω1 ) , T

for any T ∈ T h ,

(4.88)

and, as a consequence Dα (u − I h (u))Lp (T h ) ≤ K h−|α| |u|W ,p (Ω) . 3. Moreover,

u − Ibh (u)

Lp (∂Ω)

≤ K h−1/p |u|W ,p (Ω) .

(4.89)

(4.90)

The constants K, K , K depend only on p, ωT h , and the Lipschitz constant L∂Ω . Conjecture 4.40. The results in Theorems 4.35 and 4.36 are proved in Rn and for S h = Sdr (T h ) and n, r if d ≥ r2n + 1. We conjecture that the techniques of [263] can be used to generalize Theorem 4.39 as well to Sdr (T h ) and appropriate subspaces of it. Conjecture 4.41. We extend the previous Conjecture 4.40 to suitable nonpolyhedral (curved) boundaries ∂Ωhc ≈ ∂Ω with dist ∂Ωhc , ∂Ω = O(hq ), q > 2, and the corresponding triangulations Tch . We expect that Theorems 4.35–4.39 can be generalized in this case to appropriate subspaces of S h = Sdr Tch for any r if d ≥ r2n + 1, at least for n = 2. 4.2.7

Curved boundaries

In our earlier discussion we have excluded curved boundaries. We study them in this and the next subsection. We will formulate corresponding FEMs in Subsections 5.5.6, 5.5.10. The results here are valid partially for Rn , n ≥ 2, or Rn , n ≤ 3. So we consider n , d ≥ 1. Ω ⊂ Rn , n ≥ 2, with piecewise smooth ∂Ω ∈ Cpt , t ≥ 1, P = Pd−1

(4.91)

4.2. Approximation theory for ﬁnite elements

251

Figure 4.21 Admissible triangulation for a curved boundary.

To simplify the presentation, we restrict the construction to two-variate FEs in this subsection. Extensions to Rn , n ≥ 3, are theoretically pretty obvious, in contrast to the practical realization. In a ﬁrst step we deﬁne an approximating Ωh and the corresponding T h : Choose points Pj ∈ ∂Ω with distance ≤ h for neighboring points. Include all “non-smooth” points on ∂Ω ∈ Cpt into these Pj . Replace ∂Ω and Ω by ∂Ωh and Ωh : ∂Ωh is obtained by connecting the neighboring Pj ∈ ∂Ω by straight lines, thus deﬁning the new edges e ⊂ ∂Ωh . The polygonal Ωh is then the interior of ∂Ωh . Choose a nondegenerate triangulation T h for Ωh .

(4.92)

Then automatically the T ∈ T h are star-shaped. Figure 4.21 shows the original ∂Ω with an interior T h . We have to be prepared that the necessarily modiﬁed FEs will satisfy the boundary conditions only approximately. The discussion here is restricted to triangulations, possibly curved at the boundary, and to ¯ only function evaluations and no derivatives ∈ N . (4.93) K triangle, ∀ e¯ ⊂ K Note that vertices Pj ∈ e¯, see (4.98), are included. Here, we present two diﬀerent possibilities to handle Ωh and T h . For the “boundary FEs” (T, PT , NT ) with |T¯ ∩ ∂Ω| ≥ 2 we change NT or PT , thus modifying Deﬁnition 4.6. In the ﬁrst case, see Figure 4.22, we replace, along the boundary, NT by interpolation along the curved ∂Ω instead of the straight ∂Ωh .

(4.94)

Secondly, we use the original FT for T with respect to Ωh , see (4.92), Figures 4.22, 4.23, but, n , replace FT : K → T aﬃne, by F h |T ◦ FT : K → Tc , F h |T ∈ Pd−1

(4.95)

n . This for the T ∈ T h at the boundary and with the original polynomials P = Pd−1 yields the so-called isoparametric FEs. We describe in Deﬁnition 4.42 and the following pages, how these Fch , Tc are constructed. Obviously the basis functions φi ◦ (F h |T ◦ FT )−1 ∈ P are more complicated nonpolynomial functions. For a third choice see Subsection 4.2.6.

252

4. Conforming ﬁnite element methods Pe

Pi

k=2

Tc

Figure 4.22 Polynomial interpolation on ∂Ω.

∂Tc

x˜ (s)

Pe xa(s)

Pi

∂T ∂Ω 0 £ y £ ya(s)

Pc

Figure 4.23 Isoparametric interpolation on ∂Ω.

Polynomial approximation and interpolation in points of ∂Ω For the new triangulation, T h , we modify the T and NT at the boundary with |T¯ ∩ ∂Ω| ≥ 2, see (4.94): We denote the vertex points of T¯ as T¯v and deﬁne T¯v ∩ ∂Ωh = {Pi , Pe }, Pi = Pe , T¯v := {vertices of T }.

(4.96)

According to (4.94), we modify these triangles T into Tc by replacing the straight line Pi , Pe by that part of the boundary ∂Ω between Pi and Pe , see Figure 4.22.

4.2. Approximation theory for ﬁnite elements

253

The modiﬁed triangulation we denote as Tch . So we still obtain, in contrast to the isoparametric FEs below, C ¯= Tc . (4.97) Ω Tc ∈Tch

For modifying NT we proceed in two steps. Choose a smooth parametrization x = x(s) of ∂Ω between Pi and Pe with respect to the arc length s, 0 ≤ s ≤ he , such that cf. (5.307)–(5.311), with s ∈ C 2d−1 [0, he ]. Now we choose d Gauss–Lobatto points, ρ = 2 and the yjρ = yj2 there. Let ξ0 , . . . , ξd−1 , ξj := he 1 + yj2 /2, ξ0 = 0, ξd−1 = he in [0, he ] (or Gauss points in Subsubsections 5.5.5 and 5.5.6, 5.5.7). Determine, on the curved edge e = ∂T , cf. Figure 4.22, ∂Ω Pje := Pj := x(ξj ) ≈ Pjs := Pi + ξj (Pe − Pi )/(Pe − Pi )R2 , j = 0, . . . , d − 1, Pi = P0 , Pe = Pd−1 .

(4.98)

In principle, we could use any d interpolation points on e. The choice of the Gauss-type points is motivated by the applications to FEMs in Section 5.5. Now, we replace the d function evaluations in the Pjs along the straight edge Pi Pe for the original T , by the d function evaluations in the Pje = Pj , j = 0, . . . , d − 1. Thus we replace NT by a modiﬁed NTc corresponding to one curved edge, see Figure 4.22. For Gauss and Gauss–Radau points we obtain the corresponding 0 < ξ0 ≤ ξd−1 < he and ξ0 = 0, ξd−1 < he , respectively. Then discontiuous U h are generated. We discuss these cases in Subsections 5.5.6, 5.5.7. For small enough h, the edge Pi Pe will be at most O(h2 ) away from the curved part of ∂Ω between Pi and Pe , see [141], 8 ex.3. Hence PT , NTc will be unisolvent simultaneously with PT , NT for small enough h. These perturbation arguments show that the results in Subsection 4.2.4 remain essentially valid. For R3 the above Gauss-type points could be replaced by the cubature points of the best types of cubature formulas in tetrahedra in R3 . Their order of convergence is smaller than for the R2 case. This is inherited by the FEMs. We do not present those modiﬁcations here, since we consider discontinuous Galerkin methods in Chapter 7, avoiding these diﬃculties. For details we refer to Cools [208–210]. Isoparametric nonpolynomial approximation The Gauss type points decrease, as far as possible, interpolation and, for applications in FEMs, the quadrature errors along the relevant boundary part of ∂Ω. There is another method, well established in engineering applications, allowing much more freedom, but with the same order of convergence as before. In addition this method is applicable in FEMs to Rn , n ≥ 2. Particularly eﬃcient is the isoparametric polynomial approach in (4.95). As in (4.96) we assume one curved and two straight edges Pi , Pe and P0 Pi , P0 Pe for Tc . Well known results of Ciarlet and Raviart [176], Ciarlet [174] and Lenoir [471], construct a bijective, piecewise polynomial mapping F h : Ωh → Ωhc ≈ Ω ⊂ R2 , see [471] for Rn , n ≥ 2, such that F h = id away from the boundary. We present a very rough idea of this construction with the goal of showing that the isoparametric

254

4. Conforming ﬁnite element methods

FEMs ﬁt into our general discretization framework. Figure 4.23 motivates the next deﬁnition. In 3. below we need dist(F h (∂Ωh ), ∂Ω). 33 Deﬁnition 4.42. We deﬁne the triangulation Tch , the bijective isoparametric mapping F h : Ωh → Ωhc ≈ Ω, and the boundary ∂Ωhc := F h (∂Ωh ): Choose Ωh , ∂Ωh and a nondegenerate triangulation T h for Ωh as in (4.92), (4.96). Then, correlating T h with Tch , we deﬁne F h : Ωh → Ωhc := F h (Ωh ) ⊂ Rn , n ≥ 2, Tch and Tc ∈ Tch as: 1. Let (F h )|T = id|T ∀T ∈ T h with |T¯v ∩ ∂Ω| ≤ 1, see (4.96), and let F h |e¯ = id|e¯, ∀¯ e ⊂ T¯ with |¯ e ∩ ∂Ω| ≤ 1. n ∀T ∈ T h with 2. For the components of F h , the (F h )i , i = 1, 2, let (F h )i |T ∈ Pd−1 |T¯v ∩ ∂Ω| ≥ 2, see Figure 4.23, and Tch Tc := F h T ∀T ∈ T h . 3. dist (F h (∂Ωh ), ∂Ω) = O(hd ), dist (F h (Ωh ), Ω) = O(hd ). 4. (F h ) W d,∞ (Tc )←W d,∞ (T ) , ((F h ) )−1 ··· ≤ C, h independent, ∀Tc ∈ Tch . 5. Deﬁne the Tc as Tc = F h (T ) ∀ T ∈ T h with T = Tc ∀ T ∈ T h for |T¯ ∩ ∂Ω| ≤ ¯ Pi , Pe is a straight line. 1 and Tc = T for h|T ∩ ∂Ω| ≥ 2,hunless ∂Ω between h ¯ hc := ∪ Tc . 6. Deﬁne Tc := Tc for the Tc in (v), and Ω Tc ∈Tch

Note that the conditions 1.–6. are strictly local, including 3. We only describe this construction, see above, for a triangle T ∈ Tch with 2 = ¯ |Tv ∩ ∂Ω| = {Pi , Pe }, see Figure 4.23. For all other cases, (F h )|T = id|T by 1. Our chosen T ∈ T h is deﬁned by the vertices Pi , Pe , Pc . The solid line ∂Ω marks the original part of the boundary between Pi and Pe . Then F h |T : T → Tc , by 2. The isoparametric triangle Tc has the straight edges Pc , Pi , Pc , Pe and the curved n edge ∂Tc connecting Pi , Pe . By 2. the components of F h are (F h )i |T ∈ Pd−1 h with F |e¯ = id|e¯, for the two straight edges Pc , Pi Pc , Pe . By 3. we require dist ˜(s) on the ray from (F h (∂Ωh ), ∂Ω) = O(hd ). For the two points marked as xa (s), x Pc the distance in Figure 4.23 is O(hd ). We did not mark any intersection point of ∂Ω and ∂Tc , since only the distance O(hd ) and 4. are required. Finally 5., 6. yield Tc ∈ Tch . Deﬁnition 4.42 introduces an isoparametric subdivision, Tch . We proceed by introducing isoparametric FEs, U h , and, later on, an isoparametric interpolation operator, I h . We combine the aﬃne mapping F = FT : K → T ∈ T h with the polynomials F h |T : T → Tc , see Figure 4.24. The construction of this F h is a main point in this subsection. So we get nonconforming FEs violating the boundary conditions. This will be treated in Subsection 5.5.10, see Theorem 5.85. We start with the original T h deﬁned for Ωh in (4.92), the F h , Tch and Ωhc in Deﬁnition 4.42, and the points Pie on curved Tc with edges e, approximating ∂Ω, for d cf. Deﬁnition 4.1: 33 For compact A, B, the distance between A and B is deﬁned as dist (A, B) = max {max x∈A : {min {|x − y| : y ∈ B}}, maxy∈B : {min {|x − y| : x ∈ A}}}. It represents the maximal distance from any point of one set to the nearest point of the other set.

4.2. Approximation theory for ﬁnite elements

FT

K

affine

255

Fh T

Tc

Figure 4.24 Isoparametric mapping FT : K → T, F h ◦ FT : K → Tc .

d U h := uh : Ωhc = F h (Ωh ) → R : uh |Tc = αi · φi ◦ (F h |T ◦ FT )−1 , i=1

with (non)polynomials φi ◦ (F h |T ◦ FT )−1 , αi ∈ R ∀ Tc = F h (T ) ∈ Tch , (4.99) uh W k,q (Tch ) as in (4.36) with Ω, T h replaced byF h (Ωh ), Tch Ubh := uh ∈ U h : ∀e ⊂ Tc uh (Pie ) = 0, i = 0, . . . , d − 1, interpolating in e ≈ ∂Ωhc , where these P0e = Pi , Pde −1 = Pe , but Pie , i = 1, . . . , d − 2 are d − 2 points on e, unequal to those in (4.98). ¯ denote the edge mapped by the original FT : K → T onto Pi Pe . Then Let e ⊂ K ∂ΩhT := F h (FT (e)), marked as ∂Tc in Figure 4.23, only approximates that part of ∂Ω between Pi and Pe up to O(hd ). For the following interpolation process this might have the consequence that some of the interpolation points Pj ∈ ∂ΩhT , j = 1, . . . , d − 2, are Pj ∈ Ω ∪ ∂Ω and so u(Pj ) might not be deﬁned. In this case, Theorem 4.37 yields an appropriate extension ue of u, allowing the evaluation of ue (Pj ), see [141]. More appropriate is a kind of piecewise interpolation, Fc , of the above F h . The complicated construction of Fc and the following properties of Ec are presented for Rn , n ≥ 2, in [471]. We only indicate the R2 results and summarize the essential properties in (4.100) and (4.101). They allow formulating error and inverse estimates and the proofs for the FEMs below in Rn , n ≥ 2, cf. Theorem 4.43 and 5.85 ﬀ. This auxiliary Fc exactly maps Ωh → Ω. Fc = ((Fc )i )2i=1 : Ωh → Ω ⊂ Rn s.t. ∀T ∈ T h : Fc |T : T → T˜c ≈ Tc , and −1

(Fc |T )

(4.100)

: T˜c → T are C diﬀeomorphism s.t with the operator norm ||| · |||∞ , d

|||Ds (Fc |T − idT )|||∞ , |||Ds ((Fc |T )−1 − idT˜c )|||∞ ≤ Chd−s ∀s ≤ d |J(D(Fc |T ) − 1|∞ , |J(D(Fc |T )−1 ) − 1|∞ ≤ Chd and ∂Fc (Ωh ) = Fc (∂Ωh ) = ∂Ω and F h (∂Ωh ) = ∂Ω + O(hd ), and, replacing F h by Fc , this Fc satisﬁes Deﬁnition 4.42 1.; here |||Ds (Fc |T − idT )|||∞ indicates the sup-norm of the sth derivative and J(D(Fc |T ) the absolute value of the

256

4. Conforming ﬁnite element methods

Jacobian. Thus Fc has the property that in Figure 4.23 the part of ∂Ω from Pi to Pe is the image Fc (∂T ) where ∂T indicates the straight segment Pi , Pe . We introduce Ec , transforming functions deﬁned on the original Ω to those deﬁned on the approximate Ωhc = F h (Ωh ) ≈ Ω and its inverse (Ec )−1 : 1. Ec : Ω → Ωhc = F h (Ωh ) as Ec := F h ◦ Fc−1 : t ∈ Ω → x ∈ F h (Ωh ) ≈ Ω, 2. (Ec )−1 : F h (Ωh ) → Ω with t − Ec (t) = O(hd ); x − (Ec )−1 (x) = O(hd ) and 3. (Ec ) − IdΩ = O(hd−1 ), ((Ec )−1 ) − IdF h (Ωh ) = O(hd−1 ), as in (4.100), 4. Ec |T = IdT and Ec |e = Ide ∀ T, e in Deﬁnition 4.42 1.

(4.101)

With this Ec , the uh , U h and Ωhc = F h (Ωh ) in (4.99) we start with ˆh as uh : F h (Ωh ) → R, uh ∈ U h , and we deﬁne the new u h h u ˆh : Ω → R, u ˆh (t) := (uh ◦ Ec )(t), and UDbh = u ˆ : u ∈ Ubh .

(4.102)

Note that UDbh ⊂ U, but trivial Dirichlet boundary conditions are satisﬁed only approximately on the exact ∂Ω for u ˆ ∈ UDbh . Vice versa for f : Ω → R the fˇ : F h (Ωh ) → R is deﬁned as fˇ(x) := f ((Ec )−1 (x)) = (f ◦ (Ec )−1 )(x),

(4.103)

With F h in Deﬁnition 4.42 the isoparametric interpolation operator, IEh , is IEh u := (I h u) ◦ Ec ∈ Uˆh , with I h : U := {u : Ω → R} → U h := uh : Ωhc → R , I h u|T = I h u|T as in (4.16) ∀ inner T = Tc ∈ Tch with |T¯c ∩ ∂Ωh | ≤ 1, else (4.104)

h

I u|Tc =

d i=1

−1

Ni (u ◦ F |Tc ◦ FTc ) · (φi ◦ (F |Tc ◦ FTc ) h

h

)=

d

NiTc (u)φTi c .

i=1

We have replaced the aﬃne FT by isoparametric generalizations with components n , i = 1, 2, for Tc with two boundary vertices. Obviously, I h is (F h |Tc ◦ FTc )i ∈ Pd−1 a bounded linear operator as the original I h , if the norms are now deﬁned with respect to the Tch above. Furthermore, the original φi ◦ (FT )−1 and Ni (u ◦ FT ) have to be replaced by the more complicated (non)polynomial φi ◦ (F h |Tc ◦ FTc )−1 but still linear Ni (u ◦ F h |Tc ◦ FTc ). Since, in mapping a polyhedral to a smooth domain, a C 1 mapping is not appropriate, we only choose C 0 reference elements, cf. [141], p.118. Theorem 4.43. Interpolation errors: For Ω as in (4.91) deﬁne Ωh as in (4.92). Let, for 0 < h ≤ 1, the T h for Ωh satisfy Condition 4.16 for some l, d, p. We combine F h and Teh from Deﬁnition 4.42, with F h , piecewise of degree d − 1. For a C 0 reference element (K, P, N ), deﬁne U h , Ubh , I h , IEh u as in (4.99), (4.102), (4.104). Then there

4.3. FEMs for linear problems

257

exists a positive constant C = C((K, P, N ), n, d, p, χ), for χ cf. (4.30), such that for 0 ≤ s ≤ d, and ∀u ∈ W d,p (Ω), d−s h u s,p u − IE |u|W d,p (T ) , W (T ) ≤ C (diam T )

(4.105)

d−s h u s,p ))uW d,p (T ) with IE W (T ) ≤ (1 + O((diam T )

max u − IEh uW s,∞ (T ) ≤ C hd−s−n/p |u|W d,∞ (Ω) ∀ u ∈ W d,∞ (Ω), and

T ∈T h

h u s,p h ≤ C hd−s |u| d,p u − IE W (T ) W (Ω) , h u s,p h ≤ (1 + C hd−s )u d,p IE W (T ) W (Ω) .

Since P always satisﬁes P ⊆ W j,p (T ) ∩ Wql (T ), 1 ≤ q ≤ p, 0 ≤ l ≤ j, we get Theorem 4.44. Inverse estimates: Under the conditions of Theorem 4.43 and for T ∈ Tch of a quasiuniform triangulation T h let U h = uh : uh is measurable and uh ∈ U h according to (4.99) ∀Tc ∈ Tch . Then there exists a constant C = C(l, p, q, χ) such that uh W j,p (Tch ) ≤ Chl−j−(n/q−n/p) uh Wql (Tch ) ∀uh ∈ U h , ∀j ≥ l.

4.3

(4.106)

FEMs for linear problems

The usual approach in ﬁnite element methods considers the weak operator, A, G and its corresponding weak bilinear or nonlinear form, a(·, ·) as in this subsection and in Section 4.4. We will modify this approach for fully nonlinear problems in Section 5.2. In contrast to the following Chapters 5 and 7, we consider here conforming FEMs, i.e. the exact and approximate solutions and test functions are elements of the same spaces, so u0 , uh0 ∈ Ub , v, v h ∈ Vb , here with Ub = Vb = H01 (Ω) and Ub = Vb = H0m (Ω) for elliptic diﬀerential equations of order 2 in this and order 2m in Subsection 4.3.2. A straightforward modiﬁcation allows us to include these equations and systems of order 2m. Extensions to quasilinear problems are studied in Section 4.4. For conforming FEMs and our problems here, we have Ub = Vb and Ubh = Vbh , so from here on we use the notation Vb and Vbh , the standard notation. More involved discussions would be necessary to study the diﬀerent combinations of natural and induced boundary conditions, see e.g. [387] and Chapter 2. We start with the main idea for FEMs, presented for a simple example in Subsection 4.3.1. In particular, we show how the deﬁnition of FEs via the triple K, P, N can be used eﬃciently to determine the elements of the mass and stiﬀness matrices for a FEM. Next, general elliptic operators of order 2 and 2m and their corresponding coercive and elliptic bilinear forms and the induced boundary operators are recapitulated as a basis for FEMs in Subsection 4.3.2. Finally, the convergence for conforming FEMs for

258

4. Conforming ﬁnite element methods

these linear elliptic boundary value problems is formulated in Theorem 4.57, with the Aubin–Nitsche improvement in Theorem 4.60. 4.3.1

Finite element methods: a simple example, essential tools

The main idea for FEs is very simple and is demonstrated for a simple example. We choose the Helmholz operator and Dirichlet boundary conditions: Example 4.45. Homogeneous Dirichlet boundary value problem For u ∈ H01 (Ω) := {u ∈ H 1 (Ω) : u|∂Ω = 0}, f ∈ H0−1 (Ω), cf. Proposition 2.34, we have deﬁned the bilinear form a(·, ·) and the corresponding operator A as a(·, ·) : H01 (Ω) × H01 (Ω) → R and A : H01 (Ω) → H0−1 (Ω) are deﬁned by (∇ u, ∇ v)n + cuvdx =: Au, vH −1 (Ω)×H 1 (Ω) ∀v ∈ H01 (Ω),

a(u, v) := Ω

with the Euclidean product (∇u(x), ∇v(x))n = (∇u(x))T ∇v(x) in Rn . For f ∈ L2 (Ω) or f ∈ H0−1 (Ω), the exact (weak) solution u0 is then deﬁned by (4.107) u0 ∈ Ub = Vb := H01 (Ω) : a(u0 , v) = (∇ u0 , ∇ v)n + c u0 v dx Ω

= Au0 , vH −1 (Ω)×H 1 (Ω) = (f, v) or f, v∀ v ∈ Vb .

Basic idea for conforming ﬁnite elements methods For conforming FEs we replace Vb in (4.107) by a sequence of ﬁnite dimensional spaces Vbh h∈H . For nonconforming FEMs, we sometimes even choose two sequences h Ub h∈H and Vbh h∈H . The indices h satisfy 0 < h ∈ H, with inf h∈H h = 0. Deﬁnition 4.46. Conforming FEs are characterized, for second order problems, by Vbh as subsets of solution and test spaces, exactly satisfying the boundary conditions, here the trivial Dirichlet conditions, and uh , v h ∈ Vbh ⊂ Vb = H01 (Ω), ∀T ∈ T h : uh |T , v h |T ∈ P, Pd−1 ⊂ P ⊂ Pd+τ , d > 1, τ ≥ −1, thus a(uh , v h ) : Vbh × Vbh → R, and f, v h : Vbh → R for f ∈ Vb = H0−1 (Ω) are well deﬁned and Vbh ⊂ H01 (Ω) ∩ C(Ω). Conforming FEMs are deﬁned with conforming FEs. For problems of order 2m, we require Vbh ⊂ H0m (Ω). We determine the approximate and the exact solutions uh0 ∈ Vbh and u0 from ; h h : h h ∇ uh0 , ∇ v h n + c uh0 v h dx = Auh0 , v h V ×V u0 ∈ Vb : a u0 , v = (4.108) Ω

b

b

= f, v h Vb ×Vb ∀ v h ∈ Vbh ⊂ Vb = H01 (Ω) with Ah ∈ L(Vbh , Vbh ) and u0 ∈ Vb by omitting all h .

(4.109)

259

4.3. FEMs for linear problems

Remark 4.47. We want to discuss an often used notation. For practically all spaces, V, discussed here, we ﬁnd V = Vb , and V = Vb , cf. Proposition 2.34. The same holds for the discrete spaces. Nevertheless, for test functions v ∈ Vb = V and functionals f ∈ V = Vb we ﬁnd f |Vb ∈ Vb . We summarize (4.110) V = Vb ⇒ V = Vb , e.g., H −1 (Ω) = (H 1 (Ω)) = H01 (Ω) = H0−1 (Ω) f ∈ V : f, vV ×V = f, vV ×Vb ∀ v ∈ Vb ⇒ f |H01 (Ω) = f |Vb ∈ Vb = H0−1 (Ω)

and similarly V h = Vbh , V h = Vbh , f h ∈ V h = Vbh ⇒ f h |Vbh ∈ Vbh . Usually we will use the notation f h , v h V h ×V h , similarly f, vVb ×Vb , but now and b

b

then, f h , v h V h ×V h ∀v h ∈ Vbh as well. By (4.110) there is not much diﬀerence for conforming FEs, Vbh ⊂ Vb . For nonconforming FEs, Vbh ⊂ Vb , cf. Chapter 5, this will change. For general cases we will prove the convergence via the results of Chapter 3. Nevertheless, honoring Cea, we include the surprisingly simple proof for convergence u0 − uh0 H 1 (Ω) → 0 for h → 0 for his lemma. It applies not only to (4.109), but to all “conforming” FEMs for “coercive” bilinear forms. Lemma 4.48. Cea lemma: Let Vbh ⊂ Vb ⊂ V, with dim Vbh < ∞ and assume a (bounded) Vb -coercive bilinear form, hence, cf. Proposition 1.25, |a(u, u)| ≥ αu2V ≥ α u2Vb , ∀u, v ∈ Vb

|a(u, v)| ≤ CuV vV ,

(4.111)

with constants C, α, α > 0. Then, with the projector Q h in (4.117) the discrete Ah := (Q h A|Vbh )−1 ∈ L(Vbh , Vbh ) is bounded and boundedly invertible with Ah V h ←V h < C, b

b

(Ah )−1 V h ←V h < 1/α. Then unique exact solutions u0 ∈ Vb and discrete solutions b

b

uh0 ∈ Vbh of (4.107) and (4.109) uniquely exist, and satisfy the error estimate: u0 − uh0 ≤ V

C min uh − u0 V . α uh ∈Vbh

(4.112)

Proof. By (4.111) the a(., .) : Vb × Vb → R, and a(., .) : Vbh × Vbh → R, satisfy the inf–sup conditions in Theorem 2.12. Thus AVb ←Vb < C, A−1 Vb ←Vb < 1/α and Ah V h ←V h < C, (Ah )−1 V h ←V h < 1/α are bounded and boundedly invertible. So b

b

b

b

the solutions u0 ∈ Vb and uh0 ∈ Vbh of (4.107) and (4.109) uniquely exist and imply a u0 − uh0 , v h = 0 ∀ v h ∈ Vbh , and furthermore 2 by coercivity α u0 − uh0 V ≤ a u0 − uh0 , u0 − uh0 = a u0 − uh0 , u0 − uh + a u0 − uh0 , uh − uh0 with uh − uh0 ∈ V h = a u0 − uh0 , u0 − uh ≤ C u0 − uh0 V u0 − uh V by continuity.

260

4. Conforming ﬁnite element methods

This implies u0 − uh0 ≤ C u0 − uh ∀ uh ∈ Vbh . V V α Since dim Vbh < ∞ we ﬁnd, with inf = min, u0 − uh0 ≤ C min uh − u0 . V α uh ∈Vbh

This estimate is exactly the same as that obtained below via Chapter 3.

We re-interpret (4.109) by deﬁning and applying a projector Q h . For later more general applications we formulate the following Proposition for broken Sobolev spaces Vb = W01,p (T h ), Vb = W0−1,q (T h ), 1/q + 1/p = 1, Vbh ⊂ W01,p (T h ), Vbh := Vbh , and a generalized bounded fh ∈ Vb in the form, so without boundary terms, cf. Proposition 2.34 and Remark 4.47, n fi ∂ i v h dx ∀v h ∈ Vb ∪ Vbh . (4.113) f, v h := fh , v h Vb ×Vb := T ∈T h

T

i=0

Proposition 4.49. For this f = fh ∈ Vb = W0−,q (T h ) and the restriction f h = fh |Vbh the norms are asymptotically equal ∀ 0 ≤ , 1 ≤ q ≤ ∞, 1/p + 1/q = 1, f h V h = f h , ·V h ×V h ≤ f Vb = fh V h (1 + o(1)), and b

b

b

(4.114)

b

f h V h = f h , ·V h ×V h ≤ f V = fh V h (1 + o(1)). −,q For smoother f , e.g. f ∈ W0,k (T h ), k > 0, cf. (4.137), satisfying homogeneous boundary conditions, we can estimate the f Lq (T ) , f Lq (T h ) in (4.116) by f W −,q (T ) , f W −,q (T h ) and still obtain the estimate (4.114). 0,k

0,k

Particularly interesting are the f, vW −1,q (Ω)×W 1,p (Ω) = Ω (f−1 , ∇ v)n + f0 vdx. 0 0 For nonconforming FEMs in Sections 5.2, 5.5 we need the piecewise form as in (4.116). In the quadrature context, cf. Section 5.4, (4.114) applies to “smoother” f ∈ Wk−,q (T h ), k > 0, cf. Theorem 5.28 and (5.186). Proof. We start with the V = Lp (T h ) result: It is well known that |(f, v)L2 (T h ) | ≤ f Lq (T h ) vLp (T h ) , f Lq (T h ) =

1/q (|f | )dx q

Th

,

(4.115)

valid for T h and T . Combined with v h ∈ V h , dense in V we get f V h ≤ f Lq (T h ) . This is combined with Proposition 4.18. For any > 0 there is a v1 = 0 such that |f Lq (T h ) − (f, v1 )L2 (T h ) /v1 Lp (T h ) | ≤ /2.

261

4.3. FEMs for linear problems

Proposition 4.18 implies the existence of a v h such that for small enough h v1 − v h Lp (T h ) < /(2f Lq (T h ) ) ⇒ f Lq (T h ) < |(f, v h )L2 (T h ) |/v h Lp (T h ) + ≤ f V h + . Now (4.115) shows the asymptotic equality, hence (4.114) for the Lq (T h ) case. For the W0−,q (T h ) case we formulate the ﬁrst step, W0−1,q (T h ). We apply the discrete H¨older inequality (1.44) to the f ∈ W0−1,q (T h ), cf. Proposition 2.34, f, v h W −1,q (T h )×W 1,p (T h ) 0 0 = (f−1 , ∇ v h )n + f0 vdx T

T ∈T h

n

≤

i f−1 Lq (T ) ∂ i v h Lp (T ) + f0 Lq (T ) v h Lp (T )

T ∈T h i=1

⎛

≤ ⎝

(4.116)

⎞1/q ⎛

f−1 qLq (T ) + f0 qLq (T ) ⎠

⎝

T ∈T h

⎞1/p

∇ v h pLp (T ) + v h pLp (T ) ⎠

T ∈T h

= f W −1,q (T h ) v h W 1,p (T h ) ∀ v h ∈ W01,p (Ω) ∪ W01,p (T h ). 0

0

yielding (4.114).

For our special case V = H 1 (Ω), V = H −1 (Ω), Vb = H01 (Ω), Vbh ⊂ H01 (Ω), Vbh , we deﬁne for the f ∈ Vb in (4.113), but valid for f ∈ Vb = W0−m,q (Ω) as well,

as Q h f − f, v h Vb ×Vb = 0 ∀ v h ∈ Vbh ⇐⇒ Q h f − f ⊥ Vbh , Q h ∈ L Vb , Vbh

Q h f V h = b

= f V h ≤ b

sup v h ∈Vbh ,v h V =1

sup v∈Vb ,vV =1

Q h f, v h Vb ×Vb =

f, v h Vb ×Vb

sup

v h ∈Vbh ,v h V =1

f, vVb ×Vb = f Vb =⇒ Q h L(V ,V h ) ≤ 1,

and, with dense Vbh ⊂ Vb ,

b

(4.117)

b

limh→0 Q h f V h = f Vb . b

Proposition 4.50. These projectors Q h in (4.117) satisfy the conditions in Deﬁnition 3.12 for convergence of general discretization methods. Consequently (4.109) can be re-interpreted as ; : h Au0 − f , v h V ×V = 0∀ v h ∈ Vbh ⇐⇒ Q h Auh0 = Q h f. uh0 ∈ Vbh :

(4.118)

Since Vbh ⊂ Vb , the (4.107), (4.109) imply a u0 − uh0 , v h = (f, v h ) − (f, v h ) = 0 ∀ v h ∈ Vbh .

(4.119)

b

b

262

4. Conforming ﬁnite element methods

ˆz3

z3

F

Kˆ = T

K ˆ1 z z1

ˆz2

z2

Figure 4.25 Aﬃne transformations applied to a triangle.

In Subsection 4.3.3 and Section 5.5 we will call a(u0 − uh0 , v h ) the consistency error for u0 . So it vanishes for conforming FEMs.

(variational)

Reducing the computational costs By choosing a basis {φi }di=1 for P and using the aﬃne equivalence in Deﬁnition 4.6, it is possible to strongly reduce the cost for computing the (∇v, ∇w)n dxdy, vwdxdy, vdxdy, mainly for v = v h , w = wh ∈ V h (4.120) T

T

T

by computing the integrals over the reference domain K only once, cf. Proposition 5.41 and Figures 4.25 and 4.26. An analogous procedure applies to the general case and, modiﬁed, to nonconforming FEs as well. The values for i, j = 1, . . . d , ∇ uhi , ∇ uhj n dxdy and uhi , uhj dxdy, with d = dim Vbh , (4.121) T

T

are denoted as elements of the stiﬀness and mass matrices, respectively z3

z4

zˆ 4 zˆ 3 F

K

zˆ

z1

z2

zˆ 1 zˆ 2

Figure 4.26 Aﬃne transformations applied to a rhombus.

4.3. FEMs for linear problems

263

Example 4.51. We demonstrate this procedure for Example 4.11 and a subdivision into triangles or rhombuses. Obviously, it can be extended to many other cases as well. K is either the unit triangle or the square. We memorize the transformation FT , its Jacobian and its partials from (4.14), (4.19), (4.22). x = (FT )1 (ξ , η) := x1 + (x2 − x1 ) ξ + (x3 − x1 ) η y = (FT )2 (ξ , η) := y1 + (y2 − y1 ) ξ + (y3 − y1 ) η.

(4.122)

For the substitution in the integrals we need the Jacobian J = det (FT ) = (x2 − x1 )(y3 − y1 ) − (y2 − y1 )(x3 − x1 ) = 0

(4.123)

and the partials (note that J, ξx , ξy , ηx , ηy are constant for each T ) vx = uξ ξx + uη ηx , ξx = (y3 − y1 )/J, ξy = − (x3 − x1 )/J vy = uξ ξy + uη ηy , ηx = −(y2 − y1 )/J, ηy = (x2 − x1 )/J.

(4.124)

With the notation v = u ◦ FT−1 , u = v ◦ FT , w = z ◦ FT−1 , z = w ◦ FT , we obtain for the integrals in (4.120) the following relations cf. Figure 4.25 2 vx + vy2 dxdy (∇ v )2 dxdy = T

T

[(uξ ξx + uη ηx )2 + (uξ ξy + uη ηy )2 ]Jdξdη

= K

u2ξ dξdη + 2b

=a K

u2η dξdη

uξ uη dξdη + c K

(4.125)

K

with a := [(x3 − x1 )2 + (y3 − y1 )2 ]/J, b := −[(x3 − x1 )(x2 − x1 ) + (y3 − y1 )(y2 − y1 )]/J, c := [(x2 − x1 )2 + (y2 − y1 )2 ]/J. Similarly, we obtain (∇ v, ∇ w )n dxdy = (vx wx + vy wy )dxdy T

T

=

[(uξ ξx + uη ηx )(zξ ξx + zη ηx )

(4.126)

K

+ (uξ ξy + uη ηy )(zξ ξy + zη ηy )]Jdξdη uξ zξ dξdη + b (uξ zη + zξ uη )dξdη + c uη zη dξdη = a K

K

K

264

4. Conforming ﬁnite element methods

and

vwdxdy = J T

uzdξdη, K

vdxdy,

T

udξdη.

J

(4.127)

K

Whenever p q we use polynomial FEs, we are thus left with the evaluation of the integrals ξ η dξdη. K triang ξ p η q dξdη =: Ipq = p!q!/(p + q + 2)! for the unit triangle K (4.128) K

square ξ p η q dξdη =: Ipq = 1/(p + 1)(q + 1) for the unit square K. K

Finally, a basis for the polynomials in P has to be inserted into (4.125), (4.126), (4.127) to determine the system (4.109). This allows us in an eﬃcient way to compute the elements of the stiﬀness and mass matrices for our system. 4.3.2

Finite element methods for general linear equations and systems of orders 2 and 2m

These methods represent a straightforward generalization of the preceding (4.107), (4.117), (4.118), (4.119). So we do not repeat the original forms of the equations and systems, cf. Sections 2.3, 2.4, 2.6, but rather formulate the FE equations for these cases. For the convenience of the reader we recapitulate the notation: For the multiindices α, we deﬁned, see (2.73), ∂iu =

∂u ∂ |α| , ∇u = (∂ 1 u, . . . , ∂ n u) and ∂ α u = ∂1α1 . . . ∂nαn u = u, α1 ∂xi ∂x1 . . . ∂xn αn

∇0 u = u, ∇k u = (∂ α u)|α|=k , ∇u = ∇1 u = (∂ α u)|α|=1 , and vectors and reals ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn , ϑα = (ϑ1 )α1 · · · (ϑn )αn , ϑi , Θ0 ∈ R, i ≥ 0, choose the nk , Nk , s.t. Θk = (ϑα )|α|=k ∈ Rnk , (ϑα )|α|≤k ∈ RNk , Θ = Θ1 ,

(4.129)

with ∂ i u(x), ∂ α u(x) ∈ R, ∇u(x) ∈ Rn , ∇k u(x) ∈ Rnk . The FEM for a linear second order elliptic equation, see (2.160), determines uh0 such that n : ; aij ∂ i uh0 ∂ j v h dx = f, v h V ×Vb uh0 ∈ Ubh : a uh0 , v h = Auh0 , v h V ×V = b

b

Ω i,j=0

∀ v h ∈ Vbh , with aij ∈ L∞ (Ω), Vbh ⊂ Vb = H01 (Ω). The linear form f and our elliptic bilinear form are bounded, so |f, vV ×Vb | ≤ C vV , b

|a(u, v)| ≤ CuU vV ∀u, v ∈ Vb .

b

(4.130)

265

4.3. FEMs for linear problems

The principal part, ap (u, v), satisﬁes for all ϑ ∈ Rn λ|ϑ|2n ≤ λ(x)|ϑ|2n ≤ (−1)m

n

aij (x)ϑi ϑj ≤ Λ|ϑ|2n ∀ x ∈ Ω with λ > 0.

(4.131)

i,j=1

By (2.22) the principal part, ap (u, v), is Vb = H01 (Ω)-coercive, see (2.79), hence, n ap (u, u) = aij ∂ i u∂ j udx ≥ λ∗ u2U , λ∗ > 0, ∀u ∈ Ub ⊃ Ubh . (4.132) Ω i,j=1

Conforming FEMs for elliptic equations of order 2m require Vbh ⊂ Vb = H0m (Ω), hence smooth FEs, cf. (2.160) the references in Subsections 4.2.6, and 4.3.2. For linear A, determine uh0 ∈ Ubh ⊂ H0m (Ω) : ; aαβ ∂ α uh0 ∂ β v h dx = f, v h V ×Vb (4.133) a uh0 , v h = Auh0 , v h V ×V = b

b

Ω |α|,|β|≤m

b

¯ ∀v h ∈ Vbh ⊂ Vb with aαβ ∈ L∞ (Ω), and for 1 < m = |α| = |β| : aαβ ∈ C(Ω). This conformity condition Ubh ⊂ H0m (Ω) is somehow cumbersome, since C m−1 FEs are not too common. In fact, we have only given some references and the corresponding interpolation error and inverse estimates in Subsection 4.2.6. We had assumed an elliptic equation, hence a(u, v) is bounded and the principal part, ap (u, v), satisﬁes with λ > 0 and ∀ϑ ∈ Rn , see (2.79), 2m m aαβ (x)ϑα ϑβ ≤ Λ|ϑ|2m (4.134) λ|ϑ|2m n ≤ λ(x)|ϑ|n ≤ (−1) n ∀ x ∈ Ω. |α|=|β|=m

By (2.160), and Theorem 2.43, the principal part, ap (u, v), is Vb -coercive, hence ap (u, u) = aαβ ∂ α u0 ∂ β udx ≥ λ∗ u2V , λ∗ > 0, ∀u ∈ Ub ⊃ Ubh . (4.135) Ω |α|=|β|=m

For systems we have to extend and modify the above notation: We have to apply the partials ∂ l , l = 1, . . . , n, to q components uj , vi , i, j = 1, . . . , q and modify (4.129) l , ϑ α , ∂ l u(x), ∂ α u(x) ∈ Rq , and from n-vectors from reals ϑl , ∂ l u(x) ∈ R, to vectors ϑ 1 n n ∇ u(x) ∈ Rn×q , see (2.335), (2.389). ϑ = (ϑ , . . . , ϑ ), ∇u(x) ∈ R to n × q-matrices ϑ, 34 In addition to the notation in (4.129) we need: α α = ϑ , . . . , ϑα , l = ϑl , . . . , ϑl , ϑ u = (u1 , . . . , uq ), ∂ α u = (∂ α u1 , . . . , ∂ α uq ), ϑ 1 q 1 q n ), with |ϑ| = |ϑ| nq ∈ R, u(x), ∂ α u(x), ϑ l , ϑ α ∈ Rq , = (ϑ 1, . . . , ϑ ϑ

(4.136)

1, . . . , ϑ n ) ∈ Rn×q , ∂ α u = u for |α| = 0. = (ϑ ∇ u(x) = (∂ 1 u, . . . , ∂ n u)(x), Θ 34 The usual notation for θ and l , Θ, which turns u motivates the certainly not optimal notation ϑ out to be appropriate below.

266

4. Conforming ﬁnite element methods

The FEM for a linear second order elliptic system requires ansatz and test-functions uh = uh1 , · · · , uhq v h ∈ Vbh (Rq ) ⊂ Vb = H01 (Ω, Rq ), with uhj , vjh ∈ Vbh . With the Euclidean product (fk , ∂ k v h )q in Rq , see (2.340), (2.341), we obtain the FE solution uh0 ∈ Vbh (Rq ), and the exact solution u0 ∈ Vb (Rq ) by omitting all the h , as n n h h h k h Akl ∂ l uh0 , ∂ k v h q dx f , v = (fk , ∂ v )q dx = a u0 , v = Ω k=0

Ω k,l=0

∀ v h ∈ Vbh ⊂ H01 (Ω, Rq ), with Akl ∈ L∞ (Ω, Rq×q ), fk ∈ L2 (Ω, Rq×q ),

(4.137)

and f H −1 (Ω) := f H −1 (Ω,Rq×q ) := max {fk H j (Ω,Rq×q ) }, 0,j

0,j

k=0,...,n

with bounded f , v h , a( uh , v h ). The Vb -coercivity of the principal part is guaranteed for uniformly elliptic systems, satisfying the uniform Legendre condition, see (2.343), (2.345), hence 2 ≤ ∃ λ, Λ, s.t. 0 < λ < Λ < ∞ : λ|ϑ| nq

n

l )ϑ k ≤ Λ|ϑ| 2 ∀x ∈ Ω ⇒ (Akl (x)ϑ

(4.138)

k,l=1

ap ( u, u) > λ∗ u2V and a( u, u) ≥ λ∗∗ u2V − Cc u2W ∀ u ∈ Vb , W = L2 (Ω, Rq ). (4.139) For the FEM for a linear elliptic system of order 2m we choose u , v = (v1 , . . . , vq ) ∈ Vbh ⊂ Vb = H0m (Ω, Rq ), and determine uh0 ∈ Ubh such that, see (2.392), uh0 ∈ Vbh ⊂ Vb = H0m (Ω, Rq ) : f , v h = (fα , ∂ α v h )q dx = a uh0 , v h = (4.140) h

h

Ω |α|≤m

Ω |α|,|β|≤m

Aαβ ∂ β uh0 , ∂ α v h

q

dx∀ v h ∈ Vbh , Aαβ (x) ∈ L∞ (Ω, Rq×q ), fα ∈ L2 (Ω, Rq ).

We assume the strong Legendre–Hadamard condition (Aαβ (x)ϑβ η, ϑα η¯)q ≥ λ|ϑ|2m |η|2 . ∃λ > 0∀ϑ ∈ Rn , η ∈ Cq , x ∈ Ω :

(4.141)

|α|=|β|=m

By Theorem 2.104 the principal part is Ub - and Ubh -coercive h h ap ( u , u ) = (Aαβ ∂ β uh , ∂ α uh )q dx ≥ λ∗ uh 2V

(4.142)

Ω |α|=|β|=m

¯ ∀|α| = |β| = m. if additionally Aαβ ∈ C(Ω) 4.3.3

General convergence theory for conforming FEMs

For all the previous cases (4.109), (4.130), (4.133), (4.137), (4.140), the conforming FEMs have a very similar structure: we have replaced Vb by a sequence of ﬁnite

4.3. FEMs for linear problems

267

dimensional spaces Vbh h∈H . These Vbh ⊂ Vb do satisfy the boundary conditions and the appropriate regularity conditions, e.g. Vbh ⊂ Vb = H0m (Ω) ⊂ V = H m (Ω), e.g. Vbh ⊂ C m−1 (Ω). Thus a(uh , v h ) : Vbh × Vbh → R is well deﬁned. We consider the exact and the FE solutions u0 ∈ Vb = H0m (Ω), and uh0 ∈ Vbh ⊂ H0m (Ω) of equations and u0 ∈ Vb ⊂ H0m (Ω, Rq ), uh0 ∈ Vbh ⊂ H0m (Ω, Rq ) of systems. We denote the approximate solution uh0 ∈ Vbh , using the same symbol for u0 ∈ Vb = H0m (Ω, Rq ), uh0 ∈ Vbh ⊂ Vb = H0m (Ω, Rq ), q ≥ 1, : ; so uh0 ∈ Vbh ⊂ Vb : a uh0 , v h = Ah uh0 , v h V ×V = f, v h Vb ×Vb ∀ v h ∈ Vbh . (4.143) b

b

Here Ah uh is induced by the bilinear form a(uh , v h ), analogous for quasilinear problems. All these methods are special cases of the general discretization methods in Chapter 3. For the proof of convergence we have to verify the conditions in Subections 3.3–3.4 for the diﬀerent spaces and operators constituting these methods. We summarize the conditions and deﬁnitions in Summary 4.52. The wavelet and spectral methods in Chapter 9 and in [120] are conforming variational methods as well. So the following summary is applicable to these cases and to non conforming FEMs as well. Their relations are sketched in the diagram in Figure 4.27. Summary 4.52. We formulate it in a more general form to include FEMs with variational crimes, by using the necessary spaces U, Ub , V, Vb , Ubh , Vbh and extensions Ah , fh , Gh , cf. Chapter 5. For conforming methods, we have Ah = A, f = fh , Gh = G. 1. The method has to be applicable to the problem Au0 = f or Gu0 = 0, see Deﬁnition 3.12. This requires, throughout with ∀h ∈ H, h < h0 , and ﬁxed h0 , a sequence of approximating spaces Ubh , Vbh for the Ub , Vb , projectors P h , Q h , and the discrete problems Ah or Gh , with inf{0 < h ∈ H} = 0, such that

{U h , V h , P h , Q h , Gh }h∈H ,

dim Ubh = dim Vbh < ∞,

(4.144)

thus limh→0 = limh∈H,h→0 makes sense. Deﬁnition 3.12 does not require that, but for our methods the Ubh , Vbh are approximating spaces, such that ∀u ∈ U : lim dist (u, U h ) = 0, e.g. lim ||u − P h u||U h = 0, h→0

h→0

(4.145)

similarly for V, with P h = I h for conforming methods, cf. Proposition 4.18. Ub Ph

U bh

A

Φh Ah

V ′b

tested by

Vb

tested by

V bh

Q′h

V bh ′

Figure 4.27 Our methods: spaces, operators, often Ub = Vb , Ubh = Vbh .

268

4. Conforming ﬁnite element methods

2. The P h ∈ L Ub , Ubh , Q h ∈ L Vb , Vbh , are linear approximation and projection operators, cf. (4.117), Proposition 4.50 and Deﬁnitions 3.6, 3.12, such that

lim P h uU h = uU by (4.145) and lim Q h f V h = f Vb .

h→0

h→0

(4.146)

b

3. Our methods deﬁne a mapping Φh : D(Φh ) ⊂ (U → V ) → (U h → V h ), transforming Au0 = f or Gu0 = 0 into Ah uh0 = f h or Gh uh0 = 0. For nonconforming FEs, we have to extend the A, G : U → V onto the corresponding spaces, UT h , VT h , piecewise deﬁned on the subdivision, T h , cf. (4.35), (4.36), and (5.251) ﬀ., (7.57) ﬀ. A, G : U → V into Ah , Gh : U ∪ UT h → V ∪ VT h with

(4.147)

Ah , Gh |U = A, G, and Ah = A, Gh = G, for conforming methods. In fact, the A, G, Ah , Gh do not, but the unique solvability does need the boundary conditions. For our methods, Φh is deﬁned as

Ah := Φh A := Qh Ah |Ubh , and f h := Φh f := Qh fh |Ubh , hence

(4.148)

Φh (A · −f ) = Qh (Ah · −fh )|Ubh or Gh := Φh G := Qh Gh |Ubh .

4. According to Deﬁnition 3.14, we choose u, uh , related as Ah uh = Q h Ah uh = Q h Au and Gh uh = Q h Gh uh = Q h Gu. Then

Q h Ah u − Ah uh = Q h (Ah u − Ah uh ), Q h Gh u − Gh uh = Q h (Gh u − Gh uh ) (4.149) are the so-called (variational) consistency errors. 5. The (classical) consistency error or local discretization error in u is

Ah P h u − Q h Au = Ah P h u − Q h Ah u or Gh P h u − Q h Gu.

(4.150)

As a consequence of the above extension A, G to Ah , Gh we always have Au = Ah u, Gu = Gh u, cf. Theorem 4.54, (4.157). 6. The Gh are called stable in uh , and Ah stable, if for ﬁxed h0 , r, S ∈ R+ , and ∀h ∈ H uhi ∈ Br (uh ), i = 1, 2, ⇒ uh1 − uh2 U h ≤ S Gh uh1 − Gh uh2 V h . (4.151) For Ah or Ah − f h this reduces to (independent of uh , but for ∀r > 0, h < h0 )

(Ah )−1 ∈ L Vbh , Ubh exists and ||(Ah )−1 ||U h ←V h ≤ S. b

h

b

The stability of G is essentially implied by the stability of its derivative (G (u0 ))h , cf. Theorem 3.23. 7. If the exact and discrete solutions, u0 and uh0 exist, the uh0 are called convergent and convergent of order p, if (4.152) lim uh0 − u0 U = 0 and uh0 − u0 U ≤ Chp . h→0

4.3. FEMs for linear problems

8. 9. 10. 11.

269

For nonconforming FEMs uh0 − u0 U have to be replaced by uh0 − u0 U h . ˜ h , A˜h , G ˜h. For quadrature approximations the Q h , Ah , Gh are replaced by Q For the nonconforming FEMs in Chapters 5 and 7, (4.150) remains correct. For isoparametric FEMs Q h are replaced by Qch , applied to functionals deﬁned on a curved approximation Ωhc ≈ Ω. For fully nonlinear problems in strong form, F := (G, BD ), cf. (5.41), h h U , VΠ = V h × Vbh , P h , QhΠ , Qhc,Π , F h h∈H , with 0 < h ∈ H, (4.153)

replaces (4.144) and modiﬁes all the following concepts, cf. Section 5.2. Here G and BD indicate the diﬀerential and the Dirichlet boundary operators, respech h h tively, approximated h h by G and BD and QΠ projects both components (G, BD ) h to F = G , BD . 12. All our methods are so-called linear methods in the sense Φh (B + C) = Φh B + Φh C, and Φh (G + f ) = Φh G + Φh f.

(4.154)

We reformulate Theorem 3.21 in Chapter 3 for FEMs. In fact, we only have to update the previous Gh , Ubh , Vbh , P h , Q h , Gh , to the following deﬁnitions. The obvious updates for the fully nonlinear case, according to (4.153), are not explicitly formulated in the next theorem. Theorem 4.53. Unique existence and convergence of discrete solutions: Let the original problems Gu = 0 have the exact solution u0 ∈ U. Let the FE discretization method be applicable to G and yield Gh = Q h Gh |U h : U h → V h in (5.62), satisfying

1. Gh : U h → V h is deﬁned and continuous in Br (P h u0 ), r > 0, h-independent; 2. Gh is consistent with G in P h u0 ; 3. Gh is stable for P h u0 . Then the discrete problems Gh uh = 0 possess unique solutions uh0 ∈ U h near u0 for all suﬃciently small h ∈ H and uh0 converge to u0 . If Gh is consistent and consistent of order p, then uh0 converge and converge of order p, respectively: h u0 − u0 ≤ SQ h Gh (P h u0 ) h → 0 and uh0 − u0 ≤ O(hp ). (4.155) V U U b

Hence, good convergence, essentially determined by the consistency, is usually only satisﬁed if u0 is smoother than u0 ∈ U. The exact formulation for the diﬀerent nonconforming FE and other methods is delayed to Chapters 5, 7 and 8. In particular, Theorem 4.53 will be speciﬁed for the other cases. We have restricted the discussion in this chapter to conforming FEMs: the variational consistency errors vanish by (4.148) for the exact solution u0 . This is a simple consequence of Vbh ⊂ Vb , e.g. (4.143) implies : ; Au0 − Ah uh0 , v h V h ×V = a u0 − uh0 , v h = (f, v h ) − (f, v h ) = 0 ∀v h ∈ Vbh . (4.156) Theorem 4.54. Variational and classical consistency errors for conforming FEMs:

270

4. Conforming ﬁnite element methods

1. For conforming FEMs applied to linear and nonlinear problems, Au − f = 0 and Gu = 0, the variational consistency errors in u vanish, cf. (4.143), (4.149). 2. For A ∈ L(U, V ), and a nonlinear G : D(G) ⊂ U → V with Lipschitz-continuous G ∈ CL (D(G)) theclassical consistency or local discretization error is

(Φh A)(I h u) − Q h AuVb = Q h A(I h u − u)Vb ≤ CAVb ←Ub I h u − uU .

(Φh G)(I h u) − Q h GuVb = Q h (GI h u − Gu)Vb ≤ CLVb ←Ub I h u − uU . (4.157) 3. By Proposition 4.18, the classical consistency error tends to 0, according, e.g. to Q h A(I h u − u)Vb ≤ CAVb ←Ub I h u − uU → 0 for u ∈ U. Proof. For the classical consistency error we use Vbh ⊂ Vb and (Φh A)uh = Ah uh = Q h Auh . We ﬁnd Ah I h u − Q h AuV h ≤ CAVb ←Ub I h u − uU . b

We formulate for later reference the changes for nonconforming FEMs: Remark 4.55. Nonconforming variational and classical consistency errors: 1. Nonconforming FEMs applied to linear and nonlinear problems, Au − f = 0 and Gu = 0, yield nontrivial variational consistency errors, since A = Ah , G = Gh , cf. (4.149). 2. The nonconforming classical consistency error is estimated as the sum of these nontrivial variational consistency errors and with U, V, Ubh , Vbh , cf. (4.35), (4.36),

Q h Ah (I h u − u)Vb ≤ CAh V ∪V h ←U ∪UT h I h u − uUT h T

(4.158)

or Q h Gh I h u − Gh u)Vb ≤ CLI h u − uUT h , with a Lipschitz constant L for Gh : D(Gh ) ⊂ U ∪ UT h → V ∪ VT h 3. The classical consistency error tends to 0, with the variational consistency and the interpolation error. To admit nonconforming methods we again consider U, V, Ubh , Vbh , the operators P h = I h and A, Ah are already deﬁned in (4.109), (4.117), (4.130), (4.133), (4.137), (4.140). So Q h in (4.117) and Proposition 4.50 have to be generalized. The arguments (4.113)–(4.113) are still valid for f ∈ W0−m,p (Ω), 1/p + 1/p = 1, so for f, v h = f, v h Vb ×Vbh := (fα , ∂ α v h )dx ∀v h ∈ Vbh ⊂ Vb . (4.159) Ω |α|≤m

The intended Q h f ∈ Vbh implies that Q h f, v h Vb ×Vb is only deﬁned for v h ∈ Vbh :

Q h ∈ L(V , V h ) with Q h f − f, v h Vb ×Vbh = 0∀v h ∈ Vbh ⇔ Q h f − f ⊥ Vbh .

(4.160)

4.3. FEMs for linear problems

271

Lemma 4.56. For f ∈ Vb in (4.159), and the projector Q h in (4.160), induced by the approximating spaces Vbh for Vb , we get, ∀ 0 ≤ m, 1 ≤ p ≤ ∞, 1/p + 1/p = 1,

f V h ≤ f Vb = f V h (1 + o(1)), Vb = W0−m,p (T h ) and b

(4.161)

b

Q h f = f |Vbh , f V h = Q h f V h ≤ f Vb , lim Q h f V h = f Vb , b

h→0

b

b

Q V h ←V = 1 + o(1), lim Q V h ←V = 1, hence (4.146). h

b

b

h

h→0

b

(4.162)

−m,p These results remain correct for smooth f ∈ W0,k (Ω), k > 0, see (4.137) and essentially for non-conforming FEMs as well, cf. Proposition 5.55.

After these preparations, we re-interpret (4.143) as ; : h Au0 − f , v h V ×V = 0∀ v h ∈ Vbh ⇐⇒ Ah uh0 = Q h f with Ah := Q h A|Ubh . (4.163) As a consequence of Theorem 3.21, the convergence for conforming FEMs for these linear elliptic boundary value problems is proved by the following summarizing theorem. The Aubin–Nitsche improvements are formulated in Theorem 4.60. We study the FEMs in (4.130), (4.133), (4.137), (4.140), with piecewise polynomials of degree ≥ d − 1, and the spaces V, Vb , Vbh . We assume the conditions (4.131), (4.133), (4.134) for equations, and (4.137), (4.138), (4.141) for systems of orders 2, m = 1 and 2m. Note that conforming FEMs for order 2m require the smooth FEs in Subsection 4.2.6. The special (4.109) always satisﬁes (4.131). Theorem 4.57. Stability and convergence for linear and quasilinear problems: 1. Under these conditions the principal part ap (·, ·) is Ubh -coercive, and the original form a(·, ·) is Ubh -elliptic. For a boundedly invertible A : Vb → V , and under Condition 4.16, Ah = Q h A|Ub is stable by Theorem 3.29. 2. Assume an exact solution u0 ∈ H s (Ω, Rq ), q ≥ 1, m ≤ min{s, d}, and V = H m (Ω, Rq ). Then the FEM (4.163) has unique solutions uh0 ∈ Ubh , converging as u0 − uh0 ≤ C dist u0 , Ubh ≤ Chmin{s,d}−m u0 H s (Ω,Rq ) . (4.164) V 3. This remains correct for quasilinear problems Gu0 = 0 and their FEMs for a boundedly invertible G (u0 ), satifying the above conditions for A. Remark 4.58. These convergence estimates, (4.164) in Theorems 4.57, and 5.71, 5.66, are formulated for functions, e.g. u ∈ H s (Ω), s ≥ d smooth enough to yield the best order of accuracy. If this smoothness is reduced, then the interpolation errors are modiﬁed according to u ∈ H d−p (Ω) =⇒ I h u − uH m (Ω) ≤ Chd−m−p |u|H d−p (Ω)

(4.165)

or, in (5.348) for m = 1 : Ah I h u − Q h AuH m (Ω) ≤ Chd−1−p |u|H d−p (Ω) . This is a consequence of the fact that these results, including the interpolation errors, are proved with the Bramble–Hilbert Lemma 4.15. Analogous reductions are valid for all the other cases as well. We do not always refer to this remark.

272

4. Conforming ﬁnite element methods

Proof. Now we have available all the necessary results to apply the general converegence theory summarized in Theorems 4.53, 4.54. In fact, the conditions (4.144)– (4.151) are veriﬁed, ecxcept (4.151): The Vbh are admissible Galerkin approximating spaces for Vb . Their Φh is applicable to all the above problems yielding the FEMs listed preceding Theorem 4.57. By (4.150), (4.156), the consistency errors vanish. The above u0 ∈ H s (Ω), s > m, yields a consistency error as in (4.164). The discrete operator Ah deﬁned in (4.163) is continuous in Ubh . So the only missing condition, (4.151), for Theorem 4.53 is the stability of Ah . We have seen above that in all cases the principal parts ap (., .) are Vb - and Vbh -coercive by the inf–sup condition and Theorem 2.12. So these principal parts generate an invertible Ap with stable Ahp . Theorem 3.29 implies then, for a consistent Ah in u0 , and an invertible A : Ub → V , in fact a compact perturbation of Ap , a stable Ah . So these FE equations have a unique solution, uh0 , converging as in (4.164). Under conditions, which usually are satisﬁed, it is possible to increase the order of convergence with respect to a weaker norm. In addition to the original problem a(u0 , v) = (f, v)∀v ∈ Vb we need the solution, φg , of its dual problem, see (5.352), φg ∈ Vb s.t. a(w, φg ) = (g, w)∀w ∈ Ub . The proof of this lemma and the following theorem is presented for the more general case of FEMs violating continuity in Lemma 5.73 and Theorem 5.74. Lemma 4.59. Aubin–Nitsche lemma: Let a(·, ·) : Ub × Vb → R be a continuous bilinear form, and let the conditions of Theorem 4.57 be satisﬁed. Then the original and its dual problem have unique exact and FE solutions u0 , uh0 and φg , φhg . For the Banach space W ⊃ Vb with inner product and norm (g, w) and gW , respectively, let the embedding Vb → W be continuous. Then the error for the FE solution can be estimated in the W norm as u0 − uh0 h ≤ C u0 − uh0 h sup φg − φhg h /gW . (4.166) W U V 0 =g∈W

Theorem 4.60. Aubin–Nitsche convergence for u0 −uh0 L2 (Ω) : Under the conditions of Theorem 4.57 and Lemma 4.59 and for an exact solution u0 ∈ H d (Ω), the solution of the FEM equation satisﬁes u0 − uh0 2 ≤ Ch u0 − uh0 V ≤ Chd u0 H d (Ω) . (4.167) L (Ω) Instead of the Cea lemma the discrete analogue of the inf–sup Theorem 2.12 can be employed, an important tool for proving the stability of the discrete Ah . Theorem 4.61. Discrete Brezzi–Babuska or discrete inf–sup condition: Let Vbh be ﬁnite dimensional Banach spaces, Ah ∈ L(Ubh , Vbh ), and ah (·, ·) : Ubh × Vbh → R the associated operator and continuous bilinear form, deﬁned for ﬁxed uh ∈ Ubh by Ah uh , v h V h ×V h = ah (uh , v h ) ∀v h ∈ Vbh . b

b

(4.168)

4.4. FEMs for divergent quasilinear elliptic equations and systems

273

Then the four statements (4.169), (4.170), (4.171) and (4.172) are equivalent.

and C exists, s.t.(Ah )−1 V h ←U h ≤ C, (Ah )−1 ∈ L Vbh , Ubh (4.169) b

∃ , > 0 s.t and

sup 0 =v h ∈Vbh

sup 0 =uh ∈Ubh

∃ > 0 s.t and

sup 0 =v h ∈Vbh

sup

and

sup 0 =v h ∈Vbh

h

h

h

h

h

(4.170)

|ah (uh , v h )|/uh U h ≥ v h V h ∀v h ∈ Vbh ,

|ah (uh , v h )|/v h V h ≥ uh U ∀uh ∈ Ubh

0 =uh ∈Ubh

∃ > 0 s.t.

b

|a (u , v )|/v V h ≥ u U ∀u ∈ Ubh h

(4.171)

|ah (uh , v h )|/uh U h > 0 ∀v h ∈ Vbh ,

|ah (uh , v h )|/v h V h ≥ uh U h ∀uh ∈ Ubh

(4.172)

∀ v h ∈ Vbh ∃ uh ∈ Ubh : ah (uh , v h ) = 0.

For stability arguments the following constants C, , have to be valid only for suﬃciently small h < h0 , h0 > 0 and have to be independent of h, but uniformly with respect to h. Remark 4.62. We want to point out that the inf−sup conditions for A, a(·, ·), U, V, . . . in Theorem 2.12 in Chapter 2 do not imply the corresponding inf−sup conditions for the corresponding Ah , ah (·, ·). However the equivalence of the four statements is still correct. The original a(·, ·) : U h → V h or later generalizations ah (·, ·) : U h → V h have to satisfy these equations uniformly with respect to h. Hence, > 0, > 0, have to be independent of h. For stability arguments the constants C, , have to be valid only for suﬃciently small h < h0 , h0 > 0 and have to be independent of h in the discrete counterpart of Theorem 2.12. Then the previous references (4.169), (4.170), (4.171) and (4.172) in Theorem 4.61 correspond to (2.51), (2.52), (2.53) and (2.54) in Theorem 2.12.

4.4

Finite element methods for divergent quasilinear elliptic equations and systems

For divergent semilinear and quasilinear elliptic equations, FEMs are known, see, e.g. Zeidler [678]. We extend them to systems. Many of the existence, uniqueness and regularity results are only known in H0m (Ω). Generalizations to the W0m,p (Ω, Rq ) situation are possible and worthwhile. In fact linearization results as in Theorem 2.122 form the basis for numerical methods for the W0m,p (Ω, Rq ) setting for all types of nonlinear and even fully nonlinear problems. This opens the possibility for research into new problems outside of the previous scope. We use the standard notation (4.129), and again the same symbol u, uh for functions u ∈ W0m,p (Ω, Rq ), uh ∈ Vbh ⊂ W0m,p (Ω, Rq ), m, q ≥ 1, p ≥ 2,

274

4. Conforming ﬁnite element methods

of equations and systems. On the basis of Section 2.7 we restrict the discussion to 2 ≤ p. But in Section 4.5 we will study these quasilinear problems as monotone operators. Then we prove existence and convergence for W0m,p (Ω, Rq ) for 1 < p < ∞. To emphasize the similarity between orders 2 and 2m, we we use the notation (4.173) with |α| ≤ 1 for second order equations. The necessary existence, regularity and linearization results are quoted before we formulate Theorem 4.63. This includes the conditions for the W0m,p (Ω, Rq )-coercivity of the principal part. The divergent quasilinear elliptic equation or system with Dirichlet boundary conditions in the trace sense and Ω in (4.5), see Skrypnik [591], Zeidler [678] and B¨ ohmer [117], is deﬁned and solved by u0 ∈ Ub such that with ·, · := ·, ·V ×Vb and V := W m,p (Ω, Rq ) b

m, q ≥ 1 :

Dirichlet condition (∂ j u/∂ν j )|∂Ω = 0, j = 0, · · · , m − 1, hence G : D(G) ⊂ Vb :=

W0m,p (Ω, Rq )

∀v ∈ Vb : Gu0 , v = a(u0 , v) :=

→ V =

W0−m,p (Ω, Rq ), 1/p

(4.173)

+ 1/p = 1,

(Aα (·, u0 , . . . , ∇m u0 ), ∂ α v)q dx = f, v.

Ω |α|≤m

The Aα (·, u, . . . , ∇m u) : Vb → V are so-called Nemyckii operators. a(u, v) is well deﬁned under the conditions discussed in Theorems 2.63, 2.73, 2.75 and 2.76. In Section 2.7, Theorems 2.122, 2.124–2.126, we have formulated conditions for bounded G (u0 ) : Vb → V . In particular, G (u0 ) : H0m (Ω, Rq ) → H −m (Ω, Rq ) yields a H0m (Ω, Rq )-coercive principal part. This implies, for 2 ≤ p < ∞, only a H0m (Ω, Rq )coercive principal part, cf. Theorem 2.122. For 1 ≤ p < 2, in W0m,p (Ω, Rq ) the bilinear a(uh , v h ) are not even deﬁned, hence 1 ≤ p < 2 has to be excluded. This diﬀerence will be reﬂected in the convergence results in Theorem 4.63. However, we will develop in Section 4.5 a full convergence theory for quasilinear problems in W0m,p (Ω, Rq ) for all cases 1 < p < ∞. However, there is an essential diﬀerence between equations and systems. For q = 1 analytic results for the Banach space setting, G : D(G) ⊂ W0m,p (Ω, Rq ) → W −m,p (Ω, Rq ), are available, see Subsections 2.5.3, 2.5.4, 2.5.6. One of the conditions is uniform monotonicity for the quasilinear operator or its principal part, see (2.289) or (2.294). This is very loosely related to the W0m,p (Ω, Rq )-coercivity of the principal part of G (u0 ), the basic result for the stability of the FEM for the quasilinear systems. We have discussed this relation in Subsections 2.7.3 and 2.7.4. For q > 1 = m analytic results are available only for the Hilbert space setting, G : D(G) ⊂ H0m (Ω, Rq ) → H −m (Ω, Rq ); for q, m > 1 only little seems to be known, see Subsections 2.6.4, 2.6.6. The proof of numerical stability follows our standard lines. Combining the diﬀerent types of coercivity for the principal part with the compact perturbation argument in Theorem 3.29 implies the corresponding stability of the FEM for a boundedly invertible G (u0 ). For this weak form (4.173) it is straightforward to formulate a conforming FEM. As test and approximation spaces Vbh we use the standard continuous FEs vanishing

4.4. FEMs for divergent quasilinear elliptic equations and systems

275

along ∂Ω. Since Vbh ⊂ W01,p (Ω, Rq ), q ≥ 1, we simply replace in (4.173) the u, v ∈ Vb by uh , v h ∈ Vbh . In this conforming FEM we determine, for m, q ≥ 1 : ; uh0 ∈ Vbh s.t. a uh0 , v h = Guh0 , v h V ×V b b := (Aα x, uh0 , ∇uh0 , ∂ α v h )q dx Ω |α|≤m

= f, v V ×Vb = h

b

|α|≤m

fα ∂ α v h dx∀ v h ∈ Vbh ⊂ V = W0m,p (Ω, Rq ), (4.174)

Ω

cf. Theorems 2.63, 2.73, and 2.76. Again, the variational consistency error vanishes by Theorem 4.54. The generalization of Q h ∈ L(V , V h ) in (4.160) and of Lemma 4.56 to the present situation is obvious. This includes the necessary limits of the norms and Summary 4.52 applies. So we mainly need the results guaranteeing the unique existence and regularity of solutions, and the bounded linearization for the original problem, for the many diﬀerent cases included in (4.174). These results are available in the unique existence in Theorems 2.55, 2.57, 2.58, 2.61, 2.62, 2.73, 2.75, 2.76, 2.96, 2.115 or for the regularity of solutions, cf. Theorem 4.57, in Theorems 2.56, 2.60, 2.63, 2.64,2.99, 2.91, 2.101, 2.109, 2.116, and ﬁnally the bounded linearization in Theorems 2.119, 2.122, 2.99, 2.125, 2.126. Similarly we have summarized in Theorems 2.122, 2.124– 2.126 the conditions for W0m,p (Ω, Rq ), the H0m (Ω, Rq )-coercive principal part, and have to modify it as above. The FE solutions, with the same symbol u0 , uh0 ∈ H s (Ω, Rq ), converge according Theorems 4.17, 4.54; compare Theorem 4.57 and (4.164). However, we have to choose the appropriate norms. Theorem 4.63. Unique uh0 converge for quasilinear problems and 2 ≤ p < ∞ : We assume for the diﬀerent cases the conditions in the previous lines, 2 ≤ p < ∞, and a locally unique exact solution u0 ∈ W s,p (Ω), s ≥ m, for the original problems (4.173), with a bounded and boundedly invertible linearization G (u0 ) ∈ L (Vb , Vb ). Again we have to exclude 1 ≤ p < 2. 1. For the conforming FEMs in (4.174), with d ≥ m the variational consistency errors for u0 vanish, Q h Gu0 − Gh uh0 ≡ 0. 2. The principal part Gp (u0 ) of G (u0 ) ∈ L (Vb , Vb ) in Vb = W0m,p (Ω) is H0m (Ω)h coercive, by Theorem 2.122. So the discrete G (u0 ) is stable and consistency and convergence have to be and are formulated with respect to this V = H m (Ω) norm. 3. For G ∈ CL (D(G)), with the Lipschitz constant L for G, and u, I h u, u0 , I h u0 ∈ D(G) the classical consistency or local discretization error is estimated by

(Φh G)(I h u) − Q h GuV = (Φh G)(I h u0 )V ≤ CLI h u − uV .

276

4. Conforming ﬁnite element methods

4. By Theorem 4.54 the FE equations (4.174) are stable and have unique solutions uh0 ∈ Vbh . These uh0 converge as u0 − uh0 m ≤ C dist u0 , Vbh ≤ Chmin{s,d}−m u0 H s (Ω) . (4.175) H (Ω) Proof. G, hence Gh , is continuous near I h u0 . For the classical consistency error we ﬁnd, with Vbh ⊂ Vb , by the mean value Theorem 1.43,

(Φh G)(I h u) − Q h (Gu) = Q h (G(I h u) − G(u))

=Q

1 h

(4.176)

G (u + t(I h u − u))dt(I h u − u)

0

= O(I h u − uV ). The stability, and consequently consistency and convergence, is for 2 ≤ p < ∞, guaranteed by the above conditions for the linearization G (u0 ), see Theorems 3.16 and 3.23, hence the convergence by Theorem 3.21. Pousin and Rappaz [534] study from another point of view consistency, stability, convergence and a priori and a posteriori errors for Petrov-Galerkin methods applied to other nonlinear problems. The conforming FEMs are linear, cf. (4.154). Hence all the quasilinear equations in (4.174) can be solved with the methods in Section 3.7: Continuation methods h h yield good approximations h u1 for u0 . We reformulate the Newton method for the h discrete problem, G u0 = 0, based upon the mesh independence principle in Theorem 3.40 and Corollary 3.41. We start with a moderately good approximation for uh0 obtained, e.g. by continuation. Let the Newton method for the exact problem ui+1 := ui − (G(ui ) )−1 G(ui ), start with u1 . The Newton method for the discrete problem, starting with uh1 := P h u1 , yields for i = 1, . . . , −1 2 Gh uhi , with uh0 − uhi+1 V ≤ C h uh0 − uhi V . (4.177) uhi+1 := uhi − Gh uhi We require the following Lipschitz-continuity for G , G (·) ∈ CL (Br (u0 ) ∪ Br (I h u0 )), G|Vs : D(G) ∩ Vs := H s (Ω) → H s−m (Ω).

(4.178)

Then Theorem 3.40 shows, for the iterates uhi of the Newton method, a locally quadratic convergence, essentially independent of h for small enough h. For u0 ∈ H s (Ω, Rq ), s > m ≥ 1, we obtain convergence of the discrete to the exact iterates of order min{s, d} − m > 0 by (4.175). Theorem 4.64. Newton’s method for the discrete problem and mesh independence principle: In addition to the conditions in Theorem 4.63 we assume (4.178). Start the Newton process for G and Gh in (4.177) with u1 ∈ Br (u0 ) ∩ D(G) ∩ Vs and uh1 := I h u1 . Then, for small enough u0 − u1 V and h, both methods converge quadratically and the sequences ui , uhi satisfy for i = 1, . . . ,

4.5. General convergence theory for monotone operators

277

h ui+1 − uhi ≤ C uhi − uhi−1 2 and uhi − I h ui ≤ Chmin{s,d}−m ui V . s V V V (4.179)

4.5

General convergence theory for monotone and quasilinear operators

The linearization technique for the previous FEMs and all the following methods does not allow convergence results for monotone operators and quasilinear operators in W m,p (Ω) for 1 < p < 2, cf. Subsection 4.3.3 and Lemma 2.77, Section 2.7, and Chapters 4 and 5. However it yields, for 2 ≤ p < ∞, convergence of the expected order, e.g. hmin{s,d}−m , with respect to the discrete H m (Ω) norms for all types of elliptic problems. The conditions for the analytical approaches strongly vary, e.g. the condition of a strongly monotone or an elliptic operator, cf. Chapter 2. On the other hand, the monotone operator techniques yield convergence results for 1 < p < ∞ with respect to discrete W m,p (Ω) norms, with diﬀerent orders of convergence, cf. Theorem 4.67. Strongly monotone operators or quasilinear operators in H 2 (Ω) allow speciﬁc results. So we do not want to miss the chance for embedding the monotone operator approach into our general discretization theory, cf. Chapter 3. We do not aim to give a full proof for this area. Rather we combine the excellent presentation in Zeidler [678], mainly his Theorems 25.B and 26.A, cf. Theorem 2.68, with Chapter 3. Additionally, we formulate the results such that the nonconforming methods, indicated in Summary 4.52, are included. We only formulate results yielding unique existence of discrete solutions, uh0 , their convergence, and converging iteration methods for computation. The basis for the following results is the obvious Proposition 4.65. All approximation spaces U h , V h , in particular U h , V h ⊂ W m,p (Ω) or ⊂ W m,p (T h ), 1 ≤ p < ∞, in this and the next book [120], are reﬂexive Banach spaces.

We need the operators P h : Ub → Ubh and the Q h such that for f ∈ Vb and f ∈ Vb ∪ VT h B A Q h : Vb → Vbh : Q h f − f, v h h h = 0 ∀v h ∈ Vbh , for conforming, and

Q

h

:

Vb

∪

VT h

A

V

→ Vb : Q f − f, v h

h

×V

h

B

V

T

= 0 ∀v h ∈ Vbh nonconf. meth. (4.180)

h ×VT h

In Summary 4.52 we had introduced FEMs by applying a mapping Φh : D(Φh ) ⊂ (U → V ) → (U h → V h ). It transforms Au0 = f or Gu0 = 0 into Ah uh0 = f h or Gh uh0 = 0. For nonconforming FEs, we have to extend the A, G : U → V to the corresponding spaces, UT h , VT h , piecewise deﬁned on the subdivision, T h , cf. (4.35), (4.36), and (5.251) ﬀ., (7.57) ﬀ. extend A, G : U → V into Ah , Gh : U ∪ UT h → V ∪ VT h piecewise with

(4.181)

Ah , Gh |U = A, G, and Ah , Gh = A, G, for conforming FEMs.

278

4. Conforming ﬁnite element methods

These A, G, Ah , Gh , Ah , Gh do not, but the unique solvability does need the boundary conditions. With the previous Qh , Ah , Gh , Ubh , the Φh is deﬁned as

Ah := Φh A := Qh Ah |Ubh , and f h := Φh f := Qh fh |Ubh , hence

(4.182)

Φh (A · −f ) = Qh (Ah · −fh )|Ubh or Gh := Φh G := Qh Gh |Ubh . Now we restrict the discussion to the case Ubh = Vbh .

(4.183)

We modify the results in Subsection 2.5.5 to show that Theorem 2.68 in Chapter 2 applies to Gh . We formulate the properties for Gh : D(Gh ) ⊂ X h = Ubh → Ubh , with X h to emphasize the similarity to (2.242)–(2.253) The deﬁnition of Q h in (4.180) is combined with the essential properties of Deﬁnition 2.65, cf ( 2.241), and the function a : R+ → R+ , a(0) = 0, lim a(t) = ∞, e.g. a(t) = c|t|p−1 , 0 < c, 1 < p. t→∞

(4.184)

So Gh : D(Gh ) ⊂ Ubh → Ubh is an operator, deﬁned on a Banach space, X h = Ubh . The following conditions are imposed ∀uh , v h ∈ X h = Ubh . Then Gh is, cf. (2.242)– (2.253), monotone ⇔ Gh uh − Gh v h , uh − v h ≥ 0, ∀uh , v h ∈ X h = Ubh , . . . , (4.185) strictly mon. ⇔ Gh uh − Gh v h , uh − v h > 0

∀uh = v h ,

(4.186)

uniform. mon. ⇔ Gh uh − Gh v h , uh − v h ≥ a(uh − v h X h )uh − v h X h , (4.187) e.g. ≥ cuh − v h pX h , strongly mon. ⇔ Gh uh − Gh v h , uh − v h ≥ cuh − v h 2X h , c ∈ R+ ,

(4.188)

Gh uh , uh = ∞, →∞ uh X h Xh

(4.189)

coercive ⇔

lim

uh

continuous ⇔ ∀ > 0∃δ > 0 : Gh uh − Gh v h X h < ∀uh − v h X h < δ, (4.190) hemi cont. ⇔ t → Gh (uh + tv h ), w is continuous on [0, 1], stable ⇔ Gh uh − Gh v h X h ≥ a(uh − v h X h ).

(4.191) (4.192)

We claim that the properties (4.185)–(4.192) for the discrete Gh are inherited from (2.242)–(2.253) for the original G. This is a consequence of (3.18), (4.147). Combined with (2.242), we obtain for conforming methods, hence G = Gh , uh , v h ∈ Ubh ∈ Ub

∀uh , v h ∈ Ubh : Gh uh − Gh v h , uh − v h = Q h Guh − Q h Gv h , uh − v h = Guh − Gv h , uh − v h ≥ 0,

(4.193)

with the same relation for all the other (4.185)–(4.192). For extending this argument to nonconforming methods, cf. Deﬁnition 3.5 and Summary 4.52, we distinguish the cases of Ubh ⊂ Ub violating the boundary conditions,

4.5. General convergence theory for monotone operators

279

or U h ⊂ U. The ﬁrst case can be handled by combining Theorem 2.128 with the technique applied in Section 5.2.6. We do not give further details. For U h ⊂ U, we refer to the conditions for the coeﬃcient functions of G, simultaneously of Gh , e.g. in (2.262)–(2.267) and (2.284)–(2.289). Then Theorems 2.71 and 2.73 can be immediately applied to the broken Sobolev spaces, W m,p (T h ) ⊃ U h , cf. (4.35) ﬀ. They yield the desired properties for Gh by Theorem 2.68, e.g. (4.185)–(4.192). The previous ellipticity conditions for the diﬀerent equations and systems of orders 2 and 2m imply strong monotonicity. We summarize this discussion in Theorem 4.66. Existence of (Gh )−1 and converging discrete solutions: 1. The properties of monotony-stability of G are inherited by Gh for conforming and nonconforming methods, cf. (4.185)–(4.192). 2. The previous properties of monotony-stability for G and Gh , (4.185)–(4.192), are valid for equations and systems, where the uh , v h have to be replaced by uh , v h . 3. If Gh is strictly monotone, coercive and hemicontinuous, cf. (4.186), (4.189), (4.191), Theorem 2.68 3., implies the existence of the inverse (Gh )−1 : h operator h h h h h h h U → U and ∀f ∈ U of unique solutions u0 for G u0 = f . 4. If Gh is additionally uniformly and strongly monotone, respectively, cf. (4.187) and (4.188), then the inverse (Gh )−1 is continuous and Lipschitz-continuous. 5. A strongly monotone operator is coercive. Hence, if Gh is strongly monotone and Lipschitz-continuous, then the inverse (Gh )−1 exists and is Lipschitz-continuous. The classical consistency in u0 with Gu0 = 0 implies, see (3.35), (3.42)–(3.43),

Gh P h u0 − Q h Gu0 = Gh P h u0 − Q h 0 = Gh P h u0 → 0 ∀ h → 0, h ∈ H. (4.194) By Theorem 4.66 (Gh )−1 is deﬁned in and close to 0 = Gh uh0 . So we obtain P h u0 − uh0 = (Gh )−1 Gh P h u0 − (Gh )−1 0. For a vanishing consistency error in u0 and a continuous or even Lipschitz-continuous (Gh )−1 this implies the following estimate and theorem: h P u0 − uh0 h → 0 or ≤ LGh (P h u0 ) h → 0 ∀ h → 0, h ∈ H. (4.195) U

Ub

Theorem 4.67. Unique solution uh0 and convergence for all monotone operators and quasilinear equations and systems of order 2m, m ≥ 1, in W0m,p (T h ), 1 < p < ∞, and all discretization methods in this and the next book [120]: Let (2.242)–(2.253) or (2.284)–(2.288) be satisﬁed for X h , G : X h→ X h , consequently for Ubh , Gh : Ubh → Ubh as well. Here U h = X h is a sequence of approximating subspaces for monotone operators and Ubh ⊂ W0m,p (Ω) for quasilinear equations and systems or ⊂ W0m,p (T h ) for the nonconforming cases. Choose any of the conforming or nonconforming discretization methods in this or the next book [120], cf. (4.181), (4.182), yielding Gh for G. 1. Then both G(u) = 0 and Gh (uh ) = 0 have unique solutions u0 and uh0 , respectively. 2. These solutions converge according to limh→0,h∈H P h u0 − uh0 U h = 0 for conb forming methods. For nonconforming methods with m = 1, they converge as

280

4. Conforming ﬁnite element methods

Gh P h u0 U h . These classical consistency errors Gh P h u0 U h are estimated in b b Chapters 5 and 7. 3. If the function a in (4.184) has the form a(t) = c|t|p−1 , 0 < c, 1 < p, then h P u0 − uh0 h ≤ LGh (P h u0 ) h /c 1/(p−1) . (4.196) U U b

b

4. For FEMs of local degree ≥ d − 1, a Lipschitz-continuous G, hence Gh , and u0 ∈ W0d,p (Ω), d > m, cf. (4.175), (4.41), we estimate for conforming methods h P u0 − uh0 h ≤ hd−m Lu0 W d (Ω) 1/(p−1) . (4.197) U b

For the nonconforming methods in Chapters 5 and 7 this holds for m = 1. 5. If, additionally, G is strongly monotone, U h ⊂ U = H0m (Ω), and u0 ∈ H0d (Ω), d > m, we obtain h P u0 − uh0 h ≤ hd−m Lu0 H d (Ω) . (4.198) U 6. The case Ubh ⊂ Ub , violating the boundary conditions, can be handled by combining Theorem 2.128 with the technique applied in Section 5.2.6. Remark 4.68. Quasilinear PDEs cannot yet be evaluated for wavelets, hence wavelet methods are impossible. For all other methods, these results remain valid with slight modiﬁcations as well. For diﬀerence methods this is formulated in Remark 8.27. We ﬁnish with formulating the following generalized gradient iteration method for computing the uh0 , cf. Zeidler [678], Theorem 26.B. For the above

Gh : D(Gh ) ⊂ Ubh → Ubh , Ubh a Hilbert space, and Gh uh0 = 0, uh0 ∈ Ubh deﬁne uhk+1 := uhk − tLG uhk , k = 1, 2, . . . .

(4.199)

The following assumption Gh (0) = 0 makes sense, since otherwise 0 would be the desired solution. Theorem 4.69. Convergent gradient iteration: In (4.199) let Gh (0) = 0, and Gh be uniformly monotone, cf. (4.187), locally Lipschitz-continuous, hence for uh , v h ∈ D(Gh ), ∀r > 0 ∃M (r) : ∀uh U h , v h U h ≤ r : Gh uh − Gh v h U h ≤ M (r)uh − v h U h . b

Furthermore, let L be linear, bounded, self-adjoint, and strongly monotone with (Luh , uh ) ≥ buh 2U h , ∀uh ∈ Ubh , b > 0, and deﬁne the numbers with a in (4.187), F

c := LU h →U h , r0 := 1 + c/b G(0)/a, t0 := 2a/ cM (r0 )2 . b

b

Then for every t ∈ (0, t0 ), the sequence uhk in (4.199) converges to uh0 according to (4.200) lim uhk − uh0 U h = 0 and uhk − uh0 U h ≤ Gh uhk U h /a. k→∞

b

b

b

4.6. Mixed FEMs for Navier–Stokes and saddle point equations

4.6

281

Mixed FEMs for Navier–Stokes and saddle point equations

4.6.1

Navier–Stokes and saddle point equations

In Section 2.8 we presented the necessary theoretical results for the Navier–Stokes and saddle point equations, cf. (2.320)–(2.522) and (2.501). So we repeat here only those aspects necessary for the mixed FEMs in Subsection 4.6.2. This area has collected a huge amount of very substantial work. For examples, the numerical methods for the Navier–Stokes equations are listed 3,731 times in the Zentralblatt in late 2009. So the author has decided to present one of the classical approaches. This shows many of the essential ideas for these methods very clearly. The stationary Navier–Stokes equation has the form, cf. (2.320), ⎞ ⎛ n u ∂ u + grad p −νΔ u + f i i ⎠ ⎝ = in Ω G( u, p) : = i=1 0 div u pdx = 0, Ω ∈ CL1 bounded and p : Ω → R, (4.201) u = 0 on ∂Ω, Ω

u = (u1 , . . . , un ), v = (v1 , . . . , vn ), f = (f1 , . . . , fn ) : Ω ⊂ Rn → Rn , n ≤ 3. The u, p and f denote the velocity, pressure andforcing terms of an incompressn ible medium, v is a test function with div u = i=1 ∂ui /∂xi , and ν = 1/R with the Reynolds number, R. Its linearization in ( u0 ≡ 0, p0 ), applied to an increment ( u, p), u = 0, and for ν = 1, is the Stokes operator, S, cf. (2.496), −Δ u + ∇p f pdx = 0. (4.202) S( u, p) := = in Ω, u = 0 on ∂Ω, −div u 0 Ω For proving convergence for our conforming FEMs, we recall that the variational consistency errors vanish, hence the classical consistency error can be estimated by the interpolation errors. The stability results are obtained by applying the results in Chapter 3, cf. Summary 4.52 to S and its generalization, a saddle point problem, see (4.208). In Theorems 2.129, 4.70, this is shown to be stable, essentially if the well-known Brezzi–Babuska conditions are satisﬁed, see (4.214). We show that, for moderate ν, the linearized Navier–Stokes operator represents a compact perturbation of S, hence satisﬁes the conditions of Theorem 3.29. So we obtain convergence for the stationary Navier–Stokes equations as well. This seems to be the state of the art anyway. We have introduced the spaces and continuous bilinear forms, cf (2.497),(2.498), n Vb = H01 (Ω) , W = L2∗ (Ω) = p ∈ L2 (Ω) : ∫ p(x)dx = 0 , X = Vb × W (4.203) Ω

(∇ u(x), ∇ v (x))n dx with (∇ u(x), ∇ v (x))n =

a( u, v ) = Ω

and b(p, v ) = − Ω

n ∂ui ∂vi ∂x j ∂xj i,j=1

(4.204)

p(x)div v (x)dx for p ∈ L2∗ (Ω), u, v ∈ Vb ,

with bounded |a( u, v )| ≤ Ca uV · v V , |b(p, v )| ≤ Cb v V · pW .

(4.205)

282

4. Conforming ﬁnite element methods

We generalized this weak Stokes form (4.203)–(4.205), cf. (2.500), to a saddle point problem by admitting continuous linear and bilinear forms, f 1 ( v ), f 2 ( v ), a( u, v ), b(p, v ), and Hilbert spaces Vb , W: For continuous a(·, ·) : Vb × Vb → R, b(·, ·) : W × Vb → R, f = (f 1 , f 2 ) : Vb × W → R2 , determine u0 ∈ Vb , p0 ∈ W s.t. a( u0 , v ) + b(p0 , v ) = f 1 ( v )

∀ v ∈ Vb ,

b(q, u0 ) = f 2 (q) ∀ q ∈ W.

(4.206)

To obtain the situation in Chapter 3 and Summary 4.52, we introduce X := Vb × W, x := ( u, p), y := ( v , q) ∈ X and the continuous bilinear form c( x, y ) = a( u, v ) + b(p, v ) + b(q, u). Let A ∈ L (Vb , Vb ) , B ∈ L (W, Vb ) and C ∈ L(X , X ) be the linear operators induced by a(·, ·), b(·, ·) and c(·, ·), respectively. Then, cf (2.502), A B A B A u + Bp u ), C x = C := ∈ L(X , X = p Bd 0 Bd 0 B d u 8 c( x, y ) = C x, y X ×X =

A u + Bp B d u

,

9 v q X ×X

(4.207)

= A u, v Vb ×Vb + Bp, v Vb ×Vb + B d u, qW ×W = a( u, v ) + b(p, v ) + b(q, u) = c( x, y ). For solving the saddle point and the Stokes problem, cf. (2.503) and Chapter 9, determine x0 = ( u0 , p0 ) ∈ X for given f ∈ X such that

(4.208)

c( x0 , y ) = f , y X ×X for all y ∈ X ⇐⇒ C x0 = f ∈ X . In Theorem 2.131 we formulated results for the bounded invertibility of C and the existence, uniqueness and regularity of solutions x0 . 4.6.2

Mixed FEMs for Stokes and saddle point equations

We formulate the corresponding stability, consistency and convergence properties in this subsection, based upon the results in Chapter 3. In the next subsection we will combine these results with Summary 4.52. The C −1 ∈ L(X , X ), and the stability of C h in (4.211) yield the stability for the (linearized) Navier–Stokes operator, cf. Theorems 2.131 and 3.22. By (4.201) the orders of the highest derivatives of u and p are 2 and 1, respectively. This is one of the reasons for choosing the FE spaces approximating u and p diﬀerently. The FEMs are called mixed FEMs. Obviously, approximating spaces h corresponding Vb h∈H , (W h )h∈H for Vb , W yield Vbh × W h h∈H , again approximating Vb × W. They are Petrov–Galerkin approximations for X simultaneously with Vbh and W h for Vb and W. One can show that dim Vbh ≥ dim W h is a necessary criterion for the stability of the discrete operator, cf. [387]. According to [374, 387], we formulate a conforming FEM for (4.206) and (4.208): we choose approximating subspaces Vbh , W h

4.6. Mixed FEMs for Navier–Stokes and saddle point equations

283

and a right-hand side f : h Vb h∈H , (W h )h∈H ∀h ∈ H : dim Vbh ≥ dim W h and Vbh ⊂ Vb , W h ⊂ W (4.209) for f = (f 1 , f 2 ) ∈ Vb × W = X determine xh0 = uh0 , ph0 ∈ Vbh × W h = X h by a uh0 , v h + b ph0 , v h = f 1 ( v h ) ∀ v h ∈ Vbh , (4.210) h h ⇐⇒ b q , u0 = f 2 (q h ) ∀ q h ∈ W h h h c x0 , y = f ( y h ) ∀ y h = ( v h , q h ) ∈ X h ⇐⇒ C h xh0 = f h . (4.211) In Section 2.8 we introduced in (2.507) the closed subspace V0 ⊂ Vb . This V0 and the corresponding discrete spaces V0h are

V0 = { v ∈ Vb : b(q, v ) = 0∀q ∈ W}, V0h = v h ∈ Vbh : b q h , v h = 0∀q h ∈ W h ⊂ V0 . For f 2 ≡ 0 every solution uh0 of (4.209)–(4.211) uh0 ∈ V0h solves a uh0 , v h = f 1 ( v h ) ∀ v h ∈ V0h .

(4.212)

Now v h ∈ V0h ⊂ V0 usually violates div v h = 0. So this form (4.212) is, cf. [386], a nonconforming FEM for u0 ∈ V0 solves a( u0 , v ) = f 1 ( v ) ∀ v ∈ V0 .

(4.213)

This observation is the starting point for intensive research combining mixed FEMs with nonconformity. Here we use (4.209)–(4.211) directly for the conforming FEMs for our saddle point problems. According to Theorem 4.53 we need C −1 ∈ L(X , X ), see Theorem 3.29, for the unique existence of an exact solution, u0 , the consistency, a consequence of Theorem 4.54 and the approximating subspaces Vbh , W h , X h for Vb , W, X , and the stability of C h in (4.211). The latter is related to the Ladyˇzenskaja–Brezzi–Babuska condition. It has the form, [387], Theorem 12.3.6, inf sup |a( uh , v h )| : v h ∈ V0h , v h V = 1 : uh ∈ V0h , uh V = 1 = αh > 0, inf sup |b(q h , v h )| : v h ∈ Vbh , v h V = 1 : q h ∈ W h , q h W = 1 = βh > 0. (4.214) For these saddle point problems, Hackbusch’s Theorem 12.3.6 [386] yields: Theorem 4.70. problems:

Unique discrete solutions and their convergence for saddle point

1. Let Ω ∈ C 0,1 , let the a(·, ·), b(·, ·) and f 1 , f2 , in (4.206) be continuous bilinear and linear forms on Vb , W, choose the approximating spaces (X h )h∈H and the mixed FEM as in (4.209)–(4.211), and let (4.214) be satisﬁed. Then the discrete problem (4.210) is uniquely solvable.

284

4. Conforming ﬁnite element methods

2. If in (4.214) additionally αh ≥ α > 0, βh ≥ β < 0 ∀ h ∈ H, then C h in (4.211) is stable and, with Ch = Ch (α, β, Ca , Cb ) and Ca , Cb in (4.205), 2 2 1/2

1/2 1 2 + f2 2 uh0 + ph0 f ≤ C , and (4.215) h W V V W b h h h h u0 − u0 + p0 − p0 ≤ C inf u0 − v V + inf p0 − q W V

W

q h ∈W h

v h ∈Vbh

e.g. ≤ Ch( u0 H 2 (Ω) + p0 H 1 (Ω) ) by (4.220) for the approximations in Examples 4.71 and 4.72 For the speciﬁc case of the Stokes operator we only present three examples for X h = Vbh × W h , see [386], satisfying the above conditions. For other examples cf. e.g. Brenner and Scott [141], Brezzi and Fortin [144], Grossmann and Roos [374], Girault and Raviart [348, 349] and Temam [622, 623]. Let Ω ⊂ R2 be a polygon, hence Vb = H01 (Ω) × H01 (Ω) and T h a quasiuniform triangulation for Ω, such that, there exists a ﬁxed σ0 > 0 with the quotient h/ dim T ≤ σ0 for every triangle T ∈ T h . For quadrature approximations or curved ∂Ω the modiﬁcations from Sections 5.2– 5.5 can be used. Example 4.71. We deﬁne W h :=

Piecewise linear elements and bubble functions:

q h : ∀ T ∈ T h : q h |T ∈ P1 , Ω

q h dx = 0 ⊂ L2∗ (Ω).

(4.216)

To introduce the bubble functions u on T h let T˜ represent the reference triangle T˜ = {(ξη) : ξ, η > 0, ξ + η < 1} and Φ the bijective aﬃne mapping Φ : T˜ → T. We deﬁne, for every T ∈ T h and the components u of u u ˜(ξ, η) := ξ · η(1 − ξ − η) for (ξ, η) ∈ T˜, and := 0 otherwise,

(4.217) 2 uT (x, y) := u ˜(Φ−1 (x, y)) for (x, y) ∈ T and := 0 otherwise, uT ∈ H01 (Ω) and h := v h ∈ H01 (Ω) : ∀ T ∈ T h : v|hT linear combinations of P1 Vb1 h h and bubble functions on T h , Vbh := Vb1 × Vb1 , X h = Vbh × W h .

Example 4.72. h h and T h : Let Th/2 be deﬁned by replacing each 1. Piecewise linear elements on Th/2 h T ∈ T by four congruent subtriangles. Then h h h h , Vbh := Vb1 Vb1 := v ∈ H01 (Ω) : linear elements on Th/2 × Vb1 , (4.218) W h := q ∈ L2∗ (Ω) : linear elements on T h , X h = Vbh × W h .

4.6. Mixed FEMs for Navier–Stokes and saddle point equations

2. Piecewise quadratic and linear elements on T h : Let h h h Vb,2 := v ∈ H01 (Ω) : quadratic elements on T h , Vbh := Vb,2 × Vb,2 , h W h := q ∈ L2∗ (Ω :) linear elements on T h , X h = Vb,2 × W h. These approximations (4.217)–(4.219) satisfy inf v − v h V : v h ∈ Vbh ≤ hvH 2 (Ω) ∀v ∈ Vb

285

(4.219)

(4.220)

inf{p − ph W : ph ∈ W h } ≤ hpH 1 (Ω) ∀p ∈ W. The following theorem, cf. Hackbusch [387], Theorem 12.3.11, needs an Ω, such that the Poisson problem is H 2 (Ω)-regular. That means, the unique solution, u0 , satisﬁes f vdx∀v ∈ H01 (Ω) (4.221) u0 ∈ H01 (Ω) : a(u0 , v) = (∇u0 , ∇v)n dx = Ω

Ω

and f ∈ L2 (Ω) =⇒ u0 ∈ H 2 (Ω).

(4.222)

By Theorems 2.47, 2.131, this is correct if Ω is convex. We summarize [387], Theorems 12.3.11–12.3.15 and Corollary 12.3.16: Theorem 4.73. problem:

Unique discrete solutions and their convergence for the Stokes

2 1. Let Ω ⊂ R2 be a bounded polygon, hence Ω ∈ C 0,1 , let Vb = H01 (Ω) and T h be a quasiuniform triangulation for Ω and let (4.221) be satisﬁed, e.g. for a convex Ω. Then Examples 4.71, 4.72 satisfy (4.214) with αh > α > 0, β h > β > 0. 2. Then we obtain the following convergence estimates for the discrete solutions ph0 − p0 uh0 , ph0 of (4.210), (4.211), the exact solution of (4.202). For h toward 1 we use the dual norm p0 − p0 (H 1 (Ω)∩W)∗ with respect to W ∩ H (Ω). The convergence is of order 1 or 2, depending upon the smoothness of the solutions: h h u0 − u0 2 p0 − p0 1 + ≤ C ∗ h u0 (H 1 (Ω))2 + p0 L2 (Ω) 2 (L (Ω)) (H (Ω)∩W)∗ (4.223) ≤ C ∗∗ h2 u0 (H 2 (Ω))2 + p0 H 1 (Ω) h h u0 − u0 2 + h p0 − p0 L2 (Ω) ≤ C ∗∗∗ h2 u0 (H 2 (Ω))2 + p0 H 1 (Ω) . (L (Ω))2 Proof. We only show that the additional condition in [387], Theorem 12.3.15, is a consequence of (4.221). In fact, for f 2 ∈ W ∩ H 1 (Ω) we can determine a u1 ∈ W ∩ H 2 (Ω), such that div u1 = f 2 . Then u − u1 satisﬁes f 3 := Δ( u − u1 ) ∈ (L2 (Ω))2 . So we have to discuss for every f − f 3 ∈ (L2 (Ω))2 and f 2 = 0 ∈ W ∩ H 1 (Ω) the Stokes problem Δ u + ∇p = f , div u = 0.0 By (4.221) this has a solution u0 ∈ (H 2 (Ω))2 ∩ 1 2 H0 (Ω) , p0 ∈ H 1 (Ω)2 ∩ W, cf. Theorem 2.131, with u0 (H 2 (Ω))2 + p0 H 1 (Ω) ≤ C f − f 3 (L2 (Ω))2 , and we are done.

286 4.6.3

4. Conforming ﬁnite element methods

Mixed FEMs for the Navier–Stokes operator

Now we discuss the linearized Navier–Stokes operator again for Ω ⊂ Rn and follow [349, 623, 623]. Again we multiply by the test function v = (v1 , . . . , vn ) and use the Green formula to obtain, with a(·, ·), b(·, ·), Vb , W in (4.204) For given (f 1 , 0) ∈ Vb × W determine u0 ∈ Vb , p0 ∈ W such that n −νΔ u0 + ( u0 )i ∂i u0 + grad p0 , v dx Ω

i=1

(4.224)

n

= νa( u0 , v ) + d( u0 , u0 , v ) + b(p0 , v ) = f 1 ( v ) ∀ v ∈ Vb and b(p0 , v ) = 0 ∀ v ∈ Vb ; here n ui (∂i vj )wj dx. d( u, v , w) := i,j=1

Ω

For bounded Ω, and n ≤ 4, see Temam [623] Lemma 1.2, Chapter II, Section 1, d( u, v , w) is a bounded trilinear form on Vb × Vb × Vb . To linearize we consider for ﬁxed u, v and small w d( u + w, u + w, v ) − d( u, u, v ) = d( u, w, v ) + d(w, u, v ) + o(w). An analogous results holds for the nonlinear (4.201) with (4.224): We obtain ⎛ ⎞ n −νΔ w + (u ∂ w + w ∂ u ) + grad r i i i i ⎠ G ( u, p)(w, r) = ⎝ (4.225) i=1 div w and, with the a(·, ·), b(·, ·), d(·, ·) in (4.204) and (4.224). νa(w, v ) + d( u, w, v ) + d(w, u, v ) + b(r, v ) (G ( u, p)(w, r)), ( v , q) (L2 (Ω))2 := b(q, w) ∀( v , q) ∈ Vb × W = X . Now we consider the d( u, w, v ), d(w, u, v ) n ui (∂i wj )vj dx, for u, v , w ∈ Vb d( u, w, v ) = i,j=1

d(w, u, v ) =

(4.227)

Ω

n i,j=1

(4.226)

Ω

wi (∂i uj )vj dx = −

n i,j=1

wi uj ∂i vj dx.

Ω

For ﬁxed u, the d( u, w, v ) + d( u, v ) are continuous bilinear forms, corresponding w, n to the (sum of) operator(s) + wi ∂i u) in (4.225). This is, for ﬁxed u, i,j=1 (ui ∂i w v ) and linear and bounded in v , w ∈ Vb . Then the continuous bilinear forms d( u, w,

4.6. Mixed FEMs for Navier–Stokes and saddle point equations

287

d(w, u, v ) ∈ R deﬁne elements d( u, w, ·), d(w, u, ·) ∈ Vb , hence deﬁne linear continuous operators D1 , D2 ∈ L (Vb , Vb ) as D1 w := d( u, w, ·), D2 w := d(w, u, ·), for u ﬁxed.

(4.228)

By (4.224), note that u is ﬁxed, see (4.227) D1 w, v Vb ×Vb = d( u, w, v ), D2 w, v Vb ×Vb = d(w, u, v ).

(4.229)

Now the embedding I : H01 (Ω) → L2 (Ω) is continuous and compact, cf. Theorem 1.26, and (4.229) shows that D1 v = D1 I v ∀ v ∈ H01 (Ω). Hence, as a product of a compact and a continuous operator, D1 = D1 I is a compact operator. The same is correct for D2 as well. Similarly to the transformation from (4.206) to (4.208) we use here, with the factor ν in (4.201), νA B D1 + D2 0 , (4.230) , D := C := 0 0 Bd 0 and, with slightly diﬀerent x = (w, r), y = ( v , q), + D2 w νAw + Br + D1 w , (C + D) x = Bdw (C + D) x, y X ×X = νAw, v Vb ×Vb + Br, v Vb ×Vb +D1 w + D2 w, v Vb ×Vb + B d w, qW ×W . We formulate for given f ∈ X determine x0 = (w 0 , r0 ) such that (C + D) x0 , y X ×X = f , y X ×X ∀ y = ( v , q) ∈ X or equivalently (C + D) x0 = f ∈ X .

(4.231)

Again Remark 3.30 applies. So we can guarantee the existence of the inverse of C + D if −1 is not an eigenvalue of the compact operator C −1 D. The corresponding mixed FEMs are then deﬁned by straightforward modiﬁcation of the above (4.209)– (4.211) for (4.230)–(4.231). mixed FEM for the nonlinear Navier–Stokes equation requires: Find a pair The uh0 , ph0 ∈ Vbn × W h such that νa uh0 , v h + d uh0 , uh0 , v h + b v h , ph0 = f, v h for all v h ∈ Vbh , (4.232) b uh0 , q h = g, q h for all q h ∈ W h . Combining this with Summary 4.52 we obtain the following.

288

4. Conforming ﬁnite element methods

Theorem 4.74. Convergence of mixed FEMs for the (nonlinearized) Navier–Stokes equations, and solution of the nonlinear systems, cf. Theorems 4.70 and 4.73: 1. For Ω ∈ C 0.1 , the approximating spaces X h = Vbh , W h h∈H and the mixed FEM for (4.230)–(4.231) analogous to (4.209)–(4.211), let (4.214) be satisﬁed, and let G ( u, p) : X → X be boundedly invertible. Then the mixed FE equations (4.230)–(4.231) are uniquely solvable. 2. If additionally αh ≥ α > 0, β h ≥ β > 0, then (4.230)–(4.231) are stable, consistent, and the discrete solutions converge as in Theorem 4.70. 2 3. Let Ω ⊂ R2 be a bounded polygon, hence Ω ∈ C 0,1 , let Vb = H01 (Ω) and T h be a quasiuniform triangulation for Ω and let (4.221) be satisﬁed, e.g. for a convex Ω. Then Examples 4.71, 4.72 satisfy (4.214) with αh > α > 0, β h > β > 0. 4. Then we obtain the following convergence estimates for the discrete solutions uh0 , ph0 of (4.210), (4.211), and of (4.232) toward the exact solution of the Stokes, the linearized Navier–Stokes and the nonlinear Navier–Stokes equations (4.202), (4.231), and (4.224). For ph0 − p0 we use the dual norm ph0 − p0 (H 1 (Ω)∩W)∗ with respect to W ∩ H 1 (Ω). The convergence is of order 1 or 2, depending upon the smoothness of the solutions: h h p0 − p0 1 u0 − u0 2 + ≤ C ∗ h u0 (H 1 (Ω))2 + p0 L2 (Ω) 2 (L (Ω)) (H (Ω)∩W)∗ (4.233) ≤ C ∗∗ h2 u0 (H 2 (Ω))2 + p0 H 1 (Ω) h h u0 − u0 2 + h p0 − p0 L2 (Ω) ≤ C ∗∗∗ h2 u0 (H 2 (Ω))2 + p0 H 1 (Ω) . (L (Ω))2 5. The numerical solution of the high dimensional nonlinear systems in this theorem is eﬃciently computed by combining continuation and discrete Newton’s methods, cf. Theorem 3.40. Proof. We again apply Summary 4.52. The consistency is proved in Theorem 4.54. The stability follows from Theorem 4.57. Hence, Theorem 4.53 implies the unique existence of discrete solutions, their convergence and the quadratic convergence of the discrete Newton’s method, cf. Theorem 3.40. Other types of FEMs based upon nonlinear stability, bubbles, and a posteriori estimates are studied by Brezzi, Hughes, Marini, Houston, Rannacher, Russo, and S¨ uli. [145, 406, 612].

4.7 4.7.1

Variational methods for eigenvalue problems Introduction

Eigenvalue problems play an important role in many scientiﬁc and technical applications. Examples are stability, buckling and breaking, bifurcation of, e.g. mechanical structures, and originating of periodic solutions and vibration, e.g. in reaction– diﬀusion models, and biological, geophysical, ﬂuid dynamical, phenomena, cf. B¨ ohmer [120]. A detailed discussion of important applications, emphasizing linear problems, is presented by e.g., Dautray and Lions [256–261], and nonlinear problems by Zeidler [675–678].

4.7. Variational methods for eigenvalue problems

289

We study only eigenvalue problems for linear operators for two reasons. The numerical methods for nonlinear eigenvalue problems are far less developed than for the linear case. Our main motivation for including eigenvalue problems is the computation of a bifurcation as a starting point for some types of bifurcation and/or evolving dynamical scenarios. If G depends upon a parameter, these scenarios are related to the eigenvalues of its Fr´echet derivative G (u0 ), evaluated at u0 . In this section we formulate the results for eigenvalue problems in this chapter on conforming FEMs, based upon the variational characterization of the eigenvalues. However this remains correct for the diﬀerent variational methods, presented in this book and in [120]. So all the following results remain valid for conforming and nonconforming FEMs, the DCGMs, diﬀerence, wavelet, mesh-free and spectral methods. This is due to the uniﬁed treatment of these methods, based upon their similar structure. In fact Hackbusch’s [387], Chapter 11, proofs for the eigenvalue results remain valid with extremely minor changes. So we only indicate these modiﬁcations and omit the proofs. So we generalize his results from elliptic equations of order 2 and conforming FEMs to equations and systems of order 2 and 2m and all the above methods. For nonconforming FEMs and DCGMs the results are restricted to m = 1. For ﬁnite diﬀerence methods, Hackbusch [387], Chapter 11, employed several technical deﬁnitions and lemmas for proving convergence for one second order equation. Here we only refer to these results in [387], but do not formulate or even generalize them. Partly these results are included or modiﬁed in our variational methods for eigenvalues and diﬀerence methods. There are many interesting papers and books on problems with higher eigenvalues as well. Without trying to present a complete list, we give some examples: Ahues [5], Alimov and Pulatov [7], Arias [38], Aslanov [43], Banerjee and Suri [57], Bramble and Osborn [140], Osborn [521], Parlett et al. [514, 523, 524], Chatelin [167, 168], Babuska and Osborn [46–48], Ciarlet and Lions [177], Davies [262], Delic et al. [279], Dencker et al. [282], Graham et al. [365], J¨ orgens [427], Levendorskij [474], Mertins [488, 489], Saad [562,563], Schechter [570], Sun [614], Vanmaele and Zen´ısek [651], Vanmaele and Van Keer [650] Beattie [76], and Toyonaga et al. [630]. We only consider eigenvalue problems with simple eigenvalues. The study of multiple eigenvalues and their numerical methods is much less advanced than for simple eigenvalues. The main reason for our restriction here is the fact that the numerical study here yields the tools for determining bifurcation. In this context, the invariant subspaces, a generalization of eigenspaces of higher eigenvalues, are important. So this class of problems is delayed to B¨ ohmer [120]. Again we quote some papers and books: Beyn [92], Beyn et al. [93,94], Boˇsek [133],Cliﬀe et al. [179], Dieci and Friedman [287], Govaerts [357], Joshi [429]. Sebastian [579] and Schwarzer [577]. 4.7.2

Theory for eigenvalue problems

The classical EVP for elliptic PDEs is here deﬁned by a linear elliptic operator A of order 2m, m ≥ 1, and its boundary operators Bj as determine λ ∈ C s.t.∃e = 0 : Ae = λe in Ω.Bj e = 0, j = 0, . . . , m − 1, on ∂Ω.

(4.234)

290

4. Conforming ﬁnite element methods

So for a linear operator, A, we determine those nontrivial elements 0 = e ∈ V which are mapped into a multiple of themselves. This A represents either one equation or a system of q equations. For a simpliﬁed notation we usually employ the same symbols e, u, v, . . . for the functions in equations and systems and avoid the e, u, v , . . . form for systems. Similarly, W m,p (Ω) indicates W m,p (Ω) and W m,p (Ω, Rq ) as well. We include the boundary conditions in V, hence ∀v ∈ Vb ⇐⇒ Bj v = 0, j = 0, . . . , m − 1. Again we consider the bounded bilinear form a(·, ·), induced by A as in Section 2.3, and the Gelfand triple Vb ⊂ W ⊂ Vb , cf. (2.62). Then we determine λ ∈ C s.t. ∃0 = e : a(e, v) = λ(e, v)W ∀ v ∈ Vb ⊂ W ⊂ Vb , (4.235) v dx. For discussing eigenvalue problems appropriately, we usually with (u, v)W = Ω u¯ have to complexify V, W, A, cf. Deﬁnition 1.21, Theorem 1.22. The adjoint eigenvalue problem requires ¯ e)W ∀ v ∈ Vb ⊂ W ⊂ V . determine λ ∈ C s.t. ∃0 = e : a(v, e) = λ(v, b

(4.236)

The diﬀerent A and a(·, ·) are introduced for equations of order 2, in (2.16), (2.18), of order 2m in (2.76), (2.88), (2.102), (2.110), for systems of order 2 in (2.339), (2.340), and order 2m in (2.391), (2.392). In Theorems 2.20 and 2.21 we have already introduced the concepts of eigenvalues, ¯ for Ad . Note eigenfunctions, and eigenspaces E(λ) for A and for a(·, ·) , and E d (λ) d ¯ d that we use the notation E (λ) in contrast to E (λ) in Hackbusch [387]. Example 4.75. Chladny sound ﬁgures This nonlinear problem satisﬁes, with a parameter λ, cf. Example 1.3, (1.16), G(u) = Δu + λ sin u = 0 in Ω = [0, L] × [0, L] and

∂u = 0 on ∂Ω. ∂n

(4.237)

We linearize this equation with respect to u at u = 0 and apply G (u) to v, cf. (1.18), obtaining G (0)v = Δv + λv = 0 ⇐⇒ Δv = −λv in Ω with

∂u = 0 on ∂Ω. ∂n

(4.238)

The −λ, allowing nontrivial solutions of (4.238), are the eigenvalues, λ0 = −λ, of the Laplacian with the corresponding eigenfunctions, vm,n , vn,m , the eigenvalues λ0 = −(m2 + n2 )π 2 /L2 ∀ m, n ∈ N0 with the eigenfunctions, n m (4.239) v(x, y) = vm,n (x, y) := cos πx cos πy, and v(x, y) = vn,m (x, y). L L Therefore, we obtain, e.g. for (m, n) = (1, 1), (m, n) = (2, 3) and (m, n) = (5, 5), (m, n) = (1, 7) one, two and three linearly independent solutions vm,n , vn,m for the simple, double and triple eigenvalues λ0 L2 /π 2 = −2, −13, and −50, respectively. The relevant theoretical results for eigenvalue problems in elliptic PDEs are summarized in the Riesz–Schauder theory in Theorem 2.20 and the Fredholm alternative

4.7. Variational methods for eigenvalue problems

291

in Theorem 2.21, the latter formulated for the operator A. Here, we essentially use the bilinear form a(·, ·), and recapitulate the most important results. Theorem 4.76. Alternatives for eigenvalue problems: We assume a Gelfand triple, see (2.62), Vb ⊂ W = W ⊂ Vb with a continuous dense and additionally compact embedding Vb → W, a reﬂexive Banach space Vb and a Hilbert space W. For the bounded elliptic bilinear form a(·, ·) : Vb × Vb → R we get: 1. For every λ ∈ C one of the following alternatives is valid: either the induced (i) (A − λI)−1 ∈ L (Vb , Vb ); or (ii) λ is an eigenvalue of A or of a(·, ·). ¯ is an eigenvalue for (4.236). Then the ﬁnite 2. λ is an eigenvalue for (4.235) ⇐⇒ λ ¯ satisfy dim E(λ) = dim E d (λ) ¯ < ∞. dimensional eigenspaces E(λ) and E d (λ) 3. The spectrum of a(·, ·), or A, the σ(A) := {λ eigenvalue for A}, is at most countable and admits no limit points in C. 4. If a(·, ·) is symmetric, hence a(u, v) = a(v, u)∀u, v ∈ Vb , then the eigenvalues λ = ¯ = E d (λ). ¯ in (4.235) and (4.236) are real and equal and E(λ) = E d (λ) λ From here on, we proceed in line with the presentation in Hackbusch [386, 387]. As indicated above, the reason is the following. His proofs for the results for the eigenvalues of A and their numerical approximation are based upon the structural results for Vb -coercive and Vb -elliptic bilinear forms and regularity results for the solutions of linear elliptic problems. So they remain valid for our more general situation as well. Note that Hackbusch’s deﬁnition of Vb -coercive and Vb -elliptic bilinear forms is chosen according to the originally more analytic version. We have exchanged their role, according to the usual version in FEMs. So we formulate his results for our more general case of Vb -coercive and Vb -elliptic bilinear forms with Vb = W m,p (Ω, Rq ), m ≥ 1, q ≥ 1, 2 ≤ p < ∞, and again exclude 1 ≤ p < 2, cf. Sections 4.4 and 4.5. Instead of repeating Hackbusch’s proofs, we indicate which of his tools (exercises, lemmas, theorems) have to be replaced by our results. We only formulate those results, that are important for the regularity of the exact solutions, and the computation and convergence of the numerical approximations of eigenvalues and eigenfunctions. For Hackbusch’s other results, necessary for his proofs, we only indicate the corresponding replacement of his tools by ours. Note that we only formulate these results for Dirichlet, but not for natural boundary conditions. Correspondingly, our results are only valid for these cases, cf. [386, 387], Theorem 11.1.5: Theorem 4.77. Regular eigenfunctions: Assume Vb = W0m,p (Ω, Rq ), m, q ≥ 1, 2 ≤ p < ∞, and the conditions for regular solutions for diﬀerent types of problems in Theorems 2.28, 2.31, 2.32, 2.33, and 2.38,2.39, and 2.45,2.47,2.49 for equations, and 2.91, 2.92, and 2.108 for systems of order 2m and 2 and 2m, respectively Then for appropriate s > 0 we ﬁnd E(λ) ⊂ W m+s,p (Ω, Rq ), m, q ≥ 1, usually omitting , Rq . Proof. In Hackbusch’s proof we only have to replace his Corollary 9.1.19 by our generalized versions. Hackbusch’s (obvious) proof of his Corollary 9.1.19 uses his

292

4. Conforming ﬁnite element methods

Theorem 9.1.16, which is our Theorem 2.45. Similarly the other regularity results can be employed for proving Theorem 4.77. 4.7.3

Diﬀerent variational methods for eigenvalue problems

The usual approach in variational methods, applied to (4.235) and (4.236), yields determine λh ∈ C s.t. ∃0 = eh ∈ Vbh : ah (eh , v h ) = λh (eh , v h )W ∀ v h ∈ Vbh , ¯h

h∗

determine λ ∈ C s.t. ∃0 = e

∈

Vbh

h∗

¯h

h∗

: a (v , e ) = λ (v , e )W ∀ v ∈ h

h

h

∗

h

(4.240)

Vbh ,

∗

with Vbh ⊂ Vb or Vbh ⊂ Vb and (eh , v h )W , (v h , eh )W deﬁned ∀ eh , eh , v h ∈ Vbh , ¯ h for a symmetric ah (uh , v h ). There is one obvious diﬃculty: The with real λh = λ original eigenvalue problems for elliptic PDEs, (4.235) and (4.236), have inﬁnitely many eigenvalues. Their discrete counterparts, (4.240), represent (generalized) eigenvalue problems of ﬁnite dimension, and thus only have ﬁnitely many eigenvalues. Furthermore, we cannot expect dim E(λ) = dim E(λh ) for an approximation λh ≈ λ. In fact the above Example 4.75 yields for a square and a symmetric diﬀerence method approximate eigenvalues λh with multiplicities diﬀerent from the exact eigenvalues λ. So for simple or double eigenvalues λ we get dim E(λ) = dim E(λh ) = k = 1, 2, with λh , near λ. For a multiple eigenvalue λ with dim E(λ) = k > 2 we ﬁnd λhj ≈ λ, j = 1, . . . , k k ˆ h := k λh /k conwith dim E(λ) = j=1 dim E λhj . Then only the mean value λ j=1 j verges comparably well to λ as for simple eigenvalues, cf. Babuˇska and Aziz [45], p. 338. We start with the conditions in Theorem 4.76, cf. Hackbusch [387], (11.2.2a) ﬀ. We have seen in Sections 4.4, 4.5 that a numerical convergence theory for Vb = W0m,p (Ω), 2 ≤ p < ∞, has to employ the embedding Vb → H m (Ω), and yields convergence with respect to discrete H m (Ω) norms. We assume, Vb = W0m,p (Ω) → H m (Ω), 2 ≤ p < ∞, a(·, ·) : Vb × Vb → R is H0m (Ω)-elliptic, so Vb ⊂ L2 (Ω) is continuously, densely and compactly embedded.

(4.241)

Then all our FE, DCG, diﬀerence, wavelet, mesh-free and spectral approximations introduce sequences of approximating spaces with the corresponding conforming or nonconforming norms Vb approximated by Vbh ⊂ H m (T h ) with the norm v − v h Vbh := v − v h H m (T h ) (4.242) and ∀v ∈ Vb : lim inf v − v h Vbh : v h ∈ Vbh = 0. h→0

We still use the notation Vbh , standard in this chapter, although for nonconforming cases we might have Vbh ⊂ H01 (Ω). Except for conforming FE, diﬀerence and wavelet methods, we have to introduce extensions, ah (·, ·), of the original, a(·, ·), ah (·, ·) : Vb ∪ Vbh × Vb ∪ Vbh → R s.t. a(·, ·) = ah (·, ·)|Vb ×Vb and ah (·, ·) = ah (·, ·)|Vbh ×Vbh .

(4.243)

4.7. Variational methods for eigenvalue problems

293

For conforming FEMs the original a(·, ·) suﬃces, and for diﬀerence methods we directly work with the discrete ah (·, ·). So for conforming FEMs we simply omit the index h, for diﬀerence methods we only discuss the nonobvious modiﬁcations. But for all cases the Vb -ellipticity of a(·, ·) and the Vb -coercivity of its principal part implies the Vbh -ellipticity of ah (·, ·) and the Vbh -coercivity of its principal part with respect to the discrete norms v h Vbh and g h W h . For nonconforming FEMs we have to add c(uh , v h )L2 (ω) , for DCGMs the appropriate penalty terms. These are necessary as well for deﬁning the v h Vbh . Now we are ready to introduce the aλ (·, ·), aλ,h (·, ·), ahλ (·, ·) as aλ (·, ·) : Vb × Vb → C by aλ (u, v) := a(u, v) + λ(u, v)W h aλ (u, v) and ω(λ) := inf sup u∈Vb , uVb =1

(4.244)

v∈Vb , vVb =1

ahλ (·, ·) : Vbh × Vbh → C by ahλ (uh , v h ) := ah (uh , v h ) + λ(uh , v h )W , h h h aλ (u , v ). ω h (λ) := inf sup uh ∈Vbh , uh V h =1 b

v h ∈Vbh , v h V h =1 b

Note (uh , v h )W h is well deﬁned, and = (uh , v h )W , except for diﬀerence methods. Analogously to ahλ (·, ·) we deﬁne aλ,h (·, ·). Then [387], Exercise 11.2.4. remains correct, since his Lemmas 6.5.3 and 6.5.17 correspond to our Theorem 2.12 and Corollary 2.14 with Theorem 2.20 yielding his Lemma 6.5.17. His Theorem 6.5.15, our Theorem 2.20, allows relating the ω(λ) and ω h (λ) as in Remark 11.2.5 and Lemmmas 11.2.6 and 11.2.7. Here Remark 11.2.5 and Lemmma 11.2.6 are rather obvious. For the technical proof of his important Lemmma 11.2.7 we have to keep in mind that his deﬁnition of z h in his (11.2.6b) requires the well-deﬁned ahμ (z h , v h ) = (λ − μ)(u, v h )W ∀v h ∈ Vbh . The reference to his Theorem 8.2.2 following (11.2.6f) has to replaced by our convergence results for the variational methods, mentioned above. His estimate sup |aλ (u − uh , v)| ≤ CS u − uh Vb has to be replaced by sup |aλ,h (u − uh , v)| ≤ CS u − uh Vbh . This last inequality remains correct with aλ,h (u − uh , v) indicating the bounded extension of aλ (·, ·) in (4.243), (4.244), with still coercive aμ,h (uh , v h ), and the consistency of the aλ,h (uh , v h ) with the aλ (u, v). The bounds in (11.2.6a)–(11.2.6f) in [387] have to be slightly modiﬁed. For diﬀerence methods the coercive ahμ (uh , v h ), and the consistency of the ahλ (uh , v h ) with the aλ (u, v) allow maintaining these [387] arguments. Similarly his Theorem 11.2.8, Lemma 11.2.9, Theorems 11.2.10–11.2.11 and Exercise 11.2.12 remain correct. Here the proofs of Theorem 11.2.8, Lemma 11.2.9, and Theorem 11.2.10 are simple generalizations. The proof of Theorem 11.2.11 remains verbatim correct for conforming variational methods. Only the reference to [387], Theorem 8.2.2, after (11.2.6f) has to replaced by our convergence results for the variational methods. For the nonconforming FEMs and the DCGMs the [387] approximate eigenfunctions eh are no longer in Vb . Then convergence results for eigenfunctions have to be formulated with respect to · Vbh . Additionally, the anticrime transformation, E h : U h → U, cf. Lemma 5.77 for nonconforming FEMs and for the DCGMs, have to be employed.

294

4. Conforming ﬁnite element methods

Then the [387] eh are approximated by E h eh ∈ Vb and again allow the proof in [387], and consequently of Exercise 11.2.12 as well. We summarize the qualitative convergence results from the [387], Theorems 11.2.8, 11.2.10, 11.2.11, Exercise 11.2.12 in. Theorem 4.78. Convergence of discrete eigenvalues and eigenfunctions: For all cases we assume (4.241)–(4.242), or the modiﬁcations for conforming FE and diﬀerence methods. 1. Choose, for the sequence of Vbh in (4.242), the approximate eigenvalues λh in (4.240), and assume limh→0 λh = λ0 . Then λ0 is an eigenvalue of (4.235). 2. For λ0 , an eigenvalue of (4.235), there exist discrete eigenvalues λh0 of (4.240) with limh→0 λh = λ0 . 3. For eh ∈ E(λh ) in (4.240) with eh Vbh = 1 and limh→0 λh = λ0 , a subsequence hi exists such that for an eigenfunction e ∈ E(λ) in (4.235), the ehi converge, so limhi →0 ehi − eV hi = 0. b

4. If dim E(λ0 ) = 1, we get limh→0 dim E(λh ) = dim E(λ0 ), for these λh , eh , λ0 , e. 5. For dim E(λ0 ) = 1, choose the eh as in (3). and eˆh := eh /(eh , e)V if (eh , e)V ≥ 1/2 and otherwise eˆh := eh . Then limh→0 eh − eVbh = 0.

We turn to the quantitative convergence results. In [387] the purely analytic results in Lemmas 11.2.13, 11.2.14 and Exercise 11.2.15 obviously remain valid. Lemmas 11.2.16, 11.2.18 and Exercise 11.2.17 remain correct for the conforming method and have to be modiﬁed for the other methods here and below: vVb for v ∈ Vbh and aλ (·, ·) have then to be replaced by vVbh and ahλ (·, ·) or aλ,h (·, ·). In particular for Lemma 11.2.16, the proof of part (1) with its L2 (Ω) arguments remain correct, since Vbh ⊂ L2 (Ω). For all other cases the previous replacements yield the proofs. In particular, for diﬀerence methods the ahλ (·, ·) and the consistency have to be combined. For the next three theorems see [387], Theorems 11.2.19, 11.1.20. Theorem 4.79. Quantitative convergence results for discrete eigenvalues: We assume (4.241)–(4.242), an eigenvalue λ0 for (4.235) with dim E(λ0 ) = 1 and geometric equal algebraic multiplicity of λ0 , cf. [387], Lemma 11.2.13, hence dim N ((A − λ0 I)) = dim N ((A − λ0 I)2 ).

(4.245)

¯ 0 ) can be normed such that (e, e∗ )W = 1. Then span e = E(λ0 ) and span e∗ = E d (λ h Then there exist discrete eigenvalues λ of (4.240), such that |λh − λ0 | ≤ Cdist e, Vbh dist e∗ , Vbh with dist e, Vbh = inf w − eVbh . (4.246) w∈Vbh

Theorem 4.80. Convergence estimates for discrete eigenfunctions: Under the condi∗ d ¯ tions of Theorem 4.79 and e ∈ E(λ hfor the above 0) and e ∈ E (λ0 ) there exist discrete h h h,∗ h,d ¯ h approximations e ∈ E λ0 and e ∈ E λ0 such that e − eh Vbh ≤ Cdist e, Vbh and e∗ − eh,∗ Vbh ≤ Cdist e∗ , Vbh . (4.247)

4.7. Variational methods for eigenvalue problems

295

More detailed convergence results are possible for smooth eigenfunctions. In Theorem 4.77 we have formulated conditions guaranteeing E(λ0 ), E d (λ0 ) ⊂ W m+s,p (Ω, Rq ), m, q ≥ 1 for Vb = W m,p (Ω, Rq ).

(4.248)

Then the error estimates for the diﬀerent methods, e.g. Theorems 4.17, 4.43, 4.36, 7.7, and Lemma 9.7, with appropriate modiﬁcations for mesh-free and spectral methods in B¨ ohmer [120], imply ¯ 0 ). (4.249) dist u, Vbh ≤ Chs uW m+s,p (Ω,Rq ) ∀u ∈ E(λ0 ) ∪ E d (λ Theorem 4.81. Order of convergence for discrete eigenvalues and eigenfunctions: Under the conditions of Theorems 4.79, 4.80 and (4.248), (4.249) there exist λh , eh ∈ ¯ h ) such that E h (λh ) and eh,∗ ∈ E h,d (λ |λh − λ0 | ≤ Ch2s , e − eh Vbh ≤ Chs , e∗ − eh,∗ Vbh ≤ Chs .

(4.250)

For diﬀerence methods this holds for s = 1 and s = 2 for unsymmetric and symmetric formulas. Sometimes the eigenfunctions have better regularity properties than in Theorem 4.77. This allows for m = 1, cf. [387], Theorem 11.2.22, better estimates of the form e − eh Vbh ≤ C h2 , e∗ − eh,∗ Vbh ≤ C h2 . Remark 4.82. These results remain valid with slight modiﬁcations for the other methods as well. In particular, we even might consider cases, where Ubh = Vbh .

5 Nonconforming ﬁnite element methods 5.1

Introduction

In Chapter 4 we discussed conforming FEMs and their strong requirements with respect to ansatz and test functions, exactly evaluated test integrals, and restrictions to weak forms. We avoid this drawback by turning to nonconforming FEMs in this chapter. But then the proofs for consistency and stability become much more complicated. Proceding along the lines of Chapter 3 requires, for nonconforming FEMs, replacing the variational consistency errors by classical consistency errors. The variational consistency error is the usual concept in the ﬁnite element community. It is, for a linear operator A, based upon the bilinear form a(., .), and for a quasilinear operator based upon its weak form and vanishes for conforming FEMs. This contrasts with the classical consistency error, directly related to operators and very appropriate for any nonlinear problems as well. We have to discuss both concepts for the following variational crimes. We start with the most convincing example for this general approach. For two decades FEMs for general fully nonlinear elliptic problems were an open problem. They were, for the ﬁrst time and on the basis of nonconforming FEMs, published by B¨ohmer [118,119]. Before results for special equations of order 2 in R2 in Fulton [330], Oliker and Prussner [519], Dean and Glowinski [272–276], have been known. The latter formulate for Monge–Amp`ere and Pucci equations in R2 equivalent variational forms or use regularization techniques, but see below. Linear or quasilinear elliptic problems in divergence form are studied in the weak form and solved numerically this way. However, fully nonlinear and nondivergent quasilinear elliptic problems do not allow a weak form. So FEMs for these equations and systems, G(u0 ) = 0, are based upon the strong form and are studied in Section 5.2. As a consequence we do need in this chapter the full machinery of discretization methods with G : D(G) ⊂ U → V , Gh : D(Gh ) ⊂ U h → V h and possibly U = V, U h = V h .

(5.0)

Quite recently the author discovered a totally diﬀerent approach in a paper which seems to be forgotten: Barles and Souganidis [73] prove the convergence for diﬀerent discretization methods, based upon the concept of viscosity solutions going back to Crandall and Lions [217–222, 227] and Crandall et al. [215, 216, 228]. [73]

5.1. Introduction

297

gives completely new results concerning the convergence of numerical schemes for stochastic diﬀerential games. This approach is up to now not extendable to higher order equations and systems. For general fully nonlinear elliptic problems, conforming FEMs are not (yet?) possible. In fact, we need new regularity results for the FEM solutions of these Gh (uh0 ) = 0, requiring convex polyhedral domains or Ω ∈ C 1 for m = 1. Thus uh0 ∈ C 1 (Ω) will violate the boundary conditions. So we have to turn to nonconforming FEMs. The standard techniques for handling violated boundary conditions or continuity of the FEMs, cf. Section 5.5, do not apply in this case. Combining very unusual proofs for stability, consistency and Davydov’s FEs in C 1 (Ω), cf. Subsection 4.2.6, we are able to show that for a continuous G with boundedly invertible h linearization G (u0 ), the h h discrete solutions u0 for the nonlinear discretization G u0 = 0 converge to u0 in a discrete H 2 (Ω) norm. We generalize this result to elliptic equations and systems of orders 2 and 2m. Fortunately, the numerical solution methods can be reduced in this case to the well established methods for weak forms. However, the standard FEs in C(Ω) there have to be replaced by FEs in C 1 (Ω) here. In many of the preceding and following cases the integrals in (4.3) and their appropriate generalizations cannot be computed exactly. So quadrature and cubature formulas have to be employed. These approximate FEMs, the next kind of variational crimes, are studied in Section 5.4. The following Section 5.5 is devoted to other types of nonconformity. The exact boundary conditions and/or the continuity required in conforming FEs, are violated here, cf. Feistauer and Zen´ısek [317]. Since the fully nonlinear problems are discussed above and FEs ∈ C(Ω) are excluded for them, only linear and quasilinear equations are left. The convergence proofs for violated boundary conditions and/or the continuity, use, generalizing Brenner and Scott [141], the detour from the weak form to its strong counterpart. For FEMs violating these conditions, there are two options: either we use the approach via quadrature approximations along the boundaries of the subtriangles or polytopes for polynomial FEs. This allows piecewise polynomial FEs of degree d − 1 in Ω ⊂ Rn with n = 2, and converging with high order. For violated boundary conditions the second alternative is isoparametric FEs in Rn , n ≥ 2. For all these cases we extend the previously known results from linear to quasilinear elliptic equations and systems of order 2. As a consequence of the above violated continuity a direct comparison of a(·, ·) and ah (·, ·) is not possible. So, in contrast to [129, 567], we have to prove stability via coercivity of the principal part and estimate the consistency error and ﬁnally combine both to get the desired convergence. Another possibility for violated continuity and boundary conditions, the discontinuous Galerkin methods (DCGMs), combine duality arguments with penalty terms, see e.g. Rachford and Wheeler [540], Douglas and Dupont [303], Wheeler [665], Arnold [39], Riviere et al. [554, 555, 555], Arnold et al. [40]. We present them in Chapter 7 with V. Dolejˇs´ı, including hp- methods. DCGMs eliminate some of the above limitations, but impose others.

298

5. Nonconforming ﬁnite element methods

5.2

Finite element methods for fully nonlinear elliptic problems

5.2.1

Introduction

This section is certainly one of the highlights of this book. We present a FEM for the general case of fully nonlinear elliptic diﬀerential equations of second order in Rn , n ≥ 2 and its extension to order 2m and to systems of orders 2 and 2m. To the knowledge of many specialists, consulted by the author, his papers [118, 119] seem to be the ﬁrst papers solving these general cases, see below. The basis is the coercivity of the principal part of the linearized operator. This is directly available via Legendre and Legendre–Hadamard conditions. So we require these conditions. This does not work for the most general cases of nonlinear systems in the sense of Agman et al. [3]. Here other methods, e.g. as used for the NavierStokes equations, cf. Section 4.6, will have to be developed. We apply our general discretization methods in Chapter 3. The necessary quadrature approximations are studied in Subsubsection 5.4.4 and 5.4.5. For these “nonstandard” semiconforming FEMs (based upon conforming FEs and strong forms of the operator) and (fully) nonconforming FEMs, none of the standard proofs for converegence of FEMs work, see below. So we start giving a short summary of the main ideas behind the formulation and the proofs for our FEMs. For simplifying the presentation we start with one equation of order 2:

r Our FEM for general fully nonlinear elliptic diﬀerential equations requires a r r r

r r

bounded, convex, polyhedron with conforming FEs, or a C 2 domain in Rn with nonconforming C 1 FEs. Both cases are presented. Only FEMs for strong, not for weak, forms make sense. This requires FEs in C 1 , satisfying Dirichlet boundary conditions on polyhedra and violating them on C 2 domains. The classical theory of discretization methods, cf. Chapter 3, has to be applied to the diﬀerential operator and, on C 2 domains, to the diﬀerential and the boundary operator. The most diﬃcult claim, the stability of the nonlinear operator, is proved via the stability of the linearized operator, with coinciding strong and weak bilinear FE forms. Compact perturbation and a new regularity result for FE solutions of weak linear FE equations lift the “weak stability” back to the “strong stability”, cf. Theorem 3.29 and Summary 4.52. For computations the weak linear forms, coinciding with the strong forms, are used. Thus the many known powerful numerical methods apply for our FEM as well. The mesh independence principle (MIP) guarantees a quadratically convergent solution process via the Newton method for highly nonlinear systems of equations.

Our FEM is a natural translation of the nonstandard situation for fully nonlinear elliptic diﬀerential equations, including quasilinear equations not in divergence form; we abbreviate this as nondivergent quasilinear equations. As for fully nonlinear equations, weak forms are not possible. The solution has to be determined from the original strong form with derivatives up to second order. This situation is directly

5.2. FEMs for fully nonlinear elliptic problems

299

translated into our strong form of a FEM; see Subsection 5.2.2 for more details. This FEM, based upon classical discretization theory, strongly diﬀers from the earlier approaches. Diﬀerence methods are studied by Crandall and Lions [217] and Crandall et al. [216], by modifying their concept of exact viscosity solutions. Kuo and Trudinger [456] employ discrete maximum principles. For FEMs, there are only a few papers with results for special equations, e.g. the Monge–Amp`ere equations in R2 or modiﬁcations. Oliker and Prussner [519] prove, by convexity arguments, that their sequences converge monotonically to the solution. Fulton [330] studies multigrid solutions as models of nonlinear balance equations in meteorology. Glowinski with Caﬀarelli, Dean, Guidoboni, Ju´ arez, and Pan [154, 272–276, 352] reformulate it in R2 as a problem of calculus of variations involving the biharmonic (or a related) operator. A reinterpretation as a saddle-point problem for a well-chosen augmented Lagrangian functional leads to iterative methods such as that of Uzawa, Douglas, and Rachford with viscosity solutions. This method can be applied to a related problem, the Pucci equation. Other techniques are based upon operator-splitting methods. Similarly, Feng and Neilan [318] solve the MongeAmp`ere equations and prove convergence of Galerkin methods via a regularization with Δ2 . This yields a fourth order again quasilinear√equation with a solution u 0 and an error estimate, including a term c() u 0 H l hl / . Finally, [155, 230, 671] are loosely related to our problem: Cai et al. [155] present a FEM for fully nonlinear, three-dimensional water surface waves, described by standard potential theory. Their method is based on a transformation of the dynamic water volume onto a ﬁxed domain. The transformation brings about an elliptic boundary value problem. This equation is solved by a preconditioned conjugate gradient method. Numerical experiments indicate convergence of the iterative solver as well as convergence of the entire solution towards a reference solution computed by an another scheme. Extensions to parabolic equations are possible and studied, e.g. by Yang [671] and Cui [230]. The other FE approaches are based upon either a combinations of the special properties of a speciﬁc problem with the proposed numerical method or upon maximum and/or monotonicity arguments. But none of them seems to be applicable nor has been applied to the general fully nonlinear elliptic diﬀerential equations and systems as treated in this paper. This seems to indicate the need for new ideas. 5.2.2

Main ideas and results for the new FEM: An extended summary

We propose an approach totally diﬀerent from those in the above papers. As mentioned, a major part of the analysis for fully nonlinear diﬀerential equations is based upon the strong form of these equations; see, e.g. Gilbarg and Trudinger [346], Showalter [589] and Taylor [618–620]. This motivates a FEM, based upon the strong form of the diﬀerential equation. For proving existence, uniqueness, and related results for the exact solution, e.g. monotonicity, maximum principles, and the Schauder ﬁxed point theorem are needed. We avoid this, but employ, as for many other numerical approaches, these results as the basis for our method. We simplify the presentation by starting with semiconforming and the nonconforming FEMs for second order equations; cf. Subsections 5.2.4, 5.2.5 and 5.2.8 for the

300

5. Nonconforming ﬁnite element methods

general case. We prove convergence in the framework of general discretization theory, based upon consistency and stability, cf. Chapter 3, in a ﬁrst step ﬁtted to the present problem. We need the strong form for the FEM and (see below) a convex, bounded (and we assume additionally) polyhedral Ω or an Ω ∈ C 2 ; both cases are discussed here. The strong form requires C 1 FEs for an L2 evaluation of the diﬀerential equation. For a polyhedral Ω, (trivial) boundary conditions are satisﬁed exactly by the FEs, for curved domains they violate the boundary conditions. For the ﬁrst case and the FEs in C 1 we need only Theorem 4.36, for the second case Theorems 4.35–4.39. Hence, the ﬁrst case allows the full order of convergence in Theorem 4.36. For Ω ∈ C 2 , the proved results in Subsection 4.2.6 yield second order in R2 . Conjecture 4.40, probably proved soon, allows an extension to Rn , those in Conjecture 4.41 higher order as well. So we anticipate Conjecture 4.40 and correspondingly formulate the following results, sometimes hinting at Conjecture 4.41. The second step is the proof of consistency. For polyhedral Ω and their conforming FEs and a continuous diﬀerential operator this is a straightforward consequence of a trivial variational error = 0 and a vanishing interpolation error. For smooth Ω ∈ C 2 , requiring nonconforming FEs, consistency is more complicated. We formulate the results on the basis of Conjecture 4.40 and indicate modiﬁcations by Conjecture 4.41. The earlier approaches for curved boundaries, cf. e.g. Lenoir [471] and Brenner and Scott [142] for linear problems, do not seem to be applicable to fully nonlinear problems. So other nonstandard, not surprisingly similarly complicated considerations are necessary. The clue is an operator with two components, the diﬀerential and the boundary operator. We will use this idea for other problems as well, e.g. for nonlinear boundary operators in Section 5.3. This requires a more complicated version of general discretization theory. Speciﬁcally, we have to carefully introduce the necessary projectors and spaces for including the violated diﬀerential operator and the boundary conditions into the classical discretization theory. Then the “classical consistency error” for the diﬀerential and the boundary operator is estimated in (5.65) and Theorem 5.4 below. Essentially we split this error into the FE error for a semiconforming FEM on an Ωhc ≈ Ω and its complementary geometric error, due to ∂Ω = ∂Ωhc , according to a new idea of Tiihonen [629]. The third step is the most diﬃcult, the proof of stability. We did not see a direct possibility for the strong form. So we combine linearization with regularity of FE solutions. This is another diﬀerence of our FEM to the standard methods and those in the previous papers. Stability is ﬁrst proved for the linearized weak form. Unfortunately, we lose the powerful machinery of weak bilinear forms and their FEMs, unless we ﬁnd a kind of connection between the two forms. This is achieved by Proposition 5.7 and Lemma 5.10. The ﬁrst shows coinciding weak and strong FE bilinear forms in our case. The latter states, similarly to the exact solution of a “smooth” linear elliptic problem, a new regularity and stability result for our FE solutions. This and the available existence, uniqueness, and regularity results require a convex, bounded, polyhedral Ω implying satisﬁed or an Ω ∈ C 2 , implying violated boundary conditions for our FEM. We generalize our earlier compact perturbation techniques, cf. Chapter 3, Theorem 3.29, and B¨ ohmer and Sassmannshausen [128] and B¨ohmer [114, 115], to the above satisﬁed or violated boundary conditions. This allows

5.2. FEMs for fully nonlinear elliptic problems

301

lifting the stability from coercive to general elliptic bilinear forms. A combination of these results lifts the weak back to the strong stability and allows the convergence proof. We do need convex, bounded (polyhedral) or smooth Ω ∈ C 2 for existence, convergence, and the regularity lemma, Lemma 5.10. The smooth Ω ∈ C 2 requires and allows the extension operator in Theorem 4.37. We require FEs in C 1 , since for FEs in C 0 we did not see a possibility to handle the interior variational crimes. The standard interplay between the weak and strong form is impossible for fully nonlinear problems and excludes, e.g. discontinuous Galerkin methods. Thus we choose the unusual C 1 FEs, conforming for polyhedral and nonconforming for C 2 domains. Davydov [264], cf. Subsection 4.2.6, has proved complete approximation results on bounded, polyhedral Ω in Rn . This allows convergence results similarly to the the standard conforming FEMs, cf. Theorem 5.2. For Ω ∈ C 2 the situation changes. We need the new “stable splitting” of C 1 -FEs into interior and boundary elements. Presently, these results in Subsection 4.2.6 are available for modiﬁed Argyris piecewise polynomials of degree 5 on polygonal Ωh ≈ Ω ⊂ R2 . This restricts, for Ω ∈ C 2 , the order of convergence of our FEM to 2. Davydov is extending his results to Rn and to higher polynomial degrees. Our proof is formulated for including the extensions in Conjecture 4.40, and indicating those in Conjecture 4.41. The FE equations have to be set up. They are solved numerically by a quadratically converging discrete Newton method. Its local quadratic convergence is guaranteed by the MIP; see Section 3.7, Subsection 5.2.10 and Allgower and B¨ ohmer [9, 10] and Allgower et al. [17]. It starts with a good approximation, e.g. obtained by a continuation method. Again linear problems are essential to solve the highly nonlinear systems of equations. As a welcome consequence of our coinciding weak and strong FE bilinear forms, the standard solution techniques for weak linear problems are applicable to the fully nonlinear equations as well. However, the standard C 0 FEs have to be replaced by C 1 FEs. With the notation introduced below, let G be a uniformly elliptic operator on Ω, see (5.18), (5.20). Homogeneous Dirichlet conditions are obtained by the standard techniques. Violated boundary conditions for the nonconforming FEM below will play a dominant role. The exact solution, u0 , is determined by the diﬀerential equation and the boundary condition in the trace sense, cf. Subsections 5.2.4, 5.2.5. Mind 4 (Ω), cf. that G(u0 ) ∈ L2 (Ω) is satisﬁed for a smooth solution u0 ∈ Us , e.g. Us = H+ Example 3.26 and Proposition 3.27. Smoothness will be required for good convergence anyway. u0 ∈ D(G) ∩ Us ⊂ H 2 (Ω) : G(u0 )(x) := Gw (x, u0 (x), ∇u0 (x), ∇2 u0 (x)) = 0

(5.1)

∀ a.e. x ∈ Ω, G(u0 ) ∈ L (Ω) ⇐⇒ (G(u0 ), v)L2 (Ω) = 0 ∀v ∈ L (Ω) = (L (Ω)) , 2

2

and u0 |∂Ω = 0 ⇐⇒ (u0 |∂Ω , vb )L2 (∂Ω) = 0 2

2

∀vb ∈ L2 (∂Ω),

with Gw : D(Gw ) ⊂ Ω × R × Rn × Rn → R, Ω, Ωhc ⊂ Ω (see (5.16)). We assume G(uh )(x) := Gw (x, uh (x), ∇uh (x), ∇2 uh (x)) to be deﬁned as well, cf. Proposition 3.27. This condition has to be proved for any speciﬁc G, e.g. the Monge-Ampere equations.

302

5. Nonconforming ﬁnite element methods

a.e. for x, u4 (x), ∇2 u4 (x) ∈ D(Gw ). Since H 3/2 (∂Ω) is dense in L2 (∂Ω) only testing for all vb ∈ L2 (∂Ω) is correct. For a systematic study of the diﬀerential and the boundary operator, we abbreviate (5.1) into an equivalent form, determining u0 ∈ Us ∩ D(G) : 0 = F (u0 ) := (G(u0 )|Ω = 0, u0 |∂Ω = 0) ⇐⇒ (G(u0 ), v)L2 (Ω) + (u0 |∂Ω , vb )L2 (∂Ω) = 0

∀v ∈ L2 (Ω),

(5.2) vb ∈ L2 (∂Ω).

The diﬀerence between quasilinear and fully nonlinear problems for a second order equation is discussed in Subsection 5.2.3. This motivated our new FEM. The standard ellipticity condition for nonlinear systems via linear systems of order 2 has to be modiﬁed for equations and systems of higher order. They have to imply the coercivity of the linearized principal part, see Subsections 5.2.3, 2.4.4, 2.6.3, 2.6.5. There we formulated existence, uniqueness and regularity results as a basis for the corresponding FEMs for fully nonlinear equations and systems of orders 2 and 2m in Subsection 5.2.8. We introduce the C 1 FE spaces of local degree d ≥ n2 + 1 in Rn , the Sd1 Tch , for polyhedral, and for C 2 domains. For a uniﬁed notation we assume that they are deﬁned on a triangulation Tch of the original polyhedral Ω = Ωhc , or a curved Ωhc ≈ Ω ∈ C 2 . In this subsection we use the same notation Tch , Ωhc for all cases; cf. Subsections 4.2.6 and 5.2.5 ﬀ. for S h ⊂ Sd1 Tch : U h := S h ⊂ Sd1 Tch and Ubh := V h := uh ∈ S h : uh |∂Ωhc = 0 ⊂ U h = V h . (5.3) For convex polyhedral Ω, these FEs uh ∈ U0 ⊂ S h ⊂ H 2 (Ω) satisfy the trivial boundary conditions exactly. So they deﬁne a semiconforming FEM. For Ω ∈ C 2 they violate the boundary conditions, so they yield a nonconforming FEM. For convex polyhedral Ω, the FE solution, uh0 ∈ Ubh , is determined by testing only the diﬀerential equation with respect to v h ∈ V h , again in the L2 sense: G : Ubh ∩D(G) → L2 (Ω) determine uh0 ∈ Ubh = V h s.t. G uh0 , v h L2 (Ω) = 0 ∀v h ∈ V h . (5.4) For this semiconforming FEM, uh ∈ Ubh = V h ⊂ H 2 (Ω) ∩ H01 (Ω). 2 h For smooth Ω ∈ C 2 we have to test both operators. For Ω ∈h C , comparing u0 and 2 2 u0 requires extension operators, e.g. Ec : H (Ω) → H Ω ∪ Ωc , cf. Subsection 4.2.6, Theorem 4.37. A simpliﬁed notation is possibel by (realistically) assuming G in (5.1) deﬁned on D(G) ⊂ Us ∩ Uc → Vc with Uc := H 2 Ω ∪ Ωhc , Vc := L2 Ω ∪ Ωhc . (5.5) So the FEM, cf. (5.2), deﬁnes F h , and determines, under the Condition H, see below, the FE solution uh0 ∈ U h , cf. Theorems 4.39, 5.6, from uh0 ∈ U h : 0 = F h uh0 := Gh uh0 , uh0 |∂Ωhc = 0 ⇐⇒ h h G u0 , v L2 (T h ) + uh0 |∂Ωhc , vbh L2 (∂Ωh ) = 0 c

c

∀v h ∈ V h , vbh ∈ Vbh := uh |∂Ωhc : uh ∈ U h .

(5.6)

5.2. FEMs for fully nonlinear elliptic problems

303

Our FEM transforms F (u0 ) = 0 in (5.2) into F h uh0 = 0 in (5.6). It is a classical discretization method in the sense of Chapter 3 and Summary 4.52, cf. e.g. Stetter [596], Stummel [607–609], Vainikko [644], Keller [441], Reinhardt [547], Zeidler [676– 678] and B¨ ohmer [113–115]. In particular, Theorems 3.21, 3.25 provide the tools for showing convergence, if consistency and stability can be proved. Thus Subsections 5.2.4–5.2.7 are the core of the section. The test integrals for nonlinear operators, see (5.6), can be exactly computed only for special cases. So usually they have to be approximated. We extend our results to the necessary quadrature and cubature approximations, see Subsections 5.4.4 and 5.4.5: We prove the corresponding convergence results for quadrature approximated FEMs. In the general case, we will update the H 1 (Ω), H 2 , H01 and C 1 FEs into H m (Ω, Rq ), 2m H , H0m and C 2m−1 FEs, where 2m indicates the order and q the number of equations of the operators. Accordingly, (5.2) and (5.6) have to be generalized. Finally, the boundary condition u|∂Ω = 0 has to be replaced by (∂ k u)/(∂ν)k |∂Ω = 0, k = 0, . . . , m − 1. For our unusual FEM, we summarize the main ideas. Here, we formulate the results for the nonconforming FEM for Ω ∈ C 2 . The corresponding reduction to the semiconforming FEM for polyhedral Ω is obvious, cf. Subsection 5.2.4. 1. The FEMs, transforming (5.1) into (5.4), and (5.2) into (5.6), are general discretization methods:

r The existence, uniqueness, and regularity results for the original problems in Subsection 5.2.3 are the basic results for a general discretization method.

r The fully nonlinear elliptic problem (5.1), (5.2) is approximated by the strong form (5.4), (5.6), requiring FEs in C 1 . For (5.4) identical ansatz and test functions, Ubh = V h , are possible. For (5.6) we choose as ansatz functions U h = S h and test the diﬀerential operator with v h ∈ V h = S h ∩ H01 Ωhc and the boundary operator with the restrictions to the boundary, vbh ∈ Vbh = S h |∂Ωhc ; see (5.3) and Subsection 4.2.6 for the relevant extension results in Theorem 4.37. r The FEM is a general linear discretization method; cf. Chapter 3 and Subsections 5.2.4 and 5.2.5 ﬀ. r In Subsections 5.2.5–5.2.7 we reformulate the FEM, introduce the necessary projectors, and verify the required conditions. Thus the FEMs are convergent if they are stable and consistent; see Theorem 5.2. r For a “smooth” problem, G, the semiconforming FEMs for a polyhedron and the nonconforming FEMs for Ω ∈ C 2 are consistent; see Theorems 4.54 and 5.4. 2. Stability for the nonlinear strong equation (5.4), (5.6), is the most diﬃcult problem; cf. Subsection 5.2.7: r The nonlinear stability is a consequence of the stability for the linearized strong problem, again under the conditions (5.4), (5.5), Condition H, see (5.7) and Theorem 5.6. For given uh0 and f, φ, determine the FE solution uh1 from uh1 ∈ U h s.t. (F h ) uh0 uh1 = G uh0 uh1 , uh1 |∂Ωhc = (f, φ). (5.7)

304

5. Nonconforming ﬁnite element methods

For this linearized problem, two FEMs complement each other; see below:

r The ﬁrst component of the linearized strong and weak bilinear forms coincide: h h h ; : G u0 u , v L2 (Ωh ) = G uh0 uh , v h c

∀ uh ∈ U h ,

v h ∈ V h , (5.8)

with ·, · = ·, ·H −1 (Ωh )×H 1 (Ωh ) ; see Proposition 5.7.

r For (extended) (f, φ) ∈ H −1 Ωhc × H 1/2 ∂Ωhc , the linearized weak problem 0

c

0

c

; : (F h ) uh0 uh1 = (f, φ) ⇔ G uh0 uh1 − f, v h + uh1 − φ |∂Ωhc , vbh L2 (∂Ωh ) = 0 c

∈ is stable, essentially if u0 is an isolated solution of for all v ∈ V F (u0 ) = 0 with boundedly invertible F (u0 ). This is a consequence of the discussion at the end of Subsection 5.2.3 and the compactness arguments in Theorem 5.9. r By the new Lemma 5.10 the regularity results for the exact solution of a linear elliptic problem are valid for the corresponding FE solutions. In the weak linearized problem, again under the conditions (5.4), (5.5), (F h ) uh0 uh1 = (f, φ) ∈ H −1 Ωhc × H 1/2 ∂Ωhc , we determine uh1 ∈ U h for the smooth (f, φ) ∈ L2 Ωhc × H 3/2 ∂Ωhc ; see (5.4), (5.6), (5.8). The stability of the weak problem implies the stability of the strong problem h h G u0 u1 − f, v h L2 (Ωh ) + uh1 − φ |∂Ωhc , vbh L2 (∂Ωh ) = 0 h

h

, vbh

Vbh

c

∀v ∈ V , h

h

vbh

∈

c

Vbh .

This allows estimates for discrete solutions with respect to apiecewise H 2norm 1 2 h 3/2 ∂Ωhc instead of the usual H norm for a right-hand side in L Tc × H h h −1 1/2 Tc × H ∂Ωc . This requires smooth coeﬃcients, instead of the usual H necessary for good convergence, and a convex, bounded, polyhedral Ω, or Ω ∈ C 2. 3. Summary: Essentially simultaneously with a unique u0 , the FE solutions uh0 of (5.4) and (5.6) uniquely exist, for small pnenough h, and converge, under the conditions (5.4), (5.5), and if G : (D(G) ∩ U h ) ⊂ H 2 (Ω ∩ Ωhc ) → L2 (Ω ∩ Ωhc ) is continuous; see (5.9), Condition H, and Theorems 5.2 and 5.13. 4. Setting up the FE equations (5.4), (5.6), and solving them; see Subsection 5.2.10: r Setting up the nonlinear system (5.4), (5.6), is formulated in (5.153). r As a consequence of 1.–2. the MIP is valid. So the Newton method for the discrete equation (5.4), (5.6), with a good enough initial guess uh1 = P h u1 , e.g. obtained by a continuation method, converges quadratically to uh0 equally well as the Newton–Kantorovich method, started in u1 , for the original problem (5.1); see Theorem 5.21 and (5.10). r The linearization of G is elliptic. Thus the eﬃcient solvers for linear elliptic problems in weak form (see (5.7)) can be applied in the continuation and Newton processes, replacing C 0 FEs by C 1 FEs. Results of Davydov [263] and Davydov and Stevenson [268] for Ωh allow extensions to nested sequences of triangulations T1h T2h · · · , hence even multiresolution techniques.

5.2. FEMs for fully nonlinear elliptic problems

305

5. Quadrature approximations for (5.4), (5.6), (5.7), see Subsections 5.4.4, 5.4.5 : r The exact equations (5.4), (5.6), (5.7) usually have to be approximated by good enough quadrature to maintain convergence, see Theorems 5.33 and 5.40. We formulate the convergence of the FE and the discrete Newton method: let (5.1), for smooth enough G and Ω, have a smooth isolated solution u0 ∈ H (Ω) with bounded (F (u0 ))−1 and choose FEs of local degree d in C 1 with 0 ≤ − 1 ≤ d, d ≥ n2 + 1; cf. (4.62). Then the discrete solutions uh0 of (5.6) converge as (see Theorem 5.13) Ec u0 − uh0 h 2 h ≤ Chmin{−2,p} u0 H (Ω) H (Ω )

for large enough , d.

(5.9)

c

For convex, bounded, polyhedral domains and semiconforming FEMs we only need the results from Theorem 4.36 in Rn , cf. Subsection 4.2.6, and we get p = − 2. For Ω ∈ C 2 and the nonconforming FEMs we obtain p = 2 for polyhedral Ωhc in R2 , by Theorem 4.39, in Rn , by Conjecture 4.40 and with p > 2 for future Ωhc by Conjecture 4.41 with appropriate curved boundary approximations. Davydov has fully proved the case R2 , d = 5, p = 2, cf. Subsection 4.2.6, and he is working on higher Rn , p > 2. The proofs in this paper are based upon these generalizations in Conjecture 4.40. The Newton method for the discrete problems (5.4), (5.6) has, for (5.6), the form (5.10) and converges locally quadratically; see Theorem 5.21: uhν+1 ∈ U h : (F h ) uhν uhν+1 − uhν = −F h uhν

2 h h =⇒ uhν+1 − uhν H 2 (Ωh ) ≤ C uhν − uhν−1 H 2 (Ωh ) . (5.10) c

c

Finally, we indicate, but do not explicitly formulate, two further extensions. In Subsection 5.2.7 the linearized weak problem turns out to be a compact perturbation of a coercive bilinear form. Hence, the usual Fredholm alternative results apply. The above FEM can easily be extended to parameter dependent problems and for studying bifurcation scenarios. Some general results prove the convergence of the bifurcation characteristics of the discrete FEM to those of the original problem, starting with Brezzi et al. [147–149], and, e.g. B¨ ohmer [113–115] and Cliﬀe et al. [180]. Extensions of these results to convergence of the characteristics of center manifolds for the discrete FEM to those for the original problem are, for the ﬁrst time, studied in B¨ ohmer [115, 118], cf. [120]. Remark 5.1. Except for DCGMs and wavelet methods these results remain valid with slight modiﬁcations for the other methods as well. 5.2.3

Fully nonlinear and general quasilinear elliptic equations

We recapitulate the standard notation for derivatives, with (−1)j>0 := 1, j = 0, and := −1 for j ≥ 1, ∂iu =

∂u , ∇u = (∂ i u)ni=1 , ∇2 u = (∂ i ∂ j u)ni,j=1 , ∂ 0 u := u. ∂xi

(5.11)

306

5. Nonconforming ﬁnite element methods

We formulate the conditions for the domain and introduce four Hilbert spaces with W0 for trivial Dirichlet boundary conditions: Ω is an (open) bounded domain in Rn with Lipschitz-continuous ∂Ω and

U := H (Ω) ⊂ W := H (Ω) ⊂ V := L (Ω) = V , VD := H 2

1

2

3/2

(∂Ω), WD := H

1/2

(5.12) (∂Ω),

W0 := H01 (Ω) := {v ∈ H 1 (Ω) : v = 0 on ∂Ω}, U0 := W0 ∩ U, W := H −1 (Ω), with the standard Sobolev norms, seminorms, inner products for U, W, V, and Sobolev–Sloboditskii norms for VD , WD ; cf. (1.60), Theorem 1.30 [387, 518]. For a quasilinear elliptic equation in divergence form a weak semiconforming FEM determines the FE solution uh0 ∈ V h = Ubh as n : h h; Gu0 , v W ×W := aj ·, uh0 , ∇uh0 ∂ j v h dx = f, v h W ×W ∀v h ∈ V h . (5.13) Ω j=0

For a general quasilinear or a fully nonlinear G a weak form is impossible: u0 ∈ U0 : Gu0 − f =

n

ai,j (·, u0 , ∇u0 )∂ i ∂ j u0 − f (·) = 0 ∈ V, or

(5.14)

i,j=0

u0 ∈ U0 : G(u0 ) = G(·, u0 (·), ∇u0 (·), ∇2 u0 (·)) = 0 ∈ V.

(5.15)

Consequently, a weak form of a FEM is impossible, thus motivating our FEM. The general theory of discretization methods used in this section, cf. Chapter 3, can be applied if the original problem has two properties: It admits a locally unique solution u0 , strongly related to the stability by the condition of a boundedly invertible G (u0 ), and u0 is smooth enough, implying the consistency and good enough convergence of our method, cf. Condition H. So we refer to the existence and regularity results for fully nonlinear elliptic equations in Subsection 2.5.7. We assume (5.12) for Ω ⊂ Rn , a slightly ¯ ⊂ Ω , Ωh ⊂ Ω , where Ωh indicates later approximations larger open Ω ⊂ Rn with Ω c c h h for Ω for the u ∈ U . A real valued function Gw is deﬁned such that 2

Gw : w = (x, z, p, r) ∈ D(Gw ) = Ω × Γ → R, Γ ⊂ R × Rn × Rn , Ω , Γ open, (5.16) with r restricted to symmetric real valued n × n matrices. We consider the strong elliptic BVP, cf. (5.1), (5.15), (5.16), often with φ = 0 and w(x) := wu (x) := (x, u(x), ∇u(x), ∇2 u(x)), u ∈ D(G) := {u ∈ U : wu (x) ∈ D(Gw )

u → G(u) := Gw (wu ) for (5.17)

∀x ∈ Ω a.e., and G(u) ∈ V},

the solution satisﬁes u0 ∈ D(G) : G(u0 ) = 0 a.e. on Ω,

u0 = φ on ∂Ω. (5.18)

Ellipticity is deﬁned by the linear G (u0 ) in w0 := wu0 applied to u: G (u0 )u =

n n ∂Gw ∂Gw ∂Gw (w0 )u + (w0 )∂ i u + (w0 )∂ i ∂ j u∀u, u0 ∈ D(G). ∂z ∂p ∂r i ij i=1 i,j=1

(5.19)

5.2. FEMs for fully nonlinear elliptic problems

307

This will be important below for local uniqueness and stability. The operator G is called uniformly elliptic in u0 ∈ D(G) and all of D(G) if its linearization G (u) is uniformly elliptic for u = u0 and for all u ∈ D(G), respectively. In the ﬁrst case its principal part, ap (u, v) (see (5.21)), satisﬁes, for 0 < λ < Λ, and w0 = wu0 in (5.17), n ∂Gw (w0 )ϑi ϑj ≤ Λ|ϑ|2 0 < λ|ϑ| ≤ − ∂r ij i,j=1 2

∀0 = ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn ,

(5.20)

sometimes only for a.e. x ∈ Ω in w0 (x). For fully nonlinear second order elliptic equations for the smooth u, u0 ∈ D(G) this implies the W0 -coercivity of the bilinear form induced by the principal part, ap (u, v) of G (u0 ). By the standard ϑi = ∂ i u(x), integrating over Ω, and nusing partial integration we obtain, with the equivalence in W0 of the energy norm ( i,j=1 ∂ i u2V )1/2 and · H 1 (Ω) and again with 0 < λ1 < Λ1 ∈ R, n ∂Gw (w0 )∂ i u∂ j udx ≤ Λ1 u2H 1 (Ω) ∀u ∈ W0 . λ1 u2H 1 (Ω) ≤ ap (u, u) := Ω i,j=1 ∂rij (5.21) Therefore we choose the Hilbert space setting, W = H 1 (Ω). A Banach space setting, W 1,p (Ω), might be appropriate if in addition to (5.20) some monotonicity conditions were considered. Furthermore, since W 2,p (Ω) is continuously embedded into H 2 (Ω), for p ≥ 2, our new FEM even works for fully nonlinear equations in W 2,p (Ω), e.g. G(u) = (Δu)p + f (u); however it yields only discrete H 2 (Ω) estimates. In Example 2.78, some of the most important examples are listed, the Monge– Amp`ere equation and the equation for a surface with prescribed Gauss curvature, see (2.308), (2.310). The existence of surfaces with prescribed Gauss curvature is known for appropriate given curvature. Speciﬁc properties would become more ﬂexibly available via our FEMs than by our ﬁnite diﬀerence methods in Chapter 8, and e.g. Crandall and Lions [227] and Barles and Souganidis [73]: our FEM allows C 1 FEs on quasiuniform grids and yields, for the forthcoming better boundary approximations, higher orders of convergence than the diﬀerence methods in [227], which are only deﬁned on equidistant grids. We have previously observed that we need the following conditions H: Condition H: Essential conditions for well-deﬁned and diﬀerentiable G: These very important properties (5.1), (5.4), (5.5) have to be discussed in more detail. We have realized that for the Monge-Ampere equation, cf. Example 3.26, the G(u) is not deﬁned for u ∈ U, but only for u ∈ D(G), a subset of smooth functions u ∈ Us ﬁtting to the growth properties of G: Smooth solutions u0 are necessary anyways for good convergence. For the Monge-Ampere equation Proposition 3.27 shows that G(uh ) ∈ L2 (Ω), easily extended to L2 (Ω ∪ Ωhc ). More generally, we require, cf. (5.5) (H1 ) Assume G in (5.1) is well deﬁned on G : D(G) ⊂ Us ⊂ Uc = H 2 (Ω ∪ Ωhc ) → Vc = L2 (Ω ∪ Ωhc ), so G(uh ) is well deﬁned.

308

5. Nonconforming ﬁnite element methods

(H2 ) Below, G (u0 ) is important for stability. So we will assume G : Br (u0 ) ⊂ D(G) ⊂ Us → L2 (Ω ∪ Ωhc ), G ∈ C 1 (Br (u0 )) w.r.t. the · Us , s.t. ∀u ∈ Br (u0 ) : G (u) ∈ L(H 2 (Ω), L2 (Ω)), and can be extended as ∈ L(Uc , Vc ) and ∈ L(Wc , Wc ), e.g. by imposing (5.19)–(5.21), or in more detail (5.73)–(5.75). These properties have to be veriﬁed for each combination of G, U, V, U h , V h . Existence, uniqueness and regularity results are summarized in Theorems 2.79, 2.80, 2.81, and 2.83. They are obtained by methods, usually independent of linearization. However, preparing the next subsections, we assume additionally smooth D(G), compare Condition H for F : D(G) ⊂ U → V × VD , in (5.2), with a locally unique solution u0 of F (u0 ) = (0, φ), let F (u0 ) ∈ L(U, V × VD ) be boundedly invertible.

(5.22)

For systems of q fully nonlinear equations of order 2m of the form (5.111), (5.112) below, nearly nothing seems to be known, for the order 2 some results exist, see Subsection 2.6.8. Now, a combination of our FEMs below and continuation techniques allows studying this class of problems. 5.2.4

Existence and convergence for semiconforming FEMs

In Subsection 5.2.2 we have seen that the standard FEM approaches, based on weak forms, do not seem to be possible for the general case of fully nonlinear elliptic equations. We formulated the new FEM for fully nonlinear equations of second order with trivial Dirichlet boundary conditions, see (5.4), (5.6) and introduced the necessary approximating FE spaces in Subsection 4.2.6. We apply the general theory of discretization methods, cf. Chapter 3 and Summary 4.52, with its standard “consistency and stability implies convergence” result to our FEMs, here the semiconforming and the nonconforming FEMs in the next subsection. Semiconforming FEMs are based upon strong forms and conforming FEs. We use the notation in (5.11), (5.16), and (5.17). Let (5.18) have a locally unique solution (see Subsection 5.2.3) with boundedly invertible derivative G (u0 ). We recall that our program for proving convergence requires several steps, summarized in Subsection 5.2.2 1.–4.: Until now we have worked through the ﬁrst two bullets of 1., and we turn, in this and the next subsection, to the last three bullets of 1. The conforming FEs, deﬁned on a triangulation with straight edges, T h , require ¯ = ∪T ∈T h T¯. Ω convex, bounded and polyhedral with Ω

(5.23)

In Subsection 5.2.5 we will return to nonconforming FEs, deﬁned on Ω ∈ C 2 . This yields the more complicated case of violated diﬀerential and boundary equations. The original problem and its semiconforming discretization We need the following spaces, note no Gelfand triple: U = H 2 ⊂ V = V = L2 ,

U0 = U ∩ H01 all deﬁned on Ω.

(5.24)

5.2. FEMs for fully nonlinear elliptic problems

309

G and its exact solution u0 ∈ D(G) ∩ U0 satisfy, cf. Condition H G : D(G) ⊂ U → V and G(u0 ) = 0 ∈ V ⇐⇒ (G(u0 ), v)V = 0 ∀v ∈ V.

(5.25)

h

We choose the discrete G for G as in (5.4). The sequences of approximating subspaces U h = V h = S0h ⊂ H 2 (Ω) = U in (5.3), deﬁned on the original Ω in (5.23), satisfy the boundary conditions on the exact boundary. So we call this FEM a semiconforming FEM. More precisely we need

{U h , V h , P h , Qh , Gh }h∈H , inf{0 < h ∈ H} = 0, Sd1 (T h ) ⊃ U h = V h = S0h .

(5.26)

For these spaces with identical elements we require (asymptotically) nonequivalent norms and scalar products, deﬁned as those of the original spaces, cf. (5.24), e.g. uh U h := uh U , v h V h := v h V .

(5.27)

So we might call this method a Petrov–Galerkin method as well. By Theorem 4.36 these sequences approximate the Banach spaces U, V. For the ﬁnal FE version of (5.4) we need Gw , wuh (x), D(Gw ) in (5.16), (5.18), (5.25): D(Gh ) := {uh ∈ U h ∩ D(G) : wuh (x) ∈ D(Gw )

∀x a.e. in Ω, G(uh ) ∈ V}. (5.28)

So we deﬁne the FE solution uh0 ∈ U h ∩ D(Gh ) ⊂ U such that Gh : D(Gh ) → V h , and Gh uh0 = 0 ⇐⇒ G uh0 , v h V = 0 ∀v h ∈ V h ⊂ V. (5.29) The necessary projectors Hence, we introduce sequences of projectors, {P h , Qh }h∈H , as appropriate tools for our FEM. The P h have to satisfy, cf. Theorem 4.36, P h ∈ L(U, U h ) : ∀u ∈ U lim P h uU h = uU , we choose P h u := I h u. h→0

(5.30)

Here we have chosen I h in (4.63) as in Theorem 4.36, cf. Davydov [264]. Hence, interpolating values of functions or derivatives or other kinds of quasi- or averaginginterpolation or best approximation operators could be used, sometimes called Cl´ement interpolation; cf. e.g. Cl´ement [178], de Boor [270], Scott and Zhang [578]. the Qh ∈ L(V, V h ), More delicate than P h are the projectors to the image spaces, motivated by Gh . We test G(u0 ) = 0 as in (5.25) and Gh uh0 = 0 in (5.29). We keep the concepts for the original and the discrete problem as close as possible and use the function spaces and norms in (5.24), (5.26), (5.27). In (5.25) and (5.29) we have satisﬁed the diﬀerential equation and its discrete counterpart by testing with V and V h . We deﬁne, 35 cf. (4.73)–(4.75) and Theorem 4.38, for all f ∈ V = V ,

Qh ∈ L(V, V h ) : Qh f − f, v h (V h ∪V )×V h = 0 ∀v h ∈ V h ; lim Qh f V h = f V . h→0

(5.31) 35 Systematically we use the following identiﬁcation: For U ⊂ V = V , for all u ∈ U , for all v ∈ V ⊂ U , and the corresponding discrete terms, we obtain (u, v)V = u, v U×U = u, v U ×U , similarly for , . . . , and W ⊂ V = V ⊂ W below. This is applied for the following projectors VD ⊂ Vb = Vb ⊂ VD h Q , . . . as well.

310

5. Nonconforming ﬁnite element methods U Ph Uh

G Φh Uh

V

tested by

V = V′

tested by

V h ≠V h ′

Qh ′ Vh′

Figure 5.1 Semiconforming FEMs for fully nonlinear problems.

The last limit is obtained as a consequence of (4.64). By (4.82), (4.83) with V h = V h it makes sense that for a function, e.g. f ∈ V or G(uh ) ∈ V h , the Qh f ∈ V h is not a h h function. We did not identify V = V for emphasizing this situation more clearly. The choice of P h , Qh , for the given U h , V h , and problem (5.25) certainly is not unique. The combination U, V, U h , V h , P h , Qh , Gh should be chosen appropriately to yield classical consistency (see below) with the highest possible order; see (4.150). Projectors and discretization For the nonlinear operator, G : D(G) ⊂ U → V, in (5.25), the previously introduced projectors P h , Qh allow a shorthand notation of the corresponding discrete operator, h h h G : D(G ) ⊂ U → V h ; see (5.29). So we determine the approximate solution uh0 ∈ U h such that : h h; G u0 , v V×V h = 0 ∀v h ∈ V h ⇐⇒ Gh uh0 = 0 with Gh := Qh G|U h . (5.32) This brings about a mapping

D(Φh ) := {G in (5.25)} and Φh deﬁned by Φh (G) := Gh := Qh G|U h .

(5.33)

h

This Φ is even linear in the sense Φh (c1 G1 + c2 G2 ) = c1 Φh (G1 ) + c2 Φh (G2 )

∀c1 , c2 ∈ R, ∀G1 , G2 ∈ D(Φh ). (5.34)

Linearity is essential for the MIP in Subsection 5.2.10, allowing the proof of a quadratically convergent discrete Newton method. Our general discretization theory in Chapter 3, in contrast to Stetter [596] and Vainikko [644], emphasizes the testing with the V, V h : summarizing yields Figure 5.1 with uniformly bounded P h , Qh . In Summary 4.52 we have recalled discretization methods for FEMs for G(u) = 0. Essential are consistency, consistency of order k and stability in u0 . Our semiconforming FEM is consistent for a continuous or Lipschitz-continuous G, by Theorem 4.54. For G ∈ CL , the consistency order is the same as the order of the interpolation error, I h u0 − u0 H 2 (Ω) , cf. Theorem 4.36, (4.66). It is nearly obvious how to simplify the highly technical proof for the stability of the nonconforming F h uh0 in Subsection 5.2.7. So we refer to this modiﬁcation yielding stability for the semiconforming Gh uh0 in Theorem 5.12. With Theorem 3.21, we get the following Theorem: Theorem 5.2. Existence and convergence for the semiconforming FEM for convex polyhedral domains Ω in Rn : Let the original problem (5.25) be deﬁned on Ω in (5.23), satisfy Condition H and (5.22) and have the exact isolated solution u0 ∈ H (Ω), − 1 ≤ d with a boundedly invertible G (u0 ). Let G be Lipschitz-continuous

5.2. FEMs for fully nonlinear elliptic problems

311

in Br (u0 ) ∩ D(G) as G(u) − G(v) ≤ Lu − vU , recall U = H 2 , V = L2 (Ω). Then its discretization Gh = Qh G|U h : U h → V h in (5.32) with Sd1 (T h ) ⊃ U h = V h = S0h in (5.26) has the following properties:

1. The projectors P h and Qh are (equi-)bounded and satisfy (5.30) and (5.31). 2. Gh : U h → V h is deﬁned and continuous in Br (P h u0 ), r > 0, h-independent. 3. Gh is consistent with G in P h u0 , so, by (4.66),

Gh P h u0 − Qh Gu0 V h ≤ Lu − I h uH 2 (Ω) ≤ KLh−2 |u|H (Ω) . h

(5.35)

h

4. G is stable for P u0 . Therefore the discrete problem Gh (uh ) = 0 possesses a unique solution uh0 ∈ U h near u0 for all suﬃciently small h ∈ H, and uh0 converges to u0 , according to h u0 − P h u0 h ≤ SKLh−2 |u|H (Ω) . (5.36) U

We will come back to these conditions 1.–4. in Theorem 5.2 in Remark 5.3 below. 5.2.5

Deﬁnition of nonconforming FEMs

Now we return to nonconforming FEs. We choose a triangulation, Tch , possibly with curved edges near ∂Ω and require and deﬁne, for Ω cf. (5.16), Ω ∈ C 2 , a triangulation, Tch , with curved edges, s.t. Ωhc := ∪T ∈Tch T (5.37) and a sequence of open Ωh0 s.t. Ω ⊃ Ωh0 ⊃ Ωhc ∩ Ω and dist ∂Ωh0 , ∂Ω ≤ Chp , with p = 2 for the situation in Conjecture 4.40, and possibly p ≥ 2 for Conjecture 4.41. As a consequence of Theorems 4.37 and 4.38 for the U, W, V = V , G, B, G , . . . extensions from Ω to Ωh0 are possible. The following estimates then have to be multiplied by the constant C in Theorems 4.37 and 4.38, C independent of h. We maintain the notation U, W, V = V , G, B, G , . . . deﬁned on Ω, and only refer to Ωh0 if necessary. Again, the ﬁrst two paragraphs in Subsection 5.2.4 remain valid. The U, W, V = V , deﬁned on the original Ω, are now approximated by sequences of nonconforming subspaces U h , W h , V h , deﬁned on the approximate Ωhc . This implies violated boundary conditions on the exact boundary; cf. (5.38). The FEM changes into a so-called nonconforming Petrov–Galerkin method, requiring a more complicated treatment than in Subsection 5.2.4. The original problem and its nonconforming discretization We need, cf. (5.24), U = H 2 ⊂ W = H 1 ⊂ V = V = L2 ⊂ W = H −1 , W0 = H01 all deﬁned on Ω (5.38) 1 and Uc = Hc2 ⊂ Wc = Hc1 ⊂ Vc = L2c ⊂ Wc = Hc−1 , Wc,0 = Hc,0 all deﬁned on Ωhc .

We have indicated above that the realization of the diﬀerential equation and the boundary condition and their violation is a deciding step of this method. Hence, the discretization method has to care about both aspects. This is achieved by the new operator F = (G, BD ), with the diﬀerential operator and the boundary operator as

312

5. Nonconforming ﬁnite element methods

its two components; see (5.1), (5.2). The boundary operator, BD , maps, by the Trace Theorem 1.38, BD : U → VD := H 3/2 (∂Ω) = BD U, BD u := u|∂Ω , VD ⊂ Vb = Vb := L2 (∂Ω). (5.39) We consider here mainly trivial Dirichlet conditions. Thus we have to characterize u|∂Ω = 0 ⇔ v , u|∂Ω V

D ×VD

= 0 ∀v ∈ VD ⇔ (u|∂Ω , v)L2 (∂Ω) = 0 ∀v ∈ Vb

(5.40)

, a dense (compact) embedding. for u ∈ U, with VD = H 3/2 (∂Ω) → Vb = Vb → VD The transformation from nontrivial to trivial Dirichlet conditions will be combined in Theorems 5.4, 5.9, and 5.13 obtaining stability and convergence for the inhomogeneous case as well. F = (G, BD ) is called elliptic simultaneously with G, cf. (5.20). F and its Fr´echet derivative F (u) map D(G) and U into and onto, respectively, the Cartesian product V × VD : ; F := (F1 , F2 ) := (G, BD ) : D(G) ⊂ U → VΠ := V × VD ⊂ VˆΠ := V × Vb = VˆΠ

(5.41)

see (5.17). Since VD is dense in Vb , the smooth exact solution u0 is hence characterized by F (u0 ) = 0 ∈ VΠ ⇐⇒ (F (u0 ), vΠ )VˆΠ = 0 ∀vΠ := (v, vb ) ∈ VˆΠ

(5.42)

⇐⇒ (G(u0 ), v)V = 0 ∀v ∈ V, and (BD u0 , vb )Vb = 0 ∀vb ∈ Vb , abbreviated ⇐⇒ (G(u0 ), v)V + (BD u0 , vb )Vb = 0 ∀v ∈ V,

vb ∈ Vb .

For the standard weak form for F (u) in Subsection 5.2.7, we need, cf. (5.39), (5.40) BD : W → WD := H 1/2 (∂Ω) = BD W, u ∈ W and BD u = 0 ⇐⇒ (u|∂Ω , v)L2 (∂Ω) = 0

u → BD u = u|∂Ω , with

(5.43)

∀v ∈ Vb dense in WD .

BD : U → VD and BD : W → WD are continuous with respect to the corresponding Sobolev– Sloboditskii norms, cf. (1.60), Theorem 1.30. Similarly to (5.38), we introduce Vc,D := H 3/2 ∂Ωhc ⊂ Wc,D := H 1/2 ∂Ωhc ⊂ Vc,b := L2 ∂Ωhc and (5.44) Bc,D : Uc → Vc,D , uc → Bc,D uc := uc |∂Ωhc ; similarly, Bc,D : Wc → Wc,D . To apply the general discretization theory in Chapter 3, we introduce the discrete h ), F h for F , and the corresponding projectors P h ∈ L(U, U h ), and QhΠ ∈ L(VΠ , VΠ h h Qc,Π ∈ L(Vc,Π , VΠ ); see (5.62). This yields the standard “consistency and stability implies convergence” result, (5.37); note S0h = S h ∩ H01 (Tch ) and the stable splitting in Theorem 4.36: h h = WD = Vbh := Sbh |∂Ωhc . Sd1 Tch ⊃ U h = W h = S h = S0h ⊕ Sbh , V h = S0h , VD (5.45)

5.2. FEMs for fully nonlinear elliptic problems

313

Again these spaces require nonequivalent norms and scalar products. With the norms of the original spaces, now deﬁned on Ωhc and ∂Ωhc , cf. (5.38), (5.44), let uh U h := uh Uc , wh W h := wh Wc , v h V h := v h Vc ,uh |∂Ωhc V h := uh |∂Ωhc V , c,D D h h h h w |∂Ωh h := w |∂Ωh , v |∂Ωhc V h := v |∂Ωhc V . (5.46) c W c W D

c,D

b

c,b

By Theorems 4.36–4.38 these sequences approximate the Banach spaces and subsets, cf. (4.73), Ec U, Ec W, Ec V, Ec VD , Ec WD , Ec Vb , Ec D(G) deﬁned on Ωhc and ∂Ωhc with respect to the corresponding norms. The uh ∈ S h , and v h ∈ Vbh , violate the original boundary conditions. Since Ec maintains smoothness properties, we can choose this Ec s.t. (5.5) is maintained to these Ec U, Ec W, . . . . The product operator F : D(G) ⊂ U → V × VD and the corresponding F h : D(G) ∩ h h cause some unpleasant technicalities in the deﬁnition of QhΠ , Qhc,Π , U → V h × VD compared to Subsection 5.2.4. Subsequently, the consistency theory becomes relatively simple again; see Figure 5.2 and Subsection 5.2.6. These technicalities are not surprising compared to some other approaches for violated boundary conditions, e.g. in Lenoir [471] and Bernadou [85–88]. The unavoidable complications are piled up in Subsection 5.2.7. The terms uh , G(uh ), and uh |∂Ωhc are, by (5.17), deﬁned on Ωhc and ∂Ωhc instead of the original Ω and ∂Ω. Thus we start updating the preliminary FE version (5.6). With Gw , D(Gw ), wuh (x) in (5.16), (5.17), (5.18), and U h in (5.45), let D(Gh ) := uh ∈ U h : wuh (x) ∈ D(Gw ) ∀x ∈ Ω ∪ Ωhc , G(uh ) := Gw (wuh ) ∈ Vc . (5.47) This condition has to be veriﬁed similarly to Proposition 3.27, cf. Condition H. The extended deﬁnition of Gw was the reason for introducing the Ω in (5.16). We replace the diﬀerential equation G(u0 ) = 0 ∈ V, u0 ∈ D(G), in (5.42) by the FE equations in equivalent formulations; see (5.6), (5.41), (5.46): deﬁne the sequence F1h uh0 = Gh uh0 = 0 ⇐⇒ G uh0 , v h V = 0 ∀v h ∈ V h ⊂ Vc . c

(5.48) Similarly, the boundary conditions (5.40) are tested (see the equivalence (5.6)) as h h h u = uh |∂Ωhc = 0 ∈ VD ⇔ vbh , uh V h = 0 ∀vbh ∈ Sbh , uh ∈ U h : F2h (uh ) = BD b (5.49) with ∪h∈H Sbh dense in Vc,D . Consequently we introduce, cf. F in (5.41), h with D(F h ) := D(Gh ). F h = F1h , F2h = Gh , BD (5.50) The nonconforming projectors Again, projectors represent an appropriate tool for F h = F1h , F2h . They have to handle the discrepancy between uh , G(uh ), and uh |∂Ωhc , by (5.17), deﬁned on Ωhc and ∂Ωhc instead of the original Ω and ∂Ω. We combine the extension operator Ec introduced in Theorem 4.37 with inverse estimates and the estimates for

314

5. Nonconforming ﬁnite element methods

interpolation errors on Ω in Theorems 4.36 and 4.38. The sequence of P h has to satisfy P h ∈ L(U, U h ) : ∀u ∈ U lim P h uU h = uU , we choose P h u := I h Ec u. (5.51) h→0

Ec is unnecessary for the Argyris FEs but possibly needed for other schemes. Again more delicate than P h are the projectors on the product space, QhΠ ∈ h L(VΠ , VΠ ). They live on the original and discrete image spaces, listed in (5.52)–(5.54), with their duals, scalar products, and pairings; cf. (5.41), (5.42). For the reader’s convenience, we summarize and introduce the spaces deﬁned on Ω and Ωhc ; cf. (4.82), (4.83), Theorem 4.36, (5.38), (5.40), (5.45), (5.48), (5.49). We mark functions and functionals, according to their deﬁnition, by obvious indices. For functions deﬁned on Ωhc we use c , for ∂Ω we use b , for ∂Ωhc we use c,b , for the images of BD on ∂Ωhc we use c,D , and on the product of the domain and boundary on Ω and Ωhc we use Π or c,Π : VΠ := V × VD ⊂ VˆΠ := V × Vb , with U = H 2 (Ω), V = L2 (Ω) = V ,

(5.52)

Vb = L2 (∂Ω) = Vb ⊃ VD = BB U = H 3/2 (∂Ω), deﬁned on Ω, with (u, ub ), (v, vb ) V := (u, v)V + (ub , vb )VD , (u, ub ), (v, vb ) Vˆ := (u, v)V + (ub , vb )Vb . Π

Π

Similarly, we deﬁne the spaces and scalar products on the curved approximations Vc = L2 Ωhc , Vc,D = BB Uc = H 3/2 ∂Ωhc , Vc,Π , Vc,b , . . . , deﬁned on Ωc . (5.53) Correspondingly, the discrete analog spaces and their scalar products are

h h h h h := V h × VD := S0h × Sbh |∂Ωhc , V h = V h , VΠ = V h × VD := S0h × Sbh |∂Ωhc = VΠ , VΠ B

A h VˆΠ := V h × Vbh , uh , uhb , v h , vbh := uh , v h V h ×V h + uhb , vbh V h . h h VΠ ×VΠ

D

(5.54)

In (5.42), (5.48) and (5.40), (5.49) we have satisﬁed the diﬀerential equation and boundary condition or their discrete counterparts by testing with V and Vb or (see the equivalence (5.6)) with V h and Vbh . The two components (5.42), (5.40) and (5.48), (5.49) of F and F h , deﬁned on Ω and Ωhc , respectively, are now combined with the extension operator Ec in (4.73) for motivating the appropriate deﬁnitions of the projection operators. For the discrete spaces V h = V h and Vbh = Vbh the FE equations are always tested with v h ∈ V h and vbh ∈ Vbh , but we use for the boundary condition the equivalence (5.49). Thus corresponding to the product spaces h ) VΠ and Vc,Π we have to introduce two projectors QhΠ := (Qh , Qhb ) ∈ L(VΠ , VΠ h ). We start with the component of the difand Qhc,Π := (Qhc , Qhc,b ) ∈ L(Vc,Π , VΠ ferential operator, (5.48) (note V = V , Vb = Vb ), and deﬁne (see (4.73)–(4.75) and Theorem 4.38)

Qh ∈ L(V, V h ) for f ∈ V by Qh f, v h V h ×V h − (Ec f, v h )Vc = 0 ∀v h ∈ V h .

(5.55)

315

5.2. FEMs for fully nonlinear elliptic problems

The approximate domains require B A Qhc ∈ L(Vc , V h ) for fc ∈ Vc : Qhc fc , v h

V h ×V h

− (fc , v h )Vc = 0

∀v h ∈ V h .

Both projectors yield the norm limits, the last limit as a consequence of (4.64): lim Qh f V h = f V , and lim Qhc fc

h→0

= lim fc Vc .

V h

h→0

(5.56)

h→0

Since the S0h and S0h form a pair of dual spaces (see Theorem 4.36), we alternatively formulate corresponding projectors (see Theorem 4.38) and use them, e.g. in (5.103): Qhd

∈ L(V, V ), by f ∈ V : h

lim Qhd f

:=

m0

Qh f, sj V h ×V h sj ∈ V h , again with

j=1

Vh

h→0

Qhd f

= f V ; similarly, e.g. lim Qhc,d fc h→0

Vh

= lim fc Vc . h→0

(5.57)

We turn to the projectors for the traces on the boundary. We always have to recall that the ub ∈ VD and uhb ∈ Vbh ⊂ Vb have the forms ub = u|∂Ω , u ∈ U and uhb = uh |∂Ωhc , uh ∈ V h , respectively. It suﬃces to test vanishing boundary conditions with respect to the L2 scalar product (·, ·)Vc,b . This and the dense embeddings VD ⊂ Vb and Vc,D ⊂ Vc,b motivate the following projectors: h Qhb ∈ L VD , VD

for ub := u|∂Ω ∈ VD ⊂ Vb : Qhb ub − (Ec u)|∂Ωhc , vbh

h , uc,b := u|∂Ωhc ∈ Vc,D : Qhc,b uc,b − uc,b , vbh Qhc,b ∈ L Vc,D , VD

Vc,b

Vc,b

= 0,

= 0 ∀vbh ∈ Vbh , (5.58)

h and by (5.45) we obtain for all vbh ∈ Vbh = VD that vbh = v h |∂Ωhc for an appropriate v h ∈ V h and vbh = Qhc,b v h |∂Ωhc . The convergence results along ∂Ωhc in (4.66), (5.6), and Theorems 4.36 and 4.38 imply the limits of the boundary norms (5.59) lim Qhc,b uc,b h = lim uc,b Vc,D , lim Qhb ub h = lim ub VD . VD

h→0

h→0

h→0

VD

h→0

We summarize the product projectors and norms in (5.55)–(5.59) obtaining

Qhd,Π

h h = L VΠ , VΠ , (5.60) QhΠ := Qh , Qhb ∈ L V × VD , V h × VD

h , with (f, u|∂Ω )VΠ = f V + u|∂Ω VD , and := Qhd , Qhb ∈ L VΠ , VΠ lim QhΠ (f, u|∂Ω ) h = lim Qhd,Π (f, u|∂Ω ) h = (f, u|∂Ω )VΠ . h→0

VΠ

h→0

VΠ

316

5. Nonconforming ﬁnite element methods

The approximate domains yield

h h Qhc,Π := Qhc , Qhc,b ∈ L Vc × Vc,D , VΠ = L Vc,Π , VΠ , lim Qhc,Π fc , Ec u|∂Ωhc h = lim fc , Ec u ∂Ωhc V . VΠ

h→0

h→0

(5.61)

c,Π

Again, the choice of P h , QhΠ , Qhc,Π , . . . is not unique. Nonconforming projectors and discretization We need only a few changes compared to the corresponding semiconforming subsubsec tion. The operator, spaces, and projectors there, G : D(G) ⊂ U → V, Qh , have to be replaced by the product terms, F = (F1 , F2 ) = (G, BD ) : D(F ) ⊂ U → VΠ = V × VD , QhΠ , Qhc,Π . Again we obtain a short deﬁnition of the corresponding discrete operator, F h : U h → V h × Vbh , and a linear mapping, Φh ; see (5.47), (5.48) and (5.40), (5.49), (5.52)–(5.54), (5.60), (5.33). We ﬁnally get

(5.62) F h (uh ) = Gh (uh ), uh |∂Ωhc = Qhc G(uh ), Qhc,b uh |∂Ωhc . We determine the approximate solution uh0 from h h h = 0 ∀vΠ = v h , vbh ∈ VΠ (5.63) uh0 ∈ U h s.t. F h uh0 = 0 ⇐⇒ F uh0 , vΠ Vc,Π ⇐⇒ G uh0 , v h V h + uh0 , vbh V h = 0 ∀v h ∈ S0h = V h ⊂ Vc ∀vbh ∈ Sbh = Vbh . b

By (5.62), (5.63) the new FEM is deﬁned for the F = (G, BD ) in (5.41), cf. (5.44), as

D(Φh ) := {F = (G, BD )} : Φh (F ) := F h = Qhc,Π F |U h = Qhc G|U h , Qhc,b Bc,D |U h . (5.64) F

U Ph Uh

Φh F

h

VΠ = V ×VD

tested by

ˆ = V × V , cf. (5.42 ) V Π b

tested by

ˆ = V h × V h , cf. (5.62 ) V Π b

QhΠ′ V hΠ′ = V h ′ × V hD

Figure 5.2 Nonconforming FEMs for fully nonlinear problems.

We have thus deﬁned a nonconforming Petrov–Galerkin method applicable to every F in (5.41). With uniformly bounded P h , QhΠ , Qhc,Π we modify the above Figure 5.1 into Figure 5.2. We reformulate only the not quite trivial consistency:

h h F P u − QhΠ F u h = Qhc G(P h u) − Qh G(u), P h u ∂Ωhc − Ec u∂Ωh h . VΠ

c

VΠ

(5.65) Remark 5.3. 1. The above changes will modify Theorem 5.2. We still obtain convergence, and consistency of reduced order, stability, on modiﬁes domains.

317

5.2. FEMs for fully nonlinear elliptic problems

2. We need for our approach only equicontinuous operators and the limits of the norms as listed in Deﬁnition 3.12 and (3.24). They are satisﬁed according to (5.51) and (5.55) ﬀ. Stetter [596] requires for his theory linear bounded operators P h and Qhc,Π := Qhc , Qhc,b in (5.51) and (5.60). The convergence properties for the spaces in (5.45) are not required here, since usually the consistency, hence Condition 3. in Theorem 5.2, is impossible without them. 3. Condition 2. is correct if F or G are continuous in Br (P h u0 ) or Br (u0 ). 4. The consistency condition 3. will be veriﬁed in Theorem 5.4 with reduced order. Due to the careful preparations this is a relatively simple proof. 5. Then the stability condition 4. is the only missing and hard condition in Theorems 5.2 and 5.13. In our approach the stability of the linearized problem and its regularity are the essential tools for the proof of stability; see Theorem 5.6. This makes sense only if the original F is continuously diﬀerentiable. Then F is Lipschitz-continuous as well. 5.2.6

Consistency for nonconforming FEMs

Verifying the missing conditions analogous to Theorem 5.2 we start with the easier consistency. Similarly to the Ciarlet, Raviart, Lenoir, Bernadou and Boisserie approaches (see [85–89, 174, 176, 471]) we again use the extension operator, Ec in Theorem 4.37. It allows estimates of the interpolation errors on Ω, ∂Ω, and Ωhc and ∂Ωhc , cf. (4.76), (4.77), and thus is essential for the following proof. Theorem 5.4. Consistency for nonconforming FEM: We require the conditions for Theorems 4.36, with Ω in C q , q = 2 and 2 < q ≤ 5 for future curved approximations ∂Ωhc for ∂Ω as in Conjecture 4.41, an exact solution u0 ∈ H (Ω), > 2, with Condition H satisﬁed, a Lipschitz-continuous Gw in (5.16) with a global constant L, Gw (·) ∈ CL (Ωhc ∪ Ω) × Γ with (Ωhc ∪ Ω) × Γ := {(x, z, p, r) : (5.66) x ∈ Ωhc ∪ Ω, (z, p, r) = (Ec u)(x), ∇(Ec u)(x), ∇2 (Ec u)(x)) with u ∈ Br (u0 ) ∩ D(G)}, and U h ⊂ Sd1 Tch with d ≥ n2 + 1. Then F h is consistent with F in P h u0 and h h F P u0 − QhΠ F u0

h VΠ

= Qhc G(P h u0 ), −Ec u0 |∂Ωhc

h VΠ

(5.67)

≤ CLhmin{−2,q} u0 H (Ω) . Remark 5.5. d ≥ n2 + 1 (see (5.3)), and − 1 ≤ d are related by (4.62). For inhomogeneous boundary conditions as in (5.39), a modiﬁed estimate (5.67) remains valid. Proof. We estimate the ﬁrst component in (5.67) with (5.55), L in (5.66), G(u0 ) ≡ 0, and (4.76):

318

5. Nonconforming ﬁnite element methods

h Qc G(P h u0 ) − Qh G(u0 ) h V h h = Qc G(P u0 ) − 0 Vh h ≤ Qc G(P h u0 ) − Qhc G(Ec u0 ) + Qhc G(Ec u0 ) Vh Vh ≤ CLP h u0 − Ec u0 U h + Qhc G(Ec u0 ) h by (4.65) V h −2 ≤ CLh u0 H (Ω) + Qc G(Ec u0 ) h .

(5.68)

V

We test Qhc G(Ec u0 ) − Qh G(u0 ) with v h ∈ V h , combine it with (5.16), (5.55), and get A B h Qc G(Ec u0 ), v h (5.69) V h ×V h % & h G(Ec u0 )(x) v (x)dx ∀v h ∈ V h = Ωhc % % & h & h G(Ec u0 )(x) v (x)dx − 0 ≡ G(u0 )(x) Ec v (x)dx ≤ Ωhc Ω G(Ec u0 )(x)v h (x)dx, since − = 0, ≤ Ωh c \Ω

Ωh c

Ω

Ωh c ∩Ω

h h 2 By %where G(Ec u0&), v h∈ L Ωc are well deﬁned; cf. (5.5), and Condition H, (5.47). 0 ≡ G(u0 )(x) Ec v (x), we do not need to explicitly deﬁne the extension Ec v h . Now (5.66) and u0 ∈ H (Ω), > 2, imply |G(Ec u0 )(x)| ≤ CLu0 H (Ω) . This is combined with dist Ωhc , Ω ≤ C hq , cf. Conjecture 4.41, and the bounded S∂Ω := measure of the surface (∂Ω) < ∞ yielding G(Ec u0 )(x) satisﬁes G(Ec u0 ) ≡ 0 in Ω =⇒ G(Ec u0 )L2 (Ωhc \Ω)

(5.70)

≤ CLu0 H (Ω) 1L2 (Ωhc \Ω) ≤ CLu0 H (Ω) S∂Ω C hq ≤ C ∗ hq u0 H (Ω) . We combine (5.68), (5.69), (5.70) with the Cauchy–Schwarz inequality h Qc G(P h u0 ) h ≤ CLh−2 u0 H (Ω) + Qhc G(Ec u0 ) h V V

(5.71)

≤ CLh−2 u0 H (Ω) + C hq u0 H (Ω) ˆ min{−2,q} u0 H (Ω) ; ≤ Ch thus we have estimated the ﬁrst component in (5.67). To simplify estimating the second component in (5.67), we assume n < 2( − 1). Then the Sobolev embedding Theorem 1.26, implies the compact embedding of H (Ω)

5.2. FEMs for fully nonlinear elliptic problems

319

¯ Then u0 ∈ H (Ω) ∩ H 1 (Ω) implies C(∂Ω) u0 |∂Ω ≡ 0. Now choose any into C 1 (Ω). 0 h P ∈ ∂Ωc , a ray through P ⊥ ∂Ωhc , the intersection point, Pb of this ray with ∂Ω, and the directional derivative ∂P Ec u0 of Ec u0 from Pb to P . Then P |Ec u0 (P )| = 0 + ∂P Ec u0 ≤ u0 H (Ω) C hq . Pb So we obtain for the second component in (5.67) Ec u0 L2 (∂Ωhc ) ≤ C hq u0 H (Ω) . For inhomogeneous boundary conditions, cf. (5.39), we employ the extension of φ, necessarily here φ ∈ H −1/2 (∂Ω), to φ ∈ H (Ω); we still have G(u0 ) = 0, and so (5.68), (5.69) remain correct. Boundaryconditions for the exact and the discrete problem are (u0 − φ)|∂Ω ≡ 0 and uh0 − P h φ |∂Ωhc ≡ 0 with P h φ extended to Ωhc . We replace the exact boundary conditions by (u0 − P h φ)|∂Ω ≡ 0. Then the error estimates in Theorem 4.38 and the stability results for linear elliptic operators with respect to the right-hand side (f, φ) ∈ V × VD , cf. Theorem 2.45, Hackbusch [387] Theorem 9.1.16, again yield (5.67). 5.2.7

Stability for the linearized operator and convergence

Fortunately, there is a standard result strongly simplifying the proof of stability to that for the derivative of an operator; see Theorem 3.23, and e.g. Stetter [596] Theorem 1.2.5, Stummel [607–609] and Keller [441]: we impose the condition of a continuously diﬀerentiable F h in Br (P h u), u ∈ D(G). This slightly enforced condition is usually satisﬁed by the equations considered here. It is even simpliﬁed in our case, since F ∈ C 1 (Br (u)) implies for smooth u, F h ∈ C 1 (Br−δ (P h u)) with a small δ > 0 for h0 > h ∈ H. We formulate it for our F h . Theorem 5.6. From linear to nonlinear stability: Assume Condition H, so the nonlinear discrete F h , is continuously diﬀerentiable in Br (P h u) for u ∈ D(G), and h be stable at P h u, satisfying let the sequence of linearized (F h ) : U h → VΠ −1 ≤ S uniformly ∀h ∈ H, h0 > h. (5.72) h (F h ) (P h u) h U ←VΠ

Then the nonlinear F h is stable at P h u as well with stability bound and threshold, 2S and r0 = r/(2S), respectively. According to Subsection 5.2.2 3., Theorems 5.2, 5.4, and 5.6 and Remark 5.3 show that the only missing claim for convergence is the stability for the linearized operator for uh = P h u, u ∈ D(G), here u = u0 . So we have to elaborate the program in Subsection 5.2.2 in 3. We indicate only the semiconforming FEM on convex polyhedral domains in (5.23) and mainly discuss the nonconforming FEM on C 2 domains in (4.5). For the conforming FEs it suﬃces to discuss the diﬀerential operator G. Then the boundary operators, BD , the extension, Ec , and all the product machinery can be avoided. This would considerably simplify the proofs, presented here only for the nonconforming FEMs. The interested reader can easily verify this claim. For our F (u)

320

5. Nonconforming ﬁnite element methods

with F (u1 )u = (G (u1 )u, u|∂Ω ) the linearization of G is essential. Thus a major part of the following discussion is concerned with G (u1 ). We obtain the claimed stability, −1 V h ←U h ≤ C, by the end of this subsection. (F h ) (uh1 )|U h →V h Π

Π

The linearized operator We start with G : D(G) ⊂ U → V = L2 (Ω), assuming a continuously diﬀerentiable operator G or its deﬁning function Gw ; see (5.17), (5.18). Here we sometimes integrate the boundary conditions into the spaces U0 , W0 and Ubh , Wbh . In the literature the same notation G (u0 ) : U → V and G (u0 ) : W → W is used for the strong and weak forms of the linearized operators, Gs (u0 ) = G (u0 ) and G (u0 ), respectively. For a smooth enough situation, the strong form is just the restriction of the weak form, and so G (u0 )|U = Gs (u0 ) : U → V under the condition (5.73). The proofs in this subsection fundamentally depend upon a systematic interplay between these strong and weak forms of the linearized operators. To avoid misunderstandings, it is necessary to distinguish at least the strong and weak bilinear forms as as (·, ·) and a(·, ·). Later on, we even use the notation A = G (u0 ), As = Gs (u0 ) and Ah , Ahs for the weak and strong linear operators. Correspondingly, we formulate two diﬀerent FEMs for the linearized problem. To its strong form, as (·, ·), we apply the FEM (5.48), (5.78); to its weak form, a(·, ·), the standard FEM; see (5.79) below. We will see that for the linear case both bilinear forms coincide, and so as (uh , v h ) = a(uh , v h ) for all uh ∈ U h = W h ⊃ Ubh = V h v h ; see (5.77) and Proposition 5.7. We called the original nonlinear problem G uniformly elliptic in u0 if its derivative G (u0 )u in (5.19) satisﬁes (5.20). In contrast to G, it is possible to transform the linearization in (5.19) into the standard weak form. We start with the strong operator G (u0 ) : U → V (see (5.12)) and introduce new coeﬃcients aij . Condition This yields the strong bilinear form as (·, ·) and conditions for the partials of Gw ; see (5.12), (5.16), (5.17). We already have included the (5.73), (5.74) into Condition H. G (u0 ) : U → V and as (u, v) := (G (u0 )u, v)V G (u0 )u =

n

∀ u ∈ U, v ∈ V with

(5.73)

(−1)j>0 ∂ j (aij ∂ i u) and w0 (·) = (·, u0 (·), ∇u0 (·), ∇2 u0 (·)) :

i,j=0

and assume

aij = aij (u0 ) =

∂Gw ∂Gw (w0 ), (w0 ) for i, j ≥ 1, a00 = a00 (u0 ) = ∂rij ∂z

ai0 = ai0 (u0 ) =

n ∂Gw ∂Gw (w0 ) + ∂j (w0 ), ∂pi ∂rij j=1

a0i = 0 for i ≥ 1;

aij = aij (w0 (·)) ∈ W 1−δ0j ,∞ (Ω), so this strong form G (u0 ) is bounded.

Partial integration or Green’s formula now combined with v ∈ W ∩ H01 (Ω) yields the (standard) weak problem or the coinciding bounded weak and strong bilinear form, a(·, ·) : W × W → R, and the induced weak, G (u0 ) : W → W (see (5.12), (5.13)), as

5.2. FEMs for fully nonlinear elliptic problems

a(u, v) :=

n

Ω i,j=0

321

aij ∂ i u∂ j vdx =: G (u0 )u, vW ×W ∀v ∈ W0 = W ∩ H01 (Ω)

∀u ∈ W, a(·, ·), G (u0 ) bounded for aij = aij (u0 ) = aij (w0 (·)) ∈ L∞ (Ω) for i, j = 0, . . . , n, and with as (u, v) = a(u, v)

∀u ∈ U, ∀v ∈ W0 .

(5.74)

The principal part ap (u, v) of a(u, v) is W0 coercive, by (5.20), and, since uH01 (Ω) and uW are equivalent norms on W0 , n ap (u, u) ≥ λ (∂ i u)2 dx = λu2H 1 (Ω) ≥ αu2W ∀u ∈ W0 . (5.75) 0

Ω i=1

The original a(u, v), inducing A = G (u0 ), is the sum of ap (u, v), inducing B, and its complement c(u, v), inducing C, a compact perturbation of A = B + C, with A, B, C ∈ L(W, W ): a(u, v) = ap (u, v) + c(u, v) = G (u0 )u, vW ×W = (B + C)u, vW ×W . (5.76) B is boundedly invertible on W0 by (5.75). Thus G (u0 ) satisﬁes the Fredholm alternative on W0 . This compact perturbation C ∈ L(W, W ) will play a dominant role for the following stability results. In addition to the above approximating spaces (see (5.45), (5.46), (5.54)), we need for W0 for U0 ⊂ W0 , (5.77) U h = W h ⊃ V h and Ubh = Wbh := V h = U h ∩ H01 Ωhc again with the corresponding nonequivalent norms · U h , · W h , · V h . We want to contrast the two diﬀerent FEMs for the linear problem. We start with (5.48) and then turn to the standard FEM based upon the weak bilinear form a(·, ·). In both cases we have to determine, cf. (5.73), uh1 ∈ Ubh and uh1 ∈ Wbh s.t. ⎞ ⎛ n h h h h h as u1 , v = G P u0 u1 , v V := ⎝ (−1)j>0 ∂ j aij (P h u0 )∂ i uh1 , v h ⎠ c

h

= (Ec f, v )Vc

i,j=0

∀v ∈ V , with f ∈ V, aij ∈ W h

h

Vc 1−δ0j ,∞

(Ω).

(5.78)

Similarly as Gw we assume the aij (P h u0 ) to be well deﬁned on the extended Ω ∪ Ωhc . The standard FEM for the weak bilinear form a(·, ·) with its solution uh1 ∈ Wbh is n ; : aij (P h u0 )∂ i uh1 ∂ j v h dx = G (P h u0 )uh1 , v h W ×W uh1 ∈ Wbh : a uh1 , v h = Ωh c i,j=0

c

c

= Ec f, v h Wc ×Wc or = Ec f, v h Vc for f ∈ V∀v h ∈ Wbh , with aij ∈ L∞ (Ω).

(5.79)

As in (5.48) for G, we use here the same notation for the discrete as (uh , v h ), a(uh , v h ) and the original as (u, v), a(u, v). These bilinear forms are, for small enough h,

322

5. Nonconforming ﬁnite element methods

simultaneously bounded and elliptic. Equations (5.78), (5.79) can and will be similarly reinterpreted by using the above projectors and deﬁning new ones. Weak and strong linear operators coincide on U h × V h The techniques in this and the next subsubsections are strongly related to the standard approach in spectral methods for collocation (see B¨ ohmer [120] and Canuto et al. [158]) and to regularity for diﬀerence equations; see Chapter 8 and, e.g. Hackbusch [387], Theorem 9.2.26. Proposition 5.7. Choose the above U h , . . . , ⊂ Sd1 Tch in (5.77) for d ≥ 2n + 1 as approximating spaces for U, . . . , and let aij ∈ W 1−δ0j ,∞ (Ω). Then for all uh ∈ U h , v h ∈ V h = Ubh , as (uh , v h ) = (G (P h u0 )uh , v h )Vc = a(uh , v h ) = G (P h u0 )uh , v h Wc ×Wc . (5.80) Proof. We apply partial integration for every T ∈ Tch with Ωhc = ∪T ∈Tch T¯. We use the notation ν = νT for the outer unit normal vector of T and Ba and u for the boundary operator, induced by A = G (u0 ); see, e.g. [387]: Ba u =

n

νj aij ∂ i u +

i,j=1

n

νj a0j u, e.g. Ba u = ∂u/∂ν

for As u = −Δu. (5.81)

j=1

All ∂T have a positive orientation with respect to T . Hence in (5.83) every interior edge e ∈ T¯ will be obtained twice in opposite directions. Edges e ⊂ ∂Ωhc will appear once. Let ν = νT be one of the normal vectors for an interior edge e ⊂ Tr . It is oppositely oriented for neighboring Tr , Tl ∈ Tch with e ⊂ Tr ∩ Tl and ν the outer normal 36 for e ⊂ ∂Ωhc . To consider the transition from a triangle Tl to its neighboring Tr , we introduce the restriction of v to Tl , Tr , ∂Ωhc and the standard notation; see [665]: vl = v|Tl ,

vr = v|Tr ,

[v] := vl |e − vr |e ,

and {v} := (vl |e + vr |e )/2,

(5.82)

for the corresponding jumps and arithmetic means of v across an interior e, and [v] := {v} := v|∂Ωhc along e ∈ ∂Ωhc , and

v arbitrary in Rn \Ωhc .

The next transformation via the n-dimensional Green’s formula is the deciding & % reason for the of C 1 FEs. Only for this choice do the following jump contributions, Ba uh = 0, vanish (the [v h ] already vanish for C FEs v h ):

36

h h h We will use the notation e ∈ ∂Ωh c , although mostly e ⊂ ∂Ωc ; similarly, e ∈ Tc \ ∂Ωc .

323

5.2. FEMs for fully nonlinear elliptic problems

a(uh , v h ) = G (Ec u0 )uh , v h Wc ×Wc n = aij ∂ i uh ∂ j v h dx T i,j=0

T ∈Tch

=

(−1)j>0 ∂ j (aij ∂ i uh )v h dx +

e∈Tch \∂Ωh c

vlh Ba uh ds

∂Ωh c

T i,j=0

T ∈Tch

+

n

& % {v h } Ba uh + [v h ]{Ba uh } ds =

e

(G (Ec u0 )s uh )v h dx

Ωh c

= as (uh , v h ) since uh ∈ U h ⊂ C 1 Ωhc , v h ∈ V h ⊂ H01 Ωhc ; hence, a(uh , v h ) = as (uh , v h )

∀uh ∈ U h , v h ∈ V h .

(5.83)

Discontinuous Galerkin methods, cf. Chapter 7, would certainly be possible for a(uh , v h ). But they would violate the necessary condition a(uh , v h ) = as (uh , v h ) by their penalty terms. Stability of the weak linear operator The previous subsubsections play the role of preparing this stability proof. We had essentially conﬁned the discussion there to the diﬀerential equation and indicated the boundary conditions either by the index 0 or using v h ∈ V h . Instead of the previous strong and weak forms, As : U → V and A : W → W (see (5.38), (5.41)–(5.43)), we need their two component versions: AU := (As , BD ) : U → V × VD ⊂ VˆΠ = V × Vb , AU ∈ L(U, V × VD ) and

AW := (A, BD ) : W → W × WD ⊂

WΠ

(5.84)

= W × Vb , AW ∈ L(W, W × WD ).

With the dense (compact) embedding WD → Vb = L2 (∂Ω) we test the W × WD ⊂ by the following WΠ . We emphasize that the stable splitting S h = S0h + Sbh in WΠ Theorem 4.36 plays a dominant role. We choose, cf. (4.65), (5.6), (5.52)–(5.54), = W × Vb , with Vb ⊃ WD and W × WD ⊂ WΠ := W × Vb = WΠ

(5.85)

(u , ub ), (v, vb )W ×WΠ := u , vW ×W + (ub , vb )Vb , and on Π

Ωhc

the Wc,Π = Wc × Vc,b = Wc,Π with (u , ub ), (v, vb )W

c,Π ×Wc,Π

.

The discrete counterparts are

h h h h = W h × Vbh := S h × Sbh |∂Ωhc = VΠ , WΠ := W h × Vbh := S h × Sbh |∂Ωhc = VΠ , WΠ B

A and uh , uhb , v h , vbh := uh , v h W h ×W h + uhb , vbh V h . (5.86) h h WΠ ×WΠ

b

The exact and discrete linear boundary conditions remain nearly unchanged, except replacing VD by WD , and so they can be treated as in (5.40), (5.49); see (5.58), (5.59).

324

5. Nonconforming ﬁnite element methods W Ph Wh

AW

′ =W′×WD WΠ

tested by

W Π= W × Vb

tested by

W hΠ = W h × V hb

Q hΠ′

Φ hW AhW

W hΠ′ = W h ′ × W hD

Figure 5.3 Nonconforming FEMs for linear problems.

For the weak diﬀerential operator, we introduce a modiﬁed Ritz operator. The above extension operator Ec in Theorem 4.37 is continuously extended to f ∈ H k (Ω), k < 0. We deﬁne the projectors, Qh , Qhc , and so on, indicating the weak form by the exponent h , in contrast to the strong projectors Qh and so on with the exponent h in (5.60); cf. (4.64), (4.66), (4.65), (5.6):

Qh ∈ L(W , W h ) for f ∈ W by Qh f, v h W h ×W h − (Ec f ), v h Wc ×Wc = 0, (5.87)

B A − fc , v h Wc ×Wc = 0 ∀v h ∈ W h . Qhc ∈ L Wc , W h : fc ∈ Wc : Qhc fc , v h h h W

×W

This allows formulating the product projectors, and we obtain

h h , Qhc,Π := Qhc , Qhc,b ∈ L Wc,Π . , WΠ , WΠ QhΠ := Qh , Qhb ∈ L WΠ

(5.88)

For both types of projectors we obtain the necessary norm limits: lim Qh f W h = f W , and lim Qhc fc W h = lim fc Wc = lim Ec fc W , h→0

lim

h→0

h→0

QhΠ (f, u1 |∂Ω )W h Π

h→0

h→0

= (f, u1 |∂Ω )WΠ , and

lim Qhc,Π fc , (Ec u)|∂Ωhc W h = lim fc , (Ec u)|∂Ωhc W

h→0

Π

h→0

c,Π

= lim (Ec fc , (Ec u)|∂Ω )WΠ .

(5.89)

h→0

Similarly to the above F h = Qhc,Π F |U h we use the projectors for reformulating (5.78) and (5.79), cf. (5.84), for aij ∈ CL Ω ∪ Ωhc . For the strong AU and AhU we get AhU = Qhc,Π AU |U h : uh0 ∈ U h : AhU uh0 = Qhc,Π As uh0 , uh0 |∂Ωhc = QhΠ (f, φ). (5.90) , Similarly, the weak linear operators AW and AhW yield, for (f, φ) ∈ WΠ AhW = Qhc,Π AW |W h : uh0 ∈ W h : AhW uh0 = Qhc,Π Auh0 , uh0 |∂Ωhc = QhΠ (f, φ). (5.91)

We summarize the new situation with uniformly bounded Ah , P h , Qhc,Π in Figure 5.3. For proving the stability for AhW in Theorem 5.9 assume the principal part induces the (W0 -coercive) operator B; cf. (5.76). We use the notation and assume AW = (A, BD ), BW = (B, BD ), the principal part B is W0 -coercive, CW = (C, BD ), C compact, and

h h AhW , BW , CW

(5.92)

are consistent with AW , BW , CW .

325

5.2. FEMs for fully nonlinear elliptic problems

Theorem 5.8. Stability and convergence for the principal part: Assume (5.74), the ellipticity (5.75), the approximate spaces in (5.77) with the norm limits in (5.89) and h as in (5.84), (5.92). Then BW is boundedly invertible, and (5.60), and BW and BW h h h and uh1 of BW u1 = BW is stable. The unique solutions u1 of BW u1 = (f, φ) ∈ WΠ h h h h QΠ (f, φ) ∈ WΠ exist and u1 converge such that u1 − Ec u1 W h → 0 for h → 0. Proof. Equation (5.74) shows that ahp (uh , v h ) is bounded on W h . The ellipticity condition (5.75) implies with the usual arguments, cf. (5.75), that n h h h (∂ i uh )2 dx = λuh 2H 1 (Ωh ) ≥ αuh 2W h ∀uh ∈ Wbh , (5.93) ap (u , u ) ≥ λ 0

Ωh c i=1

c

h in Wbh . We reduce the in homogeneous problem hence the stability of BW

h h h u1 = QhΠ (f, φ) = Qh f ∈ W h , Qhb φ ∈ Vbh ∈ WΠ BW

to Wbh : As usual, we determine uh2 ∈ W h with uh2 − Qhb φ |∂Ωhc = 0 ∈ Vbh . Then

h h h u1 − uh2 = Qh f − B h uh2 ∈ W h , 0|∂Ωhc ∈ Vbh ∈ WΠ . BW

This guarantees, by (5.93) and Theorems 5.2 and 5.4, the stability, consistency, and hence existence of the unique discrete solution uh1 − uh2 and its convergence to the corresponding exact solution u1 − u2 , and hence of uh1 as well.

Theorem 5.9. Stability of AhW = (A = G (u0 ), BD )h ∈ L W h , W h × Vbh : Under the conditions of Theorem 5.8, assume a boundedly invertible AW in (5.84). Then AhW in (5.91) is stable. Proof. This is a generalization to nonconforming FEMs of the B¨ ohmer and Sassmannshausen results (see [114, 115, 567]) for conforming FEMs. For an arbitrary u ∈ W = W × WD . By assumption, unique exact and discrete choose vw := CW u ∈ WΠ h solutions, u ˆ and u ˆ , exist for the equations

h h h ˆ = vw and BW u ˆ = Qhc,Π BW |W h u ˆh = QhΠ BW u ˆ = QhΠ vw ∈ WΠ . BW u

(5.94)

With the notation

h −1 h −1 Tw := BW ∈ L (WΠ , W) and Twh := BW QΠ ∈ L WΠ , Wh .

Theorem 5.8 implies that Ec u ˆ−u ˆh W h = Ec Tw − Twh CW uW h → 0 for h → 0 and ∀u ∈ W. Since, by assumption, CW is compact and Ec Tw − Twh are equibounded, this implies Ec Tw − Twh CW → 0 for h → 0. (5.95) W←W Now let uh ∈ W h ⊂ Wc . Because AW has been extended to yield = Wc × Wc,D still boundedly invertible with W h ⊂ Wc , AW : Wc → Wc,Π

(5.96)

326

5. Nonconforming ﬁnite element methods

we can estimate AW uh Wc,Π = A−1 BW (I + Tw CW )uh Wc,Π uh W h ≤ A−1 W W h ←Wc,Π W W h ←Wc,Π h ≤ A−1 (5.97) ←W h (I + Tw CW )u W h ; hence, W W h ←W BW Wc,Π c,Π

(I + Tw CW )uh W h ≥ A−1 W W h ←W

c,Π

BW Wc,Π ←W h

−1

uh W h .

h h h The QhΠ BW u is deﬁned for every u ∈ W. We apply the stability of BW to wh := BW u : h −1 h ) uh W h ≤ (BW · w Wh . W h ←W h Π

Π

Furthermore h −1 h I + Twh CW uh h = BW BW I + Twh CW uh W h W h h −1 h BW I + Twh CW uh h ≤ BW W ←W h W Π

Π

implies h 1 BW I + Twh CW uh h ≥ −1 WΠ h B W

· I + Twh CW uh W h .

(5.98)

h W h ←WΠ

We combine (5.94)–(5.98) and use the fact that AW , BW , CW are extended to W h , such that Qhc,Π AW uh is well deﬁned for estimating h h AW u h WΠ = Qhc,Π AW |W h uh h = Qhc,Π AW uh h ∀uh ∈ W h WΠ

= Qhc,Π BW (I + Tw CW )uh

WΠ

h WΠ

(5.94) h = BW (I + Tw CW )uh W h Π

h h I + Twh CW uh W h − BW Ec Tw − Twh CW uh W h ≥ BW Π

h −1 ≥ I + Twh CW uh W h / BW

(5.98)

h W h ←WΠ

h Ec Tw − Twh CW uh − BW

h −1 ≥ (I + Tw CW ) uh W h − Ec Tw − Twh CW uh W h / BW h Ec Tw − Twh CW W h ←W h uh W h − BW Π (5.97) 1 ≥ h A−1 h (B h )−1 B h h

W

W ←Wc,Π

W WΠ ←W

− O (Ec Tw − Twh )CW W h ←W h Π

W

uh W h .

h W h ←WΠ

h W h ←WΠ

h WΠ

5.2. FEMs for fully nonlinear elliptic problems

327

The last estimate shows the following: the A−1 is the deciding factor if W W h ←Wc,Π h h BW has a moderate condition number. Because of (5.95) and the stability of (BW )h∈H there exists a positive constant K, independent of h, such that for all h ≤ h0 , h h h h AW u h = Q A u ≥ Kuh W h ∀uh ∈ W h . c,Π W W

Wh

Qhc,Π AW |W h

is invertible for h ≤ h0 . Moreover, we This and dim W < ∞ show that h −1 obtain (Qc,Π AW |W h ) W h ←W h ≤ 1/K; i.e. AhW |W h is stable. h

Regularity for FE solutions Our next task is lifting the preceding stability result with respect to the weak forms back to the strong form. This is achieved by the regularity for FE solutions in this subsubsection. Until now, regularity results for the solutions of FEMs seem to be known only as oscillation results in the sense of De Giorgi, Nash, and Moser; see Aguilera and Caﬀarelli [4]. They are not applicable to our problem of stability. Here we need, for the solution of a FEM applied to a diﬀerential equation of order 2, results analogous to those for the exact solution. These are estimates of the H 2 (Ω) instead of the usual H 1 (Ω) norm for a right-hand side in L2 (Ω) × H 3/2 (Ω) instead of the usual H −1 (Ω) × H 1/2 (Ω). To the author’s knowledge, the following type of lemma was not known for FEMs. It is a transformation to our FEM of Hackbusch’s regularity result [387], Theorem 9.2.26, for diﬀerence methods. We employ inverse and interpolation error estimates (4.57), (4.64), (4.76), valid under the conditions of Theorems 4.43, 4.36, and 4.38. The lemma is based upon a regularity result for the exact solution u1 of the corresponding linear problem AW u1 = (f, φ) ∈ V × VD ⊂ VˆΠ , u1 ∈ W0 . Indeed, known regularity results, e.g. in Hackbusch [387], Theorems 9.1.22, 9.1.16, show that, for a convex bounded or smooth domain and smooth coeﬃcients, the restriction AU = AW |U ∈ L(U, V × VD ) is boundedly invertible for a boundedly invertible AW ∈ L(W, W × WD ); hence we assume Ω bounded, and either convex, polyhedral or Ω ∈ C 2 , −1

(AW )

Lemma 5.10.

∈ L(W × WD , W),

ai,j ∈ W

1−δj0 ,∞

(Ω)

∀0 ≤ i, j ≤ n.

(5.99) (5.100)

Regularity for FE solutions:

1. We assume (5.99), the bounded elliptic AW ∈ L(W, W × WD ) and AU ∈ L(U, V × VD ) to be boundedly invertible, e.g. by (5.100). Choose the above FEs, (5.90) and (5.91), of local degree d ≥ 2, on a quasiuniform triangulation; see (5.3), (5.77). Then AhW is regular with respect to stability and convergence; hence, there exist unique solutions u1 for AW u1 = (f, φ), uh1 , and a C > 0, independent of h, such that (see (4.64), (4.76)) AhW uh1 = Qhc,Π (f, φ), (f, φ) ∈ V × VD ⇒ uh1 U h ≤ C (f V + φVD ) (5.101) with Ec u1 − uh1 U h ≤ C (Ec u1 − P h u1 U h + Ec φ − P h u1 VDh ). (5.102) ) and AU ∈ L(U, VˆΠ ) simultaneously satisfy the Fredholm 2. AW ∈ L (W, WΠ alternative.

328

5. Nonconforming ﬁnite element methods

U Ph Uh

AU

AW

W

ˆ =V ˆ′ V Π Π

Ph Wh AhU

QhΠ ˆ h ≠V ˆ h′ V Π Π

W ′Π QhΠ′ W hΠ′

AhW

Figure 5.4 Strong and weak FEMs for linear problems.

Remark 5.11. Hence, the FE error in the strong norm Ec u1 − uh1 U h is estimated h . The FE spaces considered here by the interpolation errors of u1 in U h and VD need a quasiuniform triangulation guaranteeing inverse estimates in the global form, v h hH j (Ωh ) ≤ C hl−j v h hH l (Ωh ) ; see (4.57). In Subsection 4.2.5 we have summarized c c inverse estimates for Sd1 (Tch ) on speciﬁc degenerate triangulations, based upon the results of Graham, Hackbusch and Sauter [360–364]. The previous lemma remains valid for this case. However, since the smooth FEs in Subsection 4.2.6 still need nondegenerate triangulations, we require nondegenerate triangulations in this Section. Under the conditions of the lemma, (5.101) holds even for noninvertible AW with (f, φ)VˆΠ replaced by (f, φ)VˆΠ + u1 W ; see [387]. For Figure 5.4 and the following proof we contrast the strong and weak operators AU ∈ L(U ∪ Uc , VˆΠ ) and AW ∈ L (W ∪ Wc , WΠ ), extended as in (5.96); see (5.52)– (5.54), (5.85), (5.86) with the FE versions in (5.60), (5.88). The horizontal arrows in the ﬁrst line of Figure 5.4 indicate compact embeddings and in the second line the h h ⊂ WΠ . The −→ between decreasingly weaker norms in the spaces U h = W h and VˆΠ ˆ W, VΠ is missing, since only W −→ V would be appropriate. Proof. For a boundedly invertible AW ∈ L(W, WΠ ), Theorem 5.9 implies the stability h h h and thus the bounded invertibility of AW = Qc,Π AW |W h ∈ L(W h , WΠ ). This yields, with the consistency in Theorem 5.4, the existence of a discrete solution uh1 for h by Theorem 5.2. The solution uh1 ∈ W h for (f, φ) ∈ WΠ AhW uh1 = QhΠ (f, φ) ∈ WΠ h satisﬁes the estimate u1 W h ≤ C(f, φ)WΠ . Now we use the same technique as in the proof of Theorem 5.8 for reducing the inhomogeneous to trivial Dirichlet boundary conditions. We have assumed φ ∈ VD = H 3/2 (∂Ω) as a restriction of φ ∈ V = H 2 (Ω). This is possible by combining Theorems 4.37, 1.37, 1.38, cf. [678], p. 1030. The last lines of the proof of Theorem 5.4 can then be updated to obtain the (5.101), (5.102) results for φ ≡ 0 if they are correct for φ ≡ 0. So we restrict the discussion for the remaining proof to trivial Dirichlet boundary conditions. h h = (v h , 0) ∈ VˆΠ , cf. (5.96), (5.52)–(5.54), We ﬁnd for P h = I h Ec , AU , and the vΠ h h h −1 h h h ˆh P h A−1 U : VΠ ⊂ Vc,Π → U or P AU vΠ is well deﬁned ∀vΠ = (v , 0) ∈ VΠ . h 2,γ (Ωhc ) by standard V h ⊂ C 1 (Ωhc ) ⊂ C γ (Ωhc ) for a 0 < γ < 1 implies that A−1 U vΠ ∈ C regularity results, cf. Theorems 2.38, 2.39, and, e.g. [677], Chapter 6.3, Problem 6.8,

329

5.2. FEMs for fully nonlinear elliptic problems

h h h ˆh p. 259. So the P h A−1 U vΠ is well deﬁned for all vΠ = (v , 0) ∈ VΠ . Alternatively, we h choose interpolation via the Bramble–Hilbert lemma, Theorem 4.15. Thus P h A−1 U vΠ h is well deﬁned for derivatives up to second order for P in (4.83), e.g. for the Argyris FEs. Higher derivatives deﬁning P h require quasi-interpolants as, e.g. in Theorem 4.36, Scott and Zhang [578], and Davydov [264]. For the estimate in (5.107) we need the embedding IW U : W h → U h , the bounded transformation, cf. (5.57), and [264]:

h ,0 h 0 := V h × {0} ⊂ VΠ → Vc,Π := Vc × {0} ⊂ Vc,Π , (v h , 0) → Qhd v h , 0 . IVh0 : VΠ

(5.103) We assumed a smooth (f ∈ V, φ ≡ 0) ∈ VˆΠ . By (5.58) and φ ≡ 0 we restrict the discussion to uh ∈ W h ∩ S0h = Ubh = Wbh = V h . For the equalities between weak and strong problems we use special spaces and (5.52)–(5.55), (5.80), (5.85)–(5.89):

h,0 h ,0 h ˆ h,0 , W h ,0 . := (v h , 0) ∈ VˆΠ := V h × {0}, VˆΠ , W ∀uh , v h ∈ Ubh = Wbh = V h , ∀vΠ Π Π

We ﬁnd for f ∈ V and the deﬁnition of A, As in Ω ∪ Ωhc that : h h h; ; : h AW u , vΠ W h ,0 ×W h,0 = Auh , v h Wc ×Wc = (As uh , v h )Vc = AhU uh , vΠ h ,0 h,0 , VΠ ×VΠ Π Π B B A A h h (Ec f, v h )Vc = QhΠ ,0 (Ec f, 0), vΠ = Qh,0 . Π (Ec f, 0), vΠ h ,0 h,0 h ,0 h,0 WΠ

×WΠ

VΠ

×VΠ

(5.104) This implies coinciding weak and strong bilinear forms, since for all v h ∈ V h , : h h h; ; : h AW u1 , vΠ W h ,0 ×W h,0 = As uh1 , v h V = AhU uh1 , vΠ = (Ec f, v h )Vc . V h ,0 ×V h,0 Π

Π

c

Π

Π

(5.105) Hence, a weak solution uh1 solves the strong problem as well:

h ,0 AhW uh1 = QΠ (f, 0),

0 (f, 0) ∈ VΠ := V × {0} =⇒ AhU uh1 = Qh,0 Π (f, 0).

(5.106)

For proving stability and consistency for the strong form, we start with the identity 6 5 h −1 (5.107) − (I ) AU U h ←V h ,0 = PUhh ←Uc,0 A−1 h h W U 0 Ub ←Wb U Uc,0 ←Vc,Π b Π b 6 % & −1 −1 AU U ←V 0 IV 0 ,V h ,0 . × AhU W h ←W h ,0 AhU P h − Qhc,Π AU W h ,0 ←U b

Π

Π

c,0

c,0

c,Π

c,Π

Π

The inverse and the error estimates (4.57) and (4.64), (4.76), the approximation property (4.76), for the local degree d ≥ 2, show the three inequalities IW U Ubh ←Wbh ≤ C1 h−1 , P h Ubh ←Uc,0 ≤ C2 , P h u1 − Ec u1 W h ≤ C3 hu1 U . (5.108)

330

5. Nonconforming ﬁnite element methods

With (4.76), the solution of AU u1 = (f, 0) ∈ VΠ , equal weak and strong bilinear forms, h,0 h ai,j , cf. (5.73), deﬁned in Ω ∪ Ωhc and the special vΠ = (v h , 0) ∈ VΠ , we get by (4.75) : ; h h h AU P u1 − QhΠ AU u1 , vΠ h ,0 h,0 W ×W Π

Π

= |(As P h u1 − Qh f, v h )Vc | = |(As P h u1 − Ec f, v h )Vc | = |(As P h u1 − Ec As u1 , v h )Vc | ⎛ ⎞ n = ⎝ aij (∂ i P h u1 − Ec ∂ i u1 ), ∂ j v h ⎠ + Chu1 U v h W h i,j=0 Vc

≤ (C3 + C)hu1 U v h W h

h,0 h ∀vΠ = (v h , 0) ∈ VΠ .

With (5.106), this consistency, and the stability of AhW , the (AhW )−1h we estimate the ﬁrst term in (5.107): −1 % & h AU W h ←W h ,0 AhU P h − Qhc,Π AU W h ,0 ←Uc,0 Π

b

,0

h Wb ←WΠ

≤ C4 ,

Wbh ←Uc,0

Π

−1 % & = AhW W h ←W h ,0 AhU P h − Qhc,Π AU W h ,0 ←U c,0 Π

b

Wbh ←Uc,0

Π

≤ C4 AhU P h − QhΠ AU W h ,0 ←U c,0 Π : ; h h h AU P u − AU u, vΠ h ,0 h,0 WΠ ×WΠ ≤ C4 sup sup ≤ C3 C4 h. v h W h uU 0 =u∈Uc,0 0 =v h ∈W h b

0 ≤ C5 and a bound With (5.108), (5.107) and a boundedly invertible A−1 U Uc,0 ←Vc,Π

for IV 0

h ,0 c,Π ,VΠ

V 0

h ,0 c,Π ←VΠ

−1 h AU

,0

h Ubh ←VΠ

h ,0 ≤ C6 , this yields the stability of AhU : Ubh → VΠ , since

5 ≤ P h U h ←U b

c,0

−1 A U

0 Uc,0 ←Vc,Π

+ (IW U )U h ←W h b

−1 % & h AU W h ←W h ,0 AhU P h − Qhc,Π AU W h ,0 →Uc,0 I

b

,0

0 ,V h Vc,Π Π

Π

Π

,0

0 ←V h Vc,Π Π

Wbh ←Uc,0

b

−1 A U

& % ≤ C2 C5 + C1 h−1 C3 C4 hC5 C6 ≤ C.

6

0 Uc,0 ←Vc,Π

×

The combination of these inequalities yields the claim, since −1 −1 h h h h h u 1 h = QΠ h ,0 A Q (f, 0) ≤ f V ≤ Cf V . AU h U Π h ,0 U V ←V 0 Uh

Ub ←VΠ

Π

Π

The convergence in (5.102) is, by Theorem 5.2, an immediate consequence of the stability in (5.101) and the consistency in Theorem 5.4.

5.2. FEMs for fully nonlinear elliptic problems

331

Stability for the strong linear operator and convergence Combining Theorem 5.9, Proposition 5.7, and Lemma 5.10 we obtain: Theorem 5.12. Stability for semiconforming and nonconforming FEMs: For the linearized operator As = G (u0 ) ∈ L(U, V) in (5.73), assume (5.20), a quasiuniform Sd1 (T h ), d ≥ 2n + 1, cf. (5.3), and let the corresponding As ∈ L(U0 , V), cf., for the domain (5.100), and, hence, AU := (As , BD ) ∈ L(U, V × VD ) be boundedly invertible. Then for a convex polyhedral Ω in Rn and linear Ahs uh1 = Qh As |U0 uh1 = Qh f, f ∈ V, 2 the semiconforming FEM (5.78) is stable. For an Ω ∈ C , the nonconforming FEM (5.90), AhU uh1 = Qhc,Π AU uh1 = QhΠ (f, φ), (f, φ) ∈ V × VD , is stable as well. We discussed the relation between the bounded invertibility of As = G (u0 ) ∈ L(U0 , V) and AU = (As , BD ) ∈ L(U, V × VD ) and the existence of a locally unique solution of G(u0 ) = 0 and F (u0 ) = 0 at the end of Subsection 5.2.3. Proof. The boundedness of the strong AU := (As , BD ) ∈ L(U, V × VD ) and its weak counterpart AW ∈ L(W, W × WD ) under the conditions in (5.100), (5.73), (5.74) is straightforward. We want to show that the bounded invertibility of AU implies that of AW ∈ L(W, W × WD ) as well. Indeed, by (5.73)–(5.74), the corresponding G (u0 ) ∈ L(U, V ) induces a strong bounded bilinear form, by Proposition 5.7 simultaneously a W-elliptic bilinear form, and thus ﬁnally AW ∈ L(W, W × WD ). Thus the Fredholm alternative applies: if AW ∈ L(W, W × WD ) were not boundedly invertible, each of the inﬁnitely many solutions of AW u1 = (f, φ) ∈ V × VD for appropriate (f, φ) would be u1 ∈ U by (5.99) and the well-known regularity results; see, e.g. Hackbusch [387], Chapter 9. So AU could not be boundedly invertible either. Now we apply Theorem 5.9 to guarantee the stability for the strong form: the h ) and Lemma 5.10 guarantee the bounded invertibility of QhΠ AW |W h ∈ L(W h , WΠ h h h stability of QΠ AU |U h ∈ L(U , VΠ ) and thus the claim. Theorems 5.2, 5.4 and 5.6 and the last subsection combined with the last proof show that we have proved stability and convergence for the FEM in (5.62) if G is continuously diﬀerentiable near a locally unique smooth solution u0 , indicated by A = G (·) ∈ C(Br,W (u0 )) with boundedly invertible F (u0 ) cf. Condition H. This is guaranteed, again in the interplay between the weak and strong form of the derivative, e.g. with Br,W (u0 ) the ball with respect to u − u0 W < r, by aij (w(·)) ∈ C 1−δ0j ,∞ (Br,W (u0 )) and aij (w(·)) ∈ C 1−δ0j ,∞ (Br,U (u0 )),

(5.109)

yield C(Br,W (u0 )) A = G (·) : W → W and C(Br,U (u0 )) As = G (·) : U → V , where i, j = 0, . . . , n, aij (w(·)) are deﬁned in (5.17). Alternatively, we might require aij (·) ∈ C 1−δ0j (D(Gw )), with w, w0 ∈ D(Gw ) as in (5.16). Theorem 5.13. Existence and convergence for the nonconforming FEM: Let the nonlinear elliptic problem in (5.42) satisfy Condition H. In particular, let G(u0 ) = 0, for u0 ∈ D(G) ∩ U0 , or F (u0 ) = (G, BD )(u0 ) = 0, for u0 ∈ Br (u0 ) ∩ D(G) ⊂ U, have an isolated solution, with boundedly invertible G (u0 ) : U0 → V, or F (u0 ) : U → VˆΠ , let G be continuously diﬀerentiable in Br (u0 ) ∩ D(G), e.g. by (5.109), and let the

332

5. Nonconforming ﬁnite element methods

domain satisfy (5.99) with Ω ∈ C 2 . Choose U h = Sd1 Tch , d ≥ 2n + 1 (see (5.3)), on a quasiuniform triangulation. 1. Then the nonconforming FEM (5.62), so

F h = Qhc,Π F |U h : D(F h ) ⊂ U h ∩ D(G) → V h × Vbh , uh0 ∈ D(F h ) s.t. F h uh0 = 0, is stable and consistent in P h u0 , and convergent. It has, for small enough h, a unique solution uh0 near u0 . We obtain, cf. (5.67), for u0 ∈ H (Ω), > 2, for the nonconforming FEM, and C p , 2 ≤ p ≤ 5, domains, h P u0 − uh0 h ≤ Chmin{−2,p} u0 H (Ω) , (5.110) U respectively, with p = 2 for polyhedral Ωh and p > 2 for better curved Ωhc approximations for Ω ∈ C 2 , cf. Conjectures 4.40 and 4.41. 2. F (u0 ) ∈ L(W, W × WD ) and F (u0 ) ∈ L(U, V × VD ) simultaneously satisfy the Fredholm alternative. 3. For quadrature approximations, exactly reproducing polynomials of degree < k ˜h0 = 0 satiswith k > 2(d − 1) > d > 2, the solution u ˜h0 of an approximate F˜ h u ﬁes the same estimate as in (5.110); see Subsection 5.4.4 and 5.4.5 for m > 1 or for systems. Remark 5.14. For the above F (see (5.18), (5.20), (5.73)), and d are related by (4.62). With the ellipticity assumption (5.20), the previous (5.73), (5.74), (5.100) are implied by (5.109). Ω convex polyhedral or in C 2 are necessary for the regularity and the extension operators.

5.2.8

Discretization of equations and systems of order 2m

The essential ideas have been presented above for equations of second order. So we only have to generalize our FEM formulated in (5.6) to equations and systems of orders 2 and 2m. The basic facts for success are the existence of C 2m−1 FEs with local support, combined with an unusual splitting. In Theorems 4.36–4.39 we have formulated and completely proved that for m = 1 in R2 on polyhedral domains. An extension to systems q > 1 is obvious. Extensions to m > 1, n ≥ 2, are the goal for future research in Davydov’s For the time being, we have formulated these results for smooth group. splines, Sdr Tch , needed for m > 1, for the case n ≥ 2, in Conjecture 4.40. Any smooth Ω ∈ C 2m have to be approximated now by polyhedral Ωh in Rn . For future results this is extended in Conjecture 4.41 to approximating curved boundaries, Ωhc ≈ Ω ∈ C 2m in Rn . The obvious diﬀerence in changing 1 to m > 1, is the change from the H 1 (Ω), H 2 (Ω) setting to the H m (Ω, Rq ), H 2m (Ω, Rq ) setting and the updated product spaces. We again split F into the diﬀerential and a more complicated boundary operator. The less obvious diﬀerence is the fact that for elliptic systems and equations of higher order, the simple equivalence of ellipticity (2.136) for m = q = 1 and W0 -coercivity of the principal part, see Theorem 2.43 2., is no longer correct for the general case

5.2. FEMs for fully nonlinear elliptic problems

333

m, q ≥ 1. This has been discussed in Subsections 2.4.4, 2.6.3, 2.6.5, Theorems 2.43, 2.89, 2.104. We will repeat the essential conditions for ellipticity for m, q ≥ 1 below, when we introduce the derivative of F in (5.128). Secondly, the conditions for the regularity of the exact solutions of linear systems considerably diﬀer for the diﬀerent cases, see the summary in (5.140) ﬀ., and Subsections 2.4.4, 2.6.3, 2.6.5, Theorems 2.45, 2.47, 2.91–2.93, 2.95, 2.100, 2.101, 2.108. Finally we generalize the semiconforming FEMs for convex polyhedral domains from Subsection 5.2.4 to m = 1, q ≥ 1. In fact, for these domains only for m = 1, q ≥ 1 is the regularity of the FE solutions correct. Compared to (5.32) only the functions uh0 have to be replaced, for q ≥ 1, by uh0 . So we reformulate the semiconforming FEM for m = 1 < q in (5.149). For the other cases we have to impose speciﬁc conditions, in particular Ω ∈ C 2m , cf. (5.140) ﬀ. These three aspects have to be considered in the following discussion. We present the cases m, q ≥ 1 all in parallel. The diﬀerences in this and the beginning of the next subsection are minimal. A direct comparison of the following diﬀerences in, e.g. (5.132) and (5.140) ﬀ. is more informative this way, than it would be for separate formulations. So we only deﬁne the updated diﬀerential and boundary operators, the original and approximating spaces, projectors, FEMs in the diﬀerent equivalent forms and their solutions. The proofs of consistency and stability, now based upon the updated conditions, require only the minor modiﬁcations reported below. So we strongly restrict the presentation here. Although we have presented the results for second order equations, we include them here once more as the case m = q = 1. This allows directly comparing all diﬀerent cases. Fully nonlinear elliptic systems and their discretization In the following we use the notation u, v , . . . , for deﬁning Rq valued functions. We need the following operators, extensions, cf. (5.5), and spaces deﬁned on Ω and Ωhc : G( u(·)) := Gw (·, u, ∇ u, . . . , ∇2m u)(·), G : D(G) ⊆ U ∪ Uc → V ∪ Vc , (5.111) j ∂ u(x) |∂Ω ∀0 ≤ j ≤ m − 1 (5.112) BD : U → VD or BD : W → WD , BD u(x) := ∂ν j with U = H 2m (Ω, Rq ) ⊂ W = H m ⊂ V = L2 , W0 := { u ∈ W : BD u = 0}, U0 := W0 ∩ U, W = H −m , VD = BD U ⊂ WD := BD W ⊂ Vb = (L2 (∂Ω, Rq ))m and similarly, deﬁned on Ωhc : Uc = H 2m Tch , Rq ⊂ Wc ⊂ Vc , Wc,0 , Uc,0 , m . (5.113) Wc , Vc,D ⊂ Wc,D ⊂ Vc,b = L2 ∂Ωhc , Rq As a consequence of the consistent notation for m = q = 1 and m, q ≥ 1, the product operator F = (G, BD ), with the diﬀerential and the boundary operators, requires only obvious changes: There are essentially two diﬀerences: Functions u, . . . , have to be replaced by u(x) ∈ Rq , . . . , and BD u(x) ∈ Rm×q . This remains correct for all other operators as well, e.g. the projectors in (5.55), . . . F is called elliptic simultaneously

334

5. Nonconforming ﬁnite element methods

with G, and maps D(G) ⊂ U onto the Cartesian product VΠ , cf. (5.111), (5.112), F := (G, BD ) : D(G) ⊂ U → VΠ := V × VD ⊂ V × Vb ; F ( u0 ) := G( u0 ), BD u0 = 0, (5.114) with a locally unique solution, u0 , and a boundedly invertible F ( u0 ) ∈ L(U, V × VD ). We mainly consider trivial right-hand sides, 0 = ( 0, 0), characterizing vanishing G( u0 ) and BD u0 by testing with elements v ∈ V and vb ∈ Vb , respectively. For applying general discretization methods, see Chapter 3, we choose the approxh , . . . , combine F h with the corresponding projectors P h ∈ imating spaces, U h , VΠ h h L(U, U h ), and QhΠ ∈ L(VΠ , VΠ ), Qhc,Π ∈ L(Vc,Π , VΠ ) to get the sequences h U h , VΠ = V h × Vbh , P h , QhΠ , Qhc,Π , F h , with inf{0 < h ∈ H} = 0 for lim . h∈H

h→0

Theorems 4.36 and 4.38 show that the sequences, deﬁned on Ωhc , see (4.77), (5.115) U h = W h = Sd2m−1 Tch , Rq = Sd2m−1 Tch = S0h ⊕ Sbh ; h m h 2m−1 h h h Tc ∩ Wc,0 , Vb := Sb |∂Ωhc , or V = S0 = Sd h h h := BD Sb , d ≥ (2m − 1)2n + 1, h ∈ H, VD

with the corresponding norms, cf. (5.27), approximate the Banach spaces U, W, V, Vb deﬁned on Ω and ∂Ω. The v h ∈ Vbh violate the original boundary conditions. Again, the terms uh , G( uh ), cf. (5.5), and BD uh are deﬁned on Ωhc and ∂Ωhc instead of the original Ω and ∂Ω, cf. (5.5), motivating the choice our projectors. P h ∈ L(U, U h ) : ∀ u ∈ U lim P h uU h = uU , we choose P h u := I h Ec u. (5.116) h→0

h For the discretization we deﬁne analogously new Gh , BD in (5.117), (5.118). This will be combined in Theorems 5.15, 5.18 to obtain stability and convergence for the inhomogeneous case as well. than P h are the projectors to the image spaces, e.g. the QhΠ ∈ More hdelicate h L VΠ , VΠ , motivated by F . Thus we start updating the previous FE version by the FE equations in equivalent formulations, see (5.114): deﬁne for uh0 ∈ U h the (5.117) sequence Gh uh0 = 0 ⇐⇒ G uh0 , v h Vc = 0 ∀ v h ∈ V h ⊂ Vc ⇐⇒ V h ⊥Vc Gh uh0 = G uh0 V h with uh0 ∈ D(Gh ) ⊂ Uc .

Similarly, the boundary conditions (5.114) are tested, see the equivalence (5.6), as j h ∂ u0 h h h h | h ∀0 ≤ j ≤ m − 1 = 0 ⇔ vbh , BD u0 V = 0∀ vbh ∈ Vbh . BD u0 := c,b ∂ν j ∂Ωc (5.118) Projectors and discretization So we update the spaces and norms on Ω and Ωhc , omitting Rq , cf. (5.38), (5.113):

5.2. FEMs for fully nonlinear elliptic problems

335

h cf. (5.113)–(5.115), (5.46) for · U h , For U, W, V, U0 , VD , U h , W h , V h , Ubh , VD

VΠ := V × VD ⊂ V × Vb , VD = BB U, Vb := (L2 (∂Ω))m with V = V , Vb = V ⊃ VD m h Vc,Π := Vc × Vc,D ⊂ Vc × Vc,b , Vc,D = BD Uc , Vc,b := L2 ∂Ωhc , Vc = Vc , . . . , with ( u, ub ), ( v h , vb ) V := ( u, v )V + ( ub , vb )Vb and ( u, ub ), ( v h , vb ) V versus Π c,Π m h h h VΠ = V h × Vbh := S0h × Sbh |∂Ωhc , VΠ = V h × Vbh := S0h × Vbh = VΠ with B

A V h = V h and uh , uhb , v h , vbh := uh , v h V h ×V h + uhb , vbh V h . h h VΠ ×VΠ

b

(5.119) The two components (5.117), (5.118) of F h motivate the appropriate deﬁnitions of the projectors. Compared to the original form (5.55), . . ., we only replace functions u, . . . , by u ∈ Rq , . . . So we do not repeat these formulas, except Qhc,b , similarly Qhb , for demonstrating BD u ∈ Rm×q , e.g. by

h h h Ec u or := BD u ∈ Vc,D as Qhc,b uc,b , vbh Qhc,b ∈ L Vc,b , Vbh , uc,b := BD h h h h − uc,b , vbh V = 0 ∀ vbh ∈ Vbh , now with Qhc,b BD u = BD u ∀ uh ∈ U h c,b and lim Qhc,b uc,b V h = lim uc,b Vc,b , lim Qhb ub V h = lim ub Vb . h→0

h→0

b

h→0

b

h→0

Vc,b

(5.120) (5.121)

The above changes yield the updated deﬁnition of product projectors, see (5.52), e.g.

h , similarly Qhc,Π , QhΠ := Qh , Qhb ∈ L V × VD , V h × Vbh = L V × VD , VΠ (5.122) with the corresponding limits of norms. The corresponding discrete operators, F h : U h → V h × Vbh , see (5.48)–(5.50), (5.52), (5.60) deﬁne the FE solutions uh0 ∈ U h as : h; uh0 ∈ D(G) ∩ U h : F h uh0 = 0 ⇔ F h uh0 , vΠ V h ×V h c,Π

Π

h h h = Fh uh0 , vΠ = 0 ∀ vΠ ∈ VΠ Vh

c,Π

h h h ⇐⇒ G uh0 , v h V + BD u0 , vb V h = 0 ∀ v h ∈ V h ⊂ Vc ∀ vbh ∈ Vbh , where c

b

F : D(F ) = D(G ) ∩ U → V h × Vbh with (5.123) h h h h F h ( uh ) = Gh ( uh ), BD u = Qhc,Π Fh ( uh ) = Qhc,Π G( uh ), BD u . h

h

h

h

With these listed modiﬁcations the above Theorem 5.2 and Remark 5.3 remain correct, based upon the following Condition Hm: Generalized Condition H: In Condition H replace H 2 (Ω ∪ Ωhc ) by H 2m (Ω ∪ Ωh , R2q ) and (5.19)–(5.21), (5.73)–(5.75) by (5.128)–(5.132), (5.140)– (5.148), everything else remains unchanged.

336 5.2.9

5. Nonconforming ﬁnite element methods

Consistency, stability and convergence for m,q ≥ 1

By Theorem 3.21, we have to prove consistency and stability, by Theorem 5.6, only for the linearized operator. The approach and the proofs are nearly identical with m = q = 1 up to the above minor modiﬁcations. So we reformulate the theorems and indicate the changes in the proofs. They are mainly caused by the modiﬁed conditions, guaranteeing the coercivity of and regularity results for the linearized operator. Wherever appropriate we refer to the conforming FEM for m = 1, q ≥ 1, cf. Theorems 5.2 and 2.94, (2.358). Theorem 5.15. Consistency for semiconforming and nonconforming FEs: We require the conditions in Theorem 4.36, and Condition Hm, in particular an exact smooth solution u0 ∈ H (Ω, Rq ), > 2m, a Lipschitz-continuous Gw in (5.16) with a global L, Gw (·) ∈ CL (D(Gw ) ∩ (Ω × Br,U ( u0 ))) and d ≥ (2m − 1)n2 + 1.

(5.124)

1. For m = 1, q ≥ 1 and convex, polyhedral Ω, the Gh is consistent with G in P h u0 and

Gh P h u0 − Qh G u0 V h ≤ KLh−2 u0 H (Ω,Rq ) .

(5.125)

2. For m, q ≥ 1 and Ω ∈ C 2m , the F h is consistent with F in P h u0 and, with p = 2 for the polyhedral Ωh and p > 2 for curved Ωhc , cf. Conjectures 4.40, 4.41, h h F P u0 − QhΠ F u0

h VΠ

≤ CLhmin{−2m,p} u0 H (Ω,Rq ) .

Proof. Replace h−2 , . . . , by h−2m , . . . , in the proof of Theorem 5.4.

(5.126)

The linearized operator We assume a continuously diﬀerentiable Gw . Again we sometimes integrate the boundary conditions into the spaces U0 , W0 and Ubh , Wbh , see (5.113). We distinguish the strong and weak coinciding bilinear forms as (·, ·) and a(·, ·), see Proposition 5.16. We formulate two diﬀerent FEMs for the linearized problem applying the new FEM (5.127), to its strong form and the standard FEM (5.128) to the weak form. Equations and systems of order 2m with 1 ≤ m, q : We determine the solutions u0 (smooth) and uh0 by F ( u0 ) = (G( u0 )(·) = Gw (·, u0 (·), . . . , ∇2m u0 (·)), BD u0 ) = 0 ∈ V × VD , and h h F h uh0 := Gh uh0 , BD (5.127) u0 = 0 h h h h h h h = v h , vbh ∈ VΠ . ⇐⇒ G u0 , v L2 (T h ,Rq ) + BD u0 , vb V h = 0 ∀ vΠ c

b

5.2. FEMs for fully nonlinear elliptic problems

337

h h The stability uses the linearization (F h ) ( uh0 ) uh = (G (Ec u0 ) uh , BD u ), starting with h h h h h 0 := w uh and Aαβ = Aαβ (w 0 ), cf. (5.17), G (Ec u0 ) and w 0 h γ h h β , |β| ≤ 2m) w ∂ϑγ Gw (·, ϑ 0 ∂ u (5.128) G (Ec u0 ) u = |γ|≤2m

=:

(−1)|α| ∂ α Ahαβ ∂ β uh , ahs ( uh , v h ) := (G (Ec u0 ) uh , v h )Vc and

|α|,|β|≤m

ah ( uh , v h ) :=

Ωh c |α|,|β|≤m

Ahαβ ∂ β uh , ∂ α v

q

dx = G (Ec u0 ) uh , v h W ×Wc c

∀ uh ∈ U h , v h ∈ V h with Aαβ ∈ W |α|,∞ and Aαβ ∈ L∞ Tch , Rq×q . The imposed conditions Aαβ ∈ W |α|,∞ and Aαβ ∈ L∞ (Tch , Rq×q ) imply well-deﬁned continuous bilinear forms as (·, ·) and ah (·, ·), respectively Again we split a(u, v), induced by A = G (u0 ), into the sum of its principal part, ap (u, v), inducing B, and its complement c(u, v), inducing C, a compact perturbation as A, B, C ∈ L(W, W ), a(u, v) = ap (u, v) + c(u, v) = G (u0 )u, vW ×W = (B + C)u, vW ×W .

(5.129)

Here we come back to an essential diﬀerence to the case m = q = 1, with equivalent ellipticity and coercivity of the principal part. For m, q ≥ 1, we impose β )ϑ α ≥ λ|Θ| 2m , or ∈ Rn×q : (Aαβ (x)ϑ (5.130) ∀x ∈ Ω, Θ |α|=|β|=m

∀ϑ ∈ Rn , η ∈ Cq : η¯T Ap (x, ϑ)η = η¯T

Aαβ (x)ϑβ ϑα η ≥ λ|ϑ|2m |η|2 . (5.131)

|α|=|β|=m

Notice that (5.130), (5.131) reduce to (2.136), for m = q = 1. Now we require Aαβ ∈ L∞ (Ω, Rq×q ) for |α|, |β| ≤ m with Aαβ = Aij for m = 1, ¯ for |α| = |β| = m > 1, and Aαβ = aαβ for q = 1, with Aαβ ∈ C(Ω)

(5.132)

the strong Legendre condition, (5.130), for m ≥ 1, q = 1 and m = 1, q ≥ 1, and the strong Legendre–Hadamard condition, (5.131), for m > 1, q > 1. Then the principal part ap ( u, v ) is W0 -coercive and B is boundedly invertible on W0 . This is a consequence of (2.22), Theorems 2.43, 2.89, 2.104. Thus G (u0 ) satisﬁes the Fredholm alternative on W0 . In addition to (5.115), we need for W0 (5.133) U h = W h ⊃ V h and Ubh = Wbh := S0h = V h = U h ∩ H0m Tch , Rq . We contrast the two diﬀerent FEMs for the strong as (·, ·) and the weak bilinear form a(·, ·) and its standard FEM. We determine uh1 ∈ Ubh and uh1 ∈ Wbh such that uh1 ∈ Ubh : ahs uh1 , v h = G (Ec u0 ) uh1 , v h V = (Ec f , v h )Vc ∀ v h ∈ V h , f ∈ V, (5.134) c h h : ; h h h h h u1 ∈ Wb : a u1 , v = G (Ec u0 ) u1 , v = Ec f , v h ∀ v h ∈ Wbh , f ∈ W , (5.135)

338

5. Nonconforming ﬁnite element methods

with ·, · = ·, ·Wc ×Wc . Equations (5.78), (5.79) can and will be similarly re

interpreted by replacing the above projectors, QhΠ , . . . , by the previous new ones QΠh , . . . , cf. (5.120). Coinciding strong and weak bilinear form and stability of the weak form Proposition 5.16. For U h , V h in (5.119) and Aα,β ∈ W |α|,∞ (Tch , Rq×q ) we obtain ∀ uh ∈ U h , v h ∈ V h = S0h = Ubh ahs ( uh , v h ) = (G (Ec u0 ) uh , v h )Vc = ah ( uh , v h ) = G (Ec u0 ) uh , v h Wc ×Wc .

(5.136)

Proof. This is generalized by induction of the proof for Proposition 5.7.

Remark 5.17. The technique of the proof for Proposition 5.7, can be applied to quasilinear equations and systems as well. This yields instead of (5.106), and ∀ uh ∈ U h , v h ∈ V h ⊂ W0 , h h (−1)|α| ∂ α (Aα (·, uh , . . . , ∇m uh )) v h dx (G( u ), v )Vc = Ωh c |α|≤m

= G( uh ), v h Wc ×Wc := Aα (·, uh , . . . , ∇m uh )∂ α v h dx.

(5.137)

Ωh c |α|≤1

From the diﬀerential operators, we return to the two-component version of our operators AU := (As , BD ) : U → VΠ = V × VD ⊂ V × Vb , AU ∈ L(U, V × VD ) and

(5.138)

= W × WD ⊂ W × Vb , AW ∈ L(W, W × WD ), AW := (A, BD ) : W → WΠ

where the “weak spaces” W, . . . and the following projectors represent obvious gener alization of (5.85), (5.87), (5.88) to our case. Similarly to the above F h = Qhc,Π F |U h we reformulate (5.134) and (5.135) for the strong and weak linear AU and AW and ∈ W , the solution uh from determine, for (f , φ) 1 Π h h AhU = Qhc,Π AU |U h : uh1 ∈ U h : AhU uh1 = Qhc,Π Ahs uh1 , BD u1 = QhΠ (f , φ), h h h h h ⇐⇒ A u1 , BD u1 = QΠh (f , φ) AW |W h : uh1 ∈ W h : AhW uh1 = Qc,Π AhW = Qc,Π B A h h h v h Qc,Π AW uh1 − QΠh (f , φ), = 0∀ vΠ ∈ WΠ , Aα,β ∈ CL Tch ∪ Ωhc , Π h h WΠ ×WΠ

see (5.138). Figure 5.2 remains unchanged. For the bounded and equibounded AW = (A, BD ), BW = (B, BD ), CW = (C, BD ), h h , CW ∈ L(W h , W h × Vbh ) we use the notation in (5.79). Theorems 5.8 and AhW , BW and 5.9, stating stability of the weak form and convergence for the principal part and “weak” stability for an invertible AW , remain correct if the ellipticity is replaced by

5.2. FEMs for fully nonlinear elliptic problems

339

(5.132) and the approximation spaces in (5.77) by (5.133). Then both proofs remain unchanged except replacing functions, e.g. u by u. Regularity for FE solutions For lifting the preceding stability result with respect to the weak forms back to the strong form, we need modiﬁed regularity results for the exact solutions of linear equations. These are estimates of the H 2m (Tch , Rq ) instead of the usual H m norm for the right-hand side in L2 × H 2m−1/2 × · · · × H m+1/2 (Tch , Rq ) instead of the usual H −m × H m−1/2 × · · · × H 1/2 (Tch , Rq ). The basic assumptions for all cases are the bounded invertibility of AW and the coercivity of its principal part, by the conditions in (5.132). So we require for A: A strongly elliptic, W0 -coercive principal part, (AW )−1 ∈ L(W × WD , W0 ). (5.139) For the diﬀerent cases m ≥ 1, q ≥ 1 we impose, cf. Theorems 2.45, 2.46, 2.47, 2.91–2.93, 2.94, 2.108 For m = q = 1 letΩ ∈ C 2 , ai,j ∈ W 1−δj0 ,∞ Tch , 0 ≤ i, j ≤ n, (f, φ) ∈ V × VD (5.140) h 2m γ ∞ m > q = 1 let Ω ∈ C , ∀α, β with |α|, |β| ≤ m, ∂ aα,β ∈ L Tc ∀γ with (5.141) |γ| ≤ |β|, else aα,β ∈ L∞ Tch , (f, φ) ∈ V × VD , (5.142) m = 1 < q let Ω ∈ C 2 , Akl ∈ W 1−δ0k ,∞ Tch , f ∈ V ∀k, l, or m = 1 < q let n = 2, 3, Ω be convex, polyhedral, and satisfy Theorem 2.94, and Akl satisfy (2.358).

(5.143)

For systems of order 2m, we need the modiﬁed strong Legendre condition, cf.(2.401), ∃λ > 0 : ∀x ∈ Ω, ∀ϑ ∈ Rn : Aαβ ϑβ ϑα ≥ λ|ϑ|2m I, (5.144) |α|=|β|=m

with the identity matrix I.

For m, q > 1, assume (5.144), Ω ∈ C ∞ , Aα,β ∈ C ∞ Tch ∀α, β, f ∈ L2 Tch . (5.145)

These analytic regularity results show that (5.139)–(5.145) imply V . ∈ V × VD ⇒ u1 U ≤ C f V + φ V = C(f , φ) AW u1 = (f , φ) D Π

(5.146)

With these modiﬁcations Lemma 5.10 in particular (5.101), (5.102) hold. In the proof for our general case, we only have to replace h, h−1 in (5.108) by hm , h−m . Stability for the strong linear operator and convergence We are nearly done. Collecting the results from the last two subsubsections, we get the “strong” stability of the linear operator (only formulated analogously) and then ﬁnally the existence and convergence of discrete solutions for fully nonlinear problems.

340

5. Nonconforming ﬁnite element methods

After all these preparations we are ready to prove the stability of the linearized strong operator and thus the convergence of the method. We had to impose many diﬀerent conditions for the diﬀerent areas of ellipticity and W0 -coercivity, approximation, coinciding weak and strong bilinear forms and regularity. We recall the notation Aαβ = aαβ for m > 1, q = 1, Aαβ = Aij for m = 1 < q, and Aαβ = aij , i, j = 0, . . . , n for m = 1 = q. For the operator and the domain, we imposed (5.130), the strong Legendre condition for m ≥ 1, q = 1 and m = 1, q ≥ 1, and (5.131) and (5.144), the strong Legendre–Hadamard and the modiﬁed Legendre condition for regularity, for m > 1, q > 1, summarized as ∀m, q ≥ 1 : U, U h = Sd2m−1 Tch .., in (5.113), (5.115), d ≥ (2m − 1)2n + 1 (5.147) and (5.130), Ω ∈ C 2m , Aαβ ∈ W |α|,∞ Tch , Rq×q ∀|α|, |β| ≤ m, except m, q > 1, equations : m = q = 1 : aij ∈ W 1−δi,0 ,∞ (Ω), ∀0 ≤ i, j ≤ n, systems : m = 1 < q : Aij ∈ W 1−δi,0 ,∞ (Ω, Rq×q ), ∀0 ≤ i, j ≤ n, ¯ equations : m > q = 1 :, Ω ∈ C 2m , aαβ ∈ C(Ω)∀|α|, |β| = m, ¯ Rq×q )∀|α| = |β| ≤ m, systems : m, q > 1 : Ω ∈ C ∞ , (5.131), (5.144), Aαβ ∈ C ∞ (Ω, ﬁnally let m = 1 < q, n = 2, 3, Ω be convex, polyhedral, and satisfy Theorem 2.94, and let Akl satisfy (2.358). The consistency in Theorem 5.4 needed an exact solution u0 ∈ H (Ω, Rq ), > 2m, and a Lipschitz-continuous Gw in (5.16) with a global constant L, see (5.66). With the notation in (5.73), (5.147), we require continuity with respect to the variable u, via w u , see (5.17), (5.111), for Theorem 5.13 below, such that u ) ∈ C |α|,∞ (W ∩ Br ( u0 )) and Aαβ (w u ) ∈ C |α|,∞ (U ∩ Br ( u0 )), (5.148) Aαβ (w ∀|α|, |β| ≤ m, yield G (u) : W → W , and G (u) : U → V ∈ C(Br ( u0 )). Theorem 5.18. Discrete solutions uniquely exist and converge for m, q ≥ 1: Let the nonlinear elliptic problem in (5.114), F ( u0 ) = 0 with F = (G, BD ) : U → VΠ = V × VD , equivalently G( u0 ) = 0 with G : U0 → V, have an isolated solution u0 ∈ D(G) ∩ U0 . Let G and F satisfy (5.132), (5.139)–(5.145) (5.147)–(5.148), let G ( u0 ) : U0 → V, hence F ( u0 ) : U → VΠ be boundedly invertible and G, hence F as well, be continuously diﬀerentiable in Br ( u0 ) ∩ D(G), satisfying Condition Hm. In particular, deﬁne the FEM for m = 1, q ≥ 1 and convex, polyhedral Ω ∈ Rn , with n = 2, 3 for q > 1, as, cf. (5.123), uh0 ∈ D(Gh ) ⊂ Ubh s.t. Gh uh0 = 0, Gh = Qh G|Ubh : Ubh → V h , (5.149) and for the other cases in (5.147) as uh0 ∈ D(F h ) ⊂ U h s.t. F h uh0 = 0, F h = Qhc,Π F |U h : U h → V h × Vbh .

(5.150)

5.2. FEMs for fully nonlinear elliptic problems

341

1. Then these FEMs are stable in P h u0 , consistent and convergent. They have, for small enough h, a unique solution uh0 near u0 . n ⊂ S h , > 2m, cf. (4.62), 2. We get, see (5.67), for u0 ∈ H (Ω, Rq ), P−1 h P u0 − uh0 h ≤ Chmin{−2m,p} u0 H (Ω,Rq ) , (5.151) U with p = 2 for polygonal Ωh and p > 2 for curved Ωhc . 3. For m = 1, q ≥ 1 and convex, polyhedral Ω we obtain instead h P u0 − uh0 h ≤ KLh−2 u0 H (Ω,Rq ) . U

(5.152)

4. Simultaneously F ( u0 ) ∈ L(W, W × WD ) and F ( u0 ) ∈ L(U, V × VD ) satisfy the Fredholm alternative. 5.2.10

Numerical solution of the FE equations with Newton’s method

We have indicated this problem for the (highly) nonlinear systems, cf. (5.29), (5.63), (5.149), (5.150), already in Subsection 5.2.2 4. We studied it for general operators and discretization methods in Section 3.7. So we discuss here only the speciﬁcs for the fully nonlinear case. We have to determine the FE solution uh0 ∈ U h ⊂ C 1 (Ωhc ) such that : h h h ; F u0 , vΠ = 0 ⇐⇒ G uh0 , v h V = 0 ∀v h ∈ V h , uh0 , vbh V = 0 ∀vbh ∈ Vbh . c

c,b

(5.153) We restrict the presentation here to second order equations. Let Condition H be satisﬁed The general notation, introduced in (5.113), allow a relatively easy generalization to the other cases. In this context again local and stable bases and stable splittings for S h are important; see Theorems 4.36 and 4.36: / . m0 m1 h h h h μ μ (5.154) cμ s + cμ s . S = S0 + Sb = u = μ=1

μ=m0 +1

We insert ∈ S into (5.153) and test with v h = sj , j = 1, . . . , m0 , and the h i i+m0 , i = 1, . . . , m1 − m0 . So we obtain a nonlinear boundary term with vb = sb = s system of m equations for the m unknowns cμ , μ = 1, . . . , m1 . This allows formulating the following algorithm: uh0

h

G(uh0 )

Algorithm 5.1. Nonstandard FEM for elliptic equations INPUT: G, Tch , Sd1 Tch = S0h + Sbh (Diﬀerential operator, subdivision and a stable splitting for the FE space of dimension m1 on Ωh c ); COMPUTATION: FOR j = 1, . . . , m0 , i = 1, . . . , m1 − m0 /* Determine the m1 equations in (5.153) */ : h j i ; h j h i compute the Fh uh 0 , vΠ = s , sb V h ×V h = G u0 , s Vc + u0 , sb Vc,b ; Π Π : h; OUTPUT: Fh uh 0 , vΠ V h ×V h , j = 1, . . . , m0 , i = 1, . . . , m1 − m0 Π

Π

(Nonlinear system of m1 equations in (5.63)).

342

5. Nonconforming ﬁnite element methods

For computing the terms in (5.153) the multilevel results of Davydov [263] and Davydov and Stevenson [268], with multiresolution techniques can be used. Sparse nonlinear systems After calculating this system (5.153) has to be solved. The most important tools are continuation and discrete Newton methods. They require the solution of the corresponding linear sparse FE equations with the defects for the nonlinear FE equations, F h uhν , uhν ≈ uh0 as a special input, see below. For these systems, we can employ the standard eﬃcient solvers for weak formulations byProposition 5.7. But the usual FEs in C 0 (Ωh ) have to be replaced by FEs in C 1 Ωhc . Nevertheless the question is interesting, whether (5.153) represents a sparse nonlinear system, with only a few nontrivial terms for the unknowns cμ in each equation. Special conditions guarantee this, see Proposition 5.19. K has the special k-linear algebraic property if K(u) := g(a1 (u), . . . , as (u)) with k -linear a (u1 , . . . , uk ) : Ωhc → R, k ≤ k , (5.155) uj : Ωhc → R, = 1, . . . , s, a (u1 , . . . , uk ) = 0 if supp u1 ∩ · · · ∩ supp uk = ∅, with k := maxs=1 {k } ≤ k , a (u) := a (u, . . . , u) and a special algebraic function g : Rs → R, deﬁned by sums of constants, the a (u) and roots. For Proposition 5.19 and Sd1 Tch we introduce CS1 :=

(5.156)

max { number of triangles T ∈ Tch s.t.supp sμ ∩ T = ∅}

μ=1,...,m1

CSk :=

max

νi =1,...,m1

{ number of k − tuples (ν1 . . . , νk ) s.t.

(5.157)

supp sν1 ∩ · · · ∩ supp sνk = ∅ for νi = νj for i = j, and i, j = 1, · · · , k} k Proposition 5.19. For Sd1 Tch we get the upper bound CSk ≤ CS1 . If K has the k+1 special k-linear algebraic property (5.155), (5.156) and if CS1 m1 , then (5.153) is a sparse nonlinear system. For k = 1, this yields sparse linear systems. Remark 5.20. In some sense, it is disappointing that this proposition only discusses one special case. This is compensated by the sparse linear FE equations in continuation and discrete Newton methods, obtained by linearizing (5.6) into equations (5.19). Thus it does not matter too much that (5.155), (5.156) is a strong condition nor did we try to get the sharpest possible results. Products are excluded, since a1 a2 would be (k1 + k2 )-linear. But Proposition 5.19 admits at least important examples: (2.308), for appropriate f and (2.310) are included. Generalizations are certainly possible. k Proof. Obviously CSk ≤ CS1 and by testing a linear K(uh ) with a ﬁxed sj , we ﬁnd at most CS1 m1 nontrivial coeﬃcients, thus we obtain a sparse linear system. We restrict the proof to k = 2; k > 2 is then clear. A bilinear a1 evaluated in x is: h

h

a1 (u )(x) = a1 (u =

m1 μ=1

μ

h

cμ s , u =

m1 ν=1

ν

cν s )(x) =

m1 μ,ν=1

∗

cμ cν a1 (sμ , sν )(x), (5.158)

5.2. FEMs for fully nonlinear elliptic problems

343

where ∗ indicates that, for a ﬁxed x ∈ Ω, we only need those terms with x ∈ supp sμ ∩ 2 supp sν = ∅, so at most CS2 = CS1 terms. For all other x, the a1 (sμ , sν )(x) = 0 by (5.155), (5.156). Now we test with a ﬁxed sj and obtain

h

j

a1 (u )(x)s (x)dx = Ωh c

m1

∗

cμ cν a1 (sμ , sν )(x)sj (x)dx.

(5.159)

Ωh c μ,ν=1

Each of the CS1 triangles T , with supp sj ∩ T = ∅, contributes nontrivially at most μ 1 ∩ for T ∈ supp sμ ∩ supp sν = ∅. The at most CS2 triangles in ∪m μ,ν=1 supp s 3 ν 3 1 supp s = ∅, yield at most CS = CS nontrivial terms in (5.159). Solving the nonlinear systems with Newton’s method, MIP In Section 3.7 we have presented continuation methods and the mesh independence priniple for Newton’s method. So we give only a very short summary here. Equation (5.153) is solved by the well-known continuation and Newton methods. We formulate the discrete Newton method and its quadratic convergence as special case of the preceding mesh independence principle (MIP), cf. Theorem 3.40 and Corollary 3.41. Both methods require the solution of the corresponding linearized sparse FE equations, with the defect or residuum, F h uhν , uhν ≈ uh0 . The uh |∂Ωhc = 0 and Proposition 5.7 allow standard eﬃcient solvers with FEs in C 0 (Ωh ) replaced by h 1 FEs C Ωc . For using these methods for diﬀerential equations and systems of order 2m, they can be transformed into larger equations andsystems of order 2. Otherwise the FEs in C 0 (Ωh ) have to be replaced by C 2m−1 Ωhc and appropriate solvers have to be used. Eﬃcient strategies combining continuation with the MIP on diﬀerent discretization levels have been studied by Allgower and B¨ ohmer [9, 10]. Currently, multiresolution techniques are available for polyhedral Ω in Rn and d ≥ 2n + 1; see Davydov’s papers, e.g., [263, 267, For Ω ∈ C 2 and the necessary Ωhc with curved boundaries for h268]. 1 general Sd Tc this is still an open problem. For the speciﬁc cases in Proposition 5.19, (5.153) represents a sparse nonlinear system. The MIP, the Newton–Kantorowich method, and the convergence of our FEM require a unique solution u0 , such that F (u0 ) is boundedly invertible. So we assume F (u0 ) = 0 with F (·) ∈ CL (Br (u0 )), (F (u0 ))−1 ∈ L(V, U).

(5.160)

Starting near uh0 ≈ u0 , we apply Newton’s method to F h . By Theorem 5.12, the h h F ui : U h → V h are equiboundedly invertible; hence −1 F h uhi , uhi+1 := uhi − F h uhi

i = 1, . . . , is well deﬁned. (5.161)

In addition to (5.160), the MIP, cf. Subsection 3.7.2, and Allgower and B¨ ohmer [9, 10] and Allgower et al. [17], requires a linear FEM as in (5.34), satisﬁed for our FEMs. Then the Newton method converges quadratically; cf. Theorem 3.40. Equations (5.110) and (5.151) show that for u0 ∈ Us = H (Ω), > 2, the FEM converges with orders p = − 2m > 0 and p := min{ − 2m, p} > 0. Then, with the

344

5. Nonconforming ﬁnite element methods

h above projectors P h = I h Ec : U → U h , QhΠ : VˆΠ → VΠ (see (5.51), (5.60)), we impose the condition (5.162) F |Us : D(F ) ∩ Us := H (Ω) → H −2m (Ω).

Theorem 5.21. MIP; see Theorem 3.40: In addition to the conditions in Theorems 5.13 and 5.18, in particular Condion H, with convergence in (5.110) of order p = − 2m > 0 or p = min{ − 2m, p} > 0 we assume (5.160), (5.162). Start the Newton processes for F , by omitting all upper indices h in (5.161), and for F h with the suﬃciently good starting values u1 ∈ Br (u0 ) ∩ D(F ) ∩ Us and uh1 := P h u1 . Then, for h ≤ h0 = (min{r, (SCL)−1 }/(2Sc0 ))1/p , cf. the following proof, both Newton methods h converge quadratically to u0 and u0 , respectively, and h uν+1 − uhν h ≤ C uhν − uhν−1 h 2 and U U h ui − P h ui h ≤ Chp ui U , i = 1, . . . , s U h h (5.163) F ui − QhΠ F (ui ) ≤ Chp ui Us , i = 1, . . . , V h ui − uh0 − (P h (ui − u0 )) h ≤ Chp ui U , i = 1, . . . . s U ¯ ¯ = h(, ¯ u0 ) such that for all h ∈ (0, h] Under these conditions there exists h h min i ≥ 1 : ui − P h u0 h < − min {i ≥ 1 : ui − u0 < } ≤ 1. (5.164) U U This last result is the reason for the notation as MIP and our remark concerning the essential h independence of the convergence. Proof. We have to show that the conditions in Theorem 3.40 are satisﬁed. By h h Subsection 5.2.5, P h = I h Ec ∈ L(U, U h ), QhΠ ∈ L(VˆΠ = VˆΠ , VΠ = VΠ ) are bounded h and linear, and F h = Qhc,Π F |U h : D(F h ) = D(F ) ∩ U h → VΠ , similar for m > 1. As a consequence of (5.162) and U h , we obtain the uniform Lipschitz-continuity of our (F h ) : ≤ CLuh − v h U h (F h ) (uh ) − (F h ) (v h )VΠh ←U h = Qhc,Π (F (uh ) − F (v h )) h h VΠ ←U

for all uh , v h ∈ Br (P h u0 ). As a consequence of F (·) ∈ CL (Br (u0 )) and the isolated solution u0 of F (u) = 0 with boundedly invertible F (u0 ) we obtain stability of (F h (uh )) , and hence ((F h ) (uh ))−1 U h ←VΠh ≤ S for all uh ∈ Br/2 (P h u0 ) with a slightly reduced r/2; see Theorem 5.6. Finally we need the consistent diﬀerentiability ∀u, v ∈ Br (u0 ) ∩ Us :

h QΠ F (u) − Qhc,Π F (P h u)

and QhΠ F (u)v − Qhc,Π F (P h u)P h v

h VΠ

h VΠ

≤ Chp uUs ≤ c0 hp ,

(5.165)

≤ Chp (1 + uUs )vUs ≤ c1 hp vUs ,

an immediate consequence of F (·) ∈ CL (Br (u0 )) and of (5.162).

5.3. FE and Other Methods for Nonlinear Boundary Conditions

345

For (5.164), the only missing conditions (see (3.25) in [17]) require the existence of a constant δ > 0 such that lim inf P h uU h ≥ δuU h>0

∀u ∈ Us .

This is an immediate consequence of (4.64).

5.3

FE and other methods for nonlinear boundary conditions

The techniques in the preceding section open up the possibility for formulating FEMs and prove their convergence for nonlinear boundary conditions in elliptic problems. Compared to the previous situation only few analytical existence and even less regularity results are available, however, cf. Zeidler [678], p. 537 ﬀ. A convergence analysis for numerical methods seems to be totally missing. We demonstrate this analysis only for FEMs, although all the other methods, considered in this and the next book, allow the same results. We assume the nonlinear boundary conditions, studied here, in a form such that the linearization yields either Dirichlet boundary conditions or the more general form of a special Dirichlet system, cf. (2.80) ﬀ., (2.137) ﬀ., (2.158), and Theorem 2.50. We formulate a nonlinear version of a Dirichlet boundary condition; extensions to special Dirichlet systems are obvious bj (∂ j−1 u/∂ν j−1 ) = 0 with appropriate nonlinear bj , j = 0, . . . m − 1, (5.166) with obvious extensions to special Dirichlet systems. For m = 1, we admit natural boundary condition, cf. (2.123). For example, the boundary operators, induced by quasilinear operators as in (5.286), might be imposed as nonlinear boundary operators BG uh := BG (x, uh , ∇uh ) :=

n

νi Ai (x, uh , ∇uh ).

(5.167)

i=1

Generalizations to other boundary conditions are possible, but are not studied here. We formulate the convergence result for problems of order 2m. Theorem 5.22. FE and other methods for nonlinear boundary conditions: Let the nonlinear elliptic problem in (5.42), F (u0 ) = (G, B)(u0 ) = 0, u0 ∈ U, have an isolated solution, with boundedly invertible F (u0 ) : U → VΠ . We admit nonlinear G and/or B and assume G and B to be continuously diﬀerentiable in Br (u0 ), e.g. by (5.109), and an analogous condition for B. Let the domain satisfy (4.5). Let Gh and B h be consistent of the same order, usually min{s, d} − m, for the following combinations. 1. If G is linear or quasilinear, choose U h , V h as one of the above families of conforming or nonconforming FEs of local degree d − 1, such that dist(∂Ω, ∂Ωh ) ≤ Chd . Assume an exact solution u0 ∈ H s (Ω), m < min{s, d}. Then the FEM for G is stable, consistent in P h u0 , and convergent. Then the same is valid for the FEM for F with the same order of convergence. So this F h (uh0 ) = 0 has, for small

346

5. Nonconforming ﬁnite element methods

enough h, a unique solution uh0 near u0 , usually converging as u0 − uh0 ≤ Chmin{s,d}−m u0 H s (Ω) . V

(5.168)

2 2. For a fully nonlinear h G and na convex polyhedral or a C domain Ω in (5.100), h 1 choose U = Sd Tc , d ≥ 2 + 1 (see (5.3)), on a quasiuniform triangulation. Then the nonconforming FEM in (5.47) ﬀ., now for nonlinear B,

F h = Qhc,Π F |U h : U h → V h × Vbh , uh0 ∈ D(F h ) ⊂ U h s.t. F h uh0 = 0, is stable and consistent in P h u0 , and convergent. It has, for small enough h, a unique solution uh0 near u0 . We obtain, cf. (5.67), for u0 ∈ H (Ω), > 2, for m = 1, the nonconforming FEM, and convex polyhedral or C p , 2 ≤ p ≤ 5, domains, h P u0 − uh0 h ≤ Chmin{−2,p} u0 H (Ω) , (5.169) U respectively, with p = 2 for polyhedral Ω and p ≥ 2 for polyhedral or curved Ωhc ≈ Ω ∈ C p. 3. F (u0 ) ∈ L(W, W × WD ) and F (u0 ) ∈ L(U, V × VD ) simultaneously satisfy the Fredholm alternative. Remark 5.23. 1. For the above F (see (5.18), (5.20), (5.73)), and d are related by (4.62). With the ellipticity assumption (5.20), the previous (5.73), (5.74), (5.100) are implied by (5.109), except Ω convex polyhedral or in C 2 . The latter are necessary for the regularity and the extension operators. The conforming FEM in (5.26) ﬀ. does not apply, since nonlinear boundary conditions will be violated. 2. For quadrature approximations, exactly reproducing polynomials of degree < k uh0 ) = 0 satiswith k > 2(d − 1) > d > 2, the solution u ˜h0 of an approximate F˜ h (˜ ﬁes the same estimate as in (5.110); see Subsection 5.4.4 and 5.4.5 for m > 1 or for system. 3. Quadrature approximations for these problems follow the lines of Section 5.4. Proof. The proof follows essentially the lines of the last section with some straightforward modiﬁcations. The consistency in Theorem 5.4 has to be modiﬁed for quasilinear G if nonconforming FEMs are used. In any case these results have to be applied to B as well. For the stability we apply the technique to the product operator F = (G, B), but need the regularity result in Lemma 5.10 only for fully nonlinear G. The remaining part of the proof remains essentially unchanged.

5.4 5.4.1

Quadrature approximate FEMs Introduction

In many cases it is impossible to evaluate exactly the linear, bilinear and higher nonlinear forms deﬁning the FE equations. So we consider in this section, quadrature

5.4. Quadrature approximate FEMs

347

and cubature or other approximations, e.g. diﬀerence approximations, for the diﬀerent terms in these FE equations. They are applied to elliptic problems of order 2m, m ≥ 1 in Rn . These quadrature approximations will be chosen such that, the order of a given FEM is maintained in its quadrature approximate form. So the quadrature approximate forms have to be stable and consistent with the exact forms. These two conditions will sometimes have diﬀerent implications, see e.g. Theorem 5.31. Simplifying the notation, we assume throughout that the functionals, coeﬃcients, nonlinear functions, f, aij , Aαβ ; Gw are already extended to Ω ∪ Ωhc , so (f, aij , Aαβ , Gw ) = (Ec f, Ec aij , Ec Aαβ , Ec Gw ). e.g. by the extension operator Ec , cf. ˜h (u, v h ) for Theorem 4.37. Thus, e.g. a ˜h (uh , v h ) in (5.181) is deﬁned, in contrast to a u ∈ U, where the necessary function values of u might not be available. In particular, for the fully nonlinear problems as in Section 5.2 and Subsection 5.4.4 we sometimes omit the extension operator Ec . This is not always possible for functions, u, and the solution, u0 , e.g. Theorem 5.33. All the following quadrature formulas are deﬁned piecewise on every Tc ∈ Tch , admitting broken Sobolev spaces for the diﬀerent terms. Extensions to several neighboring Tc ∈ Tch , similarly to Subsections 4.2.5, and 4.2.6 would be possible, but are not considered here. The FEs studied here are again piecewise polynomials with n n ⊂ P ⊂ Pd−τ , τ ≥ −1. For including the violated boundary conditions, studied Pd−1 already in Section 5.2 and later on in Section 5.5, we use the notation Tc ∈ Tch indicating that we admit either a polyhedral or a curved approximation Ωhc ≈ Ω triangulated by Tch with straight or curved edges along the boundary ∂Ω. For the isoparametric FEMs in Subsection 5.5.10 we will need other strategies anyway. For the second type of variational crimes, studied in Subsection 5.5.7 and caused by violated continuity, the a(uh , v h ) is not, but the piecewise ah (·, ·) ∀Tc ∈ Tch in (5.179) is deﬁned, cf. (5.78) above, due to uh ∈ U, v h ∈ V. We will in Subsection 5.5.9 study the total error introduced by the combination of quadrature and these nonconformity errors. We denote the original linear, bilinear forms, linear, nonlinear operators as f (·), a(·, ·), A, G, their extensions and modiﬁcations due to variational crimes to broken Sobolev spaces deﬁned on Ωhc ∪ Ω as fh (·), ah (·, ·), Ah , Gh , and their restriction to the approximating spaces U h , V h , . . . , as f h (·), ah (·, ·), Ah , Gh . Their quadrature ˜ h , respectively Other, e.g. diﬀerence ˜h (·, ·), A˜h , G approximations are denoted as f˜h (·), a approximations of order ka , and combinations with quadrature of order k, we indicate ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜h ˜ ˜h (·, ·), A˜h , G , respectively Explicitly, we will as f h (·), ah (·, ·), Ah , Gh and f˜h (·), a study and formulate the quadrature approximations in this Section. For the other ˜ ˜ ˜h ˜ appropriate approximations, a ˜h (·, ·), A˜h , G with the new order k := min{k, ka }, the quadrature results essentially remain correct. According to Theorem 3.21, we need two diﬀerent results for proving convergence. ˜h (·, ·) has to satisfy a discrete inf– For a linear problem with stable A˜h , the inducing a sup condition. Usually this is a consequence of the corresponding property for ah (·, ·), see e.g. Theorem 5.30. For nonlinear problems its derivative, G (u0 ) = 0, has to satisfy this condition. Further we do need consistency results for the discretized equations. h ˜ h , F˜ h are usually not deﬁned It causes some problems that the a ˜h (·, ·), f, ·˜ , A˜h , G for the original u ∈ U, v ∈ V, since quadrature formulas require u(Pj ) or even ∂ i u(Pj ),

348

5. Nonconforming ﬁnite element methods

see (5.170), (5.311). This is sometimes circumvented by comparing the exact solution u0 with uh0 , solving the exact FE equation or the interpolant I h u0 . Then the stability and estimates for a ˜h (uh , v h ) − ah (uh , v h ) or nonlinear analogues, see e.g. (5.188), ˜h0 U h . The standard approach (5.204), allow the ﬁnal error estimates for u0 − u based upon the Strang lemmas and a neighboring u0 ≈ z h ∈ U h only works for linear problems. Alternatively, extensions as discussed in Theorem 4.36 could be used. For the proofs of the following Theorems 5.30, 5.33, 5.36 5.37 and 5.40, we apply the results of the general discretization theory in Chapter 3, here as Summary 4.52, in Subsection 5.2.5. We will prove in this section that the a ˜h (·, ·), . . . are stable and h consistent in P u, simultaneously with the original or the modiﬁed ah (·, ·), . . . , if the quadrature formulas, introduced in Subsection 5.4.2, are good enough and h is small enough. By Theorem 3.21, we thus get convergence, via stability and consistency of the quadrature approximate FEMs, see Deﬁnition 4.46. The standard argumentation for the following convergence results is: the quadrature approximations represent discretizations satisfying all the conditions for the chosen spaces, their approximations, the projectors, the operators and their dis˜ h , A˜h , G ˜ h , or F˜ h , guaranteeing a unique solution, u ˜h, Q ˜h0 , and its cretizations, Q convergence to the unique exact solution, u0 . In Subsection 5.4.3 we start with the spaces in (5.177), combined with the exact projectors and FEMs in (4.117), (4.132), or (4.135) for linear diﬀerential operators of order 2, and extend it to fully nonlinear equations of order 2, see (5.52), (5.60), (5.62) in Subsection 5.4.4. The next generalization to linear and nonlinear equations and systems of order 2m as in (4.133)–(4.140), (4.163), and (5.111)–(5.114), (5.123), is presented in Subsection 5.4.5. As a consequence of the perturbation or consistency results, e.g. in Theorem ˜ h , the linear and nonlinear A˜h , ˜h, Q 5.28, the approximate quadrature projectors, Q h h ˜ ˜ G , F simultaneously satisfy these conditions with the exact Qh , Ah , Gh , F h . ˜ h , F˜ h are continuous near P h u0 and consistent and stable In particular, A˜h and G h in P u0 . Remark 5.24. These results remain valid with slight modiﬁcations for the other methods as well. 5.4.2

Quadrature and cubature formulas

The main tools in this section are the following quadrature formulas in higher dimensions. Usually these quadrature formulas reproduce the integral of polynomials of a certain degree. For tensor product FEs, e.g. bilinear FEs, see Subsection 4.2.2, or the serendipity class, see [141], modiﬁcations are possible. Proposition 5.25. Assume the following quadrature formula f (x)dx ≈ qThc (f ) := Tc

Pj,Tc ∈Tc

wj,Tc f (Pj,Tc )

(5.170)

5.4. Quadrature approximate FEMs

349

to be exact for polynomials of degree k − 1, and let diam Tc ≤ h. Then ∀k ∈ N , k > n/p , there exists C = Ck ∀ 0 < h < 1, 1 ≤ p ≤ ∞, f ∈ W k,p (Tc ) ∩ C(Tc ), such that f (x)dx − wj,Tc f (Pj,Tc ) ≤ Ck hk+(1−1/p ) |f |W k,p (Tc ) . (5.171) P ∈T Tc

j,Tc

c

Proof. By k > n/p and the Sobolev embedding Theorem 1.26, f ∈ C(Ω), so all the f (Pj,Tc ) are well deﬁned. Now we proceed as in Proposition 5.62, adding and n for f in (5.171). Then (5.170) subtracting the averaged Taylor polynomial Qk f ∈ Pk−1 k,p implies with f ∈ W (Ω), the H¨ older inequality (1.45) and the Bramble–Hilbert lemma 4.15 f (x)dx − wj,Tc f (Pj,Tc ) Tc Pj,Tc ∈Tc k k f (x)dx − Q f + Q f dx − wj,Tc f (Pj,Tc ) ≤ Tc Pj,Tc ∈Tc k ≤ 1 · (f (x) − Q f )dx with H¨ older and Bramble–Hilbert T

≤ Ck h1−1/p |f − Qk f |Lp (Tc ) ≤ Ck hk+1−1/p |f |W k,p (Tc ) .

Remark 5.26.

1. In these quadrature approximations (5.170) the quadrature points Pj,Tc are usually chosen independently of the interpolation points for the FEs. 2. The Pj,Tc ∈ Tc , Tc ∈ Tch in (5.170) are often determined as Pj,Tc = FTc (Pj ), Pj ∈ K. Then the computations are performed along the lines of Examples 4.11 and 4.51. So let all the Tc ∈ Tch be aﬃnely equivalent to the reference element K, see Deﬁnition 4.6, hence, FTc (K) = Tc , FTc = ATc x + bTc , ATc nonsingular. Then qThc (f ) = Pj,T ∈Tc wj,Tc f (Pj,Tc ) so we obtain the wj,Tc := det(ATc )wj , Pj,Tc := c FTc (Pj ), from the quadrature formula on K, q h (f ) = Pj ∈K wj f (Pj ). For Tc at the boundary slight modiﬁcations are necessary. We apply the above qThc to the Tc ∈ Tch , to yield a quadrature formula q h (f ) for Ω or Ωhc . Note that for conforming and nonconforming FEs, satisfying the boundary conditions, we have Ω = Ωhc and, hence, Ωh f (x)dx = Ω f (x)dx, otherwise we use the c

extension fh : Ω ∪ Ωhc → R, cf. (5.178), (5.5), cf. (5.170) for qThc (fh ), in qThc (fh ) ≈ fh (x)dx ≈ f (x)dx. q h (fh ) := Tc ∈Tch

Ωh c

Ω

(5.172)

350

5. Nonconforming ﬁnite element methods

Often we additionally ﬁnd, sometimes with C = 1, if all weights are positive, h qT (f ) ≤ C meas (Tc )f L∞ (T ) ∀Tc ∈ Tch , (5.173) c c admitting modiﬁed relations. We estimate the total quadrature error in (5.172) with Proposition 5.25 and for p = 1 as 37 h |q (fh ) − f (x)dx| ≤ Ck hk |fh |W k,1 (Tc ) = Ck hk |fh |W k,1 (Tch ) . (5.174) Ωh c

Tc ∈Tch

Some of the following estimates are applied to fh = uh v h , uh ∈ W k,p , v h ∈ W k,p . Proposition 5.27. For uh ∈ W k,p Tch , v h ∈ W k,p Tch , 1/p + 1/p = 1 we get uh v h (x)dx| ≤ Ck hk uh W k,p (Tch ) v h W k,p (Tch ) . (5.175) |q h (uh v h ) − Ωh c

Proof. The standard and discrete H¨ older inequalities (1.45), (1.44) yield with p = 1 in (5.171) and again with Ck indicating diﬀerent constants, depending upon k, h h h uh v h (x)dx| |q (u v ) − Ω

≤ Ck hk

or

Tc ∈Tch

≤ Ck hk

Ωh c

|uh v h |W k,1 (Tc )

∂ β uh ∂ α−β v h L1 (Tc )

Tc ∈Tch α,β≥0 |β+α|=k

≤ Ck hk

∂ β uh Lp (Tc ) ∂ α−β v h Lp (Tc )

Tc ∈Tch α,β≥0 |β+α|=k

⎛ ≤ Ck hk ⎝

Tc ∈Tch |β|=k

⎞1/p ⎛ ∂ β uh pLp (Tc ) ⎠

⎝

Tc ∈Tch |α|=k

⎞1/p

∂ α v h pLp (T ) ⎠ c

≤ Chk uh W k,p (Tch ) v h W k,p (Tch ) . 5.4.3

Quadrature for second order linear problems

Here we consider weak forms of second order problems, hence we choose U = V. The basic condition, compare the Sobolev Embedding Theorem 1.26 is a bounded Ω ⊂ Rn with Lipschitz boundary , n ≥ 1 and f, aij ∈ C(Ω).

(5.176)

u, u0 , v ∈ C 1 (Ω), e.g., W k,p (Ω) ⊂ C j (Ω), 1 ≤ p < ∞, k > j + n/p for Ω, see the modiﬁcations below for less smooth u, u0 , v ∈ C 1 (Ω). 37

For some cases p = 1 is the optimal choice: It gives, in combination with (1.44), hp v h H 1 (T h )

with largest p, e.g. p = ∞ would yield at most hp−1 v h H 1 (T h ) . c

c

5.4. Quadrature approximate FEMs

351

The piecewise polynomials uh , v h are continuously diﬀerentiably extendable to each Tc . For a Pj,Tc1 ∈ Tc1 ∩ Tc2 the values uh (Pj,Tc1 ) and uh (Pj,Tc2 ) might diﬀer. But it is no problem to exactly evaluate the uh , v h , ∂ i uh , ∂ j v h in the points Pj and to exactly integrate their products and the ah (·, ·). Usually the f, v h have to be approximated h by an f, v h˜ . So there are good reasons to study both cases. Sometimes other forms

of a ˜h (·, ·) : U h × V h → R are deﬁned by combining the exact uh , v h , ∂ i uh , ∂ j v h with appropriate means of the aij , f . We do not discuss those methods here, cf. Grossmann, Roos and Stynes [374–376]. The spaces here and their approximating spaces are, cf. (4.35), (4.36) U = V ⊂ H 1 (Ω), Ubh , Vbh ⊂ L∞ Tch : ∀Tc ∈ Tch : uh |Tc ∈ P , Pd−1 ⊂ P ⊂ Pd+τ , dim Ubh = dim Vbh , usually τ = −1, and norms · U h := · V h := · H 1 (Tch ) . (5.177)

These Ubh , Vbh indicate the FE spaces, satisfying the boundary conditions either exactly or approximately, corresponding to the chosen FEM. If we use U h , V h we omit boundary conditions, often for the bilinear forms or operators. Sometimes U ⊂ W 1,p (Ω), V ⊂ W 1,p (Ω) or U ⊂ W m,p (Ω), V ⊂ W m,p (Ω), m > 1, are appropriate. This can be handled similarly to below via the Sobolev Embedding Theorem 1.26. ˜h (·, ·), A˜h we need the approximate In addition to the quadrature versions f˜h , a ˜ h , cf. Propositions 4.49, 4.50. All versions are obtained by applying projectors Q the quadrature formulas (5.172) to diﬀerent combinations, f, f v h , aij uh v h , . . . As in Section 5.2, we denote, e.g. the original and the sometimes necessary extensions, note (5.5) omitting Ec , extensions of f, a(·, ·), A, G, to Ω ∪ Ωhc as fh , ah (·, ·), Ah , Gh , a.s.o., so (5.178) ⎛ ⎞ n ⎝ ah (uh , v h ) := Ah uh , v h := aij ∂ i uh ∂ j v h ⎠ dx ∀uh ∈ U ∪ U h , Tc

Tc ∈Tch

h˜h

a ˜h (uh , v h ) := A˜h uh , v :=

i,j=0

⎛ qThc ⎝

Tc ∈Tch

fh , v h :=

Tc ∈Tch

h˜h

fh , v :=

Tc ∈Tch

Tc

n

fi ∂ i v h

n

⎞ aij ∂ i uh ∂ j v h ⎠ , and

(5.179)

i,j=0

dx ∀v h ∈ V ∪ V h , fj ∈ H k Tch orf ∈ Hk−1 Tch ,

i=0

qThc fh v h =

Tc ∈Tch

Pj,Tc ∈Tc

wj,Tc

n

fi ∂ i v h

(Pj,Tc )

i=0

˜ h f − f, v h˜h = ((Q ˜ h f − fh )v h )h˜ = 0 ∀v h ∈ Vbh . ˜ h : V → V h , Q Q

(5.180)

Note that for applying the quadrature formulas, we have to impose the additional conditions f or f = (fj )nj=0 , aij ∈ H k (Ω), or f ∈ Hk−1 (Ω), k large enough for the

352

5. Nonconforming ﬁnite element methods Ub Ph U hb

G

V b′

tested by

V

˜ ′h Q

˜h Φ ˜h G

V hb ′

appropx. tested by

V hb

Figure 5.5 Quadrature approximate FEMs.

required function evaluations, cf. (5.184), (5.185). Similarly to (5.5) we get ah (uh , v h ) := Ah uh , v h := Ah uh , v h Ubh ×Vbh = ah (uh , v h ) h

Ub ×Vbh

,

a(u, v) = ah (u, v)|Ub ×Vb = Au, v, and (5.181) h h h ˜h (uh , v h ) Ubh ×Vbh , f˜h , v h˜ := fh , v h˜ , a ˜h (uh , v h ) := A˜h uh , v h˜ := a h V

h

h

for u, v ∈ H k (Ω) : a ˜h (u, v) := A˜h u, v˜ := a ˜h (u, v)|U ×V := A˜h u, v˜ |U ×V . The approximate solutions u ˜h0 are deﬁned for these approximations uh0 ∈ Ubh :

; h h h h h h h Ah u ˜0 , v = f h , v h˜ or = A˜h u ˜0 , v ∀v ∈ Vbh . ˜h0 , v h = ah u ˜h0 , v h˜ = a ˜h u (5.182)

Reformulating with the exact and quadrature projectors yields ˜ h f, and A˜h = Q ˜ h Ah |U h , A˜h u ˜ h f. Ah = Q h Ah |Ubh , Ah u ˜h0 = Q ˜h0 = Q b

(5.183)

Again a slight modiﬁcation of the general discretization theory in Chapter 3, see Figure 5.5, cf. Figures 4.27, 5.2, yields convergence for a consistent and stable method ˜ h , since P h = I h or I h Ec is again uniformly in P h u0 with uniformly bounded Q bounded, note Summary 4.52. h We will show in Theorems 5.28, 5.30 that a ˜h (·, ·), fh , ·˜ in (5.182) are stable and

consistent in P h u, simultaneously with ah (·, ·), fh , ·h and the above P h and the new ˜ h are uniformly bounded for smooth enough data, hence if ˜h, Q Q Π n h n for fh , ·˜ , f = fj ∈ Hk Ω ∪ Tch j=0 ∈ Hk−1 (. . .) (5.184) k > max d + τ − 1, 2 k > 2(d + τ − 1) for a ˜h (·, ·), aij ∈ W k,∞ Ω ∪ Ωhc , τ ≥ −1, usually τ = −1. (5.185)

Then, by the Sobolev Embedding Theorem 1.26, the quadrature formulas in (5.179)– (5.180) are well deﬁned. We assume (5.184), (5.185), but do not show (5.185) to be necessary for a stable A˜h ; analogously to (4.137) we deﬁne f H −1 (T h ) . c k Hence k is often pretty large. Solutions u0 and general functions u will usually ∈ H k+1 (Ω). So we combine in (5.190) the Ec u with I h u. If u is smooth enough, the quadrature formula can be applied to u directly. This is not a necessary condition, but is satisﬁed, e.g. if u ∈ H s (Ω) with s > n/2, and the quadrature formula requires only function values, cf. Theorem 1.26. FEMs for fully nonlinear problems, their linearization and isoparametic FEs do need the extension operator Ec .

5.4. Quadrature approximate FEMs

353

Quadrature approximate FEMs for nonconforming FEs are delayed to Section 5.5 Subsection 5.5.9 we omit the Ec here, except for fully nonlinear equations. ˜ h , consistent a ˜h (uh , v h ): Theorem 5.28. Convergence of the f˜h and Q 1. Assume (5.176), and for the quasiuniform subdivision T h the Condition 4.16. Choose a quadrature formula as in (5.172), (5.174), satisfying (5.184) and h (5.185) for fh , ·˜ and a ˜h (·, ·), with fj ∈ H k (Ω) or f ∈ Hk−1 (Ω), aij ∈ W k,∞ (Ω), and let h > 0 be small enough. 2. Then the following consistency results hold, with u ∈ H d−p (Ω), p, ≥ 0, in (5.190), notice the diﬀerent norms in (5.188), (5.189), h |fh , v h − f˜h , v h˜ | ≤ Chk−(d+τ −1) f H −1 (T h ) · v h V h , k

c

(5.186)

˜ h f h ≤ Chk−(d+τ −1) f −1 h , lim Q ˜ h f h = f V , Q h f − Q V V H (T ) k

c

h→0

(5.187) ˜ (u , v )| |a (u , v ) − a ≤ Chk−2(d+τ −1) Maij uh V h , for v h = 0 h v V h h

h

h

h

h

h

(5.188)

≤ Chk−(d+τ −1) · Maij uh H d+τ (T h ) ; Maij n

:= max aij W k,∞ (Ωhc ) . i,j=0

(5.189)

˜h (I h u, v h )| |ah (u, v h ) − a ≤ C hd−1−p + hk−(d+τ −1) Maij uH d−p (Ω) . (5.190) h v V h Remark 5.29. The condition of a quasiuniform subdivision, T h , in this and the following theorems, can be slightly relaxed with the results in Subsection 4.2.5. This requires the restriction to the most important cases for FEMs, n = 2, 3, and requiring Assumptions 4.21 and 4.26. Then in the sum in (5.191) the terms have to be estimated for each T ∈ T h separately. The results in Theorems 4.30–4.33 allow appropriate estimates for continuous and L2 FEs. We leave the details to the interested reader. Proof. We start using (5.172),(5.174) estimating for f ∈ H k (Ω) n h h h h˜ j h h j h ˜ fj ∂ v dx − qTc (fj ∂ v ) |fh , v − f , v | = Tc ∈T h j=0 Tc c

≤ Ck hk

n Tc ∈Tch

= Ck hk

n j=0

|fj ∂ j v h |W k,1 (Tc )

j=0

|fj ∂ j v h |W k,1 (Tch ) .

(5.191)

354

5. Nonconforming ﬁnite element methods

We estimate these derivatives and use (5.184) |fj ∂ j v h |W k,1 (Tc ) = (fj ∂ j v h )(α) L1 (Tc ) |α|=k

≤

()

Cα fj (∂ j v h )(α−) L1 (Tc ) by (5.312)

|α|=k 0≤≤α

≤C

()

fj L2 (Tc ) (∂ j v h )(α−) L2 (Tc ) by (1.44)

(5.192)

|α|=k 0≤≤α

⎛ ≤C⎝

⎞1/2 ⎛

() fj 2L2 (Tc ) ⎠

|α|=k 0≤≤α

⎝

⎞1/2

(∂ j v h )(α−) 2L2 (Tc ) ⎠

|α|=k 0≤≤α

= Cfj H k (Tc ) ∂ v H d+τ (Tc ) = Cfj H k (Tc ) v h H d+τ −(1−δj0 ) (Tc ) . j h

Combining (5.191), (1.44) with the usual δij yields, cf. (4.137), and [141], 8.x.15, h h |fh , v h − fh , v h˜ | = |fh , v h − f˜h , v h˜ |

n

Tc ∈Tch

j=0

(1.44) ≤ Ck hk

(5.193)

fj H k (Tc ) v h H d+τ −(1−δj0 ) (Tc ) ≤ Ck hk−(d+τ −1) f H −1 (T h ) v h H 1 (Tch ) . k

c

thus (5.186) by the inverse estimate (4.47) for the quasiuniform Tch . Relation (5.187) is an immediate consequence of (5.186). Similarly we have to estimate the quadrature errors for the weak bilinear form, replacing (5.184) by (5.185). For modifying (5.192), (5.193), we use (5.185) and the Cauchy–Schwarz inequality, (1.48). Then |(aij ∂ i uh ∂ j v h )|W k,1 (Tc ) ≤ Caij W k,∞ (Tch ) ∂ i uh H k (Tc ) ∂ j v h H k (Tc )

(5.194)

≤ Caij W k,∞ (Tch ) ∂ i uh H d+τ −(1−δi0 ) (Tc ) ∂ j v h H d+τ −(1−δj0 ) (Tc ) . uh , v h are always ∈ H k (Tch ). This implies with the discrete H¨ older inequality i h j h h i h j h a ∂ u ∂ v dx − q (a ∂ u ∂ v ) (5.195) ij Tc ij Tc ∈T h Tc c ∂ i uh H d+τ −(1−δi0 ) (Tc ) ∂ j v h H d+τ −(1−δj0 ) (Tc ) ≤ Ck hk aij W k,∞ (Ωhc ) Tc ∈Tch

≤ Ck hk aij W k,∞ (Tc ) ∂ i uh H d+τ −(1−δi0 ) (Tch ) ∂ j v h H d+τ −(1−δj0 ) (T h ) c

with uH k+1 (Ω) for ∂ u H d+τ −(1−δi0 ) (Tch ) if u is replaced by u. i h

h

5.4. Quadrature approximate FEMs

355

The analogous summation as in (5.193) and the inverse estimate (4.47) again yield (5.188), (5.190), since by (5.177), (5.180). (5.185), (5.195), we ﬁnd ˜h (uh , v h )| |ah (uh , v h ) − a n aij ∂ i uh ∂ j v h dx − qThc . . . = Tc ∈T h i,j=0 Tc c ⎛ ⎞ n n ∂ i uh H d+τ −(1−δi0 ) (Tc ) ∂ j v h H d+τ −(1−δj0 ) (Tc ) ⎠ ≤ Chk max aij W k,∞ (Ωhc ) ⎝ i,j=0

Tc ∈Tch i,j=0

n

≤ Chk max aij W k,∞ (Ωhc ) · ∇uh H d+τ −1 (Tch ) ∇v h H d+τ −1 (Tch ) i,j=0

+ ∇uh H d+τ −1 (Tch ) v h H d+τ (Tch ) + uh H d+τ (Tch ) ∇v h H d+τ −1 (Tch ) + uh H d+τ (Tch ) v h H d+τ (Tch ) ;

(5.196)

note that ∇v h indicates ∂ j v h ∈ P d+τ −1 , j = 1, . . . , n. We employ (4.47) for uh , v h , e.g. ∇v h L2 (Tch ) ≤ v h H 1 (Tch ) and verify (5.188) by (5.185). If we replace uh by a smooth enough u, such that the quadrature formula is deﬁned for u, we end up with (5.190). If only u ∈ H d−p (Ω), p, ≥ 0, we use I h u. Then ah (u − I h u, v h ) causes the hd+τ −p term by the interpolation error, the quadrature error, ˜h (I h u, v h ) remains unchanged by (5.189). ah (I h u, v h ) − a Theorems 5.30 and 5.31 imply stability and a convergent quadrature approximate FEM, see Deﬁnition 4.46. Theorem 5.30. Stability of A˜h : Under the conditions of Theorem 5.28 the ah (·, ·) and a ˜h (·, ·) are simultaneously bounded on U h × V h . Let ah (·, ·) be Ubh -coercive or, e.g. for Ubh = Vbh , satisfy a discrete inf–sup condition and (4.145) for U h and V h , hence Ah is stable and let k > 2(d + τ − 1). Then, for small enough h, A˜h and a ˜h (·, ·) have the same property. Proof. We only consider the case of a discrete inf–sup condition for ah (·, ·). This implies with Theorem 5.28 uh U h ≤ ≤ ≤

inf

|ah (uh , v h )|/v h U h

inf

|˜ ah (uh , v h ) + (ah (uh , v h ) − a ˜h (uh , v h ))|/v h U h

inf

|˜ ah (uh , v h )|/v h U h

0 =v h ∈Vbh 0 =v h ∈Vbh 0 =v h ∈Vbh

+

sup 0 =v h ∈Vbh

≤

inf

0 =v h ∈Vbh

|(ah (uh , v h ) − a ˜h (uh , v h ))|/v h U h

|˜ ah (uh , v h )|/v h U h + Chuh U h ,

(5.197)

356

5. Nonconforming ﬁnite element methods

so ﬁnally ( − Ch)uh U h ≤

inf

0 =v h ∈Vbh

|˜ ah (uh , v h )|/v h U h .

(5.198)

Analogously the inequalities for the inf 0 =uh ∈Ubh are handled. This shows the discrete coercivity or discrete inf–sup condition for a ˜h (·, ·) for suﬃciently small h > 0. Theorem 5.31. Stability and convergence for diﬀerent quadrature approximations: 1. Let ah (·, ·) induce a stable Ah and assume the conditions of Theorem 5.28. Then, for small enough h, simultaneously a ˜h (·, ·) induces a stable A˜h . If furh thermore (5.184) is correct, then a ˜h (˜ uh , v h ) = f, v h˜ ∀v h ∈ V h , deﬁnes a stable 0

b

and convergent discretization for Au0 = f . For the other approximations, men˜ ˜ ˜h , tioned above, this result remains correct if the a ˜h , A˜h are replaced by ah , a ˜ ˜ Ah , A˜h . n 2. For conforming FEMs and u0 ∈ H s (Ω), Maij = max aij W k,∞ (Ω) , we obtain i,j=0

u 0 − u ˜h0

Uh

≤ Chmin{s−1,k−(d+τ −1)}

(5.199)

(Maij + 1)u0 H min{s,k+1} (Ω) + f H −1 (Ω) .

k

h

˜h (uh , v h ) are used in (5.182), 3. If only approximations for f, v h˜ and none for a then the term (1 + Maij )u0 H min{s,k+1} (Ω) in (5.199) is dropped. Proof. This theorem is an immediate consequence of (5.187), (5.190). The original (conforming) FEM is based upon an appropriate combination of spaces, projectors ˜ h ∈ L(H −1 (T h ) ⊂ V , V h ) Q h , linear and bilinear forms. Then (5.187) shows that Q c k and with (5.190) we obtain for conforming FEMs and u0 ∈ H k+1 (Ω) h h h ˜0 , v − a(u0 , v h ) = f˜h , v h˜ − f, v h a ˜h u h h =⇒ a ˜h u ˜0 − u0 , v h = a(u0 , v h ) − a ˜h (u0 , v h ) + f˜h , v h˜ − f, v h h =⇒ u ˜0 − u0 U h ≤ Chk u0 H k+1 (Ω) + Chk−(d+τ −1) (Maij · u0 H k+1 (Ω) + f H −1 (Ω) ). k

For u0 ∈ H s (Ω), s < k + 1, we combine the interpolating FE I h u0 with bounded and u0 − I h u0 H 1 (Tch ) < Chs−1 u0 H s (Ω) , I h u0 H s (Tch ) < Cu0 H s (Ω) < ∞ cf. (4.42), or (4.105), with Theorem 4.17 and 4.61, the discrete Brezzi–Babuska

5.4. Quadrature approximate FEMs

357

condition (4.169), h h ˜ 0 − I h u0 , v h ˜h0 U v h cV h ≤ a ˜ u I h u 0 − u h h h ˜0 , v − a ˜h (I h u0 , v h ) + (ah (I h u0 , v h ) = | a ˜ u − ah (I h u0 , v h )) + (a(u0 , v h ) − a(u0 , v h ))| h h h ≤ f˜h , v h˜ − f, v h + a ˜ (I u0 , v h ) − ah (I h u0 , v h ) + ah (I h u0 , v h ) − a(u0 , v h ) ≤ C hk−(d+τ −1) (f H −1 (T h ) + Maij · I h u0 H d+τ (T h ) ) + u0 − I h u0 U · v h V h k

c

≤ C hk−(d+τ −1) (f H −1 (T h ) + Maij u0 H s (Ω) ) + hs−1 u0 H s (Ω) · v h V h

k

c

With u0 − u ˜h0 U ≤ u0 − I h u0 U + I h u0 − u ˜h0 U we have proved the claim.

5.4.4

Quadrature for second order fully nonlinear equations

We do not formulate the results for second order quasilinear problems, since the general case in Subsection 5.4.5 is nearly the same. Second order fully nonlinear equations require U = V. Since the boundary conditions remain unchanged, we start with the quadrature diﬀerential operator. The approximate solution has to satisfy cf. (5.3) h h h h ˜0 , D˜ ˜0 , v = 0∀v h ∈ Vbh uh0 , D2 u ˜h0 , v h˜) = qThc G u u ˜h0 ∈ Ubh : (Gw ·, u Tc ∈T h

(5.200) ˜h = Q ˜ hc G|U h . We need the spaces, approximate projectors Q ˜ h , the thus deﬁning G nonlinear operators and the generalizations, see (5.77)–(5.79), and their linearization, ˜hs (uh , v h ), cf. (5.73) with the aij , and the strong bilinear forms, a (5.201) U ⊂ H 2 (Ω), V ⊂ L2 (Ω), Ubh = S h ⊂ Sd1 Tch , Vbh = S h ∩ C01 Ωhc , Uc ⊂ H 2 Ωhc , . . . , · U h := · H 2 (Tch ) , · V h := · L2 (Tch ) , the linearization ⎛ ⎞ n ⎝ (−1)j>0 ∂ j (aij ∂ i uh ) v h ⎠ dx with (As,h uh , v h ) = as,h (uh , v h ) = Tc ∈Tch

Tc

A˜hs uh , v h := a ˜hs (uh , v h ) =

i,j=0

Tc ∈Tch

⎛ qThc ⎝

n

⎞ (−1)j>0 ∂ j (aij ∂ i uh ) v h ⎠.

i,j=0

In Theorems 5.32 and 5.33 we will show the consistency and stability of our approximate FEM (5.200). The stability is equivalent to showing under which conditions a ˜hs (·, ·) inherits the boundedness and discrete inf–sup condition from as (·, ·) h on Ub , Vbh . The proofs of these results are very similar to those in Subsection 5.4.3.

358

5. Nonconforming ﬁnite element methods

Note the consequences of the diﬀerent norms in (5.202) and testing in the L2 sense in (5.200) in contrast to the weak form in (5.182), · V h := · L2 (Tch ) compared to · V h := · H 1 (Tch ) in the last subsection. So the exponents in the consistency estimates, e.g. k − d in (5.202) compared to k − (d + τ − 1) in (5.186), diﬀer by 1. n ⊂ S h ⊂ Sd1 , is relevant: Furthermore, the − 1 in P−1 Theorem 5.32. Consistency for diﬀerent quadrature approximations of fully nonlinear problems: Assume (5.176), and for the quasiuniform subdivision T h the Condition 4.16. Choose a quadrature formula as in (5.172), (5.174), satisfying (5.184) h k Ω ∪ Ω → H Ω ∪ Ωhc , and (5.185). Let the fully nonlinear extension G : H k+2 c Ω ∪ Ωhc , see (5.73), f, fj ∈ be diﬀerentiable, Gh (u0 ) be bounded with aij ∈ W k+1,∞ k h H Ω ∪ Ωc , and let a u ∈ H (Ω), yield Ec u ∈ H Ω ∪ Ωhc , e.g. for u = u0 . Choose n ⊂ S h ⊂ Sd1 , d ≥ 2n + 1, as in (5.201), the approximating spaces Ubh , Vbh ⊂ S h , P−1 with k > 2(d − 1), hence k > d, and let h > 0 be small enough. Finally, let uh be extended into uh ∈ H k+2 Ω ∪ Ωhc . n Then the following consistency results hold with Maij = max aij W k+1,∞ (Ω) i,j=0

h |f, v h − f˜h , v h˜ | ≤ Ck hk−d f H k (Tch ) v h V h , k > d,

(5.202)

˜ h f h ≤ Ck hk−d f −1 , lim Q ˜ h f h = f −1 , Qh f − Q V V H (Ω) H (Ω) k

h→0

(5.203)

k

˜

|(G(uh ), v h ) − (G(uh ), v h )h | ≤ Chk−d G(uh )H k (Tch ) v h V h , and h ˜ hc G(uh ) h ≤ Chk−d G(uh )H k (T h ) , or for Ec u, I h u : Qc G(uh ) − Q (5.204) V c h ˜ hc G(I h u) h ≤ Chk−d G(uh )H k (T h ) , Qc G(Ec u) − Q V c ˜hs (uh , v h ) ≤ Chk−2(d−1) Maij uh U h v h V h (5.205) and for Gh : ahs (uh , v h ) − a |ah (Ec u, v h ) − a ˜h (I h u, v h )| ≤ Chmin{k−d,−2} Maij uH min{k+2,} (Ω) v h V h . These inequalities only estimate the quadrature errors for the diﬀerential and not for boundary operators. For the complete consistency errors the terms in Theorem 5.4 and L× the interpolation errors have to be added, cf. Theorem 5.33. L denotes the Lipschitz constant for G. Proof. The above conditions d ≥ 2n + 1 > 2, n ≥ 2, for Sd1 Tch ⊂ C 1 (Ω) and k > 2(d − 1), imply k > 2(d − 1) ≥ 2n+1 > n/2, n ≥ 2, k > d. Hence, by the Sobolev ¯ Thus embedding theorem the W k,∞ (Ω), H k (Ω) are compactly embedded into C(Ω). all the quadrature formulas for as and (G(uh ), v h )), see e.g. (5.200), are well deﬁned. we use the interpolating I h u0 . Since the If u or u0 should h notk be hsmooth enough, h h 1 h u , v ∈ Sd Tc ⊂ H Tc ∩ C Tc , the estimates in (5.174) can be applied to (5.201)–(5.200) under the above conditions to diﬀerent choices of f or fh . Without

5.4. Quadrature approximate FEMs U

F

VΠ= V× VD

VΠ= V× Vb

tested by

Ph

˜h Φ

˜ hΠ Q

Ub

Fh

V hΠ′ = V h ′ × V hb

359

approx. tested by

V hΠ = V hb × V hb

Figure 5.6 Nonconforming quadrature FEMs for fully nonlinear problems.

repeating all the details we get

n |fh , v h − f˜h , v | = fj ∂ j v h dx − qThc (fj ∂ j v h ) Tc ∈T h j=0 Tc h˜h

c

≤ Ck hk

n

|fj ∂ j v h |W k,1 (Tc )

(5.206)

Tc ∈Tch j=0

≤ Ck hk−d f H −1 (T h ) v h V h . k

c

As in the proof of Theorem 5.28, the last inequality follows by the inverse estimates (4.57) for the piecewise polynomials v h of degree ≤ d with d > 2, the discrete H¨older inequality and fh v h L1 (Tch ) ≤ v h V h fh L2 (Ωhc ) . Again (5.203) is an obvious consequence of (5.202). We get for k > d ˜

|(G(uh ), v h ) − (G(uh ), v h )h | ≤ Chk G(uh )H k (Tch ) v h H d (Tch )

(5.207)

≤ Ck hk−d G(uh )H k (Tch ) v h V h . The other estimates follow by similar modiﬁcations compared to above.

Now we consider the fully nonlinear problem, including the approximate boundary conditions. In particular, we update the original product projectors and the FEM, see ˜h (5.62),(5.63), into the quadrature version, based upon Qc and the unchanged boundary h h projector Qb and Fh = G, BD , cf. Figure 5.6.

h h h ˜ hc,Π Fh |U h u ˜ hc,Π := Q ˜ hc , Qhb : Vc × Vc,D → VΠ ˜0 := Q ˜0 = 0. Q ,u ˜h0 ∈ Ubh : F˜ h u Theorem 5.33. Convergence for diﬀerent quadrature approximations of fully nonlinn ⊂ ear problems: Under the conditions of Theorem 5.32 (k > 2(d − 1), d ≥ 2n + 1, P−1 h h S , > 2) with G ∈ CL (Br (P u0 )) and boundedly invertible F (u0 ), this quadrature h is stable and consistent in P h u0 , and has a unique approximate FEM F˜ h : Ubh → VΠ solution u ˜h0 for F˜ h (˜ uh0 ) = 0. It converges, for u0 ∈ H (Ω) with Ec u0 ∈ H (Ω ∪ Ωhc ), p = 2 and > 2 for polyhedral Ωh and curved Ωhc , and uh U h = uh H 2 (Tch ) , cf. (5.67)(5.110), (5.205), Conjectures 4.40, 4.41, Ec u0 − u ˜h0 U h ≤ Chmin{−2,p,k−d} u0 H min{,p+2,k−d+2} (Ω) , (5.208) for the nonconforming, and with p, p + 2 deleted in the exponents in (5.208) for the conforming FEM.

360

5. Nonconforming ﬁnite element methods

Remark 5.34. The combination of the inequalities for k and d requires large values ˜ h is proved for k, e.g. k > 8 for n = 2 and k > 16 for n = 3. If the stability of G via inverse estimates, as in (5.205), this is unavoidable. Otherwise, the quadrature formulas could be applied to subtriangles of the Tc ∈ Tch and k could be reduced. ˜hs (·, ·) are simultaneously bounded and, for Proof. We show that the as (·, ·) and a small enough h, satisfy the discrete inf–sup condition on Ubh × Vbh with equiboundedly ˜ h (u0 )) : invertible A˜h : Ubh → V h . This implies the stability of the corresponding (G ˜ h : U h → V h in P h u0 . By Lemma 5.10 we can assume the Ubh → V h , and hence of G b discrete inf–sup condition for as (·, ·) and obtain uh U h ≤ = ≤

inf

|as (uh , v h )|/v h V h ∀uh ∈ Ubh

inf

h h h a ˜s (u , v ) + (as (uh , v h ) − a ˜hs (uh , v h )) /v h V h

inf

h h h a ˜s (u , v ) /v h V h

0 =v h ∈Vbh

0 =v h ∈Vbh

0 =v h ∈Vbh

+

sup 0 =v h ∈Vbh

≤

inf

0 =v h ∈Vbh

(5.209)

as (uh , v h ) − a ˜hs (uh , v h ) /v h V h

h h h a ˜s (u , v ) /v h V h + Chuh U h ,

with (5.205) and k > 2(d − 1), so ﬁnally ( − Ch)uh U h ≤

inf

0 =v h ∈Vbh

h h h a ˜s (u , v ) /v h V h .

(5.210)

The inequalities for the inf 0 =uh ∈Ubh are handled analogously. This shows the discrete inf–sup condition for a ˜hs (·, ·) for suﬃciently small h > 0. The classical consistency error, see (5.205), is combined with the preceding (5.204) yielding with k > d, u0 ∈ H k+2 (Ω) and Theorem 5.13 h h ˜ hΠ F u0 h F˜ P u0 − Q VΠ h ˜ c G(P h u0 ), −Qhb BD u0 h = Q V Π

h ˜ G(P h u0 ) − Qh GP h u0 + Qh GP h u0 , −Qh BD u0 ≤ Q c c c b

h VΠ

≤ Chk−d u0 H k+2 (Ω) + CLhmin{,k+2,p+2}−2 u0 H k+2 (Ω) ≤ C hmin{,p+2,k−d+2}−2 u0 H min{,k+2} (Ω) . and hence, the claimed convergence.

5.4. Quadrature approximate FEMs

361

We ﬁnally discuss the implications of the two inequalities for k and d. d ≥ 2n + 1 =⇒ Sd1 Tch ⊂ C 1 (Ω) and k ≥ 2(d − 1) + 1 =⇒ inf–sup condition for a ˜hs (uh , v h ), both =⇒ k ≥ 2n+1 + 1 > 0. This indeed requires the large values for k mentioned above. 5.4.5

Quadrature FEMs for equations and systems of order 2m

Starting with linear and quasilinear cases we generalize the above set of spaces and approximations as in (5.177), with the notation in (4.129), (4.136), cf. Figure 5.5, U = V ⊂ W m,p (Ω, Rq ), Ubh , Vbh ⊂ L∞ Tch , Rq : ∀Tc ∈ Tch : uh |Tc ∈ P, uh vanishes exactly or approximately along ∂Ωhc , Pd−1 ⊂ P ⊂ Pd+τ , and · U h := · V h := · H m (Tch ,Rq ) , with dim Ubh = dim Vbh and

(5.211)

p = 2 and p ≥ 2 for linear and quasilinear cases, respectively. cf. Theorem 5.38 3. for 1 < p < ∞. The generalized weak linear FEM form and the ˜ h are projectors Q (5.212) For Aαβ (x) ∈ H k,∞ (Ω, Rq×q ), fα ∈ H k (Ω, Rq×q ) deﬁne ⎛ ⎞

˜h ∈ U h s.t. ˜h , ∂ α v h ⎠ =: Ah u ˜h , v h˜h and determine u Aαβ ∂ β u qThc ⎝ 0 0 0 b Tc ∈Tch

|α|,|β|≤m

q

⎛ ⎞

h ˜h , v = a ˜h , v h = f h , v h˜ = ˜h u qThc ⎝ (f α , ∂ α v h )q ⎠ ∀ v h ∈ Vbh , A˜h u 0 0 h˜h

Tc ∈Tch

|α|≤m

˜ h f − f , v h˜h = ((Q ˜ h f − f ) v h )h˜ = 0 ∀ v h ∈ Vbh . ˜ h : V → V h , Q and Q The previous Theorem 5.28 is generalized to these new linear problems. As a consequence of the modiﬁed spaces in (5.211) the exponent d + τ − 1 in Theorem 5.28 has to be replaced by d + τ − m here. So we update (5.184) and (5.185) for order 2m and systems in Rq , usually with τ = −1, as h n for f h , ·˜ , f = (f α ∈ H k (. . .))|α|≤m ∈ Hk−m Ω ∪ Tch , Rq k > max d + τ − m, 2 (5.213) k > 2(d + τ − m) for a ˜h (·, ·), Aαβ ∈ W k,∞ Ω ∪ Ωhc , Rq×q , |α|, |β| ≤ m. On this basis Theorems 5.30 and 5.31 remain verbatim and again guarantee convergence of the quadrature approximation. Theorem 5.35. Consistency of diﬀerent quadrature approximations of linear equations and systems of order 2m: 1. Assume (5.176), (5.211), and for the quasiuniform subdivision T h the Condition 4.16. Choose a quadrature formula as in (5.172), (5.174), satisfying (5.213) for

362

5. Nonconforming ﬁnite element methods

h vector valued functions, f , ·˜ and a ˜h (·, ·), with (extended) f ∈ H k Ω ∪ Tch , Rq , Aα,β ∈ W k,∞ Ω ∪ Tch , Rq×q , u ∈ H s (Ω, Rq ), s ≥ d, in (5.217). 2. Then, for small enough h > 0, the following consistency results hold ˜ ˜ |(f h , v h ) − (f h , v h )h | ≤ Chk−(d+τ −m) f H −m (Ω,Rq ) · v h V h ,

(5.214)

k

˜ h f h ≤ Chk−(d+τ −m) f −m Q h f − Q V H (Ω,Rq ) , lim k

h→0

˜ h f h = f V , Q V

(5.215)

|ah ( uh , v h ) − a ˜h ( uh , v h )| ≤ Chk−2(d+τ −m) MAα,β uh V h , for v h = 0 and h v V h (5.216) ≤ Chk−(d+τ −m) MAα,β uh H s (Tch ,Rq ) with MAα,β := and

max

|α|,|β|≤m

Aα,β W k,∞ (Ω,Rq×q ) ,

˜h (I h u, v h )| |ah ( u, v h ) − a ≤ C hd−1 + hk−(d+τ −1) MAα,β uH s (Ω,Rq ) . h v V h (5.217)

Theorem 5.36. Convergence for diﬀerent quadrature approximations of linear equations and systems of order 2m: 1. Let ah (·, ·) induce a stable Ah and assume (5.212) and the conditions in Theorem 5.35. Then, simultaneously a ˜h (·, ·) induces a stable A˜h . If furthermore (5.213) ˜h , v h ) = f , v h˜h ∀v h ∈ V h , deﬁnes a stable and convergent is correct, then a ˜h ( u 0 b discretization for A u0 = f . For the other approximations, mentioned above, this result remains correct if the a ˜h , A˜h are replaced by good enough approximations ˜ ˜ ˜ ˜h ˜ h h h ˜ ,A ,A . a ,a 2. For conforming FEMs, the above MAα,β , and C ∗ := C(1 + MAα,β ), u0 ∈ H s (Ω), we obtain ˜h h ≤ C ∗ hmin{s−m,k−(d+τ −m)} ( u0 min{s,k+m} u0 − u H (Ω,Rq ) + f H k (Ω,Rq ) ). 0 U h 3. If only approximations for f , v h˜ and none for a ˜h ( uh , v h ) are used, then ˜h h ≤ Chk−(d+τ −m) f H k (Ω,Rq ) for k > d + τ − m. u0 − u 0 U

We turn to quasilinear systems with a “quasibilinear” a ˜h ( uh , v h ), where

Aα (·, uh , . . . , ∇m uh ) : W m,p (Ω, Rq ) → Lp (Ω, Rq ), 1/p + 1/p = 1, (5.218) ⎛ ⎞ : ; h ⎝ a ˜h uh0 , v h = qThc Aα ·, uh0 , . . . , ∇m uh0 , ∂ α v h q ⎠ =: Gh uh0 , v h˜ . Tc ∈Tch

|α|≤m

5.4. Quadrature approximate FEMs

363

The previous estimate (5.214) is not needed, but (5.215) is still needed here, and (5.216), (5.217) have to be reformulated for the quasilinear operator and its linearized form, providing consistency and stability, hence convergence. Theorem 5.37. Consistency of diﬀerent quadrature approximations of quasilinear systems of order 2m in W s,p (Ω), p ≥ 2: 1. Assume (5.176), the updated (5.211), and for the quasiuniform subdivision T h the Condition 4.16. Choose a quadrature formula as in (5.174), satisfying (5.213), (5.212), (5.218), a diﬀerentiable Gh , k > m + n/2, s ≥ d, m, and 1 ∂Aα 1 + = 1, ∀ u ∈ W s,p (Ω) h := (·, u0 , .., ∇m u0 ) ∈ W k,p/(p−2) (Ω), p p ∂ϑβ

(5.219)

with u − u0 W m,p (Ω) < δ and h < h0 . 2. Then the following consistency results hold ˜ h

˜h (I h u, v h )| = |Gh ( u), v h − Gh (I h u), v h | ≤ C v h V h (5.220) |ah ( u, v h ) − a d−m h uW s,p (Ω,Rq ) + hk−(d+τ −m) · max Aα (·, u, . . . , Dm u)W k,p (Ω,Rq ) , |α|≤m

and for uh1 − I h u0 V h < δ : : ; h (Gh ) uh1 uh , v h − (Gh ) uh1 uh , v h˜ ≤ Chk−2(d+τ −m) ·

(5.221)

. / ∂Aα h h h ·, u1 , .., ∇m uh1 max k,p/(p−2) h · u V h v V h . |α|,|β|≤m ∂ϑβ W (T ) c

Theorem 5.38. Convergence of diﬀerent quadrature approximations for quasilinear systems of order 2m in W s,p (Ω), p ≥ 2: 1. Under the conditions of Theorem 5.37, and for a locally unique exact solution, ˜ h ( uh ) = u0 , of G( u) = 0, with boundedly invertible G ( u0 ), the FE equation G h h h ˜ Gh |U h in (5.218) has a unique solution u converging to u0 . ˜ =Q 0, G 0 2. With MAα := max Aα (·, . . . , Dm u0 )W k,p (Ω,Rq ) and for conforming FEMs we |α|≤m

get with u0 ∈ W s,p (Ω, Rq ) ˜h h ≤ Chmin{s−m,k−(d+τ −m)} u0 min{k,s},p u0 − u W (Ω,Rq ) (1 + MAα ). (5.222) 0 U 3. With Theorem 4.67 and (4.194), (4.195) analogous results are possible for the monotone operator approach for quasilinear problems even for 1 < p < ∞ and discrete W m.p (T h ) norms.

364

5. Nonconforming ﬁnite element methods

˜ h . For the Proof. Relation (5.221) implies the simultaneous stability of Gh and G ˜ h with G in P h u0 we obtain by (5.174) consistency of G : h ; : ; Aα ·, u0 , . . . , Dm uh0 , ∂ α v h q dx − qThc Aα ·, uh0 , . . . , Dm uh0 , ∂ α v h q Tc

≤ Ck hk Aα ·, . . . , Dm uh0 W k,p (T ) v h W k,p (Tc ) , c

(5.223)

hence, (5.220). For the stability we have to linearize our quasilinear operator, see (2.459), and get for uh1 , uh , v h ∈ Vbh ⊂ W m,p Tch , Rq , and with ·, · := ·, ·V h ×V h , 3 4 9 8 ∂Aα β h α h ∂Gh uh1 h h m h x, .., ∇ ∂ = u , v u u , ∂ v q dx. 1 ∂ uh ∂ϑβ Tc h Tc ∈Tc

|α|,|β|≤m

(5.224) With the above condition (5.219), we get 8 9 ∂A β h α h α m h ·, .., ∇ u1 ∂ u , ∂ v ∂ϑβ q W k,1 (T ,R) c ∂Aα β h α h ≤C ∂ϑβ k,p/(p−2) h q×q ∂ u W d+τ −|β|,p (Tc ,Rq ) ∂ v W d+τ −|α|,p (Tc ,Rq ) W (T ,R ) c

We modify (5.223) and estimate 8 9 8 9 ∂Aα β h α h ∂A α ∂ u , ∂ v dx − q h ∂ β uh ∂ α v h (5.225) β Tch ∂ϑβ ∂ϑ q q k ∂Aα ∂ β uh W d+τ −|β|,p (Tc ) ∂ α v h W d+τ −|α|,p (Tc ) ≤ Ch β ∂ϑ W k,p/(p−2) (T h ) h c

Tc ∈Tc

∂Aα m h ·, .., ∇ ≤ Ck hk u ∂ β uh W d+τ −|β|,p (Tch ) ∂ α v h W d+τ −|α|,p (Tch ) . 1 ∂ϑβ k,p/(p−2) h W (T ) c

The inverse estimates show, with (5.212), the stability, hence the claim (5.222).

Considering the many cases of semilinear equations as special cases of the preceding quasilinear problems with appropriately simpliﬁed conditions, the only missing class of elliptic equations are the nondivergent quasilinear and the fully nonlinear equations and systems of orders 2 and 2m. So we recall the spaces, the extension operators, Ec , ˜ h , the strong bilinear forms, nonlinear operators and the the approximate projectors Q generalizations, see (5.77), (4.73), (5.78), (5.47), (5.128), cf. Figure 5.6. We replace in (5.200), (5.201), the 2m = 2 and q = 1 by general m and q and obtain, note U = V, from here on U ⊂ H 2m (Ω, Rq ), V ⊂ L2 (Ω, Rq ), Ubh = S h = S0h ⊕ Sbh ⊂ Sd2m−1 Tch , Rq , (5.226) Vbh = S0h = S h ∩ C0m−1 Tch , Rq , Vbh = Sbh , ·U h := ·H 2m (Tch ,Rq ) , ·V h := ·L2 .

5.4. Quadrature approximate FEMs

365

˜h has to satisfy the quadrature approximate diﬀerential operator The FE solution u 0

˜h , v h˜h = ˜h ∈ U h s.t.Gw ·, . . . , D2m u ˜h v h = 0∀ v h ∈ V h . qThc G u u 0 b 0 0 b q Tc ∈Tch

(5.227)

For the stability we need the linearized form A% & h γ h hB β , |β| ≤ 2m) w ∂ϑγ Gw (·, ϑ qThc 0 ∂ u , v q . a ˜hs ( uh , v h ) = Tc ∈Tch

(5.228)

|γ|≤2m

So we replace the original approximation Gh = Qhc G|U h , by the quadrature approxi˜ hc G|U h in (5.227). We start as in Subsection 5.4.4 with G ˜h, Q ˜h, Q ˜ hc : ˜h = Q mation G Theorem 5.39. Consistency of diﬀerent quadrature approximations for fully nonlinear systems of order 2m: 1. For m, q ≥ 1, k > 2(d − m), hence, k − d > 0, let h β , |β| ≤ 2m) w 0 ∈ W k+|γ|,∞ (Ω × RN2m ), ∀|γ| ≤ 2m : ∂ϑγ Gw (·, ϑ cf (5.73), (5.17) (5.5), ∩ H k+2m (Ω) → H k (Ω) and G : D(G) ∩ G : hD(G) k+2m h k Ω ∪ Ωc → H Ω ∪ Ωc . Choose the approximating Ubh = S h ⊃ Vbh = H h S0 , d ≥ (2m − 1)2n + 1 as in (5.226). Let the quadrature formula satisfy (5.172), (5.174), and let h > 0 be small enough. Again let, p = 2 for the conforming FEM, and for the nonconforming FEM 2 ≤ p ≤ 5, with p = 2 for polyhedral Ωh and p > 2 for better curved Ωhc approximations for Ω ∈ C 2m , cf. Conjectures 4.40 and 4.41. 2. Then (5.202) holds and ˜ h f h ≤ Ck hk−d f −m , lim Q ˜ h f h = f −m , Qh f − Q V V H (Ω) H (Ω) k

h→0

h |G( uh ), v h Vc − G( uh ), v h˜ | ≤ Chk−d G( uh )Hck (Tch ) v h V h , or h ˜ hc G( uh ) ≤ Chk−d G( uh )H k (T h ) , and Qc G( uh ) − Q V c c h h h h h h k−2(d−m) h as ( u , v ) − a ˜s ( u , v ) ≤ Ch u U h v h V h h β , |β| ≤ 2m) w × max ∂ϑγ Gw (·, ϑ 0 W k+|γ|,∞ (Tch ) . |γ|≤2m

(5.229)

k

(5.230)

(5.231)

3. Again, these inequalities only estimate the quadrature errors for the diﬀerential and not for boundary operators. For the complete consistency errors the terms in Theorem 5.15 and L× the interpolation errors have to be added, cf. Theorem 5.40. L denotes the Lipschitz constant for G. Proof. Since uh , v h ∈ Sd2m−1 Tch ⊂ Hck Tch , the estimates in (5.174) again can be applied to diﬀerent choices of fc . All the above estimates follow as in the proof of Theorem 5.32, by replacing d − 1 by d − m and updating the coeﬃcients.

366

5. Nonconforming ﬁnite element methods

The above conditions d ≥ (2m − 1)2n + 1 > 2, n ≥ 2, for Sd2m−1 Tch ⊂ C 2m−1 (Ω) and k > 2(d − m), imply k > 2(d − m) ≥ 2n+1 > n/2, n ≥ 2, k > d. Hence, by the ¯ Sobolev embedding theorem the W k,∞ (Ω), H k (Ω) are compactly embedded into C(Ω). So all the quadrature formulas, even for a u ∈ W k,∞ (Ω), H k (Ω), are well deﬁned. Now we add the (unchanged) boundary conditions, cf. (5.118), the m−1 ∂ j uh0 h h h h h h = 0 ⇔ vbh , BD u0 V = 0 ∀ vbh ∈ Vbh ⇔ Qhc,b BD u0 = 0. BD u0 := j c,b ∂ν ∂Ωh c

j=0

˜h = Replacing the previously introduced projectors Qhc,Π = (Qhc , Qhc,b ), by Q c,Π ˜h ∈ U h as ˜ hc , Qh ), we deﬁne the discrete quadrature operator and the FE solution u (Q 0 c,b b

h ˜ h ˜h ˜h = 0 ∈ V h . ˜ hc,Π Fh |U h u ˜ hc , Qhb : Vc × Vc,D → VΠ ˜ hc,Π := Q , F u0 := Q Q 0 Π (5.232) Theorem 5.40. Convergence of diﬀerent quadrature approximations for fully nonlinear systems of order 2m: n ⊂ S h, d ≥ 1. Assume the conditions in Theorem 5.39, e.g. k > 2(d − m), P−1 n (2m − 1)2 + 1, a locally unique solution u0 ∈ H (Ω), > 2m, and boundedly invertible F ( u0 ). Choose for the conforming FEMs for m = 1 and a convex polyhedral domain, cf. (5.67), and for the nonconforming FEM for domains in C p , 2 ≤ p ≤ 5, the p = 2 for polyhedral approximation Ωh and p > 2 for better curved Ωhc approximations for Ω ∈ C 2 , cf. Conjectures 4.40 and 4.41. h ˜ h ˜h , F ( u0 ) = 0 in (5.232) 2. Then the quadrature approximate FEM F˜ h : Ubh → VΠ ˜ h has a unique solution u0 , is stable and consistent in P h u0 , hence convergent, such that, see (5.67),(5.110), Ec u0 − uh0˜ h ≤ Chmin{,p+2m,k−d+2m}−2m u0 ∈H min{,k+2m} (Ω) . (5.233) U

Proof. The proof of Theorem 5.32 remains essentially unchanged. The uh are turned into uh , etc., and the last inequality, with k > 2(d − m), d > (2m − 1)2n , has to be modiﬁed into h h ˜ hΠ F u0 h ≤ C hmin{,p+2m,k−d+2m}−2m u0 H min{,k+2m} (Ω) . F˜ P u0 − Q V Π

We ﬁnally discuss the implications of the two inequalities for k and d. We consider m ≥ 1 and obtain d ≥ (2m − 1)2n + 1 to guarantee Sd2m−1 Tch ⊂ C (2m−1) (Ω) and (5.234) k ≥ 2(d − m) + 1, m ≥ 1 =⇒ inf–sup condition for a ˜hs ( uh , v h ), consequently k − d

(5.235)

≥

d − 2m + 1

(5.234)

>

(5.235)

0.

(5.236)

This indeed requires the large values for k mentioned above.

5.4. Quadrature approximate FEMs

5.4.6

367

Two useful propositions

At the end of this section we want to present two results which are important for integrations and for applications in the ﬁnal stability proof for nonconforming FEMs. Sometimes we have to rescale the integral over an interval [−1, 1] to [0, h] or a more general reference domain K to T . Examples are (5.171), (5.172) and (5.173). We prove the following result on scaled Sobolev norms and apply it to the interpolation basis. Let P = span{φ1 , φ2 , . . . , φd } be a basis dual to N , hence Ni (φj ) = δij and FT : K → T the aﬃne bijection. Then, see Deﬁnition 4.8, φTi = φi ◦ FT−1 = φi ◦ ϕ : T → R, i = 1, . . . , d, ∀ T ∈ T h with the aﬃne ϕ := FT−1 : T t → x := Bt/h + b ∈ K, B ∈ Rn×n , b ∈ Rn .

(5.237)

is called an interpolation basis for Ubh or Vbh . It is not necessarily a basis for Ubh or Vbh , but for a ﬁxed T ∈ T h the φTi , i = 1, . . . , d are a basis for PT . Proposition 5.41. Transformation theorem: Let K ⊂ Rn be a reference domain, see Deﬁnition 4.1, 0 < h ≤ 1 and ϕ in (5.237) an aﬃne scaling mapping. Choose u ∈ W s,p (K) and α ∈ Nn a multi-index with |α| ≤ s. Then n −1 (u ◦ ϕ) = h |det(B)| u , and T

K

|∇α (u ◦ ϕ)|p

p1

|∇α u|p

= hn/p−|α|

T

p1 , for B = I, else

K

|∇α (u ◦ ϕ)|p

p1

≤ hn/p−|α| B|α| |det(B)|−1/p

T

|∇α u|p

p1 ,

(5.238)

K

speciﬁcally |u ◦ ϕ|W s,p (T ) ≤ C hn/p−s (max{1, B})s |det(B)|−1/p |u|W s,p (K) , u ◦ ϕW s,p (T ) ≤ C hn/p−s (max{1, B})s |det(B)|−1/p uW s,p (K) and T φi s,p ≤ C hn/p−s (max{1, B})s |det(B)|−1/p φi W s,p (K) , W (T ) d with φTi H 1 (Ωh ) ≤ C hn/2−1 max{φj H 2 (K) }, j=1

so the φT

i H 1 (T h ) are bounded for n ≥ 2. Deﬁning a quadrature formula as in (5.172) for T , we start with a quadrature formula for K with the corresponding wj , Pj . Then the Pj,T := FT (Pj ) and the wj,T := hn |det(B)|−1 wj .

We omit the straightforward proof, but prove that for f h ∈ V h there exists an appro priate f0h ∈ V such that f h = Q h f0h . By Theorem 2.35, (2.110f), (2.108) we know that f h has the form (5.239) with bounded fj Lp (Tch )=Lp (Ω) , so f ∈ W −1,p (Ω) = V . We need this property for the later proof of stability of nonconforming FEMs for

368

5. Nonconforming ﬁnite element methods

f h = Ah uh , for Q h cf. (5.294). n h h i h f , v := dx ∀v h ∈ V ∪ Vbh , f h ∈ W −1,p Tch , fi ∂ v Tc ∈Tch

Tc

(5.239)

i=0

fj ∈ Lp Tch , with Q h : V → V h , Q h f − f, v h = 0 ∀v h ∈ Vbh ∈ W 1,p (Ω). Proposition 5.42. Assume the local and nonlocal interpolation operators I h in Deﬁnition 4.8 (4.67) are well deﬁned and allow the corresponding interpolation bases, the φTi in (5.237) and let V = W 1,p (Ω). Then

∀f h in (5.239), ∃ feh ∈ W −1,p (Ω) ⊂ V , 1 ≤ p ≤ ∞, s.t. f h = Q h feh , so : h h; fe , v = f h , v h ∀v h ∈ Vbh , lim f h V h = Q h feh V h = feh V . (5.240) h→0

−1,p (Ω) ⊂ V , 1 ≤ p, p ≤ ∞, 1/p + 1/p = 1 for f h in Proof. We aim for feh∈ W h 1,p h Tc . With the restrictions v h |Tc and feh |Tc , we have (5.239) with V ⊂ W v h ∈ V h ⊂ W 1,p Tch , f h ∈ W −1,p Tch ⇒ ∂ i v h |Tc ∈ Lp (Tc ), fi |Tc ∈ Lp (Tc ), ∀T ∈ Tch , i = 0, . . . , n. For this type of f h ∈ W −1,p Tch ⊂ V h the fi |Tc ∈ Lp (Tc ) imediately extend to an fi,e ∈ Lp (Ω) s.t. fi,e |Tc ∈ Lp (Tc ). Then obviously the extended n : h ; i fi,e ∂ v dx ∈ V fe , v := Ω

i=0

satiﬁes (5.240).

5.5 5.5.1

Consistency, stability and convergence for FEMs with variational crimes Introduction

Similarly to the conforming FEMs in Section 4.3 we restrict the discussion here to U = V, Ub = Vb , Ubh = Vbh . These we had exactly evaluated the weak bilinear and higher nonlinear forms, e.g. a(·, ·) : V h × V h → R. So they are well deﬁned and exactly available. However, V = H 1 (Ω) for m = 1 requires V h ⊂ C(Ω) (and V = H 2 (Ω) for m = 2 even V h ⊂ C 1 (Ω)), so often too restrictive conditions. We have to allow violations of these conditions. The Vbh ⊂ Vb , admit violated boundary conditions and Vbh ⊂ V violated continuity (and diﬀerentiability) across the boundary edges, e of T ∈ T h . These are the so-called nonconforming FEs, see cases (1), (2) in Subsection 5.5.2. Except partially in Subsection 5.5.7, we consider T h with curved triangles at the boundary. So we again use the notation Tch . As for quadrature approximations, the goal here is ﬁtting the violated boundary and continuity conditions, such that the original convergence rate is not invalidated. This will motivate the appropriate

5.5. Consistency, stability and convergence for FEMs with variational crimes

369

choice of parameters for these methods. We will come back to this point in (5.325) 2 2 ⊆ P ⊆ Pd+τ with τ ≥ −1. This allowed including, ﬀ. below. Before we discussed Pd−1 as the most important case, the tensor product FEs Qnd , cf. (4.13) with τ = d − 2. The following convergence results would remain correct for τ > −1, or even for τ = d − 2 only if unrealistic conditions were imposed. So we restrict the FEs, studied here, to 2 ¯ = ∪T ∈T h T ∀ T ∈ Tch , Ω V h ⊂ v : Ω ⊂ R → R2 : v|T ∈ P = Pd−1 c

(5.241)

the piecewise polynomials of degree d − 1. For reasons discussed below, we restrict the discussion in this section to R2 to, except for isoparametric FEs. We unify the presentation again by using the same symbols u, uh , . . . ∈ H 1 (Ω), . . . instead of ∈ H 1 (Ω, Rq ) and, as coeﬃcients, aij instead of Aij , for real and vector valued functions as in (5.285). Already in the last Sections 5.2–5.4 we studied diﬀerent types of nonconforming FEMs. For fully nonlinear problems the available C 1 FEs are not able to satisfy the boundary conditions exactly. In fact, we had to introduce an Ωhc ≈ Ω for the new FEs. We will similarly discuss violated boundary conditions in Subsection 5.5.6 ﬀ., and isoparametric FEMs in Subsection 5.5.10. In the Subsections before 5.5.10 we will, however, only study polynomial FEs, deﬁned on a subsdivision of the original domain Ω or a curved approximation Ωc . For more complicated data the integrals in the FE equations have to be computed by quadrature or more general approximations as in the previous Section 5.4. The interplay between the weak and strong operators yields an essential tool for handling nonconforming FEMs. Here we start with linear operators, estimating the violation of Vbh ⊂ Vb or Vbh ⊂ V. Later on we will extend this approach to nonlinear problems as well. We systematically indicate by appropriate indexing, the weak and strong operators as A, Ah and As , Ahs , and the corresponding bilinear forms as a(·, ·), ah (·, ·), ah (·, ·) and as (·, ·), as,h (·, ·), ahs (·, ·). The following technique in Subsections 5.5.6 and 5.5.7 is a generalization of Brenner and Scott [141]. It extends their results for the Laplacian in R2 to general second order linear and quasilinear equations and systems. We systematically develop high accuracy employing the high order Gauss-type quadrature formulas along the one-dimensional edges. They exist no longer for the two-dimensional edges, e, of T ∈ T h ⊂ R3 . So even for the few examples of two-dimensional cubature rules of higher order, the corresponding FEMs lose much of their higher order convergence. So we omit them here. The R2 results are closely related to the conditions for patch tests as discussed by Stummel, Shi et al. [583–585, 587, 611]. High accuracy FEMs with variational crimes are important for path-following of parameter-dependent solutions of nonlinear problems and the study of turning and bifurcation points, in particular for avoiding spurious solutions. Furthermore, we will use a similar approach for the spectral methods in [120]. Other possibilities are the DCGMs and the direct tensor product versions for rectangular subdivisions and collocation due to Bialecki and Fairweather [95], not discussed here.

370

5. Nonconforming ﬁnite element methods

We start with the Helmholz operator, Example 5.44 in Subsection 5.5.2, see Example 4.45. This simple case motivates deriving the necessary extensions for bilinear forms and operators and are discuss the consequences of the nonconformity for the bilinear form. We elaborate the impact of variational crimes via the diﬀerence of the weak and strong bilinear forms and prove the basic Strang Lemma 5.52. In Subsection 5.5.3 we generalize these results to second order linear and quasilinear problems as appropriate general discretization concepts. These are again generalized Petrov–Galerkin methods. Subsection 5.5.4 shows the coercivity of the principal part as a ﬁrst step towards stability. It formulates, as a guideline for the further subsections, the variational consistency error for linear problems. The variational consistency errors are estimated via the quadrature errors of quadrature formulas applied to the edges of neighboring triangles. So we introduce in Subsection 5.5.5 diﬀerent types of high order Gauss quadrature formulas. Now we are ready to estimate, in Subsections 5.5.6 and 5.5.7, the classical and variational consistency for violated bondary conditions and continuity in linear and quasilinear problems. In Subsection 5.5.7 and for linear problems, the Aubin–Nitsche trick for increasing the order of convregence by one with respect to the L2 (Ω) norm is applied. By combining the results from the last three subsections, we prove stability for boundedly invertible linear operators with consistent discretization in Subsection 5.5.8. We use here an anticrime transformation for a modiﬁcation of the original proof in Section 3.5. The next subsection, 5.5.9, proves convergence for the exact FEMs and their quadrature approximations for linear and quasilinear problems. For eﬃciently solving the nonlinear equations, we again apply the mesh independence principle for our discrete problems. The last Subsection 5.5.10 considers isoparameteric FEMs and strongly extends them to quasilinear problems. In this section we assume that Ω ⊂ R2 is a bounded domain, and the solution is u0 ∈ H 2 (Ω).

(5.242)

For nonsmooth u0 ∈ H 2 (Ω) or nonconvex Ω, singular adapted FEs are added V h . Then the strong regularity conditions can be reduced and still the usual high accuracy can be guaranteed, for details cf., e.g. Blum and Rannacher [98].

5.5.2

Variational crimes for our standard example

Often, these nonconforming FEMs are only studied for cases, particularly interesting for applications. This holds, e.g. for the results of Graham, Hackbusch and Sauter, [361–364] Their results, relevant for our approach, are summarized in Subsection 4.2.5. On this basis it is possible, generalizing the following results for nonconforming FEMs on quasiuniform to more general triangulations. Remark 5.43. Again, the condition of a quasiuniform subdivision, T h , in this and the following theorems, can be relaxed, to more general triangulations cf. Subsection 4.2.5. Here we study only FEMs in R2 , and require Assumptions 4.21 and 4.26. Again in the messy sums below, T ∈T h , the terms have to be estimated for each T ∈ T h separately.

5.5. Consistency, stability and convergence for FEMs with variational crimes

371

The results in Theorems 4.30–4.33 allow appropriate estimates for continuous and L2 FEs. We leave the details to the interested reader. Here we intend a discussion of general nonconforming FEMs, where we impose Condition 4.16. For presenting the essental ideas more clearly, we start studying, for our simple Example 4.45, the impact of violated boundary conditions and continuity. Later on, we have to generalize linear and bilinear forms, operators and projectors, similarly to Sections 5.2 and 5.4. Necessary extensions towards variational crimes Example 5.44. For Dirichlet boundary conditions, u ∈ H01 (Ω), f ∈ H −1 (Ω), and the Helmholtz operator we memorize, see (2.2) ﬀ, (4.107), (4.109), (5.242), the bounded operators and the corresponding bilinear forms and obtain by partial integration A : V := H 1 (Ω) → H −1 (Ω), As : H 2 (Ω) → L2 (Ω) bounded, and for u ∈ H 2 (Ω) : a(u, v) := ∇ u, ∇ v + cuvdx =: Au, vH −1 (Ω)×H 1 (Ω) (5.243) Ω

(−Δu + cu)vdx ∀v ∈ Vb := H01 (Ω).

= as (u, v) := (As u, v)L2 (Ω) = Ω

The weak and FE solutions u0 ∈ Vb = H01 (Ω) and uh0 ∈ Vbh ⊂ Vb are deﬁned by a(u0 , v) = f, vH −1 (Ω)×H01 (Ω) ∀v ∈ Vb ∇ uh0 , ∇ v h 2 + c uh0 v h dx = f, v h ... ∀ v h ∈ Vbh ⊂ Vb , a uh0 , v h =

(5.244) (5.245)

Ω

with u0 ∈ H 2 (Ω) for f ∈ L2 (Ω), Ω ∈ C 2 , and the inner product (·, ·)2 in R2 . Indeed, by Theorems 2.43 and 2.45 u0 , uh0 uniquely exist for c ≥ 0. Obviously the integral is well deﬁned for conforming FEs Vbh ⊂ H01 (Ω). But here we want to study nonconforming ﬁnite elements. Nonconforming ﬁnite elements 1. Finite elements violating the boundary conditions: In Section 5.2 we have discussed this problem of nonconforming FEs already for fully nonlinear problems. It is always possible to apply this technique to linear and quasilinear operators and their nonlinear forms. However, here we aim for the higher accuracy obtainable with polynomial FEs on curved domains in Subsection 4.2.7. For these FEs we recall (4.97), Ω = ∪T ∈Tch T and restrict the discussion to homogeneous Dirichlet boundary conditions. So the FEs uh : Ω → R satisfy uh (Pj ) = 0 only at certain points Pj ∈ ∂Ω or even only close to ∂Ω for curved boundaries by (4.98). For homogeneous Dirichlet boundary conditions Bv = v|∂Ω = 0 , this implies Vbh = {uh ∈ V h ⊂ H 1 (Ω) = V : B(uh ) ≈ 0}, H01 (Ω) = Vb ⊃ Vbh ,

(5.246)

372

5. Nonconforming ﬁnite element methods

This implies that (5.243) is violated if u, v are replaced by uh , v h , yielding an estimate for the error due to violated boundary conditions in the corresponding FEM. For natural boundary conditions, only speciﬁc test functions v h ∈ Vb ⊃ Vbh ⊂ V will cause problems. But the original bilinear a(·, ·) and linear forms f, · are deﬁned in V h × V h and V h . We postpone isoparametric FEs to Subsection 5.5.10. 2. Finite elements violating the continuity conditions: Let Vbh ⊂ H 1 (Ω), e.g. we have discontinuous FEs, or more generally, cf. (5.300) Vbh ⊂ V e.g. discontinuous across the edges e.

(5.247)

This happens for nonconforming ﬁnite elements, e.g. the Crouzeix–Raviart FE, see Figure 4.10 [135,141]. So, the uh are continuous only in all the, say k, aﬃnely equivalent points Pje ∈ e ⊂ T ∈ Tch on every e. Here k is so small that uhl (Pj ) = uhr (Pj ), j = 1, . . . , k, allows (uh |Tl )|e ≡ (uh |Tr )|e . Usually, even for curved ∂Ω, the number of points Pj ∈ e ⊂ T on the edges interior to Ω and along ∂Ω is the same. Then violated continuity implies violated boundary conditions as well. In these so-called nonconforming FEMs, (1), (2), we study polynomial FEs, FE spaces with Vbh ⊂ Vb or Vbh ⊂ V,

(5.248)

uh |T , v h |T ∈ P ∀uh , v h ∈ V h , T ∈ Tch with Pd−1 = P, of ﬁxed degree, with a sequence Vbh h∈H indicated by h, 0 < h ∈ H, inf h∈H h = 0. We combine these crimes with approximations for projectors, operators and linear, bilinear and higher nonlinear forms, mainly by quadrature, cf Section 5.4. In this subsubsection we mainly study the impact of nonconforming FEs onto necessary modiﬁcations of the linear and bilinear forms. For the following results we memorize, see Subsection 4.2.4: the v h ∈ V h usually are discontinuous across common edges e ⊂ T ∩ T1 of neighboring T, T1 ∈ Tch , and hence v h ∈ H 1 (Ω), [174], pp. 207–208, or for short v h ∈ C(Ω) ⇒ v h ∈ H 1 (Ω) (but C(Ω) ⊂ H 1 (Ω)).

(5.249)

The original linear, bilinear and higher nonlinear forms, e.g. f, · and a(·, ·), as (·, ·), have to be modiﬁed for the nonconforming FEs. This implies that the modiﬁed (5.243) is violated for uh , v h . We will estimate the errors due to nonconforming FEMs, caused by Vbh ⊂ H01 (Ω) or ⊂ H 1 (Ω). As in Section 5.4 the f, ·, a(·, ·), as (·, ·), A, As , . . . , have to be deﬁned piecewise according to (5.251). We start the discussion, assuming Ω = Ωh = ∪T ∈Tch T . We summarize these cases, cf. (5.178), under the notation v h ∈ V h ⊂ W 1,p Tch , fh ∈ W −1,p Tch , . . . , with Lp (Ω) = Lp Tch . For the important cases f ∈ V , fh ∈ W −1,p Tch , including Ah uh , we obtain the general form

5.5. Consistency, stability and convergence for FEMs with variational crimes

373

for f ∈ V , and

Ω = Ωh = ∪T ∈Tch T , 1/p + 1/p = 1, V = W 1,p (Ω), f ∈ V ⊂ W −1,p Tch : f, vW −1,q (Ω)×W 1,p (Ω) = (f−1 , ∇ v)2 + f0 v dx, cf. Propos. 2.34, extended Ω

fh , v W −1,q (Tch )×W 1,p (Tch ) := h

T ∈Tch

(f−1 , ∇ v h )2 + f0 v h dx

(5.250)

T

for v h ∈ V + V h , fh ∈ V + V h , with f = fh |V , f h := fh |V h . Correspondingly to f , we deﬁne the following bounded extensions of the weak and strong bilinear forms and the operators ah (uh , v h ), Ah uh , and as,h (uh , v h ), As,h . For a simpliﬁed presentation we use the Hilbert–Sobolev version with p = 2 for linear problems: (∇ uh , ∇ v h )2 + c uh v h dx =: Ah uh , v h H −1 (Tch )×H 1 (Tch ) ah (uh , v h ) := T ∈Tch

T

∀uh , v h ∈ V + V h , with a(·, ·) = ah (·, ·)|V×V , ah (·, ·) := ah (·, ·)|V h ×V h and

as,h (uh , v h ) :=

T ∈Tch

T

(5.251) (−Δ uh + c uh )v h dx =: (As,h uh , v h )L2 (Tch ) (5.252)

∀uh ∈ H 2 (Ω) ∩ V h , ∀v h ∈ L2 (Ω) + V h with as (·, ·) = as,h (·, ·)|H 2 (Ω)×L2 (Ω) , ahs (·, ·) = as,h (·, ·)|V h ×V h . Remark 5.45. We prevent a well-motivated misunderstanding: obviously the ah (u, v h ), as,h (u, v h ) are well deﬁned for u ∈ Vb . Now, ah (u, v h ) = Au, v h H −1 (Tch )×H 1 (Tch ) ∀ v h ∈ Vbh for the variational crimes with Vbh ⊂ Vb . This inequality will be speciﬁed in Proposition 5.47. Note that Au, v h H −1 (Tch )×H 1 (Tch ) always makes sense for u ∈ H 1 (Ω): the f := Au ∈ V according to (5.243), is deﬁned by testing ∀ v ∈ Vb . This f induces, by (5.250), an fh , now tested ∀ v h ∈ Vbh ∪ Vb and a f h , now tested ∀ v h ∈ Vbh . So for u ∈ H 1 (Ω) the Ah u is certainly well deﬁned by ah (u, v h ) = Ah u, v h H −1 (Tch )×H 1 (Tch ) , but for variational crimes

(5.253)

= Au, v h H −1 (Tch )×H 1 (Tch ) ∀ v h ∈ Vbh or Q h Au = Q h Ah u,

for the projectors Q h introduced in (5.294) below. This situation is essential for the case of the later classical consistency errors. Impact of variational crimes for the standard example The relation between strong and weak bilinear forms is the essential tool for studying the impact of variational crimes in FEMs. They are reﬂected by the discontinuity

374

5. Nonconforming ﬁnite element methods

of the FEs crossing the interior edges e ⊂ T ∈ Tch and violated boundary conditions along the boundary edges e ⊂ ∂Tch . Generalizing (5.245), we determine the (weak) discrete solution as : ; uh0 ∈ Vbh : ah uh0 , v h = Ah uh0 , v h H −1 (T h )×H 1 (T h ) = f h , v h ··· ∀ v h ∈ Vbh , c

with

Vbh

⊂

c

H01 (Ω)

or ⊂ H 1 (Ω).

(5.254)

cf. (5.248). For natural boundary conditions only Vbh , ⊂ H 1 (Ω) is problematic. Similarly as in Section 5.2 we now systematically study the relation between weak and strong operators and forms, here for our standard example. We apply the ﬁrst Green’s formula (2.23) to every T or T = Ω ﬁnding ∂u (5.255) −v Δ u dx = (∇u, ∇ v)2 dx − v ds. T T ∂T ∂ν This implies for the exact bilinear forms, see (5.243) for c = 0 as well, as (u, v) = a(u, v) ∀ u ∈ H 2 (T ), and either ∂uh /∂ν|∂T = 0 or v ∈ H01 (T ). (5.256) Hence, (5.244), and a corresponding strong problem as (u0 , v) = (f, v)∀ v ∈ Vb would have identical solutions u0 ∈ H 2 (Ω), e.g. for f ∈ L2 (Ω), Ω ∈ C 2 . This is satisﬁed for either natural or Dirichlet boundary conditions. For the extended bilinear forms ah (u, v) and as,h (u, v) in (5.251) and (5.252), now for c = 0, we apply the Green’s formula (5.255) for every T ∈ Tch . With the surface integrals e · · · ds, e ∈ T¯ and νe indicating the outer normal vector for T , perpendicular to e, we get ∀u ∈ (V + V h ) ∩ H 2 (Tch ), v ∈ V + V h ,

−vΔudx = (∇v, ∇u)2 dx − v(∇u, νe )2 ds as,h (u, v) = T ∈Tch

T

= ah (u, v) −

T ∈Tch

T ∈Tch e∈T¯

T

e∈T¯

e

v(∇u, νe )2 ds

e

with BΔ u : = (∇u, νe )2 = ∂u/∂νe .

(5.257)

Obviously (5.257) reduces to (5.243) for u ∈ (V ∩ H (Ω)), v ∈ Vb : In (5.257) the surface integral e for every interior edge e ∈ T will be obtained twice with opposite normal directions for neighboring Tr , Tl ∈ Tch and e ⊂ Tr ∩ Tl . The edges 38 e ∈ ∂Ω will be summed up once. We consider the transition from a triangle Tl to its neighboring Tr . Let ν = νl = νe be the outward normal for e as the face of Tl . Then νr = −νl is the outward normal for e as the face of Tr . We denote the restriction of v, analogously for ∇u, (∇u, νe )2 , to Tl , Tr , and their extension to Tl , Tr as vl , vr , ∇ul , (∇u, νe )2 . We introduce the standard notation, see [665], 2

vl = v|Tl , vr = v|Tr , [v] := vl |e − vr |e and {v} := (vl |e + vr |e )/2,

(5.258)

38 We use the notation e ∈ ∂Ω, although either e ⊂ ∂Ω or even less for curved bouindaries, similarly e ∈ Tch \ ∂Ω.

5.5. Consistency, stability and convergence for FEMs with variational crimes

375

for the corresponding jumps and arithmetic means of v across an interior e, and [v] := {v} := v|∂Ω along ∂Ω, and

v arbitrary in R2 \Ω.

For (5.257) we need, with e ∈ Tch \ ∂Ω or e ∈ ∂Ω, cf. [665], p. 153, [141], p. 206, v∂u/∂νe ds = v(∇u, νe )2 ds (5.259) T ∈Tch e∈T

e

=

e

(vl (∇ul , νe )2 − vr (∇ur , νe )2 ) ds +

e∈Tch \∂Ω

=

e

(vl (∇ul , νl )2 + vr (∇ur , νr )2 ) ds +

e∈Tch \∂Ω

=

T ∈Tch e∈T¯

e

e

e∈∂Ω

e

([v]({∇u}, νe )2 + {v}([∇u], νe )2 ) ds + e

e∈Tch \∂Ω

v(∇u, νe )2 ds

e∈∂Ω

v(∇u, νe )2 ds

e∈∂Ω

v(∇u, νe )2 ds,

e

since ([v]{∇u} + {v}[∇u])

(5.260)

1 1 (vl − vr )(∇ul + ∇ur ) + (∇ul − ∇ur )(vl + vr ) = (vl ∇ul − vr ∇ur ). 2 2 This yields the following relation between the weak and strong bilinear forms ([v]({∇u}, νe )2 + {v}([∇u], νe )2 ) ds as,h (u, v) = ah (u, v) − =

−

e∈∂Ω

e∈T \∂Ω

e

v(∇u, νe )2 ds ∀u ∈ (V + V h ) ∩ H 2 (Ω), v ∈ V + V h .

(5.261)

e

We will mainly estimate the absolute value of the last messy sums for u, v ∈ Vbh , see below and Subsubsections 5.5.2 and 5.5.4. So we can choose any one of the two diﬀerent νT = νTl or νT = νTr . The following relations for the as (u, v), a(u, v) and ahs (u, v), ah (u, v) are basic to study the impact of variational crimes. This (5.261) implies several cases: for u ∈ Vb ∩ H 2 (Ω), v ∈ Vb we ﬁnd again as (u, v) = a(u, v). More interesting are the uh , v h ∈ Vbh and we obtain from (5.261) Proposition 5.46. 1. For our standard Example 5.44, we obtain for the bounded ah (uh , v h ) and as,h (uh , v h ) in (5.251) and (5.252), respectively, now with ν = νe , h h h h [v h ]({∇uh }, ν)2 + {v h }([∇uh ], ν)2 ds as,h (u , v ) = ah (u , v ) − −

e∈∂Ω

e∈T \∂Ω

e

v h (∇uh , ν)2 ds ∀uh ∈ (V + V h ) ∩ H 2 (Tch ), v h ∈ V + V h .

e

(5.262)

376

5. Nonconforming ﬁnite element methods

2. More generally, if the Vbh satisfy exact Neumann and Dirichlet boundary conditions, then v h (∇uh , ν)2 = 0 ∀ e ∈ ∂Ω, uh , v h ∈ Vbh and the last h respectively, term e∈∂Ω e v (∇uh , ν)2 ds in (5.262) vanishes. Three special cases are important. Relation (5.262) simpliﬁes by (1.67) as v h (∇uh , ν)2 ds (5.263) ah (uh , v h ) = ahs (uh , v h ) − e∈∂Ω

e

for v h ∈ Vbh ⊂ H 1 (Ω) and uh ∈ Vbh ∩ H 2 (Ω), v h ([∇uh ], ν)2 ds ah (uh , v h ) = ahs (uh , v h ) − −

e∈∂Ω

h

h

h

a (u , v ) =

e∈Tch \∂Ω

(5.264)

e

v h (∇uh , ν)2 ds for v h ∈ V h ⊂ H 1 (Ω),

e

ahs (uh , v h )

for v h ∈ Vbh ⊂ H01 (Ω), uh ∈ Vbh ⊂ H 2 (Jch ).

(5.265)

For the diﬀerent types of variational crimes we combine (5.262)–(5.265), with the V. nonconformity Vbh ⊂ Vb and Vbh ⊂ Our ﬁnal goal is an estimate for u0 − uh0 V h for the exact and discrete solutions u0 and uh0 , respectively. For general discretization methods, the classical and variational consistency errors are If ah (·, ·) satisﬁes an inf–sup condition, this is achieved relevant. h h by estimates for ah u0 − u0 , v or ah (u − uh , v h ), cf. Subsection 5.5.4, based upon the following proposition. Proposition 5.47. For FEs with violated boundary conditions and/or continuity choose an arbitrary u ∈ H 2 (Ω), fu := Au ∈ L2 (Ω) and the corresponding discrete solu tion Ah uh = Q h Au, e.g. exact and the discrete solutions u0 ∈ H 2 (Ω) and uh0 for (5.244) and (5.254). Then we obtain, in particular for (u, uh ) = u0 , uh0 , Ah u, v h H −1 (Tch )×H 1 (Tch ) − Au, v h H −1 (Tch )×H 1 (Tch ) = Ah u − Au, v h ... = ah (u − u , v ) = h

h

e∈Tch \∂Ω

[v ]∇u · νds + h

e

e∈∂Ω

v h (∇u, ν)2 ds.

(5.266)

e

Remark 5.48. For variational crimes (5.263)–(5.265)and (5.266) imply ahs (uh , v h ) = ah (uh , v h ) and ah (u0 − uh0 , v h ) = 0. So for small errors, hence a close relation between the ah (·, ·) and ahs (·, ·) : Vbh × Vbh → R, the FEs have to be chosen such that along the common edges e the [v h ] and on ∂Ω the v h disappear suﬃciently often. Choosing these zeros among appropriate quadrature points, the corresponding quadrature rules applied to the edges e disappear and consequently the integrals, e.g. h h v ([∇u ], ν)2 ds, are very small. Extensions to R3 are possible using results as in e Cools [208–210]. Since these formulas have only lower order of converge, and since we consider DCGMs in Chapter 7, we do not discuss these modiﬁcations here.

5.5. Consistency, stability and convergence for FEMs with variational crimes

377

Remark 5.49. In Section 5.5.4 we will estimate the consistency errors introduced in Deﬁnition 3.14. The standard form, for appropriate p, k, q, will be of the form sup {|ah (P h u − u, v h )|/v h V h } = C μ hp uWqk (Ω) .

(5.267)

0 =v h ∈Vbh

Now this u ∈ Wqk (Ω) is much smoother than the usual u ∈ V = H 1 (Ω) ⊃ Wqk (Ω). Nevertheless we get vanishing consistency errors for u ∈ H 1 (Ω) as well. For a given u ∈ V = H 1 (Ω) and any > 0 we have to show that |ah (P h u − u, v h )|/v h V h < ∀h < h . We choose us ∈ Wqk (Ω) such that u − us V < C δ and determine δ such that the previous < is satisﬁed. We combine (5.267), the boundedness of ah and (4.42) to get |ah (P h u − u, v h )| ≤ |ah (P h us − us , v h )| + |ah (P h u − P h us , v h )| + |ah (us − u, v h )| ≤ |ah (P h us − us , v h )| + C P h u − P h us V h v h V h + C us − uV h v h V h ≤ C μ hp us Wqk (Ω) + C (CC + 1)δ v h V h ≤ v h V h for suﬃciently small h, δ. Due to the testing of the exact and discrete solutions u0 , uh0 by v ∈ Vb , v h ∈ Vbh with Vbh ⊂ V, Vb , this argument is not suﬃcient for variational crimes. Therefore, violated continuity and boundary conditions and approximations imply more or less complicated errors. The corresponding estimates for these nonconforming FEMs are highly technical. We will study the variational consistency errors in Subsections 5.5.4 ﬀ. A ﬁrst step in this direction is to update Proposition 5.47 for the generalized elliptic operators considered in (2.135). We only have to replace the ∇u0 · ν in (5.269) by Ba u0 . This yields Proposition 5.50. For FEs with violated boundary conditions and/or continuity the exact solution u0 ∈ H 2 (Ω) and the discrete (weak) solutions uh0 ∈ Vbh are related by h h h ah u0 − u0 , v = [v ]Ba u0 ds + v h Ba u0 ds ∀v h ∈ Vbh . (5.268) e∈Tch \∂Ω

e

e∈∂Ω

e

Remark 5.51. For evaluating these ∂u/∂ν|∂Ω or Ba u|∂Ω , and formulating the nonconforming FEMs, the situation is more complicated. An arbitrary u ∈ H 1 (Ω) does not allow a meaningful restriction to ∂Ω or to e to obtain ∂u/∂ν ∈ L2 (∂Ω) and ∂u/∂ν ∈ L2 (e). We omit the reference to Ba u|∂Ω . There are two ways out of this dilemma. Either Theorems 1.37, 1.39 are used for replacing the above u ∈ H 1 (Ω) by

2 u ∈ H 3/2+ (Ω) with any > 0. Then ∂u/∂ν| ∂Ω ∈ H (∂Ω) ⊂ L (∂Ω) or ∂u/∂ν|e ∈ 2 L (e) is valid. Hence, ∂Ω v∇udS, and e v∇udS, are well deﬁned, cf. (7.42). Or u ∈ H 1 (Ω) is a function in the domain, e.g. of the Laplacian; then again e v∇udS are well deﬁned. This problem of boundary traces has attracted some attention recently, cf. Taylor [618], Chapter 4, Proposition 4.5, and Jonsson and Wallin [425], Jerison,

378

5. Nonconforming ﬁnite element methods

and Kenig [419], Schwab [575], and Grisvard [373]. Readers not so familiar with the H 3/2+ (Ω) results can in this chapter nearly everywhere replace H 3/2+ (Ω) by H 2 (Ω), with ∂u/∂ν|∂Ω ∈ L2 (∂Ω) and ∂u/∂ν|e ∈ L2 (e), thus requiring a bit more than necessary. Proof. For u ∈ H 2 (Ω) we modify (5.262) and use e {v h }([∇u], ν)2 ds = 0 along edges e ∈ Tch ﬁnding Ah u − Au, v h ...

= (5.251)

=

Ah u, v h ... − Au, v h ... with Ah uh = Q h Au − ah (uh , v h )

ah (u, v h )

u incorrectly – uh correctly tested by v h ∈ Vbh = (5.254)

=

ah (u − uh , v h ) (∇ u ∇ v h + c u v h )dx − ah (uh , v h ) T ∈Tch

(5.262)

=

T

T ∈Tch

+

(−Δ u + c u )v h dx

T

({v h }([∇u], ν)2 + [v h ]({∇u}, ν)2 )ds

e∈Tch \∂Ω

+

e∈∂Ω u∈H 2 (Ω)

=

e

v h (∇u, ν)2 ds − ah (uh , v h )

e

(f, v h )L2 (Ω) − (f, v h )L2 (Ω) + [v h ]({∇u}, ν)2 ds + v h (∇u, ν)2 ds, e∈Tch \∂Ω

hence the claim.

(5.269)

e

e∈∂Ω

e

In Subsection 5.5.4 we will estimate the terms in the last sum for general problems and thus estimate ah u0 − uh0 , v h . That these estimates are crucial is shown in the following Strang lemma, generalizing the Cea Lemma 4.48, see [135, 141]. For general cases we will proof convergence via the results of the general discretization theory in Chapter 3. Nevertheless, we include Strang’s proof for the convergence limh→0 u0 − uh0 H 1 (Ω) → 0. It essentially applies to “coercive” bilinear forms. Lemma 5.52. Strang lemma: Violated boundary and/or continuity conditions: Let Vbh ⊂ Vb , and/or Vbh ⊂ V, hence the v h , uh ∈ Vbh violate the Dirichlet boundary conditions v h |∂Ω = 0 and the Neumann boundary conditions ∇uh |∂Ω = 0, and/or are discontinuous. Let the bounded a(·, ·) and for more general cases let a(·, ·) satisfy an inf-sup condition and ah (·, ·) a uniform inf-sup condition. Let u0 ∈ Vb and uh0 ∈ Vbh

5.5. Consistency, stability and convergence for FEMs with variational crimes

379

be the exact and the unique approximate solution, deﬁned by a(u0 , v) = f, vV ×V ∀ v ∈ Vb , and ah uh0 , v h = f, v h V h ×V h ∀ v h ∈ Vbh . (5.270) Then, with C independent of h, and the broken Sobolev norms, · V h u0 − uh0

Vh

≤C

inf u0 − u V h + h

uh ∈Vbh

sup

ah u0 − uh0 , v h

0 =v h ∈Vbh

v h V h

. (5.271)

Proof. For any uh ∈ Vbh , u0 − uh0

Vh

≤ u0 − uh V h + uh − uh0 V h (triangle inequality) h ah u − uh0 , v h 1 h sup in (4.170) ≤ u0 − u V h + vh ∈V h \{0} v h V h b

= u0 − u V h h

ah (uh − u0 , v h ) + ah u0 − uh0 , v h 1 sup + vh ∈V h \{0} v h V h b

≤ u0 − uh V h +

1 |ah (uh − u0 , v h )| sup vh ∈V h \{0} v h V h

(5.272)

b

ah u0 − uh0 , v h 1 sup (triangle inequality) + vh ∈V h \{0} v h V h b

C h u − u0 V h continuity in Vbh × Vbh ah u0 − uh0 , v h 1 sup + vh ∈V h \{0} v h V h

≤ u0 − uh V h +

b

=

C 1+

u0 − u V h h

(5.273)

ah u0 − uh0 , v h 1 sup + vh ∈V h \{0} v h V h b

Note that, by continuity, ah u0 − uh0 , v h v h V h

≤ C u0 − uh0 V h

(5.274)

so that u0 − uh0

Vh

1 ≥ C

sup v h ∈Vbh \{0}

ah u0 − uh0 , v h v h V h

.

(5.275)

380

5. Nonconforming ﬁnite element methods

Combining (5.272) and (5.275), we ﬁnd / . ah u0 − uh0 , v h 1 h sup (5.276) , inf u0 − u V h max C vh ∈V h \{0} v h V h uh ∈Vbh b ≤ u0 − uh0 V h ah u0 − uh0 , v h C 1 h inf u0 − u V h + sup ≤ 1+ . uh ∈Vbh vh ∈V h \{0} v h V h b

Remark 5.53. 1. Inequality (5.276) indicates that the sum of approximation terms inf uh ∈Vbh u0 − uh V h , and nonconformity terms supvh ∈Vbh \{0} |ah (u0 − uh0 , v h )|/ v h V h correctly reﬂects the size of the discretization error u0 − uh0 V h . The same result for the general cases is obtained by the general discretization results in Chapter 3. Indeed by (3.33), the classical consistency error is estimated by the variational consistency and the interpolation error for the exact solution. For a ah (uh , v h ), satisfying a uniform discrete inf-sup condition, or a stable Ah this again yields (5.271), cf.(3.33). 2. As a consequence of (5.266), the second term on the right-hand side of (5.271) would be zero if Vbh ⊆ Vb , compare (4.156) and Theorem 4.54. Therefore, it measures the eﬀect of nonconformity in Vbh ⊆ Vb or Vbh ⊆ V. 5.5.3

FEMs with crimes for linear and quasilinear problems

In Subsection 5.5.2 we have studied the impact of FEMS with crimes for a special linear equation, the Helmholtz equation. Here we generalize these results to general linear and quasilinear equations and systems of order 2 in R2 . Since there are strong similarities we do not repeat all details for (5.251)–(5.261). The corresponding Propositions 5.46–5.47 and Lemma 5.52 will be formulated for all these cases simultaneously. Extensions to Rn , n ≥ 3 can be worthwhile, whenever the available cubature formulas for Rn−1 , n ≥ 3, cf. Cools [208–210] are good enough for the speciﬁc problem. Compared to (5.243)–(5.245) with the original Δu and ∂u/∂ν we replace this special diﬀerential and boundary operator by the general forms Au and Ba u in the following (5.278), (5.279). An essential tool is again Green’s formula (2.9): for v, w ∈ H 1 (Ω), we obtain by partial integration and with the outer normal ν = (ν1 , ν2 )T for ∂Ω w∂v/∂xi dx + v∂w/∂xi dx = ∂(vw)/∂xi dx = vwνi ds. (5.277) Ω

Ω

Ω

∂Ω

5.5. Consistency, stability and convergence for FEMs with variational crimes

381

We ﬁnd the bounded bilinear forms and operators for Ω ⊂ R2 ⎛ ⎞ 2 ⎝ a(u, v) = Au, vV ×V = aij ∂ i u ∂ j v ⎠ dx with aij ∈ W 1−δj0 ,∞ (Ω), Ω

⎛

⎝

= Ω

2

⎞ (−1)j>0 ∂ (aij ∂ u)⎠ vdx + j

(Ba u)vds

(5.278)

(Ba u)vds ∀v ∈ V = H 1 (Ω), u ∈ V ∩ H 2 (Ω),

(As u)vdx +

=:

i

∂Ω

i,j=0

i,j=0

Ω

∂Ω 2

with

aij ξi ξj ≥ |ξ|22 , > 0, for elliptic operators, and

i,j=1

Ba u =

2

νj aij ∂ i u +

i,j=1

2

νj a0j u, and

j=1

Ba u = ∂u/∂ν for aij = δi,j ∀i, j = 0, 1, 2, e.g. As u = −Δu + cu. It possible to transform the problem such that the above Ba u is replaced by Ba u = is 2 i i,j=1 νj aij ∂ u, always satisﬁed for Ba u = ∂u/∂ν. Then sometimes one obtains a higher order of consistency and convergence by the additional factor h. Obviously, a(uh , v h ) is deﬁned ∀ uh , v h ∈ Vbh ⊂ V independent of the boundary conditions. For FEs with crimes uh , v h ∈ Vbh ⊂ Vb , V, we again use the extended fh , ·W −k,p (Tch )×W k,p (Tch ) , 1/p + 1/p = 1, ah (·, ·), and the broken Sobolev norms · W k,p (Tch ) and obtain, cf. (5.250)–(5.252), (5.248), (5.254), ah (uh , v h ) := Ah uh , v h H −1 (Tch )×H 1 (Tch ) :=

=

T ∈Tch

+

T ∈Tch

⎡

⎣(As,h uh )v h :=

T

%

{v } Ba u

=: as,h (uh , v h ) + e∈∂Ω

2

⎛

2

⎝

⎞ aij ∂ i uh ∂ j v h ⎠ dx

T i,j=0

⎤

(−1)j>0 ∂ j (aij ∂ i uh ) v h ⎦ dx

(5.279)

i,j=0

e∈Tch \∂Ω

+

e

h

e

e∈Tch \∂Ω

v h Ba uh ds

h

&

+ [v ]{Ba u } ds + h

h

e∈∂Ω

v h Ba uh ds

e

& % {v h } Ba uh + [v h ]{Ba uh } ds

e

∀ uh ∈ (Vb ∩ H 2 (Ω)) + Vbh ⊂ H 2 Tch ,

382

5. Nonconforming ﬁnite element methods

v h ∈ Vb + Vbh ⊂ H 1 Tch , with ah (·, ·) := ah (·, ·)|V h ×V h , a(·, ·) = ah (·, ·)|V×V and ahs (·, ·) := as,h (·, ·)|H 2 (Tch )×L2 (Ω) , as (·, ·) = as,h (·, ·)|H 2 (Ω)×L2 (Ω) , with ellipticity as in (4.138). The exact and FE solutions u0 and uh0 are deﬁned by u0 ∈ Vb : Au0 , vH −1 (Ω)×H 1 (Ω) = a(u, v) = f, vV ×V ∀ v ∈ Vb , (5.280) ; : uh0 ∈ Vbh : Ah uh0 , v h H −1 (T h )×H 1 (T h ) = ah uh0 , v h = f h , v h V h ×V h ∀ v h ∈ Vbh . c

c

These relations (5.278)–(5.280) are easily extendable to elliptic linear second order systems or quasilinear equations and systems. We use the notation in (4.136) and ansatz and test functions uh = uh1 , · · · , uhq , v h ∈ Vbh := Vbh (Rq ), now with Vbh ⊂ Vb = H01 (Ω, Rq ), so violating the boundary conditions or below the continuity, Vbh ⊂ H 1 (Ω, Rq ). We start with linear systems, and, because of the high similarity with (5.278) ﬀ., we only formulate the FE equations: again f, a(·, ·), fh , ah (·, ·) and A, Ah are bounded linear and bilinear forms and operators.

Ah uh , v h H −1 (Tch )×H 1 (Tch ) = ah ( uh , v h ) :=

T ∈Tch

∀ uh , v h ∈ Vb + Vbh ,

with

2

(Akl ∂ l uh , ∂ k v h )q dx

T k,l=0

Akl ∈ L∞ (Ω, Rq×q ), cf.(5.254) for Vbh .

(5.281)

The Vb -coercivity of the principal part follows from the uniform Legendre condition, see (2.343), (2.345), (4.138), (4.139), hence with W = L2 (Ω, Rq ), cf. Subsection 5.5.4, 2 ≤ ∃ λ, Λ, s.t. 0 < λ < Λ < ∞ : λ|ϑ| nq

2

l , ϑ k )q ≤ Λ|ϑ| 2 ∀x ∈ Ω, (Akl (x)ϑ

(5.282)

k,l=1

=⇒ ap ( uh , uh ) > λ∗ uh 2V and a( uh , uh ) ≥ λ∗∗ uh 2V − Cc uh 2W ∀ uh ∈ Vb + Vbh . For relating ah (·, ·), Ah and as,h (·, ·), As,h , we require Akl ∈ W 1−δk0 ,∞ (Ω, Rq×q ) :

ah ( u , v ) := Ah u , v H −1 (Tch )×H 1 (Tch ) := h

h

h

h

T ∈Tch

⎡ = ⎣(As,h uh , v h )L2 (Ω,Rq ) :=

T ∈Tch

+

e∈Tch \∂Ω

e

2

2

T k,l=0

Akl ∂ l uh , ∂ k v h q dx ⎤

(−1)k>0 (∂ k (Akl ∂ l uh ), v h )q dx⎦

T k,l=0

(5.283) h % & h ({ v }, Ba uh )q + ([ v h ], {Ba uh })q ds + vl , Ba uh q ds e∈∂Ω

e

5.5. Consistency, stability and convergence for FEMs with variational crimes

=: as,h ( uh , v h ) +

+

e∈Tch \∂Ω

e∈∂Ω

383

& % ({ v h }, Ba uh )q + ([ v h ], {Ba uh })q ds

e

( v h , Ba uh )q ds ∀ uh ∈ (Vb ∩ H 2 (Tch , Rq )) + Vbh ⊂ H 2 Tch , Rq ,

e

2 2 νk Akl ∂ l u + νk Ak0 u, v h ∈ Vb + Vbh ⊂ H 1 Tch , Rq with Ba u := k,l=1

l=1

again with a (·, ·) := ah (·, ·)|V h ×V h , a(·, ·) = ah (·, ·)|V×V and h

ahs (·, ·) := as,h (·, ·)|H 2 (Tch )×L2 (Ω) , as (·, ·) = as,h (·, ·)|H 2 (Ω)×L2 (Ω) . The exact and FE solutions u0 and uh0 are deﬁned by (5.284) u0 ∈ Vb : A u0 , v H −1 (Ω)×H 1 (Ω) = a( u, v ) = f , v V ×V ∀ v ∈ Vb , ; : uh0 ∈ Vbh : Ah uh0 , v h H −1 (T h )×H 1 (T h ) = ah uh0 , v h = f h , v h V h ×V h ∀ v h ∈ Vbh . c

c

We generalize these relation between ah (·, ·) and as,h (·, ·). to quasilinear equations and systems. For avoiding a double formulation as for the linear case, we use the standard notation (4.129), but the same symbols for all functions u, u0 ∈ Vb = W01,p (Ω, Rq ), uh , uh0 ∈ Vbh ⊂ W01,p (Tch , Rq ), q ≥ 1, p ≥ 2,

(5.285)

for equations and systems. Only (., .)q indicates the possibility for systems, q ≥ 1. Again we only formulate the FE equations, cf (4.173), (4.174) and use (5.277). The necessary conditions for the Nemickii operators Ai (x, uh , ∇uh ), e.g. for bounded forms and operators, ah (uh , v h ) and Gh , are formulated in Sections 2.5.4, 2.5.6, 4.4 in (4.174) and 4.5 in (4.193). We additionally modify them, such that G is Lipschitz-continuous. With smooth enough Ai : Ω × Rq × Rn×q → Rq , we consider Vbh as in (5.254), and ∀ uh ∈ (Vb ∩ W 2,p (Ω, Rq )) + Vbh ⊂ W 2,p Tch , Rq , v h ∈ Vb + Vbh ⊂ W 1,p Tch , Rq . Then we deﬁne the weak and strong forms, ah (uh , v h ) and as,h (uh , v h ), related by ah (uh , v h ) := Gh uh , v h W −1,p (Tch )×W 1,p (Tch ) 2

:=

T ∈Tch

(5.286)

(Ai (x, uh , ∇uh ), ∂ i v h )q dx =

T i=0

5

= as,h (uh , v h ) := (Gs,h uh , v h )L2 (Ω,Rq ) :=

2 T ∈Tch

T i=0

(−1)i>0 (∂ i Ai (x, uh , ∇uh ), v h )q dx

6

384

5. Nonconforming ﬁnite element methods

+

e∈Tch \∂Ω

& % ({v h }, BG uh )q + ([v h ], {BG uh })q ds

e

+

e∈∂Ω

(v h , BG uh )q ds

e

and BG uh := BG (x, uh , ∇uh ) :=

2

νi Ai (x, uh , ∇uh ) with

i=1

ah (·, ·) := ah (·, ·)|V h ×V h , a(·, ·) = ah (·, ·)|V×V and ahs (·, ·) := as,h (·, ·)|H 2 (Tch ,Rq )×L2 (Ω,Rq ) , as (·, ·) = as,h (·, ·)|H 2 (Ω,Rq )×L2 (Ω,Rq ) . Again the exact and FE solutions u0 and uh0 are deﬁned by (5.287) u0 ∈ Vb : a(u0 , v) = Gu0 , vV ×V = 0 ∀ v ∈ Vb , : ; uh0 ∈ Vbh : ah uh0 , v h = Gh uh0 , v h W −1,p (T h ,Rq )×W 1,p (T h ,Rq ) = 0 ∀ v h ∈ Vbh , c

c

often for Ub = Vb , Ubh = Vbh . We summarize the results for second order linear and quasilinear equations and systems and use the notation as in (5.285): Proposition 5.54. 1. For either Dirichlet or natural boundary conditions and the bilinear or nonlinear forms ah (uh , v h ), as,h (uh , v h ) and the operators Ah , As,h , Gh , Gs,h in (5.279), (5.283) and (5.286), with the previously mentioned conditions for the Ai (·, ·, ·), are continuous. We ﬁnd, cf. (5.256), with Bu := Ba u, p = 2 for linear, and Bu := BG u, 2 ≤ p < ∞ for quasilinear problems with q ≥ 1 as (u, v) = a(u, v) ∀ u ∈ W 2,p (Ω, Rq ), v ∈ W 1,p (Ω, Rq ), and vBu|∂Ω = 0. (5.288) Note that Vb = W01,p ∩ W 2,p (Ω, Rq ) for Dirichlet boundary conditions. 2. Valid for anyuh ∈ Vbh + (Vb ∩ H 2 (Ω, Rq )) ⊂ W 2,p Tch , Rq and v h ∈ Vbh + Vb ⊂ W 1,p Tch , Rq we get, cf. (5.258), 2 ≤ p < ∞, 1 ≤ q, ah (uh , v h ) = as,h (uh , v h ) ({v h }, [Buh ])q + ([v h ], {Buh })q ds + e∈Tch \∂Ω

+

e∈∂Ω

e

(5.289)

e

(v h , (Buh ))q ds ∀ uh ∈ (Vb ∩ H 2 (Ω, Rq )) + Vbh , v ∈ Vb + Vbh .

5.5. Consistency, stability and convergence for FEMs with variational crimes

385

For important special cases (5.289) simpliﬁes, cf. (5.249), (5.263)–(5.265), as (5.290) ah (uh , v h ) = ahs (uh , v h ) for Vbh ⊂ W01,p (Ω, Rq ), Vbh ⊂ W 2,p (Ω, Rq ), v h (Buh )ds for only violated boundary cond. ah (uh , v h ) = ahs (uh , v h ) + e

e∈∂Ω

and Vbh ⊂ C(Ω, Rq ) ∈ W 2,p (Ω, Rq ), Vbh ⊂ C 1 (Ω, Rq ) ∈ W 1,p (Ω, Rq ), (5.291) h h h h h h v h [Buh ]ds (5.292) a (u , v ) = as (u , v ) + +

e∈∂Ω

e

e∈Tch \∂Ω

v h (Buh )ds for Vbh ⊂ C(Ω, Rq ) ∈ W 1,p (Ω, Rq ).

e

Simplifying and shortening these formulas we use the notation Ah u, v h H −1 (Tch ,Rq )×V h := Ah u, v h H −1 (Tch ,Rq )×H 1 (Tch ,Rq ) for v h ∈ V h , . . .

(5.293)

We generalize the sequences of projectors Q h in (4.117). For variational crimes we have chosen for the f used in (5.244) the general form (5.250). Their obvious extensions in (5.250) are deﬁned for v ∈ V and simultaneously for v h ∈ Vbh . Here we explicitely reformulate Propositions 4.49, 4.50 for the Vb in (5.285). We only need the f, fh in (5.250). We deﬁne the generalized projectors Q h by testing as

Q h ∈ L W −1,p Tch , Vbh : Q h fh − fh , v h W −1,p (Tch )×V h = 0∀v h ∈ Vbh . (5.294) Nearly analogous to (4.117) we ﬁnd here with (5.296)

Q h fh − fh ⊥ Vbh , with

Q h fh V h =

sup v h ∈Vbh ,v h V h =1

fh , v h W −1,p (Tch )×V h

= fh V h = f V (1 + o(h))

=⇒ lim Q h fh V h = lim fh V = f V , h→0

f

h

h→0

:= Q fh |V h , h

lim Q h L(V ,V h ) = lim Q h L(V h ,V h ) = 1. (5.295)

h→0

h→0

Proposition 5.55. For the extension fh ∈ V + W −1,p (Tch ) in (5.250), of f ∈ V ⊂ W −1,p (Ω) with W 1,p (Tch ) ⊃ V h , f h = fh |V h , and the projectors Q h ∈ −1,p h h (Tc ), V ) in (5.294), the norms are asymptotically equal ∀ 1 ≤ p ≤ ∞, 1/p+ L(W 1/p = 1,

f h V h = fh W −1,p (Tch ) (1 + o(1)) = f V (1 + o(1)), f h := Q h fh |V h ,

lim Q h fh V h = f V , lim Q h L(V ,V h ) = lim Q h L(V h ,V h ) = 1.

h→0

h→0

h→0

Particularly interesting are the fh , vW −1,p (Tch )×V h with fh = Ah uh .

(5.296)

386

5. Nonconforming ﬁnite element methods G

U Ph

U′

←→

Q′h

Φh Gh

Uh

U

tested by

tested by

U h with ⊄ U b or ⊄ U

←→

U h′

Figure 5.7 Nonconforming FEMs: Spaces and operators.

We summarize: with the projectors Q h in (5.294), and the operators Ah and Gh , extended to V + Vbh in (5.254), (5.279), (5.283), (5.286), we determine the discrete (weak) solutions uh0 ∈ Vbh from the following equations, : ; Ah uh0 − fh , v h H −1 (T h )×V h = 0 ∀ v h ∈ Vbh ⇔ Ah uh0 := Q h Ah uh0 = f h = Q h fh , c ; : and Gh uh0 , v h W −1,p (T h )×V h = 0 ∀ v h ∈ Vbh ⇔ Gh uh0 := Q h Gh uh0 = 0. (5.297) c

Reformulating, the discrete operators Ah and Gh have the form

Φh A := Ah := Q h Ah |Vbh = Q h Ah , Φh G := Gh := Q h Gh |Ubh , with the discrete solutions Ah uh0 − f h = 0, Gh uh0 = 0.

(5.298)

Again we sketch the diﬀerent spaces and operators constituting the method in the diagram in Figure 5.7. The necessary consistency errors for the nonconforming FEMs require, cf. Lemma 5.52, estimates for the following ah (u − uh , v h ). Proposition 5.56. 1. For FEs with violated boundary conditions and/or continuity, choose an arbitrary u ∈ H 2 (Ω), fu := Au ∈ L2 (Ω), e.g. the exact solution u0 ∈ H 2 (Ω), fu0 := Au0 , with the discrete solution uh0 , cf. (5.279), (5.283), (5.286). Then we obtain for u and the discrete solution uh of Ah uh = Q h Au

ah (u − uh , v h ) = Q h Ah u − Ah uh , v h H −1 (Tch )×V h = Ah u − Au, v h ... (5.299) = ([v h ], {Bu})q ds + (v h , (Bu))q ds e∈Tch \∂Ω

e

e∈∂Ω

e

∀ u ∈ H 2 (Ω, Rq ), v h ∈ Vbh . 2. For quasilinear (5.286), the A, Ah , H0−1 Tch , Rq , have to be replaced by G, Gh , W0−1,p Tch , Rq , p ≥ 2. Proof. The proof of Proposition 5.50 remains correct.

Remark 5.57. A solution u0 of Au0 = f with u0 ∈ V = H 1 (Ω) implies f ∈ H −1(Ω). Requiring u0 ∈ 2 H (Ω) implies f ∈ L2 (Ω), so f, vH −1 (Ω)×H 1 (Ω) = (f, v)L2 (Ω) = Ω f vdx.

5.5. Consistency, stability and convergence for FEMs with variational crimes

5.5.4

387

Discrete coercivity and consistency

In Proposition 5.56 we have determined, not yet estimated, the variational errors for linear and quasilinear equations and systems of order 2. As in (5.285), we use the same symbols u, uh , . . . ∈ H 1 (Ω), . . . instead of ∈ H 1 (Ω, Rq ) and, as coeﬃcients, aij instead of Aij , for real and vector valued functions. In fact, the same arguments are valid for all the linear problems in the previous Subsection. For quasilinear problems slight modiﬁcations are necessary, e.g. in Lemma 5.65. The general convergence theory in Chapter 3 requires uniform discrete inf-sup conditions, equivalently the stability, of the discrete linear and linearized operators and estimates for the variational discretization errors. These estimates will be discussed in Subsections 5.5.6, 5.5.7. Towards stability, we do need for a(·, ·), ah (·, ·) the Vbh coercivity for Dirichlet or natural boundary conditions. We prove this, as a ﬁrst step, for compactly perturbed principal parts, ap (·, ·) + (·, ·)L2 (Ω) , of arbitrary elliptic bilinear (weak) forms. In Subsection 5.5.8 we will extend these results to the general case of a consistent compact perturbation A = B + C, Ah = B h + C h with boundedly invertible A, B and stable B h ; e.g. the original operator, A, induced by a(·, ·) is a compact perturbation of the operator B := Ap + I induced by ap (·, ·) + (·, ·)L2 (Ω) . Thus the results of Subsection 5.5.8 show the stability of Ah . For the FEs with crimes in this section the standard approach requires Ubh = Vbh . For Ub = H01 (Ω) or = H 1 (Ω) ⊂ L2 (Ω) = L2 (Ω) ⊂ Ub = H −1 (Ω) we assume ∀ uh ∈ Ubh = Vbh ∃v h ∈ Vbh s.t uh − v h V h < βuh V h or vice versa ∀ v ∈ h

Vbh

=

Ubh

∃u ∈ h

Ubh

(5.300)

s.t u − v V h < βv V h , h

h

h

here β is an arbitrarily small constant and dim Ubh = dim Vbh , with the same norms in U h , V h , e.g. uh U h = uh V h = uh H 1 (Tch ) . Remark 5.58. Beyond the usual case Ubh = Vbh , this condition is satisﬁed, if Ubh and Vbh are deﬁned via the same interpolation or transition points along the interior and boundary edges e ∈ Tch . Diﬀerent degrees in Ubh and Vbh are still allowed. The nonlocal smooth approximations in Subsection 4.2.6 have this property as well, unless the application to fully nonlinear problems requires the splitting there. A similar condition will be satisﬁed for the general case of Gelfand triples: let Ub , U, X be Hilbert spaces and let X = X be identiﬁed such that Ub is dense in X and I : Ub → X , Iv := v ∈ X for all v ∈ Ub is continuous. Then (5.300) can be generalized as Ub ⊂ X = X ⊂ Ub and Ub ⊂ U is densely and continuously embedded. The convergence theory for mesh free methods in [120] will allow to generalize this (5.300) for FEMs as well. Obviously the linear and bilinear forms studied in the last subsection are uniformly continuous, such that there exists a constant, C, independent of h, with vV = vH 1 (Ω) , uh V h = uh H 1 (Tch ) , . . . |ah (uh , v h )| ≤ Cuh V h · v h V h ∀ uh , v h ∈ V + V h , |fh , v | ≤ Cfh V h v V h ∀ v ∈ V + h

h

h

Vbh , fh

(5.301) h

∈V +V .

388

5. Nonconforming ﬁnite element methods

The following theorem applies to violated boundary conditions and continuity as discussed in Subsections 5.5.6, 5.5.7 and with the necessary machinery in Subsection 5.5.10 and Section 5.4 as well. For the general case beyond ap (·, ·) + (·, ·)L2 (Ω) the stability proof is delayed to Subsection 5.5.8. Theorem 5.59. Stability for modiﬁed principal parts: 1. Let this implies A, a(·, ·), Ah , ah (·, ·) and Dirichlet and/or natural boundary conditions be given as in (5.278)–(5.280), (5.284) and > 0. Let ap (·, ·), ap,h (·, ·) be the principal parts. Then the ap,h (·, ·) + (·, ·)L2 (Ω) is bounded, say by M = C in (5.301), and Vb + Vbh -coercive, hence a constant α > 0 exists such that ap,h (uh , uh ) + (uh , uh )L2 (Ω) ≥ α(uh V h )2 ∀uh ∈ Vb + Vbh .

(5.302)

2. If for Ubh = Vbh , (5.300) is satisﬁed with βM/ < 1, then the uniform discrete inf-sup conditions (2.52), (2.53) are valid, |ap,h (uh , v h ) + (uh , uh )L2 (Ω) |/v h V h ≥ uh U h ∀uh ∈ Ub + Ubh ,

sup 0 =v h ∈Vb +Vbh

3. In particular, for Dirichlet boundary conditions Ap and Ahp are boundedly and equiboundedly invertible, hence Ahp is stable. Note that the inf–sup condition for a(u, v) does not include the discrete inf–sup condition. This is possible under the conditions of compact perturbation results, e.g. in Theorem 5.78. Theorem 5.59 implies the unique existence of the exact solution u0 and of the discrete solutions uh0 for the special case of ap (·, ·) + (·, ·)L2 (Ω) , if the consistency results in Subsections 5.5.6 and 5.5.7 are valid. For isoparametric FEs for curved boundaries, see Subsection 5.5.10, this result is available in Theorem 5.84. Proof. The principal part of the above linear elliptic operators or bilinear forms, ⎛ ⎞ 2 ⎝ aij ∂ i u ∂ j v ⎠ dx ap (u, v) := Ω

i,j=1

satisﬁes, cf. (4.131) ﬀ., 2

aij ξi ξj + η 2 ≥ |ξ|22 + η 2 , > 0,

i,j=1

with ξj , η ∈ R, and |ξ|2 the Euclidean norm in R2 . We have chosen twice the same = as in (4.131). Now let ξj := ∂j uh (x), η := uh (x) and obtain 2

aij ∂i uh ∂j uh (x) + (uh (x))2 ≥ |∇uh (x)|22 + (uh (x))2

i,j=1

and by integration

5.5. Consistency, stability and convergence for FEMs with variational crimes

ap,h (uh , uh ) + (uh , uh )L2 (Ω) =

T ∈Tch

≥

T

T ∈Tch

T

⎛ ⎝

2

389

⎞ aij ∂i uh ∂j uh + (uh )2 ⎠ dx

i,j=1

|∇uh |22 + (uh )2 dx ≥ (uh V h )2

∀uh ∈ Vb + Vbh . For the inf-sup conditions we need (5.300). Let Ubh ⊂ / Ub , Vbh ⊂ / Vb , Ubh = Vbh . By h h h h (5.300) we know that for any u ∈ V we may choose a u ∈ U with uh − v h V h < βuh V h . with the notation bh (uh , uh ) := ahp (uh , uh ) + e (uh , v h )L2 (Ω) we ﬁnd bh (uh , v h ) − bh (uh , uh ) = bh (uh , uh − v h ) ≤ M (uh V h uh − v h V h ) since bh (·, ·) is continuous. With bh (uh , uh ) ≥ uh 2H 1 (Ω) this implies bh (uh , v h ) ≥ bh (uh , uh ) − bh (uh , v h ) − bh (uh , uh ) ≥ (uh V h )2 − M (uh V h uh − v h V h ) % & ≥ uh V h uh V h − (M/ )βuh V h hence, going back to v h and for (M/ )β < 1 the inf-sup condition is proved. For systems, with (4.131) replaced by (4.138), the proof remains nearly unchanged. As the next step, we prepare estimating the variational consistency errors as in Subsections 5.5.6 and 5.5.7. In contrast to classical consistency errors, we always have to assume here that the discrete solution uh0 exists. So, we assume the existence of the exact and approximate solutions u0 and uh0 of ∃u0 : a(u0 , v) = (f, v) ∀v ∈ Vb or ∃uh0 : ah uh0 , v h = (f, v h )h ∀v h ∈ Vbh . (5.303) For nonconforming FEs we obtain as a special case of Proposition 5.56 the following results for the variational consistency error. Variational consistency error for violated boundary conditions: (5.304) v h Ba u0 ds for u0 ∈ H 2 (Ω), ∀ v h ∈ Vbh ⊂ H 1 (Ω). ah u0 − uh0 , v h = ∂Ω

Variational consistency error for violated continuity, see (5.258): (5.305) [v h ]Ba u0 ds for u0 ∈ H 2 (Ω), ∀v h ∈ Vbh ⊂ H 1 (Ω). ah u0 − uh0 , v h = e∈Tch

e

Variational consistency error for general nonconforming FEs [v h ]Ba u0 ds + v h Ba u0 ds ah u0 − uh0 , v h = e∈Tch

e

e∈∂Ω

for u0 ∈ H 2 (Ω), ∀ v h ∈ Vbh ⊂ Vb , Vbh ⊂ H 1 (Ω).

(5.306)

390

5. Nonconforming ﬁnite element methods

In Subsections 5.5.6 ﬀ. we will aim for the classical consistency errors as well. So we generalize (5.304), (5.305), (5.306) from u0 , uh0 to general u and its corresponding discrete counterpart uh , deﬁned by Ah uh = Q h Au, cf. (5.297), (5.298). Note that the classical consistency error is independent of the existence of a (unique) discrete solution, in contrast to the variational consistency error. We require, beyond Condition 4.16, speciﬁc approximating spaces: Condition 5.60. Conditions for nonconforming FEs: Choose piecewise polynomial 2 1 2 in R2 . Recall that P d−1 = Pd−1 and Pd−1 = Pd−1 denote FEs locally in P = Pd−1 the uni-and bi-variate polynomials of degree d − 1, respectively Every T ∈ Tch at the boundary has at most one curved side, see Figures 4.21–4.23 above. Let ∂Ω be piecewise smooth. Nonsmooth points of ∂Ω are used as vertices of subtriangles, T ∈ Tch , compare Condition 5.64. Let the subdivision be quasiuniform, or generalizd nondegenerate, see Deﬁnition 4.12. This condition of quasiuniformity can be relaxed in R2 by applying the results in Subsection 4.2.5. Extensions to Rn , n ≥ 3, are possible using results as in Cools [208–210]. We only discuss R2 .

5.5.5

High order quadrature on edges

These are an important tool for estimates of all types errors. The of nonconformity main idea is very simple: the variational errors ah u0 − uh0 , v h , listed in (5.304)– (5.306), are small if the [v h ] disappear often enough along interior edges e and v h along the boundary ∂Ω. Then in R2 the Gauss-type approximations along edges e and the boundary ∂Ω vanish for the right-hand sides of (5.304)–(5.306). Consequently the corresponding right-hand side integrals, and hence the variational errors ah u0 − uh0 , v h , are small. Similarly to the techniques in Section 5.4, this is shown with inverse estimates, costing some powers of h. There quadrature is applied to the bilinear or higher nonlinear forms, compare (5.179) and (5.306), (5.330), on the T ∈ Tch in R2 . Here we apply quadrature to boundary forms on edges of the triangles. Since the degree of freedom for a polynomial increases with the dimension n, cf. (4.9), this concept works well for Gauss-type formulas on edges and is still not too bad for high order cubature on triangles, but seldom for Rn , n ≥ 4. We often have to distinguish interpolation and quadrature errors and points. Univariate Gauss–type quadrature formulas For one-dimensional intervals or edges, we use three types of Gauss formulas, see Lemma 5.61 below. They are distinguished by the requirement that either none, one or two boundary points of the interval are included in the corresponding quadrature grid points. They are deﬁned for Legendre and Chebyshev polynomials with respect to 1 the weight functions w ≡ 1 and w = (1 − x2 ) 2 , respectively, on the interval [−1, 1]. We introduce three types of univariate polynomials, Qρd ∈ P d := Pd1 , ρ = 0, 1, 2, based on the Legendre and Chebyshev polynomials Pd ∈ P d and Td ∈ P d , respectively, deﬁning quadrature formulas of the type of

5.5. Consistency, stability and convergence for FEMs with variational crimes

Gauss: ρ = 0 : Q0d := Pd or Q0d := Td and for both cases Gauss–Radau: ρ = 1 : Q1d := Q0d + aQ0d −1 , with a such that Q1d (−1) = 0, Gauss–Lobatto: ρ = 2 : Q2d := Q0d + aQ0d −1 + bQ0d −2 , with a, b

391

(5.307)

such that Q2d (±1) = 0. This implies d ≥ max{1, ρ}. The d quadrature points yjρ := yj ∈ [−1, 1], j = 0, . . . , d − 1 for the quadrature formulas are deﬁned as the roots of the polynomials (5.308) Qρd yjρ = 0 for j ∈ {0, . . . , d − 1}, ρ = 0, 1, 2, max{1, ρ} ≤ d , with y0ρ = −1 for ρ = 1, 2 and ydρ −1 = 1 for ρ = 2. The corresponding weights are then computed via the Lagrange elementary polynomials, pρj , as

pρj ∈ P d : pρj (ylρ ) = δj,l ∀ 0 ≤ j, l ≤ d − 1 as 1 wj : = wjρ := pρj (x)w(x)dx, ρ = 0, 1, 2, d ≥ max{1, ρ}, −1

with the above weight function, w. The following proposition is proved in many textbooks, e.g. [158]. Proposition 5.61. 1. Let −1 ≤ y0 < · · · < yd −1 ≤ +1 be the above roots of the Qρd , for ρ = 0, 1, 2, and d wj := wjρ the corresponding weights. Then the quadrature approximation qw (u) is deﬁned as 1 d −1 ρ ρ d d qw (u) := u yj wj ≈ u(x)w(x)dx and q d (u) := qw≡1 (u), −1

j=0

for u ∈ C[−1, 1], ρ = 0, 1, 2, max{1, ρ} ≤ d ∈ N, and satisﬁes d (p) qw

=

d −1

p yjρ wjρ =

j=0

1

−1

p(x)w(x)dx, for all p ∈ P 2d −ρ−1 .

(5.309)

2. This implies for scalar products, (u, v)w , with respect to the weight function w that (u, v)w =

(u, v)dw

:=

d qw (u

· v) =

d −1

(u · v) yjρ wjρ ∀u ∈ P d −ρ , v ∈ P d −1 .

j=0

(5.310)

d 3. So (u, v)dw = qw (u · v) exactly reproduces scalar products for u, v ∈ P d −1 for ρ = 0, 1. For ρ = 2 we have to require at least one of the u or v ∈ P d −2 .

The next proposition is applied for proving Theorems 5.66, ﬀ.

392

5. Nonconforming ﬁnite element methods

Proposition 5.62. For all d ∈ N with max{1, ρ} ≤ d , hence 2d − ρ ≥ d ≥ 1, there exists C = Cd such that ∀0 < h < 1, 1 ≤ q ≤ ∞, f ∈ W 2d −ρ,q [0, h] ∩ C[0, h], h d −1 f (x)dx − h/2 wj f (hξj ) ≤ Cd h2d −ρ+(1−1/q) |f (2d −ρ) |Lq (0,h) . (5.311) j=0 0

With ρ = 0, 1, 2, for Gauss, Gauss–Radau and Gauss–Lobatto quadrature, respectively, the points hξj := hξjρ := h yjρ + 1 /2 and yjρ are zeros of the Legendre polynomials Qρd of degree d , see (5.308). Proof. We use, for any T ⊂ R2 , the H¨ older inequality, see (1.45), ⎛ ⎞1/p ⎛ ⎞1/q f (x)g(x)dx ≤ ⎝ |f (x)|p dx⎠ ⎝ |g(x)|q dx⎠ T

T

(5.312)

T

for all 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1 and f ∈ Lp (T ), g ∈ Lq (T ). Now we add and subtract the Taylor polynomial Q2d −ρ f for f in (5.311). Then (5.309) 2d −ρ,q (0, h), and the Bramble–Hilbert–Lemma 4.15 implies with f ∈ W h d −1 f (x)dx − h/2 wj f (hξj ) j=0 0 h d −1 2d −ρ 2d −ρ f +Q f dx − h/2 wj f (hξj ) ≤ f (x) − Q j=0 0

h ≤

1 · |f (x) − Q2d −ρ f |dx ≤ Cd h1−1/q |f − Q2d −ρ f |Lq (0,h)

0

≤ Cd h2d −ρ+1−1/q |f (2d −ρ) |Lq (0,h) .

5.5.6

Violated boundary conditions

In this Subsection we estimate the consisitency errors of continuous FEs violating Dirichlet boundary conditions. Natural boundary value problems without prescribed boundary conditions do not play any role in this context. We do need here the classical and variational consistency error. With the discrete solution uh of Ah uh = Q h Ah uh = Q h fu = Q h Au they are introduced in Deﬁnition 3.14 as Q h Au − Ah P h u and Q h Ah u − Ah uh = Q h Ah u − Q h Ah uh , respectively (Note that for conforming FEMs with Ah = A the variational consistency error, Ah uh − Q h Au vanishes!) By

5.5. Consistency, stability and convergence for FEMs with variational crimes

393

Chapter 3, (3.33), they are, for a linear operator, A, related as

Q h Ah u − Ah uh = Q h Ah (u − P h u) + Ah P h u − Q h Au with Ah uh = Q h Au < => ? => ? < => ? < var.cons.error interp.error –class.cons.error (5.313) For the ﬁrst term, the variational consistency error, tested with v h ,

Q h Ah u − Ah uh , v h V h ×V h = Ah u − Ah uh , v h V h ×V h = ah (u − uh , v h ), we have an explicit form in (5.299). With the interpolation operator, P h = I h , the middle term Q h Ah (u − P h u) = Q h Ah (u − I h u) can be estimated by the interpolation error. So the classical consistency error can be estimated as

Ah P h u − Q h AuV h ≤ Q h Ah u − Ah uh V h + Q h Ah (u − P h u)V h

(5.314)

≤ (1 + o(h))AV ←V P h u − uV + Q h Ah u − Ah uh V h . This (5.314) has to be slightly modiﬁed for quasilinear operators. Again, we introduce fu := Gu and uh , the discrete solution of Gh uh = Q h fu = Q h Gu. From the possibly many solutions uh for the nonlinear Gh we choose that with small uh − u. We ﬁnd with Gh = Q h Gh |Vbh in (5.287),

Q h Gh u − Gh uh = Q h (Gh u − Gh P h u) + (Gh P h u − Q h Gu).

(5.315)

So we have to estimate these variational consistency errors, cf. (5.299),

ah (u − uh , v h ) = Ah u − Ah uh , v h V h ×V h ∀v h ∈ Vbh , with Ah uh = Q h Au and

for nonlinear G : Gh u − Gh uh , v h V h ×V h ∀v h ∈ Vbh , for Gh uh = Q h Gu. (5.316) Explicit formulas for linear and quasilinear problems are listed in Proposition 5.56 and the G, Gh modiﬁcation of (5.299). We will update it for violated boundary conditions.

Proposition 5.63. Let Q h and Ah , Gh be deﬁned in (5.294)–(5.297), and choose an arbitrary u ∈ Vb ∩ H 2 (Ω) and the corresponding uh from Ah uh = Q h Au or Gh uh = Q h Gu. Then the violated boundary conditions are reﬂected in the variational consistency error ∀ v h ∈ Vbh as, cf. Proposition 5.56, h h h h h h h h v h Buds. (5.317) Q Ah u − A u , v V h ×V h or Q Gh u − G u , v V h ×V h = ∂Ω

We start with linear problems and Ba , for nonlinear problems with B = BG cf. Theorem 5.68. According to (5.317), an estimate for the consistency error for v h ∈ Vbh ⊂ Vb , hence v h |∂Ω ≡ 0, depends on how well v h approximates v h |∂Ω = 0 and how this implies small | ∂Ω v h Ba u0 ds|. The following steps involve quadrature errors and inverse estimates. First, we use Proposition 5.61 estimating | ∂Ω f ds − d ,∞ (Ω), cf. (5.308) ﬀ. Here, Pj := Pjρ ∈ ∂Ω indicate Pj ∈∂Ω wPj f (Pj )| for f ∈ W the quadrature points along one or all edges on ∂Ω, or on a subsection of ∂Ω or its approximation. The wj indicate the corresponding quadrature weights. For

394

5. Nonconforming ﬁnite element methods

FEMs we only consider Legendre polynomials with w ≡ 1. The Gauss-type points are used for nonconforming FEs to specify v h ∈ Vbh . We choose for the boundary Ω a parametrization ∂Ω = {xb (s) : s ∈ [0, sb ], s = arc length, sb = arc length of ∂Ω, xb = linear on straight parts of ∂Ω}.

(5.318)

We delay the isoparametric case to Subsection 5.5.10. There we will impose Dirichlet boundary conditions in d points close to ∂Ω. Now we proceed as in Subsubsection 4.2.7. For each edge e with endpoints Pi , Pe ∈ e ∩ ∂Ω such that Pi = xb (se ), Pe = xb (se + he ) we deﬁne a parametrization e = {xe (s) : s ∈ [se , se + he ]}; xe := xb |[se ,se +he ] .

(5.319)

Any derivatives of this global and local parametrization in (5.318) and (5.319), respectively, are determined solely by ∂Ω and can be bounded independently of h. With the ξj in (5.311) we deﬁne the additional boundary nodes on the (curved) boundary, cf. (4.98), e Pjρ,e := Pje := xe (se + he ξj ) ∈ e, j = 0, 1, . . . , d − 1 e ∩ ∂Ω}| = 2 hξj := h yj2 + 1 /2 ∀ e ∈ Tch with |{¯

(5.320)

often with Gauss–Lobatto quadrature ρ = 2 and Pi = P0e , Pe = Pde −1 , cf. Proposition 5.61. This allows deﬁning the Vbh ⊂ V h , cf. (5.241), as Vbh := v : Ω → R : v ∈ V h , ∀e ∈ Tch , v Pje = 0∀Pje in (5.320) , Ω = ∪T ∈Tch T. (5.321) We apply (5.311) to the special case of f = (v h Ba u) ◦ xb with the above parametrization xb for ∂Ω. We use (1.67) for p = ∞, and v h Ba u ∈ W k+1,∞ (Ω), (5.278) and the product and chain rule estimating ((v h Ba u) ◦ xb )(k) L∞ (∂Ω) ≤ Cxb uW k+1,∞ (Ω) v h W min{k,d−1},∞ (Ω) k+2 ∀u ∈ W∞ (Ω), v h ∈ V h ;

(5.322) (j)

this Cxb depends on powers of order ≤ k of diﬀerent derivatives xb , j = 0, . . . , k, for the above parametrization. Finally, inverse estimates yield for (5.311) and Condition 5.60, cf. e.g. Theorems 4.17, 4.19 and Proposition 4.18 with n = 2, v h W j,p (Ω) ≤ C hl−j−(2/q−2/p) v h W l,q (Ω) for FEs and 0 ≤ l ≤ j ≤ d − 1, 1 ≤ q ≤ p ≤ ∞ ∀v h ∈ V h . We get estimates for the interpolation errors with Condition 5.60 and v ∈ W

(5.323) d,p

(Ω):

v − I h vW s,p (Ω) ≤ C hd−s |v|W d,p (Ω) and v − I h vW s,p (T ) ≤ C (diam T )d−s |v|W d,p (T ) for 0 ≤ s ≤ d.

(5.324)

5.5. Consistency, stability and convergence for FEMs with variational crimes

395

We have to consider the consequence of (5.320), (5.321) for imposing Dirichlet boundary conditions in d points on the edges, e. In R2 we require for ρ = 0, 1, 2 choose max{1, ρ} ≤ d hence 2d − ρ ≥ d ≥ 1.

(5.325)

This would still allow many diﬀerent orders of convergence in the following results and rather complicated case studies yielding only few really interesting results. For avoiding these technicalities, we choose the other way round. We exclude all nonoptimal cases, and prove the results only for the optimal cases. Exotic cases violate d − 1 ≤ 2d − ρ. Furthermore, as already indicated in the introduction 5.5.1, we want to maintain the order of convergence, d − 1, of the original FEM in its violated form as well. For d − 1 ≤ 2d − ρ a detailed discussion would yield as order of convergence, with 2d − ρ − (d − 1) ≥ 0, the min{2d − ρ − (d − 1) + 1/2, d − 1} aimed to be ≥ d − 1. So we assume d − 1 ≤ 2d − ρ and min{2d − ρ − (d − 1) + 1/2, d − 1} ≥ d − 1.

(5.326)

We distinguish two cases. For polygonal boundaries the trivial boundary conditions can be satisﬁed exactly. This is no longer correct in combination with discontinuous FEs, see Subsection 5.5.7. So we include polygonal boundaries here, too, mainly with ρ = 0. For curved boundaries, Dirichlet boundary conditions usually cannot be satisﬁed exactly by polynomial FEs. Polygonal ∂Ω violating boundary conditions, mainly for discontinuous FEs: The piecewise linear xb , see (5.318), and v h |T ∈ Pd−1 yield (v h ◦ xb )|e ∈ Pd−1 . Now d > d − 1 boundary points with v h (Pj ) = v h (xb (se + he ξj )) = 0, j = 0, . . . , d − 1 imply (v h ◦ xb )e ≡ 0, hence the Dirichlet boundary conditions are satisﬁed exactly. So violated Dirichlet boundary conditions imply d ≤ d − 1. Condition 4.16, 6. assumes the same number of functionals on every edge by requiring aﬃnely equivalent (K, P, N ) and (T, PT , NT ) ∀T ∈ Tch . In this sense, max{1, ρ} ≤ d ≤ d − 1 is even equivalent to violated continuity conditions, see Subsection 5.5.7. The second part of (5.326) yields, for ρ = 2, d ≥ d and, for ρ = 0, 1, d ≥ d − 1. With the above d ≤ d − 1, this excludes ρ = 2 and allows ρ = 0, 1 with d = d − 1 and max{1, ρ} = 1. So we require for polygonal and allow curved ∂Ω also, so for polygonal and general ∂Ω : ρ = 0, 1, and 1 ≤ d = d − 1.

(5.327)

Curved ∂Ω with violated boundary conditions: In this situation a d > d + τ and v h ◦ xb |e ∈ P d+τ does not imply v h ◦ xb |e ≡ 0 as above. For τ cf. Condition 4.16. Thus, violated Dirichlet boundary conditions usually do not imply violated continuity. However, if we assume the same number of interpolation points on each e, including curved boundary edges as well, the above arguments for a polygonal boundary apply again and yield (5.327), mainly with ρ = 0. Another possibility, combining curved boundaries and violated continuity via ρ = 0, 1, 2 is discussed in the introduction of Subsection 5.5.7. Finally, (5.326) implies, for ρ = 2, d ≥ d. Since by (5.326) a d > d would not increase the convergence rate,

396

5. Nonconforming ﬁnite element methods

the only interesting case is the standard case for curved ∂Ω : ρ = 2 ≤ d = d, Pd−1 = P.

(5.328)

We summarize: Condition 5.64. Conditions for boundary and boundary quadrature points: Let ∂Ω be a piecewise smooth, e.g. a curved boundary and introduce the points Pj = Pje along ∂Ω as in Subsubsection 4.2.7, see (4.92) and (5.311), (5.318)–(5.320). Let the segments e ⊂ ∂Ω, see (5.318),(5.319), be deﬁned by Pi = Pj , Pe = Pj+1 . Choose a quadrature approximation according to Propositions 5.61 and 5.62, along the above closed (curved) segments e of the piecewise smooth ∂Ω ⊂ R2 . Impose Dirichlet boundary conditions in d points Piρ,e ∈ ∂Ω, i = 0, . . . , d − 1 on each edge, e. We require ρ = 0, 1 ≤ d = d − 1 and ρ = 2 ≤ d = d only for curved ∂Ω, see Proposition 5.62 and (5.320),(5.321), (5.327), (5.328). In practical computations only ρ = 0, 2 are used. The Piρ,e , cf. (5.319), (5.320), are deﬁned by the Gauss, Gauss–Radau and Gauss– Lobatto points according to the chosen quadrature approximation ρ = 0, 1, 2 along the segment e, parametrized by xe with respect to the arc length, see (5.319). For the proof of Theorem 5.66 we need Lemma 5.65, discussing the estimates for violated boundary values along a single edge e. The following notation allows a simultaneous formulation for the two cases (5.327), (5.328). . (d − ρ, 2d − ρ − 1) for (5.327): ρ = 0, 1 ≤ d = d − 1, 0 (5.329) (ν0 , ν ) := (d, 2d − 1) for (5.328): ρ = 2 ≤ d = d. Lemma 5.65. Under Conditions 5.60 and 5.64 we choose a linear or a quasilinear diﬀerential operator, A or G, and the corresponding boundary operator B = Ba or B = BG , satisfying (5.335), homogeneous Dirichlet boundary conditions as in (5.278) and 0 0 0 an arbitrary u ∈ W ν +1,∞ (Ω) ∩ W ν +1,∞ (∂Ω), or u ∈ H ν +2 (Ω), cf. Remark 5.67. 39 Furthermore, we deﬁne Vbh as in (5.321) and choose e ⊂ T¯, T ∈ Tch . Then there exists a constant C = C(d, χ) such that v h Bu ds ≤ C hν0 u ν 0 ,∞ |v h |H 1 (T ) , ∀ v h ∈ Vbh ⊂ H01 (Ω). (5.330) W (T ) e

Proof. The (5.311), (5.322) and (5.323) with l = 1, j = d − 1, p = ∞, q ≥ 2 are applied e ∩ ∂Ω}| ≥ 2. We use the parametrization to an edge e ⊂ T¯ ∈ Tch with length he and |{¯ xe for e, see (5.319). Now we combine q d ((v h Ba u) ◦ xe ) = 0 ∀v h ∈ Vbh , Proposition 5.62 for q = ∞ with (5.321), (5.322), Theorem 4.19 with (5.323), and (1.67), requiring 39

0

The transition from line 1 to 2 in (5.331) requires more than u ∈ W ν +1,∞ (Ω), since Theorem 0 1.37 is wrong for p = ∞. This is not surprising since ∂Ω has measure 0 in R2 So u ∈ W ν +1,∞ (Ω) 0 does not imply any form of u ∈ W ν +1,∞ (∂Ω). So, motivated by Remark 5.67 and Theorem 1.26, we 0 0 0 impose u ∈ W ν +1,∞ (Ω) ∩ W ν +1,∞ (∂Ω) or u ∈ H ν +2 (Ω).

5.5. Consistency, stability and convergence for FEMs with variational crimes

397

u ∈ W μ +2,∞ (Ω) with μ := 2d − ρ, for (5.331). We estimate for B = Ba se +he h h v Ba uds = (v Ba u) ◦ xe ds ≤ Chμe +1 v h Ba uW μ ,∞ (e) e se

≤ Chμe +1 uW μ +1,∞ (T ) v h W d−1,∞ (T ) with min{μ , d − 1} = d − 1

≤ Chμ +1 h1−2/q−(d−1) uW μ +1,∞ (T ) v h Wq1 (T ) or

(5.331)

≤ Chμ +2−d uW μ +1,∞ (T ) · v h H 1 (T ) , for q = 2,

≤ Chμ +2−d uW μ +1,∞ (T ) · |v h |H 1 (T ) , where we have replaced v h H 1 (T ) by |v h |H 1 (T ) in (5.331). This is correct, since v h (Pj ) = 0 for at least one point Pj ∈ e. So we can use a modiﬁed Poincar´e–Friedrich inequality of the form uL2 (T ) ≤ h|u|H 1 (T ) ∀u ∈ H 1 (T ) with u(Pj ) = 0 for at least one Pj ∈ e. The additional changes for quasilinear problems with Lipschitz-continuous G are obvious by (5.286). Theorem 5.66. Consistency for FEMs violating boundary conditions and convergence under coercivity conditions: 1. Under Conditions 5.60 and 5.64 we choose A, a(·, ·), homogeneous Dirichlet boundary conditions as in (5.278), ρ = 0, 2, and an arbitrary u ∈ W 2d,∞ (Ω) ∩ W 2d,∞ (∂Ω), or u ∈ H 2d+1 (Ω), cf. Remark 5.67. Furthermore, we deﬁne Vbh as in (5.321) and choose the FEMs in (5.278)–(5.280), (5.294)–(5.298). 2. Then Ah = Q h Ah |Vbh is consistent with A in u. Let uh , deﬁned by Ah uh =

Q h Au , exist. Then the variational and classical consistency errors vanish, 40 cf. (5.329) and Remark 5.67, such that, with an h-independent C = C(d,n,ρ,xb ) , sup 0 =v h ∈Vbh

1 |ah (u − uh , v h )| ≤ Chν0 − 2 uW 2d−1,∞ (Jch ) , and, by (5.329), h v H 1 (Tch )

Ah I h u − Q h AuV h ≤ Chd−1 uW 2d−1,∞ (Ω) for u ∈ W 2d,∞ (Ω).

(5.332)

3. Let ah (·, ·) be Vbh -coercive, cf. (5.301) and Theorem 5.59. Then, for ρ = 0, 2, the solutions uh exist uniquely and converge to u according to uh − uH 1 (Ω) ≤ Chd−1 uW 2d−1,∞ (Ω) .

(5.333)

4. For ρ = 1 in (5.327) the convergence is only ≤ Chd−3/2 uW 2d−2,∞ (Ω) , u ∈ W 2d−1,∞ . 40

In this and the next subsection we often use the pair u, uh with Ah uh = Q h Au instead of the exact solution u0 , uh 0 . This is motivated by the classical consistency error. It is deﬁned not only for u0 but for general u as well.

398

5. Nonconforming ﬁnite element methods

Remark 5.67. 1. Here we consider Gauss, Gauss–Radau, and Gauss–Lobatto points. For violated continuity we restrict the discussion to Gauss points. Consequently, diﬀerent techniques for the proofs are necessary. 2. An alternative result to (5.330), (5.332) is possible. Replace the ≤ Chμe +1 2(d−1) v h Ba uW μ ,∞ (e) by ≤ Che |([v h ]Ba u)2(d−1)) |L1 (e) in the ﬁrst line of (5.331), and combine it with (1.68). Then the uW 2d−1,∞ (Ω) in (5.330), (5.332) can be replaced by uH 2d (Ω) . The analogous result to (5.350) is incorrect for curved boundaries, hence Theorems 4.60 does not hold either. Proof. This collects the preceding results in Proposition 5.63, Lemma 5.65 estimating (5.332). Relation (5.317) shows that we have to estimate the sum over all edges e with |{e ∩ ∂Ω}| ≥ 2 along ∂ Ω,

∂ Ω

h v Ba uds ≤ := v Ba uds with

h

e

e

e

e:|{e ∩∂Ω}|≥2

μ +1−(d−1)−1/2

≤ Ch

uW μ +1,∞ (Ω)

h h1/2 e |v |H 1 (T )

e

≤ Chμ −(d−1)+1/2 uW μ +1,∞ (Ω) |v h |H 1 (Ω) , since, by the H¨older inequality and (4.36),

h h1/2 e |v |H 1 (T )

≤

e

he

1/2

e

≤

e

1/2 |v h |2H 1 (T )

e

1/2 he

|v h |H 1 (Ω) and

he ≤ 2 × (arc length of ∂Ω)

e

for small enough h. So we obtain (5.332). Relation (5.333) is obtained either by combining the stability result in Theorem 5.78 with the consistency estimate in (5.332) or by a combination with Lemma 5.52, see (5.271). The last claim immediately follows, since, e.g. H 2d+1 (Ω) is dense in H 1 (Ω). So it is u − u is small enough. always possible to choose a u ˆ ∈ H 2d+1 (Ω), such that ˆ For quasilinear problems the previous terms in (5.330)–(5.332), u ∈ W 2d,∞ (Ω), uW 2d−1,∞ (Ω) , uW d,∞ (Ω) < ∞,

u ∈ W 2d −ρ+2,∞ (T ), uW 2d −ρ+1,∞ (T ) < ∞,

(5.334)

5.5. Consistency, stability and convergence for FEMs with variational crimes

399

have to be replaced, with Bu = BG u in (5.286) and (5.330) in Lemma 5.65, by Bu ∈ W 2d−1,∞ (Ω), BuW 2d−2,∞ (Ω) , BuW d−1,∞ (Ω) < ∞,

Bu ∈ W 2d −ρ+1,∞ (T ), BuW 2d −ρ,∞ (T ) < ∞.

(5.335)

These estimates for the Bu in terms of u are possible with the results in Subsections 2.5.4 and 2.5.6, based upon the second line of (5.334). The linearized forms of these quasilinear problem are studied in Section 2.7, Subsections 2.7.3 ﬀ., e.g. Theorem 2.122 and have to require p ≥ 2 in (5.286). Furthermore, we already mentioned and require the necessary conditions for the Ai (x, uh , ∇uh ), e.g. for bounded forms and operators, ah (uh , v h ) and Gh , formulated in Sections 2.5.4, 2.5.6, 4.4 and (4.174), modiﬁed such that G is Lipschitz-continuous. Theorem 5.68. Violated boundary conditions for quasilinear problems: 1. Let the FEM in (5.286) for the quasilinear problem in Section 4.4, (4.174), satisfy Conditions 5.60 and 5.64, and deﬁne Vbh as in (5.321). Furthermore let, for 0 0 0 u ∈ W ν +1,∞ (Ω) ∩ W ν +1,∞ (∂Ω), or u ∈ H ν +2 (Ω), (5.334) imply (5.335), cf. Remark 5.67 and let the previous conditions be satisﬁed. Then all the consistency results in this subsection remain correct for these quasilinear problems as well. 2. Let the bilinear form of the linearized quasilinear problem be bounded, Vbh coercive. Notice that we only consider W 1,p (Ω) for 2 ≤ p < ∞ and the corresponding stability and convergence results as in Theorem 4.63. with respect to the discrete H 1 (T h ) norm. In this sense, the stability of the corresponding Gh and the existence of uh0 are secured. Then the solutions uh0 converge to u0 according to (5.333). Proof. The claims concerning consistency are an immediate consequence of Chapter 3, Theorem 3.21, and Lemma 5.65. It remains correct, if (5.334) implies (5.335). Finally, the linearized principal part of the linearized quasilinear operator is H01 (Ω)coercive for 2 ≤ p < ∞ or satisﬁes an inf-sup condition. If the quasilinear operator is bounded, implying its discretization to be equibounded, and consistent, cf. (5.286), we obtain the claim. 5.5.7

Violated continuity

We consider nonconformity for piecewise polynomial V h ⊂ V, including violated continuity and boundary conditions. We have already seen that, for the case of polygonal ∂Ω and the same number of functionals on every edge for FEs as in Deﬁnitions 4.6, 4.8, violated boundary conditions are equivalent to violated continuity. For polygonal ∂Ω, violated boundary conditions are directly included in the following discussion. In contrast to [141], we allow curved boundaries and discontinuous FEs by combining the diﬀerent results and techniques here with those in Subsubsection

400

5. Nonconforming ﬁnite element methods

5.5.6 for the case ρ = 0. Isoparametric FEs in Subsubsection 5.5.10 require other approaches. Other combinations would be possible as well: Curved boundaries with ρ = 2 and discontinuous FEs with ρ = 0 require ρ = 1 for discontinuous FEs on the edges of the triangulation with exactly one boundary point. In this case the aﬃne equivalence of the FEs, cf. Deﬁnition 4.6, would have to be replaced by three diﬀerent types of FEs on triangles with none, one and more than one point on the boundary ∂Ω. The whole theory would essentially remain correct. We leave its formulation with the many technical details to the interested reader. So we only formulate the results with the same value of ρ = 0 for violated boundary conditions and continuity. We restrict, as in Subsubsection 5.5.6, the presentation here to the only practically relevant, standard case, cf. (5.327), required for the interior inner and the boundary edges, standard discontinuous FEs with straight edges: 0 = ρ < d = d − 1.

(5.336)

For FEs Vbh ⊂ V, we have already introduced the generalized fh , ·, in (5.250), the ah (·, ·) in (5.289) ﬀ. and norms · V h , deﬁned in (4.36). We study the impact of discontinuous Vbh ⊂ V onto the consistency estimates, cf. (5.279): ⎛ ⎞ 2 ⎝ aij ∂ i uh ∂ j v h dx⎠ ∀ u, v ∈ Vb + Vbh (5.337) ah (uh , v h ) = T i,j=0

T ∈Tch

= as,h (uh , v h ) +

+

e∈∂Ω

e∈Tch \∂Ω

({v h }[Ba uh ] + [v h ]{Ba uh })ds e

v h (Ba uh )ds,

e

with uh ∈ Vb ∩ H 2 (Ω) for as,h (uh , v h ) and with Ba as in (5.278). Again we use the uniﬁed notation in (5.285). The exact and discrete solutions satisfy, cf. (5.280) a(u0 , v) = f, vV ×V ∀v ∈ Vb , and ah uh0 , v h = f h , v h V h ×V h ∀v h ∈ Vbh . (5.338) In this two diﬀerent approaches are possible for estimating the nonconformity errors. The μ-and σ-case are based on diﬀerent proofs in Lemmas 5.65 and 5.70 The σ-case yields slightly inferior consistency orders for less smooth solutions u0 . The μ-case allows slightly higher consistency orders but often needs much higher smoothness of u0 . For example, we get for the σ-case hd−1 |u|H d (Ω) instead of hd |u|H 2d (Ω) for the μ-case. Condition 5.69. Conditions for interpolation points on edges: We require Condition 5.60 and modify Condition 5.64. In (5.336) we replace the Dirichlet conditions by interpolation conditions on all straight inner and (curved) boundary edges, e ∈ Tch . The uh , v h ∈ V h , interpolate prescribed values v h (Pi ) = yi , i = 0, . . . , d − 1, ∀e ∈ Tch

5.5. Consistency, stability and convergence for FEMs with variational crimes

401

in the above Pi = Piρ=0,e ∈ e, i = 0, . . . , d − 1, see (5.320). This implies continuity of the FEs in the Pi , but not in e, even less in Ω. For problems with natural boundary conditions we do not have to impose boundary conditions. For Dirichlet boundary conditions we additionally impose the original Condition 5.64 along the boundary ∂Ω again with ρ = 0. Accordingly we deﬁne for Ω = ∪T ∈Tch T , see (5.321), possibly with curved e ∈ ∂Ω, (5.339) Vbh := v : Ω → R : v|T ∈ Pd−1 ∀ T ∈ Tch , [v](Pi ) = 0, ∀ e ∈ Tch v(Pi ) = 0, ∀ e ∈ ∂Ω ∀ i = 0, . . . , d − 1, Pi = Piρ=0,e }, v h V h := v h H 1 (Tch ) . The second condition, v(Piρ=0,e ) = 0, e ∈ ∂Ω, only applies to Dirichlet boundary conditions; for natural boundary conditions it has to be omitted. Except these new Vbh in (5.339), instead of (5.321) the whole formal procedure in this and the preceding subsection is the same. Diﬀerences occure in the following formulas, estimates and theorems due to the discontinuity in the many interior edges here, compared to only violated boundary conditions in the few boundary edges before. Correspondingly, we only recall the important concepts, Q h and discrete linear and quasilinear equations and operators are, cf. (5.294) ﬀ.:

Q h ∈ L W −1,p Tch , Vbh : Q h fh − fh , v h W −1,p (Tch )×V h = 0∀v h ∈ Vbh , (5.340) ; : ah uh0 , v h − f h , v h V h ×V h = Ah uh0 − fh , v h H −1 (T h )×V h = 0 ∀ v h ∈ Vbh c h h : ; h h h and a u0 , v = Gh u0 , v W −1,p (T h )×V h = 0 ∀ v h ∈ Vbh . (5.341) c

Ah := Q h Ah |Vbh , and Gh := Q h Gh |Vbh ⇒ Ah uh0 − f h = 0, and Gh uh0 = 0.

(5.342)

The variational discretization error for u ∈ H 2 (Ω) with the discrete solution uh of the linear Ah uh = Q h Au and the quasilinear generalization, see (5.305), (5.299), is h h h [v ]Buds + v h Buds ∀u ∈ H 2 (Ω). (5.343) ah (u − u , v ) = e

e∈Tch

e∈∂Ω

e

The classical consistency errors are, cf. (5.313), (5.339), (3.35),

Ah I h u − Q h AuV , Gh I h u − Q h GuV , and we need |ah (u − u , v )| sup = sup v h V h h v =0 h

h

|

e∈Tch

e

[v h ]Buds +

(5.344)

e∈∂Ω

v h V h

e

v h Buds| .

The next lemma could be proved by the same technique as the previous Lemma 5.65. However, we neglect the fact that along the straight edges e the parametrization

402

5. Nonconforming ﬁnite element methods

is aﬃne. This allows modiﬁed estimates compared to Lemma 5.65 and a reduced smoothness conditions for u. Since the interpolation error dominates the consistency error, cf. (5.348), we only discuss this updated form. Lemma 5.70. 1. Under Conditions 5.60,5.69 and with the notation in (5.258), (5.318), we choose A, a(·, ·), ah (·, ·), G, Gh as in (5.278) ﬀ., the Vbh as in (5.339), and u ∈ H d (Ω). Let T1 , T2 ∈ Tch be two neighboring or T1 a boundary triangle with diam Ti ≤ h, i = 1, 2 and a joint or a boundary straight edge and let Ue := interior (T1 ∪ T2 ) for e ⊂ T1 ∩ T2 or Ue := T1 , for |T1 ∩ ∂Ω| ≥ 2. Then there exists a constant C = C(d, χ) such that, with |v h |Hh1 (Ue ) := |v h |Hh1 (T1 ∪T2 ) , [v h − v]Ba u ds ≤ C hd−1 |u|H d (U ) v h − vH 1 (U ) , ∀v ∈ H 1 (Ue ) (5.345) e e h e h = [v ]Ba u ds ≤ C hd−1 |u|H d (Ue ) |v h |Hh1 (Ue ) for v = 0. e

2. This remains correct for a straight boundary edge e ⊂ ∂Ω with [v h − v], . . . , replaced by v h − v, . . . , but not for curved boundaries. 3. For quasilinear problems under the condition (5.335) for Theorem 5.68, the previous estimates in (5.345) remain correct with Ba replaced by B = BG . Proof. We apply a technique similar to Lemma 5.65 to our problem. For every straight edge e ∈ T¯ ∈ Tch with length he , the parameterization xe is aﬃne, see (5.319). This allows the following modiﬁcation. We combine q d ([v h ]w) ◦ xe = 0 ∀v h ∈ Vbh , Propositions 5.61, 5.62 for q = 2 in (5.309), (5.311), with (5.339) to yield

se +he

se

([v ]w) ◦ xe ds = h

se +he

([v h − v](w − p2 )) ◦ xe ds

(5.346)

se

∀v ∈ H 1 (Ω) or ∀c1 ∈ R ∀ p2 ∈ P d−2 , [c] = 0 = [v], [v h ] = [v h − v − c1 ], s +h for these c1 , v. Note v h p2 ∈ P 2d−3 , hence see e ([v h − v]p2 ) ◦ xe ds = 0. We choose in (4.34), Q1 v h = c1 , Qd−1 Ba u = p2 . This allows estimating

5.5. Consistency, stability and convergence for FEMs with variational crimes

[v h ]Ba uds = [v h − v]Ba uds e e se +he h = ([v − v − c1 ](Ba u − p2 )) ◦ xe ds se

403 (5.347)

≤ Cxe L∞ (e) [v h − v − c1 ]L2 (e) (Ba u − p2 )L2 (e) by (1.67)

1/2 ≤ C [v h − v − c1 ]L2 (Ue ) [v h − v − c1 ]Hh1 (Ue )

(Ba u − p2 )L2 (Ue ) (Ba u − p2 )Hh1 (Ue )

1/2

1/2 ≤ C hv h − v2H 1 (Ue ) (hd−1 Ba uH d (Ue ) hd−2 uH d (Ue ) )1/2 with (4.34),(4.41) and choosing in (5.346) the inﬁmum for c1 , p2 , ≤ Chd−1 |v h |Hh1 (Ue ) |Ba u|H d−1 (Ue ) for v = 0 and ≤ Chd−1 v h − vHh1 (Ue ) |Ba u|H d−1 (Ue ) ∀v ∈ H 1 (Ω), hence, (5.345). The v h − v modiﬁcation is an immediate consequence of (5.346) and the use of v h H 1 (Tch ) instead of |v h |H 1 (Tch ) . With the above change of [v h − v], . . . , into v h − v, . . . , for a boundary triangle T2 ∩ ∂Ω = ∅ this result (5.345) remains valid as well. For quasilinear problems (5.347) remains correct for the nonlinear Bu = BG u. Theorem 5.71. Consistency for FEMs violating continuity conditions and convergence for coercive forms: 1. Let Ω and the FEs satisfy Conditions 5.60 and 5.69. With the notation in (5.258), (5.318), we choose A, a(·, ·), Ah , Ah , ah (·, ·), Vbh , Q h as in (5.278) ﬀ., (5.339) ﬀ., and natural or Dirichlet boundary conditions. 2. Then Ah= Q h Ah |Vbh and Gh= Q h Gh |Vbh are consistent with A and G in u ∈ H d (Ω), respectively. The variational and classical consistency errors can be estimated for straight edges and polygonal ∂Ω, with an h-independent C = C(d,n,ρ,xb ) as sup 0 =v h ∈Vbh

|ah (u − uh , v h )| ≤ Chd−1 uH d (Ω) , and v h V h

(5.348)

Ah I h u − Q h AuV h , Gh I h u − Q h GuV h ≤ Chd−1 uH d (Ω) . 3. Let ah (·, ·) be Vbh -coercive, e.g. the principal part, ahp (·, ·). Then the solutions uh0 for Ah uh0 = Q h Au0 exist and converge to u0 according to h u0 − u0 1 ≤ Chd−1 u0 H d (Ω) . (5.349) H (Ω)

404

5. Nonconforming ﬁnite element methods

4. For curved boundary edges in a curved ∂Ω, (5.349) has to be combined with ρ = 0 in 5.64 has to be imposed. Then (5.349) is replaced by and Condition h(5.332), d−1 u0 − u0 1 ≤ Ch u 0 H 2d (Ω) . For the ρ = 0, 1, 2 combination indicated in H (Ω) ≤ Chd−3/2 u0 H 2d (Ω) . the introduction, we only have uh0 − u0 1 H (Ω)

5. This remains correct for quasilinear problems under the conditions previous to Theorem 5.68, if the coercivity of ah (·, ·) is replaced by the coercivity of the bilinear form of the linearized operator G (u0 ). 6. As in Lemma 5.70 the ﬁrst estimate in (5.348) remains correct for v h − v, sup

|ah (u − uh , v h − v)| ≤ Chd−1 uH d (Ω) , ∀ v ∈ H 1 (Ω). v h − vV h

(5.350)

Proof. Similarly to Theorem 5.66 the consistency errors are immediate consequences of Lemma 5.70. We sum over all e ∈ Tch in Condition 5.60, where Ue is deﬁned in Lemma 5.70. We use | e [v h ]Ba u ds| ≤ C hd−1 |u|H d (Ue ) |v h |Hh1 (Ue ) , see (5.306) and (5.345) estimating 41 |ah (u − uh , v h )| = [v h ]Ba u0 ds + v h Ba u0 ds e∈∂Ω e∈T h e c |v h |Hh1 (Ue ) |u|H d (Ue ) (5.351) ≤ Chd−1 e∈Tch ∪∂Ω

≤ Chd−1

|v h |H 1 (T ) |u|H d (T ) (by H¨ older)

T ∈Tch ∪∂Ω

≤ Chd−1 |v h |V h · |u|H d (Ω) . The analogous v h − vV h extension of Lemma 5.70 is veriﬁed as there.

Remark 5.72. Lemma 5.70 is strictly local with the local step size h and the locally appropriate, e.g. uH d (Ue ) valid for all u ∈ H d (Ue ). This can be used in two directions. Either locally large Bu and discretization errors are compensated by mesh reﬁnements or the so-called hp-methods are employed. For locally smooth solutions u0 and Bu0 , high orders of convergence d − 1 and large he , for locally unpleasant u0 and Ba u0 , small orders d − 1 and small he have to be combined. These are the so-called adaptive FEMs, introduved in Chapter 6. The hp viewpoint is discussed for DCGMs in Section 7.14. We generalize the Aubin–Nitsche Lemma 4.59 and the corresponding convergence result in Theorem 4.60 to the case of variational crimes, with straight edges e, see, e.g. [135]. Here, we need the extension ah (·, ·) : Vb + Vbh × Vb + Vbh → R instead of Vbh × Vbh → R. In addition to the original problem a(u0 , v) = (f, v)∀v ∈ Vb we need the solution, φg , of its dual problem, φg ∈ Vb s.t. a(w, φg ) = (g, w)∀w ∈ Vb .

(5.352)

41 For 0 ≤ a ≤ a , 0 ≤ b ≤ b we have (a + a )(b + b ) ≤ a b + 3a b and every T ∈ T h con1 2 1 2 1 2 1 2 1 1 2 2 c tains only a few edges, e.g. three for triangulations.

5.5. Consistency, stability and convergence for FEMs with variational crimes

405

Lemma 5.73. Generalized Aubin–Nitsche lemma: Let a(·, ·) : Vb × Vb → R and its corresponding ah (·, ·) : Vb + Vbh × Vb + Vbh → R be continuous bilinear forms with ah (·, ·)|Vb ×Vb = a(·, ·), see (5.279). Assume that the original and its dual problem satisfy the conditions for unique exact and FE solutions u0 , uh0 and φg , φhg . For the Banach space W ⊃ Vb with inner product and norm (g, w) and gW , respectively, let the embedding Vb → W be continuous. Then the error for the FE solution can be estimated in the W norm as u0 − uh0 h ≤ sup (1/gW ) C u0 − uh0 h φg − φhg h (5.353) W V V 0 =g∈W

+ ah u0 − uh0 , φg − u0 − uh0 , gh + ah u0 , φg − φhg − fh , φg − φhg . Proof. We employ the dual norm of an element w of a Banach space in (1.32) and combine the solutions u0 , uh0 and φg , φhg of both problems to ﬁnd, see [135], u0 − uh0 , gh = ah (u0 , φg ) − ah uh0 , φhg = ah u0 − uh0 , φg − φhg + ah uh0 , φg − φhg + ah u0 − uh0 , φhg % & = ah u0 − uh0 , φg − φhg − ah u0 − uh0 , φg − u0 − uh0 , gh & % − ah u0 , φg − φhg − fh , φg − φhg . For the equality in the second and fourth lines we use the linearity of the ah (·, ·) and split and replace the linear functionals in the [..] by the corresponding a(·, ·), ah (·, ·), respectively. Finally, use (1.32), and the continuity of ah (·, ·) to obtain (5.353). We combine Lemma 5.73 with Theorem 5.71, but not with Theorem 5.66: Theorem 5.74. Aubin–Nitsche theorem for nonconforming FEMs: Under the conditions of Theorem 5.71 and Lemma 5.73 we allow standard FEs and FEs violating the continuity in (5.336) and assume u ∈ H d (Ω). Then u0 − uh0 2 ≤ Chd uH d (Ω) . (5.354) L (Ω) Proof. We apply Lemma 5.73 to our situation: we choose W = L2 (Ω) and Vb = H01 (Ω). This is justiﬁed since, by uL2 (Ω) ≤ uH 1 (Ω) ∀ u ∈ H01 (Ω), the H01 (Ω) is continuously embedded into L2 (Ω). We estimate the terms in (5.353). For the ﬁrst term in the second line of (5.353) we get, with the unusual deﬁning relations for φg φhg and (5.347), (5.350), [ah u0 − uh0 , φg − u0 − uh0 , g ] = [a(u0 , φg ) − (u0 , g)] − [ah uh0 , φg − uh0 , g ] & % & % = − ah uh0 , φg − ah uh0 , φhg = − ah uh0 , φg − φhg % & = − ah u0 − uh0 , φg − φhg .

406

5. Nonconforming ﬁnite element methods

Similarly we ﬁnd for the second term in the second line of (5.353), with the deﬁning relations for u0 and uh0 , [ah u0 , φg − φhg − f, φg − φhg ] & % = [a(u0 , φg ) − (f, φg )] − ah u0 , φhg − f, φhg & % (5.355) = − ah u0 , φhg − ah uh0 , φhg & % & % = − ah u0 − uh0 , φhg = − ah u0 − uh0 , φhg − φg , again with (5.347), and (5.350), now applied to the dual problem (5.352). We combine the (5.347) with the continuity of the ah (., .) and the estimates for u0 − uh0 1 h ≤ Chd−1 u0 H d (Ω) , H (Tc ) φg − φhg 1 h ≤ Chφg H 2 (Ω) ≤ ChgL2 (Ω) . H (T ) c

This allows, with (5.355) and by the continuity of ah (., .), the estimates h ah u0 , φg − φhg = ah uh0 − u0 , φg − φhg ≤ C u0 − uh0 V h φg − φhg V h ≤ Chd u0 H d (Ω) φg H 2 (Ω) ≤ Chd u0 H d (Ω) gL2 (Ω) . In fact the second and the following lines in this inequality can be used for the term in the ﬁrst line of (5.353) and by (5.355) the two last terms coincide. So estimating the three terms in (5.353) yields, with W = L2 (Ω) and V = H 1 (Ω), u0 − uh0 2 ≤ sup (1/gL2 (Ω) ) C u0 − uh0 V h φg − φhg V h L (Ω) 0 =g∈L2 (Ω)

+ ah uh0 , φg − φhg + ah uh0 − u0 , φg − φhg ≤ sup (1/gL2 (Ω) ) Chd u0 H d (Ω) gL2 (Ω) , 0 =g∈L2 (Ω)

the claim. 5.5.8

Stability for nonconforming FEMs

We turn to proving the stability for FEMs with variational crimes. Chapter 3 shows that we can conﬁne the proof to the linear problems. By Theorem 3.23, stability is implied by the stability of the linearized Ah and some technical conditions. In Section 4.3 we could directly apply the results in Chapter 3. In Sections 5.2 and 5.4 speciﬁc ideas allowed us to overcome the diﬃculties with variational crimes in the fully nonlinear and quadrature approximate FEMs. Here we will procede in a similar way. We have already proved in Theorem 5.59, assuming (5.300), see Remark 5.58, the stability of the above nonconforming FEMs for the principal parts, Ap , or speciﬁc

5.5. Consistency, stability and convergence for FEMs with variational crimes

407

compact perturbations, B, yielding a stable B h . Here A was a linear operator for any of our above elliptic equations or systems of order 2. Furthermore, we have proved in Theorems 5.66 and 5.71 the consistency for general linear operators and extended it to quasilinear operators in Theorems 5.68 and 5.71, 5. We have seen in Sections 4.3 and 5.2 that the existence of a bounded inverse of a linear A is related to, but not suﬃcient to ensure that the discrete Ah is stable, see [128, 567]. The next Theorem 5.78 guarantees the stability for nonconforming FEMs studied in this section for all types of elliptic equations and systems of order 2. It shows that a linear invertible B with stable B h inherits the stability to Ah , for a linear invertible compactly perturbed A, such that Ah is consistent with A. This consistency is always valid for conforming FEs and is assumed for general discretizations in Theorem 3.29. It is not too surprising that the perturbations by the nonconformity of the FEs have to be small. Good convergence of the discrete solutions is only to be expected for consistency of higher order, so only for higher order methods and smooth enough u0 . For the following considerations we recall the diﬀerent consistency errors

Q h Ah u − Ah uh = Q h Ah (u − P h u) + Ah P h u − Q h Au, with Ah uh = Q h Au. < => ? => ? < => ? < var.cons.err. class.cons.err. interp.err. (5.356) Condition 5.75. Choose Vbh ⊂ V as in (5.321), or (5.339) and require Condition 5.60, (5.300), and P h = I h and Q h as in Proposition 5.55, (5.294), (5.295). Choose the original linear problems and the FEMs as listed in (5.278)–(5.280), (5.281)– (5.284), (5.297)–(5.298). Lemma 5.76. Under Condition 5.75, let Ap : Vb → V be a linear principal part, cf. (5.278) or (5.281). Then Ap u = f ∈ H −1 (Ω), and for small enough h0 > h ∈ H, Ahp uh = Q h f as well, have unique solutions, and uh0 converge to u0 as h u0 − u0 h ≤ uV , for h < h0 , hence lim uh0 − u0 h = 0. (5.357) V V h→0

For general A, u ∈ Vb , P h = I h , with Ah uh = Q h Au, the preceding consistency errors converge to 0, as

lim Q h Ah u − Ah uh V h = 0,

h→0

lim Q h Au − Ah P h uV h = 0.

h→0

(5.358)

Proof. By Proposition 5.54, all the bilinear forms ah (·, ·) are bounded and continuous. By Theorem 5.59, Ahp is stable in P h u. In Theorems 5.66, 5.71, we have formulated the smoothness of the u guaranteeing the optimal order of consistency. So we choose, e.g. ˜ we have convergence, u ˜ ∈ H d (Ω), near u. Then Theorem 4.53 implies that for u and u ˜h = Q h Ap u ˜, cf. (5.357), (5.358), hence, e.g. for Ahp u uH d (Ω) < uV /3 for h < h0 . ˜ u−u ˜h V h ≤ Chd−1 ˜ For estimating u − uh V h ≤ u − u ˜V h + ˜ u−u ˜h V h + uh − u ˜h V h < ,

408

5. Nonconforming ﬁnite element methods

we handle the ﬁrst and the last term, u − u ˜V h = u − u ˜V and uh − u ˜h V h . We h ˜ such that with the following employ the deﬁnition of Ap , Ap cf. (5.279), and choose u bounded maximum, −1 h h ˜)V < uV /3, Q A max 1, Ahp h p Vb ←Vb (u − u h V ←V Vb ←Vb

b

b

hence h h Ap (u − u ˜h )V h = Q h Ap (u − u ˜)V h =⇒ −1 h u − u ˜h h ≤ Ahp Q h h V

Vbh ←Vbh

Vb ←Vb Ap Vb ←Vb (u

−u ˜)V ≤ uV /3,

hence (5.357). For the general Ah this argument yields (5.358).

We combine the form of the consistency errors in Proposition 5.56, the stability of the principal parts in Theorems 5.59, and the consistency, speciﬁed in Theorems 5.66 and 5.71, to obtain the convergence of the discrete to the exact solution. This can be used deﬁning an anticrime transformation from Vbh to Vb , the essential tool for proving the following stability result in Theorem 5.78, cf. Proposition 5.42. Lemma 5.77. Anticrime transformation: Let A0 ∈ L (Vb , Vb ), e.g. A0 = Δ be bound edly invertible, Ah0 : Vbh → V h be stable and Ah0 be consistent with A0 for every u ∈ Vb . For a given uh ∈ Vbh deﬁne fˆh := Ah0 uh ∈ H −1 Tch , cf. (5.250), (5.279). This fˆh ∈ H −1 Tch deﬁnes, as proved in Proposition 5.42, by its components f−1 , f0 via (5.250) an fh ∈ H −1 (Ω), such that fh H −1 (Ω) ≤ |Ah0 |V h ←V h uh V h . Deﬁne E h : Vbh → Vb b

by uh := E h uh := A−1 E h uh H 1 (Ω) ≤ C1 uh H 1 (Tch ) . Then we obtain 0 fh ∈ Vb with h convergence as: ∀ > 0 ∃h0 = h0 u H 1 (Tch ) : ∀h < h0 : E h uh − uh H 1 (Tch ) < uh H 1 (Tch ) , hence, lim E h uh − uh H 1 (Tch ) = 0. h→0

(5.359)

For fh ∈ L2 (Ω), we even get E h uh − uh V h ≤ Chfh L2 (Ω) for h → 0 implying convergence of order 1. Theorem 5.78. Compactly perturbed invertible and consistent discretizations inherit stability: Under the Condition 5.75, let A ∈ L (Vb , Vb ) be boundedly invertible or be one of the linear(ized) operators for equations or systems, studied in this section, see (5.278), or (5.281), their extensions (5.279), or (5.283), and the corresponding FE equations (5.280), or (5.284). Finally, let B ∈ L (Vb , Vb ), B h be stable and Ah , B h , hence C h , be consistent with A, B, C, for u ∈ Vb . Then, for small enough h, A−1 ∈ L(V , Vb ) =⇒ Ah (or its quadrature approx. A˜h ) is stable in P h u. (5.360) Proof. 1. We employ the anticrime transformation, E h : Vbh → H01 (Ω), introduced in Lemma 5.77. We derive several relations and inequalities. The proof is similar to that of Theorem 5.9. The diﬀerences, mainly from (5.369) ﬀ., are caused by the

5.5. Consistency, stability and convergence for FEMs with variational crimes

409

necessary P h E h uh . As discussed it suﬃces to show that A−1 ∈ L(V , Vb ) implies the stability of Ah with A, B, C ∈ L(Vb , V ), C compact, A =: B + C, with B, B h boundedly invertiblestable, and Ah , B h , C h consistent with A, B, C in u ∈ H 1 (Ω) or ∈ H 2 (Ω).

(5.361)

In fact, for all nonconforming FEMs in this section, the consistency of the B h , C h is as correct as that of Ah . We use the notation in Proposition 5.55, (5.294)– (5.298), Ah , Bh , Ch : Vb + Vbh → V h , Q h : (V + V h ) → V h ,

Ah = Q h Ah |Vbh : Vbh → V h , . . . with Φh A = Ah , . . . , cf. (5.298). The bounded invertibility and stability of B and B h and consistency with B in u imply the unique existence of the following solutions u ˆ and u ˆh , deﬁned, for an arbitrary u ∈ Vb and v := Cu by

ˆh = Q h v . Bu ˆ = v and B h u

With equicontinuous B h , and Q h , and Φh (B · −v ) = B h · −Q h v , let

T := B −1 and T h := (B h )−1 Q h ∈ L V + V h , Vbh .

(5.362)

Then the convergence of the u ˆh to u ˆ can be written as ∀u ∈ Vb : lim ||(T − T h )Cu||H 1 (Tch ) = 0.

(5.363)

h→0

This implies, since C is compact and the T − T h are equibounded, ||(T − T h )C||H 1 (Tch )←Vb → 0, for h → 0.

(5.364)

The limit (5.363) remains correct if u ∈ Vb is replaced by any bounded uh = E h u ∈ Vb , depending upon h, with ||uh ||V = E h uh H 1 (Ω) ≤ C2 uh H 1 (Tch ) . Indeed the consistency estimate, cf. Remark 5.49, and the stability of B h can be applied to any bounded uh ∈ Vb as well: ||(T − T h )CE h uh ||H 1 (Tch )←Vbh ≤ C3 ||(T − T h )C||H 1 (Tch )←Vb uh H 1 (Tch ) . (5.365) We combine uh ∈ Vbh with E h . With the boundedly invertible A we estimate uh V h ≤ 2||E h uh ||V ≤ 2||A−1 ||Vb ←V ||AE h uh ||V = 2||A−1 ||Vb ←V ||B(I + T C)E h uh ||V −1

≤ 2||A

(5.366)

||Vb ←V ||B||V ←Vb ||(I + T C)E u ||V , h h

hence, ||(I + T C)E h uh ||V ≥ uh V h /(2||A−1 ||Vb ←V ||B||V ←Vb ).

(5.367)

410

5. Nonconforming ﬁnite element methods

With the stability of B h we get (I + T h C h )uh V h = (B h )−1 B h (I + T h C h )uh V h ≤ (B h )−1 V h ←V h B h (I + T h C h )uh V h . b

This implies B h (I + T h C h )uh V h ≥ (I + T h C h )uh V h / (B h )−1 V h ←V h . (5.368) b

The linearity of FEM with variational crimes implies Φh (A) = Ah = Φh (B + C) = B h + C h

= B h (I + (B h )−1 C h ) = B h (I + T h C h ) : Vbh → V h . For the following estimate we refer to (5.359), and Theorem 4.17 and obtain with (5.357), uh − P h E h uh H 1 (Tch ) ≤ (E h uh − P h E h uh )H 1 (Tch )

(5.369)

+ (uh − E h uh )H 1 (Tch ) ≤ 2uh ||V h . 2. For the ﬁnal stability estimates we return to the (B h )−1 Ah uh = (I + T h C h )uh term. The identities Q h |V h = IV h , T h |V h = (B h )−1 Q h |V h = (B h )−1 , the equiboundedness of C h and T h , the consistency of C h , and the indicated error terms, are combined with the triangle inequality estimating

(B h )−1 V h ←V h ||Ah uh ||V h = (B h )−1 V h ←V h ||B h (I + (B h )−1 Q h C h )uh ||V h b

h −1

b

b

b

≥ (B ) ||B (I + T C )u || (5.370)

≥ ||(I + T C)E h uh ||V − ||uh − E h uh ||V h − ||T h C h uh − T CE h uh ||V h ≥ ||(I + T C)E h uh ||V − ||uh − E h uh ||V h − ||(T h − T )CE h uh ||V h

− ||T h (C h uh − CE h uh )||V h ≥ ||(I + T C)E h uh ||V − ||uh − E h uh ||V h − ||(T h − T )CE h uh ||V h

− ||T h (C h P h E h uh − CE h uh )||V h − ||T h (C h − C h P h E h )uh ||V h Vbh ←Vbh

h

h

h

h

V h

5.5. Consistency, stability and convergence for FEMs with variational crimes

411

≥ || (I + T C)E h uh ||V − || (uh − E h uh ) ||V h − 2|| (T h − T )C ||Vbh ←Vb ||uh ||V h < < => ? < => ? => ? by (5.367)

− ||T h

FEM error (5.369)

by (5.364)

(C h P h E h uh − CE h uh )||V h < => ? class.consistency of C,cf.Theorems 5.66, 5.71

− ||T h C h

(uh − E h uh ) + (E h uh − P h E h uh ) ||V h . < < => ? => ?

FEM error (5.369)

interpol.error (4.41)

We choose h small enough, such that all the preceding errors in (5.369), (5.364), (5.358), (4.41) are < ||uh ||V h with < 1/ 4(3 + ||T h ||V h ←V h + 2||T h C h ||Vbh ←Vbh )||A−1 ||Vb ←V ||B||V ←Vb . b

b

This implies

−1 ||Ah uh ||V h ≥ ||uh ||V h / 4||A−1 ||Vb ←V ||B||V ←Vb /||B h ||V h ←V h b

≥ K ||u ||V h , h

i.e. (Ah )h∈H is stable.

(5.371)

Theorem 5.78 yields a criterion for the stability of discretizations of operators, which are compact perturbations of coercive operators. An important class of compactly arding perturbed coercive operators are the A ∈ L (Vb , Vb ) satisfying a so-called G˚ inequality. Hence Theorem 3.32 can directly be generalized to our FEMs with variational crimes. 5.5.9

Convergence, quadrature and solution of FEMs with crimes

Now we have proved all necessary results guaranteeing consistency, stability, and convergence for the FEMs with variational crimes studied in this section. It would be boring to repeate all the diﬀerent complex conditions for the many diﬀerent cases in this section. We prefer the formulation of a summarizing theorem. It collects the diﬀerent desired results. Theorem 5.79. FEMs for elliptic problems of second order, violating homogeneous Dirichlet boundary conditions and/or continuity: 1. These are stable under the conditions in Theorems 5.59 (coercivity), 5.78 (stability), consistent for violated Dirichlet conditions and/or continuity in Theorems 5.66 and 5.71. Hence all these methods converge with the same estimates as their consistency, multiplied by a stability constant S. 2. Extensions to quasilinear problems in Theorems 5.68 and 5.71, 5. imply the corresponding convergence results. 3. An improvement of the convergence rate for linear problems with violated continuity and with respect to the L2 (Ω) norm is formulated in Theorem 5.74.

412

5. Nonconforming ﬁnite element methods

The quadrature approximations are illustrated in Figure 5.5. Now Vbh are the FEs violating boundary conditions and/or continuity. Formulating with the quadrature ˜ h in (5.180), and Ah in Theorem 5.66 and in projectors yields, cf. (5.183), with the Q (5.342), ˜ h Ah |V h , A˜h u ˜ h f. ˜h0 = Q A˜h = Q

(5.372)

˜ h Ah |V h is proved in Theorem 5.30. So according to Theorems The stability for A˜h = Q 3.21, 4.53 and the continuity of A, we only have to show the classical consistency of A˜h with A in the exact solution u0 yielding convergence in Theorem 5.80. The corresponding smoothness of Sobolev norms is collected in Cˆ = C σ or Cˆ = C μ . We require, for the coeﬃcients aij and the inhomogeneity f , k aij ∈ W∞ (Ω), i, j = 0, . . . , n, f ∈ H k (Ω).

(5.373)

We impose the condition k > max{2(m + τ − 1), 2} for a quadrature version of the same order of convergence as the conforming FEM. u0 ∈ H μˆ (Ω) is motivated by Theorem 5.71 and Remark 5.72. We have proved in Theorem 5.30 that for k > 2(d − 2) ˜h (·, ·) are simultaneously bounded and deﬁne stable Ah and A˜h . the ah (·, ·) and a Theorem 5.80. Quadrature for nonconforming FEs: 1. Assume n = 2, nonconforming FEs and quadrature formulas, exact for piecewise polynomials of degree k − 1 with k > max{2(d − 2), 2}. Let (5.176),(5.373) for the aij , f , and Conditions 5.60 and 5.69 be satisﬁed. Let ah (·, ·) induce a stable Ah . Then h h h ˜0 , v = f, v h˜ ∀v h ∈ V h , (5.374) a ˜h u deﬁnes a stable and convergent discretization for Au0 = f . For the other approximations, mentioned above, this result remains correct if the a ˜h , A˜h are replaced ˜ ˜ ˜ ˜h ˜ h h h ˜ ,A ,A . by a , a 2 2. For u0 ∈ H d−p (Ω), p ≥ 0, we obtain (M := Maij := max aij W k,∞ (Ωhc ) ) : u 0 − u ˜h0

Vh

i,j=0

≤ Chk+2−d (f V + M P h u0 H d−1 (Tch ) ) + Chd−1−p u0 H d−p (Ω) . (5.375)

3. For violated boundary conditions on curved boundaries, hd−1 u0 H d (Ω) has to be replaced by hd−1 u0 W 2d−1,∞ (Ω) . For isoparametric FEs the P h u0 has to be combined with Ec u0 . h 4. If only approximations for f, v h˜ and none for a ˜h (uh , v h ) are used in (5.374), then the term M u0 H min{s,k+1} (Ω) in (5.375) is dropped. Proof. A˜h is stable by Theorems 5.59 and 5.30 for k > 2(d − 2) and the conditions in the theorem. So it suﬃces to estimate the classical consistency errors

˜ h Au0 h ≤ A˜h P h u0 − Ah P h u0 h A˜h P h u0 − Q V V ˜ h f h . +Ah P h u0 − Q h Au0 V h + Q h f − Q V

5.5. Consistency, stability and convergence for FEMs with variational crimes

413

The terms on the right represent the quadrature errors for Au0 and Q h f and the nonconforming consistency errors of the exact FEM for Ah in u0 . Estimates are proved in Theorems 5.66, 5.71, and 5.28, cf. Remark 5.67. We get ≤ Chk+2−d (f H −1 (T h ) + Maij P h u0 H d−1 (Tch ) ) + Chd−1 du0 H d (Ω) . (5.376) k

c

If only u0 ∈ H d−p (Ω), p ≥ 0, the consistency error is reduced to Chd−1−p du0 H d−p (Ω) . Now the above k > 2(d − 2) implies k + 2 − d ≥ d − 1. So the term Chd−1−p d u0 H d−p (Ω) dominates in (5.376) and we do not specify the other terms. Theorem 5.68 for violated boundary conditions and quasilinear problems in the last subsection remains nearly unchanged. Theorem 5.81. Quadrature approximate FEMs with crimes for quasilinear problems: For the quasilinear problem in Section 4.4, (4.174) let its FEM in (5.286) satisfy the conditions of Theorems 5.68 and 5.71. In particular we require the necessary conditions for the Nemickii operators Ai (x, uh , ∇uh ), e.g. for bounded forms and operators, ah (uh , v h ) and Gh , as formulated in Sections 2.5.4, 2.5.6, 4.4 and (4.174). We additionally modify them, such that G is Lipschitz-continuous. Then for a boundedly invertible linearized operator the convergence and quadrature approximation results in this subsection remain correct. We discuss the solution of the nonlinear equations here for all the previous FEMs with crimes, including quadrature approximations. These FEMs with crimes are linear, cf. ( 4.154). Hence all the quasilinear equations and their quadrature approximations can be eﬃciently solved with the methods in Section 3.7, combining continuation with Newton’s methods and the mesh independence priciple in Theorems 5.21, 3.40. By (5.375) and for u0 ∈ H d−p (Ω, Rq ), d − p > 1, we obtain convergence of order d − p − 1 > 0. We impose the following Lipschitz-continuity for G , (5.377) G (·) ∈ CL (Br (u0 ) ∪ Br (P h u0 )), h h d−p d−p d−p−1 d−p−1 Ωc →H Ωc . (Ω) ∪ H (Ω) ∪ H G|Vd−p : D(G) ∩ Vd−p := H Theorem 5.82. Newton’s method for the FE equations: In addition to the conditions in Theorems 5.80 and 5.81 we assume (5.377). Start the Newton process for G and ˜ h with u1 ∈ Br (u0 ) ∩ D(G) ∩ Vd−p and uh := P h u1 . Then, for small enough Gh or G 1 u0 − u1 V and h, both methods converge quadratically and the sequences ui , uhi satisfy, independend of h, for i = 1, . . . , h ui+1 − uhi

Vh

2 ≤ C uhi − uhi−1 V h and uhi − P h ui V h ≤ Chd−p−1 ui Vd−p . (5.378)

We have presented the proof and futher details for fully nonlinear problems in Theorems 5.21 and 4.64.

414

5. Nonconforming ﬁnite element methods

5.5.10

Isoparametric FEMs

In this subsection we generalize isoparametric FEMs from the standard second order linear elliptic equations to second order linear and quasilinear elliptic equations and systems. Even extensions to order 2m, m ≥ 1, in Rn , would be possible, if the interpolation theory in Subsection 4.2.7 could be approporiately generalized to C m−1 reference elements. The essential tool is, as in the case of fully nonlinear problems, the general discretization theory in Chapter 3 and Summary 4.52. Motivated by Proposition 5.63 we only discuss Dirichlet boundary conditions. For simplicity we again formulate the computations for n = 2 and the Laplacian, and indicate the generalizations to linear equations and systems of order 2 and the modiﬁcations for the quasilinear case and n ≥ 2. For the consistency estimates we recall the diﬀerent tools introduced in Deﬁnition 4.42, (4.99), (4.100), (4.101). We recall the deﬁnition of the original and approximating polygonal domains Ω and Ωh , the isoparametric FEs and the mappings in (4.102), (4.103), (4.100), (4.101). The isoparametric FEM uses Vbh = Vbh . F h : Ωh → Ωhc := F h (Ωh ), Fc : Ωh → Ω, and Ec := F h ◦ Fc−1 : Ω → Ωhc , uh , v h : Ωhc = F h (Ωh ) → R, u ˆh = (uh ◦ Ec ) : Ω → R, h h Dbh = V Dbh := u ˆ : u ∈ Vbh , and Vbh = Vbh , V fˇ : Ωhc → R, fˇ(x) := f ((Ec )−1 (x)) = (f ◦ (Ec )−1 )(x) with

(5.379)

t − Ec (t) = O(hd ) =⇒ (Ec ) − IdΩ = O(hd−1 ), ((Ec )−1 ) − IdΩhc = O(hd−1 ). The isoparametric FEM with the above Vbh are formulated directly for the general, including the quaislinear case. We use the notation aij ∂ i u ∂ j v, redeﬁning them for systems and the matrices aij = Aij as aij ∂ i uh , ∂ j v h = (Aij ∂ i uh , ∂ j v h )q . For deﬁning the extensions ah (·, ·), . . . , Gh , as above, we assume aij , f, Ai deﬁned on Ω∪ := Ω ∪ Ωhc for Ai in the sense of (5.5).

(5.380)

We denote the linear and quasilinear cases and the projectors as ah (·, ·), ah (·, ·), Ah , Ah , Gh , Gh , and Q h and Qch ., cf. (5.286), Proposition 5.55. For a uniﬁed notation we use W 1,p (Ω∪ ) = H(Ω∪ ) with A = G, p = 2 for 42 the linear case and D(G) = ∩ni=0 D(Ai ) ⊂ Vb . Then we consider A : H 1 (Ω) = V → H −1 (Ω) = V or Vb → Vb and G : D(G) ⊂ V → V or ⊂ Vb → Vb

42

h h h For simplicity often only f ∈ L2 Ωh c . In this case the deﬁnition (f, v ) := Ωh f (x)v (x)dx c

is used. J(Ec ) (t) denotes the absolute value of the determinant of the Jacobian of (Ec ) (t).

5.5. Consistency, stability and convergence for FEMs with variational crimes A, G

Ub P

U′

tested by

←→

Ub

tested by

U bh ⊄ U

415

Q c′ h

h

h

Φ

U bh

Ah, Gh

Uh′

←→

Figure 5.8 Nonconforming FEMs: Spaces and operators

Ah : H 1 (Ω∪ ) → H −1 (Ω∪ ) : Ah u, vH −1 (Ω∪ )×H 1 (Ω∪ ) := ∀u, v ∈ H 1 (Ω∪ ), ⎛ ⎞ n ⎝ aij ∂ i u ∂ j v ⎠ (x)dx, and ah (u, v) : = Ω∪

i,j=0

ah (uh , v h ) = Gh uh , v h W −1,p ×W 1,p (Ω∪ ) =

n

ah (·, ·) := ah (·, ·)|W −1,p (Ωhc )×Vbh ,

and

(Ai (x, uh , ∇uh ), ∂ i v h )q dx

Ω∪ i=0

a(·, ·) = ah (·, ·)|W −1,p (Ω)×Vb

Q h : H −1 (Ω) → V h : Q h f − fˇ, v h H −1 (Ωhc )×Vbh = (Q h f − fˇ)(x)v h (x)dx for f ∈ L2 (Ω)

∀v h ∈ Vbh (5.381)

Ωh c

h (Q f − f )(t)ˆ v h (t)J(Ec ) (t)dt = 0 and

= Ω

Qch : H −1 Ωhc → V h : Qch fˇ − fˇ, v h H −1 (Ωhc )×Vbh

Qch fˇ − fˇ (x)v h (x)dx = 0 for fˇ ∈ L2 Ωhc =

∀v h ∈ Vbh and

Ωh c

f h , v h := f, v h h := fˇ, v h H −1 (Ωhc )×H01 (Ωhc ) = Q h f, v h H −1 (Ωhc )×Vch and

Ah := Qch Ah |Vbh and Qch Gh |Vbh : Vbh → V h ,

with the

above ah (uh , v h ) = Ah uh , v h V h ×V h and Gh uh , v h V h ×V h . Here the spaces Vbh ⊂ V are deﬁned on Ωhc , and thus cannot satisfy the boundary conditions, cf. Figure 5.8. Theorem 5.83. Summary for isoparametric FEMs for linear and quasilinear second order equations and systems, cf. the following proofs:

1. Let A ∈ L(Vb , Vb ) be a boundedly invertible linear elliptic diﬀerential operator and f ∈ L2 (Ω). Then the exact solution u0 of Au0 = f uniquely exists, with u0 ∈ H 2 (Ω) for convex Ω. For the quasilinear G assume a unique solution u0 ∈ H 2 (Ω) with boundedly invertible G (u0 ). Finally, let Ω ⊂ R2 be bounded and have a piecewise smooth boundary.

416

5. Nonconforming ﬁnite element methods

2. Then the isoparametric FEMs with the equations Ah uh0 = f h and Gh uh0 = 0, see Theorems 5.84 ﬀ. for the necessary conditions, are consistent and stable and (locally) unique discrete solutions uh0 exist and converge to u0 ∈ H d (Ω) u 0 − u 1 (Ω) . ˆh0 H 1 (Ω) ≤ Chd−1 u0 H d (Ω) + u0 W∞ 3. Quadrature approximations as in Theorems 5.80 and 5.81 yield corresponding convergence. 4. For isoparametric FEs the above nonlinear Ec , see (5.379), determines the discretization. However, since (Ec ) − id = O(hd ), see (4.101), the discrete Ah and Gh for the isoparametric case is still k-times consistently diﬀerentiable with A and G in u0 , respectively. This concept will play an essential role for the mesh independence principle and in B¨ ohmer [120] for numerical methods in bifurcation and center manifolds. In a ﬁrst step we prove the discrete coercivity, cf. Theorems 5.59 and 4.61: Theorem 5.84. Isoparametric coercivity: We choose the above Ω ⊂ R2 , the isopara h metric FEs and the ah (·, ·), f, · and Q h . If a(·, ·) is Vb -coercive, see (5.302), then ah (·, ·) is Vbh -coercive. Proof. We present it for the Laplace operator using the substitution rule in h h h ∇uh (x) · ∇v h (x)(dx) a (u , v ) =

(5.382)

Ωh c

=

∇uh (Ec (t)) · ∇v h (Ec (t)) det(Ec ) (t)dt.

(5.383)

Ω

The usual | det(Ec ) (t)| is replaced by det(Ec ) (t), since, with the identity id, (Ec ) (t) = id + O(hd−1 ),

hence, det(Ec ) (t) = 1 + O(hd−1 ) > 0

(5.384)

for suﬃciently small h. For the above u ˆh (t) = uh (Ec (t)) and uh (x) the derivatives ti t xi x ˆh and x ∂ , ∇ and ∂ , ∇ are always understood with respect to the variable t for u h for u , respectively. Then we ﬁnd with the unit vector ei ∂iu ˆh (t) = ∂ ti u ˆh (t) = ∂((uh ◦ Ec )(t))/∂ti = ∇x uh (Ec (t))∂ i Ec (t) = ∇x uh (x)(Ec ) (t)ei , with x = Ec (t) and ∇ˆ uh (t) = ∇t u ˆh (t) = ∇x uh (Ec (t)) (Ec ) (t) = ∇x uh (x)(Ec ) (t) or uh ◦ (Ec )−1 )(x))/∂xi = ∇t u ˆh (t)∂ xi (Ec )−1 (x) ∂ i uh (x) = ∂ xi uh (x) = ∂((ˆ ˆh (t)((Ec )−1 ) (x)ei and = ∇t u ∇uh (x) = ∇x uh (x) = ∇t (ˆ uh ◦ (Ec )−1 (x)) ((Ec )−1 ) (x) ˆh (t)((Ec )−1 ) (x), with t = (Ec )−1 (x). = ∇t u

(5.385)

5.5. Consistency, stability and convergence for FEMs with variational crimes

417

Applied to the above (5.383) we ﬁnd with (4.101) −1 h (Ec ) (t)T ∇ˆ u (t) · ∇ˆ v h (t)((Ec ) (t))−1 det(Ec ) (t)dt ah (uh , v h ) = Ω

uh H 1 (Ω) · ˆ v h H 1 (Ω) . = a(ˆ uh , vˆh ) + O(hd−1 )ˆ

(5.386)

By (5.384), ˆ v h H 1 (Ω) = v h V h (1 + O(hd−1 )). So with (5.386) and Vˆbh ⊂ Vb , the Vb h coercivity of a(ˆ uh , u ˆh ) implies of ah (uh , uh ). Hence, there exists a h the Vb -coercivity h h h h h h unique solution u0 for a u0 , v = (f, v ) ∀ v ∈ Vbh , see (5.381). For the general case in (5.381) we ﬁnd similarly, for smooth enough aij n h h h h h h A u , v V h ×V h = a (u , v ) = aij (x)∂ i uh ∂ j v h (x) b

Ωh c i,j=0

uh H 1 (Ω) ˆ v h H 1 (Ω) . = a(ˆ uh , vˆh ) + O(hd−1 )ˆ

(5.387)

An immediate consequence is sup 0 =v h ∈Vbh

|ah (uh , v h ) − a(ˆ uh , vˆh )| / v h H 1 (Ω) ≤ Chd−1 uH d (Ω) ,

yielding the Vbh -coercivity of ah (uh , v h ) for uh = v h .

(5.388)

As a second step we consider the consistency and convergence, starting with linear h problems. With ah (·, ·), f, · , Q h , Qch as introduced in (5.381), we solve ; : uh0 ∈ Vbh : Ah uh0 , v h V h ×V h = ah uh0 , v h = f, v h h ∀ v h ∈ Vbh . (5.389) b

Theorem 5.85. Isoparametric FEMs for linear problems: 1. For a bounded, piecewise smooth Ω ⊂ Rn , our isoparametric FEs, a boundedly invertible A and for u ∈ H d (Ω), d ≥ 1, let the discrete solution uh for (5.389) with f := Au exist. Then Ah is consistent with A in u, and the classical and variational consistency errors for u vanish of order d − 1. They can be estimated, with C, independent of h, and for h suﬃciently small, by sup 0 =v h ∈Vbh

|ah (uh , v h ) − a(u, vˆh )|/v h H 1 (Ω) ≤ Chd−1 uH 1 (Ω) ,

Ah I h u − Q h AuV h ≤ Chd−1 uH d (Ω) , sup 0 =v h ∈Vbh

(5.390) (5.391)

|(f, v h ) − (fˇ, vˆh )|/v h H 1 (Ω)

≤ Chd−1 f H −1 (Ω) ≤ Chd−1 uH 1 (Ω) .

(5.392)

d 2. For an invertible A, and a stable Ah , let u0 ∈ W∞ (Ω) and uh0 be the exact and discrete solutions of Au0 = f and (5.389), respectively. Then u0 − u 1 (Ω) . ˆh0 H 1 (Ω) ≤ Chd−1 u0 H d (Ω) + u0 W∞

418

5. Nonconforming ﬁnite element methods

Proof. By (5.381) the I h , Q h , Qch are equibounded linear operators with, e.g.

lim Q h f H −1 (Ωhc ) = f H −1 (Ω) .

h→0

Obviously (5.388) implies (5.390). The error term for f is estimated as: h h h h −1 ˇ f (t)ˆ v (t) − f (t)ˆ v (t)dt det((Ec ) ) (t) dt f v dx = f, vˆ − Ωh Ω c v h (t)dt ≤ f (t)(1 − det((Ec )−1 ) (t))ˆ Ω

v h L2 (Ω) ≤ Chd−1 f H −1 (Ω) ˆ ≤ Ch

d−1

(5.393)

uH 1 (Ω) ˆ v L2 (Ω) for f = Au. h

The proof of (5.391) is delayed to the next theorem for the quasilinear case.

The next theorem, some conditions, and its proof are very similar to Theorem 5.4, and e.g. (5.66), however with the there replaced by d here. Theorem 5.86. Isoparametric FEMs for quasilinear problems: 1. We require the conditions for Theorems 4.43, 4.44, 5.84, 5.85, a Lipschitzcontinuous quasilinear operator G with a global constant L, G, Ai ∈ CL (D(G)), i = 0, . . . n.

(5.394)

2. Then for u ∈ H (Ω) ∩ D(G), d > 1, G is consistent with G in I u and Gh I h u − Q h GuV h = Qch Gh I h u − Q h Gu h ≤ CLhd−1 uH d (Ω) . (5.395) d

h

h

VΠ

Π

3. If G(u) = 0 posesses a unique solution u0 ∈ H d (Ω) with boundedly invertible G (u0 ), then the Gh uh0 = 0 are locally uniqely solvable as well and converge according to u0 − u ˆh0 1 ≤ CLhd−1 u0 H d (Ω) . H (Ω)

Proof. We estimate the left-hand side with (5.381), (4.105), and L in (5.394): h (5.396) Qc Gh I h u − Q h Gu h − Qch Gh Ec u − Q h Gu h V V ≤ Qch Gh I h u − Qch Gh Ec u ≤ CLI h u − Ec uV h ≤ CLhd−1 uH d (Ω) . Vh

We test Qch Gh Ec u − Q h Gu with v h , and combine it with (5.379), (5.380), (5.381). For the estimate we need the set of triangles with at most one boundary point,

Tih := T ∈ Tch in Deﬁnition 4.42 1. : Ωb := Ω \ ∪T ∈Tih T ,

Ωhb := Ωhc \ ∪T ∈Tih T .

5.5. Consistency, stability and convergence for FEMs with variational crimes

419

Then we obtain with Gh (Ec u)v h − Gu(v h ◦ Ec ) ≡ 0 in Ωhi := Ωhc ∩ ∪T ∈Tih T = Ωi A B h h h Qc Gh Ec u − Q h Gu, v h = G E uv dx − Guˆ v dx ∀v h ∈ Vbh h c h h V ×V Ωh Ω c h h ≤ Gh (Ec u)v dx − Gu(v ◦ Ec )dx . Ωh Ωb b

(5.397) ∩ Ωb the integrands are Gh (Ec u)v In (5.394). Furthermore, the remaining strips and the integrands are equibounded. So The convergence claim is, for a coercive and (5.395). Ωhb

− Gu(v ◦ Ec ) = O(h ) by (4.101) and Ωhb \ Ωb and Ωb \ Ωhb have a width O(hd ) (5.394), (4.100), (4.101) imply (5.395). G (u0 ), a consequence of Theorem 5.84

h

h

d

6 Adaptive ﬁnite element methods, by W. D¨ orﬂer 6.1

Introduction

The ﬁnite element method (FEM) has become a ﬂexible tool to solve partial diﬀerential equations numerically. It is especially useful in program packages where the user typically adds his geometry. This is comfortable even for the nonexpert. However, such use can be dangerous if no controls exist. For example, in 1991, the oﬀshore oil platform Sleipner broke down and this failure could be traced back to an underestimation of the shear stresses in some part of the structure because of insuﬃcient grid resolution [51], Ch. 1.2. Thus the remedy would be to set up an algorithm that provides suﬃcient reﬁnement in an eﬃcient way and that provides a measure for the quality of the solution. The task of this chapter is to present techniques to overcome these deﬁciencies by a posteriori error estimation and local mesh reﬁnement. The theory of a posteriori error estimation for partial diﬀerential equations came up about 1980 [49, 50]. Combined with methods of local mesh reﬁnement (or coarsening) it is called the adaptive ﬁnite element method (AFEM). Its applicability has now been demonstrated in many circumstances and can be viewed as a standard tool nowadays. These techniques are now widespread, concerning the type of error estimation and mesh reﬁnement techniques. In this short overview we will present the main ideas only and we will further restrict to the Poisson problem with Dirichlet boundary conditions (6.1), (6.2). This equation can be seen as a simpliﬁed model of linear elasticity which played a role in the example mentioned above (Section 6.1.1). We explain that solutions of such equations may exhibit singular behavior and the consequences for the ﬁnite element method (Section 6.1.2). A heuristic argument shows that local mesh reﬁnement can overcome this problem (Section 6.1.4). We develop the theory of a posteriori error estimation with the residual error estimator (Section 6.2). This states that the adaptive algorithm can be designed to converge as an iterative method (Section 6.2.6) and that it leads in fact to the optimal grid in a certain sense (Section 6.2.7). Only brieﬂy will we mention some alternatives to the residual error estimator for estimating the energy norm in Section 6.2.8 and refer to the literature. In the ﬁnal Section 6.3 we report on the methodology of estimating errors in quantities of interest. Finally we mention a few books and overview articles on all these methods such as, for example, [6, 51, 77, 499, 655].

6.1. Introduction

6.1.1

421

The model problem

Let Ω ⊂ R3 be the domain occupied by an elastic body. The continuum mechanical model yields the equations 1

−∇· C ∇u + (∇u)d = f in Ω 2 for the deformation ﬁeld u : Ω → R3 and the elasticity tensor C. For simpliﬁcation we study the equation −∇·(a∇u) = f

in Ω

(6.1)

¯ → R for given a : Ω ¯ → R+ and f : Ω ¯ → R. To state the boundary to obtain u : Ω conditions we assume a disjoint decomposition of ∂Ω ∂Ω = ∂D Ω ∪ ∂N Ω and we require the solution of (6.1) to satisfy u∂ Ω = g D , a∇u · n∂ D

6.1.2

NΩ

= gN .

(6.2)

Singular solutions

Let us consider the boundary value problem (6.1), (6.2). Our question is: are solutions ¯ as the formulation would require? In general the answer is of this equation in C 2 (Ω) no, if ∂Ω is not smooth. Corner singularities in 2D A point x at the boundary ∂Ω is called a corner point if the boundary arc is continuous and one-sided diﬀerentiable in x, but not diﬀerentiable. It can be shown that such a situation can be locally described by that of a solution in a suitable angular domain with angle θ and tip at 0. Special solutions for the case a = 1, f = 0 are of the form φ , r := |x|, φ = arg(x), u(r, φ) = rγ sin mπ θ in the case of homogeneous Dirichlet boundary values for φ = 0 and φ = θ. Insertion into −Δu = 0 yields 2 φ 2 2π γ−2 =0 −γ + m 2 r sin mπ θ θ for all r ∈ (0, ∞) and φ ∈ (0, θ). This equation for γ is solved by γ 2 = m2

π2 . θ2

The smallest positive solution for γ is obtained for m = 1: φ π/θ . sin π u(r, φ) = r θ

(6.3)

422

6. Adaptive ﬁnite element methods

For γ < 0, the solution would not satisfy our least requirement that u has ﬁnite energy, i.e. Ω |∇u|2 exists. In the case of homogeneous Neumann conditions at {φ = 0} or {φ = 0, π} we have to take πφ φ γ γ u(r, φ) = r cos π , u(r, φ) = r sin 2θ θ respectively. Note that we would get in the ﬁrst case πφ . u(r, φ) = rπ/(2θ) sin 2θ For the slit domain (θ = 2π), this yields the worst cases u(r, φ) = r1/2

or

u(r, φ) = r1/4 .

We will study the regularity of these solutions in Sobolev function spaces. Recall the deﬁnitions in Section 1.4.3:

1/p m,p (Ω) := v : Ω → R : ||v||W m,p (Ω) := |∇α v|p <∞ , W |α|≤m

Ω

H m (Ω) := W m,2 (Ω). Moreover, recall that this deﬁnition can extended to m ∈ R, cf. Theorem 1.30 and [668]. We may consider u as above on the sector Ω = {(r, φ) : r ∈ (0, 1), φ ∈ (0, θ)}. For u that behaves like rγ near 0 we ﬁnd that 1 1 ! ||u||pW m,p (Ω) ≤ C |rγ−m |p r dr = C r(γ−m)p+1 dr < ∞ 0

0

holds if (γ − m)p + 1 > −1 ⇔ γ > m − p2 . For u ∈ H m (Ω) we need the bound m < 1 + γ = 1 + π/θ, respectively 1 + π/(2θ). Solutions of Dirichlet boundary value problems are thus at least in H 3/2 (Ω); for mixed Dirichlet and Neumann conditions at least in H 5/4 (Ω). If we consider the existence in W 2,p spaces, this allows arbitrary p ∈ [1, ∞] 2 for γ ∈ (0, 2). For γ < 1 we will have W 2,p regularity for some for γ ≥ 2 but p < 2−γ p ∈ [1, 2). Especially, u ∈ W 2,1 (Ω) holds in general. Only for γ ≥ 1 do we have H 2 regularity. The general structure of singular solutions u of elliptic partial diﬀerential equations is as follows. Given m ≥ 2, there exists nm ∈ N, smooth functions wj that are locally supported near the singular point, e.g. 0, of ∂Ω, and singular functions ηj (often ηj (x) = |x|αj for some αj ∈ R) such that u(x) =

nm j=1

for w ∈ W m,2 (Ω) [255, 372, 446, 455].

wj (x)ηj (x) + w(x)

6.1. Introduction

423

Corner and edge singularities in 3D A well studied example of a corner point in three dimensions is the Fichera corner in 0 for Ω := (−1, 1)3 \ [0, 1)3 . The solution of Δu = 0 near 0 behaves like x → |x|γ for γ = 0.45 . . . . The problem of ﬁnding γ leads to a quadratic eigenvalue problem (λ2 A + λB + C)u = 0 for λ ∈ R, u = 0 [35]. Here A, B, C are suitable operators. Besides the corners, there are additional singular solutions along the edges [212]. Problems with jumping coeﬃcients We consider Equation (6.1) for a that is piecewise constant in the four quadrants of the coordinate system with values a1 , . . . , a4 . This problem could appear for ﬂow in porous media where large jumps in the permeability a may in fact occur. Again we try an ansatz of the form u(r, φ) = rγ s(φ) to get in each quadrant a(s − γ 2 s)rγ−2 = 0, !

that is, the following eigenvalue problem for γ π π

for j = 0, 1, 2, 3. s = γ 2 s in j , (j + 1) 4 4 The solution must have a continuous ﬂux (a∇u) · n along the normal n of the interfaces, thus we can require periodicity of s and aj s j π4 − = aj+1 s j π4 + for j = 0, 1, 2, 3 [533]. In the case ofFa1 = a3 = 1 and a2 = a4 ≈ 100 one gets a value γ = 0.1. Thus u(x) behaves like 10 |x| near 0! Moreover, u ∈ H 1.1 (Ω) [444, 498]. Boundary layers We consider the case of, cf. [559] −εΔu + b · ∇u + cu = f u=0

in Ω, on ∂Ω,

with constant coeﬃcients ε, b, c, where b = 0 or c = 0 and 0 < ε |b| + |c|. For example, the typical behavior of solutions near outﬂow points of ∂Ω (i.e. where b · n > 0) is |∇l u| ∼ 1/εl . 43 At other parts of the boundary, a weaker dependence on 1/ε will occur. Although u might be smooth, it appears as a nonsmooth function unless the discretization method resolves the scale ε, which will lead to prohibitive work on uniform meshes. 6.1.3

A priori error bounds

In the following, we will simplify (6.1) further by assuming a = 1. The weak form of −∇·∇u0 = −Δu0 = f u=0 43

in Ω, on ∂Ω

a ∼ b :⇔ ∃c1 , c2 : 0 < c1 < c2 such that c1 a ≤ b ≤ c2 .

424

6. Adaptive ﬁnite element methods

is given by

∇u0 · ∇v = Ω

Ω

for all v ∈ C0∞ (Ω).

fv

(6.4)

As previously in this book, we denote solutions of speciﬁc problems, e.g. of (6.4) and (6.5), as u0 and uh0 , respectively For given f ∈ L2 (Ω), there exists a solution u0 ∈ H01 (Ω) of this problem (cf. Theorem 2.2 and Theorem 2.43 in Chapter 2). By density of C0∞ (Ω) in H01 (Ω) we can equally well state “for all v ∈ H01 (Ω)” in (6.4). Let V h be a ﬁnite dimensional subspace of V := H01 (Ω). Then the Galerkin approximation uh0 ∈ V h is deﬁned by ∇uh0 · ∇v h = f v h for all v h ∈ V h . (6.5) Ω

Ω

It is well known that both problems are uniquely solvable and that (Cea’s lemma, cf. Lemma 4.48 in Section 4.3) ||∇ u0 − uh0 ||L2 (Ω) ≤ C inf ||∇(u0 − v h )||L2 (Ω) . (6.6) v h ∈V h

If we have a continuous mapping P h : H01 (Ω) → V h , we can further estimate the righthand side of (6.6) to get ||∇ u0 − uh0 ||L2 (Ω) ≤ C||∇ u0 − P h u0 ||L2 (Ω) . Let V h be a ﬁnite element space that at least contains the linear ﬁnite elements. Then there exists a continuous interpolation operator P h : H01 (Ω) −→ V h with the property ||∇(v − P h v)||L2 (Ω) ≤ Chmax ||∇2 v||L2 (Ω) for all v ∈ H 2 (Ω). Here, hmax is the maximal diameter of the mesh cells in the underlying discretization. This is a special case of Theorem 4.17 in Section 4.2 of Chapter 4. However, u ∈ H 2 (Ω) is not always true as we have seen in Section 6.1.2. In fact, we get u ∈ H 1+s (Ω) for s > 0 only. Theorem 6.1. Interpolation operator: Let V h be a ﬁnite element space that at least contains the linear ﬁnite elements. Then there exists a continuous interpolation operator P h : H01 (Ω) −→ V h with the property ||∇(v − P h v)||L2 (Ω) ≤ Chsmax ||∇v||H s (Ω)

for all v ∈ H 1+s (Ω)

and suitable s ∈ (0, 1]. Here, hmax is the maximal diameter of the mesh cells in the underlying discretization and C depends on properties of the mesh [141], Ch. 14.3. This leads to the following error bound. Theorem 6.2. A priori error bound: The error for the ﬁnite element solution of (6.5) on V h as in Theorem 6.1 obeys the a priori error bound ||∇ u0 − uh0 ||L2 (Ω) ≤ Chsmax ||∇u0 ||H s (Ω)

6.1. Introduction

425

for some s ∈ (0, 1]. Here, hmax is the maximal diameter of the mesh cells in the underlying discretization and C depends on properties of the mesh (cf. Section 6.1.7). The meaning of this result is that a lack of regularity (smaller s) leads to a low convergence rate for the error in terms of the maximal mesh size in the underlying discretization. Or, in other words, for a ﬁxed mesh, there is a signiﬁcant loss in accuracy if one approximates less regular functions. 6.1.4

Necessity of nonuniform mesh reﬁnement

As we have observed in Section 6.1.3, there are cases where solutions of (6.4) are in H 1+s (Ω) for s ∈ (0, 1]. By Theorem 6.2 the H 1 error behaves like hs on, say, a uniform mesh of mesh size h. In this case we have N h = O(1/hn ) unknowns for linear ﬁnite elements in a bounded domain in Rn . In terms of N h the H 1 error thus behaves like hs ∼ (N h )−s/n . We discuss this, as an example, in the case n = 2.

r Ω ⊂ R2 and s = 1 (which is the full regular case). Now the error behaves like (N h )−1/2 and to achieve a prescribed tolerance TOL, one needs (N h )−1/2 ∼ TOL, or, N h ∼ TOL−2 . Thus, if we start with an error of size 1 and if we want to reduce the error by a factor of 100, we have to increase N h by a factor of 104 . r Ω ⊂ R2 and s = 1/2 (the case of the “slitted domain” Ω = B1 (0) \ {x ∈ R2 : x1 < 0, x2 = 0}). Now the error decreases like (N h )−1/4 and to achieve a prescribed tolerance TOL we need N h ∼ TOL−4 unknowns. That means, in the same situation as above we now need about 108 unknowns!

6.1.5

Optimal meshes – A heuristic argument

(See [78].) Let Ω ⊂ Rn be bounded and let Ω be decomposed into simplices or rectangles. We assume that we are given a mesh-size function h : Ω −→ R+ with h(x) describing the local mesh size near x (i.e. x ∈ T ⇒ h(x) ≈ diam(T )). Let the function σ denote a geometry dependent function, σ : Ω −→ R+

with σ(xT ) =

h(xT )n . vol(T )

For linear ﬁnite elements we have approximately one unknown per simplex. Therefore the total number of unknowns N (h) for the linear ﬁnite element space is approximately given by σ(xT ) vol(T ) σ(x) = 1= vol(T ) ≈ dx. (6.7) N (h) ≈ n n vol(T ) h(x ) h(x) T Ω h h T

T ∈T

T ∈T

Thus we deﬁne for a given mesh-size function h the number of unknowns as σ N (h) := . n Ω h Assume that we have a representation of the local error in the form |∇ u0 − uh0 (x)| ≈ h(x)α E(x),

(6.8)

426

6. Adaptive ﬁnite element methods

where α is the convergence order and E does not depend on h. For linear ﬁnite elements and u0 ∈ H 2 (Ω) we have for example α = 1, E(x) = |∇2 u0 (x)|. If the solution u0 , we 2 (Ω), then the global error E(h) is are trying to approximate, is in Hloc

12 E(h) := (hα E)2 . Ω

We further assume that the total work to establish the approximate solution is proportional to N (h) (e.g. use a multigrid method cf. [141]). Our aim is to ﬁnd a meshsize function h that gets an error of TOL with lowest possible number of unknowns. In other words, we have to ﬁnd a mesh-size function h such that N (h) is minimal while E(h) = TOL. This is a constraint optimization problem! To solve it we introduce the Lagrange function (6.9) L(h, λ) = N (h) − λ TOL2 −E(h)2

σ = (6.10) − λ TOL2 − h2α E 2 . n Ω h Ω The point (h, λ) we are looking for is a stationary point of L. Thus for all φ ∈ C ∞ (Ω) we get nσ − n+1 + 2αλh2α−1 E 2 φ. 0 = ∂1 L(h, λ)[φ] = h Ω This requires the pointwise condition −nσ/hn+1 + 2αλh2α−1 E 2 = 0 and leads to n σ h2α E 2 = in Ω. 2αλ hn To satisfy the constraint n n σ N (h) h2α E 2 = = TOL2 = E(h)2 = n 2αλ Ω Ω 2αλ h λ should be taken as λ=

n N (h) . 2α TOL2

Insertion into the equation for h yields h2α E 2 =

TOL2 σ N (h) hn

or

1 n+2α 2 TOL2 h . E = σ N (h)

(6.11)

The left-hand side depends on x, but the right-hand side is independent of x. Hence our result is to distribute hn+2α E 2 /σ equally on the mesh. For any Tin the discretization, let ET2 (h) := T h2α E 2 be the local error on T and NT (h) := T σ/hn be the local number of degrees of freedom in T . Then we have by (6.11) TOL2 TOL2 σ 2 2α 2 NT (h). ET (h) = h E = = N (h) T hn N (h) T

6.1. Introduction

427

In our setting, so far, NT is a constant depending on the ﬁnite element method. Thus our result is that we have to equidistribute the error over the decomposition of Ω. From (6.11) we get the optimal mesh size h∗ and if Ω |E|2n/(n+2α) < ∞, we obtain from this N (h∗ ) ∼ TOL−n/α . This recovers the result we would get for the ﬁnite element method for the full regular problem (see Section 6.1.4)! 6.1.6

Optimal meshes for 2D corner singularities

We apply the previous result to the case of a typical corner singularity in two space dimensions for −Δu = 0 as in Section 6.1.2. For simplicity, we assume that the corner point is located at 0 and that Ω is near 0 a sector of the unit circle with angle θ = βπ, β ∈ (1, 2). Then the solution (6.3) behaves like |∇l u0 (x)| ∼ r1/β−l for r = |x| ∈ (0, 1), l ≥ 0. Let us assume a discretization with a uniform mesh size h(x) := h0 for all x ∈ Ω. In Ω ∩ B h (0) the error is given by ||∇ u0 − uh0 ||2L2 (Bh

0

(0))

≈ ||∇u0 ||2L2 (Bh

0

(0))

h0

∼

2

2

r β −2 r dr ∼ h0β ,

0

while in the remaining domain, by (6.8), ||∇ u0 − uh0 ||2L2 (Ω\Bh

(0)) 0

∼ ||h0 ∇2 u0 ||2L2 (Ω\Bh

(0)) 0

1

∼ h20

2

2

r β −3 dr ∼ h20 h0β

−2

2

∼ h0β .

h0

1 Thus ||∇ u0 − uh0 ||L2 (Ω) ∼ h0β and the requirement

! ||∇ u0 − uh0 ||L2 (Ω) ∼ E(h0 ) = TOL leads to the relation, cf. (6.7), N (h0 ) ∼ h−2 0 =

1

h0β

−2β

= TOL−2β .

We already noticed this relation for β = 2 in Section 6.1.4. But, what does the optimal mesh-size distribution h look like, and what can we achieve with it? Let us assume that we established the optimal mesh-size distribution. We observed in (6.11) that hn+2α |∇2 u0 |2 /σ = TOL2 /N (h) =: K. For a radial symmetric mesh-size distribution it follows from this that (recall n = 2, α = 1) 2

h(r)4 r β −4 ∼ K

or

1

1

h(r) ∼ K 4 r1− 2β .

(6.12)

428

6. Adaptive ﬁnite element methods

The constants in this estimate depend on σ, but we assume σ ∼ 1. So the error for such a choice of h will be 1 1 1 2 1 1 1 h2 |∇2 u|2 ∼ K 2 r2− β r β −4 rdr = K 2 r β −1 dr (6.13) TOL2 = Ω

0

0 1

∼ K2 =

TOL 1

N (h) 2

,

(6.14)

and so we ﬁnally achieve N (h) ∼ TOL−2 , independent of β. We thus recover the complexity for the full regular case as stated in Section 6.1.4 (these statements are only valid in the sense that they show the same asymptotic behavior for large N (h); the constants have not been considered). Compared with the second example in Section 6.1.4, we notice that the number of unknowns on the optimal mesh is a factor TOL2 less than on a uniform mesh, that is a factor 10−4 for TOL = 0.01 for this two-dimensional problem. 6.1.7

The ﬁnite element method–Notation and requirements

For the reader’s convenience we summarize some deﬁnitions for the conforming ﬁnite element method for Equation (6.4) for simplicial discretizations, cf. Section 4.2. Let Ω ⊂ Rn , an open bounded set and polygonally bounded, be decomposed into a set of simplices T h , so that C Ω= T. T ∈T h

In this chapter, simplices are assumed to be closed sets, T ◦ ∩ D◦ = ∅ for all T, D ∈ T h , T = D and each vertex of T is also a vertex of a neighbor of T . T h is then called a regular discretization of Ω (or triangulation if n = 2). For T ∈ T h let hT := diam(T ) and let h : Ω → R+ deﬁned by hT = hT be the mesh-size function. Furthermore let ρT be the radius of the largest inscribed ball in T , σ h the piecewise constant function with σT := hT /ρT ≥ 1, the shape factor or chunkiness parameter of T . Let N h be the set of all vertices, E h the set of all edges. Boundary vertices and edges are N h,∂ and E h,∂ , respectively, and for the interior sets we deﬁne E h,◦ := E h \ E h,∂ and N h,◦ := N h \ N h,∂ . The space of piecewise polynomials of degree p ∈ N is Pp (T h ) := v : Ω → R : v T ∈ Pp for all T ∈ T h . The space of conforming ﬁnite elements of degree p is Sph := Pp (T h ) ∩ C 0 (Ω) ⊂ H 1 (Ω).

429

6.1. Introduction

In the following we will work with V := H01 (Ω) and its subspace V h := Sph ∩ H01 (Ω) = v ∈ Sph : v ∂Ω = 0 .

(6.15)

For each z ∈ N h,◦ , there is a unique function φz ∈ S1h such that φz (z ) = δzz

for all z ∈ N h,◦ , z ∈ N h .

Each v h ∈ S1h , the space of linear conforming ﬁnite elements, can be written as vh = αz φz . z∈N h,◦

Inserting such an ansatz into the equation ∇uh0 · ∇v h = f h vh Ω

for all v h ∈ V h

Ω

yields a linear system of equations A u = b

with Azz := Ω ∇φz · ∇φz , bz := Ω f h φz for z, z ∈ N h,◦ and the vector u = [αz ]z∈N h,◦ of the representation uh0 = αz φz . z∈N h,◦

In contrast to the interpolation estimates cited before in Theorem 6.1 the following are stated for nonuniform meshes. Theorem 6.3. Interpolation operator: There exists a continuous interpolation operator P h : L2 (Ω) → Sph that has for all v ∈ H m (Ω) with m = 0, 1, . . . , p + 1 and l = 0, 1, . . . , min(p, m) the following properties: ||∇l P h v||L2 (Ω) ≤ C||∇l v||L2 (Ω)

(6.16)

and ||∇l (v − P h v)||L2 (T ) ≤ Chm−l ||∇m v||L2 (ST ) T l− 12

||v − P h v||L2 (E) ≤ ChE ||∇l v||L2 (SE )

for all T ∈ T h ,

(6.17)

for all E ∈ E h ,

(6.18)

where ST or SE are the domains composed of all triangles that touch T ∈ T h or E ∈ E h , respectively. If v ∈ H01 (Ω), then P h v ∈ H01 (Ω). The constants only depend on m, p and σT for T ⊂ ST , respectively T ⊂ SE . Proof. See [141], Ch. 4.8 or Theorem 4.17 in Section 4.2 of Chapter 4 and 7.7, Corollary 7.8 in Section 7.7 of Chapter 7.

430

6. Adaptive ﬁnite element methods

6.2

The residual error estimator for the Poisson problem

6.2.1

Upper a posteriori bound

Deﬁnition 6.4. Jumps and residuals: 1. Let E ∈ E h,◦ . The edge jump [v]E of a piecewise continuous function v on T h along the edge E is deﬁned by [v]E (x) := lim {v(x + snE ) − v(x − snE )}

for x ∈ E,

s0

where nE is a ﬁxed unit normal vector on E. 2. Let uh0 be the solution of the discrete equation (6.5) in V h as in (6.15). Deﬁne the element residual on T ∈ T h

RT := Δuh0 + f h and the edge residual RE := [∂n uh0 ]E

on E ⊂ E h,◦ .

Note that this deﬁnition does not depend on the choice of nE . Theorem 6.5. Upper a posteriori error estimate: Let u0 , uh0 be the solutions of the continuous equation (6.4) in V = H01 (Ω) and the discrete equation (6.5) in V h as in (6.15), respectively. Then there are sets of positive constants {cT }T ∈T h , {cE }E∈E h,◦ such that cT h2T ||RT ||2L2 (T ) + cE hE ||RE ||2L2 (E) ||∇ u0 − uh0 ||2L2 (Ω) ≤ T ∈T h

E∈E h,◦

+CP (Ω)||f −

f h ||2L2 (Ω) .

cT , cE depend on σ h (the shape constant) and p (the polynomial degree of the ﬁnite element space). CP (Ω) is Poincar´e’s constant for Ω (cf. (1.57) and the residuals RT , RE have been deﬁned in Deﬁnition 6.4. Proof. Let u0 , uh0 be the respective solutions and let v ∈ V, v h ∈ V h be arbitrary. Then by (6.4) and (6.5) and partial integration ∇ u0 − uh0 · ∇v Ω

f v − ∇uh0 · ∇(v − v h ) − f h v h

=

Ω

(f − f h )v + f h (v − v h ) − ∇uh0 · ∇(v − v h )

= Ω

(f − f h )v +

= Ω

(f − f )v + h

Ω

T ∈T

h

f h + Δuh0 (v − v h ) −

T

T ∈T h

=

T

∂ h uh0 (v − v h ) ∂T

h

f +

Δuh0

(v − v ) − h

E∈E h,◦

E

[∂ h uh0 ]E (v − v h ). (6.19)

6.2. The residual error estimator for the Poisson problem

431

Equation (6.19) is called the error representation formula. Now we choose v h = P h v with P h as in Theorem 6.3 and use the interpolation estimates to get (note that cT , cE will change from estimate to estimate) ∇ u0 − uh0 · ∇v ≤ cT hT ||RT ||L2 (T ) ||∇v||L2 (ST ) Ω

T ∈T h

+

1/2

cE hE ||RE ||L2 (E) ||∇v||L2 (SE ) + ||f − f h ||L2 (Ω) ||v||L2 (Ω)

E∈E h,◦

⎛ ≤⎝

T ∈T

c2T h2T ||RT ||2L2 (T ) +

h

⎞1/2 c2E hE ||RE ||2L2 (E)⎠ ||∇v||L2 (Ω)

E∈E h,◦

+ CP (Ω)||f − f h ||L2 (Ω) ||∇v||L2 (Ω) . Here we used the Cauchy–Schwarz inequality for sums and that ∪E∈E h,◦ SE and ∪T ∈T h ST give a ﬁnite coverage of Ω with a constant that only depends on σ h . Since v is arbitrary, the bound will follow dividing by ||∇v||L2 (Ω) and taking the supremum over all v ∈ V. Remark 6.6. 1. This estimate in Theorem 6.5 is in principle computable. However, the constants cT , cE have to be estimated. There are some attempts to get explicit or tight bounds, e.g. [164]. 2. The estimate in Theorem 6.5 shows the expected dependence on h for smooth u0 , since ⎞1/2 ⎛ ⎝ h2T ||RT ||2L2 (T ) ⎠ = O(hmax ) T ∈T h

and ⎛ ⎝

⎞1/2 hE ||RE ||2L2 (E) ⎠

= O(hmax )

E∈E h,◦

because of [∂n uh0 ]E ≈ ∂n2 u0 hE and 2 & % hE || ∂n uh0 E ||2L2 (E) ∼ h4E ∂n2 u0 ∼ h2T ||∇2 u0 ||2L2 (T ) for T with E ⊂ ∂T \ ∂Ω. Note that E∈E h,◦ h2T ∼ T ∈T h |T | = |Ω|. 3. Note that the estimate in Theorem 6.5 does not require any higher regularity than u0 ∈ V. Thus it covers also the cases when u0 ∈ H 2 (Ω), like in Section 6.1.2.

432

6. Adaptive ﬁnite element methods

6.2.2

Lower a posteriori bound

Deﬁnition 6.7. Bubble functions: If φ1 , φ2 , φ3 are the linear basis functions at the vertices of a triangle T , then βT := φ1 φ2 φ3 /h3T deﬁnes a cubic function on T called an element bubble. Note that βT can be extended by 0 to a function in V with support in T . For each edge E = T1 ∩ T2 = ∅ (T1 , T2 ∈ T h ), say E = z1 z2 for some z1 , z2 ∈ N h , we deﬁne a piecewise quadratic polynomial θE := φz1 φz2 /h2E on ωE := T1 ∪ T2 called an edge bubble. Note that θE is continuous on E and it can be extended by 0 to a function in V with support in ωE . The following theorem gives a lower bound for the actual error of uh0 . Theorem 6.8. Lower a posteriori error estimate: Let u0 , uh0 be as in Theorem 6.5. Then there is a constant c that depends only on σ h and p, such that for each T ∈ T h and for RT , RE and ωE as in Deﬁnitions 6.4 and 6.7, respectively, 2 2 2 c cE hE ||RE ||L2 (E) + cT hT ||RT ||L2 (E) h

T ⊂ωE

≤ ||∇ u0 − u0 ||2L2 (ωT ) + ||h(f − f h )||2L2 (ωT ) . The constants cT , cE are as in Theorem 6.5. Proof. Let E ⊂ E h,◦ and T be any of the adjacent triangles, and let v := βT RT . Thus v ∈ V and the error representation formula (6.19) specializes to ∇ u0 − uh0 · ∇(βT RT ) = T βT RT2 + (f − f h )βT RT . T

Since RT and βT RT are polynomials, there is a constant c, depending on σT and p, such that 1/2

||RT ||L2 (T ) ≤ C||βT RT ||L2 (T ) and furthermore we have the inverse estimate −1 ||∇(βT RT )||L2 (T ) ≤ Ch−1 T ||βT RT ||L2 (T ) ≤ ChT ||RT ||L2 (T ) .

Hence we can derive the bound ||RT ||2L2 (T ) ≤ C βT RT2 = C {∇ u0 − uh0 · ∇(βT RT ) + (f − f h )βT RT } T

h

T

≤ C ||∇ u0 − u0 ||L2 (T ) ||∇(βT RT )||L2 (T ) + ||f − f h ||L2 (T ) ||RT ||L2 (T )

h h || ||∇ u − u + ||f − f || ≤ C h−1 2 2 0 0 L (T ) L (T ) ||RT ||L2 (T ) T so that

hT ||RT ||L2 (T ) ≤ C ||∇ u0 − uh0 ||L2 (T ) + ||h(f − f h )||L2 (T ) .

6.2. The residual error estimator for the Poisson problem

433

For the edge term there is a similar trick: we extend RE onto R2 by a constant extension normal to E [654] and deﬁne as a test function v := θE RE . Clearly, v ∈ V and the error representation yields h 2 ∇ u0 − u0 · ∇(θE RE ) = θ E RE + RT θE RE + (f − f h )θE RE . ωE

E

T ⊂ωE

T

ωE

Similar to the above, the inequality 2 |RE |2 ≤ C θ E RE E

E

≤ C ||∇ u0 − uh0 ||L2 (ωE ) ||∇(RE θE )||L2 (ωE )

+

1/2 ||RT ||2L2 (T )

T ⊂ωE

≤C

h−1 E ||∇

u0 −

uh0

||θE RE ||L2 (ωE ) + ||f − f h ||L2 (ωE ) ||θE RE ||L2 (ωE )

||L2 (ωE ) +

1/2 ||RT ||2L2 (T )

T ⊂ωE

+||f − f h ||L2 (ωE ) ||RE ||L2 (ωE ) 1/2

leads, with ||RE ||L2 (ωE ) ≤ ChE ||RE ||L2 (E) (recall the deﬁnition of RE on ωE ), to

1/2

hE

1/2

|RE |2 E

⎛

h

≤ C ⎝||∇ u0 − u0 ||L2 (ωE ) +

⎞

1/2 h2T ||RT ||2L2 (T )

T ⊂ωE

+ ||h(f − f h )||L2 (ωE ) ⎠ .

Now observe that the second term on the right has already been estimated in the ﬁrst step of the proof in terms of the other two terms. This yields the result. 6.2.3

The a posteriori error estimate

We summarize the results of the last two sections in the following theorem. Theorem 6.9. A posteriori error estimate: Let u0 , uh0 be the solutions of the continuous equation (6.4) in V = H01 (Ω) and the discrete equation (6.5) in V h as in (6.15), respectively. For each E ∈ E h,◦ we deﬁne the local error indicator 2 ηE := cE hE ||RE ||2L2 (E) + cT h2T ||RT ||2L2 (E) (6.20) T ⊂ωE

with constants cT , cE from Theorem 6.5, and residuals RT , RE from Deﬁnition 6.4. Then there are constants c, C, both depending only on σ h and the polynomial degree p, such that

434

6. Adaptive ﬁnite element methods

⎛ ceﬀ ⎝

⎞1/2

2⎠ ηE

≤ ||∇ u0 − uh0 ||L2 (Ω) + ||h(f − f h )||L2 (Ω)

E∈E h,◦

and

h

⎛

||∇ u0 − u0 ||L2 (Ω) ≤ ⎝

⎞1/2 2⎠ ηE

+ CP (Ω)||f − f h ||L2 (Ω) .

E∈E h,◦

Proof. Use Theorem 6.5 and Theorem 6.8 (sum over E ∈ E h,◦ ). Remark 6.10. A posteriori error estimate:

1. The upper bounds were ﬁrst proved in [49, 50], the lower bound ﬁrst appeared in [653]. 2. Usually, f h is not givenexplicitly, but is implicitly deﬁned by using quadrature rules for the integrals T f ϕ, where ϕ is a basis function. In general, this quadrature rule should be as cheap as possible but at least converge as fast as the error or one degree higher (e.g. for linear ﬁnite elements choose a quadrature rule of order 1 or 2). This same approximation f h to f should also be taken into account when we compute RT and ||f − f h ||L2 (T ) . 3. The edge residuals RE can be avoided in one dimension since v − v h in the proof of Theorem 6.5 can be chosen to vanish in the vertices. The jump residuals RE are unavoidable and even dominant in more than one dimension [165]. This is, for example, easy to see if p = 1 and f = 0 (for nonhomogeneous Dirichlet data). Note that globally Δuh0 is not a function but a distribution. Its regular part is zero, its singular part is [∂n uh0 ]E . For ϕ ∈ C0∞ (ωE ) and for some E ∈ E h,◦ we obtain h h h h −Δu0 [ϕ] = ∇u0 · ∇ϕ = [∂n u0 ]E ϕ − Δu0 ϕ = RE ϕ. ωE

E

T1 ∪T2

E

4. The upper bound Theorem 6.5 is global while the lower bound Theorem 6.8 is local. The upper bound cannot be local since a local error source will have global inﬂuence. Only a part of the error can be judged by a local interpolation error and local data error, while other parts of the error have its source somewhere else. This part is also called pollution error. 5. Such residual error estimates can also be derived for nonlinear equations. For this see [655]. 6.2.4

The adaptive ﬁnite element method

The macro discretization We need a ﬁrst decomposition of Ω into simplices that resolves the geometric details. This is often a very diﬃcult problem, especially in 3D. A major diﬃculty is to avoid simplices T with large shape constant σT .

6.2. The residual error estimator for the Poisson problem

435

Initial discretization The macro-discretization may not resolve the data required by the problem. In order to judge and improve data resolution we deﬁne a data error estimator (take ||f − f h ||L2 (Ω) and perform mesh reﬁnement until a certain coarse tolerance is met), e.g. ⎛ ⎞1/2 δT2 ⎠ with δT2 := ||f − f h ||2L2 (T ) . δT h := ⎝ T ∈T h

The adaptive iteration 1. Solve for uh0 (e.g. by a multigrid method). We assume in the following that uh0 is the exact ﬁnite element solution. 2. Estimate: Compute ηE as in (6.20) for all E ∈ E h,◦ . Stop the iteration if ⎞1/2 ⎛ 2⎠ ηE ≤ TOL ηT h := ⎝ E∈E h,◦

is satisﬁed (or use TOL ||∇uh0 ||L2 (Ω) on the right-hand side in the case of relative errors). 3. Mark a set of edges F h ⊂ E h,◦ for reﬁnement. Possible strategies are:

r Maximum strategy. Let

E ∈ F h ⇐⇒ ηE ≥ (1 − θ)ηmax with ηmax := maxE∈E h,◦ ηE and some θ ∈ [0, 1]. Note that we get F h ≈ ∅ for θ ≈ 0 and F h ≈ E h,◦ for θ ≈ 1. r Fixed energy fraction strategy. For θ ∈ [0, 1] seek a minimal set F h ⊂ E h,◦ such that 2 ηE ≥ θ2 ηT2 h . (6.21) E∈F h

Note that we get F h ≈ ∅ for θ ≈ 0 and F h ≈ E h,◦ for θ ≈ 1. 4. Mesh reﬁnement. Aim: create a new regular mesh from T h and the set of marked triangles Ah := {T ∈ T h : T ⊂ ωE , E ∈ F h }.

(6.22)

r 1D problems. Interval bisection: bisect T ∈ Ah in its center. r 2D problems. A reﬁnement of one triangle introduces a hanging node (a vertex that is not shared with a neighboring triangle [655], Sect. 4.1) and enforces the reﬁnement of neighboring triangles if we have to work on regular meshes. On the other hand, our reﬁning algorithm has to stay local. Arbitrary bisection of triangles may lead to very small angles over several reﬁnement steps and as a consequence the shape constant will grow. We will study this topic in Section 6.2.5.

436

6. Adaptive ﬁnite element methods

Figure 6.1 Left: Red reﬁnement. Right: Blue reﬁnement of the right triangle.

Remark : If we are bound to rectangular decompositions we cannot avoid working with hanging nodes. r 3D problems. The principal problems are the same as in 2D but solutions to it are more involved. Now go back to 1. 6.2.5

Stable reﬁnement methods for triangulations in R2

The aim of this section is to describe some methods that return a (reﬁned) regular mesh for a given regular mesh T h and a set Ah of marked triangles to be reﬁned. Such a method is called geometrically stable if repeated application of the method leads to a sequence of meshes such that the corresponding sequence of shape constants is uniformly bounded. The following methods will fulﬁll this requirement. Red–green (–blue) reﬁnement [655], Sect. 4.1 In case T ∈ Ah we will use red reﬁnement as shown in Figure 6.1, left. Note that all new triangles have the same angles as their original triangle. For the neighbors that are not in Ah , we enforce bisection to remove the hanging node, regardless of the size of the new angles. These so-called green reﬁnements, Figure 6.1, right, will be coarsened before such a triangle will be reﬁned in a subsequent step. If the opposite angle that is divided by bisection is very small, one might use bisection methods that avoid these (like the one in the next subsection). In this context, this reﬁnement is called blue reﬁnement. Triangles reﬁned in this way need not be removed in later steps. Newest node bisection [67] We store the nodes P1 , P2 , P3 of triangle T in a counterclockwise sense: nodes [T, : ] = [P1 ,P2 ,P3 ]. If T is marked, we plan to bisect T from P1 towards the opposite edge as in Figure 6.2. The two new triangles must again have oriented nodes – entries with Pnew in the ﬁrst place. However, the neighboring triangle has to be added to Ah if Pnew is a hanging node. In a practical implementation one actually only reﬁnes simple situations: • two triangle share the marked edge; • the marked edge is a boundary edge;

6.2. The residual error estimator for the Poisson problem

P1

P′1 Pnew

437

Pnew

Figure 6.2 Left: Bisection in the interior. Middle: Bisection near the boundary. Right: Reﬁnement with interior node property.

as seen in Figure 6.2. It can be shown that Algorithm 6.1, which is based on these ideas, stops after ﬁnitely many steps and results in a locally reﬁned mesh. The reﬁnement method is geometrically stable if the macro-triangulation T0 has a “correct” distribution of marked edges. We only have to avoid all edges emanating from one vertex being marked edges. A newest node bisection with m-times repeated bisection is as follows. The marking procedure assigns to each T a number m(T ) ∈ N. Then Ah := {T ∈ T h : m(T ) > 0}. After each bisection, we decrease m(T ) until we obtain 0, see Algorithm 6.1. Algorithm 6.1. Repeated newest node bisection while any m(T ) > 0 for T with m(T ) > 0 identify neighbor T opposite P1 situation Figure 6.2 left/right?; Yes : bisect T (and T ); m(T ) := max{m(T ) − 1, 0}; (respectively m(T ) := max{m(T ) − 1, 0}); No : m(T ) := max{m(T ), 1}

Longest node bisection Always reﬁne towards the longest edge in T and only in one of the situations as in Figure 6.2. For that, work on a copy of the coordinate ﬁeld with a random distortion (to assure that the longest edge in T is unique almost surely). The important question is now whether this ends after ﬁnitely many steps in one of the mentioned situations? The answer is aﬃrmative since following these choices the length of the largest edge increases and thus will end at an edge of locally maximal length or at the boundary. The algorithm is like Algorithm 6.1, but a special ordering in the nodes ﬁeld is not necessary.

438 6.2.6

6. Adaptive ﬁnite element methods

Convergence of the adaptive ﬁnite element method

The following description is very special to the case of linear ﬁnite elements. Nevertheless, the method works for arbitrary polynomial degree with some modiﬁcations, see Remark 6.15. Deﬁnition 6.11. Interior node property: A reﬁnement method has the interior node property, if all marked triangles of the coarse mesh are reﬁned in such a way that all their edge centers are vertices of the reﬁned mesh and that each such triangle contains in its interior a vertex of the reﬁned mesh. An example for such a method is the repeated newest node bisection with m(T ) = 3, Algorithm 6.11. Theorem 6.12. Computable a posteriori estimates: Let T H be a triangulation, V H H the linear ﬁnite element space over T H and uH the ﬁnite element solution to 0 ∈V H h the discrete Poisson problem (6.5) in V . Let T be a reﬁnement of T H , V H ⊂ V h the reﬁned ﬁnite element space and uh0 the discrete solution in V h . Assume that both approximations f H and f h of f are piecewise constant. If F H ⊂ E H,◦ is any set of reﬁned edges and if all triangles in AH as in (6.22) have been reﬁned by a reﬁnement method with the interior node property (see Figure 6.2), then H 2 2 h H 2 u0 ≤ ||∇ uh0 − uH ηE ceﬀ 0 ||L2 (Ω) + ||H(f − f )||L2 (Ω) E⊂F H H h with some constant function H ceﬀ depending only on σ , σ . Here, H is the mesh-size H . for T and ηE u0 is the error indicator (6.20) with respect to uH 0

Proof. [499], Lemma 4.2. We can assume the following situation. Let E ∈ E H,◦ and T1 , T2 ∈ T H with E = T1 ∩ T2 . Both T1 and T2 have been reﬁned and T h has vertices xE (the center of E) and xi in the interior of Ti for i ∈ {1, 2}. Let ϕE , ϕi be the (maximally coarse) basis functions related to these vertices in V h with ωi := supp(ϕi ) ⊆ Ti and ωE := supp(ϕE ) ⊆ T1 ∪ T2 . For all ϕ with ωϕ := supp(ϕ) ⊆ T1 ∪ T2 % & h · ∇ϕ = ∂n u H ∇ uh0 − uH f ϕ − 0 0 Eϕ ωϕ

ωϕ

ωϕ ∩E

(f h − f H )ϕ +

= ωϕ

fHϕ −

ωϕ

%

∂n u H 0

ωϕ

& E

ϕ.

Let ϕ = ϕi for i ∈ {1, 2}. Since f H is constant and ϕi E = 0 we have H · ∇(f ϕi = f H f H ϕi = ∇ uh0 − uH ϕ ) − (f h − f H )f H ϕi |f H |2 i 0 ωi

ωi

H

ωi

ωi

≤ ||∇ uh0 − u0 ||L2 (ωi ) ||∇(f H ϕi )||L2 (ωi ) + ||f h − f H ||L2 (ωi ) ||f H ϕi ||L2 (ωi ) . Using inverse estimates on ϕi (see Theorem 4.19) and considering the nodal scaling ϕi (xi ) = 1 yields

h H H Hi ||f H ||2L2 (ωi ) ≤ C Hi−1 ||∇ uh0 − uH 0 ||L2 (ωi ) + ||f − f ||L2 (ωi ) ||f ||L2 (ωi )

6.2. The residual error estimator for the Poisson problem

439

with a constant C that depends on σTi . Similar, we derive the bound for the constant edge residual taking the test function ϕ = RE ϕE 2 2 ||RE ||L2 (E) ≤ C |RE | ϕE = C RE (RE ϕE ) E

E

H

∇ uh0 − u0

=C

ωE

≤ C ||∇

uh0

−

uH 0

· ∇(RE ϕE ) −

f h R E ϕE ωE

−1/2 ||L2 (ωE ) hE ||RE ||L2 (E)

1/2 + ||f H ||L2 (ωE ) + ||f h − f H ||L2 (ωE ) hE ||RE ||L2 (ωE ) . ||f H ||L2 (ωE ) can be estimated with the ﬁrst step of the proof and our stated result follows readily. Deﬁnition 6.13. Saturated data approximation: Let T H be a triangulation and T h H a reﬁnement of T H , uH the solution of the discrete Poisson problem (6.5) in 0 ∈V H V . We say that the data approximation is saturated with a factor μ > 0, if δHh := max CP (Ω)||f − f H ||L2 (Ω) , CP (Ω)||f − f h ||L2 (Ω) , ||H(f H − f h )||L2 (Ω) (6.23) ≤ μηT H uH 0 . δHh is called the data approximation error. Note that this bound is quite natural for small h, since generally δHh will be of lower order compared with ηT H (uH 0 ) (for example, if f is piecewise smooth, then ||f − f H ||L2 (Ω) = O(h2max ). However, the following results will depend on the quantitative bound (6.23) and not on a vage higher order argument. h Theorem 6.14. Convergence of adaptive ﬁnite element method: Let T H , V H , uH 0 , u0 , H h H and f , f be as in Theorem 6.12. Assume that we deﬁne F by the ﬁxed energy fraction strategy (6.21) for some θ ∈ (0, 1) and that we reﬁne T H to get T h as required in Theorem 6.12. Assume that

δHh ≤ μηT H holds for a suﬃciently small μ > 0, only depending on σ H , σ h and θ. Then there is a constant c that again depends only on σ H , σ h , such that √ ||∇ u0 − uh0 ||L2 (Ω) ≤ 1 − cθ2 ||∇ u0 − uH 0 ||L2 (Ω) as long as ηT H > TOL. Thus, for a given tolerance TOL > 0 and a geometrically stable reﬁnement algorithm, this method reaches ηT H < TOL after ﬁnitely many steps. Proof. We can achieve δHh ≤ μηT H by reﬁning the mesh. We could even establish δHh ≤ μ TOL for the given μ, so our requirement would hold as long as ηT H > TOL. Since the ﬁnite element approximations to f converge on reﬁned meshes, this can be achieved in ﬁnitely many steps. (See [499], where δHh and ηT H are treated within one iteration.)

440

6. Adaptive ﬁnite element methods

We get for the lower bound from Theorem 6.12 with the deﬁnition of F H ceﬀ ηF H ≤ ||∇ uh0 − uH 0 ||L2 (Ω) + μηT H which leads with the marking strategy (6.21) to (ceﬀ θ − μ)ηT H ≤ ceﬀ ηF H − μηT H ≤ ||∇ uh0 − uH 0 ||L2 (Ω) . We use the approximate Galerkin orthogonality (since V H ⊂ V h !) = Ω (f − f h ) uh0 − uH ∇ u0 − uh0 · ∇ uh0 − uH 0 0 Ω

and conclude from this h h H 2 ∇ u0 − u0 · ∇ u0 − u0 ≥ −2||f − f h ||L2 (Ω) ||uh0 − uH 0 ||L2 (Ω) Ω

≥ −2CP (Ω)||f − f h ||L2 (Ω) ||∇ uh0 − uH 0 ||L2 (Ω) ≥ −2μηT H ||∇ uh0 − uH 0 ||L2 (Ω) 2 1 ≥ −2μ2 ηT2 H − ||∇ uh0 − uH 0 ||L2 (Ω) . 2

Using this inequality and Theorem 6.12, we get 2 2 h h H ||∇ u0 − uH 0 ||L2 (Ω) = ||∇ u0 − u0 + u0 − u0 ||L2 (Ω)

= ||∇ u0 −

uh0

||2L2 (Ω)

+ ||∇

uh0

−

uH 0

||2L2 (Ω)

∇ u0 − uh0 · ∇ uh0 − uH 0

+2 Ω

h

1 2 2 2 ≥ ||∇ u0 − u0 ||2L2 (Ω) + ||∇(uh0 − uH 0 )||L2 (Ω) − 2μ ηT H 2 1 ≥ ||∇(u0 − uh0 )||2L2 (Ω) + (ceﬀ θ − μ)2 − 2μ2 ηT2 H 2 2 ≥ ||∇(u0 − uh0 )||2L2 (Ω) + cθ2 ||∇(u0 − uH 0 )||L2 (Ω)

for a suﬃciently small μ, depending on ceﬀ and θ, and this yields the assertion F ||∇(u0 − uh0 )||L2 (Ω) ≤ 1 − cθ2 ||∇(u0 − uH 0 )||L2 (Ω) . Remark 6.15. Convergence of adaptive ﬁnite element method: 1. The ﬁrst convergence proof was in [54] (n = 1 with no rate). The method of Theorem 6.14 was developed in [300] and [498]. A more operator oriented approach was [96].

6.2. The residual error estimator for the Poisson problem

441

H = ΔuH 2. For p > 1 the error indicator has the element residual RT uH 0 0 +f . The ﬁrst step towards convergence is to prove (compare Theorem 6.12) H 2 u0 ηE E∈E H,◦

2 H 2 h 2 ≤ C ||∇(uh0 − uH 0 )||L2 (Ω) + ||H(f − f )||L2 (Ω) + ||h(f − f )||L2 (Ω) ,

where one has to assume that marked edges and triangles in AH have been reﬁned in a suitable way. More generally, one has to prove the inequality RT (uH )v h H chT ||RT (u0 )||L2 (T ) ≤ sup T h 0 ||v ||L2 (T ) v h ∈XT h for some constant c depending on σ and p. Here XT is the surplus space on T : h H XT = V T \V T . A similar inequality has to hold for RE [302]. 3. The error decreases geometrically for ﬁxed θ. If we miss the tolerance TOL slightly in one step, we have to perform an additional step and we will very likely end up with an iterate that is more accurate than necessary. Note that the last step determines the total work if the number of elements increases geometrically. However, we can use the error decay to lower θ near the tolerance level. Assume that the error satisﬁes 2 . Ek2 = (1 − cθ2 )Ek−1

From this we may eliminate the unknown constant c by

2

2 ηk Ek 1 1 c = θ2 1 − Ek−1 ≈ θ2 1 − ηk−1 , where the error Ek is approximated by our guess ηk := ηTk . When the next error is presumably below the tolerance, that is if (1 − cθ2 )Ek2 < TOL2 , we use a safety factor 0.9 and deﬁne a new value θnew such that 2 2 2 2 (0.9 TOL)2 = 1 − cθnew Ek ≈ 1 − cθnew ηk . This yields the explicit formula 2 θnew = θ2

1 − (0.9 TOL)2 /ηk2 2 1 − ηk2 /ηk−1

under the assumption that both nominator and denominator are positive [300]. 4. How do we compute F H ? We may sort {ηE }E∈E H,◦ by size and then put E ∈ F H beginning from the largest value until ηF H meets the requirement (6.21). The complexity of this algorithm is NE log(NE ) in the number NE of edges in T H . Alternatively, we may do only a rough sort and keep the linear complexity in NE , Algorithm 6.2 [300].

442

6. Adaptive ﬁnite element methods

Algorithm 6.2. Fixed energy fraction marking sum := 0; τ := 1; ν = 0.1; F H := ∅; while sum < θ 2 ηT2 H τ := τ − ν; for all E ∈ E H,◦ if E ∈ F H ; 2 2 > τ 2 ηmax if ηE F H := F H ∪ {E}; 2 sum := sum + ηE

6.2.7

Optimality

Although the convergence of the algorithm is an important result, it does not guarantee that we found the most eﬃcient method. In this section we introduce optimal approximation spaces and show that the proposed algorithm does indeed produce meshes of the optimal asymptotic complexity. This description follows [600]. If u0 ∈ H 2 (Ω) and if we choose a linear ﬁnite element method on a uniform mesh (hT ≡ h for all T ∈ T h ), we know that ||∇(u0 − uh0 )||L2 (Ω) ∼ h is optimal with respect to the exponent in h. For Ω ⊂ R2 we have the relation h ∼ N −1/2 with the number of elements (and thus number of unknowns) N . To achieve ||∇(u0 − uh0 )|| ∼ TOL, we need N ∼ TOL−2 degrees of freedom and this is again optimal with respect to the exponent, cf. Section 6.1.4. If u0 ∈ H 1+r (Ω) for some r ∈ (0, 1], then we know ||∇(u0 − uh0 )||L2 (Ω) ∼ hr 2

and this yields that N ∼ TOL− r . We say that u0 ∈ Bs , for some s > 0, if the best approximation error of u0 in an N -dimensional linear ﬁnite element space behaves 1 asymptotically as N −s . Thus N = O(TOL− s ) is the necessary dimensionality to achieve the approximation error TOL for u0 . We now deﬁne the function space . . // B s :=

v ∈ H01 (Ω) : sup ε ε>0

s

inf

T c : inf vT c ∈VT c ||∇(v−vT c )||

L2 (Ω)

≤ε

{(T c − T0c ) } < ∞

,

where T denotes the number of triangles in a triangulation T . The inﬁmum is taken over all conforming triangulations T c that can be achieved from the given initial mesh T0c by newest node bisection, Section 6.2.5. VT c is the linear ﬁnite element space over T c . The space B s is equipped with the norm ||v||Bs := ||∇v||L2 (Ω) + [v]Bs .

6.2. The residual error estimator for the Poisson problem

443

By construction, v ∈ Bs guarantees that there is a reﬁnement T c of T0c with 1

T c − T0c ≤ O(ε− s ) such that inf

vT c ∈VT c

||∇(v − vT c )||L2 (Ω) ≤ ε.

As seen above (for r = 2s), a function in H 1+2s (Ω) fulﬁlls this requirement, hence H 1+2s (Ω) ⊂ Bs , but B s is deﬁnitely a larger set. This kind of space is related to Besov spaces for which a detailed analysis is well known [97]. Based on this deﬁnition we will formulate the following notion of optimality. Deﬁnition 6.16. An adaptive method is of optimal complexity if it realizes a mesh T c and a numerical approximation uT c of u0 such that 44 . ||∇(u0 − uT c )||L2 (Ω) ≤ ε 1

with a number of O(ε− s ) operations for every solution u0 ∈ Bs of (6.4). The adaptive algorithm, for which we sketch the optimality proof, is the one we proposed before with ﬁxed energy fraction marking (6.21) for some θ > 0 and a reﬁnement as required in Theorem 6.12. It is important to control the amount of the actual reﬁnement that is caused by our marking strategy. For this, the reﬁnement step is split into two steps, REFINE and MAKECONFORM. Theorem 6.17. Complexity of MAKECONFORM: The reﬁnement edges on T0c can be arranged so that the following holds true. Construct a sequence of meshes {Tkc }k≥1 by % Reﬁne marked triangles only, Tk+1 is nonconforming Tk+1 = REFINE (Tkc , fk , uk ); % Reﬁne further to get a conforming mesh c = MAKECONFORM (Tk+1 ); Tk+1 Here, fk is the representation of f and uk the discrete solution of (6.5) on Tkc . If REFINE reﬁnes according to the requirements in Theorem 6.12 and if MAKECONFORM uses the nodes of newest node bisection, then for n ∈ N Tnc − T0c ≤ C

n

c Tk − Tk−1 ,

(6.24)

k=1

where C only depends on T0c . Proof.

[96], Lemma 2.1, Theorem 2.4.

Let T be a triangulation that is a reﬁnement of T0c and ET◦ the set of interior edges in T . For the following arguments we will require that f is piecewise constant on T0c , hence we can take f = fT for any reﬁnement T of T0c . For vT ∈ VT and E ∈ ET◦ we 44

h h To avoid over-indexing, we abbreviate our usual uh 0,T c to uT c , and below u G , u0,k to uTG , uk . 0,T

444

6. Adaptive ﬁnite element methods

have the error indicator, see (6.20),

ηE (T , f, vT )2 := cE hE ||[∂ h vT ]E ||2L2 (E) +

cT h2T ||f ||2L2 (T ) .

T ∈T : T ⊂ωE

The global error estimate is then given by ηE (T , f, vT )2 . η(T , f, vT )2 := ◦ E∈ET

We recall what we know about a posteriori error estimation in our present setting from the previous chapters: if u0 and uT are continuous and discrete solutions, then we have the upper bound (note: f is assumed to be constant!) ⎛ ⎞1/2 ||∇(u0 − uT )||2L2 (Ω) ≤ c1 ⎝ ηE (T , f, uT )2 ⎠ (6.25) ◦ E∈ET

and the lower bound c2 η(T , f, uT ) ≤ ||∇(u0 − uT )||L2 (Ω) ,

(6.26)

see Theorem 6.9. Note that it is not assumed that T is a conforming triangulation. Here in this case one deﬁnes interpolation equations in the hanging nodes uT (xE ) = 1/2(uT (p1 ) + uT (p2 )) if xE is the center of the edge p1 p2 to get VT ⊂ H 1 (Ω). We now study a reﬁnement of the bounds (6.25), (6.26). Lemma 6.18. Modiﬁed upper and lower bounds: G 1. Let T c be given and TG be a reﬁnement. Let F be a set of edges E for which T ∈ T exists such that ωE ∩ T = ∅. Then F ≤ C TG − T c and with c1 as in (6.25) 1/2 c 2 ηE (T , f, uT c ) . (6.27) ||∇(uTG − uT c )||L2 (Ω) ≤ c1 E∈F

2. Let T be given and F be a set of edges. For each edge E ∈ F reﬁne the two adjacent triangles by the reﬁnement rule as required in Theorem 6.12. On the resulting triangulation TG we have TG − T c ≤ CF for some universal constant C and with c2 as in (6.26) 1/2 ηE (T c , f, wT c )2 ≤ ||∇(uTG − wT c )||L2 (Ω) (6.28) c2 c

E∈F

for arbitrary wT ∈ VT c . c

Proof. The proof of the ﬁrst inequality is as in Theorem 6.5, we only have to note that the chosen functions wTG , wT c can be taken to be equal on triangles in T c ∩ TG . The proof of the second inequality is as in Theorem 6.12. This relates the number of reﬁned triangles to the regularity of u0 .

6.2. The residual error estimator for the Poisson problem

445

Lemma 6.19. Complexity of REFINE: Let T0c be a macro-triangulation as in Theorem 6.17, f ∈ P0 (T0c ), and u0 ∈ Bs be a solution of the continuous problem (6.4) for some s > 0. Let T c be a reﬁnement of T0c , uT c the solution of the discrete problem (6.5) on T c , and T = REFINE (T c , f, uT c ) with some θ ∈ (0, c2 /c1 ) in (6.21). Then −1

1

T − T c ≤ C||∇(u0 − uT c )||L2s(Ω) [u0 ]Bs s . The constant C is independent of s for s ≥ s0 > 0. Proof. Fix θ ∈ (0, c2 /c1 ), choose λ ∈ (0, 1) with (c2 /c1 )2 (1 − λ2 ) ≥ θ2 and let Tˇ be any reﬁnement of T c (Tˇ is not conforming!) such that ||∇(u0 − uTˇ )||L2 (Ω) ≤ λ||∇(u0 − uT c )||L2 (Ω) . Then, by Lemma 6.18, there is a set Fˇ ⊂ ET c and a constant C > 0 such that Fˇ ≤ C(Tˇ − T c ) and, with (6.27), Galerkin orthogonality, and (6.26), ηE (T c , f, uT c )2 ≥ ||∇(uTˇ − uT c )||2L2 (Ω) c21 ˇ E∈F

= ||∇(u0 − uT c )||2L2 (Ω) − ||∇(uTˇ − u0 )||2L2 (Ω) ≥ (1 − λ2 )||∇(u0 − uT c )||2L2 (Ω) ≥ c22 (1 − λ2 )η(T c , f, uT c )2 . By the choice of λ it follows that ηE (T c , f, uT c )2 ≥ θ2 η(T c , f, uT c )2 . ˇ E∈F

This set of edges Fˇ thus fulﬁlls the inequality (6.21). But since the constructed set of marked edges F by REFINE is minimal with this property, we have T − T c ≤ CF ≤ CFˇ ≤ C(Tˇ − T c ). Now let TD be the minimal reﬁnement of T0c by REFINE such that uTD satisﬁes ||∇(u0 − uTD )||L2 (Ω) ≤ λ||∇(u0 − uT c )||L2 (Ω) and by this assumption 1 − 1 TD − T0c ≤ λ||∇(u0 − uT c )||L2 (Ω) s [u0 ]Bs s .

446

6. Adaptive ﬁnite element methods

Now we can assume that Tˇ is the smallest common reﬁnement of T c and TD . Thus we have Tˇ − T c ≤ TD − T0c and we conclude

1 −1 T − T c ≤ C(Tˇ − T c ) ≤ C TD − T0c ≤ C||∇(u0 − uT c )||L2s(Ω) [u0 ]Bs s

with C depending on λ−1/s hence on θ and any lower bound s0 > 0 for s.

Algorithm 6.3. Algorithm SOLVE, with f ∈ P0 (T0c ) assumed ! [Tkc , uTkc ] = SOLVE (T0c , f, ε) Solve for uT0c ; k := 0; while c1 η Tkc , f, uTkc ≥ ε Tk+1 := REFINE Tkc , f, uTkc ; c := MAKECONFORM (Tk+1 ); Tk+1 c ; Solve for uTk+1 k =k+1

We summarize the method in the routine SOLVE in Algorithm 6.3. The next theorem proves the optimality of this procedure according to Deﬁnition 6.16. Theorem 6.20. Complexity of SOLVE: T0c macro-triangulation, f ∈ P0 (T0c ), u0 ∈ B s solution of the continuous problem (6.4) for some s > 0 and uT c the solution of the discrete problem (6.5) on T c . Then the adaptive algorithm (T c , uT c ) = SOLVE (T0c , f, ε) (see Algorithm 6.3) terminates after ﬁnitely many steps with ||∇(u0 − uT c )||L2 (Ω) ≤ ε and 1

T c − T0c ≤ Cε− s [u0 ]Bs . Proof. The Galerkin orthogonality yields

c c ||∇ u0 − uTkc ||2L2 (Ω) = ||∇ u0 − uTk+1 ||2L2 (Ω) + ||∇ uTk+1 − uTkc ||2L2 (Ω) . By the previous estimates (6.28) and (6.25)

2 c − uTkc ||2L2 (Ω) ≥ (c2 θ)2 η Tkc , f, uTkc ||∇ uTk+1

and so with

δ :=

1−

2

||∇ u0 − uTkc ||2L2 (Ω) .

≥

c2 θ c1

2 1/2

c2 θ c1

<1

6.2. The residual error estimator for the Poisson problem

we arrive at

447

c ||L2 (Ω) ≤ δ||∇(u0 − uTkc )||L2 (Ω) ||∇ u0 − uTk+1

(this is the simpliﬁed version of Theorem 6.14). From this it follows that the algorithm SOLVE terminates after ﬁnitely many steps. It remains to show optimality. Let n be the index with

c c ≥ε , f, uTn−1 c1 η Tn−1 and

c1 η Tnc , f, uTnc < ε.

With Lemma 6.19 we get −1

1

Tk+1 − Tkc ≤ C||∇(u0 − uTkc )||L2s(Ω) [u0 ]Bs s for k = 0, . . . , n − 1. From our results on newest node bisection, Lemma 6.19, Tnc − T0c ≤ C

n−1

{Tk+1 − Tkc } ≤ C

k=0

≤C

n−1

n−1

−1

1

||∇(u0 − uTkc )||L2s(Ω) [u0 ]Bs s

k=0 1

−1

1

c (δ n−1−k ) s ||∇(u0 − uTn−1 )||L2s(Ω) [u0 ]Bs s

k=0

− 1s 1 1 −1 c c c ≤ C||∇(u0 − uTn−1 )||L2s(Ω) [u0 ]Bs s ≤ Cη Tn−1 , f, uTn−1 [u0 ]Bs s 1

1

≤ Cε− s [u0 ]Bs s .

This is the ﬁrst part of the proof in [600]. It continues with the general case of variable f and uTkc being replaced by an approximate solution of the discrete system. Using interior loops, one establishes a right-hand side fT c and an approximate discrete solution wT c so that ||f − fT c ||H −1 (Ω) + ||∇(uT c − wT c )||L2 (Ω) ≤ ωη(T c , fT c , wT c ) (for ω suﬃciently small, not depending on the mesh size and the data). If the discrete approximate solution can be established with optimal complexity, then this adaptive algorithm is of optimal complexity with regard to the regularity properties of u0 and f . 6.2.8

Other types of estimators

There are several possibilities to obtain alternative a posteriori error estimates in the energy norm. The objective of these variants is to get formulas that are easy to implement or that allow us to better quantify the constants (cT , cE in (6.20)). However, from the mathematical viewpoint it is decisive to have the upper and lower bounds as

448

6. Adaptive ﬁnite element methods

in Theorem 6.9 that would allow us to prove convergence of the adaptive method and optimality. Actually, these estimates mean that these error indicators are equivalent to the residual error estimator [655], Sect. 1.3, and these extensions are straightforward (besides the technical details). Subdomain residual method One (approximately) solves a local error equation ∇χz · ∇w = ωz f h w − ∇uh0 · ∇w

for all w ∈ H01 (ωz ),

(6.29)

ωz

where ωz is a patch of triangles around the node z and Vz is an enhanced ﬁnite element space with support in ωz . The local error estimators, given by ||∇χz ||L2 (ωz ) , are equivalent to the residual error estimator [6], Ch. 3.2, [655], Sect. 1.3. Gradient recovery The idea is to replace ||∇(u0 − uh0 )||L2 (Ω) by ||G(uh0 ) − ∇uh0 ||L2 (Ω) , where G uh0 is a h h smoothed gradient linear, ∇uh0 is piecewise h of u0 . For example, if u0 is piecewise h constant and G u0 is deﬁned as a projection of ∇u0 into the space of linear ﬁnite elements. The method is easy and works well in practise [6], Ch. 4. Hierarchical error estimate The idea is to deﬁne an enhanced discrete space (V h )∗ and write the equation on (V h )∗ as a system with respect to the decomposition (V h )∗ = V h ⊕ W h . W h is called the surplus space. With some simpliﬁcations one can approximate the error on V h by an equation on W h that is easy to solve. For this method to work it is decisive that the data error is well resolved [6], Ch. 5. Equilibrated residual method In order to work more locally than the previous method it is necessary to solve local Neumann problems. To obtain such bounds one needs to carefully choose the ﬂuxes that deﬁne these local boundary problems. There are several approaches to constructing such ﬂuxes [6], Ch. 6. 6.2.9

hp ﬁnite element method

The method so far will work for mesh reﬁnement (h method) for ﬁxed polynomial degree. If one is interested in variable polynomial degrees, one faces the problem of deciding whether to reﬁne the mesh (for ﬁxed p) or increase the polynomial degree for ﬁxed h. The optimal choice of h and p will lead to exponential (instead of algebraic) decay of the error with the number of unknowns. For the error estimation one may also use the residual error estimator, however, the explicit dependence of the error indicator from the polynomial degree is important [575]. A convergence proof for such a method is only known in one space dimension [301]. Many numerical issues can be found in [280, 281].

6.3. Estimation of quantities of interest

6.3 6.3.1

449

Estimation of quantities of interest Quantities of interest

In the following considerations the functions mostly, but not always, are exact and discrete solutions. So we use the notation u and uh instead of the previous u0 and uh0 . So far we have been interested in discrete solutions uh of −Δu = f and looked for estimates for ||∇(u − uh )||2 . In practical problems, however, one has often more speciﬁc questions. For example, we could ask u(x0 ) = ? , ∂1 u(x0 ) = ? for some x0 ∈ Ω, ∂n u = ? for some curve Γ ⊂ Ω, ||u − uh ||Ω for Ω ⊂⊂ Ω. Γ

A mesh to answer these speciﬁc questions might look very diﬀerent from those we considered before. Therefore we will pose the generalized question Q(u) = ? , where Q : V → R is a continuous operator. However, the examples Q(u) := u(x0 ), Q(u) := ∂1 u(x0 ) or Q(u) := Γ ∂n u do not satisfy this if u ∈ V = H 1 (Ω) is assumed! Hence we have to regularize these operators. Note that solutions are usually more regular than just in V, so that Q will have some meaning unless very special situations occur (e.g. ∂1 u(x0 ) and x0 is a corner point). For example, we let 1 u for x0 ∈ Ω and suﬃciently small > 0. Q (u) := |B (x B (x0 ) 0 )| Since in this case Q ≡ Q is linear, we can write Q(u) − Q(uh ) = Q(u − uh ) = Q(e). Our aim is to deﬁne a computable quantity η(uh ) for which one can show cη(uh ) ≤ |Q(e)| ≤ Cη(uh ) for positive constants c, C. 6.3.2

Error estimates for point errors

Again we need a reasonable error representation. For this we use the duality argument, to represent Q( . ) as a scalar product in V. We choose u∗ ∈ V to be the solution of ∗ ∇v · ∇u = Q(v) = − v ≈ v(x0 ) = δx0 [v] for all v ∈ V, Ω

B (x0 )

where δx0 is the Dirac measure in x0 . This means that u∗ is an approximate Green’s function u∗ (x) ≈ G(x; x0 ),

that is, G( . ; x0 ) is the solution of −ΔG( . , x0 ) = δx0 , G( . , x0 )∂Ω = 0. Now we can localize the error using ∗ ∗ ∇e · ∇u = ∇e · ∇(u∗ − wh ). e(x0 ) = δx0 [e] ≈ Q(e) = Ω

Ω

450

6. Adaptive ﬁnite element methods

The last equality uses the deﬁnitions of u and uh . Now we can also solve the discrete dual problem ∇v h · ∇(uh )∗ = Q(v h ) for all v h ∈ V h Ω

and by taking v = (uh )∗ we obtain h ∗ h ∗ ∇(u − u ) · ∇(u − (u ) ) = ∇e · ∇e∗ . Q(e) = h

Ω

(6.30)

Ω

Thus, the dual error ∇(u∗ − (uh )∗ ) can be viewed as a weight for the error ∇(u − uh ) of the primal problem. To proceed we transform the ﬁrst integral in (6.30) as in the well-known error representation (6.19) to get ∗ h ∗ RT (u − (u ) ) + RE (u∗ − (uh )∗ ), Q(e) = T ∈T h

T

E∈E h,◦

E

with RT ≡ RT (uh ) and RE ≡ RE (uh ) as in Deﬁnition 6.4. Proceeding as in Theorem 6.5 leads to ⎛ ⎞ 3/2 ⎝h2 ||RT ||L2 (T ) + hE ||RE ||L2 (E) ⎠ ζT |Q(e)| ≤ C E⊂∂T \∂Ω

T ∈T h

−3/2 ∗ ∗ h ∗ with ζT := max h−2 ||u − (uh )∗ ||L2 (∂T ) and some positive T ||u − (u ) ||L2 (T ) , hT constant C that depends only on the shape regularity σ h and the polynomial degree p. Unfortunately, we do not know u∗ and it is additional work to compute (uh )∗ .

Heuristic approach 1 log(|x − x0 |) + w(x) where w is In R2 , we know that u∗ has the form u∗ (x) = 2π smooth. Therefore we take hT ζT ≈ . |x − x0 |2 + 2 3/2 Deﬁning ηT := hT ||f ||2;T + E⊂∂T \∂Ω hE ||RE ||2;E , we obtain with = hT |Q(e)| ≤ C

T ∈T h

h2T ηT , |xT − x0 |2 + h2T

where xT denotes the center of gravity of T and C is as above. Thus we have derived a computable estimate for |Q(e)| in terms of the local contribution and we can now proceed with the marking and reﬁnement strategies of Section 6.2.4. Estimates using the discrete dual solution One computes the dual solution (uh )∗ on the same mesh for simplicity. Using diﬀerence quotients or gradient recovery (see Section 6.2.8), one can replace ||u∗ − (uh )∗ ||L2 (T ) by h2T ||∇(˜ uh )∗ ||2;T or by ||u∗h,recover − (uh )∗ ||L2 (T ) . Another possibility is to compute dual

6.3. Estimation of quantities of interest

451

solutions on a ﬁne (h) and a coarse (H) mesh and let ||u∗ − (uh )∗ ||L2 (T ) ≈ ||(uH )∗ − (uh )∗ ||L2 (T ) . 6.3.3

Optimal meshes–A heuristic argument

First recall Section 6.1.5, where we deﬁned an optimal mesh-size distribution as the minimizer of a constraint As before, we let N (h) := Ω σ/hn and 2 2 optimization problem. Q(e) = E(h) = Ω h E , but now with E 2 (x) = |∇2 u(x)| |∇2 u∗ (x)|, motivated by the !

previous section. Observe that our error criterion E(h) = TOL leads to the Lagrange function σ − λ TOL − h2 E 2 , L(h, λ) = N (h) − λ TOL −E(h) = n Ω h Ω which diﬀers from the former deﬁnition (6.9) by the exponent of TOL and E(h), and by the formula for E 2 . Therefore we will conclude, instead of (6.11), that h2 E 2 =

TOL σ . N (h) hn

Extracting h and inserting this into the deﬁnition of N (h) we ﬁnd the relation N (h) ∼ TOL−n/(2α)

if the integral Ω E 2n/(n+2α) exists. To compare this to our previous situation we consider ﬁrst a smooth solution on a uniform mesh. Since for linear ﬁnite elements ||u − uh ||L2 (Ω) = O h20 , that is α = 1, the requirement h20 ∼ TOL gives N (h0 ) ∼ TOL−n/2 as above. If we establish a solution by the adaptive ﬁnite element method using the energy estimator (6.20) with tolerance TOL, we need N (h) ∼ TOL−n (cf. Section 6.1.4). However, these results do not come with a reliable a posteriori error estimate for the point values. A rigorous a posteriori error bound, that would very likely give the optimal complexity, has been given by [513]. A theory of convergence and optimality is the subject of current research. A recent result is [495]. 6.3.4

The general approach

Now we assume that the problem is given by: seek u ∈ V such that A(u)[φ] := A(u), φV ×V = 0 for all φ ∈ V. Here, V is some Hilbert space and A : V → V has two derivatives A and A . For example, the nonlinear Poisson problem −Δu = f (u) in Ω with homogeneous Dirichlet boundary conditions leads to ∇u · ∇φ − f (u)φ , A (u)[φ, ψ] = ∇ψ · ∇φ − f (u)ψφ A(u)[φ] := Ω

Ω

A (u)[φ, ψ, χ] =

Ω

− f (u)ψφχ .

452

6. Adaptive ﬁnite element methods

Now let V h ⊂ V be a ﬁnite element space and assume that integration is performed exactly. Then the discrete problem reads: seek uh ∈ V h such that A(uh )[φh ] = 0 for all φh ∈ V h . Our quantity of interest may be given by some nonlinear, twice diﬀerentiable functional Q : V → R. We deﬁne the error functional by 'Q(uh ) := Q(u) − Q(uh ). As before, we seek a representation formula for 'Q. We start with a Taylor series (or twice applying the Mean Value Theorem 1.43): with e := u − uh we get 1 A (uh + se)[e, . ](s − 1) ds (6.31) A(u)[ . ] = A(uh )[ . ] + 0

= A(uh )[ . ] + A (uh )[e, . ] −

1

A (uh + se)[e, e, . ](s − 1) ds

0

and

1

Q(u) = Q(u ) + Q (u )[e] − h

h

Q (uh + se)[e, e](s − 1) ds

0

which gives 'Q(uh ) = Q (uh )[e] −

1

Q (uh + se)[e](s − 1) ds.

0

Now let u∗ ∈ V be the solution of the dual problem A (uh )[φ, u∗ ] = Q (uh )[φ]

for all φ ∈ V

and (uh )∗ ∈ V h be the solution of its discrete version A (uh )[φh , (uh )∗ ] = Q (uh )[φh ]

for all φh ∈ V h .

Taking φ = e we obtain by deﬁnition of u and uh 1 'Q(uh ) = A (uh )[e, u∗ ] − Q (uh + se)[e, e](s − 1) ds 0 ∗

= A(u)[u ] − A(u )[u∗ ] 1

A (uh + se)[e, e, u∗ ] − Q (uh + se)[e, e] (s − 1) ds + h

0

≡ A(u)[u∗ ] − A(uh )[u∗ ] + R(uh , e, u∗ ) = −A(uh )[u∗ − φh ] + R(uh , e, u∗ )

(6.32)

6.3. Estimation of quantities of interest

453

for all φh ∈ V h . Hence

|'Q(uh )| ≤ inf A(uh )[u∗ − φh ] + O ||e||2V . φh ∈V h

This generalizes Equation (6.30). In the linear case the higher order term R vanishes. In the example from the beginning of this section, the nonlinear Poisson problem, and with a linear Q, we ﬁnd the dual problem ∇φh · ∇u∗ − f (uh )φu∗ = Q[φ] for all φ ∈ V, Ω

whose strong form is −Δu∗ − f (uh )u∗ = Q[ . ] in Ω, u∗ ∂Ω = 0. Since 'Q(uh ) = Q[e], 1 h ∗ h h ∗ h Q[e] ≤ inf ∇u · ∇(u − φ ) − f (u )(u − φ ) + f (uh + se)e2 u∗ . φh ∈V h

0

Ω

Ω

By partial integration in the ﬁrst term we obtain ⎫ ⎧ ⎬ ⎨ % & ∂n uh E (u∗ − φh ) . f (uh )(u∗ − φh ) + ⎭ ⎩ T h E⊂∂T \∂Ω

T ∈T

The development of the a posteriori error estimate and the practical aspects are as before in Section 6.3.2. The special case of A=L’ and Q=L This holds in many physical applications where L is an “energy” and the weak equation is the extremal condition L (u)[v] ≡ A(u)[v] ≡ 0 for all v ∈ V. Our quantity of interest may be the value of the energy Q(u) := L(u). Hence 'Q(uh ) := Q(u) − Q(uh ) = L(u) − L(uh ). Then (6.32) translates to 1 h L (u)[ . ] = L (u )[ . ] + L (uh + se)[e, e, . ](s − 1) ds 0

1

= L (u )[ . ] + L (u )[e, . ] − h

h

L (uh + se)[e, e, . ](s − 1) ds

0

and instead of (6.32) we have the expansion 1 1 'Q(uh ) = L (uh + se)[e] ds = sL (uh + se)[e] − 0

0

1

sL (uh + se)[e, e] ds

0

1 1 1 1 (s2 − 1)L (uh + se)[e, e, e] ds = − (s2 − 1)L (uh + se)[e, e] + 2 0 0 2 1 1 1 2 = − L (uh )[e, e] + (s − 1)L (uh + se)[e, e, e] ds. 2 2 0 We now observe that we do not need to solve a dual problem since it now suﬃces to use −1/2 e as a test function in the expansion of L (u)[ . ] to eliminate − 21 L (uh )[e, e]

454

6. Adaptive ﬁnite element methods

and so we get

1 1 1 2 (s − 1)L (uh + se)[e, e, e] ds 'Q(uh ) = − L (uh )[e, e] + 2 2 0 1 1 1 = L (uh )[e] − s(1 − s)L (uh + se)[e, e, e] ds 2 2 0 1 = L (uh )[e − φh ] + O ||e||3V 2 1 = L (uh )[u − φh ] + O ||e||3V 2

and thus

1 'Q(uh ) = inf L (uh )[u − φh ] + O ||e||3V . h h 2 φ ∈V

We repeat that this holds without a dual solution and that the second term vanishes if L is quadratic! This is especially useful if we consider eigenvalue problems [398].

7 Discontinuous Galerkin methods (DCGMs), with V. Dolejˇs´ı 7.1

Introduction

Discontinuous Galerkin methods (DCGMs) represent a class of FEMs. In contrast to the usual FEMs, piecewise polynomials of varying degrees in diﬀerent elements K are used. Continuity between neighboring elements and boundary conditions are not required. These violations are punished by a penalty function, see (7.41), (7.61), increasingly unpleasant for ﬁner mesh sizes h. In some sense, DCGMs are therefore nonconforming FEMs. So, many of the remarks in the introduction to Chapter 4 remain valid for DCGMs as well. Again we use the general discretization theory in Chapter 3. The proofs for convergence via stability and consistency techniques are again based upon linearization. The original DCGM was introduced by Reed and Hill [546] for the solution of the neutron transport equation. It was analyzed for the ﬁrst time by Le Saint and Raviart [472]; later improvements were achieved by Wheeler [665] and Johnson and Pitk¨ aranta [426]. These methods were also applied to purely elliptic problems. Examples are the original method of Bassi and Rebay [75] the stabilized version of the original Bassi– Rebay method studied by Brezzi et al. [146], and its generalization, called the local discontinuous Galerkin methods (LDGMs), introduced by Cockburn and Shu [194], and further studied by Castillo et al. [166, 190]. In the 1970s, Galerkin methods for elliptic and parabolic equations using discontinuous ﬁnite elements were independently proposed and a number of variants introduced and studied, cf. Arnold [39], Baker [56] and Wheeler [665]. These DCGMs were called interior penalty (IP) methods, and their development remained independent of the development of the DCGMs for hyperbolic equations. A uniﬁed analysis and comparison of various discontinuous Galerkin techniques for elliptic problems was developed by Arnold et al. [40]. For a survey see, e. g. Cockburn et al. [182,192], summarizing the development of DCGMs for elliptic, parabolic and hyperbolic problems. Finally, we mention the so-called hybridized discontinuous Galerkin methods proposed and developed by Cockburn and co-workers in [163, 183–189]. This approach belongs among the mixed techniques where the hybridization reduces the globally coupled unknowns.

456

7. Discontinuous Galerkin methods (DCGMs)

Here and throughout this book, we restrict the discussion to elliptic problems. In this chapter, we present a detailed study of a class of DG methods for second-order elliptic problems which includes most of the above-mentioned methods. Many papers (e.g. [166, 297, 319, 517]) consider the linear diﬀusive operator −Δu or ∇ · (A∇u), where A is a symmetric matrix independent of u. In addition, lower order terms are allowed, cf. (7.13). Several generalizations to nonlinear diﬀusion operators were studied. Rivi`ere and Wheeler [552] consider the nonlinear diﬀusion operator −∇ · (a(x, u)∇u),

a : Ω × R → R,

(7.1)

where the function a(x, u) is Lipschitz-continuous with respect to its second variable and there exist constants γ and γ( such that 0 < γ ≤ a(x, u) ≤ γ(

∀(x, u) ∈ Ω × R.

(7.2)

Extensions and improved energy estimates with applications to single phase ﬂow in porous media are described by Wheeler et al. [554, 555]. Houston, Robson and S¨ uli considered in [407], and extended in Houston, S¨ uli and Wihler [412], the quasilinear diﬀusive operator in the form −∇ · (μ(x, |∇u|)∇u) , with|∇u| = |∇u|n ,

(7.3)

where μ ∈ C(Ω × [0, ∞)) is a function satisfying the assumption mμ (t − s) ≤ μ(x, t)t − μ(x, s)s ≤ Mμ (t − s),

t ≥ s ≥ 0, x ∈ Ω,

(7.4)

with positive constants mμ and Mμ . Here we start with εΔu plus a low order ﬂux term in (7.13) and proceed to more general cases. For quasilinear systems, e.g. the diﬀusion term takes the form −∇ · R(x, u, ∇ u),

: Ω × Rq × Rn×q → Rq . R

(7.5)

Among several discontinuous Galerkin techniques two approaches, SIPG (symmetric interior penalty Galerkin) and NIPG (nonsymmetric interior penalty Galerkin), are very popular. The main idea, applied to a linear diﬀusion problem, here to −Δu, is the following. By applying Green’s theorem, we obtain, for every inner face e and its unit normal vector ν, an integral ({∇u}, ν)n [v] dS , (7.6) e

appropriately modiﬁed for boundary faces. The symbols {·} and [·] denote an average value and jump of a function beyond the face e, cf. (7.32), (7.33). Then we add, by formally exchanging the regular function u and the test function v, an integral in the form (7.7) θ ({∇v}, ν)n [u] dS , e

7.1. Introduction

457

where θ = 1 for SIPG and θ = −1 for NIPG. Obviously, the integrals (7.7) vanish for a regular function u ∈ H 1 (Ω). Moreover, the so-called interior penalty, σ[u][v] dS , σ>0 (7.8) e

again vanishing for a regular function, is included in the interior penalty variants of DCGMs. In Antonietti et al. [33] a diﬀerent type of stabilization of DCGMs is presented. Moreover, Burman and Stamm [153] propose a method which requires stabilization only on a part of the skeleton. The antisymmetry of the sum of terms in (7.6) and (7.7) (with θ = −1) implies the coercivity of an extended bilinear form plus the penalty term, cf. (7.57), (7.161), of the NIPG technique for any penalty parameter σ > 0. On the other hand, for ensuring the coercivity of this bilinear form for the other DCGMs it is necessary to choose the parameter σ suﬃciently large. However, this pays for the SIPG type discretizations. It yields a symmetric discrete problem. This is important for numerical methods and allows optimal error estimates in the L2 norm. The SIPG and NIPG techniques can be directly extended to nonlinear diﬀusion terms as in (7.1)–(7.3). The face integrals corresponding to (7.6) and (7.7) are ({a(u)∇u}, ν)n [v] dS , ({μ(x, |∇u|)∇u}, ν)n [v] dS (7.9) e

and

e

θ

({a(u)∇v}, ν)n [u] dS ,

e

({μ(x, |∇u|)∇v}, ν)n [u] dS ,

θ

(7.10)

e

respectively. The incomplete replacement of u by v from (7.9) to (7.10) maintains the essential linearity with respect to v. The linearity with respect to u is lost anyway. This technique is no longer possible for general quasilinear diﬀusion terms of the form (7.5). Hence, SIPG as well as NIPG methods are not suitable for the discretization of quasilinear terms ∇ · R(x, u, ∇ u), unless R(x, u, ∇ u) is linear in ∇ u, or speciﬁc conditions as for (7.3) are imposed. Applying Green’s theorem, the face integral of type (7.6) and/or (7.9) yields, diﬀerently from before, u, ∇u)}, ν)n [v] dS . (7.11) − ({R(x, e

Exchanging the arguments u and v as in (7.7) would cause the resulting integral to become nonlinear with respect to v. Therefore we employ the so-called incomplete interior penalty Galerkin (IIPG) method studied by Dawson et al. [269, 615, 616]. Our method here for elliptic problems has been afterwards, without reference, modiﬁed and generalized to the parabolic case in [295]. We obtain a nonsymmetric diﬀusive form and have to choose the penalty parameter suﬃciently large. As for FEMs with variational crimes, the interplay between the strong and the weak form of the problem plays an essential role. So the original strong problem is

458

7. Discontinuous Galerkin methods (DCGMs)

transformed into its weak form. This excludes, for our approach, general fully nonlinear problems. For smooth enough functions, the strong forms of our operators can be easily determined, cf. (7.1)–(7.5). The corresponding weak forms we systematically deﬁne via testing in the appropriate spaces, cf. e.g. (7.19)–(7.22). We start discussing these ideas for the important model problem (7.13)–(7.15). For u ∈ H 2 (Ω) ∩ L∞ (Ω), (7.13) is transformed into its weak form (7.23), valid for u ∈ H 1 (Ω) ∩ L∞ (Ω). These convection–diﬀusion problems are among the most interesting applications of the DCGMs. Here the diﬀusion term is multiplied by a small diﬀusion coeﬃcient ε > 0. In most of the chapter, n we deal with convection–diﬀusion problems. For a reaction term f (u) instead of s=1 ∂fs (u)/∂xs , (7.13) represents a reaction–diﬀusion problem. Sometimes a reaction term is added to the equation, e.g. for obtaining better properties. However, it is not present in the equations of ﬂuid dynamics. The terms in (7.6)–(7.11) are combined with the piecewise or broken Sobolev norms in (7.29)–(7.31) on the triangulation T h for u ∈ H k (T h ) for all k ≥ 0 for convergence results. Penalty terms, e.g. (7.8), measure the transition errors along the interior and boundary edges for broken bilinear and nonlinear forms and penalty terms, cf. (7.41). Thus the original problem, e.g. (7.23), with its boundary condition, is translated into the discrete (7.70) or (7.71), cf. (7.57)–(7.61). For the strong form, (7.13), with u ∈ H 2 (Ω), v ∈ L2 (Ω), this often requires u ∈ L∞ (Ω) for the nonlinearity. For the transition to the weak form, and the DCGMs, we have to impose conditions u ∈ H 1 (T h ) or u, v ∈ H 3/2+ (T h ). This up and down between u, v ∈ H 2 (T h ), u, v ∈ H 3/2+ (T h ), and u, v ∈ H 1 (T h ), all u ∈ L∞ (Ω), sometimes complicates the presentation, cf. Remarks 7.1 and 2.1, (2.156), and Theorem 7.5. As indicated, we extend the DCGMs from the semilinear model problem (7.13) to more general quasilinear systems with diﬀusive terms in (7.5). Furthermore, we study, besides the standard numerical ﬂux for convection diﬀusion equations, other types of ﬂuxes as well, cf. (7.45)–(7.50). As nonconforming FEMs, the DCGMs are possible for linear to quasilinear, but not for fully nonlinear problems. The modiﬁcation from our second order equations to the diﬀerent types of semilinear or quasilinear systems is totally analogous to that for the previous FEMs. So we do not formulate the DCGMs for all these types of elliptic problems, but only for some order 2 including general quasilinear cases. The results are presented in the Hilbert space setting, although extensions to the Banach space setting would be essentially possible. Extensions to order 2m are problematic. The main tool for second order problems, the appropriate relation between the strong an weak forms of the problem, would become highly technical. So we omit problems of order 2m, m > 1. Recall that the following problems are only discussed for FEMs: General convergence theory for monotone operators, cf. Section 4.5, variational methods for eigenvalue problems, cf. Section 4.7, methods for nonlinear boundary conditions, cf. Section 5.3, quadrature approximate methods, cf. Section 5.4. They remain valid for DCGMs as well. Our techniques for fully nonlinear elliptic problems, cf. Section 5.2, are not possible for DCGMs. However, the monotone operator approach, cf. Section 4.5, allows DCGMs for problems of order 2m, m ≥ 1.

7.2. The model problem

7.2

459

The model problem

Let Ω ⊂ Rn (here n = 2 or 3) be a bounded polyhedral domain with closure Ω and boundary ∂Ω = ∂ΩD ∪ ∂ΩN , ∂ΩD ∩ ∂ΩN = ∅. We consider the following boundary value problem, for the exact solution u0 in Ω : u0 : Ω → R, u0 ∈ H 1 (Ω) ∩ L∞ (Ω), Ω ⊂ Rn , Ω ∈ C 0,1 , open bounded, (7.12) G(u0 ) :=

n ∂fs (u0 ) s=1

∂xs

− εΔu0 − g = 0

in Ω,

(7.13)

u0 |∂ΩD = uD on ∂ΩD ,

(7.14)

∂u0 |∂ΩN = gN on ∂ΩN , ∂ν

(7.15)

cf. Remark 7.1, with an appropriate ﬂux term. We require the following conditions: (a) fs ∈ C 1 (R), fs (0) = 0, s = 1, . . . , n;

(7.16)

(b) ε > 0; (c) g ∈ L2 (Ω), or below g ∈ H −1 (Ω) with g, v := g, vH −1 (Ω)×H 1 (Ω) ; (d) uD is the trace of some u∗ ∈ H 1 (Ω) ∩ L∞ (Ω) on ∂ΩD ; (e) gN ∈ L2 (∂ΩN ). This u∗ ∈ H 1 (Ω) ∩ L∞ (Ω) allows trivial boundary conditions by substituting u := u − u∗ and updating g. We avoid that here to demonstrate the role of the violated boundary conditions in DCGMs more clearly, cf. (7.41). Equation (7.13) shows that the assumption fs (0) = 0, s = 1, . . . , n, does not cause any loss of generality. The functions fs , called ﬂuxes, represent convective terms, ε > 0 is the diﬀusion coeﬃcient. The diﬀusion term can be more complicated, in some cases even nonlinear, see Section 7.5. For our model problem in (7.13)–(7.15), we consider mixed Dirichlet–Neumann boundary conditions. The same technique is possible for the following general cases as well. There we only present the Dirichlet condition on the whole boundary. A suﬃciently regular function, u0 ∈ C 2 (Ω), satisfying (7.13) – (7.15) pointwise is called a classical solution. Let Vb = {v ∈ H 1 (Ω), v|∂ΩD = 0}.

(7.17)

Multiplying (7.13) with v ∈ Vb and partial integration yields, via the ﬁrst Green’s formula, (2.9), (2.10), and with (7.15), and the Euclidean product (., .)n in Rn ,

Ω

(∇ u, ∇ v)n dx −

(−Δ u) vdx = Ω

∂ΩD

∂u vdS − ∂ν

gN vdS. ∂ΩN

(7.18)

460

7. Discontinuous Galerkin methods (DCGMs)

Now we have to discuss, e.g. the ∂ΩD ∂u/∂ν vdS, vanishing for v ∈ Vb . The restriction v|∂Ω for v ∈ H 1 (Ω) yields v|∂Ω ∈ H 1/2 (∂Ω) in the trace sense. This is, for our Ω in (7.12), an immediate consequence of Theorems 1.37, 1.39. Remark 7.1. For Neumann conditions (7.15), and for the evaluation of the ∂u/ ∂ν|∂ΩD in (7.18) and the formulation of the DCGMs, the situation is more complicated. An arbitrary u ∈ H 1 (Ω) does not allow a meaningful restriction to ∂Ω or to e for ∂u/∂ν ∈ L2 (∂Ω) or ∂u/∂ν ∈ L2 (e). There are two ways out of this dilemma. Either Theorems 1.37, 1.39 are used for replacing the above u ∈ H 1 (Ω) by u ∈ H 3/2+ (Ω) with any > 0. Then ∂u/∂ν| ∂Ω ∈ H (∂Ω) ⊂ L2 (∂Ω) or ∂u/∂ν|e ∈ L2 (e) is valid. Hence, the ∂Ω v∇udS, and e v∇udS, are well deﬁned, cf. (7.42). Or u ∈ H 1 (Ω) is a function in the domain, e.g. of the Laplacian, then again e v∇udS are well deﬁned. This problem of boundary traces has attracted some attention recently, cf. Taylor [618], Chapter 4, Proposition 4.5, and Jonsson and Wallin [425], Jerison and Kenig [419], Schwab [575] and Grisvard [373]. Readers not so familiar with the H 3/2+ (Ω) results can in this chapter nearly everywhere replace H 3/2+ (Ω) by H 2 (Ω), with ∂u/∂ν|∂Ω ∈ L2 (∂Ω) and ∂u/∂ν|e ∈ L2 (e), thus requiring a bit more than necessary. So we assume u ∈ H 3/2+ (Ω), hence ∂ΩD ∂u/∂ν vds is well deﬁned, and ∂u/∂ν vdS = 0 for our v ∈ Vb . This allows the transition, motivated by (7.18), ∂ΩD to the following well-deﬁned forms in their appropriate spaces: a(u, v) = ε (∇u, ∇v)n dx, u ∈ U := H 1 (Ω) ∩ L∞ (Ω), v ∈ Vb cf. (7.17): (7.19) Ω

n ∂fs (u) b(u, v) = v dx, Ω s=1 ∂xs

u ∈ U, v ∈ L2 (Ω), cf. (7.16) (a),

(v) = (g, v)L2 (Ω) − (gN , v)L2 (∂ΩN ) ,

v ∈ L2 (Ω), v|∂Ω ∈ L2 (∂Ω).

(7.20) (7.21)

These a(u, v), b(u, v), (v) allow introducing the nonlinear operator G : G : u ∈ D(G) ⊂ U → U : G(u), v − b(u, v) − a(u, v) + (v) = 0 ∀ v ∈ Vb .

(7.22)

A weak solution, u0 , of (7.13) – (7.15) satisﬁes the weak equation u0 ∈ U : G(u0 ), v = 0

∀ v ∈ Vb and u0 |∂ΩD = uD on ∂ΩD

(7.23)

∂u0 |∂ΩN = gN on ∂ΩN . ∂ν A regular solution u0 ∈ H 2 (Ω) of (7.23) solves (7.13)–(7.15) as well. In Chapter 2, we discussed existence and uniqueness results for a large class of problems, generalizing (7.13), (7.14), (7.23). A classical solutions solves (7.23) for ∀ v ∈ Vb . Vice versa, if (7.23) has smooth input data and a smooth enough solution, u0 ∈ H 2 (Ω), then its solves (7.13),(7.14). So we assume a unique solution u0 of (7.23) with boundedly invertible G (u0 ). A similar discussion applies to the more general cases, (7.74), (7.78), (7.79) and (7.90), (7.91).

7.3. Discretization of the problem

7.3 7.3.1

461

Discretization of the problem Triangulations

We formulate the DCGM on a polyhedral domain Ω ⊂ Rn , n ≥ 2, bounded polyhedral with boundary ∂Ω ∈ C 0,1 .

(7.24)

Generalizations to curved domains Ω are possible, admitting Ω ⊂ Rn , n ≥ 2, bounded curved domain with boundary ∂Ω ∈ C 0,1 ,

(7.25)

if we require that, except at the edges of ∂Ω, the solution u0 ∈ H k (Ω) can be extended into u0 ∈ H k (Ωe ) with an Ωe ⊃ Ω. Under this condition, we can employ the technique in Section 5.2, Theorem 5.4, for extending DCGMs on polyhedral to curved domains. We can obtain at most second order accuracy, if we choose a sequence of approximating polygonal Ωh for Ω, s.t. dist(∂Ω, ∂Ωh ) ≤ Ch2 .

(7.26)

Higher order versions would require Theorems 7.6 and 7.5 also for curved elements. Theorem 7.6 is proved for this case; for Theorem 7.5 we did not succeed. So we restrict the formulation of the following DCGMs to polyhedral domains. Let T h (h > 0) be a partition of the domain Ω into a ﬁnite number of open n-dimensional (convex or nonconvex) mutually disjoint polyhedra K. In contrast to triangles T ∈ T h , for the standard FEMs, we denote the often more general elements here as K. In 2D problems we usually choose K ∈ T h as triangles or quadrilaterals, or as in Figure 7.1. In 3D, K ∈ T h can be, e. g. tetrahedra, pyramids or hexahedra. But we can construct even more general elements K, as dual ﬁnite volumes from [314]. In the analysis carried out in Section 7.7 it is important to assume star-shaped elements K ∈ T h . Then the standard error and inverse estimates in Theorems 7.6, 7.7 are valid, cf. Theorems 4.17, 4.19, and [141], p. 104. The exact assumptions on T h will be given in Section 7.7.1. We modify the notation for a triangulation, T h , of Ω as ∀K ∈ T h : K is open, ∀K1 = K2 ∈ T h : K1 ∩ K2 = ∅, Ω = ∪K∈T h K, T

h

(7.27)

= {K}K∈T h with hK = diam(K), h = maxK∈T h hK and {e}e∈T h ,

the set of all faces of T h . Then for any e ∈ T h there are two possibilities: (i) e ∈ T h \ ∂Ω is an inner face, so there exist two elements Kl , Kr ∈ T h , Kl = Kr sharing e, i.e. e ⊂ K l ∩ K r . 45 If K l ∩ K r is not a straight face we split it into a minimal possible number of straight faces, see Figure 7.1, showing a possible 2D situation. It is arbitrary which one of the elements sharing the inner face is denoted as Kl and as Kr , but this choice should be kept. (ii) e ⊂ ∂Ω, often denoted as e ∈ ∂Ω, is a boundary face. Then there exists an element Kl ∈ Th such that e ⊂ ∂Kl . By νe := ν := (ν1 , . . . , νn ) we denote the unit normal on the face e, which is the outer normal to the element Kl sharing e (see Figure 7.1). Moreover, the symbols 45

Subscripts

l

and

r

indicate “left” and “right” elements sharing e.

462

7. Discontinuous Galerkin methods (DCGMs)

Kl

e1

e2

Kr

νc1 νc2

Figure 7.1 Neighboring elements Kl , Kr sharing two faces e1 , e2 ∈ T h .

|K| and |e| mean the n-, and (n − 1)-dimensional Lebesgue measure of K, and of e, respectively, (i. e. the length and the area of e, if n = 2 and n = 3), and diam(e) its diameter. It is obvious that for K ∈ T h and its face e ⊂ K we have ≤ C1 hn−1 , d(e) := diam(e) ≤ hK ≤ C1 h. (7.28) |K| ≤ C1 hnK ≤ C1 hn , |e| ≤ C1 hn−1 K 7.3.2

Broken Sobolev spaces

We recapitulate the broken Sobolev space, norms and seminorms over the triangulation T h only for the case of Hilbert spaces, see (4.35), (4.36). H k (T h ) = {v : v|K ∈ H k (K) ∀ K ∈ T h }, ⎞1/2 ⎛ vH k (T h ) = ⎝ v2H k (K) ⎠ ,

(7.29) (7.30)

K∈T h

⎛ |v|H k (T h ) = ⎝

⎞1/2 |v|2H k (K) ⎠

.

(7.31)

K∈T h

For v ∈ H 1 (T h ), we recall jumps and means, cf (5.82): for a face e ∈ T h ∩ K we extend v to K; for a boundary face e ∈ ∂Ω, or an inner face with e = K l ∩ K r , let vl = v|K l , vr = v|K r , [v] = [v]e = vl |e − vr |e , {v} = {v}e = (vl |e + vr |e )/2, (7.32) [v] = [v]e = {v} = {v}e = v|∂Ωh along e ∈ ∂Ωh , v arbitrary in Rn \Ωh .

(7.33)

Obviously, the value [v]e depends upon the setting of which of the sharing element is Kl and which is Kr . On the other hand, the interesting term νe [v]e is independent of the choice of Kl and Kr . In the following, omitting the index, e , we agree [v]({∇u}, ν)n means [v]e ({∇u}e , νe )n etc. e∈T h

e

e∈T h

e

7.3. Discretization of the problem

7.3.3

463

Extended variational formulation of the problem

In order to derive the discrete problem, we start from the strong solution u0 ∈ H 2 (Ω) ∩ L∞ (Ω) of (7.13). We consider (7.13) on each K, multiply it by an arbitrary v ∈ H 1 (T h ), integrate over each K ∈ T h , apply Green’s theorem and summarize over K ∈ T h . Note that we no longer require ∈ H01 (T h ). Whenever deﬁned, we obtain the identity − (f (u0 ), ∇v)n dx + (f (u0 ), ν)n v dS (7.34) K∈T h

K

∂K

K

with

(∇u0 , ν)n v dS − ∂K

(f (u), ν)n =

n

g v dx = 0 ∀v ∈ H 1 (T h )

(∇u0 , ∇v)n dx − ε

+ε

fs (u) νs , and (f (u), ∇v)n =

s=1

K n

fs (u) ∂s v.

s=1

We combine (7.32), (7.33) with (5.259), obtaining for the sum of the second boundary integrals in (7.34), whenever deﬁned, (∇u, ν)n v dS = (7.35) −ε K∈T h

−ε

∂K

[v]({∇u}, ν)n + {v}([∇u], ν)n dS − ε v(∇u, ν)n dS.

e∈T h \∂Ω

e

e∈∂Ω

e

Similarly, we obtain as in (5.259), (5.260), (f (u), ν)n v dS K∈T h

=

(7.36)

∂K

[v]({f (u)}, ν)n + {v}([f (u)], ν)n dS + v(f (u), ν)n dS.

e∈T h \∂Ω

e

e∈∂Ω

e

We have to discuss the question of when the terms in (7.34), (7.35), (7.36) are deﬁned. The extended variational form (7.34) is certainly not deﬁned for the original u0 ∈ U as in (7.19). Instead (7.34) motivates spaces u0 , u ∈ Ueo = H 3/2+ (Ω) for the original problem, or even an extension to T h , allowing ∂u/∂ν|e , ∂v/∂ν|e ∈ L2 (e). Similarly we will choose Ve = Ue for (7.42) ﬀ. We use Vb in (7.17) and introduce, always restricting U, and for simplicity the V as well, to ∩L∞ (Ω), U = V = H 1 (Ω) ∩ L∞ (Ω), U ⊂ H −1 (Ω), Vb , Ub = H01 (Ω) ∩ L∞ (Ω),

(7.37)

Ueo := Veo := H 3/2+ (Ω) ∩ L∞ (Ω), Ue := Ve := H 3/2+ (T h ) ∩ L∞ (T h ), . . . , and for the strong forms Us = H 2 (Ω) ∩ L∞ (Ω), Us,e := H 2 (T h ) ∩ L∞ (Ω). We are going to introduce DCGMs. They are deﬁned on S h ⊂ H 3/2+ (T h ) ∩ Ue . So a combination now of the exact solution u0 ∈ Ue , with v ∈ Ue , the Neumann condition

464

7. Discontinuous Galerkin methods (DCGMs)

(7.15), and (7.34), (7.35), (7.36), yields the following equation: o (f (u0 ), ∇v)n dx u0 ∈ Ue ⊂ Ue : −

+

K∈Th

−ε

e∈∂ΩD

[v]({f (u0 )}, ν)n + {v}([f (u0 )], ν)n dS + v(f (u0 ), ν)n dS

K

e∈T h \∂Ω

e

v(∇u0 , ν)n dS −

e

[v]({∇u0 }, ν)n + {v}([∇u0 ], ν)n dS

g v dx − ε Ω

e

e∈∂Ω

(∇u0 , ∇v)n dx − ε

(7.38)

K

e

e∈T h \∂Ω

+ε

K∈T h

gN v dS = 0 ∀v ∈ U. ∂ΩN

There are diﬀerent ways for further modifying the diﬀusion terms: The unchanged ε e∈T h e [v]({∇u}, ν)n dS, based upon the vanishing ([∇u0 ], ν)n , is used by Wheeler [665], p. 153, extended to the case of nonlinear diﬀusion in (7.1). Here we use the fact that for the exact solution u0 ∈ H 3/2+ (Ω) ⊂ Ue the terms [f (u0 )], [∇u0 ] and (u0 − uD )|∂ΩD in (7.38) vanish. Furthermore, preparing the DCGMs, we ﬁrst omit the terms {v}([f (u0 )], ν)n , {v}([∇u0 ], ν)n . Then we add to (7.38), for e ∈ T h \ ∂Ω, 0 = θε({∇v}, ν)n [u0 ], and, for e ∈ ∂ΩD , 0 = θε(∇v, ν)n (u0 − uD ). This is used in the integration over the edges e in (7.38). The standard choices for θ, are, see below, θ = −1, or θ = 1, or θ = 0, sometimes even θ ∈ [−1, 1]. For the DCGMs we need the function σ, deﬁned by, cf. (7.28), σ:

C

e → R : σ|e =

e∈T h

cw , with d(e) = diam(e), cw > 0. d(e)

(7.39)

Another possibility is to deﬁne σ, e.g. as σ|e =

cw max{diam(Ke1 ), diam(Ke2 )}

or

σ|e =

cw min{|Ke1 |, |Ke2 |} , d(e)

(7.40)

where Ke1 and Ke2 are neighboring elements sharing face e, and Ke1 = Ke2 for e ⊂ ∂Ω. These choices of interior penalties lead to equivalent results. Moreover, we should note that cw may depend on the degree of the polynomial approximation. Finally, we add the following penalty terms to the last two lines of (7.38). σ [u] [v] dS, ε σ (u − uD ) v dS. (7.41) ε e∈T h \∂Ω

e

e∈∂ΩD

e

This allows the desired coercivity results for DC piecewise polynomial test functions v h ∈ S h , cf. (7.63), Theorems 7.16–7.19. They are based upon the penalty norm

7.3. Discretization of the problem

465

vJH 1 (T h ) = v, see (7.153). But the original vH 1 (T h ) and the new vJH 1 (T h ) are nonequivalent norms on H 1 (T h ) for h → 0, see Theorem 7.12. We will formulate the corresponding consistency results in Theorems 7.21–7.33, tested by v h ∈ S h , linear and bounded with respect to vJH 1 (T h ) . All the terms in (7.38), (7.41), including those evaluating u, ∇u0 , v, ∇v along edges, e, are well deﬁned ∀u, v ∈ Ue , e ∈ T h , and many of them vanish or equal uD for e ∈ T h , for the exact solution u0 ∈ H 3/2+ (Ω), cf. Theorem 1.37 on trace operators. This would be incorrect for a u0 ∈ U. We use (7.32), (7.33), (7.35), and collect e∈T h \∂Ω and e∈∂ΩD into e∈T h \∂ΩN , now for u0 , v ∈ H 3/2+ (Ω) = Ueo . Then we obtain o u0 ∈ Ue : − (f (u0 ), ∇v)n dx + [v]({f (u0 )}, ν)n dS (7.42)

+ε

K∈Th

⎛

K

−

e∈T h \∂ΩN

e

σ[u0 ] [v] dS − e

[v]({∇u0 }, ν)n + θ({∇v}, ν)n [u0 ] dS,

e∈∂ΩD

gN v dS − εθ

g v dx + ε Ω

e∈T h \∂ΩN

e

e∈T h

(∇u0 , ∇v)n dx − ε

+ ε⎝

K

K∈T h

∂ΩN

⎞ σuD v dS ⎠

e

e∈∂ΩD

(∇v, ν)n uD dS

= 0 ∀v ∈ Ue .

e

For the exact solution u = u0 of (7.23), satisfying the boundary conditions (7.14), and the test functions v ∈ H 3/2+ (Ω) ⊂ Ue we obtain [u0 ] = [u0 ]e = 0, {∇u0 } = {∇u0 }e = ∇u0 |e ,

and similarly for v,

(7.43)

thus we get the weak form (7.23), and the strong form (7.13) for u = u0 ∈ Us . For the “convective” face integrals [v]({f (u0 )}, ν)n dS, (7.44) e∈T h

e

the more sophisticated concept of a numerical ﬂux in DCGMs is employed. This is motivated, e.g. by ﬁnite volume techniques for conservation laws. There the use of the numerical ﬂux ensures the stability of numerical schemes using explicit time discretization, see, e.g. [313, 315]. For the diﬀerent choices of numerical ﬂuxes the present type of problems plays a crucial role. For example, the center numerical ﬂux ({f (u0 )}, ν)n is unconditionally unstable for explicit discretization of time dependent problems. However, for implicit time discretization we have no stability restriction, but we have to solve a system of nonlinear algebraic equations at each time step. Our results below show stability and convergence for elliptic problems.

466

7. Discontinuous Galerkin methods (DCGMs)

In our context the numerical ﬂux emphasizes the evaluation of the term (f (u), ν)n appearing as the integrand in the ﬁrst line of (7.34). We consider the edge, e, again as the face of Kl and Kr , e ⊂ K l ∩ K r given by (7.27) ﬀ., (7.32) with the outer normal, ν. The numerical ﬂux H(ul |e , ur |e , ν) approximates the evaluation along e as the face of Kl in the ﬁrst argument of H(·, ·, ·) and indicates the impact of e as the face of Kr in its second argument. A famous example is the Lax–Friedrich numerical ﬂux in the form

1 P (ul |e , ν) + P (ur |e , ν) − λ(ur |e − ul |e ) , λ > 0, (7.45) H(ul |e , ur |e , ν) = 2

1 (f (ul |e ), ν)n + (f (ur |e ), ν)n − λ(ur |e − ul |e ) (e.g.) = (7.46) 2

1 = ({f (u|e )}, ν)n − λ[u|e ] (7.47) 2 hence, P (u, ν) := (f (u), ν)n , for (7.46). Another example is the upwind scheme given by . n (f (ul |e ), ν)n = s=1 fs (ul |e )νs , if A > 0 H(ul |e , ur |e , ν) = n (f (ur |e ), ν)n = s=1 fs (ur |e )νs , if A ≤ 0,

(7.48)

where A=

n

fs ({u}e )νs = (f ({u}e ), ν)n .

(7.49)

s=1

The face integrals (7.44) correspond to the central numerical ﬂux H(ul |e , ur |e , νe ) = ({f (u|e )}, νe )n

∀e ∈ T h .

(7.50)

For further examples of numerical ﬂuxes we refer to, e.g. [313]. We shall assume that the numerical ﬂux has the following properties: Assumptions (H): 1. H(u, v, ν) is deﬁned in R2 × B1 , where B1 = {ν ∈ Rn ; |ν| = 1}, and Lipschitzcontinuous with respect to u, v: |H(u, v, ν) − H(u∗ , v ∗ , ν)| ≤ C2 (|u − u∗ | + |v − v ∗ |),

(7.51)

u, v, u∗ , v ∗ ∈ R, ν ∈ B1 . 2. H(u, v, ν) is ﬂux-consistent: H(u, u, ν) = (f (u), ν)n ,

u ∈ R, ν ∈ B1 .

(7.52)

u, v ∈ R, ν ∈ B1 .

(7.53)

3. H(u, v, ν) is conservative: H(u, v, ν) = −H(v, u, −ν),

467

7.3. Discretization of the problem

Moreover, (7.51) and (7.52) imply that the function f is Lipschitz-continuous with constant Lf = 2C2 and from (7.16) (a) and (7.52) we see that H(0, 0, ν) = 0 ∀ ν ∈ B1 .

(7.54)

Obviously, numerical ﬂuxes given by (7.45), (7.48) and (7.50) satisfy Assumptions (H). So we obtain in the ﬁrst line of (7.34), (7.42) with (7.33) the terms (f (u), ν)n v dS ∂K

K∈T h

=

H(ul |e , ur |e , νe )vl |e dS

+

(7.55)

e

e∈∂Ω

H(ul |e , ur |e , νe )(vl |e − vr |e ) dS +

e∈Th \∂Ω

=

H(ul |e , ur |e , νe )vl |e + H(ur |e , ul |e , −νe )vr |e dS

e

e∈Th \∂Ω

=

e

H(ul |e , ur |e , νe )vl |e dS

e∈∂Ω

e

H(ul |e , ur |e , νe )[v]e dS, ∀v ∈ H 1 (T h ).

e

e∈Th

Consequently (7.42) has the form u0 ∈ Ue : − (f (u0 ), ∇v)n dx + H(u0,l |e , u0,r |e , νe )[v]e dS K

K∈T h

+ε

K∈Th

K

−

ε

σuD v dS − εθ

e

[v]({∇u0 }, ν)n + θ({∇v}, ν)n [u0 ] dS

g v dx − Ω

gN v dS ∂ΩN

e∈∂ΩD

(∇v, ν)n uD dS

−ε

= 0 ∀v ∈ Ue .

e

On the basis of equations (7.42) and (7.56) we introduce: (∇u, ∇v)n dx ∀u, v ∈ Ue : ah (u, v) := ε K∈Th

(7.56)

e

e

e∈T h \∂ΩN

σ[u0 ] [v] dS −

e∈∂ΩD

e

e∈T h \∂ΩN

(∇u0 , ∇v)n dx − ε

+ε

e∈Th

(7.57)

K

e∈T h \∂ΩN

e

({∇u}, ν)n [v] + θ({∇v}, ν)n [u] dS

468

7. Discontinuous Galerkin methods (DCGMs)

bh (u, v) = −

K∈T

ch (u, v) := −

h

({f (u)}, ∇v)n dx +

K

K∈T h

e∈T

({f (u)}, ∇v)n dx +

K

:=

e∈T h \∂ΩN

(7.58)

H(ul , ur , ν)[v] dS

(7.59)

e

(∇v, ν)n uD dS

(7.60)

e

σuD v dS with

e

e∈∂ΩD

Jhσ (u, v)

e∈∂ΩD

({f (u)}, ν)n [v]dS

e

h

e∈Th

h (v) := g, v + (gN , v)L2 (∂ΩN ) − εθ +ε

σ[u] [v] dS, and e

Jh1 (u, v)

:=

e∈T h \∂ΩN

[u] [v] dS.

(7.61)

e

In the discrete problem the form Jhσ represents a penalty function for interior discontinuities along the edges e. The boundary errors are collected in the terms ε e∈∂Ω e σ(uD v − u v)dS obtained from εJhσ (u, v) − h (v). These terms replace the requirement of continuity and boundary conditions of the approximate solution in conforming FE methods. Note that for increasingly ﬁner T h the σ in (7.39) increases. Thus violated conditions are punished more strongly. For some of the later discussions we mainly consider uh , v h ∈ S h , with bounded Jhσ (v, v), e.g. those well approximating u, v ∈ U. This is achieved by introducing the so-called penalty norms, cf. Deﬁnition 7.11. So we have modiﬁed the above (7.38). These (7.57)–(7.61) allow formulating diﬀerent DCGMs. As mentioned above, the standard choices for θ, are θ = −1, or θ = 1, or θ = 0, sometimes even θ ∈ [−1, 1],

(7.62)

with θ = −1 for the so-called nonsymmetric form of the interior penalty Galerkin method, NIPG cf. Rivi`ere et al. [552, 553, 555], θ = 1 for the symmetric form, SIPG, cf. Arnold [39], θ = 0 for the incomplete form, IIPG, cf. Dawson et al. [269, 615, 616], and −1 ≤ θ ≤ 1 for the general form for the general penalty Galerkin method, GIPG, cf. Houston et al. [407]. Hence ah (u, v) = ah (v, u) or, for θ = 1, the ah (u, v) = ah (v, u) is obtained. Our analysis includes all four cases. Standard numerical methods essentially use θ = −1, 0, 1. For θ = −1, we only have to impose cw > 0. For θ = 0, 1, and θ ∈ (−1, 1] we have to observe lower bounds for cw > 0, cf. Theorem 7.16 ﬀ. There we will come back to the discussion of appropriate choices of the cw > 0. These diﬀerent choices have “pros and cons”: The symmetric form maintains the symmetry of the previous and later a(u, v) = a(v, u) with desirable consequences for numerical methods. However, the cw in (7.39) has to be chosen, for the symmetric form, the incomplete form and for θ ∈ (−1, 1] according to (7.164), (7.180), (7.194). This implies coercive principal parts of the corresponding linearized discrete operators. This is valid for the unsymmetric form for any cw > 0.

7.3. Discretization of the problem

469

The symmetric bilinear form ah (·, ·) allows deriving optimal a priori error estimates in the L2 norm using the well-known Aubin–Nitsche trick, cf. Lemma 5.73 and Theorem 5.74 for FEMs and [40] for DCGMs and the references therein. Nevertheless, the methods with θ = 1 give in some situations optimal experimental order of convergence, namely for odd degrees of polynomial approximations when (almost) equidistant partitions are employed. On the other hand, for ensuring the coercivity for the symmetric form it is necessary to choose the penalty parameter σ, in particular the cw , more carefully than in the nonsymmetric case, cf. Theorems 7.16, 7.18, 7.19. For general quasilinear problems we only formulate this last type of IIPGM below. 7.3.4

Discretization

Now we introduce the discrete problem approximating (7.56) via the DCGM. We deﬁne the space of discontinuous piecewise polynomial functions S h := S d−1,−1 (T h ) := {v : T h → R; v|K ∈ Pd−1 (K) ∀ K ∈ T h }, d ≥ 2,

(7.63)

where Pd−1 (K) denotes the space of all polynomials on K of degree ≤ d − 1 ∈ N. We derive the discrete problem in such a way that u0 in (7.56) is approximated by uh0 ∈ S h and tested by v h ∈ S h . In the open K ∈ T h , ∇uh0 , ∇v h and their restrictions to e ⊂ K are well deﬁned for uh , v h ∈ S h ⊂ H 3/2+ (T h ) ⊂ H 1 (T h ). Hence the forms ah (u, v), ch (u, v), bh (u, v), h (v), Jhσ (u, v), cf. (7.57)–(7.61) make sense for u, v ∈ Ue and u = uh , v = v h ∈ S h . We have seen that extended analytical problems require u, v ∈ Ue . So it is surprising and advantageous that for the DCGMs a combination of Jhσ (v, v) and |v|2H 1 (T h ) is suﬃcient for proving the boundedness, and consistency of the previous ah (·, ·), bh (·, ·), ch (·, ·), h (·), and the coercivity of the principal part of and the stability for ah (·, ·) with respect to such a norm. This so-called penalty norm, cf. (7.61), and Deﬁnition 7.11, is 1/2 . v := vV h := vJH 1 (T h ) := Jhσ (v, v) + |v|2H 1 (T h )

(7.64)

This norm is chosen for U h = V h = S h , cf (7.153) uh , v h ∈ U h = V h = S h with v h U h = v h V h := v h JH 1 (T h ) = v.

(7.65)

It is not equivalent to the broken Sobolev norm for h → 0, cf. Theorem 7.12. Before we formulate the DCGMs, we discuss the relations between the original and discrete operators, G and Gh . This is essentially the same idea for all the following problems. So we demonstrate it for our model problem. With the a(u, v), . . . in (7.19)–(7.21), the ah (u, v), . . . , Jhσ (u, v) in (7.57)–(7.61), and, e.g. the Gh (u), v = Gh (u), vUe ×Ue , we deﬁne, for a given ﬁxed u or uh , the weak operators G, Gh , Gh , by testing with v or v h . We indicate this in (7.66), e.g. by u, ∀v ∈ U. For G we use the b(u, v) in (7.20), and in Gh we allow replacing ch (u, v) by bh (u, v) in (7.66). For G we have two possibilities: either we impose the Dirichlet condition directly as in

470

7. Discontinuous Galerkin methods (DCGMs)

(7.66) (b), or we enforce them for h → 0 as in (7.66) (c) by the penalty terms.

(a) G : D(G) ⊂ U → U , Gh : D(Gh ) ⊂ Ue → Ue , Gh : D(Gh ) ⊂ U h → U h ,

(7.66)

(b) G(u), v − a(u, v) − b(u, v) + (v) = 0 ∀ v ∈ Vb , (u − uD )|∂ΩD = 0, u ∈ U, (c) G(u), v − ah (u, v) − bh (u, v) − εJhσ (u, v) + h (v) = 0, u, ∀ v ∈ Ueo , (d) Gh (u), v − ah (u, v) − bh (u, v) − εJhσ (u, v) + h (v) = 0, u, ∀ v ∈ Ue , (e) Gh (uh ), v h − ah (uh , v h ) − bh (uh , v h ) − εJhσ (uh , v h ) + h (v h ) = 0, uh , ∀v h ∈ S h , ah (uh , v h ), bh (uh , v h ) := ah (uh , v h ), bh (uh , v h )|S h ×S h , h (v h ) := h (v h )|S h , or we may replace in (c)–(e) the bh by ch . In (c) we either have to consider u, ∀v ∈ Ueo , or u, ∀v ∈ U, with [u] = [v] = 0 for interior faces e and such that u|∂Ω , v|∂Ω ∈ L2 (∂Ω). For relating G, Gh , and Gh we use the previous Ueo = H 3/2+ (Ω) ∩ L∞ (Ω), cf. (7.37). For all u, v ∈ Ueo , many terms in Gh cancel: u, v ∈ Ueo and [u]e = [v]e = 0, for all interior faces e ∈ T h , imply

(∇u, ν)n v + θ(∇v, ν)n u dS (∇u, ∇v)n dx − ε ah (u, v) := ε K

K∈Th

bh (u, v) = −

K∈T h

ch (u, v) := −

K∈T h

(f (u), ∇v)n dx +

K

(f (u), ∇v)n dx +

K

h (v) := g, v + (gN , v)L2 (∂ΩN ) − εθ Jhσ (u, v) :=

(f (u), ν)n vdS

e∈∂Ω

e

e∈∂Ω

e

H(ul , ur , ν)v dS

e∈∂ΩD

e∈T h

e

e∈∂ΩD

(∇v, ν)n uD dS + ε

e

e∈∂ΩD

σuD v dS

e

σ[u] [v] dS.

e

We collect these terms in Gh (u), v and ﬁnd ∀ u, v ∈ Ueo : Gh (u), v = ah (u, v) + bh (u, v) + εJhσ (u, v) − h (v) (f (u), ν)n vdS = G(u), v +

e∈∂Ω

e

(∇u, ν)n v + θ(∇v, ν)n u dS + ε σu v dS

−ε

e∈∂ΩD

+ εθ

e

e∈∂ΩD

e

(∇v, ν)n uD dS − ε

e∈∂ΩD

e∈∂ΩD

σuD v dS

e

= G(u), v∀ u ∈ Ueo , (u − uD )|∂ΩD = 0 ∀ v ∈ Ub .

e

(7.67)

471

7.3. Discretization of the problem

Note that we used for the last equality in (7.67) the ({f (u)}, ∇dxv)n b(u, v) = bh (u, v) = − K∈T h

+

K

e∈∂ΩN

(f (u), ν)n vdS ∀ v ∈ Ub .

(7.68)

e

This allows summarizing these relations as Gh (u), v = ah (u, v) + bh (u, v) + εJhσ (u, v) − h (v)∀ u, v ∈ Ueo ,

(7.69)

= G(u), v = a(u, v) + b(u, v) − (v) ∀ u : (u − uD )|∂ΩD = 0 ∀ v ∈ Ub , = G(u), v ∀ u ∈ U : (u − uD )|∂ΩD = 0, ∀ v ∈ Ub , so, with o Ue,b := {u ∈ Ueo , (u − uD )|∂ΩD = 0} , Ub := {u ∈ U, (u − uD )|∂ΩD = 0} : o o = G|Ue,b = G ↑Ub . Gh = Gh |S h , and Gh |Ue,b o o = G|Ue,b = G ↑Ub , indicates the Remark 7.2. This last equality in (7.69), Gh |Ue,b o o , and the extension of G, deﬁned in Ue,b , to G restriction from Gh , deﬁned in Ueo to Ue,b o h h deﬁned in the original Ub ⊃ Ue,b . Note that we have to test G = Gh |S h by v ∈ S , Gh u by v ∈ Ue , Gh |Ueo = G|Ueo by v ∈ Ueo , and G ↑Ub by v ∈ Ub . We observe an analogous situation for all the following problems as well.

We deﬁne the nonstandard DCGM and its discrete approximate solution as a function uh0 : T h → R satisfying the conditions, cf. (7.66), (7.69), ; : (7.70) uh0 ∈ S h ; Gh uh0 , v h = 0 ∀ v h ∈ S h . Similarly, we deﬁne the discrete approximate solution of a standard DCGM, by replacing bh in (7.58), (7.69) by ch in (7.59), for a function uh0 : Ω → R satisfying (7.71) (a) uh0 ∈ S h , s.t. ∀ v h ∈ S h , : h h h; h h h h h h h h σ h h (b) G u0 , v := a u0 , v + c u0 , v + εJh u0 , v − (v ) = 0. Many terms in (7.71) cancel, in particular Jhσ (u, v) = 0, and we have obtained Gh |Ueo = G|Ueo = G ↑U , in (7.66). Proceeding as in (7.67), (7.68), (7.69), the exact solution u0 ∈ U, obviously satisﬁes the original (7.23) as G(u0 ), v = ah (u0 , v) + ch (u0 , v) + εJhσ (u0 , v) − h (v) = 0 ∀ v ∈ Vb . (7.72) With this regular exact solution u0 we ﬁnd the identity Gh (u0 ), v h := ah (u0 , v h ) + ch (u0 , v h ) + εJhσ (u0 , v h ) − h (v h ) = 0 ∀v h ∈ S h (7.73)

472

7. Discontinuous Galerkin methods (DCGMs)

by (7.57). This implies the so-called Galerkin orthogonality property of the error. This means that, for this and the following cases, ; : : h h h; G u0 , v = 0 = Gh (u0 ), v h ∀v h ∈ S h ⇒ Gh uh0 − Gh (u0 ), v h = 0 ∀v h ∈ S h .

7.4

General linear elliptic problems

Now we are ready to formulate more general problems. For simplicity, we consider only the Dirichlet boundary condition on the whole boundary of Ω in this and the following sections, i.e. ∂Ω = ∂ΩD . A possible generalization to a case with Neumann boundary conditions on a part of the boundary can be carried out simply as in the previous section. We have fully formulated FEMs for elliptic linear or semi- and quasilinear second order equations and systems. So we restrict the presentation here to linear equations and to semilinear equations and quasilinear systems in Section 7.5. The linear boundary value problem and its DCGMs are discussed brieﬂy. We give more details for the following semilinear second order equations. Since there are strong similarities we do not repeat all details for extending (7.13), (7.14) ﬀ. to general linear equations. Compared to (7.13), we replace the original Δu, ∂u/∂ν by the general Au, (Ba ∇u, ν)n or Au, (Ba ∇0 u, ν)n in the following (7.74), (7.75) and omit the ﬂux terms, and the restriction to L∞ (Ω) in U, . . . , in (7.37). As usual, multiplying the strong operator, As , with v ∈ U, we obtain by partial integration and Green’s formula, the relation to the weak operator, A, cf. (5.278): Compute 46 u0 ∈ U := H01 (Ω) : a(u0 , v) = Au0 , vV ×V = f, vV ×V ∀v ∈ U, with (7.74) ⎛ ⎞ n ⎝ a(u, v) = Au, vV ×V = aij ∂ i u ∂ j v ⎠ dx with max{|aij (x)|} ≤ C0 , Ω

=

⎛ ⎝

Ω

n

x∈Ω

i,j=0

⎞ (−1)j>0 ∂ (aij ∂ u)⎠ vdx +

i,j=0

j

(Ba ∇0 u, ν)n vds

i

∂Ω

(Ba ∇0 u, ν)n vds

=: (As u, v)L2 (Ω) + ∂Ω

∀v ∈ U, u ∈ Us = H 2 (Ω), if aij ∈ W 1−δj0 ,∞ (Ω), with (Ba ∇0 u, ν)n :=

j=1,...,n

νj aij ∂ i u, and

i=0,...,n n

aij (·)ξi ξj ≥ λ|ξ|2n , λ > 0, for our elliptic operators.

i,j=1 46 We require max ∞ x∈Ω {|aij (x)|} ≤ C0 instead of the usual aij ∈ L (Ω), since we need bounds for aij L∞ (e) along e ∈ T h as well, not available for aij ∈ L∞ (Ω).

7.4. General linear elliptic problems

473

This (Ba ∇0 u, ν)n depends upon ∇0 u = (∂ i u)ni=0 , e.g. As u = −Δu yields (Ba ∇0 u, ν)n = (Ba ∇u, ν)n = ∂u/∂ν = (∇u, ν)n . Again we replace H k (Ω) by H k (T h ) and obtain with Green’s formula as for (7.34) ﬀ., with U h , V h in (7.63), (7.65), and with (7.32), (7.33), Ah u, vH −1 (T h )×H 1 (T h ) ⎛ ⎞ n ⎝ := aij ∂ i u∂ j v ⎠ dx K i,j=0

K∈T h

=

⎡

⎣(As,h u)v :=

K

K∈T h

+

e∈∂Ω

n

⎤ (−1)j>0 ∂ j (aij ∂ i u) v ⎦ dx

(7.75)

i,j=0

e∈T h \∂Ω

+

∀u, v ∈ Ue ⊃ U h

& % [v]{(Ba ∇0 u, ν)n } + {v} (Ba ∇0 u, ν)n ds

e

v(Ba ∇0 u, ν)n ds

e

=: as,h (u, v) +

e∈T h

& % [v]{(Ba ∇0 u, ν)n } + {v} (Ba ∇0 u, ν)n ds.

e

As above, for preparing the DCGMs, we consider several terms, which vanish for the exact solution u0 = u ∈ Ue , and v h = v ∈ V h . We omit, for all e ∈ T h , the {v h }[(Ba ∇0 u, ν)n ]. We add, for all e ∈ T h \ ∂Ω, 0 = θ{(Ba ∇0 v, ν)n }[u], and for e ∈ ∂Ω, 0 = θ(Ba ∇0 v, ν)n (u − uD ). Again the nonsymmetric, symmetric, incomplete and general form are deﬁned by θ = −1, θ = 1, θ = 0 and θ ∈ [−1, 1], respectively. For all u, v ∈ Ue we use the Jhσ (u, v) in (7.61), and introduce, cf Remark 7.2: ⎛ ⎞ n i j ⎠ ⎝ ah (u, v) := {(Ba ∇0 u, ν)n }[v] aij ∂ u ∂ v dx − (7.76) K∈Th

K i,j=0

e∈T h

e

+ θ{(Ba ∇0 v, ν)n }[u] dS for u, v ∈ Ue 0 h (v) := g, v − θ ((Ba ∇ v, ν)n ) uD dS + σuD v dS and e∈∂Ω

e

e∈∂Ω

e

Ah u, v − ah (u, v) − Jhσ (u, v) + h (v) = 0 for a ﬁxed u ∈ Ue , ∀ v ∈ Ue , with ah (·, ·) := ah (·, ·)|U h ×V h , a(·, ·) = ah (·, ·)|U ×V , ahs (·, ·) := as,h (·, ·)|U h ×L2 (Ω) , as (·, ·) = as,h (·, ·)|Us ×L2 (Ω) h (·) := h (·)|V h , (·) = h (·)|U , and again, Ah u, v = Au, v ∀u, v ∈ Ue and = Au, v ∀u, v ∈ U.

474

7. Discontinuous Galerkin methods (DCGMs)

This yields for linear elliptic equations the DCGM and its discrete approximate solution as a function uh0 ∈ S h satisfying the conditions ; : (7.77) uh0 ∈ S h : Ah uh0 , v h = 0 ∀ v h ∈ S h = V h . The Galerkin orthogonality results in (7.73) remain correct with the obvious modiﬁcations. Another interesting type, the advection–diﬀusion–reaction problem, is studied by Antonietti and Houston [34]

7.5

Semilinear and quasilinear elliptic problems

7.5.1

Semilinear elliptic problems

Next we discuss semilinear boundary value problems, from now on only with Dirichlet boundary conditions. Determine the exact solution u0 : Ω → R such that G(u0 ) :=

n

∂ s fs (u0 ) − ε

s=1

n

(∂ j aij (x, u0 )∂ i u0 − g(x, u0 ) = 0 in Ω,

(7.78)

i,j=1

u0 |∂Ω = uD on ∂Ω.

(7.79) n We choose, similarly to (7.13), i,j=1 instead of an equally possible i,j=0 , n . and ∂ΩD = ∂Ω. We do not formulate the obvious modiﬁcations for i,j=0 This generalizes (7.13), (7.14). Its linearized second order principal part of the form n j i a (x, u (x))∂ ∂ u) is a special case of (7.75). The special case 0 i,j=1 ij n

∇ · a(x, u)∇u =

n

∂ i a(x, u) ∂ i u

i=1

of (7.78) has been discussed by Rivi`ere and Wheeler [552] for parabolic problems. The ellipticity of (7.78) is deﬁned by its linearized principal part, cf. (7.80) (e). We assume the following conditions in a form that is simply to handle, e.g. (c): (a) fs ∈ C 1 (R), fs (0) = 0, s = 1, . . . , n,

(7.80)

(b) ε > 0, (c) aij : Ω × R → R, i, j = 1, . . . , n, are bounded by |aij (x, u)| ≤ Λa , (d) aij are Lipschitz-continuous with respect to u, i.e. ∃L > 0 ∀x ∈ Ω, u1 , u2 ∈ R : |aij (x, u1 ) − aij (x, u2 )| ≤ L|u1 − u2 |,

i, j = 1, . . . , n,

(e) Ellipticity: For u ∈ U, with u − u0 U ≤ ρ there exist λ, Λ ∈ R+ such that 0 < λ|ϑ|2 ≤

n i,j=1

aij (x, u(x))ϑj ϑi ≤ Λ|ϑ|2

∀x ∈ Ω, ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn ,

7.5. Semilinear and quasilinear elliptic problems

475

(f) aij (·, u(·)) ∈ C 1 (Ω) for u ∈ C 2 (Ω) in (7.78), and ∈ L∞ (Ω) for u ∈ U in (7.81), (g) g : Ω × R → R, g(·, u(·)) ∈ L2 (Ω) or ∈ U for u ∈ U, (h) g ∈ U is Lipschitz-continuous with respect to u, i.e. ∃L > 0 ∀x ∈ Ω ∀u1 , u2 ∈ R : |g(x, u1 ) − g(x, u2 )| ≤ L |u1 − u2 |, (i) uD is the trace of some u∗ ∈ U on ∂Ω. A classical solution u0 ∈ Us satisﬁes (7.78)–(7.79) pointwise. For the weak solutions, u, v ∈ U, we use the notation, cf (7.19), (7.20), (7.74): n i j aij (x, u)∂ u∂ v dx = ε (Ba (u)∇u, ∇v)n dx, with (7.81) a(u, v) = ε Ω i,j=1

Ba (u)∇u :=

n

Ω

n

aij (x, u)∂ i u

,

i=1

b(u, v) =

n

j=1

∂ s fs (u) v dx, with boundary condition (7.79) u, v ∈ U.

(7.82)

Ω s=1

Thus we update the above ∇u, and a(u, v) in (7.19), by Ba (u)∇u, and a(u, v) in (7.81). A weak solution of (7.78) – (7.79) is deﬁned by the updated (7.23). 7.5.2

Variational formulation and discretization of the problem

In analogy to (7.34), we obtain ∀K ∈ T h , by replacing ∇u by Ba (u)∇u, − (f (u), ∇v)n dx + (f (u), ν)n v dS + ε (Ba (u)∇u, ∇v)n dx K∈T h

K

∂K

(Ba (u)∇u, ν)n v dS −

−ε ∂K

(7.83)

K

g(x, u) v dx

= 0 ∀v ∈ H 1 (T h ),

K

g(x, u) v dx by g(x, u), v H −1 (K)×H 1 (K) . Further considerations follow (7.38), (7.41)–(7.42): Here, we again omit the terms {v}([f (u0 )], ν)n , {v}([Ba (u0 )∇u0 ], ν)n in (7.83) and add (7.84) ε σ [u] [v] dS, θε ({Ba (u)∇v}, ν)n [u] dS, for e ∈ T h \ ∂Ω and or we replace

e

K

e

σ (u − uD )v dS, θε

ε e

(Ba (u)∇v, ν)n (u − uD ) dS, for e ∈ ∂Ω, e

with θ = −1 θ = 1, θ = 0, θ ∈ [−1, 1] for the NIPG, SIPG, IIPG, GIPG methods, respectively.

476

7. Discontinuous Galerkin methods (DCGMs)

So we obtain, instead of (7.42), again with (7.32), (7.33), and u0 replacing u, u0 ∈ Ueo = H 3/2+ (Ω) : − (f (u0 ), ∇v)n dx + [v]({f (u0 )}, ν)n dS +ε

K∈Th

K

K∈T h

(Ba (u0 )∇u0 , ∇v)n dx − ε

K

e∈T h

e

({Ba (u0 )∇u0 }, ν)n [v]

e∈T h

(7.85)

e

+ θ({Ba (u0 )∇v}, ν)n [u0 ] dS + ε σ[u0 ] [v] dS g v dx − ε

− Ω

e

e∈T h

σuD v − θ(Ba (u0 )∇v, ν)n uD dS = 0 ∀v ∈ Ueo ,

e

e∈∂Ω

cf. (7.37). Now we return to the exact solution u = u0 ∈ Ue , of (7.78), satisfying the boundary conditions, (7.79), and the test functions v ∈ Ueo ∩ Ub . Along the edges, e, we have [u0 ]e = 0, {Ba (u0 )∇u0 }e = Ba (u0 )∇u0 |e ∀e ∈ T h \ ∂Ω and {Ba (u0 )∇u0 }e = Ba (u0 )∇u0 |e , [u0 ]e = u0 |e = uD ∀e ∈ ∂Ω; similarly for v, we end up again with the weak form, the modiﬁed (7.23). On the basis of the above considerations and notation, we introduce, similarly to (7.42), (7.56), the strong and a provisional and the ﬁnal weak discrete semilinear forms, ˆh (·, ·), and ah (·, ·), by including the transition terms, as as,h (·, ·), a as,h (u, v) := −ε

ah (u, v) := ε − ε

∂ j (aij (x, u)∂ i u vdx,

5

(Ba (u)∇u, ∇v)n =

K∈T h

K

K∈T h

K

(7.86)

K i,j=1

K∈T h

a ˆh (u, v) := ε

n

n

aij (x, u)∂ i u ∂ j v

6

dx

i,j=1

(Ba (u)∇u, ∇v)n dx

(7.87)

({Ba (u)∇u}, ν)n [v] + θ({Ba (u)∇v}, ν)n [u] dS,

e∈T h

e

h (u, v) := (g(x, u), v)L2 (Ω) + ε

σuD v − θ(Ba (u)∇v, ν)n u dS

e∈∂Ω

(7.88)

e

∀ u, v ∈ Ue ⊃ S h , and ch (u, v), Jhσ (u, v) as in (7.59), (7.61). ˆh (u, v) even require u ∈ H 2 (T h ), v ∈ L2 (T h ), and u, v ∈ The as,h (u, v) and a 1 h H (T ). Or (g(x, u), v) may be replaced by g(x, u), v = g(x, u), v H −1 (T h )×H 1 (T h ) .

477

7.5. Semilinear and quasilinear elliptic problems

This yields as,h (u, v) = as (u, v)∀u ∈ Us , v ∈ L2 (Ω). Totally analogously to (7.66) we deﬁne the operators G, Gh , Gh , by replacing those ah (u, v), h (v) in (7.58), (7.59), (7.61), by the new terms in (7.87), (7.88). We do not explicitly update (7.66), cf (7.89) for Gh . We deﬁne the standard DCGM in NIPG, SIPG, IIPG, and GIPG forms and its discrete approximate solution as a function uh0 : Ω → R satisfying the conditions (7.89) (a) uh0 ∈ S h , s.t. ∀ v h ∈ S h : h h h; (b) G u0 , v := ah uh0 , v h + ch uh0 , v h + εJhσ uh0 , v h − h uh0 , v h = 0 with ah (uh , v h ), ch (uh , v h ), h (uh , v h ) := ah (uh , v h ), ch (uh , v h ), h (uh , v h )|S h ×S h . The nonstandard DCGM in NIPG, form is again deﬁned SIPG, IIPG, and GIPG by replacing in the last equation ch uh0 , v h by bh uh0 , v h . A particularly interesting problem is the stationary incompressible Navier-Stokes equation. This has to be discretized by mixed formulations of DCGMs, namely by the so-called LDG, the local discontinuous Galerkin method. It could be treated similarly to Section 4.6 on the basis of the Ladyˇzenskaja–Brezzi–Babuska condition, cf. (4.214), for the Stokes operator and for DCGMs, e.g. [191, 434]. 7.5.3

Quasilinear elliptic systems

Previously we discussed equations. The following quasilinear problems are formulated as systems of q equations of second order. Again we start with the standard divergent quasilinear systems in strong form in the Hilbert space setting. We distinguish the direct generalization of (7.78) and (7.79) in (7.90), (7.91) and its further generalization in (7.93). We determine the solution u0 ∈ D(G) ⊂ H 2 (Ω, Rq ) ∩ L∞ (Ω, Rq ) of the problem in its strong form, such that, G( u0 ) :=

n s=1

∂ s fs ( u0 ) − ε

n

∂ k Ak (·, u0 , ∇ u0 ) − g(·, u0 ) = 0 ∈ Rq in Ω,

(7.90)

k=1

u0 |∂Ω = uD on ∂Ω, with smooth enough Ak : Ω × Rq × Rn×q → Rq .

(7.91)

In this case we maintain the conditions for f , ε, g in (7.80). The precise condition for the Ak will be discussed below. For our quasilinear problem we introduce, with an appropriate D(G), the spaces, cf. (7.37), U = H 1 (Ω, Rq ) ∩ D(G), Vb = H01 (Ω, Rq ), Ue := Ve := H 3/2+ (T h , Rq ) ∩ D(G), Ub = H01 (Ω, Rq ) ∩ D(G), U ⊂ H −1 (Ω, Rq ), . . . ,

, and for the

strong forms Us = H (Ω, R ) ∩ D(G), Us,e := H (T , R ) ∩ D(G). 2

q

2

h

q

(7.92)

478

7. Discontinuous Galerkin methods (DCGMs)

The general quasilinear weak form is n Ak (x, u0 , ∇ u0 ), ∂ k v q dx = 0 ∀ v ∈ Vb , u0 ∈ U : G( u0 ), v := K∈T h

K k=0

u0 |∂Ω = uD on ∂Ω.

(7.93)

For equations we admitted u0 ∈ D(G) ⊂ W 2,α (Ω) in Chapter 2, Section 4.4 and the following results would have to be modiﬁed accordingly. The necessary conditions for the Nemickii operators Ai (x, uh , ∇uh ), e.g. for bounded forms and operators, We have studied ah (uh , v h ) and Gh , are formulated in Sections 2.5,2.6, 4.4 and (4.174). q growth conditions for systems for Ak (x, u, ∇ u) = aik (x, u, ∇ u) i=1 in Subsection 2.6.4, cf. e.g. (2.361) ﬀ., (2.368) ﬀ. and (2.372). They allow for the existence, uniqueness and regularity results for the solutions of our quasilinear systems, e.g., Theorems 2.89, 2.96–2.101. Their bounded linearizations with the Legendre condition and the coercive principal part are discussed in Section 2.7.4, cf. Theorem 2.125. For the Ak , we modify (7.80) into (7.94) on the basis of these results. As an example we consider the so-called natural growth conditions, cf. (2.380). The controllable growth conditions in (2.379) can be treated in a similar way. We summarize one type of these natural growth conditions. The ellipticity of (7.93) is deﬁned by the ellipticity of its linearized principal part. This can be guaranteed by (7.94) (e). With u = (u1 , . . . , un )T ∈ Rn×q , we need the partials of Ak (x, u, u) with k , ϑ l ∈ Rq , respect to x ∈ Rn , u, ul ∈ Rq , evaluated in (x, u, ∇ u) and, e.g. applied to ϑ l l q l k l ∈ R or (∂Ak /∂u )(x, u, ∇ u)ϑ ,ϑ ∈ R, or ∂Ak /∂u = so the (∂Ak /∂u )(x, u, ∇ u)ϑ q 1 n q n×q → Rq , and Ak : (∂Ak /∂u , . . . , ∂Ak /∂u ). We do not distinguish Ak : Ω × R × R 1 q −1 q H (Ω, R ) → H (Ω, R ), and omit the details in the norms, e.g. write ∂Ak /∂ul , instead of ∂Ak /∂ul Rq ←Rq2 . (a) Ak : w := (x, u, u) ∈ Ω × Rq × Rn×q → Rq , k = 0, 1, . . . , n,

(7.94)

(b) Ak : u ∈ D(G) ⊂ D(Ak ) ⊂ H 1 (Ω, Rq ) → Ak ( u) := Ak (·, u, ∇ u) ∈ H −1 (Ω, Rq ), (c) Ak , ∂Ak /∂x, ∂Ak /∂ u ≤ L(1 + u), ∂Ak /∂ u ≤ L ∀k > 0, (d) A0 , ∂A0 /∂x, ∂A0 /∂ u ≤ L(1 + u2 ), ∂A0 /∂ u ≤ L(1 + u), ∀ u ≤ M, all terms in c), d) are evaluated in w = (x, u, u), (e) Ellipticity: For u ∈ H 1 (Ω, Rq ), with u − u0 H 1 (Ω,Rq ) ≤ ρ ∃λ, Λ ∈ R ∀x ∈ Ω, n ∂Ak 2≤ k , ϑ l < Λϑ 2 ∀0 = ϑ = (ϑ 1, . . . , ϑ n ) ∈ Rn×q , 0 < λϑ (x, u , ∇ u ) ϑ ∂ul q k,l=1

(f) uD is the trace of some u∗ ∈ U on ∂Ω. 7.5.4

Discretization of the quasilinear systems

Now we start formulating the DCGMs for these divergent quasilinear equations of second order in (7.93), cf. the results for conforming FEMs in Theorem 5.81. We

479

7.5. Semilinear and quasilinear elliptic problems

choose d ≥ 2, generalizing (7.63), U h = V h = S h := Sqd−1,−1 (T h ) := {v : T h → Rq ; v|K ∈ Pd−1 (K) ∀K ∈ T h }.

(7.95)

By the standard multiplication of (7.93) with v h , partial integration over K, and summation, we relate the strong and weak discrete quasilinear forms as,h (·, ·), and a ˆh (·, ·) and the ﬁnal weak discrete form, ah (·, ·), in (7.99). We deﬁne these strong and weak forms for (7.96) ∀ uh , v h ∈ Ue with uh ∈ Us,e , v h ∈ L2 (Ω, Rq ) for as,h ( uh , v h ), n (−1)k>0 ∂ k Ak (x, uh , ∇ uh ), v h q dx, as,h ( uh , v h ) := (Gs,h ( uh ), v h ) := K∈T h

a ˆh ( uh , v h ) := Gh ( uh ), v h :=

K k=0

n K∈T

h

(Ak (x, uh , ∇ uh ), ∂ k v h )q dx

K k=0

where we use the abbreviations (Gs,h ( uh ), v h ) = (Gs,h ( uh ), v h )L2 (Ω,Rq ) , Gh ( uh ), v h = Gh ( uh ), v h V ×Ve . Since the summation in (7.96) starts with k = 0 in K∈T h K , we e choose the above notation for a ˆh ( uh , v h ) in contrast to the semilinear case. This yields the relation for n ∀ uh ∈ Us,e , ∀ v h ∈ Ue with BA ( uh , ∇ uh ) := Ak (x, uh , ∇ uh ) k=1 ∈ Rn×q , (7.97) ({BA ( uh , ∇ uh )}, ν)n , [ v h ] q ˆh ( uh , v h ) − as,h ( uh , v h ) = a e∈T h \∂Ω

e

(BA ( uh , ∇ uh ), ν)n , v h q dS. + ([BA ( uh , ∇ uh )], ν)n , { v h } q dS − e∈∂Ω

e

cf. (5.286). All the previous face integrals e disappear for uh = u ∈ H 3/2+ (Ω, Rq ), v h = v ∈ H01 (Ω, Rq ) implying for the exact solution Gs,h ( u0 ) = 0 and Gh ( u0 ) = 0. We proceed as from (7.83)–(7.88): we replace Ba (u)∇u by BA ( u, ∇ u), (Ba (uh )∇uh , ν)n v h by (BA ( uh , ∇ uh ), ν)n , v h q etc. We omit the ﬂux terms, and e { v }, ([BA ( u0 , ∇ u0 )], ν)n q , vanishing in (7.97) for uh = u0 , and add the penalty terms in (7.84). We only consider the IIPG with θ = 0, otherwise θ ({BA ( v h , ∇ v h )}, ν)n , [ uh ] q would cause a nonlinearity with respect to v h . Further more, we collect − e∈T h \∂Ω and − e∈∂Ω into − e∈T h e . So (7.87), (7.88), is generalized, with uh , v h as in (7.96), as n h h k h ({BA ( uh , ∇ uh )}, ν)n , [ v h ] q dS (Ak (x, u , ∇ u ), ∂ v )q dx − K∈Th

K k=0

⎛

+⎝

e∈T h

e∈T h

e

σ([ u], [ v ] )q dS −

e∈∂Ω

e

e

⎞

σ( uD , v )q dS ⎠ = 0.

(7.98)

480

7. Discontinuous Galerkin methods (DCGMs)

Again the exact solution u = u0 ∈ H 3/2+ (Ω, Rq ), of (7.93), (7.91), with the test functions v ∈ H01 (Ω, Rq ) satisfy (7.98). With this notation, we introduce, similarly to (7.57)–(7.61), the quasilinear, linear and bilinear forms, ah (·, ·), h (·), and Jhσ ( u, v ), cf. (7.97), and rewrite (7.98), cf. (7.96), as h h h h ({BA ( uh , ∇ uh )}, ν)n , [ v h ] q dS, ˆh ( u , v ) − (7.99) ah ( u , v ) := a h ( u) :=

e∈∂Ω

e∈T h

e

σ( uD , v )q dS and Jhσ ( u, v ) :=

e

e∈T h

σ([ u], [ v ])q dS. (7.100)

e

Again analogously to (7.66) we deﬁne G, Gh , Gh , by replacing those ah (u, v), ch (u, v), bh (u, v), h (v), Jhσ (u, v) in (7.58), (7.59), (7.61), by the new terms ah ( u, v ), h ( v ), Jhσ ( u, v ), ch ( u, v ) = bh ( u, v ) = 0, in (7.99), (7.100). Instead of explicitly updating (7.66), we refer to (7.89) for Gh . We only consider the IIPG form, so θ = 0, of the DCGM for the quasilinear problem (7.93), (7.91). Its discrete approximate solution uh0 : Ω → Rq is deﬁned as (a) uh0 ∈ S h , cf. (7.95), (7.101) : h h h; (b) G u0 , v := ah uh0 , v h + Jhσ uh0 , v h − h ( uh , v h ) = 0 ∀ v h ∈ S h with ah (·, ·) := ah (·, ·)|S h ×S h , and h (·) := h (·)|S h . A direct generalization of the NIPG, SIPG, GIPG form of the DCGM would yield problematic forms, nonlinear in v. Possible linearizations seem to be pretty artiﬁcial. The quasilinear problem and their DCGMs, studied by Houston et al. [407] somehow represent a combination of our approaches for semilinear and quasilinear problems. As for semilinear problems they consider all types of DCGM with θ = −1, = 0, = 1, ∈ [−1, 1], but for a quasilinear equation of the special form G(u) := −∇ · (μ(x, |∇u|)∇u) := −

n

∂ i μ(x, |∇u|)∂ i u = g(x, u).

(7.102)

i=1

Here μ ∈ C(Ω × [0, ∞)) is a real function satisfying the assumption, with mμ < Mμ ∈ R+ , mμ (t − s) ≤ μ(x, t)t − μ(x, s)s ≤ Mμ (t − s), t ≥ s ≥ 0, x ∈ Ω

(7.103)

and for t > s = 0 =⇒ 0 < mμ ≤ μ(x, t) ≤ Mμ . In addition, we obtain μ(·, |∇u(·)|) ∈ L∞ (Ω) for u ∈ W 1,∞ (Ω). Garret and Liu [338] 2 proved that (7.103) implies, for symmetric matrices, X, Y = (Yij )ni,j=1 ∈ Rn , or for x, y ∈ R, |μ(x, |Y |)Y − μ(x, |X|)X| ≤ C1 |Y − X|, and C2 |Y − X|2 ≤ μ(x, |Y |)Y − μ(x, |X|)X, (Y − X) n2 ,

(7.104)

7.5. Semilinear and quasilinear elliptic problems

481

n with |Y | the Euclidean (Frobenius) scalar product and norm, hence |Y | = i,j=1 = 1/2 Yij2 . This includes, via diagonal matrices, Y = (Y1 , . . . , Yn ) ∈ Rn . This can be used in two ways: linearization of the weak form of G(u) yields

(G (u0 )w, v) =

n i=1

+

n

μ(x, |∇u0 |)∂ i w∂ i vdx

(7.105)

Ω

μt (x, |∇u0 |)∂ j u0 |∇u0 |−1/2 ∂ j w∂ i v, with μt (x, t) := (∂μ/∂t)(x, t),

Ω

i,j=1

so for |μt (x, |∇u0 |)||∇u0 |1/2 small compared to mμ , a coercive principal part of the linearized operator and hence stability of the DCGM as in Theorem 7.18. Or Houston et al. [407] use (7.103) for the discrete form of the weak Gh , cf. (7.109). They prove that Gh is Lipschitz-continuous and strongly monotone with respect to uh . to (7.78), here we have chosen ε = 1. Then the lower order ﬂux terms Compared n s ∂ f (u ) are no longer dominant. Hence we omit them here. For g(x, u) we s 0 s=1 impose the conditions in (7.80) (g), (h). In this special quasilinear equation in (7.102), cf. [407], in (7.78), the second order term is replaced by ∇ · (μ(x, |∇u|)∇u). Hence we obtain, cf. (7.3), (7.23), a(u, v) =

n

μ(x, |∇u|)∂ i u∂ i v dx =

Ω i=1

(Ba (u)∇u, ∇v)n dx =

Ω

g(x, u)vdx Ω

(7.106) Ba (u)∇u := μ(x, |∇u|)∇u, (Ba (u)∇u, ∇v)n = μ(x, |∇u|)

n

∂iu ∂iv

∀u, v ∈ U,

i,=1

and determine the weak solution u0 ∈ U : a(u0 , v) = Ω g(x, u0 )vdx ∀v ∈ U. The DCGMs for (7.102) is similar to (7.89). We choose ε = 1 and replace the ah (uh , v h ) there by (7.106). We avoided, in (7.99), the θ terms θ ({Ba (u)∇v}, ν)n [u] dS = θ ({μ(x, |∇u|)∇v}, ν)n [u] dS e∈T h

e

e∈T h

e

by choosing θ = 0. To obtain a monotone operator, Houstan et al. [407] employ instead the form ({μ(x, h−1 |[u]|)∇v}, ν)n [u]dS θ e∈T h \∂Ω

+θ

e

e∈∂Ω

e

({μ(x, h−1 |u − uD |)∇v}, ν)n (u − uD )dS.

482

7. Discontinuous Galerkin methods (DCGMs)

Correspondingly, we reformulate ah (u, v), cf. (7.87), for −1 ≤ θ ≤ 1 as ({Ba (u)∇u}, ν)n [v] dS (Ba (u)∇u, ∇v)n dx − ah (u, v) := K

K∈T h

+θ

e∈T h

e∈T h \∂Ω

e

({μ(·, h−1 |[u]|)∇v}, ν)n [u] dS.

(7.107)

e

Similarly, we modify h (7.60) and use the Jhσ (7.61): 2 ({μ(x, h−1 |u − uD |)∇v}, ν)n (u − uD )dS h (u, v) := (g(·, u), v)L (Ω) − θ +

e∈∂ΩD

e∈∂Ω

e

σuD v dS

(7.108)

e

Finally, we determine the discrete solution uh0 ∈ S h , such that : h h h; G u0 , v := ah uh0 , v h + Jhσ uh0 , v h − h uh0 , v h = 0 ∀ v h ∈ S h .

(7.109)

As for equations the generalization to systems is considered by Houston et al. [407]. Again either the monotony or the stability of the linearization can be used. Then the uh0 , v h , μ(x, |∇u|) in (7.109) or additionally in (7.105) with v = (v1 , . . . , vn ) are replaced for systems by uh0 , v h , μ(x, |e( u)|), e( u) :=

1 ∇ u + (∇ u)T or e( u) := (∂1 v1 , . . . , ∂n vn ) 2

with Ba ( v )e( v ) := μ(x, |e( u)|)e( v ) and ﬁnally, : h h h; G u0 , v := ah uh0 , v h + Jhσ uh0 , v h − h uh0 , v h = 0 ∀ v h ∈ S h .

(7.110)

(7.111)

Again (7.103), (7.104) implies, cf. Houston et al. [407], Lemmas 2.2 and 2.3, that Gh is Lipschitz-continuous and strongly monotone with respect to uh .

7.6

DCGMs are general discretization methods

For these methods, stability and consistency implying convergence play a crucial role. For realizing this interpretation, we relate the original and discrete function spaces with the corresponding operators. We only distinguish the spaces H 3/2+ (Ω) and H 3/2+ (Ω, Rq ), . . ., where it is necessary, e.g. in the consistency results for ah ( uh , v h ) below, in Theorem 7.33. Most of the other results are valid componentwise, with obvious modiﬁcations, for both cases, e.g. Theorems 7.5–7.19. Directly comparing G and Gh , similarly A and Ah , required a detour via the Gh . This has been shown for the model problem in (7.13)–(7.15), (7.66), and indicated for the other problems as well. We recall the situation here for nonlinear operators. Previously we studied U = V, U h = V h but will allow, for possible generalizations, in

7.6. DCGMs are general discretization methods

483

this section, U = V, or U h = V h , as well. So we start with the original G : D(G) ⊂ U, e.g., U = H 1 (Ω) → V , e.g. V = H −1 (Ω), extended as Gh : D(Gh ) ⊂ Ueo , e.g., = H 3/2+ε (Ω) →

(Veo )

(7.112)

, e.g. = H −3/2−ε (Ω), extended as

Gh : D(Gh ) ⊂ Ue , e.g., = H 3/2+ε (T h ) → Ve , e.g. = H −3/2−ε (T h ), restricted as

Gh : D(Gh ) ⊂ U h , e.g., U h = S h → V h , e.g. V

h

= S h , in (7.63), (7.95).

For a nonlinear G the restriction to D(G) ⊂ U is important. This D(G) has to be chosen such that G(u) is well deﬁned and has the desired properties, e.g. listed in (7.16), (7.51) ﬀ. for (7.13), (7.14), or (7.80) for (7.78), (7.79), or (7.94) for (7.90), (7.91). Other possibilities are the conditions imposed for nonlinear operators and their linearization discussed in Chapter 2, here Sections 2.5.3 ﬀ. and 2.7.2. We combine U, V with the discrete spaces U h , V h , e.g. S h via the operators P h := h I ∈ L(U, U h ), and Q h ∈ L(V , V h ). We may choose the P h relatively arbitrarily, here the approximation operator P h introduced in Theorem 7.7. The P h , Q h have to be combined such that we obtain optimal consistency order. In particular, the Q h has to be chosen to ﬁt the above variational method. Similarly to (7.112), we need local pairs of Peh , Qeh . The diﬀerence is the piecewise deﬁnition, with Peh the local form on the K ∈ T h in Theorem 7.7. For Qeh , we have to include the integrals along edges and boundaries. Correspondingly, see (5.294),(5.295), we deﬁne Q h , Qeh . We start with the local version Qeh and return to Q h in (7.115). ; : Qeh ∈ L Ve , V h : Qeh fh − fh ⊥ V h ⇔ Qeh fh − fh , v h = 0 ∀v h ∈ V h

(7.113)

with ·, · = ·, ·Ve ×Ve . The general form of these fh is, cf. (2.107): e for f−1 ∈ L2 (Ω, Rn ), f0 ∈ L2 (Ω, R), f−1 , f0e ∈ L2 (∂T h , R), deﬁne fh , vVe ×Ve := e f−1 fh , v := ((f−1 , ∇ v)n + f0 v) dx + (∇v, ν)n + f0e v dS, (7.114) K∈T h

K

e∈T h

e

fh ∈ Ve , ∀v ∈ Ve ⊃ V + Veo + V h , with ∂T h := {e ∈ T h }, f = fh |V , f h := fh |V h . Examples are the previous cases, fh := Ah uh , uh ∈ Ue = H 3/2 (T h ), and fh := Ah u, u ∈ Ueo , induced by the ah (uh , v h ) in (7.57), (7.76), (7.87), (7.99). In contrast to these ah (·, ·), in the original a(·, ·) = ah (·, ·)|U ×V , in (7.19), (7.74), (7.81), the e are missing. The DCGMs for the previous problems have shown that it is impossible: neither evaluating Auh for a uh ∈ U h , nor testing it nor the corresponding f := Au, u ∈ U, with v h ∈ V h ⊂ Ve . So we deﬁne, for A and G, (Q h A) U h := Qeh Ah h , Φh A := (Q h A) U h := Qeh Ah h , and (7.115) U U (Q h G) U h := Qeh Gh , Φh G := (Q h G) U h := Qeh Gh , Uh

Uh

484

7. Discontinuous Galerkin methods (DCGMs) G:

U

Φ

P

→

→ U eo

h

→

Ah,Gh

→

(V eo)′

P eh

↓ Uh

→

→

Ah,Gh

Ve′

Ah,Gh

→

V′

Q′ h

h

Ue Φh G= G h

A,G

Q′eh →

tested by

←→

V

tested by

←→

V go

tested by

←→

Ve

tested by

Vh

↓ V ′h

←→

Figure 7.2 Original and discrete spaces and operators in DCGMs.

cf. Figure 7.2. The combination of Peh and Qeh * in this ﬁgure indicate the ranges Peh Ue = U h , Qeh Ve = V h . Note that by (7.115), the Q h ∈ L(V , V h ) are only deﬁned via Qeh ∈ L(Ve , V h ) in (7.113),(7.115), including the e . The uniformly bounded P h , Q h and Peh , Qeh have the properties required for the general discretization theory, cf. Deﬁnition 3.12, (3.24), Theorems 7.7, Corollary 7.8:

for d ≥ 2 : lim P h u = u ∀ u ∈ U, lim Q h f hV h = f V ∀f ∈ V , h→0

and : lim

h→0

h→0

Peh uh

= uh ∀ uh

h ∈ Ue , lim Qeh fh V h = fh Ve ∀fh ∈ Ve ,

(7.116)

h→0

with the local degree d − 1, cf. (7.63), (7.95). Thus, the U h are approximating spaces for U, Ue , and their subspaces, similarly for V h , V, Ve . Summarizing, we have assigned to T h with h, 0 < h ∈ H ⊂ R, inf h∈H h = 0, the sequence of spaces {U h , V h }h, 0
(7.117)

In contrast to (5.2), we enforce the continuity and boundary conditions in Ub via penalty terms and penalty norms for U h , cf. Theorems 7.12, 7.13. This procedure deﬁnes a mapping, Φh , applied to linear or nonlinear operators, Φh : A ∈ L(U, V ) → L(U h , V h ) or Φh : G ∈ N L(D(G), V ) → N L(D(Gh ), V h ), according to the chosen DCGM. It allows reinterpreting (7.70), (7.71), (7.89), (7.101): We determine the approximate solution uh0 ∈ V h as ; : Gh uh0 = Qeh Gh uh0 = 0 ⇔ Gh uh0 , v h = 0 ∀v h ∈ V h ⇔ Gh uh0 ⊥ V h .

(7.118)

Next, we study the relations between operators and their discretizations. Here the penalty norms for Ue , mentioned before, will play an essential role. In particular, we will prove stability and consistency.

7.6. DCGMs are general discretization methods

485

Stability is a kind of perturbation insensitivity, cf. Deﬁnition 3.19. For a nonlinear operator Gh : D(Gh ) ⊆ U h → V h , and uh ∈ D(Gh ) we assume Br (uh ) = {v h ∈ U h : v h − uh U h < r} ⊂ D(Gh ) ∀h < h0 , h ∈ H. Furthermore, let h0 , r, S ∈ R+ be ﬁxed constants, such that, uniformly in h ∈ H, h < h0 : uhi ∈ Br (uh ), i = 1, 2, =⇒ uh1 − uh2 U h ≤ SGh uh1 − Gh uh2 V h . (7.119) Then Gh are called stable in uh . For a linear Ah or for Ah − f h this reduces to (Ah )−1 ∈ L(V h , U h ) exists and ||(Ah )−1 ||U h ←V h ≤ S. Stability is proved using Theorems 3.23 and 3.29 in Chapter 3. It suﬃces to study stability for the linearized operator. It is here always a compact perturbation of a coercive operator, the latter deﬁning a stable discretization. We assumed the linearized operator to be boundedly invertible. Then the stability of the linearized operator follows as in Section 5.5 by proving the coercivity for a perturbed principal part. This follows with the penalty norm from our Theorems 7.16–7.19. Combined with a nearly verbatim copy of the arguments in Lemma 5.77 and Theorem 3.29, it yields stability for (G (u0 ))h with respect to the penalty norm v, cf. Theorem 7.12, (7.64), (7.153), for general boundedly invertible linearizations, G (u0 ). This is left as an exercise for the interested reader. We will summarize the exact conditions and results in Theorem 7.40. The classical consistency error has to converge to 0. It has, for a linear or nonlinear operator, A or G, the following general form, cf. (5.313), (3.34), Chapter 3, (3.33). These Gh P h u − (Q h G)u, cf. (7.117), have to be tested by v h ∈ S h :

Ah P h u − (Q h A)u and Gh P h u − (Q h G)u tested by 0 = v h ∈ S h yield

(7.120)

|Gh P h u − (Q h G)u, v h | → 0 or ≤ C(u)hk with C(u) ∈ R+ , k = k(u) ∈ N; v h these C(u), k = k(u) depend on the regularity of u and the degree d − 1 of the chosen polynomials in V h , cf Theorem 7.40. The classical consistency error can be estimated by combining (7.71) or the corresponding generalizations (7.77), (7.89), (7.101), with the Lipschitz-continuity of G and Gh and the penalty norm, cf. Theorem 7.21 ﬀ. The combination of (7.71) with Q h in (7.113) yields for our ﬁrst example (7.23) for any u ∈ U and v h ∈ S h , cf. (7.118), Gh P h u − Gh u, v h = ah (P h u, v h ) + bh (P h u, v h ) + εJhσ (P h u, v h ) (7.121) −h (P h u, v h ) − ah (u, v h ) + bh (u, v h ) + εJhσ (u, v h ) −h (u, v h ) ∀ v h ∈ S h . For our next examples (7.74), (7.78), and (7.93) we simply replace ah (P h u, v h ), h (P h u, v h ) in (7.57), (7.60) by the updated forms in (7.76), (7.87), (7.88), and in (7.99), (7.100). The bh (P h u, v h ) is skipped in (7.74), (7.93), (7.99). All these diﬀerences

486

7. Discontinuous Galerkin methods (DCGMs)

of corresponding pairs can be estimated by combining the interpolation errors in Theorem 7.7 and Corollary 7.8 on the K and ∂K with speciﬁc properties of the ah (P h u, v h ), . . . , h (P h u, v h ) in Theorems 7.21–7.33. Continuity, consistency plus stability imply the unique existence of the discrete solution and its convergence. For the reader’s convenience we reformulate the essential Theorem 3.21 in the following theorem. It applies to the original problems, and their discretizations, (7.13), (7.14), (7.70), (7.71), and (7.74), (7.77); (7.78), (7.79), and (7.89); (7.93), and (7.101). Theorem 7.3. Unique existence and convergence of discrete solutions: Let the origi nal problem have the exact solution u0 . Let Gh : D(Gh ) ⊂ U h → V h be its discretization, cf. (7.118), and Deﬁnitions 3.5, 3.6, 3.12 (here nonlinear Φh are allowed) and satisfy the following conditions

1. Gh = Φh (G) : D(Gh ) ⊂ U h → V h is deﬁned and continuous in Br (P h u0 ) ⊂ D(Gh ) with r > 0 independent of h; 2. Gh is (classically) consistent with G in P h u0 , in the sense of (7.120); 3. Gh is stable for P h u0 with respect to · . Then the discrete problem Gh (uh ) = 0 possesses a unique solution uh0 ∈ D(Gh ) ⊂ U h near u0 for all suﬃciently small h ∈ H and uh0 converges to u0 . If Gh is consistent and consistent of order p, then uh0 converges and converges of order p, hence uh0 − u0 ≤ SGh (P h u0 )V h and ≤ O(hp ), respectively

(7.122)

For all the cases considered here, Gh and Gh are diﬀerentiable simultaneously with G. They even satisfy consistent diﬀerentiability, cf. (5.165), so ∀u, v ∈ Br (u0 ) ∩ Us :

(Qh G (u))v − Qh G (P h u)P h vV h ≤ C2 hp (1 + uUs )vUs ,

(7.123)

with a “smooth” norm vUs . This is important for the mesh independence principle. It guarantees the quadratic convergence of the Newton method for solving the corresponding discrete nonlinear systems, essentially independent of the mesh size, cf. Section 7.13. So the program for the next sections is the proof for stability, consistency, and consistent diﬀerentiability for our preceding problems.

7.7

Geometry of the mesh, error and inverse estimates

In this section we shall be concerned with the necessary approximation results. The advantage of the approach in this section compared to the usual FE approach is the avoided condition of quasiuniform subdivisions. For FEMs this is possible by modifying the results for boundary integral methods of Graham et al. [362], cf. Chapter 4. For DCGMs some unusual conditions upon the geometry of the subdivision have to be imposed. Then a multiplicative trace inequality is combined with inverse estimates.

7.7. Geometry of the mesh, error and inverse estimates

7.7.1

487

Geometry of the mesh

Let us consider a system {T h }h∈(0,h0 ) , h0 > 0, of partitions of the domain Ω, i.e. T h = {K ∈ T h }. We assume throughout the chapter that the system {T h }h∈(0,h0 ) has the following properties: (A0) T h is a covering partition of the domain Ω = ∪K∈T h K. (A1) Each element K ∈ T h , h ∈ (0, h0 ), is a star-shaped domain with respect to at least one interior point xK = (xK1 , . . . , xKn ) of K. We assume that (i) there exists a constant κ > 0 independent of K and h such that maxx∈∂K |x − xK | ≤κ minx∈∂K |x − xK |

∀K ∈ T h , h ∈ (0, h0 );

(ii) each K can be divided into a ﬁnite number of closed simplices: C K= S ;

(7.124)

(7.125)

S∈S(K)

˜ independent of K, S and h such that there exist positive constants C3 , κ hS ≤ C3 ρS

∀S ∈ S(K)

(shape regularity)

(7.126)

where hS is the diameter of S, ρS is the radius of the largest n-dimensional ball inscribed into S and, moreover, 1≤

hK ≤κ ˜<∞ hS

∀S ∈ S(K).

(7.127)

(A2) There exists a constant C4 > 0 such that hK ≤ C4 d(e)

∀K ∈ T h , e ⊂ ∂K, h ∈ (0, h0 ).

(7.128)

Remark 7.4. The conditions (A1) and (A2) imply a bound on the maximum number of edges, i.e. there exists a constant ce such that #(e ∈ K) ≤ ce for all K ∈ T h . Moreover, it follows that the number of hanging nodes for each element is bounded. Let us note that these properties can be veriﬁed, e.g. in the case of dual ﬁnite volumes constructed over a regular simplicial mesh. Furthermore, nondegenerate subdivisions in FEMs satisfy these conditions as well, see [314]. Figure 7.1 satisﬁes (A0)–(A2), it is star-shaped and the K are nonconvex. Figure 7.3 shows a type of mesh used in [314]. For practical computation these grids are not very suitable. We cite some technical results formulated as Theorems 7.5–7.7. 7.7.2

Inverse and interpolation error estimates

Under the above assumptions, (A1), (A2) the following results can be established.

488

7. Discontinuous Galerkin methods (DCGMs) ‘dmesh’ 0.4

0.2

0

–0.2

–0.4 –0.2

0

0.4

0.2

0.6

0.8

1

1.2

1.4

Figure 7.3 Dual mesh.

Theorem 7.5. Multiplicative trace inequality: Assume (A1), (A2). Then there exists a constant C5 > 0 independent of v, h and K such that 2 (7.129) v2L2 (∂K) ≤ C5 vL2 (K) |v|H 1 (K) + h−1 K vL2 (K) , K ∈ T h , v ∈ H 1 (K), h ∈ (0, h0 ). Proof. Using the approach from [296] and [297], we prove the theorem in two steps. (i) Let K ∈ Th and S ∈ S(K) be arbitrary but ﬁxed. We denote by xS the center of the largest d-dimensional ball inscribed into S. Without loss of generality we suppose that xS is the origin of the coordinate system. We start from the following relation, a consequence, e.g. of (2.9), v 2 x · ν dS = ∇ · (v 2 x) dx, v ∈ H 1 (S). (7.130) ∂S

S

Let νe be the unit outer normal to S on the face e ⊂ ∂S. Then x · νe = |x||νe | cos α = |x| cos α = ρS

x ∈ e,

since ρS is the distance of xS from the face e, see Figure 7.4. From (7.131) we have v 2 x · ν dS = v 2 x · νij dS ∂S

e⊂∂S

=

e⊂∂S

e

v dS ≥ ρS 2

te e

e⊂∂S

e

v 2 dS = ρS v2L2 (∂S) .

(7.131)

(7.132)

7.7. Geometry of the mesh, error and inverse estimates

489

xS a x ρS

S

e

ne

Figure 7.4 Simplex S with its face e and the distances te .

Moreover,

v 2 ∇ · x + x · ∇v 2 dx

∇ · (v 2 x) dx = S

S

(7.133)

vx · ∇v dx ≤

2

=d

v dx + 2 S

dv2L2 (S)

|vx · ∇v| dx.

+2

S

S

With the aid of the Cauchy inequality the second term of (7.133) is estimated as

|vx · ∇v| dx ≤ 2 sup |x|

2 S

x∈S

|v||∇v| dx ≤ 2hS vL2 (S) |v|H 1 (S) .

(7.134)

S

Then (7.130), (7.132), (7.133) and (7.134) give 6 1 5 2hS vL2 (S) |v|H 1 (S) + dv2L2 (S) ρS 0 ( d ≤ C3 2vL2 (S) |v|H 1 (S) + v2L2 (S) , hS

v2L2 (∂S) ≤

(7.135)

where C3 is the constant given by (7.126) and is independent of S, K, v and h. (ii) Let K ∈ Th and S ∈ S(K). Then (7.135) yields

2 v2L2 (∂S) ≤ C vL2 (S) |v|H 1 (S) + h−1 S vL2 (S) , S ∈ S(K), v ∈ H 1 (K),

(7.136)

490

7. Discontinuous Galerkin methods (DCGMs)

where C = C3 max{2, d}. Using the inclusion ∂K ⊂ (7.127) and the Cauchy inequality, we ﬁnd that v2L2 (∂K) ≤ v2L2 (∂S)

@ S∈S(K)

S∈S(K)

≤C

⎧ ⎨

2 vL2 (S) |v|H 1 (S) + h−1 S vL2 (S)

⎩

S∈S(K)

.

≤C

1/2 v2L2 (S)

S∈S(K)

∂S, relations (7.136) and

⎫

⎬ (7.137)

⎭

1/2 |v|2H 1 (S)

/ +

2 κ ˜ h−1 K vL2 (K)

S∈S(K)

2 = CM vL2 (K) |v|H 1 (K) + h−1 K vL2 (K) ,

where CM = κ ˜C = κ ˜ C3 max{2, d}.

Theorem 7.6. Inverse inequality: Under the conditions (A1), (A2), there exists a constant C6 = C6 (d − 1) > 0 independent of v, h and K such that |v|H 1 (K) ≤ C6 h−1 K vL2 (K) ,

v ∈ Pd−1 (K), K ∈ T h , h ∈ (0, h0 ).

(7.138)

The proof of Theorem 4.35 remains valid for Theorem 7.6 as well. According to Schwab [575], the constant C6 is O((d − 1)2 ). Important are the approximation properties of the space S h constructed over the mesh T h . The following theorem is for the FE situation, a consequence of Deﬁnition 4.8 and Theorem 4.17; for the DCG situation it is proved by Verf¨ urth [656]. Theorem 7.7. Error estimates: Assume (A1), (A2). Then there exists a constant C7 > 0 independent of v and h and a mapping P h : H 1 (K) → Pd−1 (K) such that ∀ k = 0, 1, 2, v ∈ H s (T h ), K ∈ T h , h ∈ (0, h0 ), min{d,s}−k

|P h v − v|H k (K) ≤ C7 hK ⎛ |P h v − v|H k (T h )

|v|H min{d,s} (K) , with | · |H 0 (K) = | · |L2 (K) ⎞1/2

2 min{d,s}−k hK ≤ C7 ⎝ |v|H min{d,s} (K) ⎠ (7.139) K∈T h

By Veeser and Verf¨ urth [652], and Thrun [628], (7.139) remain correct, if the H k , H min{d,s} are replaced by W k,∞ , W min{d,s},∞ . If in K, polynomials of degree dK are employed we replace in Theorem 7.21 ﬀ. ⎛ ⎞1/2 2(d −d) hmin{d,s} |u|H min{d,s} (Ω) by h{min d,min s} ⎝ hK K |u|2H min{dK ,sK } (K) ⎠ . K∈T h

(7.140)

7.8. Penalty norms and consistency of the Jhσ

491

The precise conditions for these modiﬁed results, used in adaptive and hp- DCGMs, are described in Section 7.14. Proof. According to [656], we denote by πK ϕ ≡

1 |K|

ϕ dx

(7.141)

K

the mean value of any integrable function ϕ on K. For any v ∈ H s (K) we recursively deﬁne polynomials pd−1,K (v), pd−2,K (v), . . . , p0,K (v) in Pd−1 (K) by 1 xα πK (Dα v) pd−1,K (v) ≡ (7.142) (d − 1)! d−1 α∈N |α|=d−1

and, for k = d − 1, d − 2, . . . , 1 pk−1,K (v) ≡ pk,K (v) +

α∈N d−1 |α|=k−1

1 xα πK (Dα (v − pk,K (u))) . (d − 1)!

(7.143)

Finally, we set P h v ≡ p0,K (v).

(7.144)

For the proof of the approximation properties (7.139) we refer to [656]. As a consequence of the Trace Theorem 1.38 we obtain Corollary 7.8. There exist a constant C8 > 0 independent of v and h0 such that min{d,s}−k−1/2

|P h v − v|H k (∂K) ≤ P h v − vH k (∂K) ≤ C7 hK

|v|H min{d,s} (K) ,

(7.145)

for all v ∈ H s (K), K ∈ T h , 0 ≤ k ≤ s, and h ∈ (0, h0 ), h0 < 1.

7.8

Penalty norms and consistency of the Jhσ

The need for the penalty norms has been mentioned several times before. Their form here diﬀers from the energy norms in Chapters 2 and 4. So we maintain the notation of penalty norms for the DCGMs here. The coercivity, boundedness and even more so the consistency properties in the next two sections are not possible with respect to broken Sobolev norms, but only with respect to these penalty norms. For the following considerations we assume, cf. (7.39), σmin := min {σ|e = 1/d(e)} ≥ 1. e∈T h

(7.146)

The consistency equation (7.121) and the estimates in the following theorems show that a successful convergence approach requires carefully monitoring the Jhσ (uh , v h ). Beginning with Theorem 7.16, a naive approach to the following coercivity and

492

7. Discontinuous Galerkin methods (DCGMs)

consistency estimates would yield unbounded results with respect to broken Sobolev norms, vH 1 (Th ) . This is caused by the Jhσ (v h , v h ), strongly growing for arbitrary v h and increasingly ﬁner meshes, T h . We compare, e.g. (7.121) with the consistency results for FEMs with boundary crimes in (5.348). This motivates, e.g. for the ah (uh , v h ) in (7.57), nonlinear in uh , but linear and bounded in v h , a condition |ah (P h u, v h ) − ah (u, v h )| ≤ C(u)dist(u, S h )v h ? ,

(7.147)

with appropriate norms in dist(u, S h ) and v h ? and C(u) usually depending upon u. dist(u, S h ) is well estimated by the interpolation errors in Theorem 7.7, Corollary 7.8. However it neglects the violated continuity and boundary conditions. For V h = S h the v h ? will be chosen as a combination of the standard broken Sobolev norm and Jhσ (v h , v h ), allowing the necessary (7.147). In our general discretization approach we are mainly interested in those uh , v h ∈ Ue , approximating the elements u, v in a bounded subset of H 3/2+ε (Ω). By Proposition 7.9 and Theorem 7.12, for smooth v and the interesting v h ≈ P h v ∈ S h or Ue the above v h ? is very close to a standard broken Sobolev norm. Proposition 7.9. Essential properties of Jhσ : Choose a bounded B ⊂ H s (Ω), with s ≥ 1, and the local polynomial degree, d − 1 > 0. Then the Jhσ (·, ·) : B × B → R is continuous. With eT h = maxK∈T h {#(e ∈ K)}, d = minK∈T h {dK }, a constant C14 := 22 eT h cw C4 C72 > 0 exists, such that ∀u, v ∈ B σ h Jh (P u − u, P h v − v) ≤ C14 h2 min{d,s}−2 |u|H min{d,s} (Ω) |v|H min{d,s} (Ω) . (7.148) Proof. Corollary 7.8, the Cauchy–Schwarz inequality, and (7.128) imply for e ⊂ ∂K σ [u − P h u] [v − P h v]dS ≤ σ|e 22 u − P h uL2 (∂K) v − P h vL2 (∂K) (7.149) e

2 min{d,s}−1/2 |u|H min{d,s} (K) |v|H min{d,s} (K) ≤ σ|e 2C7 hK 2 min{d,s}−2

≤ cw C4 22 C72 hK |u|H min{d,s} (K) |v|H min{d,s} (K) . The ﬁnal e∈T h , the discrete Cauchy–Schwarz inequality, and a local degree dK = d show, with C14 = cw C4 22 C72 maxK∈T h {#(e ∈ K)}, 2 min{d ,s}−2 σ h K Jh (P u − u, P h v − v) ≤ C14 hK |u|H min{dK ,s} (K) |v|H min{dK ,s} (K) K∈T h

≤ h2 min{d,s}−2 C14

|u|H min{d,s} (K) |v|H min{d,s} (K)

K∈T h

with d = min {dK } K∈T h

hence (7.148) with (7.150).

(7.150)

7.8. Penalty norms and consistency of the Jhσ

493

s h Theorem 7.10. Consistency estimates for the Jhσ : We √ obtain for u ∈ H (Ω), v ∈ s h H (T ), the local degree, d − 1 > 0, and with C15 := 2 C5 cw C7 . σ h Jh (P u, v h ) − Jhσ (u, v h ) ≤ hmin{d,s}−1 C15 |u|H min{d,s} (Ω) v h ∀h ∈ (0, h0 ). (7.151)

where P h u is the previous S h -interpolant of u. Proof. From (7.61), the Cauchy inequality, (7.149)–(7.150), the multiplicative trace inequality (7.129) and (7.139) (but (7.317) for hp methods) we have σ Jh (u, v h ) − Jhσ (P h u, v h ) (7.152) ≤ σ|[u − P h u]| |[v h ]| dS e∈T h

e

⎛ ≤ Jhσ (v h , v h )1/2 ⎝

e∈T

≤ v h

h

⎞1/2 σ[u − P h u]2 dS ⎠

e

h 2 cw h−1 K u − P uL2 (∂K)

1/2

K∈T h

⎛ ⎞1/2 F −2 h h h 2 ⎠ ≤ C5 cw v h ⎝ h−1 K u − P uL2 (K) |u − P u|H 1 (K) + hK u − P uL2 (K) K∈T h

≤

F 2 2 2C5 cw v h CA C7 1/2 1 min{d,s} min{d,s}−1 1 2 min{d,s} 2 |u|H min{d,s} (K) × h hK + 2 hK hK K hK K∈Th

F 2 min{d,s}−1 ≤ 2 C5 cw C7 hK

1/2 |u|2H min{d,s} (K)

v h ,

K∈Th

thus (7.151) with C15

√ := 2 C5 cw C7 .

Before we discuss DCGMs, we introduce the announced penalty norm. It penalizes elements v ∈ H s (T h ), violating continuity and boundary conditions. Deﬁnition 7.11. For σ in (7.39), (7.146), we introduce the penalty norms for 1 ≤ s ∈ R, 1/2 , ∀v ∈ H s (T h ), and for (7.153) vJH s (T h ) := Jhσ (v, v) + |v|2H s (T h ) s = 1 : v := vJH 1 (T h ) used as v h V h := v h in S h , and ﬁnally 1/2 vJH s (T h ) := Jhσ (v, v) + v2H s (T h ) , used e.g. in subsection 7.11.4.

494

7. Discontinuous Galerkin methods (DCGMs)

Theorem 7.12. Banach spaces with respect to v: For s ≥ 1 and σ in (7.146), we obtain ∃ce > 1 : (ce )−1 vJH 1 (T h ) ≤ vJH 1 (T h ) ≤ ce vJH 1 (T h ) ∀v ∈ H 1 (T h ),

(7.154)

hence, equivalent norms in H s (T h ), a Banach-space with respect to both norms. In particular, we obtain for s = 1, · = vJH 1 (T h ) . The original · H s (T h ) and the new · JH s (T h ) are nonequivalent. Proof. Obviously · JH s (T h ) are norms for H s (T h ), nonequivalent with the original J norm. A Cauchy sequence {vν }∞ ν=1 with respect to · H s (T h ) is Cauchy sequence with respect to · H s (T h ) as well, hence has a limit element v0 ∈ H s (T h ) with respect to · H s (T h ) . Since Jhσ is continuous in H s (T h ), this v0 is limit element with respect to · JH s (T h ) as well, hence H s (T h ) is a Banach space with respect to both norms. The equivalence of both norms vJH s (T h ) and vJH s (T h ) is an immediate consequence of Proposition 1.25. Theorem 7.13. Interpolation errors with respect to · JH k (T h ) and the equivalence of these norms: 1. There exists a constant C16 > 0 independent of v and h, such that P h v − vJH k (T h ) ≤ C16 hmin{d,s}−(1/2+k) |v|H min{d,s} (T h ) , k = 0, 1, 2 ≤ s

(7.155)

for all v ∈ H s (T h ) or v ∈ H s (Ω) and h ∈ (0, h0 ). 2. For a given u ∈ H s (Ω), d ≥ s ≥ 1, deﬁne Buh ∈ S h as Buh := {v h ∈ H s (T h ) ∩ S h : u − v h JH k (T h ) ≤ 1/(4|u|H s (T h ) ), k = 0, 1, 2 ≤ s} with small enough h. Then all the norms · JH s (T h ) , and · H s (T h ) are equivalent in Buh , and limh→0 P h uJH s (T h ) = uH s (Ω) . Proof. Inequality (7.155) is a consequence of Theorem 7.7, and Corollary 7.8. Claim 2 follows from Proposition 7.9 and (7.155).

7.9 7.9.1

Coercive linearized principal parts Coercivity of the original linearized principal parts

Existence and uniqueness results for the original previous problems are available by the Fredholm alternative, if the linearized operators are compact perturbations of bounded coercive operators, cf. Chapter 2. This is essential for the proof of stability as well. So we summarize these results. The bilinear form for the Laplacian is, cf. (7.19), (7.156) c(u, v) := a(u, v) = ε (∇u, ∇v)n dx for u, v ∈ U, Ω

7.9. Coercive linearized principal parts

495

or a(u, v) in (7.74) for the general bilinear form. For the more general forms of a(u, v), nonlinear in u, in (7.81), and (7.96) ﬀ., the linearized principal parts, the ∂ap / ∂u, evaluated in u in the neighborhood of u0 and applied to w, v, are n ∂ap (u) (w, v) = Gp (u)w, v = ε aij (x, u)∂ i w ∂ j v dx and c(w, v) := ∂u Ω i,j=1 c(w, v ) :=

(7.157) n ∂Ak ∂ap l k ( u) (w, v ) = Gp ( u)w, v q = (x, u , ∇ u )∂ w, ∂ v ∂ u ∂ul Ω q k,l=1

(7.158) for u, w, v ∈ U and u, w, v ∈ H 1 (Ω, Rq ). The corresponding conditions are listed in (7.80) and (7.94), in particular the ellipticity conditions (e). Theorem 7.14. Coercivity of the original (linearized) principal parts: For a polygonal Ω the bilinear principal forms c(·, ·) in (7.156)–(7.158), and its generalization to (7.74), are bounded and coercive with respect to the energy (semi-) norm. For the nonlinear problems in (7.81) and (7.96) ﬀ., we have to impose the additional conditions (7.80) and (7.94): c(v, v) ≥ Cε|v|2U , |c(u, v)| ≤ C ε|u|U |v|U ∀u, v ∈ U = H 1 (Ω) c( v , v ) ≥

C| v |2U ,

(7.159)

|c( u, v )| ≤ C | u|U | v |U ∀ u, v ∈ U = H (Ω, R ). 1

q

For u, v ∈ H01 (Ω) and u, v ∈ H01 (Ω, Rq ) this bounded coercivity (7.159) holds for the norm · H01 (Ω,Rq ) , q ≥ 1, the equivalent standard Sobolev norm. 7.9.2

Coercivity and boundedness in V h for the Laplacian

Stability for the previous discrete problems can be guaranteed by two properties. We need the bounded invertibility of the corresponding linearized operator and the bounded coercivity in V h of its linearized principal part Ah (u, v) + Jhσ (u, v) or Ah (u; w, v) := ∂ah /∂u(u) (w, v) + Jhσ (w, v) in the following subsections. Remark 7.15. Boundedness and coercivity in V h always means equiboundedness and equicoercivity with respect to h. For the above diﬀerent principal parts c(u, v), we deﬁne the corresponding ch (u, v), and Ah (u, v) := ch (u, v) + εJhσ (u, v) with Jhσ (·, ·) in (7.61).

(7.160)

v ) indicate the corresponding discrete For the nonlinear cases, let ch (w, v), ch (w, linearizations analogous to c(w, v), c(w, v ) in (7.157), (7.158). Then, cf. (7.99), v ) := ch (w, v ) + Jhσ (w, v ). (7.161) Ah (u; w, v) := ch (w, v) + εJhσ (w, v) and Ah ( u; w,

496

7. Discontinuous Galerkin methods (DCGMs)

As above, we discuss the three types of DCGMs, the nonsymmetric interior, NIPG, with θ = −1, the symmetric interior, SIPG, with θ = 1, and the incomplete interior penalty Galerkin, IIPG, methods with θ = 0. For completeness, we extend it to the general interior penalty Galerkin, GIPG, method with −1 ≤ θ ≤ 1, cf. Houston et al. [407]. We prove the coercivity and boundedness of these Ah (·, ·) with respect to · For our model problem, these bilinear forms can be written in the form {∇u}, ν)n [v] (7.162) (∇u, ∇v)n dx − ε ah (u, v) = ε K∈Th

K

e∈T h \∂ΩN

e

+ θ({∇v}, ν)n [u] dS ∀u, v ∈ Ue for − 1 ≤ θ ≤ 1. The multiplicative trace and the inverse inequality (7.129) and (7.138) yield 2 h v2L2 (∂K) ≤ C5 (1 + C6 )h−1 K vL2 (K) ∀v ∈ V .

(7.163)

Theorem 7.16. Bounded and coercive ah (u, v) for the Laplacian: Choose an arbitrary constant cw > 0 in (7.39) for the NIPG method. For the SIPG, IIPG, and GIPG methods (with θ = 1, 0, ∈ [−1, 1]) choose cw ≥ C5 (1 + C6 )(1 + θ)2

(7.164)

where C5 and C6 are given by (7.138), respectively. Then the corresponding bilinear forms Ah (·, ·) given by (7.160), (7.162), are bounded and coercive with respect to the norm, v, such that Ah (v, v) ≥ cc εv2 , |Ah (u, v)| ≤ cb εu v ∀u, v ∈ V h .

(7.165) (7.166)

Remark 7.17. Theoretically it would be possible to determine the constants, deﬁning cw in (7.164): Before starting computations, check all elements K ∈ Th and ﬁnd the ˜ , C4 appearing in (7.124)–(7.128). This is combined with maximal values of κ, C3 , κ the C5 , C6 in (7.129), (7.138). However, we doubt that anybody uses this approach. However, Epshteyn and Rivi`ere [309] present computable lower bounds of the penalty parameters in the SIPG method. In particular, they derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements. Usually, a suitable value of cw is sought empirically. Furthermore, a too high value of cw leads to bad computational properties of the corresponding discrete problem. Some numerical experiences for the choice of cw for the system of the compressible Navier-Stokes equations, cf. (7.343), are given in [294]. Houston et al. use a similar approach, see their [407], text between relations (2.4) and (2.5). They have an explicit relation containing constants from a multiplicative trace inequality (Cd ) and the inverse inequality (C3 ).

7.9. Coercive linearized principal parts

497

Proof. 1. Coercivity of the NIPG method: Inequality (7.165) for Ah immediately follows from the deﬁnition (7.160), (7.162) with θ = −1, cc = 1. 2. Coercivity of the SIPG, IIPG, and GIPG methods: By (7.160) and (7.162) we have {∇v}, ν)n [v] dS + εJhσ (v, v). Ah (v, v) = ε|v|2H 1 (T h ) − ε(1 + θ) e

e∈T h \∂ΩN

(7.167)

This, the Cauchy inequality and Young’s inequality in the form, ab ≤ ca2 + b2 /(4c) for a, b, c ∈ R, c > 0,

(7.168)

imply that for c = 1 and any δ > 0, ⎛ 1 Ah (v, v) ≥ ε|v|2H 1 (T h ) + εJhσ (v, v) − ε(1 + θ) ⎝ δ ⎛

× ⎝δ

e∈T

h \∂Ω

N

≥ εv2 − εω − ε

e

e∈T h \∂ΩN

⎞1/2 d(e){∇v}2 dS ⎠

e

⎞1/2 1 [v]2 dS ⎠ d(e)

(7.169)

δ σ J (v, v), cw h

where ω=

(1 + θ)2 4δ

e∈T h \∂ΩN

d(e)|{∇v}|2 dS.

(7.170)

e

Further, from (7.129), (7.163), we get for v ∈ V h

2 (7.171) hK ∇v2L2 (∂K) ≤ C5 hK |v|H 1 (K) |∇v|H 1 (K) + h−1 K |v|H 1 (K) K∈Th

K∈Th

≤ C5 (1 + C6 )|v|2H 1 (T h ) and thus ω≤

C5 (1 + C6 )(1 + θ)2 2 (1 + θ)2 |v|H 1 (T h ) . hK ∇v2L2 (∂K) ≤ 4δ 4δ

(7.172)

K∈Th

Now let us choose δ = C5 (1 + C6 )(1 + θ)2 /2.

(7.173)

498

7. Discontinuous Galerkin methods (DCGMs)

Then it follows from (7.164), (7.167) and (7.169)–(7.173) that ε ε C5 (1 + C6 )(1 + θ)2 σ Ah (v, v) ≥ εv2 − |v|2H 1 (T h ) − Jh (v, v) 2 2cw

ε 1 ≥ εv2 − ε |v|2H 1 (T h ) + Jhσ (v, v) = v2 , 2 2 which proves (7.165) with cc = 1/2. 3. Boundedness: In view of the deﬁnition of the form Ah , we can write Ah (u, v) = ϑ1 + ϑ2 + ϑ3 , (∇u, ∇v)n dx ϑ1 = ε K∈Th

(7.175)

K

ϑ2 = −ε

(7.174)

e∈T h \∂ΩN

{∇u}, ν)n [v] + θ({∇v}, ν)n [u] dS,

e

ϑ3 = εJhσ (u, v), where θ = −1 for NIPG, θ = 1 for SIPG, θ = 0 for IIPG, and θ ∈ [−1, 1] for GIPG methods. 4. Using the Cauchy inequality, we get the bound for ϑ1 and ϑ3 |ϑ1 | ≤ ε|u|H 1 (T h ) |v|H 1 (T h ) ≤ ε u v, and |ϑ3 | ≤ ε (Jhσ (u, u))

1/2

1/2

(Jhσ (v, v))

(7.176)

≤ ε u v.

For estimating ϑ2 , we apply the Cauchy inequality, with |θ| ≤ 1 yielding ⎞1/2 ⎛ ⎞1/2 ⎛ d(e) cw [v]2L2 (e) ⎠ (7.177) ∇u2L2 (e) ⎠ ⎝ |ϑ2 | ≤ ε ⎝ c d(e) w h h e∈T \∂ΩN

⎛

e∈T \∂ΩN

+ε⎝

e∈T h \∂ΩN

ε ≤ √ cw

ε +√ cw

⎞1/2 ⎛ d(e) ∇v2L2 (e) ⎠ ⎝ cw

e∈T h \∂ΩN

⎞1/2 cw [u]2L2 (e) ⎠ d(e)

1/2 hK ∇u2L2 (∂K)

K∈Th

Jhσ (v, v)1/2 1/2

hK ∇v2L2 (∂K)

Jhσ (u, u)1/2 .

K∈Th

We combine (7.171), (7.177), and the discrete Cauchy–Schwarz inequality 1/2 2 1/2 r1 + r22 , s1 , s2 , r1 , r2 ∈ R s1 r2 + s2 r1 ≤ s21 + s22

7.9. Coercive linearized principal parts

to obtain

7

|ϑ2 | ≤ ε 7 ≤ε

499

C5 (1 + C6 ) × |u|H 1 (T h ) Jhσ (v, v)1/2 + |v|H 1 (T h ) Jhσ (u, u)1/2 cw

1/2

1/2 C5 (1 + C6 ) |v|2H 1 (T h ) + Jhσ (v, v) × |u|2H 1 (T h ) + Jhσ (u, u) cw

7 ≤ε

C5 (1 + C6 ) u v. cw

(7.178)

Now F from (7.175), (7.176), (7.178) we immediately get (7.166) with cb = 2 + ε C5 (1 + C6 )/cw . 7.9.3

Coercivity and boundedness in V h for the general linear and the semilinear case

As a consequence of (7.89), we omit the lower order compact perturbations terms bh , ch , h and conﬁne the discussion here to the principal part in (7.179). Let, for u, w, v ∈ Ue , and Dirirchlet boundary conditions on Ω, n Ah (u; w, v) = εJhσ (w, v) + ε aij (x, u) ∂ i w ∂ j v dx (7.179) K∈Th

−ε

K i,j=1

n

{aij (x, u) ∂ i w ν j } [v] + θ{aij (x, u) ∂ i v ν j } [w] dS

e∈T h

e i,j=1

with θ = −1, 1, 0, ∈ [−1, 1] for the NIPG, SIPG, IIPG, and GIPG methods and Jhσ in (7.61). For general linear problems, we replace the aij (x, u) by aij (x) in (7.179). Theorem 7.18. Bounded and coercive ah (u, v) for general linear and semilinear problems: For the Λa , λ in (7.74), (7.80), and λ = min(1, λ), we choose cw arbitrarily positive for the NIPG and, for the SIPG, IIPG or GIPG methods, as cw ≥

C5 (1 + C6 )(1 + θ)2 Λ2a n2 λ

2

(7.180)

with C5 and C6 in (7.129) and (7.138), respectively. Then the corresponding bilinear forms Ah (·, ·), given by (7.179) or by replacing in (7.179) the aij (x, u) by aij (x), are bounded and coercive with respect to the norm, v, such that Ah (u, v, v) ≥ cc εv2 , |Ah (u, w, v)| ≤ cb εw v

(7.181) ∀u, w, v ∈ V h .

(7.182)

500

7. Discontinuous Galerkin methods (DCGMs)

Proof. We formulate the proof only for semilinear problems. 1. The coercivity of the NIPG method again is a consequence of (7.179) with θ = −1 and (7.80) (e): n aij (x, u) ∂ i v ∂ j v dx (7.183) Ah (u, v, v) = εJhσ (v, v) + ε K∈Th

K i,j=1

≥ ε Jhσ (v, v) + λ|v|2H 1 (T h ) , which proves (7.181) with cc := min(1, λ). 2. Coercivity of SIPG, IIPG, and GIPG methods: Let ρ = (ρ1 , . . . , ρn ), ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn and aij (·, ·) satisfy assumption (7.80) (c). Then using an appropriately reordered discrete Chebyshev inequality (1.43), and the Cauchy– Schwarz inequality (1.44), for p = q = 2, we obtain ⎞1/2 ⎛ ⎞1/2 ⎛ n n n n n fi gj = fi gj ≤ n ⎝ fj2 ⎠ ⎝ gj2 ⎠ = n|f |n |g|n = n|f ||g|. i=1 j=1 i,j=1 j=1 j=1 (7.184) This yields

n n aij (·, ·)ρi ϑj ≤ aij (·, ·)ρi ϑj ≤ Λa |ρi | |ϑj | i,j=1 i,j=1 i,j=1 n

≤ nΛa

n

1/2 ρ2i

i=1

n

(7.185)

1/2 θi2

= nΛa |ρ| |ϑ|.

i=1

From (7.179), (7.80) (e) and (7.185) we get, with θ ∈ (−1, 1], n Ah (u, v, v) = εJhσ (v, v) + ε aij (x, u) ∂ i v ∂ j v dx K∈Th

− (1 + θ)ε

n e∈T

h

K i,j=1

{aij (x, u) ∂ i v ν j } [v] dS

e i,j=1

≥ εJhσ (v, v) + ε λ|v|2H 1 (T h ) − n(1 + θ)ε Λa

e∈T h

|∇v||[v]| dS.

e

Applying a similar technique as in the proof of Theorem 7.16, relations (7.167) – (7.174) yield Ah (v, v) ≥ λεv2 − εω − ε

δ σ J (v, v), cw h

7.9. Coercive linearized principal parts

where for v ∈ V h

(1 + θ)2 Λ2a n2 d(e)|{∇v}|2 dS 4δ e h

ω=

501

(7.186)

e∈T

≤

C5 (1 + C6 )(1 + θ)2 Λ2a n2 2 |v|H 1 (T h ) . 4δ

Now, putting δ=

C5 (1 + C6 )(1 + θ)2 Λ2a n2 2λ

(7.187)

then (7.167) and (7.169)–(7.173) imply that ελ 2 ε C5 (1 + C6 )(1 + θ)2 Λ2a n2 σ |v|H 1 (T h ) − Jh (v, v) 2 2λ cw

ελ λε 2 |v|H 1 (T h ) + Jhσ (v, v) = v2 , ≥ ελv2 − (7.188) 2 2

Ah (v, v) ≥ ε λ v2 −

which proves (7.182) with cc = ελ/2. 3. Boundedness: In view of the deﬁnition of the form Ah (7.179), we can write Ah (u, w, v) = ϑ1 + ϑ2 + ϑ3 , n aij (x, u) ∂ i w ∂ j v dx ϑ1 = ε K i,j=1

K∈Th

ϑ2 = −ε

(7.189)

n

e∈T h

e i,j=1

× {aij (x, u) ∂ i w ν j } [v] + θ{aij (x, u) ∂ i v ν j } [w] dS, ϑ3 = εJhσ (w, v). Using relation (7.185), we get |ϑ1 | ≤ ε Λa n|w|H 1 (T h ) |v|H 1 (T h ) . In order to estimate ϑ2 , we again use (7.185) and obtain |ϑ2 | ≤ ε Λa n |{∇w}| [v] + θ|{∇v}| [w] dS. e∈T h

(7.190)

(7.191)

e

With the same technique as in the proof of Theorem 7.16, (7.177)–(7.178), we ﬁnd F |ϑ2 | ≤ ε Λa n C5 (1 + C6 )/cw w v. (7.192) Now from (7.189), F (7.190), (7.192) and (7.176) we immediately get (7.182) with cb = Λa n(1 + C5 (1 + C6 )/cw ) + 1.

502 7.9.4

7. Discontinuous Galerkin methods (DCGMs)

V h -coercivity and boundedness for quasilinear problems

We have to prove the coercivity and boundedness of the linearized principal part for the IIPG method, so the summation starts with k, l = 1, instead of k, l = 0 above n ∂Ak σ l k v ) := Jh (w, v ) + (x, u, ∇ u)∂ w, ∂ v (7.193) Ah ( u; w, ∂ul K q K∈Th

k,l=1

n ∂Ak l , [ v ] (x, u , ∇ u )∂ w · ν dS, for u, w, v ∈ Ue . − k ∂ul e q h k,l=1

e∈T

Theorem 7.19. Bounded and coercive ah (u, v) for quasilinear systems: With the constant λ from the assumption (7.94) (e), let λ = min(1, λ). We choose the IIPG method (θ = 0), and a constant cw in (7.39) as cw ≥

C5 (1 + C6 ) (nqL) 2 λ

2

,

(7.194)

where C5 , C6 and L are given by (7.129), (7.138) and (7.94) (c), respectively. Then the corresponding bilinear form Ah (·, ·) given by (7.193) is bounded and coercive with respect to w, v and the norm, v , such that Ah ( u, v , v ) ≥ cc v 2 ,

(7.195)

|Ah ( u, w, v )| ≤ cb w v ∀ u, v , w ∈ H 3/2+ (T h , Rq ). Proof. Coercivity: Similarly as in (7.185) we have with L in (7.94) (c) n n ∂Ak ∂Ak (·, ·, ·)∂ l v , νk [ v ] ≤ (·, ·, ·)∂ l v , νk [ v ] l l ∂u k,l=1 ∂u q q k,l=1 ≤ nqL

n

(|∂ v , |νk [ v ]|)q ≤ nqL l

k,l=1

n

(7.196)

(7.197)

(|∂ l v |, |[ v ]|)q ≤ nqL|∇ v |q×n |[ v ]|q ,

k,l=1

for the n components of the ∂ l v , νk [ v ] ∈ Rq , their absolute values, |∂ l v |, |νk [ v ]| ∈ Rq and their norms, |∂ l v |q , |∇ v |q×n , |νk [ v ]|q ∈ R. Relations (7.94), (7.193) and (7.197), imply n ∂Ak l k (x, u , ∇ u )∂ v , ∂ v Ah ( u, v , v ) = Jhσ ( v , v ) + ∂ul K q K∈Th

k,l=1

n ∂Ak l − (x, u, ∇ u)∂ v · νk , [ v ] dS, ∂ul e k,l=1 q e∈T h σ 2 |{∇ v }|q×n |[ v ]|q dS. ≥ Jh (v, v) + λ| v |H 1 (T h ) − nqL e∈T h

e

7.10. Consistency results for the ch , bh , h

503

With very minor changes the remaining proof of coercivity as well as boundness for this theorem follows that of Theorem 7.18.

7.10

Consistency results for the ch , bh , h

For the general discretization theory we have to prove the consistency of the diﬀerent terms, the ah , ch , bh , h . We formulate the estimates with respect to global norms and step sizes. Local versions are possible as well, cf. Section 7.14. Depending upon the diﬀerent choices of ch or bh we obtain the standard or nonstandard version, for θ = −1 or θ = 1 or θ = 0 or θ ∈ [−1, 1] in the NIPG or SIPG or IIPG or GIPG version of the DCGMs. So the following results again cover all these cases. Since the consistency for the ch , bh , h is much easier, at least for the case ∂ΩD = ∂Ω, we delay the consistency for the ah , ah to the next section. The complementary case ∂ΩN = 0, requires for ch , bh a much more complicated proof, cf. [394]. The exact result is presented in Theorem 7.22. The forms ch , bh are independent of θ. h depends on θ. So the results in Section 7.10.2 are valid for θ ∈ [−1, 1]. 7.10.1

Consistency of the ch and bh

We start with the standard numerical ﬂux ch for ∂ΩD = ∂Ω. The results for the more general ch in (7.59) are valid for its special case bh in (7.58) as well. Let (7.16) (a), (A1), (A2) in Subsection 7.7.1 and, for ch , Assumptions (H) in Subsection 7.3.3 be satisﬁed. We shall be concerned with the Lipschitz-continuity of the form ch . The reason for requiring ∂ΩD = ∂Ω is the last inequality in (7.202). Jhσ is deﬁned in (7.61) as a sum over interior and Dirichlet boundary edges but not the Neumann edges, i.e. [v]2 [v]2 [v]2 dS ≤ 2 dS ≤ 2Jhσ (v, v) = 2 dS. e d(e) e d(e) e d(e) h h h K∈T

e⊂∂K

e∈T \∂ΩN

e∈T

Proposition 7.20. Under Assumption (H) and for ∂ΩD = ∂Ω, there exist constants C8 > 0, C9 > 0 such that for ch , and bh , ⎛ 1/2 ⎞ ⎠, hK u − u2L2 (∂K) |ch (u, v) − ch (u, v)| ≤ C8 v ⎝u − uL2 (Ω) + K∈Th

∀ u, u ∈ Ue , v ∈ H 1 (T h ), h ∈ (0, h0 ). For ∀uh , uh , v h ∈ S h , the last term in (7.198) is replaced by 1/2 hK u − uL2 (K) u − uH 1 (K) .

(7.198)

(7.199)

K∈T h

These estimates remain valid for |bh (u, v) − bh (u, v)|, |bh (uh , v h ) − bh (uh , v h )| as well, if f is Lipschitz-continuous.

504

7. Discontinuous Galerkin methods (DCGMs)

Proof. By (7.59), for u, u, v ∈ H 1 (T h ), f (u) − f (u), ∇v dx ch (u, v) − ch (u, v) = − < +

K∈T h

K

=>

(7.200)

?

:=σ1

H(ul , ur , ν) − H(ul , ur , ν) [v] dS .

e∈T h

<

e

=>

?

:=σ2

From the Lipschitz-continuity of the function f , we have |u − u| |∇v| dx ≤ Lf u − uL2 (Ω) |v|H 1 (T h ) . |σ1 | ≤ Lf K∈T

(7.201)

K

h

The Lipschitz-continuity (7.51) of H, the Cauchy inequality, d(e) in (7.28) imply 9 8 |σ2 | ≤ C2 |[v]| |ul − ul | + |ur − ur | dS (7.202) e∈T h

≤ C2

e

3

K∈T h

≤ 2C2

K∈T h

4

e⊂∂K

3

|[v]| |u − u| dS

e

1/2 1/2 4 [v]2 2 dS d(e) (u − u) dS e d(e) e

e⊂∂K

⎛ ≤ 4C2 Jhσ (v, v)1/2 ⎝

e⊂∂K

⎞1/2 hK u − u2L2 (∂K) ⎠

.

K∈T h

Now from (7.201) and (7.202) we get (7.198) with C8 = max(Lf , 4C2 ). We want to prove (7.199). In (7.198) we set v := v h ∈ S h . Using the multiplicative trace and the inverse inequalities (7.129) and (7.138), 1/2 hK u − u2L2 (K) + u − u2H 1 (K) . (7.203) |σ2 | ≤ CJhσ (v h , v h )1/2 K∈T h

From (7.200), (7.201) and (7.203) we get the ﬁnal result (7.198). In this proof we only have to replace C2 by Cf for obtaining the estimates for |bh (u, v h ) − bh (u, v h )|. We do need the Lipschitz-continuity for f. Theorem 7.21. Consistency of the ch , and bh : Under Assumption (H) and for ∂ΩD = ∂Ω, there exists a constant C10 > 0 such that for ch , and bh , |ch (u, v h ) − ch (P h u, v h )| ≤ C10 hmin{d,s} v h |u|H min{d,s} (Ω) , u ∈ H s (Ω), v h ∈ S h , h ∈ (0, h0 ),

(7.204)

7.10. Consistency results for the ch , bh , h

505

or (7.140) with dK ≥ d and where P h u is the S h -interpolant of u from Theorem 7.7. This estimate remains valid for |bh (u, v h ) − bh (P h u, v h )|, as well. Proof. We start from (7.198) with u ∈ H p+1 (Ω), u := P h u and v := v h ∈ S h . Using the multiplicative trace inequality (7.129), the discrete Cauchy inequality and the approximating properties (7.139), we obtain hK u − P h u2L2 (∂K) (7.205) K∈T h

≤

C5 hK u − P h uL2 (K) u − P h uH 1 (Ki )

K∈T h

≤

K∈T

min{dK ,s} min{dK ,s}−1 hK |u|2H min{dK ,s} (K)

C5 C72 hK hK

h

⎛ ≤ C5 C72 h2 min{d,s} ⎝

⎞1/2 2(dK −d)

hK

|u|2H min{dK ,s} (K) ⎠

K∈T h

≤ 2C5 C72 h2 min{d,s} |u|2H min{d,s} (Ω) . √ This estimate, (7.139) and (7.198) yield (7.204) with C10 = C7 C8 ( 2C5 + 1).

In the following theorem, we recall results presented in [394], where the consistency of the forms ch and bh , both are independent of θ, is proved in the case when ∂ΩN = 0. Theorem 7.22. Consistency of the ch , and bh : Under Assumption (H) and for > 0 such that for ch , and bh , ∂ΩN = 0, there exists a constant C10 hmin{d,s}−1/2 v h L2 (Ω) (1 + v h )|u|H min{d,s} (Ω) , |ch (u, v h ) − ch (P h u, v h )| ≤ C10

u ∈ H s (Ω), v h ∈ S h , h ∈ (0, h0 ),

(7.206)

or (7.140) with dK ≥ d and where P h u is the S h -interpolant of u from Theorem 7.7. This estimate remains valid for |bh (u, v h ) − bh (P h u, v h )|, as well.

Proof. See [394], Lemma 3. 7.10.2

Consistency of the h

For the model problem (7.13), (7.14), h (v) in (7.60) is independent of u and has the form, with θ = 1, = −1, 0, and θ ∈ [−1, 1], for the SIPG, NIPG, IIPG, and GIPG methods, respectively uD θ(∇v, ν)n + σ v dS, h (v) = g, v + (gN , v)L2 (∂ΩN ) − ε e∈∂ΩD

h (v h ) = h (v h ) .

e

(7.207)

506

7. Discontinuous Galerkin methods (DCGMs)

For all the other problems (7.77), (7.78), (7.79), (7.100), we have ∂ΩD = ∂Ω, and omit the (gN , v)L2 (∂ΩN ) term and, for (7.78), we replace g(·) by g(·, u): h (u, v) = g(·, u), v(·) − ε θ(Ba (u)∇v, ν)n u + σuD v dS. (7.208) e

e∈∂ΩD

The h ( u, v ) in (7.100) is h ( v h ) =

e∈∂Ω

σ( uD , v h )q dS, h ( v h ) = h ( v h ) .

(7.209)

e

These h (v h ) = h (v h ), h ( v h ) = h ( v h ) show by (7.121), the following: Theorem 7.23. Consistency of the h : For the problems (7.13), (7.14), (7.77), with h (v) = h (v), in (7.207), and the quasilinear problem in (7.90), (7.100) with h ( v ) = h ( v ) in (7.209), the consistency errors vanish for the SIPG, NIPG, IIPG, and GIPG methods. However for the more general semilinear problem (7.78), (7.79) we have to consider the h (u, v) in (7.208), with (Ba (u)∇v, ν)n as in (7.81) and g(x, u), v = g(x, u), v H −1 ×H 1 (T h ) : h (u, v) = ε σuD v − θ(Ba (u)∇v, ν)n udS + g(x, u), v . (7.210) e∈∂Ω

e

Theorem 7.24. Consistency for the semilinear h : For the h (u, v h ) in (7.88), (7.208), with properties (7.80), a constant, C13 = (n + 1) max{L, Λa }C7 (1 + F 2C5 (C6 + 1)) > 0, exists such that ∀h ∈ (0, h0 ), h0 < 1, ⎛ |h (P h u, v h ) − h (u, v h )| ≤ C13 ⎝

min{d,s}−1

hk

|u|H min{d,s} (K)

2

⎞1/2 ⎠

(7.211)

K∈T h

(1 + uL∞ (∂Ω) )v h H 1 (T h ) ∀u ∈ H s (T h ), v h , P h u ∈ S h . Proof. We combine the Lipschitz-continuity of the gj (x, u), j = 0, . . . , n, and aij (x, u), i, j = 1, . . . , n, with ∓aij (x, P h u)u = −aij (x, P h u)u + aij (x, P h u)u = 0 = |h (P h u, v h ) − h (u, v h )| n gj (x, P h u) − gj (x, u) ∂ j v h dx ≤ K∈T h K j=0 => ? < :=σ9

n

h h h i h aij (x, P u)P u ∓ aij (x, P u)u − aij (x, u)u ∂ v νj dS . +ε e∈∂Ω e i,j=1 => ? < :=σ10

7.11. Consistency properties of the ah

507

Obviously we obtain with the Lipschitz constant, L, for the gj |σ9 | ≤ (n + 1)LP h u − uL2 (T h ) v h H 1 (T h ) ⎛ ⎞1/2

2 min{d,s} hk ≤ (n + 1)LC7 ⎝ |u|H min{d,s} (K) ⎠ v h H 1 (T h ) . K∈T h

We estimate by combining (7.129) and (7.138) hK ∇v h 2L2 (∂K) ≤ C5 hK ∇v h L2 (K) |∇v h |H 1 (K) + C5 ∇v h 2L2 (K)

(7.212)

≤ 2C5 (C6 + 1)∇v h 2L2 (K) hence, 1 hK ∇v h 2L2 (∂K) ≤ 2C5 ∇v h 2L2 (K) = 2C5 |v h |2H 1 (T h ) . C6 + 1 h h K∈T

(7.213)

K∈T

Then the Lipschitz constant, L, and the bound Λa for the aij , yield |σ10 |/ n(Λa + LuL∞ (∂Ω) ) % & ∇v h | u − P h u | dS ≤ e∈∂Ω

⎛ ≤ ⎝

e

e∈T

(7.214)

h

e

⎞1/2 ⎛ hK |∇v h |2 dS ⎠

⎞1/2

⎝

e∈T

h

h 2 ⎠ h−1 K ([u − P u]) dS

e

⎛⎛ ⎞1/2 ⎛ ⎞1/2 ⎞ ⎜ h 2 ⎠ ⎟ ≤ ⎝⎝ hK ∇v h 2L2 (∂K) ⎠ ⎝ h−1 ⎠ K (u − P u)L2 (∂K) K∈T h

≤

F

K∈T h

⎛ 2C5 (C6 + 1)C7 |v h |H 1 (T h ) ⎝

min{d,s}−1

hk

|u|H min{d,s} (K)

2

⎞1/2 ⎠

.

K∈T h

Adding the estimates for σ9 and σ10 proves the claimed (7.211).

7.11 7.11.1

Consistency properties of the ah Consistency of the ah for the Laplacian

We prove, for ∂ΩD ∩ ∂ΩN = ∂Ω, the consistency for ah in (7.215). The necessary changes for semilinear and quasilinear ah in (7.87) and (7.99) are discussed in

508

7. Discontinuous Galerkin methods (DCGMs)

Proposition 7.29 and Proposition 7.32. The ah for the model problem have the form ah (u, v) := ε (∇u, ∇v)n dx (7.215) K∈T h

K

−ε

e∈T h \∂ΩN

[v]({∇u}, ν)n + θ[u]({∇v}, ν)n dS, e

where θ = −1 for NIPG, θ = 1 for SIPG, θ = 0 for IIPG, and θ ∈ [−1, 1] for the GIPG methods. Proposition 7.25. These ah (u, v) are bounded with respect to u, v, such that u, v, u ∈ Ue |ah (u, v) − ah (u, v)| ≤ ε |u − u|H 1 (T h ) |v|H 1 (T h ) (7.216) ⎛ ⎛ ⎞1/2 1/2 ⎜ 1 1/2 + ε 2n ⎝ (Jhσ (v, v)) ⎝ hK |u − u|2H 1 (∂K) ⎠ cw h K∈T

⎛ + ⎝

K∈T

⎞1/2 ⎛ hK ∇v2L2 (∂K) ⎠

⎝

h

⎞1/2 ⎞ 2 ⎠ ⎟ h−1 ⎠. K (u − u)L2 (∂K)

K∈T h

The last line vanishes for the IIPG method. Proof. We consider u, u, v ∈ Ue as arguments of ah (·, ·) and estimate the diﬀerent terms σj in (7.217) ﬀ.: (∇(u − u), ∇v)n dx (7.217) |ah (u, v) − ah (u, v)| ≤ ε K∈T h K => ? < :=σ1

+ε ( ∇(u − u) , ν)n [v] dS e∈T h \∂ΩN e => ? < :=σ2

( ∇v , ν)n [(u − u)] dS . +ε θ e∈T h \∂ΩN e => ? < :=σ3

σ1 is estimated with both Cauchy–Schwarz inequalities (1.44), for p = q = 2, (1.45), |u − u|H 1 (K) |v|H 1 (K) ≤ |u − u|H 1 (T h ) |v|H 1 (T h ) . (7.218) |σ1 | ≤ K∈T h

7.11. Consistency properties of the ah

For σ2 the same inequalities (1.44), (1.45), and d(e) in (7.28) yield |σ2 | ≤ n |[v]| |∇u − ∇u| dS

e

e∈T h \∂ΩN

≤ 2n

1 cw

⎞1/2 ⎛ [v]2 dS ⎠ ⎝ d(e)/cw

≤ n⎝

(7.219)

e

e∈T h \∂ΩN

⎛

⎛

1/2 1/2

(Jhσ (v, v))

⎝

e∈T h \∂ΩN

d(e) cw

⎞1/2 |∇u − ∇u|2 dS ⎠

e

⎞1/2

hK |u − u|2H 1 (∂K) ⎠

.

K∈T h

σ3 vanishes for the IIPG method. Otherwise, again (1.44), (1.45) yield |σ3 | ≤ n |∇v| | [u − u] | dS e∈T h \∂ΩN

⎛ ≤ n⎝

≤ 2n ⎝

h \∂Ω

(7.220)

e

e∈T

⎛

509

N

⎞1/2 ⎛ ⎝

hK |∇v|2 dS ⎠

e

e∈T

⎞1/2 ⎛ hK ∇v2L2 (∂K) ⎠

⎝

K∈T h

h \∂Ω

N

e

⎞1/2 h−1 K ([u

− u])2 dS ⎠ ⎞1/2

2 ⎠ h−1 K (u − u)L2 (∂K)

.

K∈T h

Adding the estimates for σ1 to σ3 in (7.218)–(7.220) proves (7.216).

Proposition 7.25 implies with u = P h u, and for ∀v h ∈ S h = V h as in Theorem 7.21 Theorem 7.26. Consistency of the Laplacian ah : The ah (P h u, v h ) in (7.87) are consistent with ∀ F u ∈ Ue , ∀v h ∈ S h = V h , so a constant, C17 := √ a(u, v) in (7.81) 1/2 max{C7 , 2n 2C5 C7 ((1/cw ) + 2C5 C7 (C6 + 1)} > 0, exists, such that for the S h interpolant P h u of u in Theorem 7.7, ⎛ |ah (u, v h ) − ah (P h u, v h )| ≤ ε ⎝

min{d,s}−1

hk

K∈T

|u|H min{d,s} (K)

h

F 1/2 C7 |v h |H 1 (T h ) + 2n c−1/2 2C5 C7 /cw Jhσ (v h , v h ) w

F + 2C5 C7 (C6 + 1)|v h |H 1 (T h ) ≤ ce C17 hmin{d,s}−1 |u|H min{d,s} (T h ) v h . The 2C5

F

C7 (C6 + 1)|v h |H 1 (T h ) can be deleted for the IIPG method.

2

⎞1/2 ⎠

(7.221)

510

7. Discontinuous Galerkin methods (DCGMs)

Proof. For (7.221) and by (7.216) we get by (7.129)–(7.139). The following ﬁrst and third lines are straightforward ⎛ ⎞1/2

2 min{d,s}−1 hk |u|H min{d,s} (K) ⎠ , by (7.139), |P h u − u|H 1 (T h ) ≤ C7 ⎝ K∈T h

⎛

⎝

⎞1/2 hK |u − P h u|2H 1 (∂K) ⎠

(7.222) ⎞1/2

2 min{d,s}−1 F hk ≤ 2C5 C7 ⎝ |u|H min{d,s} (K) ⎠ , ⎛

K∈T h

⎛

⎝

K∈T h

⎞1/2 hK ∇v h 2L2 (∂K) ⎠

≤

F

2C5 (C6 + 1) |v h |H 1 (T h ) , by (7.129), (7.138),

K∈T h

⎛

⎝

K∈T

⎞1/2 h 2 ⎠ ≤ h−1 K u − P uL2 (∂K)

F

⎛ 2C5 C7 ⎝

h

min{d,s}−1 hk |u|H min{d,s} (K)

K∈T

2

⎞1/2 ⎠ .

h

For the term in the second line of (7.222) we get, by (7.129), (7.139), hK |u − P h u|2H 1 (∂K) ≤ C5 hK |u − P h u|H 1 (K) |u − P h u|H 2 (K) + C5 |u − P h u|2H 1 (K)

min{d,s}−1 min{d,s}−2 2(min{d,s}−1) ≤ C5 C7 (|u|H min{d,s} (K) )2 hK hK hK + hK

2 (min{d,s}−1) ≤ 2C5 C7 |u|H min{d,s} (K) hK implying the second line in (7.222). The third line coincides with (7.213). For the last line in (7.222) we need, with (7.129), (7.138), and (7.139), h 2 h−1 K u − P uL2 (∂K)

(7.223)

−2 h h h 2 ≤ C5 h−1 K u − P uL2 (K) |u − P u|H 1 (K) + C5 hK u − P uL2 (K)

min{d,s} min{d,s}−1 2(min{d,s}) ≤ C5 C7 (|u|H min{d,s} (K) )2 h−1 hK + h−2 K hK K hK

(min{d,s}−1) 2 ≤ 2C5 C7 |u|H min{d,s} (K) hK . Relation (7.222) is applied in this order to the diﬀerent error terms u − P h u in (7.216): ⎞1/2 ⎛

2 min{d,s}−1 hk |ah (u, v h ) − ah (P h u, v h )| ≤ ε ⎝ |u|H min{d,s} (K) ⎠ K∈T h

F F

1/2 C7 |v h |H 1 (T h ) + 2n c−1/(2 2C5 C7 Jhσ (v h , v h ) + 2C5 C7 (C6 + 1)|v h |H 1 (T h ) . w

7.11. Consistency properties of the ah

511

This shows, with (7.154) and Proposition 7.9, the estimate (7.221). Finally, Theorems 7.12 and 7.13 imply the last estimate in (7.221). 7.11.2

Consistency of the ah for general linear problems

In Section 7.4 we have introduced for general linear elliptic problems the bilinear form, ah (u, v), cf (7.74),(7.76), for θ = −1, θ = 1, θ = 0, θ ∈ [−1, 1], ⎛ ⎞ n ⎝ ({Ba ∇0 u}, ν)0n [v] (7.224) ah (u, v) := aij ∂ i u∂ j v ⎠ dx − K∈Th

K i,j=0

+ θ ({Ba ∇0 v}, ν)n [u] dS

e∈T h

e

u, v ∈ Ue

for

with (Ba ∇0 u, ∇0 v)0n :=

n

aij ∂ i u∂ j v, (Ba ∇0 u, ν)n =

i,j=0

j=1,...,n

νj aij ∂ i u.

i=0,...,n

We prove its consistency. Proposition 7.27. These bilinear ah (u, v) are bounded with respect to u, v, such that u, v, u ∈ Ue |ah (u, v) − ah (u, v)| ≤ C0 u − uH 1 (T h ) vH 1 (T h ) (7.225) ⎛ ⎛ ⎞1/2 1/2 ⎜ 1 1/2 + 2nC0 ⎝ (Jhσ (v, v)) ⎝ hK |u − u|2H 1 (∂K) + hK u − u2L2 (∂K) ⎠ cw h K∈T

⎛ +⎝

K∈T

⎞1/2⎛

hK ∇v2L2 (∂K) + v2L2 (∂K) ⎠ ⎝

h

⎞1/2 ⎞ 2 ⎠ ⎟ h−1 ⎠. K (u − u)L2 (∂K)

K∈T h

The last line vanishes for the IIPG method. Proof. The boundedness follows from (7.225) for u = 0. We consider u, u, v ∈ Ue as arguments of ah (·, ·) and use, cf. (7.74), the bound |ai,j (x)| ≤ C0 . We estimate the diﬀerent terms σj in (7.226) ﬀ.: 0 0 0 |ah (u, v) − ah (u, v)| ≤ ε (Ba ∇ (u − u), ∇ v)n dx (7.226) K∈T h K => ? < :=σ1

+ε ( Ba ∇0 (u − u) , ν)n [v] dS e∈T h e => ? < :=σ2

512

7. Discontinuous Galerkin methods (DCGMs)

( Ba ∇0 v , ν)n [(u − u)] dS . + ε θ e∈T h e => ? < :=σ3

σ1 is estimated with both Cauchy–Schwarz inequalities (1.44), for p = q = 2, (1.45), u − uH 1 (K) vH 1 (K) ≤ C0 u − uH 1 (T h ) vH 1 (T h ) . (7.227) |σ1 | ≤ C0 K∈T h

For σ2 the same inequalities (1.44), (1.45), and d(e) in (7.39) yield |σ2 | ≤ n |[v]| | Ba ∇0 u − Ba ∇0 u| dS ⎛ ≤ n⎝

(7.228)

e

e∈T h

e

e∈T h

≤ 2nC0

1 cw

⎞1/2 ⎛ ⎞1/2 d(e) [v]2 dS ⎠ ⎝ | Ba ∇0 u − Ba ∇0 u|2 dS ⎠ d(e)/cw cw e h e∈T

1/2 ⎛ 1/2

× (Jhσ (v, v))

⎝

K∈T

hK |u − u|2H 1 (∂K) + hK u − u2L2 (∂K)

1/2

⎞ ⎠.

h

σ3 vanishes for the IIPG method. Otherwise, again (1.44), (1.45) yield Ba ∇0 v | [u − u] | dS |σ3 | ≤ n

(7.229)

e

e∈T h

⎛ ≤ nC0 ⎝

⎞1/2 ⎛

e∈T

⎛ ≤ 2nC0 ⎝

h

hK (|∇v|2 + |v|2 ) dS ⎠

e

⎝

e∈T

h

e

⎞1/2 2 ⎠ h−1 K ([u − u]) dS

⎞1/2

hK ∇v2L2 (∂K) + v2L2 (∂K) ⎠

K∈T h

⎛ ×⎝

⎞1/2 2 ⎠ h−1 K (u − u)L2 (∂K)

K∈T h

Adding the estimates for σ1 to σ3 in (7.227)–(7.229) proves (7.225). Proposition 7.27 implies with u = P h u, as in Theorem 7.21:

7.11. Consistency properties of the ah

513

Theorem 7.28. Consistency for the general linear ah : The ah (P h u, v h ) in (7.76) are consistent with a(u, v) in (7.74) ∀ u ∈ Ue , ∀v h ∈S h = V h . So a constant, C18 := F √ 2 max C7 , 2n 2C5 C7 (2(1/cw )1/2 + C5 C7 (C6 + 1) > 0, exists, such that for the S h -interpolant P h u of u in Theorem 7.7, ⎛ ⎞1/2

2 min{d,s}−1 h |ah (u, v h ) − ah (P h u, v h )| ≤ C0 ⎝ |u|H min{d,s} (K) ⎠

(7.230)

k

K∈T h

F σ h h 1/2 Jh (v , v 2C C /c 2 C7 v h H 1 (T h ) + 2n c−1/2 5 7 w w + C5

The C5

F

F

C7 (C6 + 1)v h H 1 (T h )

≤ ε C18 hmin{d,s}−1 |u|H min{d,s} (T h ) v h . C7 (C6 + 1)v h H 1 (T h ) term can be deleted for the IIPG method.

Proof. The proof is nearly identical with that of Theorem 7.26. For (7.230) and by (7.225) we again use (7.231), however replacing the ﬁrst two lines by ⎛ ⎞1/2

2 min{d,s}−1 hk P h u − uH 1 (T h ) ≤ 2C7 ⎝ |u|H min{d,s} (K) ⎠ , by (7.139), K∈T h

⎛

×⎝

K∈T

F

≤2

⎞1/2 hK u − P h u2H 1 (∂K) ⎠

(7.231)

h

⎛ 2C5 C7 ⎝

min{d,s}−1

hk

|u|H min{d,s} (K)

2

⎞1/2 ⎠

,

K∈T h

(7.222), (7.231) are applied to the diﬀerent error terms u − P h u in (7.225): ⎛ ⎞1/2

2 min{d,s}−1 hk |ah (u, v h ) − ah (P h u, v h )| ≤ C0 ⎝ |u|H min{d,s} (K) ⎠ K∈T h

F 1/2 2C7 v h H 1 (T h ) + 4n c−1/(2 2C5 C7 Jhσ (v h , v h ) w + 2C5

F

C7 (C6 + 1)v h H 1 (T h ) .

This shows, with (7.225) and Proposition 7.9, the estimate (7.230). Finally, Theorems 7.12 and 7.13 imply the last estimate in (7.230).

514

7. Discontinuous Galerkin methods (DCGMs)

7.11.3

Consistency of the semilinear ah

We turn to the consistency for the semilinear ah in (7.87) where ∂ΩN = ∅. Here it is worthwhile introducing new ah (·; ·, ·), based upon Ba (u)∇w instead of the Ba (u)∇u in (7.87). For u, v, w ∈ Ue ⊃ S h , deﬁne n n i Ba (u)∇w := aij (x, u)∂ w i=1

ah (u; w, v) := ε

, and (Ba (u)∇w, ∇v)n , (Ba (u)∇w, ν)n ,

j=1

K∈T h

(7.232)

(Ba (u)∇w, ∇v)n dx

K

({Ba (u)∇w}, ν)n [v] + θ({Ba (u)∇v}, ν)n [w] dS.

−ε

e∈T h

e

Proposition 7.29. The ah (u, v) = ah (u; u, v) in (7.232) with properties (7.80) are modiﬁed Lipschitz-continuous with respect to u and linear and bounded with respect to v, such that ah (u; u, v) satisﬁes ∀ u, v ∈ Ue ∀u ∈ W 2,∞ (T h ) |ah (u, v) − ah (u, v)| = |ah (u; u, v) − ah (u; u, v)| ≤ ε nΛa |u − u|H 1 (T h ) |v|H 1 (T h ) ⎛ ⎞1/2 1/2 + ε 2nΛa /(cw )1/2 (Jhσ (v, v)) ⎝ hK |u − u|2H 1 (∂K) ⎠

(7.233)

K∈T h

⎛ + ε 2nΛa ⎝

K∈T

⎞1/2 ⎛ hK ∇v2L2 (∂K) ⎠

⎝

h

⎞1/2

h−1 K (u

−

u)2L2 (∂K) ⎠

K∈T h

+ ε nLu − uL2 (T h ) ∇uL∞ (Ω) v h H 1 (T h ) ⎛ 1/2

+ ε 2nL/(cw )1/2 uW 1,∞ (T h ) (Jhσ (v, v))

⎝

K∈T

⎛ + ε 2nLuL∞ (Ω) ⎝

K∈T h

⎞1/2⎛ hK ∇v2L2 (∂K) ⎠

⎝

⎞1/2 hK u − u2L2 (∂K) ⎠

h

⎞1/2 2 ⎠ h−1 K (u − u)L2 (∂K)

.

K∈T h

For the special case a(u, v) in (7.19) the last three lines drop out and u ∈ W 2,∞ (T h ) is replaced by u ∈ Ue . Remark 7.30. The modiﬁed Lipschitz-continuity with respect to u indicates the dependency upon uW 1,∞ (T h ) . The condition u ∈ W 2,∞ (T h ) is imposed for allowing

7.11. Consistency properties of the ah

515

∇uL∞ (e) ≤ ∇uL∞ (K) , cf. (7.202), (7.242). We will need that for u = P h u. By the W k,∞ form of (7.139), u ∈ W 2,∞ (T h ) implies P h u ∈ W 2,∞ (T h ). Moreover, P h uW 2,∞ (T h ) ≤ P h u − uW 2,∞ (T h ) + uW 2,∞ (T h ) ≤ CuW 2,∞ (T h )

(7.234)

hence ∇P h uL∞ (e) ≤ ∇P h uL∞ (K) ≤ C∇uL∞ (K) . For good convergence we will impose later on u0 ∈ H s (T h ), s ≥ 2, for the exact solution anyway. For s − 2 ≥ n/2 this implies u0 ∈ W 2,∞ (T h ), cf. Theorem 1.26, otherwise we have to assume it additionally. For the u ∈ S h with S h ⊂ W 2,∞ (T h ) it is automatically satisﬁed. Proof. We consider u, u, v ∈ Ue as arguments of ah (·; ·, ·): |ah (u; u, v) − ah (u; u, v)|

(7.235)

≤ |ah (u; u, v) − ah (u; u, v)| + |ah (u; u, v) − ah (u; u, v)|. and estimate the diﬀerent terms σj in (7.236) ﬀ. For the ﬁrst term we obtain with (7.232) (Ba (u)∇(u − u), ∇v)n dx (7.236) |ah (u; u, v) − ah (u; u, v)| ≤ ε K∈T h K => ? < :=σ4

+ ε ({Ba (u)∇(u − u)}, ν)n [v] + θ({Ba (u)∇v}, ν)n [(u − u)] dS . e∈T h e => ? < :=σ5 +θσ 5

(7.237) σ4 is estimated with the bounded aij (x, u), cf.(7.80) (c), and an appropriately reordered discrete Chebyshev inequality (1.43), and the Cauchy–Schwarz inequality, (1.44), for p = q = 2, cf. (7.185), |(Ba (u)∇(u − u), ∇v)n | dx (7.238) |σ4 | ≤ K

K∈T h

=

K∈T

h

n

≤ Λa

K∈T h

≤ nΛa

|aij (x, u)∂ i (u − u) ∂ j v| dx

K i,j=1 n

|∂ i (u − u) ∂ j v| dx

K i,j=1

|u − u|H 1 (K) |v|H 1 (K)

K∈T h

≤ nΛa |u − u|H 1 (T h ) |v|H 1 (T h ) .

516

7. Discontinuous Galerkin methods (DCGMs)

Similarly as in (7.202), we obtain for σ5 with the Chebyshev and the Cauchy–Schwarz inequality (1.43), (1.44), for p = q = 2, and d(e) in (7.28), |σ5 | ≤ nΛa

≤ nΛa

e⊂∂K

K∈T h

|[v]| |∇(u − u)| dS

e

[v]2 dS d(e)/cw ⎛

1/2 (Jhσ (v, v))

≤ 2nΛa /(cw )

1/2

(7.239)

e

K∈T h e⊂∂K

1/2

1/2

e⊂∂K

⎞1/2

⎝

d(e) |∇(u − u)|2 dS cw e

hK |u −

u|2H 1 (∂K) ⎠

.

K∈T h

The reordered discrete Chebyshev inequality (1.43), and the H¨ older inequality, (1.44) for p = q = 2, yield for σ 5 , cf. (7.220), |σ 5 | ≤ nΛa

e∈T

⎛ ≤ nΛa ⎝

h

|∇v| | [u − u] | dS ⎞1/2 ⎛

e∈T

⎛ ≤ 2nΛa ⎝

(7.240)

e

⎝

hK |∇v| dS ⎠ 2

e

h

e∈T

⎞1/2 ⎛ hK ∇v2L2 (∂K) ⎠

⎝

K∈T h

h

e

⎞1/2 h−1 K ([u

− u]) dS ⎠ 2

⎞1/2 2 ⎠ h−1 K (u − u)L2 (∂K)

.

K∈T h

For the second term in (7.235) we combine the Lipschitz-continuity of the functions aij , i, j = 1, . . . , n or Ba with (7.232), getting |ah (u; u, v) − ah (u; u, v)| (Ba (u) − Ba (u))∇u, ∇v n dx ≤ε K∈T h K => ? <

(7.241)

:=σ6

+ ε ({(Ba (u) − Ba (u))∇u}, ν)n [v] + θ({(Ba (u) − Ba (u))∇v}, ν)n [u] dS . e∈T h e => ? < :=σ7 +θσ 7

Obviously we obtain with the Lipschitz constant, L, for the aij or the Ba with (1.43), |σ6 | ≤ nLu − uL2 (T h ) ∇uL∞ (Ω) v h H 1 (T h ) .

(7.242)

7.11. Consistency properties of the ah

517

A combination with the ideas for, e.g. (7.202)–(7.203) allows the estimation of σ7 : here we need ∇uL∞ (e) ≤ ∇uL∞ (K) requiring u ∈ W 2,∞ (T h ) by Theorem 1.38. ⎛ |σ7 | ≤ 2nL ⎝

∇u2L∞ (e) e

e∈T h

⎞1/2 ⎛ ⎞1/2 n d(e) [v]2 dS ⎠ ⎝ (u − u)2 dS ⎠ d(e)/cw c w e h i=1 e∈T

⎛ 1/2

≤ 2nL/(cw )1/2 uW 1,∞ (T h ) (Jhσ (v, v))

⎝

⎞1/2

hK u − u2L2 (∂K) ⎠

.

(7.243)

K∈T h

For σ 7 we obtain, as in (7.220), |σ 7 | ≤ 2nLu

|∇v| |(u − u)| dS

L∞ (Ω) e∈T h

⎛ ≤ 2nLuL∞ (Ω) ⎝

K∈T

(7.244)

e

h

⎞1/2 ⎛ hK ∇v2L2 (∂K) ⎠

⎝

K∈T

⎞1/2 2 ⎠ h−1 K (u − u)L2 (∂K)

.

h

Adding the estimates for σ4 to σ 7 in (7.238)–(7.244) proves the claimed (7.233).

As consequence of Proposition 7.29 we obtain with u = P h u, the S h -interpolant from Theorem 7.7, but now only for ∀v h ∈ S h = V h as in Theorem 7.21, compare Remark 7.30. Theorem 7.31. Consistency for the semilinear ah : The ah (u, v h ) in (7.87) with properties (7.80) are Lipschitz-continuous with respect to u and linear and bounded with respect to v h . Furthermore the ah (P h u, v h ) are consistent with the ah (u, v h ). So constants C1∗ , C2∗ exist, cf. (7.247) such that, ∀ h < 1, u ∈ W 2,∞ (T h ), ∀v h ∈ S h = V h , and the S h -interpolant, P h u ∈ W 2,∞ (T h ), with dK ≥ d, (7.245) |ah (u, v h ) − ah (P h u, v h )| ⎛ ⎞1/2

2 min{d ,s}−1 ∗ K ≤ε ⎝ C1 + C2∗ P h uW 1,∞ (T h ) v h hk |u|H min{dK ,s} (K) ⎠ K∈T h

≤ ε hmin{d,s}−1 |u|H min{d,s} (T h ) C1∗ + 2C2∗ uW 1,∞ (T h ) v h . The term +2C2∗ uW 1,∞ (T h ) drops out for a(u, v) in (7.19) and u ∈ W 2,∞ (T h ) is replaced by u ∈ Ue . Proof. It proceeds totally analogously to that of Theorem 7.26. We essentially apply the estimates in (7.222) to the terms in (7.246) and obtain, with the equivalence of

518

7. Discontinuous Galerkin methods (DCGMs)

the penalty norms in (7.154), (1.43), |ah (u; u, v h ) − ah (P h u; P h u, v h )| ⎛ ⎞1/2

2 min{d ,s}−1 K ≤ ε⎝ hk |u|H min{dK ,s} (K) ⎠ K∈T h

× nΛa C7 |v h |H 1 (T h ) + nLC7 hv h H 1 (T h ) ∇P h uL∞ (T h ) F

1/2 2C5 C7 /cw (Λa + hLP h uW 1,∞ (T h ) ) Jhσ (v h , v h )

F + 4nC5 C7 (C6 + 1)(Λa + LP h uL∞ (T h ) )|v h |H 1 (T h ) + 2n

⎛ ≤ε ⎝

min{dK ,s}−1

hk

K∈T

|u|H min{dK ,s} (K)

2

⎞1/2 ⎠

(7.246)

h

H% F & v h nΛa (C7 + 4nC5 C7 (C6 + 1))2 + 8nC5 C7 /cw

F H + P h uW 1,∞ (T h ) (1 + ce )nL C7 h2 C7 + 8h2 C5 /cw + 16C52 (C6 + 1) and with the following C1∗ , C2∗ , ⎛ ⎞1/2

2 min{d ,s}−1 ∗ K C1 + C2∗ P h uW 1,∞ (T h ) v h . ≤ ε⎝ hk |u|H min{dK ,s} (K) ⎠ K∈T h

with C1∗ := nΛa

H% F & (C7 + 4nC5 C7 (C6 + 1))2 + 8nC5 C7 /cw

and

(7.247)

F H C2∗ := (1 + ce ) nL C7 h2 C7 + 8h2 C5 /cw + 16C52 (C6 + 1) . Thus we have proved the estimate (7.245). 7.11.4

Consistency of the quasilinear ah for systems

Again we introduce a new ah (·; ·, ·), based upon BA ( u, ∇w) instead of the BA ( u, ∇ u) in (7.97), (7.99). We update the notation in (7.92), with (T h , Rq ) replacing (T h ), and deﬁne, e.g. n h ) := Ak (x, uh , ∇w h ) k=1 ∈ Rn×q , (BA ( uh , ∇w h ), ν)n ∈ Rq BA ( uh , ∇w

(7.248)

7.11. Consistency properties of the ah

519

yielding ∀ uh , w h ∈ Ue , ∀ v h ∈ U1,e := H 1 (T h , Rq ): n ah ( uh ; ∇w h , v h ) = (Ak (x, uh , ∇w h ), ∂ k v h )q dx K∈T h

−

K k=0

({BA ( uh , ∇w h )}, ν)n , [ v h ]

e∈T h

e

q

dS.

Proposition 7.32. The ah ( u; u, v ) in (7.248) with properties (7.94) are modiﬁed Lipschitz-continuous with respect to u and linear in v , and bounded with respect to v . So ∀ u, v ∈ Ve , u ∈ W 2,∞ (T h , Rq ) and uU1,e , uU1,e ≤ M this ah ( u; u, v ) satisﬁes |ah ( u, v ) − ah ( u, v )| = |ah ( u; u, v ) − ah ( u; u, v )| ≤ L| u − u|U1,e | v |U1,e

(7.249)

+ L(1 + | u|W 1,∞ (T h ,Rq ) )| u − u|U1,e | v |L2 (T h ,Rq ) ⎛ ⎞1/2 1/2 + nL/(cw )1/2 (Jhσ ( v , v )) ⎝ hK | u − u|2H 1 (∂K) ⎠ K∈T h

+ L u − uL2 (T h ,Rq ) 1 + uW 1,∞ (T h ,Rq ) + u2W 1,∞ (T h ,Rq ) v h U1,e ⎛ ⎞1/2 + nL/(cw )1/2 ⎝ hK u − u2L2 (∂K) ⎠ K∈T h

×Jhσ ( v , v ))1/2 (1 + uW 1,∞ (T h ,Rq ) ). Proof. We consider u, u, w, v ∈ Ue as arguments of ah (·; ·, ·) and estimate |ah ( u; u, v ) − ah ( u; u, v )| ≤ |ah ( u; u, v ) − ah ( u; u, v )|

(7.250)

+ |ah ( u; u, v )| − ah ( u; u, v )|. For the second term we obtain with (7.248) |ah ( u; u, v ) − ah ( u; u, v )| n k

Ak (x, u, ∇ u) − Ak (x, u, ∇ u) , ∂ v dx ≤ q K∈T h K k=0 => ? < :=σ9 =σ9k>0 +σ9k=0

(7.251)

+ ([ v ], ({BA ( u, ∇ u )}, ν)n − ({BA ( u, ∇ u )}, ν)n )q dS . e∈T h e => ? < :=σ10

520

7. Discontinuous Galerkin methods (DCGMs)

Since the conditions for the partials with respect to u of the Ak in (7.94) n for k = 0 and k > 0 are diﬀerent, σ9 and σ11 below are split. σ9k>0 , obtained by k=1 , is estimated ∇ u), cf. (7.94) (c), and the Cauchy–Schwarz with the Lipschitz-continuous Ak (x, w, inequalities (1.44) for p = q = 2, (1.45), |σ9k>0 | ≤

n

K∈T h

≤L

(7.252)

q

K k=1

K∈T h

≤L

Ak (x, u, ∇ u) − Ak (x, u, ∇ u) , ∂ k v dx

∇ u − ∇ un×q ∇ v n×q dx

K

| u − u|H 1 (K) | v |H 1 (K)

K∈T h

≤ L| u − u|U1,e | v |U1,e . For σ9k=0 , we combine (7.94) (d), with (1.44), (1.45), |σ9k=0 | ≤

K∈T h

≤L

q

K

K∈T h

≤L

A0 (x, u, ∇ u) − A0 (x, u, ∇ u) , v dx

(1 + ∇ un×q )∇ u − ∇ un×q v q dx

K

(1 + | u|W 1,∞ (K) )| u − u|H 1 (K) | v |L2 (K)

K∈T h

≤ L(1 + | u|W 1,∞ (T h ,Rq ) )| u − u|U1,e | v |L2 (T h ,Rq ) .

(7.253)

Since BA ( u, ∇ u) is independent of A0 we do not need a σ10k=0 , but only σ10 . Similarly as in (7.238), (7.252), we obtain with d(e) in (7.28), |σ10 | ≤ nL

e∈T h

[ v ] q ∇ u − ∇ un×q dS

e

⎞1/2 ⎛ ⎞1/2 d(e) [ v ] 2q dS ⎠ ⎝ (∇ u − ∇ un×q )2 dS ⎠ ≤ nL ⎝ d(e)/c c w w e e h h ⎛

e⊂T

e⊂T

⎛ 1/2

≤ nL/(cw )1/2 (Jhσ ( v , v ))

⎝

K∈T h

⎞1/2 hK | u − u|2H 1 (∂K) ⎠

.

7.11. Consistency properties of the ah

521

For the ﬁrst term in (7.250) we combine the bounds for the partials of the functions Ak , k = 1, . . . , n in (7.94) with (7.248), hence σ11 is split: |ah ( u; u, v )| − ah ( u; u, v )| n k

Ak (x, u, ∇ u) − Ak (x, u, ∇ u) , ∂ v dx ≤ q K∈T h K k=0 => ? <

(7.254)

:=σ11 =:σ11k>0 +σ11k=0

+ ([ v ], ({BA ( u, ∇ u )} − {BA ( u, ∇ u )}, ν)n )q ) dS . e∈T h e => ? < :=σ12

We obtain with the diﬀerent bounds for the partials of the functions Ak for k = 0 and k = 1, . . . , n, and for u ∈ W 1,∞ (T h , Rq ) for the σ11k>0 and σ11k=0 |σ11k>0 | ≤ L u − uL2 (T h ,Rq ) (1 + uW 1,∞ (T h ,Rq ) ) v h U1,e

(7.255)

|σ11k=0 | ≤ L u − uL2 (T h ,Rq ) (1 + u2W 1,∞ (T h ,Rq ) ) v h L2 (T h ,Rq ) . A combination of (7.94) (c) with the ideas in (7.202)–(7.203) allows the estimation of σ12 . In (7.256), we need ∇ uL∞ (e) ≤ ∇ uL∞ (K) valid for u ∈ W 2,∞ (T h , Rq ), cf. Theorem 1.38. 1/2 [ v ] 2q dS (7.256) |σ12 | ≤ nL e d(e)/cw h K∈T

×

e⊂∂K

d(e) ( u − u) 2q dS cw e

1/2 (1 + ∇ uL∞ (e) )

e⊂∂K

⎛ 1/2

≤ nL/(cw )1/2 (Jhσ ( v , v ))

⎝

⎞1/2 hK u − u2L2 (∂K) ⎠

(1 + | u|W 1,∞ (T h ,Rq ) ).

K∈T h

Adding the estimates for σ11 to σ12 in (7.252)–(7.256), proves the claim.

As consequence of Proposition 7.32 we obtain with u = P h u as in Theorem 7.31 Theorem 7.33. Consistency for the quasilinear ah : The ah ( u, v h ) in (7.248) with properties (7.94) are modiﬁed Lipschitz-continuous with respect to u, and linear and bounded with respect to v h . Furthermore the ah (P h u, v h ) are consistent with the ah ( u, v h ). So for suﬃciently small h < 1, and uU1,e , uU1,e ≤ M, a constant 1/2 exists for ah ( u, v h ), s.t. ∀ v h ∈ Sqh and C16 := L max 2 + 2n2 C5 C7 (1 + h2 )/cw , c2e

522

7. Discontinuous Galerkin methods (DCGMs)

∀ u ∈ W 2,∞ (T h , Rq ), the S h -interpolant, P h u from Theorem 7.7, cf. (7.234): |ah ( u, v h ) − ah (P h u, v h )| ≤ C16 hmin{d,s}−1 | u|H min{d,s} (T h ,Rq ) (7.257)

× 1 + uW 1,∞ (T h ,Rq ) + u2W 1,∞ (T h ,Rq ) v h . Proof. Again we apply the slightly modiﬁed error estimates in (7.222): ⎛ P h u − uU1,e ≤ C7 ⎝

min{d,s}−1

hk

| u|H min{d,s} (K,Rq )

2

⎞1/2 ⎠

K∈T h

⎛ ×⎝

K∈T

≤

⎞1/2

F

hK | u − P h u|2H 1 (∂K) ⎠

h

⎛ 2C5 C7 ⎝

min{d,s}−1

hk

| u|H min{d,s} (K,Rq )

2

⎞1/2 ⎠

K∈T h

⎛ P h u − uL2 (T h ,Rq ) ≤ C7 ⎝

min{d,s}

hk

| u|H min{d,s} (K,Rq )

2

⎞1/2 ⎠

K∈T h

⎛ ×⎝

K∈T

≤

F

⎞1/2 hK u − P h u2L2 (∂K) ⎠

h

⎛ 2C5 C7 ⎝

min{d,s}

hk

| u|H min{d,s} (K,Rq )

2

⎞1/2 ⎠

.

K∈T h

These estimates are applied to the error terms in (7.249) to get |ah ( u, v h ) − ah (P h u, v h )| F ≤ max{C7 , 2C5 C7 }hmin{d,s}−1 | u|H min{d,s} (T h ,Rq ) × L| v h |U1,e + L(1 + | u|W 1,∞ (T h ,Rq ) )| v h |L2 (T h ,Rq ) F 1/2 + nL/(cw )1/2 2C5 C7 Jhσ ( v h , v h ) + L 1 + uW 1,∞ (T h ,Rq ) + u2W 1,∞ (T h ,Rq ) v h U1,e

F 1/2 + nLh/(cw )1/2 2C5 C7 Jhσ ( v h , v h ) (1 + uW 1,∞ (T h ,Rq ) ) .

7.11. Consistency properties of the ah

523

This implies with P h u ∈ W 2,∞ (Ω, Rq ) and the equivalence of v and v JH 1 (T h ,Rq ) with ce in (7.154) F |ah ( u, v h ) − ah (P h u, v h )| ≤ max C7 , 2C5 C7 hmin{d,s}−1 | u|H min{d,s} (T h ,Rq ) F v h L 2 + | u|W 1,∞ (T h ,Rq ) + nL/(cw )1/2 2C5 C7 + Lce 1 + uW 1,∞ (T h ,Rq ) F + u2W 1,∞ (T h ,Rq ) + nLh/(cw )1/2 2C5 C7 1 + uJW 1,∞ (T h ,Rq ) 1/2 min{d,s}−1 ≤ L max 2 + 2n2 /cw C5 C7 (1 + h2 ), c2e h | u|H min{d,s} (T h ,Rq ) v h

1 + uW 1,∞ (T h ,Rq ) + u2W 1,∞ (T h ,Rq ) .

This shows the claim (7.257). Consistency of the quasilinear ah for the equations of Houston, Robson, S¨ uli, and for systems

7.11.5

We turn to consistency for the DCGMs for the quasilinear problem G(u) = 0 in Houston et al. [407], again only for the principal part. All other terms are equal to those for the semilinear problem. As an essential tool for consistency, we refer to Barett and Liu [68], Lemma 2.1: From (7.103)–(7.106) Houston et al. [407] obtain for −1 ≤ θ ≤ 1 |(Ba (u)∇u − Ba (v)∇v)(x)| ≤ C|(∇u − ∇v)(x)| ∀x ∈ Ω, ∀u, v ∈ Ue .

(7.258)

We have Proposition 7.34. The ah (u, v) in (7.107) with properties (7.103), and C in (7.258), and C ∗ := 4Mμ C5 (1 + C6 ), are modiﬁed Lipschitz-continuous with respect to u and linear and bounded with respect to v, such that ah (u, v) satisﬁes ∀ u, u, v ∈ Ue , |ah (u, v) − ah (u, v)| ≤ nC|u − u|H 1 (T h ) |v|H 1 (T h ) ⎛ 1/2

+ 2nC/(cw )1/2 (Jhσ (v, v))

⎝

K∈T

⎛ + C ∗ |v|H 1 (T h ) ⎝

(7.259) ⎞1/2 hK |u − u|2H 1 (∂K) ⎠

h

⎞1/2

2 ⎠ h−1 K |u − u|L2 (∂K)

.

K∈T h

Remark 7.35. In (7.107) we have replaced our usual ({Ba (u)∇v}, ν)n [u] dS = θ ({μ(x, |∇u|)∇v}, ν)n [u] dS θ e∈T h

e

e∈T h

e

(7.260)

524

7. Discontinuous Galerkin methods (DCGMs)

by the form [407] θ ({μ(x, h−1 |[u]|)∇v}, ν)n [u] dS + θ ({μ(x, h−1 |u − uD |)∇v} . . . . e∈T h \∂Ω

e

e∈∂Ω

e

1/2 −1 2 Our form (7.260) yields the term in the last line of K∈T h hK |u − u|H 1 (∂K) (7.259). This would reduce the consistency order by 1, unless we restrict the method 1/2 −1 2 , and thus the to θ = 0. The above form [407] yields K∈T h hK |u − u|L2 (∂K) desired order of consistency. Proof. 1. We estimate the diﬀerent terms σj in (7.261) ﬀ. and obtain with (7.107) ah (u, v) − ah (u, v)| ≤ (Ba (u)∇u − Ba (u)∇u, ∇v)n dx K∈T h K => ? <

(7.261)

:=σ4

+ ({Ba (u)∇u − Ba (u)∇u}, ν)n [v] dS e∈T h e => ? < :=σ5

μ(x, h−1 |[u]|) [u] − μ(x, h−1 |[u]|) [u] ({∇v}, ν)n dS + + θ . . .. e∈T h \∂Ω e e∈∂Ω => ? < :=θσ 5

2. σ4 and σ5 are estimated, similarly to the previous cases. With (7.258), an appropriately reordered discrete Chebyshev inequality (1.43), and the Cauchy– Schwarz inequality, (1.44), for p = q = 2, cf. (7.185), we obtain |(Ba (u)∇u − Ba (u)∇u, ∇v)n | dx (7.262) |σ4 | ≤ K∈T h

K

≤C

K∈T h

≤ nC

|∇u − ∇u ||∇v| dx

K

|u − u|H 1 (K) |v|H 1 (K)

K∈T h

≤ nC|u − u|H 1 (T h ) |v|H 1 (T h ) .

7.11. Consistency properties of the ah

525

Similarly as in (7.202), we obtain for σ5 with (7.258), the Chebyshev and the Cauchy–Schwarz inequalities (1.43), (1.44), for p = q = 2, and d(e) in (7.28), |[v]| |∇(u − u)| dS (7.263) |σ5 | ≤ nC e

K∈T h e⊂∂K

≤ nC

K∈T h

e

e⊂∂K

[v]2 dS d(e)/cw ⎛ 1/2

≤ 2nC/(cw )1/2 (Jhσ (v, v))

1/2

1/2

e⊂∂K

⎝

d(e) |∇(u − u)|2 dS cw e

K∈T

⎞1/2 hK |u − u|2H 1 (∂K) ⎠

.

h

3. We start estimating the integrals e in σ5 . Then (7.103) implies −1 −1 μ(x, h |[u]|) [u] − μ(x, h |[u]|) [u] ({∇v}, ν)n dS e

μ(x, h−1 |[u]|) [u] − μ(x, h−1 |[u]|) [u]|{∇v}| dS e ≤ hμ(x, h−1 |[u]|) h−1 [u] − μ(x, h−1 |[u]|) h−1 [u]|{∇v}| dS ≤

e

≤

hMμ h

−1

|[u] − [u]||{∇v}| dS =

e

Mμ |[u − u]||{∇v}| dS. e

Summation yields |σ 5 | ≤ Mμ |[u − u]||{∇v}| dS e∈T h \∂Ω

e

≤ 4Mμ

e∈T

⎛ ≤ 4Mμ ⎝

h \∂Ω

|u − u||∇v| dS e

e∈T

h \∂Ω

⎞1/2 ⎛ hK |∇v|2 dS ⎠

e

⎝

e∈T

⎛ ≤ 4Mμ C5 (1 + C6 )|v|H 1 (T h ) ⎝

h \∂Ω

e

⎞1/2 2 ⎠ h−1 K |u − u| dS

⎞1/2 2 ⎠ h−1 K |u − u|L2 (∂K)

.

(7.264)

K∈T h

1/2 2 ≤ C5 (1 + C6 )|v|H 1 (T h ) we For the last estimate e∈T h \∂Ω e hK |∇v| dS ({μ(x, h−1 |u − have combined (7.129), (7.138). Similarly for θ e e∈∂Ω

uD |)∇v} . . . . Adding the estimates for σ4 , σ5 and σ 5 proves the claimed (7.259).

526

7. Discontinuous Galerkin methods (DCGMs)

As a consequence of Proposition 7.34 we obtain the consistency for the quasilinear equations. These results can be directly extended to the system of the form, cf (7.102), −∇ · (μ(x, |∇ u|)∇ u) = −

n

∂ k μ(x, |∇ u|)∂ k u = g ,

(7.265)

k=1

C(Ω × [0, ∞)) is a where u = (u1 , . . . , uq ) : Ω → Rq , g ∈ L2 (Ω)q and the previous μ ∈ q real function satisfying the assumption (7.4). Here we put |∇ u|2 = i=1 |∇ui |2 . Using Barett and Liu [68], Lemma 2.1, and (7.102)–(7.106) Houston et al. [407] obtain for −1 ≤ θ ≤ 1 similarly as in (7.258) |(Ba ( u)∇ u − Ba ( v )∇ v )(x)| ≤ C|(∇ u − ∇ v )(x)| ∀x ∈ Ω, ∀ u, v ∈ Ue ,

(7.266)

where Ue is given by (7.92) and Ba ( v )∇ v = μ(x, |∇ v |)∇ v .

(7.267)

Therefore, replacing (7.258) by (7.266) we immediately obtain the consistency results represented by Proposition 7.34 and Theorem 7.36 also for system (7.265). Theorem 7.36. Consistency for the quasilinear ah in [407] and for systems: 1. The ah (uh , v h ) in (7.106) with properties (7.103) are Lipschitz-continuous with respect to u and linear and bounded with respect to v h . Furthermore the ah (P h u, v h ) = ah (P h u, v h ) are consistent with the ah (u, v h ). So a constant C1∗ exists, cf. (7.269) such that, ∀ h < 1, ∀ u ∈ Ue , v h ∈ S h = V h , |ah (u, v h ) − ah (P h u, v h )| ≤ C1∗ hmin{d,s}−1 |u|H min{d,s} (T h ) v h .

(7.268)

2. These results remain correct, if (7.102) is replaced by the system in (7.265) ﬀ., and the uh , v h , Ba (v)v in ah (u, v h ) by uh , v h , Ba ( v ) v to obtain ah ( u, v h ). Proof. The proof proceeds totally analogously to that of Theorem 7.31. With u = P h u we obtain: ⎛ |ah (u, v h ) − ah (P h u, v h )| ≤ ⎝

K∈T

nCC7 |v h |H 1 (T h ) + 2nC F

+ 4nMμ C5 (1 + C6 )

⎞1/2 min{dK ,s}−1

(hk

|u|H min{dK ,s} (K) )2 ⎠

h

F 1/2 2C5 C7 /cw Jhσ (v h , v h )

2C5 C7 |v h |H 1 (T h ) .

(7.269)

7.12. Convergence for DCGMs

⎛ ≤ C1∗ ⎝

min{dK ,s}−1

hk

K∈T h

with C1∗ := n

|u|H min{dK ,s} (K)

⎞1/2 ⎠

v h

H%

& (C 2 C72 + 8C 2 C5 C7 /cw + 32Mμ2 C53 C7 (1 + C6 )2 .

Thus we have proved the estimate (7.268).

7.12

2

527

Convergence for DCGMs

We achieve convergence in the following way. The V h -coercivity and boundedness of the discrete principal parts (Gh )p (uh ) of the linearized operators G (u) plus the penalty term with respect to the penalty norm v h , implies its stability, cf. Theorems 7.16– 7.19. Its combination with consistency allows the proof of the stability of the discrete derivatives (Gh ) (uh ), for boundedly invertible G (u0 ), and hence of the discrete nonlinear Gh . With the consistency and some technical conditions this yields the desired convergence, cf. Theorem 3.21, 3.23, 3.29. For consistency, we tested all the ah (uh , v h ), bh (uh , v h ), ch (uh , v h ), h (uh , v h ), σ h h Jh (u , v ), by v h . They are linear and bounded in v h with respect to the · , but some are nonlinear in uh . The orders of consistency are diﬀerent for these forms and the constants in the estimates depend nonlinearly upon u, cf. Theorems 7.21–7.33. The proof that a boundedly invertible compact perturbation of the previous principal parts generates a stable discretization is pretty technical. However, we apply exactly the same technique and ideas for DCGMs as for FEMs with variational crimes, only indicated here. We use the notation U for H 1 (Ω, Rq ), q ≥ 1, Ue for H 3/2+ (T h , Rq ), and Ueo for H 3/2+ (Ω, Rq ) here and below. In a ﬁrst step towards general stability we deﬁne, similarly to nonconforming FEs, an anticrime transformation from uh ∈ U h to uh = E h uh ∈ U. In Lemma 5.77 we had combined the ah (·, ·)-coercivity for speciﬁc a(·, ·), implying the stability, with its consistency, hence the convergence, however here in the opposite direction of the exact solution uh = E h uh to the chosen discrete uh ∈ U h . For nonconforming FEs we reduced continuity and boundary conditions along the full edges only to a few appropriate points along these edges. For the DCG elements we instead consider the penalty terms. Lemma 7.37. Anticrime transformation for DCGMs: We choose the boundedly invertible Laplacian, with the inhomogeneous A = Δ − g : U → U , and Dirichlet con ditions on ∂Ω. Then its IIPG form with θ = 0, the Ah : U h → U h is stable and consistent with A for every u ∈ U . For a given uh ∈ U h deﬁne fˆh := Ah uh ∈ Ue with fh Ue ≤ Cuh , cf. (7.114). Deﬁne E h : U h → U by AE h uh , v = fh , v∀v ∈ Vb and the boundary condition (uh − uD )|∂Ω = 0. Then we obtain convergence as limh→0 E h uh − uh = 0.

528

7. Discontinuous Galerkin methods (DCGMs)

Proof. In contrast to the previous nonlinear problems, we could omit here the ∩L∞ (T h ) in the deﬁnition of Ue = H 3/2+ (T h ) ∩ L∞ (Ω), etc. That the IIPG form Ah : U h → U h is stable and consistent with A is proved above. With the ah (u, v), . . . , Jhσ (u, v) in (7.57)–(7.61), ∂ΩN = ∅, and our notation, e.g. Ah u, v = Ah u, vU ×Ue , we deﬁne the weak operators A, Ah , Ah , as e

(a) A : U → U with (u − uD )|∂Ω = 0, Ah : Ue → Ue , Ah : U h = S h → U h ,

(7.270)

(b) Au, v − a(u, v) + (v) = 0, (u − uD )|∂Ω = 0 for ﬁxed u ∈ U, ∀ v ∈ Vb , (c) Ah u, v − ah (u, v) − εJhσ (u, v) + h (v) = 0, ﬁxed u, ∀ v ∈ Ue , (d) Ah uh , v h − ah (uh , v h ) − εJhσ (uh , v h ) + h (v h ) = 0, ﬁxed uh , ∀v h ∈ S h , ah (uh , v h ) := ah (uh , v h )|S h ×S h , h (v h ) := h (v h )|S h . Similarly as for nonconforming FEMs, we deﬁne, for uh ∈ S h the fˆh ∈ S h by

uh : fˆh , v h := Ah uh , v h = ah (uh , v h ) + εJhσ (uh , v h ) − h (v h )∀v h ∈ S h . (7.271) In the next step we determine uh := E h uh ∈ U from Auh = fˆh . We start with uh , v ∈ Ue ⊃ Ueo , such that the ah (uh , v), εJhσ (uh , v), h (v), fˆh , are well deﬁned. By uh ∈ Ueo , the boundary condition and v ∈ Vb ∩ Ueo , the terms with jumps, [u] = [v] = 0, and along the boundary vanish, and we end up with uh ∈ U, and v ∈ Vb . So we determine E h uh = uh ∈ U

with boundary conditions (uh − uD )|∂Ω = 0 implying

Auh , v = fˆh , v = ah (uh , v) + εJhσ (uh , v) − h (v) =ε (∇uh , ∇v)n dx K∈Th

−ε

∀v ∈ Vb

K

∇uh , ν)n v + (∇v, ν)n uh dS + ε σuh v dS

e∈∂Ω

e

− g, v − ε =ε

(7.272)

K∈Th

e∈∂Ω

e∈∂Ω

e

σuD v − (∇v, ν)n uD dS

e

(∇uh , ∇v)n dx − g, v = a(uh , v) − (v) ∀v ∈ Vb .

K

For this uh ∈ Ue we determine the discrete solution uhh ∈ S h solving, cf. (7.270), : h h; Auh , v = ah uhh , v h + εJhσ uhh , v h − h (v h ) ∀v h ∈ S h . Thisis Equation (7.271) with uhh replaced by uh . Then the coercivity of ah uhh , v h + εJhσ uhh , v h implies uh = uhh and, with the consistency, the convergence limh→0 uh − uhh = 0 to the chosen discrete uh ∈ U h .

7.12. Convergence for DCGMs

529

This allows the strategy, starting with Lemma 5.77, for proving the stability of Ah for boundedly invertible linear A as in Theorem 3.29. We reformulate this theorem for our DCGMs. Theorem 7.38. Stability criterion for DCGMs: Choose U h = V h ⊂ U as in (7.65),(7.95). Choose P h = I h and Q h as in Theorem 7.7, Corollary 7.8, and in (7.113), (7.115), and one of the DCGMs as formulated above. Let A : Ub → V = V be boundedly invertible and be one of the linear(ized) operators for equations or systems, studied in this chapter. These DCGMs for A, the Ah , are consistent with A in a smooth enough u. Then A−1 ∈ L(V , Ub ) =⇒ Ah is stable in P h u.

(7.273)

Under these conditions the linear Ah is stable. By Theorem 3.23 this stability is inherited for boundedly invertible G (u0 ) to Φh G. With our above consistency results, Theorem 4.53 implies the unique existence of the discrete approximation uh0 and its convergence to the exact solution u0 . We summarize, for the diﬀerent problems and their discrete operators. Condition 7.39. Conditions for stability and consistency for DCGMs; summary of these results for the diﬀerent previous problems: 1. Choose U h = V h = S h ⊂ U as in (7.63), (7.95) with the penalty norm uh V h = uh in (7.153) and the parameter cw as in (7.39), (7.164). For b(·, ·) we impose Assumption (H), cf. (7.51)–(7.53), and for our DCG elements S h of local degree d − 1 the conditions (A0 )–(A2 ), (7.124)–(7.128), (7.146). In particular, the S h are approximating spaces for the function spaces considered here. We always denote the S h -interpolant for u from Theorem 7.7 as P h u. 2. We consider the nonsymmetric, the symmetric or the incomplete or general interior penalty Galerkin method, the NIPG, SIPG, IIPG or GIPG method, respectively, with θ = −1, 1, 0 or θ ∈ [−1, 1]. 3. For all problems, the operator G and its linearization in u0 , the G (u0 )u = f ∈ U , has to be combined with the boundary condition (7.14), so u |∂ΩD = uD on ∂ΩD , and ∂u/∂ν |∂ΩN = φ on ∂ΩN . In particular, we require for all cases the bounded invertibility of G (u0 ) with the Dirichlet and Neumann or only Dirichlet boundary conditions, e.g. (7.14), cf. Theorem 7.14. 4. If in the consistency results a |u|H min{d,s} (Ω) is shown, we always require u ∈ H min{d,s} (Ω); for the hp-methods in Section 7.14 we replace this by |u|H min{dK ,s} (K) for u ∈ H min{dK ,s} (K), respectively Throughout this section we use the abbreviation C(h, u) := hmin{d,s}−1 |u|H min{d,s} (Ω) .

(7.274)

5. For the special convection–diﬀusion problem (7.13), (7.14) we impose (7.16), and choose a numerical ﬂux satisfying Assumption (H): (7.51)–(7.53). Deﬁne its standard DCGM as in (7.70). Then coercivity and stability is guaranteed by Theorem 7.16, (7.165), (7.166). The following consistency estimates are valid, cf.

530

7. Discontinuous Galerkin methods (DCGMs)

Theorems 7.10, 7.21, 7.22, 7.23, 7.26, |ch (u, v h ) − ch (P h u, v h )| ≤ C10 C(h, u)v h for Dirichlet boundary cd., (7.275) |ch (u, v ) − c (P u, v )| ≤ C10 C(h, u)(1 + v ) for Dir. and Neumann b.cs. h

h

h

h

h

|ah (u, v h ) − ah (P h u, v h )| ≤ ε C17 C(h, u)v h , σ ε Jh (u, v h ) − ε Jgσ (P h u, v h ) ≤ ε C15 C(h, u)v h and h (v h ) − h (v h ) = 0 ∀v h ∈ V h = S h . 6. The nonstandard DCGM for this problem (7.13), (7.14), (7.16) is deﬁned in (7.71). The coercivity and consistency estimates for the ah , bh , h are the same as for the previous ah , ch , h . 7. For the linear operator (7.74), (7.75) with its conditions we deﬁne its DCGM in (7.77). Then coercivity (and stability) and the consistency estimates are guaranteed by obvious modiﬁcations of Theorems 7.18, and 7.10, 7.23, 7.31. Modifying the previous consistency estimates requires changing the constant in (7.275), (7.276). 8. For the semilinear convection–diﬀusion operator (7.78), (7.79) we impose (7.80), and deﬁne its DCGM as in (7.89).Then coercivity (and stability) is guaranteed by Theorems 7.10, 7.18, (7.181), (7.182) for cw in (7.180). For u ∈ W 2,∞ (T h ), we obtain the following consistency estimates, cf. Theorems 7.10, 7.21, 7.24, 7.31, here (7.204), (7.211), (7.245), |ch (u, v h ) − ch (P h u, v h )| ≤ C10 C(h, u)v h for Dirichlet boundary cd., (7.276) |ch (u, v ) − c (P u, v )| ≤ C10 C(h, u)(1 + v ) for Dir. and Neumann b.cs. |ah (u, v h ) − ah (P h u, v h )| ≤ εC(h, u) C1∗ + 2C2∗ uW 1,∞ (T h ) v h , (7.277) σ h σ h h h ε Jh (u, v ) − ε Jg (P u, v ) ≤ ε C15 C(h, u)v |h (P h u, v h ) − h (u, v h )| ≤ C13 C(h, u) 1 + uL∞ (∂Ω) v h H 1 (T h ) ∀v h ∈ V h . h

h

h

h

h

9. For the quasilinear system (7.93), we impose (7.94), and deﬁne its DCGM as in (7.101). Then coercivity (and stability) is guaranteed by Theorem 7.19, (7.195), (7.196) for cw in (7.194). For u ∈ W 2,∞ (T h , Rq ), we get the following consistency estimates, cf. Theorems 7.10, 7.23, 7.33, here (7.257), |ah ( u, v h ) − ah (P h u, v h )| ≤ C16 C(h, u)

1 + uW 1,∞ (T h ,Rq ) + u2W 1,∞ (T h ,Rq ) v h , σ Jh (u, v h ) − Jgσ (P h u, v h ) ≤ C15 C(h, u)v h h (v h ) − h (v h ) = 0 ∀v h ∈ V h = S h .

(7.278)

7.12. Convergence for DCGMs

531

10. For the special quasilinear equation in (7.102), Houston et al. [407] formulate their DCGM according to (7.109); for systems they provide all necessary tools, but do not formulate (7.110). Under the conditions (7.103)–(7.105), the coercivity (and stability) is guaranteed by Theorem 7.19, (7.195), (7.196) for cw in (7.194). The following consistency estimates are valid, cf. Theorem 7.10, 7.23, 7.36. So instead of ah (·, ·), . . . , in (7.278) we now formulate for the Gh , analogous to the Gh in (7.109), (7.110), cf. (7.268), |Gh (u, v h ) − Gh (P h u, v h )| ≤ C1∗ C(h, u)v h ∀θ ∈ [−1, 1].

(7.279)

We summarize the convergence results for all DCGMs in Theorem 7.40. In particular, we have the same order of convergence for the DCGMs as for the classical FEMs. It is straightforward along the lines of the above proofs, to prove consistent diﬀerentiability as well. This is the condition for the mesh independence principle in Theorem 3.40. Theorem 7.40. Convergence for the DCGMs for the above cases 5–10: Under the previous conditions and notation (1)–(4) and for inhomogeneous Dirichlet and/or Neumann boundary conditions choose, for the problems in (5)–(10), the quoted DCGMs with the above coercivity. Hence stability results for a boundedly invertible G (u0 ), and the consistency estimates (7.274)–(7.279) are granted. Then for the G(u0 ) = 0, the application of these DCGMs deﬁne the Gh uh0 = 0. Then for suﬃciently small h, a unique solution uh0 ∈ S h exists near u0 and h u0 − u0 J 1 h ≤ CC(h, u0 ), with C(h, u0 ) := hmin{d,s}−1 |u0 |H min{d,s} (Ω) (7.280) H (T ) with C depending upon u0 for problems (8)–(10). There, we have to impose additionally u0 ∈ W 2,∞ (T h ). For the special quasilinear equation (system) in [407] we formulate: Theorem 7.41. Convergence via monotony for the DCGMs for case 10: We assume the previous conditions and notation (1), (2), (4) and (7.103), consequently (7.105). Then for inhomogeneous Dirichlet and/or Neumann boundary conditions choose the DCGMs in (7.109) and (7.110) for equations and systems, respectively, with the consistency estimates Then there exist unique solutions u0 and uh0 ∈ S h for h(7.279). h h G(u0 ) = f and G u0 = Q f , respectively These uh0 converge to u0 according to h u0 − u0 J 1 h ≤ CC(h, u0 ), with C(h, u0 ) in (7.280). (7.281) H (T ) Proof. Under the above conditions, Lemmas 2.2 and 2.3 in [407] show that Gh is Lipschitz-continuous and monotone with respect to uh in W 1,p (T h ). Therefore, Theorem 2.68 implies the unique existence of the solutions, and Theorem 4.67 (5), and Theorems 7.5 ﬀ., and Remark 7.4, show the convergence according to (7.281). Theorem 7.42. Discontinuous Galerkin methods for monotone operators, eigenvalue problems, nonlinear boundary operators, quadrature approximations: Under the conditions of Theorem 7.40, discontinuous Galerkin methods applied to the following

532

7. Discontinuous Galerkin methods (DCGMs)

problems yield unique converging solutions for variational methods for eigenvalue problems, cf. Section 4.7, methods for nonlinear boundary operators, cf. Section 5.3, quadrature approximate methods, cf. Section 5.4. For all these problems (7.280) remains valid. Remark 7.43. By the appropriate modiﬁcations, cf. Section 7.14, these three theorems remain valid for hp-methods.

7.13 7.13.1

Solving nonlinear equations in DCGMs Introduction

For solving the nonlinear DCG equations, we restrict ourselves to presenting a discrete Newton method based upon the mesh independence principle (MIP). It is formulated and proved only for quasilinear systems. Except for the modiﬁed Lipschitz condition for the derivatives, the results are the same for semilinear equations as well. For this MIP we have to determine the linearized form of the quasilinear system and its discretization, based upon the same S h as before. This derivative has to satisfy the following property, cf. (3.83). A general discretization method is diﬀerentiably consistent and of order p, if it is classically consistent and if for a Frechet diﬀerentiable nonlinear operator G,

(Gh ) (P h u)P h v − Q h (G (u)v)V h → 0 and h

(7.282)

(G ) (P u)P v − Q (G (u)v)V h ≤ Ch (1 + uUs )vUs for h → 0. h

h

h

p

These vUs are usually vW k,q (Ω) and p, k, q appropriately chosen. For our most complicated DCGM, essentially including the previous problems as special cases, we prove this property. 7.13.2

Discretized linearized quasilinear system and diﬀerentiable consistency

Hence we have to determine the Frechet derivatives of the ah (·, ·) in (7.93), (7.285). With the Ak (·, u1 , ∇ u1 ) in (7.93), we deﬁne these weak forms, cf.(7.94), (7.93),

G ( u1 ) u, v :=

n ∂Ak K∈T h

=

n K∈T h

=

K k=0

∂Ak (·, u1 , ∇ u1 ) u + (·, u1 , ∇ u1 )∇ u, ∂ k v ∂ u ∂ u

K k=0

Ak (·, u1 , ∇ u1 )∇0 u, ∂ k v q dx

BA ( u1 , ∇ u1 )∇0 u, ∇0 v

K∈T

K

0 q

dx,

dx q

(7.283)

7.13. Solving nonlinear equations in DCGMs

533

where we used and will use the notation ∂Ak (·, u1 (·), ∇ u1 (·)) u(·) ∂ u ∂Ak (·, u1 , ∇ u1 )∇ u ∈ Rq in x ∈ Ω + ∂ u n 0 Ak (·, u1 , ∇ u1 )∇0 u, ∂ k v q ∈ R in x ∈ Ω, BA ( u1 , ∇ u1 )∇0 u, ∇0 v q := Ak (·, u1 , ∇ u1 )∇0 u :=

(7.284)

k=0

and

BA ( u1 , ∇ u1 )∇0 u, ν

n

:=

n

Ak (·, u1 , ∇ u1 )∇0 uνk

∈ Rq in x ∈ Ω,

k=1

Again we omitted, in the linearized form of the & and consequently % in the nonlinear DCGM, the BA ( u1 , ∇ u1 )∇0 u , ν n , { v } q , vanishing for u = u0 and only consider the IIPG for θ = 0. We collect the e∈T h \∂Ω and e∈∂Ω into e∈T h . Compared to ( u1 , ∇ u1 )∇0 u, ν n etc. With this notation, (7.76), we replace the (Ba ∇0 u, ν)n by BA we introduce, similarly to (7.57)–(7.61), the linearized quasilinear, and linear forms, ah ( u1 ; u, v ), h ( v ), and use Jhσ ( u, v ), 0 BA ( u1 , ∇ u1 )∇0 u, ∇0 v q dx (7.285) ah ( u1 ; u, v ) := K∈T h

−

K

e∈T h

h ( v ) := g, v +

e

BA ( u1 , ∇ u1 )∇0 u , ν n , [ v ] q dS

e∈∂Ω

σ( uD , v )q dS.

e

For this discrete IIPG method (θ = 0) for the linearized quasilinear problem (7.93), (7.91) we determine the discrete approximate solution uh2 such that (a) uh2 ∈ S h , cf. (7.95), (7.286) : h h h h; h h h h h σ h h h (b) (G ) u1 u2 , v := ah u1 ; u2 , v + Jh u2 , v − ( v ) = 0 ∀ v ∈ S h . For quadratic convergence of Newton’s method, Lipschitz-continuous derivatives of the operator are necessary. The same holds for its discrete counterpart. We assume for semilinear problems, in addition to (7.80), (a) For (b) –(d) we assume |u| ≤ R0 and an L = L (R0 ) such that (b) fs ∈

CL1 (R)

(7.287)

are Lipschitz-continuous with the constant L ,

(c) (∂aij /∂u)(·, u) are Lipschitz-continuous with respect to u with the constant L , (d) (∂g/∂u)(·, u) are Lipschitz-continuous with respect to u with the constant L .

534

7. Discontinuous Galerkin methods (DCGMs)

For quasilinear problems, we avoid a formulation analogous to (7.94), by imposing conditions, similar to (7.287), in the form (a) We assume the | u| ≤ R0 and an L = L (R0 ) such that

(7.288)

(b) (∂Ak /∂ u)( u, u), (∂Ak /∂ u)( u, u), (∂A0 /∂ u)( u, u), (∂A0 /∂ u)( u, u) are Lipschitz-continuous with respect to u, u, with the constant L ∀| u| ≤ R0 . We turn to our main goal, the consistency of the terms in (7.286). This has been proved for the Jhσ ( uh , v h ) and h ( v h ) in the previous sections; only ah uh1 ; uh , v h is missing. We proceed as above in two steps. Proposition 7.44. Under the condition (7.288), and for all u1 H 1 (T h ) ≤ C , the ah ( u1 ; u, v ) are nonlinear and bounded with respect to u1 , and bilinear and bounded with respect to u, v and satisfy the estimate for all u, v , u, u1 ∈ Ue , u ∈ W 2,∞ (T h ), and with C0 := L(1 + (C )2 ), L cf. (7.94), (7.288), H 1 (T h ) = H 1 (T h , Rq ), . . . , ah ( u1 ; u, v ) − ah ( u1 ; u, v ) ≤ C0 u − uH 1 (T h ) v H 1 (T h ) (7.289) 1/2 1 1/2 (Jhσ ( v , v )) + 2nC0 cw ⎞1/2 ⎛ ×⎝ hK | u − u|2H 1 (∂K) + hK u − u2L2 (∂K) ⎠ K∈T h

+ nL u1 − u1 H 1 (T h ) uW 1,∞ (Ω) v H 1 (T h ) + 2nL /(cw )1/2 uW 1,∞ (T h ) (Jhσ ( v , v )) ⎞1/2 ⎛ ×⎝ hK u − u2L2 (∂K) ⎠ .

1/2

K∈T h

Proof. The boundedness follows from (7.289) for u = 0 and use, cf. (7.94), of the bounds for |Ak , Ak | ≤ C0 . We consider the arguments of ah ( u1 ; u, v ) as u, u, u1 , u1 , v ∈ Ue . Then |ah ( u1 ; u, v ) − ah ( u1 ; u, v )| ≤ ah ( u1 ; u, v ) − ah ( u1 ; u, v ) + ah ( u1 ; u, v ) − ah ( u1 ; u, v ) .

(7.290)

7.13. Solving nonlinear equations in DCGMs

535

The ﬁrst term is estimate via the σ19 , σ20 as 0 BA |ah ( u1 ; u, v ) − ah ( u1 ; u, v )| ≤ ( u1 , ∇ u1 )∇0 ( u − u), ∇0 v q dx K∈T h K => ? < :=σ19

0 BA ( u1 , ∇ u1 )∇ ( u − u) , ν n [ v ] q dS . + e∈T h e => ? < :=σ20

σ19 is estimated with both Cauchy–Schwarz inequalities (1.44), for p = q = 2, (1.45),

|σ19 | ≤ C0

K∈T

u − uH 1 (K) v H 1 (K) ≤ C0 u − uH 1 (T h ) v H 1 (T h ) .

h

For σ20 the same inequalities (1.44), (1.45), and d(e) in (7.39) yield |σ20 | ≤ nC0 |[ v ]|q | ∇0 u − ∇0 u|q dS e∈T h

e

⎛⎛

⎞1/2 ⎛ ⎞1/2 ⎞ ([ v ], [ v ])q d(e) ⎜ ⎟ ≤ nC0 ⎝⎝ dS ⎠ ⎝ | ∇0 u − ∇0 u|2q dS ⎠ ⎠ d(e)/c c w w e e h h e∈T

≤ 2nC0

1 cw

e∈T

1/2 ⎛ 1/2

× (Jhσ ( v , v ))

⎝

⎞1/2

hK | u − u|2H 1 (∂K) + hK u − u2L2 (∂K) ⎠

.

K∈T h

For the second term in (7.290) we introduce σ21 , σ22 as |ah ( u1 ; u, v ) − ah ( u1 ; u, v )|

0 BA ( u1 , ∇ u1 ) − BA ≤ ( u1 , ∇ u1 ) ∇0 u, ∇0 v dx q K∈T h K => ? < :=σ21

BA ( u1 , ∇ u1 ) − BA + ( u1 , ∇ u1 ) ∇0 u , ν n [ v ] dS . q e∈T h e => ? < :=σ22

536

7. Discontinuous Galerkin methods (DCGMs)

Obviously we obtain with the Lipschitz constant, L , for the BA with (1.43), |σ21 | ≤ nL u1 − u1 H 1 (T h ) uW 1,∞ (Ω) v H 1 (T h ) . A combination with the ideas in (7.212) ﬀ., allows the estimation of σ22 . Here we need ∇0 uL∞ (e) ≤ uW 1,∞ (K) , valid for u ∈ W 2,∞ (T h ) by Theorem 1.38. ⎛

|σ22 | ≤ 2nL ⎝

e∈T

⎞1/2 ([ v ], [ v ])q dS ⎠ d(e)/cw

∇0 u2L∞ (e)

h

e

⎛

⎞1/2 n d(e) ⎝ ( u − u, u − u)q dS ⎠ cw e i=1 h e∈T

⎛ ≤ 2nL /(cw )1/2 uW 1,∞ (T h ) (Jhσ ( v , v ))1/2 ⎝

K∈T

⎞1/2 hK u − u2L2 (∂K) ⎠

.

h

Adding the estimates for σ19 –σ22 proves (7.289).

Proposition 7.44 implies with u = P h u, as in Theorem 7.21 Theorem 7.45. Diﬀerentiable consistency for quasilinear DCGMs: We assume u, u1 ∈ Ue ⊂ U, u ∈ W 2,∞ (T h ), u1 ∈ D(G), with uh1 H 1 (T h ) ≤ C , uh1 L∞ (T h ) ≤ R0 and C0 := L(1 + (C )2 ), L = L (R0 ), cf. (7.94), (7.288) and these latter conditions. Then (Gh ) (P h u1 )P h u are consistent with G (u1 )u. More precisely, ah uh1 ; uh , v h are bounded, nonlinear and bilinear with respect to uh1 , and uh , v h . A constant, F C18 := C0 ce (2 + nL uW 1,∞ (Ω) ) max{C7 , 2n 2C5 C7 /cw } exists, such that for the S h -interpolants P h u, P h u1 of u, u1 in Theorem 7.7, (7.291) |ah ( u1 ; u, v h ) − ah (P h u1 ; P h u, v h )| ⎛ ⎞1/2

2 min{d,s}−1 ≤ C18 ⎝ hk | u|H min{d,s} (K) ⎠ v h . K∈T h

With the classical consistency in Theorems 7.10, 7.33, this DCGM for the quasilinear system is diﬀerentiably consistent.

7.13. Solving nonlinear equations in DCGMs

537

Proof. The proof is nearly identical to that of Theorem 7.26. For (7.291) and by (7.289) we again use (7.222), however replacing the ﬁrst two lines by ⎛ P h u − uH 1 (T h ) ≤ 2C7 ⎝

min{d,s}−1

hk

| u|H min{d,s} (K)

2

⎞1/2 ⎠

, by (7.139),

K∈T h

⎛ ×⎝

K∈T

F

≤2

⎞1/2 hK u − P h u2H 1 (∂K) ⎠

(7.292)

h

⎛ 2C5 C7 ⎝

min{d,s}−1

hk

K∈T

| u|H min{d,s} (K)

2

⎞1/2 ⎠

,

h

(7.222), (7.292) are applied to the diﬀerent error terms u − P h u in (7.289): ah ( u1 ; u, v h ) − ah (P h u1 ; P h u, v h ) ⎛ ≤ C0 ⎝

min{d,s}−1

hk

| u|H min{d,s} (K)

2

⎞1/2 ⎠

K∈T h

F 1/2 × 2C7 v h H 1 (T h ) + 4n 2C5 C7 /cw Jhσ ( v h , v h ) + nL C7 uW 1,∞ (Ω) v h H 1 (T h ) 1/2

2C5 C7 /cw h uW 1,∞ (T h ) Jhσ ( v h , v h ) F ≤ C0 ce (2 + nL uW 1,∞ (Ω) ) max C7 , 2n 2C5 C7 /cw

+ 2nL

⎛ ⎝

F

min{d,s}−1

hk

K∈T

| u|H min{d,s} (K)

2

⎞1/2 ⎠

v h .

h

This shows, with (7.225) and Proposition 7.9, the estimate (7.291). Finally, Theorems 7.12 and 7.13 imply the last estimate in (7.291). We modify Theorem 7.45 for the semilinear, in particular, the model problem. We only have to replace (7.94), (7.288) by (7.80), (7.287), and C0 by L . Theorem 7.46. Diﬀerentiable consistency for semilinear and the special quasilinear DCGMs: We assume u, u1 ∈ Ue ⊂ U, u ∈ W 2,∞ (T h ), u1 ∈ D(G), and, the model problem, (7.13), the conditions (7.80), (7.287) (a), (b), for the semilinear problem, (7.78), the conditions (7.80), (7.287), and ﬁnally for the special quasilinear problem, (7.102), the conditions (7.288). Then the (Gh ) (P h u1 )P h u are consistent with the G (u1 )u.

538

7. Discontinuous Galerkin methods (DCGMs)

F A C18 := ce L (2 + nL uW 1,∞ (Ω) ) max C7 , 2n 2C5 C7 /cw exists, such that |ah (u1 ; u, v h ) − ah (P h u1 ; P h u, v h )| (7.293) ⎛ ⎞1/2

2 min{d,s}−1 hk |u|H min{d,s} (K) ⎠ v h . ≤ C18 ⎝ K∈T h

With the classical consistency in Theorems 7.21–7.31, this DCGM for the semilinear systems, (7.13), (7.78), are diﬀerentiably consistent. For the special quasilinear are necessary. problem, (7.102), slight modiﬁcations of the constant C18 We summarize the results of Theorem 3.40 and Corollary 3.41 for our DCGMs Theorem 7.47. Newton’s method for DCGMs Gh : Let u0 ∈ H s (Ω) be an isolated solution of G(u0 ) = 0, such that G (u0 ) : U → V is boundedly invertible and let Gh indicate one of the DCGMs for nonlinear problems in Theorems 7.45 and 7.46 with the corresponding conditions. Start the Newton process with u1 ∈ H s (Ω), and the discrete Newton process for i = 1, . . . , with uh1 := P h u1 , for small enough u0 − u1 U and h. That yields −1 (7.294) Gh uhi , uhi+1 := uhi − Gh uhi cf. (3.84). The uhi+1 in (7.294) uniquely exist and converge quadratically to uh0 and to ui+1 of order min{d, s} − 1, such that h ui+1 − uhi h ≤ C uhi − uhi−1 2 h and U U h h min{d,s}−1 ui − P ui h ≤ Ch ui Us , i = 1, . . . . U The following relation between the numbers of iterations of the exact and the discrete method is the reason for calling this a “mesh independence principle”, MIP. For small enough η : min{i ≥ 1 : uhi − P h u0 h < η} − min{i ≥ 1 : ui − u0 U < η} ≤ 1. (7.295) U

7.14

hp-variants of DCGM

For many important problems the local smoothness of the solution strongly varies in Ω. It is appropriate to choose the local degrees according to this smoothness. DCGMs are based upon a discontinuous piecewise polynomial approximation. So there is no principal obstacle for using these diﬀerent degrees of polynomial approximation on diﬀerent elements. The corners of the domain and singularities of the data are particularly interesting. The real power of hp-methods, e.g. the exponential convergence, strongly depends upon a careful combination of the analytic structure of the singularities of the solution with mesh reﬁnement strategies. In Chapter 6 we have discussed these problems for FEMs. Here we only formulate the necessary tools for hp-methods.

7.14. hp-variants of DCGM

539

We list a few from the many papers published for hp DCGMs, by B. Cockburn, P. Houston, M. Luskin, I. Perugia, D. Sch¨ otzau, Ch. Schwab, C-W. Shu, E. S¨ uli, and T.P. Wihler [193, 405, 408, 410–412]. 7.14.1

hp-ﬁnite element spaces

To each K ∈ T h , we assign a positive integer sK (local Sobolev index) and a positive integer pK (local polynomial degree). Then we deﬁne a new type of vector s ≡ {sK , K ∈ T h },

p ≡ {pK , K ∈ T h }.

(7.296)

We generalize the above broken Sobolev space to the vector s and triangulation T h H s (T h ) ≡ {v : v|K ∈ H sK (K) ∀ K ∈ T h } with the norm

⎛

vH s (T h ) ≡ ⎝

K∈T

and the seminorm

⎛

K∈T

⎞1/2 v2H sK (K) ⎠

(7.298)

h

|v|H s (T h ) ≡ ⎝

(7.297)

⎞1/2 |v|2H sK (K) ⎠

(7.299)

h

for the Sobolev space H sK (K) ≡ W sK ,2 (K). If, as in (7.63), (7.95), sK = d − 1 ∀K ∈ T h , d − 1 ∈ N then we use the notation H d−1 (T h ) = H s (T h ). Obviously, (7.300) H s (T h ) ⊂ H s (T h ) ⊂ H s (T h ), where s = max{sK , sK ∈ s} and s = min{sK , sK ∈ s}. Furthermore, we deﬁne the space of discontinuous piecewise polynomial functions associated with the vector p by S hp ≡ {v : v ∈ L2 (Ω), v|K ∈ PpK −1 (K) ∀K ∈ T h },

(7.301)

where PpK −1 (K) denotes the space of all polynomials on K of degree ≤ pK − 1, K ∈ T h . For deriving a priori hp error estimates we assume that there exists a constant CP ≥ 1 such that pK ≤ CP ∀K, K ∈ T h such that K, K have a common face. (7.302) pK The assumption (7.302) may seem to be rather restrictive. However, we suppose that an application of hp-methods to practical problems is eﬃcient and accurate when the polynomial degrees of approximation on neighboring elements do not diﬀer too much. Moreover, to each K ∈ T h we deﬁne the parameter dhp (K) :=

hK , p2K

K ∈ T h.

(7.303)

Another approach was presented by Dryja et al. [304] where the harmonic average of hK and pK of neighboring elemenents was used. Let e ∈ T h be an inner face shared

540

7. Discontinuous Galerkin methods (DCGMs)

by two elements Kle and Kre of T h . Then we put dhp (e) := min (dhp (Kle ) , dhp (Kre )) .

(7.304)

For a boundary face e ∈ ∂Ω ∩ Kle we put dhp (e) := dhp (Kle ) . 7.14.2

(7.305)

hp-DCGMs

Since the essential diﬀerences for the hp-DCGMs are well demonstrated for the model problem (7.13)–(7.15), we restrict the discussion to this case. Then many of the ideas for adaptive FEMs in Chapter 6 can be applied to DCGMs as well. Similarly as in (7.70) we deﬁne the standard hp-DCGM and its discrete approximate solution as a function uh0 : T h → R satisfying the conditions (7.306) (a) uh0 ∈ S hp , h h h (b) G u0 , v := ah uh0 , v h + ch uh0 , v h + εJhσ uh0 , v h − h (v h ) = 0 ∀ v h ∈ S hp , with ah (uh , v h ) := ah (uh , v h )|S hp ×S hp , ch (uh , v h ) := ch (uh , v h )|S hp ×S hp , and h (v h ) := h (v h )|S hp and the forms ah , ch , h and Jhσ are deﬁned by (7.57)–(7.61). The only diﬀerence occurs in the deﬁnition of the penalty parameter σ: here (7.39) is replaced by cw e∈Th (7.307) σ|e = dhp (e) where dhp (e) is given by (7.304) or (7.305) and cw > 0. Similarly, the hp-DCGMs are deﬁned by a formal exchange of S h by S hp for the general linear (7.77), semilinear (7.89) and quasilinear (7.286) problems considered within this chapter. This is a great advantage for DCGM. 7.14.3

hp-inverse and approximation error estimates

For analysing the hp-variant of DCGM we have to restrict the discussion to elements, K, which are simplices and/or parallelograms. We allow (n − 1)- dimensional faces e to be shared by more than two elements, cf. (7.304). We still do not require the conforming properties, usually imposed upon FEMs, but the elements cannot be as general as in (7.27) ﬀ. In particular, a triangulation, T h = {K}K∈T h (h > 0), is a partition of Ω into a ﬁnite number of open n-dimensional mutually disjoint simplexes and/or parallelograms K, hence star-shaped elements. We assume that T h is

r locally quasiuniform, i.e. there exists a constant CQ > 0 such that hK ≤ CQ hK

∀K, K ∈ T h sharing a face e ∈ T h \ ∂Ω;

(7.308)

r shape-regular, i.e. there exists a constant CS > 0 such that, cf. (7.126), hK ≤ CS ρK

∀K ∈ T h ,

where ρK is the radius of the largest n-dimensional ball inscribed into K.

(7.309)

7.14. hp-variants of DCGM

541

The assumptions (7.308) and (7.309) mean that the diameters of two neighboring elements do not diﬀer too much and that elements do not degenerate, respectively. The deﬁnitions (7.304)–(7.305) and the assumptions (7.302) and (7.308) imply that dhp (e) ≤ dhp (K)

and

1 1 ≤ CQ CP2 dhp (e) dhp (K)

∀e ⊂ ∂K, K ∈ T h .

(7.310)

The multiplicative trace inequality (7.129) is still valid for this section. On the other hand, we need hp-variants of the inverse inequality and approximation error estimates. Theorem 7.48. Inverse inequality: There exists a constant CI > 0 independent of v, h and K such that |v|H 1 (K) ≤ CI

p2K vL2 (K) , hK

v ∈ PpK (K), K ∈ T h , h ∈ (0, h0 ).

(7.311)

ˆ be a reference triangle or square for K ∈ T h , and FK : K ˆ → K be an Proof. Let K ˆ = K. From Proposition 5.41 it follows that if the aﬃne mapping such that FK (K) ˆ and x) = v(FK (ˆ x)) ∈ H m (K) function v ∈ H m (K), N m ≥ 0, then vˆ(ˆ n/2−m

|ˆ v |H m (K) ˆ ,

(7.312)

m−n/2

|v|H m (K) ,

(7.313)

|v|H m (K) ≤ cc hK |ˆ v |H m (K) ˆ ≤ cc hK

where cc > 0 depends on CS but not on K and v. From [575], Theorem 4.76 we have 2 v L2 (K) |ˆ v |H 1 (K) ˆ ≤ cs pK ˆ ˆ ,

ˆ v ∈ PpK (K),

(7.314)

where cs > 0 depends on n but not on vˆ and pK . A simple combination of (7.312)– (7.314) proves (7.311) with CI = cs c2c . For FE interpolation, Condition 4.16 is essential: for the local results in Theorem 4.17 we know that (i): our K are star-shaped, (ii) we choose τ = −1 and any interpolation condition according to (iii) or (iv). The above shape regularity condition (7.309) implies the nondegeneracity of the T h , cf. Condition 4.16 (v), and (4.30). Note that the constants appearing in Theorem 4.17 are not indepenent of p. Then the local DCG interpolation operator, I h = P h yields: Theorem 7.49. Local approximation properties: There exists a constant CA > 0, independent of v and h, and a mapping πpK : H s (K) → Pp (K), s ≥ 1, e.g. the previous I h = P h , such that ∀v ∈ H s (K), K ∈ T h , h ∈ (0, h0 ) the following inequality is valid min(p,s)−j K hK πp v − v j ≤ C vH min(p,s) (K) , A H (K) ps−j

(7.315)

where 0 ≤ j ≤ s. Proof. See Lemma 4.5 in [53] for the case n = 2. The n = 3 arguments are completely analogous.

542

7. Discontinuous Galerkin methods (DCGMs)

Deﬁnition 7.50. Let s and p be the vectors introduced by (7.296). We deﬁne the mapping Πhp : H s (T h ) → S hp by hp Π u |K := πpKK (u|K ) ∀K ∈ T h , (7.316) where πpKK : H sK → PpK (K) is the mapping introduced in Theorem 7.49. Theorem 7.51. Global approximation properties: Let Πhp : H s (T h ) → S hp be the mapping introduced by (7.316), and v ∈ H s (T h ). Then h2 min(pK ,sK )−2q hp 2 K Π v − v 2 q h ≤ CA v2H min(pK ,sK ) (K) , 2sK −2q H (T ) p K K∈T h

(7.317)

where 0 ≤ q ≤ minsK ∈s sK . Proof. Using deﬁnition (7.316), the Cauchy inequality, and the approximation prop erties (7.315) we obtain (7.317) and CA is given by Theorem 7.49. 7.14.4

Consistency and convergence of hp-DCGMs

Here we analyze the hp–DCGM (7.306) for our model problem. We consider for simplicity ∂ΩD = ∂Ω. More general linear, semilinear, and quasilinear cases can be analyzed analogously. The proofs are very similar to h-variant (but not identical), since we have to use hp-inverse inequality or hp-interpolation error estimates. The coercivity as well as boundedness of the form Ah (u, v) := ah (u, v) + εJhσ (u, v), proved in Theorem 7.16, remain valid for u, v ∈ Shp . Let us analyze the consistency of forms bh , ah and Jhσ . h is independent of u and therefore its consistency immediately follows from Theorem 7.23. Theorem 7.52. Consistency for the ch and bh : For ∂Ω = ∂ΩD , there exists a constant C10 > 0 such that for ch and bh ⎞1/2 h2 min(pK ,sK ) K |ch (u, v h ) − ch (Πhp u, v h )| ≤ C10 v h ⎝ u2H min(pK ,sK ) (K) ⎠ , 2sK p K K∈T h ⎛

u ∈ H s (T h ), v h ∈ S h , h ∈ (0, h0 ),

(7.318)

where Πhp u is the S hp -interpolant of u from Theorem 7.51. This estimate remains valid for |bh (u, v h ) − bh (Πhp u, v h )|, as well. Proof. The ﬁrst part of the proof of Proposition 7.20 remains unchanged, except replacing d(e), hK in (7.202) by dhp (e), dhp (K) here. After (7.202) the proof continues as: Now, we insert u := Πhp u and v := v h ∈ S hp into (7.200)–(7.202) in the modiﬁed form. Using the multiplicative trace inequality (7.129), the discrete Cauchy inequality

7.14. hp-variants of DCGM

and the approximating properties (7.317), we obtain dhp (K)u − Πhp u2L2 (∂K) K∈T h

≤

543

(7.319)

C5 dhp (K)u − Πhp uL2 (K) u − Πhp uH 1 (K)

K∈T h

hK hmin(pK ,sK ) hmin(pK ,sK )−1 K K u2H min(pK ,sK ) (K) p2K psKK psKK −1 h

2 ≤ C5 CA

K∈T 2 ≤ C5 CA

h2 min(pK ,sK ) K u2H min(pK ,sK ) (K) . 2sK +1 p K K∈T h

This estimate, (7.317) and (7.198), yield |ch (u, v) − ch (Πhp u, v)| ⎛

(7.320) ⎞1/2 ⎞ ⎜ ⎟ ≤ C8 v ⎝u − Πhp uL2 (Ω) + ⎝ hK u − Πhp u2L2 (∂K) ⎠ ⎠ ⎛

K∈T h

⎛⎛

⎞1/2 2 min(pK ,sK ) h ⎜ K u2H min(pK ,sK ) (K) ⎠ ≤ C8 CA v ⎝⎝ 2sK p K K∈T h ⎛ + ⎝C5

K∈T h

⎞1/2 ⎞ ⎟ ⎠,

2 min(pK ,sK ) hK u2H min(pK ,sK ) (K) ⎠ 2sK +1 pK

√ which proves (7.318) with C10 = CA C8 ( C5 + 1) since 1/pK ≤ 1.

The consistency of the form ah is given by Theorem 7.53. Consistency for the ah : The ah (Πhp u, v h ) in (7.57) are consis := CA + tent with a(u, v) in (7.19) ∀ u ∈ Ue , ∀v h ∈ S h = V h , so a constant, C14 F √ −1/2 hp + C5 (1 + C6 )) exists, such that for the S -interpolant Πhp u of 2nCA C5 (1/cw u in Theorem 7.51, |ah (u, v h ) − ah (Πhp u, v h )| ⎞1/2 ⎛ h2 min(pK ,sK )−2 K ≤ ε C14 v h ⎝ u2H min(pK ,sK ) (K) ⎠ . 2sK −3 p K K∈T h

(7.321)

Proof. In Proposition 7.25, we consider u, u, v ∈ Ue as arguments of ah (·, ·). Again, for the generalization to hp-methods, the proof remains unchanged, except replacing d(e), hK there by dhp (e), dhp (K) here. After that the proof of Theorem 7.26 is modiﬁed as:

544

7. Discontinuous Galerkin methods (DCGMs)

Putting u := Πhp u and v := v h we have from (7.217)–(7.220) (7.322) |ah (u, v h ) − ah (Πhp u, v h )| ≤ ε |Πhp u − u|H 1 (T h ) |v h |H 1 (T h ) ⎛ ⎛ ⎞1/2 1/2 σ h h 1/2 ⎜ 1 ⎝ Jh (v , v ) + ε 2n ⎝ dhp (K)|u − Πhp u|2H 1 (∂K) ⎠ cw h K∈T

⎛ +⎝

⎞1/2 ⎞ ⎟ ⎝ dhp (K)−1 (u − Πhp u)2L2 (∂K) ⎠ ⎠ .

⎞1/2 ⎛

dhp (K)∇v h 2L2 (∂K) ⎠

K∈T h

K∈T h

Similarly as in (7.319), it is possible to show that

2 dhp (K)−1 |u − Πhp u|2L2 (∂K) ≤ C5 CA

K∈T h

h2 min(pK ,sK )−2 K u2H min(pK ,sK ) (K) . 2sK −3 p h K K∈T (7.323)

and

2 dhp (K)|u − Πhp u|2H 1 (∂K) ≤ C5 CA

K∈T h

h2 min(pK ,sK )−2 K u2H min(pK ,sK ) (K) 2sK −3 p K K∈T h (7.324)

These formulas yield, with (7.64), (7.171), (7.317), (7.319), (7.322), (7.323) and (7.324), |ah (u, v h ) − ah (Πhp u, v h )| ⎞1/2 ⎛ h2 min(pK ,sK )−2 K ≤ εCA v h ⎝ v2H min(pK ,sK ) (K) ⎠ 2sK −2 p h K K∈T

(7.325)

⎞1/2 ⎛ h2 min(pK ,sK )−2 2nε h ⎝ 2 K + √ v C5 CA u2H min(pK ,sK ) (K) ⎠ 2sK −3 cw p h K K∈T

+ 2nε

F

⎞1/2 h2 min(pK ,sK )−2 2 K C5 (1 + C6 )v h ⎝C5 CA u2H min(pK ,sK ) (K) ⎠ 2sK −3 p K K∈T h ⎛

⎞1/2 h2 min(pK ,sK )−2 K ≤ εC14 v h ⎝ u2H min(pK ,sK ) (K) ⎠ , 2sK −3 p h K K∈T ⎛

F √ −1/2 := CA + 2nCA C5 (1/cw + C5 (1 + C6 )). C14 Finally, we show the consistency of the penalty form Jhσ .

7.14. hp-variants of DCGM

545

> 0 such that Theorem 7.54. Consistency for the Jhσ : There exists a constant C15

⎞ 12 ⎛ h2 min(pK ,sK )−2 σ K Jh (u, v h ) − Jhσ (Πhp u, v h ) ≤ C15 v h ⎝ u2H min(pK ,sK ) (K) ⎠ , 2sK −1 p K K∈T h u ∈ H s (T h ), v h ∈ S h , h ∈ (0, h0 ),

(7.326)

where Πhp u is the S hp -interpolant of u from Theorem 7.51. Proof. From (7.61), the Cauchy inequality, (7.149), and the multiplicative trace inequality (7.129) and (7.317) we have σ Jh (u, v h ) − Jhσ (Πhp u, v h ) (7.327) ≤ σ|[u − Πhp u]| |[v h ]| dS e∈T h

e

⎛ ≤ Jhσ (v h , v h )1/2 ⎝

e∈T

≤ v h

K∈T h

≤

F

σ[u − Πhp u]2 dS ⎠

e

h

cw dhp (K)−1 u − Πhp u2L2 (∂K)

C5 cw v h

⎞1/2

1/2

dhp (K)−1 u − Πhp uL2 (K) |u − Πhp u|H 1 (K)

K∈T h

+ dhp (K)−1 u − Πhp u2L2 (K)

≤

F

⎛

2 ⎝ 2C5 cw v h CA

K∈T h 2 min(p ,sK )

K 1 h + 2 K sK 2 hK (pK )

1/2

min(p ,sK )

1 hK K hK psKK

min(pK ,sK )−1

hK

psKK −1

1/2 u2H min(pK ,sK ) (K)

⎛ ⎞1/2 h2 min(pK ,sK )−2 F 2 ⎝ K ≤ 2 C5 cw v h CA u2H min(pK ,sK ) (K) ⎠ , 2sK −1 p K K∈T h √ 2 := 2 C5 cw CA . which proves Theorem 7.54 with C15

For our previous problems we impose the conditions listed in Section 7.12 for the standard DCGM and the new conditions in Subsections 7.14.1–7.14.3. We deﬁne the

546

7. Discontinuous Galerkin methods (DCGMs)

hp-DCGMs by replacing the previous S h by the S hp and (7.274) by ⎞ 12 h2 min(pK ,sK )−2 K u2H min(pK ,sK ) (K) ⎠ ∀u ∈ H s (T h ), h ∈ (0, h0 ). C(hs , u) := ⎝ 2sK −1 p K K∈T h ⎛

(7.328) Previously we have proved the consistency for our model problem. For the other cases the proofs follow similar modiﬁcations. The necessary stability with respect to the · is available via the U- and U h -coercivity as in Section 7.9. Then our hp-updated consistency results can be applied for proving the general stability and convergence as in Section 7.12. We summarize Theorem 7.55. Convergence for the hp- DCGMs for the problem 5.–10. in Condition 7.39. Under these conditions 1.–4. and for inhomogeneous Dirichlet boundary conditions choose, for the problems in 5.–10. the quoted hp-DCGMs. This implies coercivity, hence stability results for a boundedly invertible G (u0 ), and the necessary consistency estimates in Theorems 7.53,7.54 and the corresponding modiﬁcations for the other methods. Then for G(u0 ) = 0, the application of these hp-DCGMs deﬁne the Gh uh0 = 0 with a uniquely existing solution uh0 near u0 converging according to uh − uJH s (T h ) ≤ C(hs , u).

(7.329)

Again a numerical solution is eﬃciently computed via Newton’s method under the conditions of and as in Theorem 7.47. This again implies quadratic convergence essentially independent of h.

7.15 7.15.1

Numerical experiences Scalar quasilinear equation

In this section we verify theoretical results presented in the previous sections by numerical experiments. Similarly as in [407], we consider the nonlinear diﬀusion equation −

2 ∂ ∂ ν(|∇u|) u = g, u : Ω → R, ∂xs ∂xs s=1

(7.330)

where ν(w) : (0, ∞) → R is chosen in the form ν(w) = ν∞ +

ν0 − ν∞ , (1 + w)γ

γ > 0.

(7.331)

7.15. Numerical experiences

The corresponding quasilinear weak form is n Ak (x, u, ∇u), ∂ k v dx G(u), v =

547

(7.332)

Ω k=0

:=

2

ν(|∇u|)

Ω k=1

∂ u ∂xk

∂ vdx − ∂xk

gvdx. Ω

These Ak (x, u, ∇u), k = 1, . . . , n = 2, satisfy the conditions (7.94) (a)–(c) and A0 ≡ 0, cf. (2.284), (2.285), (2.286), (2.293). We set ν0 = 0.15, ν∞ = 0.1, γ = 1/2, Ω = (0, 1)2 , and deﬁne the function g and the boundary conditions in such a way that the exact solution has the form u(x1 , x2 ) = 2rα x1 x2 (1 − x1 )(1 − x2 ) =r

α+2

(7.333)

sin(2ϕ)(1 − x1 )(1 − x2 ),

where (r, ϕ)(r := (x21 + x22 )1/2 ) are the polar coordinates and α ∈ R is a constant. The function u is equal to zero on ∂Ω and its regularity depends on the value of α, namely (cf. [52]) u ∈ H β (Ω)

∀β ∈ (0, α + 3),

(7.334)

where H β (Ω) denotes (in general) the Sobolev–Slobodetskii space of functions with “noninteger derivatives”. In the presented numerical tests we use the values α = 2 and α = −3/2. The value α = 2 gives a function u suﬃciently regular (∈ H β (Ω) for β < 5), whereas the value α = −3/2 gives u ∈ H β (Ω), β < 3/2. Figure 7.5 shows functions u for both values of α. The above problem is discretized by the IIPG method (7.286). The resulting system of nonlinear algebraic equations by a pseudo-time discretization, i.e. we solve a time dependent problem h ∂w0 (t) h ,v ∀ vh ∈ S h , + Gh w0h (t) , v h = 0 (7.335) ∂t where w0h (t, x) : (0, T ) × S h → R, T > 0 and Gh is an operator introduced by (7.286). We seek a steady state solution, i.e. ∂w0h (t, x) =0 ∂t

for T → ∞

and put uh0 (x) := lim w0h (T, x), T →∞

x ∈ Ω.

The system of ordinary diﬀerential equations (7.335) is solved by a three step backward diﬀerential formula, which is formally of third order, see [293]. The following numerical

548

7. Discontinuous Galerkin methods (DCGMs)

0.1 0.075 0.05 0.025 0

0

0.2

0.4

0.6

0.8

1

1 0.8 0.6 0.4 0.2 0

0.4 0.3 0.2 0.1 0 1 0.8 0

0.2

0.6 0.4

0.6

0.8

10

0.4 0.2

Figure 7.5 The exact solution (7.333) for α = 2 (above) and α = −3/2 (below).

ﬂux is used for the discretization of the convective term ⎧ 2 ⎪ ⎪ ⎪ fs (u1 )ns if A > 0, ⎪ ⎪ ⎨ s=1 H(u1 , u2 , n) = 2 ⎪ ⎪ ⎪ ⎪ ⎪ fs (u2 )ns if A ≤ 0, ⎩

(7.336)

s=1

where A=

2 s=1

f (s (u)ns ,

u=

1 (u1 + u2 ) and n = (n1 , n2 ). 2

(7.337)

Numerical experiments are carried out with the use of piecewise linear (P 1 ), quadratic (P 2 ), and cubic (P 3 ) elements on six triangular meshes Thl , l = 1, . . . , 6 having 128, 288, 512, 1152, 2048 and 4608 elements. Figure 7.6 shows the coarsest mesh and the ﬁnest one. We investigate the experimental orders of convergence (EOC) for grid

7.15. Numerical experiences

549

1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

1

0.8

0.6

0.4

0.2

0

Figure 7.6 The coarsest and the ﬁnest grids used for the computations.

pairs Thl and Thl−1 , l = 2, . . . 6, deﬁned by EOCl =

log(ehl /ehl−1 ) , log(hl /hl−1 )

l = 2, . . . , 6,

(7.338)

where ehl is the computational error obtained on mesh Thl . Tables 7.1–7.4 show computational errors in the L2 (Ω) norm and in the H 1 (Ω) seminorm and the corresponding experimental orders of convergence (EOC) for α = 2 and α = −3/2. These values together with the marked EOC are shown in Figures 7.7 and 7.8. We observe the following:

550

7. Discontinuous Galerkin methods (DCGMs)

Table 7.1: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = 2 in the L2 norm, purely diﬀusive case; notice the deﬁnition of α in (7.333) ﬀ.

Grid

1 2 3 4 5 6

h

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

P1

P2

P3

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

0.1809E−02 0.8373E−03 0.4791E−03 0.2162E−03 0.1224E−03 0.5477E−04

— 1.90 1.94 1.96 1.98 1.98

0.9906E−04 0.3102E−04 0.1354E−04 0.4202E−05 0.1839E−05 0.5832E−06

— 2.86 2.88 2.89 2.87 2.83

0.4470E−05 0.9031E−06 0.2895E−06 0.5811E−07 0.1856E−07 0.3703E−08

— 3.94 3.95 3.96 3.97 3.98

Table 7.2: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = 2 in the H 1 seminorm, purely diﬀusive case.

Grid

1 2 3 4 5 6

h

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

P1

P2

P3

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

0.5869E−01 0.3994E−01 0.3026E−01 0.2037E−01 0.1535E−01 0.1029E−01

— 0.95 0.97 0.98 0.98 0.99

0.6045E−02 0.2756E−02 0.1570E−02 0.7073E−03 0.4006E−03 0.1793E−03

— 1.94 1.96 1.97 1.98 1.98

0.3884E−03 0.1157E−03 0.4886E−04 0.1448E−04 0.6107E−05 0.1808E−05

— 2.99 3.00 3.00 3.00 3.00

Table 7.3: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = −3/2 in the L2 norm, purely diﬀusive case.

Grid

1 2 3 4 5 6

h

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

P1

P2

P3

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

0.1580E−01 0.8664E−02 0.5659E−02 0.3050E−02 0.1950E−02 0.1032E−02

— 1.48 1.48 1.52 1.56 1.57

0.5292E−02 0.2898E−02 0.1888E−02 0.1033E−02 0.6725E−03 0.3662E−03

— 1.49 1.49 1.49 1.49 1.50

0.3121E−02 0.1518E−02 0.9201E−03 0.4623E−03 0.2866E−03 0.1481E−03

— 1.78 1.74 1.70 1.66 1.63

7.15. Numerical experiences

551

Table 7.4: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = −3/2 in the H 1 seminorm, purely diﬀusive case.

Grid

1 2 3 4 5 6

P1

h

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

P2

P3

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

0.4512E+00 0.3730E+00 0.3251E+00 0.2668E+00 0.2314E+00 0.1890E+00

— 0.47 0.48 0.49 0.49 0.50

0.2098E+00 0.1662E+00 0.1415E+00 0.1133E+00 0.9707E−01 0.7822E−01

— 0.57 0.56 0.55 0.54 0.53

0.1548E+00 0.1129E+00 0.9082E−01 0.6779E−01 0.5576E−01 0.4305E−01

— 0.78 0.76 0.72 0.68 0.64

0.01 P_1

P_1

0.001

0.01 0.0001

P_2

2

P_2

1 1e-05

P_3

P_3 1e-06

3

0.001 1 1e-07 1e-08 1e-09

4

3/2

1 0.1

0.0001

1 0.1

Figure 7.7 Computational errors and the corresponding experimental orders of convergence of the P 1 , P 2 and P 3 approximations for α = 2 (left) and α = −3/2 (right), purely diﬀusive case.

(i) For a suﬃciently regular exact solution (case α = 2) we observe the optimal order of convergence O(hp+1 ) for p = 1, 2, 3 in the L2 norm (a small decrease of the experimental order of convergence for the P 3 approximation on the ﬁnest grid is caused by rounding oﬀ errors, so that we obtain a better EOC than the theoretical results). It is a rather surprising fact that the IIPG variant gives (almost) optimal order of convergence also for an even degree polynomial an and Rivi`ere approximation, i.e. P 2 , cf. Table 7.1. On the other hand, Guzm´ [380] show that the NIPG method gives only suboptimal EOC also for odd polynomial degrees when special nonuniform grids are used.

552

7. Discontinuous Galerkin methods (DCGMs)

0.1

1

P_1 0.01

P_2

1

P_1

1 0.001 P_3 P_2

2 0.0001

P_3

1 0.1

1e-05 3 1e-06

1/2

1 1

1e-07 0.1

0.1

Figure 7.8 Computational errors and the corresponding experimental orders of convergence of the P 1 , P 2 and P 3 approximations for α = 2 (left) and α = −1/2 (right), purely diﬀusive case.

(ii) For a suﬃciently regular exact solution (case α = 2) we observe the optimal order of convergence O(hp ) for p = 1, 2, 3 in the H 1 seminorm which is in good agreement with the theoretical results, cf. Table 7.2. (iii) For the case α = −3/2, we obtain an EOC equal to 3/2 in the L2 norm and an EOC equal to 1/2 in the H 1 seminorm for p = 1, 2, 3. Using the result from [312], for any β ∈ (1, 3/2) we get, cf. Table 7.3, v − Ih vL2 (Ω) ≤ C(β)hμ vH β (Ω) ,

v ∈ H β (Ω),

(7.339)

where Ih v is a piecewise polynomial Lagrange interpolation to v of degree ≤ p, μ = min(p + 1, β) and C(β) is a constant independent of h and v. The exact approximation corresponding precisely to our experimental results can be obtained with the use of the interpolation in so-called Bessov spaces. See [51], Section 3.3 and the references therein. Moreover, we consider a generalization of (7.330) given by 2 2 ∂ ∂u ∂ ν(|∇u|) u − u = g, u : Ω → R, ∂xs s=1 ∂xs ∂xs s=1

(7.340)

where ν(w) : (0, ∞) → R is given by (7.331). This problem represents the convection– diﬀusion equation with nonlinear convective as well as diﬀusive terms. The corresponding quasilinear weak form is given by (7.332) but with A0 ≡ 0. Although this form violates condition (7.94)(d) it is possible to solve it numerically. The convective term is discretized in the same way as in (7.89) where the discrete convective form ch

7.15. Numerical experiences

is deﬁned by (7.59). There the following numerical ﬂux is used ⎧ 2 ⎪ ⎪ ⎪ fs (u1 )ns if A > 0, ⎪ ⎨ s=1 H(u1 , u2 , n) = 2 ⎪ ⎪ ⎪ ⎪ fs (u2 )ns if A ≤ 0, ⎩

553

(7.341)

s=1

where A=

2

fs (u)ns ,

s=1

u=

1 (u1 + u2 ) and n = (n1 , n2 ). 2

(7.342)

Tables 7.5–7.8 show computational errors in the L2 (Ω) norm and in the H 1 (Ω) seminorm and the corresponding experimental orders of convergence (EOC) for α = 2 and α = −3/2. These values together with the marked EOC are shown in Figures 7.9 and 7.10. We observe the completely same results as for problem (7.330). Table 7.5: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = 2 in the L2 norm, convection– diﬀusion case.

Grid

1 2 3 4 5 6

h

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

P1

P2

P3

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

0.1036E−02 0.4846E−03 0.2785E−03 0.1261E−03 0.7155E−04 0.3205E−04

— 1.87 1.93 1.95 1.97 1.98

0.4985E−04 0.1552E−04 0.6724E−05 0.2052E−05 0.8767E−06 0.2669E−06

— 2.88 2.91 2.93 2.96 2.93

0.2300E−05 0.4681E−06 0.1508E−06 0.3042E−07 0.9763E−08 0.2003E−08

— 3.93 3.94 3.95 3.95 3.91

Table 7.6: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = 2 in the H 1 seminorm, convection–diﬀusion case.

Grid

1 2 3 4 5 6

h

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

P1

P2

P3

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

0.3024E−01 0.2061E−01 0.1561E−01 0.1051E−01 0.7925E−02 0.5309E−02

— 0.95 0.96 0.98 0.98 0.99

0.3098E−02 0.1415E−02 0.8074E−03 0.3640E−03 0.2063E−03 0.9239E−04

— 1.93 1.95 1.96 1.97 1.98

0.1962E−03 0.5843E−04 0.2468E−04 0.7314E−05 0.3084E−05 0.9133E−06

— 2.99 3.00 3.00 3.00 3.00

554

7. Discontinuous Galerkin methods (DCGMs)

Table 7.7: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = −3/2 in the L2 norm, convection– diﬀusion case.

P1 Grid 1 2 3 4 5 6

P2

P3

h

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

eh L2 (Ω)

EOC

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

0.3916E−02 0.2161E−02 0.1408E−02 0.7654E−03 0.4943E−03 0.2651E−03

— 1.47 1.49 1.50 1.52 1.54

0.1413E−02 0.7447E−03 0.4765E−03 0.2577E−03 0.1682E−03 0.9297E−04

— 1.58 1.55 1.52 1.48 1.46

0.6157E−03 0.3530E−03 0.2352E−03 0.1315E−03 0.8659E−04 0.4772E−04

— 1.37 1.41 1.43 1.45 1.47

Table 7.8: Computational errors and the corresponding experimental orders of convergence (EOC) of the P 1 , P 2 and P 3 approximations for α = −3/2 in the H 1 seminorm, convection–diﬀusion case.

P1 Grid 1 2 3 4 5 6

7.15.2

P2

P3

h

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

|eh |H 1 (Ω)

EOC

0.177E+00 0.118E+00 0.884E−01 0.589E−01 0.442E−01 0.295E−01

0.2068E+00 0.1699E+00 0.1475E+00 0.1206E+00 0.1045E+00 0.8534E−01

— 0.48 0.49 0.50 0.50 0.50

0.7954E−01 0.6488E−01 0.5622E−01 0.4598E−01 0.3987E−01 0.3262E−01

— 0.50 0.50 0.50 0.50 0.50

0.3507E−01 0.2988E−01 0.2650E−01 0.2220E−01 0.1947E−01 0.1607E−01

— 0.40 0.42 0.44 0.46 0.47

System of the steady compressible Navier–Stokes equations

Let Ω ⊂ Rn , n = 2, 3, be a bounded domain, and by ∂Ω we denote the boundary of Ω which consists of several disjoint parts. We distinguish inlet ∂Ωi , outlet ∂Ωo and impermeable walls ∂Ωw , i.e. ∂Ω = ∂Ωi ∪ ∂Ωo ∪ ∂Ωw . We use the following notation: ρ – density, p – pressure, e – total energy, v = (v1 , . . . , vn ) – velocity, θ – temperature, γ – Poisson adiabatic constant, Re – Reynolds number, Pr – Prandtl number. The system of Navier-Stokes equations describing steady state viscous compressible ﬂuids can be written in dimensionless form (w) − ∇ · R(w, ∇·f ∇w) = 0

in Ω,

(7.343)

where w = (w1 , . . . , wn+2 )T = (ρ, ρv1 , . . . , ρvn , e)T

(7.344)

7.15. Numerical experiences

555

0.01

0.01 0.001 1e-04

P_1

P_1

P_2

P_2

2 1e-05

1

0.001 P_3

P_3

1e-06 3 1e-07

1 1e-04

1e-08

3/2

4 1e-09 1

1

1e-10

0.1

0.1

Figure 7.9 Computational errors and the corresponding experimental orders of convergence of the P 1 , P 2 and P 3 approximations for α = 2 (left) and α = −3/2 (right), convection– diﬀusion case.

0.1 P_1

P_1 0.01

P_2

1 1

0.1

0.001

P_2

P_3 1e-04

2 1

P_3

1e-05

1e-06

3

1/2

1 1e-07

1 0.1

0.01

0.1

Figure 7.10 Computational errors and the corresponding experimental orders of convergence of the P 1 , P 2 and P 3 approximations for α = 2 (left) and α = −3/2 (right), convection– diﬀusion case.

556

7. Discontinuous Galerkin methods (DCGMs)

is the so-called state vector, and (w) = (f (w), . . . , f (w)), f 1 n with

(7.345)

T f s (w) = fs(1) (w), . . . , fs(n+2) (w) = (ρvs , ρvs v1 + δs1 p, . . . , ρvs vn + δn p, (e + p) vs )T ,

s = 1, . . . , n

the so-called inviscid (Euler) ﬂuxes. Finally R(w, ∇w) = (R1 (w, ∇w), . . . , Rn (w, ∇w)), with

(7.346)

T Rs (w, ∇w) = Rs(1) (w, ∇w), . . . , Rs(n+2) (w, ∇w) =

0, τs1 , . . . , τsn ,

n k=1

γ ∂θ τsk vk + Re Pr ∂xs

T ,

s = 1, . . . , n

are the so-called viscous ﬂuxes. The symbols ∇ and ∇· mean the gradient and divergence operators, i.e. ∂w ∂w ∈ Rn+2 × · · · × Rn+2 ,..., (7.347) ∇w := ∂x1 ∂xn and (w) := ∇·f

n ∂f s (w) ∈ Rn+2 , ∂x s s=1

(7.348)

respectively. Again, we can, but do not explicitly, formulate this as the quasilinear weak form G( u), v =

n

Ak (x, u, ∇ u), ∂ k v dx.

(7.349)

Ω k=0

These Ak (x, u, ∇u), k = 1, . . . , n, satisfy the conditions (7.94) (a)–(c); A0 (x, u, ∇u), violates (d). If we omit A0 (x, u, ∇u), we would fully satisfy (7.94) (a)–(c), the (d) would be dropped, and would have an example similar to [407]. There probably exist some physical models whose mathematical description completely satisﬁes our assumption for a quasilinear model, but is not the case of this viscous compressible ﬂow. In our context it cerainly would not make any physical sense avoiding this term, e.g. (w), A0 (x, u, ∇u) = ∇ · f is the convective ﬂux. This would mean considering a ﬂow with only with where f diﬀusion and not convection.

7.15. Numerical experiences

557

We consider a Newtonian type of ﬂuid, i.e. the viscous part of the stress tensor has the form ' n ∂vs 2 ∂vi 1 ∂vk − (7.350) + δsk , s, k = 1, . . . , n. τsk = Re ∂xk ∂xs 3 i=1 ∂xi In order to close the system, we consider the state equation for a perfect gas and the deﬁnition of the total energy p = (γ − 1) (e − ρ|v|2 /2),

e = cV ρθ + ρ|v|2 /2,

(7.351)

Figure 7.11 NACA 0012 proﬁle, computational grid around the NACA0012 proﬁles (top), details of the leading (middle) and trailing edges (bottom).

558

7. Discontinuous Galerkin methods (DCGMs)

where cV is the speciﬁc heat at constant volume which we assume to be equal to 1 in the dimensionless case. The system (7.343)–(7.351) is equipped with the following set of boundary conditions on appropriate parts of the boundary: n n γ ∂θ = 0 on ∂Ωi , (7.352) τlk nl vk + (a) ρ = ρD , v = v D , Re Pr ∂n k=1

(b)

n

l=1

τsk nk = 0, s = 1, . . . , n,

k=1

(c) v = 0,

∂θ =0 ∂n

∂θ =0 ∂n

on ∂Ωo ,

on ∂Ωw ,

where ρD and v D are given functions and n = (n1 , . . . , nn ) is a unit outer normal to ∂Ω. Another possibility is to replace the adiabatic boundary condition (7.352) (c) by (c ) v = 0, θ = θD

on ∂Ωw .

(7.353)

Although the existence of the solution of system (7.343)–(7.352) is still an open problem we solve it numerically. The question of the existence and convergence of

P1-approximation

P2-approximation

P3-approximation

Figure 7.12 Flow around NACA0012 proﬁles, isolines of the Mach number (left) with details around the leading (middle) and trailing edges (right) computed by P1 , P2 and P3 approximations.

7.15. Numerical experiences

559

discrete solutions is equally open. However, the formulation (7.343) of the problem is pretty close to the quasilinear problem (7.90). We employ the IIPG variant of the DCGM (7.286) to the solution of the system of the compressible Navier-Stokes equations. We consider a ﬂow aroundFthe NACA0012 proﬁle characterized by the inlet Mach number Min = 0.5 (= |v|/ γp/ρ at inﬁnity), angle of attack α = 0 (= arctg(v2 /v1 ) at inﬁnity) and Reynolds number Re = 5 000. The Reynolds number is near to the upper limit for steady laminar ﬂow. A characteristic feature of this ﬂow problem is the separation of the ﬂow occurring near to the trailing edge. The computations were carried out on a relatively coarse triangular grid having 1 360 elements which was obtained by the anisotropic mesh adaptation method (cf. [292, 298]), see Figure 7.11. We employ P1 , P2 and P3 polynomial approximation and the corresponding isolines of the Mach number are shown in Figure 7.12. We observe a signiﬁcantly smooth resolution for the P3 polynomial approximation although the corresponding grid is coarse. Other interesting papers, applying DCGMs to the Stokes problem and to twodimensional incompressible and to compressible Navier-Stokes equations, are due to Hartmann et al. [390–392, 409, 505].

8 Finite diﬀerence methods 8.1

Introduction

In our synopsis it is impossible to present all methods in the same generality as the dominant FEMs. Our choice for the results in ﬁnite diﬀerence methods is motivated by the following aspects: The most popular approach between 1960 and 1990, based upon M matrices, allows error estimates, e.g. for the Laplacian, of the form supx∈Ωh |u0 (x) − uh0 (x)| ≤ Ch2 u0 C 4 (Ω) . Here u0 and uh0 are the exact and the discrete solutions of the strong equation Δu0 = f ∈ C(Ω) and the corresponding strong symmetric diﬀerence equations on a grid Ωh ⊂ Ω with step size h. We apply our general discretization theory in Chapter 3, thus avoiding the “external approximation approach”. This motivates considering ﬁnite diﬀerence methods as variational methods similar to FEMs, cf. earlier approaches, discussed below. The above |u0 (x) − uh0 (x)| can then be estimated by a stronger discrete Sobolev energy norm on Ωh . More generally, convergence results and error estimates can be proved here for similarly general problems as for FEMs. However, handling boundary conditions and adaptivity is strongly restricted. These diﬃculties are discussed by Heinrich [395] with appropriate combinations of ﬁnite diﬀerence methods and ﬁnite volume techniques, on irregular grids, Ωh . For intended applications to bifurcation, ﬁnite diﬀerence methods are often used on cuboidal domains Ω = Πni=1 (ai , .bi ). So this case will play an essential role. Finally we only discuss equidistant grids Ωh . All our results are based upon the analytical results for general elliptic problems, summarized in Chapter 2. Section 8.2 is devoted to simple examples and basic tools. We will discuss the changes from unsymmetric to symmetric diﬀerence equations. Symmetric diﬀerence equations remain unchanged, if h is replaced by −h, and thus they automatically are of order two, cf. Lemmas 8.1–8.2 and Grossmann and Roos [374–376]. The corresponding convergence theory is nearly the same as for unsymmetric forms. Section 8.3 presents the facts for discrete Sobolev spaces. Section 8.4 elaborates general elliptic problems with Dirichlet boundary conditions, and their diﬀerence methods, Section 8.5 the unique existence of the discrete solutions and their convergence. For general problems, we consider unsymmetric diﬀerence methods ﬁrst and generalize them to curved boundaries. This yields methods converging of order 1. A modiﬁcation to symmetric diﬀerence methods converges of order 2. We consider for all cases the quartets of strong and weak diﬀerential and diﬀerence equations. These weak diﬀerence forms go back to the Russian literature, cf. Samarskii [564, 565], and

8.1. Introduction

561

j are used, e.g. by Temam [621] and Grossmann and Roos [375]. By combining the ∂− i h and ∂+ u , we deﬁne an unsymmetric divergence form as, e.g. in (8.44). We proceed from unsymmetric to symmetric forms in Subsection 8.4.2. The transformation of the strong forms Ahs uh = f h , essentially the classical diﬀerence equation, into the weak forms Ah uh0 = f h , and the discussion of solving Ah uh0 = f h , proceeds along the same lines for all cases. So we give the details only for second order linear equations in Subsections 8.4.2, 8.4.3. For the other cases we restrict ourselves to listing the equations in Subsections 8.4.4 ﬀ. For nonlinear operators the domains are subsets of the corresponding Sobolev spaces in (8.43), (8.47). Conditions for the coeﬃcients are listed in Summary 8.6, and Remark 8.7. The above transformation from strong to weak forms becomes more complicated for natural boundary conditions in Section 8.6, and is not applicable in Section 8.7. With one exception, the proof for stability and convergence follows “the same general procedure as” for all other discretization methods: We combine discrete coercivity, cf. (8.120), compact perturbation of a boundedly invertible linear operator and consistency. This will yield the existence of a unique solution for the diﬀerence equation, and its convergence to the exact solution u0 , and the corresponding generalizations. For equations of higher order and general domains violated boundary conditions are treatable as for FEMs. We do not reformulate these rather technical considerations. Convergence for monotone and quasilinear operators follows the lines of Section 4.5 and again yields Theorem 4.67. Except for Section 8.6, we restrict the discussion to homogeneous Dirichlet conditions. In Lemma 2.26 we have formulated conditions allowing the transformation of a general Dirichlet boundary value problem into a homogeneous form. For diﬀerential and diﬀerence equations the transformation of strong into weak forms is based upon partial integration and summation. Here the vanishing Dirichlet boundary conditions at least for the test functions, often for the solutions as well, are crucial, cf. e.g. Proposition 8.8. For nonlinear discrete problems the mesh independence principle, cf. Theorem 8.33, implies the quadratic convergence of Newton’s method for solving the diﬀerence equations, essentially independent of the step size. In Section 8.6 we formulate, for problems with natural boundary conditions, difference methods and prove existence and convergence of the discrete solutions by modifying the previous technique. In Section 8.7, we discuss diﬀerent methods of orders ≥ 1 on curved domains for special diﬀerential equations. This will require distinguishing points more or less near the boundary. These results are only summarized, but not proved. We have conﬁned the discussion to diﬀerence methods of ﬁrst or second order accuracy. By our convergence results with respect to discrete Sobolev norms, the extrapolation or defect correction methods, based upon asymptotic expansions, presented in Section 8.8, allow a very eﬃcient formulation of higher order methods anyway. In particular, our symmetric forms yield second order accuracy and the advantageous asymptotic expansions in powers of h2 . The last Section 8.9 applies Richardson extrapolation in an academic example to the von K´ arm´an equations.

562

8. Finite diﬀerence methods

Notice that the following problems are only discussed for FEMs. General convergence theory for monotone operators, cf. Section 4.5, variational methods for eigenvalue problems, cf. Section 4.7, methods for fully nonlinear elliptic problems, cf. Section 5.2, methods for nonlinear boundary conditions, cf. Section 5.3, and quadrature approximate methods, cf. Section 5.4. All the results remain correct for diﬀerence methods. For appropriate updates see Remark 8.27. Many diﬀerence methods for many problems are considered in the literature. From the huge number of publications, we only list a very small choice of books and surveys. Papers are cited in the corresponding context. Ansorge [31], Cheng [171], Ciarlet and Lions (ed.) [177] Collatz [206], Cuvelier et al. [231], Forsythe and Wasow [319], Grossmann et al. [375,376], Hackbusch [387], Heinrich [395], Isaacson and Keller [414], Marchuk [483], Quarteroni [536], Samarskii [564, 565], Sloan et al. [592], Strikwerda [604, 606], Temam [624], Thomas [626], Vainberg [643], Vainikko [644]. Xu [670] and Zeidler [677, 678].

8.2

Diﬀerence methods for simple examples, notation

We start with the simple boundary value problem −u (x) = f (x) ∀ x ∈ Ω = (0, 1), u(0) = ua , u(1) = ue ,

(8.1)

hence −u = −Δu in R1 = R. To determine an approximate solution we replace the second derivative u by divided diﬀerences in the grid points. We choose a step size h, and introduce an equidistant one-dimensional grid as, cf. Figure 8.1, G1h := G h := {xi ∈ R : xi = ih, h := 1/k, k, i ∈ Z, k > 0},

(8.2)

Ωh0 := {xi ∈ R : xi = ih, 0 < i < k} ⊂ G h , ∂Ωh := {xi = ih, i = 0, k}, Ωh := Ωh0 ∪ ∂Ωh = {xi ∈ R : xi = ih, 0 ≤ i ≤ k} ⊂ G h . We deﬁne for a function v: R → R. For v : Ωh → R, we restrict x to x ∈ Ωh or x ∈ Gh : (1)

h ∂+ v(x) := ∂+ v(x) := ∂+ v(x) := [v(x + h) − v(x)]/h, 0 ≤ x ≤ 1 − h,

(8.3)

−h v(x) := ∂− v(x) := [v(x) − v(x − h)]/h, h ≤ x ≤ 1, recursively, ∂− v(x) := ∂+

(j) (j) (j−1) (j) (j) (j−1) v(x) , ∂− v(x) := ∂−,h v(x) := ∂− ∂− v(x) , ∂+ v(x) := ∂+,h v(x) := ∂+ ∂+ (1)

e.g.

(2)

∂+ v(x) = [v(x + 2h) − 2v(x + h) + v(x)]/h2 , 0 ≤ x ≤ 1 − 2h. Ω Ωh Ωh ∪ ∂Ωh

Figure 8.1 One-dimensional grid.

563

8.2. Diﬀerence methods for simple examples, notation

The recursive deﬁnition of symmetric formulas is a bit more complicated than for (8.3), cf. Collatz [206], pp. 538 ﬀ. In diﬀerence equations, these (8.4) can be directly used for constant coeﬃcients, but will be modiﬁed below for nonconstant coeﬃcients: (0)

∂h v(x) := [v(x + h) + v(x − h)]/2,

(8.4) (1) h h ∂h v(x) := [v(x + h) − v(x − h)]/(2h) = ∂+ + ∂− v(x) /2, (2) h h ∂− v(x) = [v(x + h) − 2v(x) + v(x − h)]/h2 ∂h v(x) := ∂+ h h h h ∂+ v(x) + ∂+ ∂− v(x) /2 = ∂− (3) (1) (2) ∂h v(x) := ∂h ∂h v(x) = [v(x + 2h) − 2v(x + h) + 2v(x − h) − v(x − 2h)]/(2h3 ) (1) (2) (2) (1) with ∂h ∂h v(x) = ∂h ∂h v(x) , (4) (2) (2) ∂h v(x) := ∂h ∂h v(x)

= [v(x + 2h) − 4v(x + h) + 6v(x) − 4v(x − h) + v(x − 2h)]/(h4 ), etc. The formulas in (8.4) have the invariance property with respect to replacing h by (i) (i) −h, so ∂h v(x) = ∂−h v(x). Compared to the nonsymmetric divided diﬀerences in (8.3), the corresponding symmetric formulas in (8.4) approximate the function values and derivatives in x of order 2, instead of the previous order 1. This implies an asymptotic expansion in powers of h2 instead of h, cf. the following Lemma 8.1, cf. [387], Lemma 4.1.1, and [103]. These advantages are inherited by the diﬀerence methods. (0)

(i)

(i)

Lemma 8.1. The mean value and diﬀerence operators ∂h , and ∂+ , ∂− , ∂± , ∂h satisfy the following relations, estimates, and asymptotic expansions. For i = 0, 1, . . . , and with ci,j ∈ R, independent of u, h, we obtain (0)

(2)

∂h = (∂+ + ∂− )/2, ∂h = (∂+ − ∂− )/h = ∂+ ◦ ∂− = ∂− ◦ ∂+ , and (i) ∂± u(x) − u(i) (x) ≤ Lh for u(i) ∈ CL (Ω), and for symmetric forms

(8.5)

2 (i) ∂ u(x) − u(i) (x) ≤ L h for u(i+1) ∈ CL (Ω) or asymptotic expansions h i+1 m u(i+1) (x) hm+1 (m+1) i u h ∈ CL (Ω), (8.6) ∂+ u(x) − u (x) − ≤L (i + 1)! (m + 1)! i=1

q (i) h2q+1 u(2j+i) (x) ∂ u(x) − u(i) (x) − ≤L , for u(2q+i) ∈ CL (Ω). h2j ci,j h (2j + i)! (2q)! j=1

These results remain valid, with obvious modiﬁcations, for the multidimensional generalizations in (8.12) ﬀ., cf. Lemma 8.2.

564

8. Finite diﬀerence methods

Replacing the u (x) in (8.1) by the ∂h u(x) for x = xi ∈ Ωh0 in (8.2) yields (2)

(2)

−∂h u(xi ) = f (xi ) + O(h2 ), 0 < i < k, h :=

1 , xi = ih, u(x0 ) = ua , u(xk ) = ub . k

Omitting the O(h2 ) term we end up with the (symmetric) diﬀerence equation for a discrete function uh : Ωh → R of the form (2)

− ∂h uh (xi ) = −[uh (xi − h) − 2uh (xi ) + uh (xi + h)]/h2 = f (xi ), 0 < i < k, uh0 = uh (x0 ) = ua , uhk = uh (xk ) = ub .

(8.7)

As for all the previous discretization methods, the deciding question is: Does an approximate solution uh0 (xi ), 0 ≤ i ≤ k, uniquely exist, and do these values converge to the values of the exact solution u0 (xi )? Do small perturbations in the equation only slightly perturb the solution uh0 (xi )? This has to be generalized, and discussed in this chapter. For motivating diﬀerence methods for PDEs, we generalize our above model problem (8.1) as A u = −Δu = f ∈ C(Ω), (u − φ)|∂Ω = 0.

(8.8)

We only consider ﬁnite diﬀerence methods with equal step sizes, h, in all directions. Diﬀerent hi can be treated with essentially the same techniques. For cuboidal Ω, Dirichlet boundary conditions can be exactly discretized: Ω :=

n

(ai , bi ) ⊂ Rn s.t. ∃h > 0 with ai /h, bi /h ∈ Z, i = 1, · · · , n.

(8.9)

i=1

The following approach works as well for unions of cuboids in (8.9), connected along (n − 1)-dimensional faces, perpendicular to the coordinate axes. We extend it to curved boundaries below. For Ω ⊂ Rn in (8.9) or in (8.18) below we introduce step sizes, h, and grids: H := {h as in (8.9) or h ∈ R+ for (8.18) below} :

(8.10)

h G h := GD := {x = (x1 , . . . , xn )T ∈ Rn : xi /h ∈ Z},

Glh := {∀j = 1, . . . , n : x = (x1 , . . . , xn )T ∈ Rn : ∀i = j : xi /h ∈ Z, xj /h ∈ R}. For Ω in (8.9) we introduce Ωh0 := Ω ∩ G h , Ωh := Ω ∩ G h , ∂Ωh := ∂Ω ∩ G h = Ωh \ Ωh0

(8.11)

but we have to modify it for Ω in (8.18). We consider grid functions v h , deﬁned on Ωh , and deﬁne their multidimensional forward, backward, and normal (divided) diﬀerences for x ∈ G h , generalizing

8.2. Diﬀerence methods for simple examples, notation

565

: pointsin Ω oh : pointsin ∂Ωh

Figure 8.2 Two-dimensional grid for Dirichlet boundary conditions.

(8.3), (8.4), as, cf. Figure 8.2, i h i ∂+ v (x) := ∂+,h v(x) := [v h (x + hei ) − v h (x)]/h, ei = i-th unit vector

(8.12)

i h i v (x) := [v h (x) − v h (x − hei )]/h = ∂−,h v(x), i = 1 . . . n, ∀x ∈ G h , ∂− i αi h α h ν h v (x) := Πni=1 ∂± v (x), and ∂± v (x) := [v h (x ± hν) − v h (x)]/h, ∂±

with the normal diﬀerences, where, for x ∈ ∂Ω, Ω ⊂ Rn in (8.9), ei is replaced by the outer normal ν or by −ν. The symmetric divided diﬀerences are i ∂hi v h (x) := ∂−h v(x) := [v h (x + hei ) − v h (x − hei )]/(2h), (8.13) (2,0) (0,2) 1 1 h 1 1 2 2 h ∂+ ∂− v h (x)), ∂h v h (x) := ∂− ∂+ ∂h v h (x) := ∂− v (x) = ∂+ v (x) , and (1,1) (1,1) 1 2 1 2 ∂h v h (x) := ∂h1 ∂h2 v h (x) or ∂h v h (x) := ∂− ∂+ + ∂+ ∂− v h (x), (3,0)

:= ∂h1 ∂h

(4,0)

:= ∂h

∂h ∂h

(2,0)

(2,1)

, ∂h

(2,0)

:= ∂h2 ∂h

(2,0) (2,0) (3,1) ∂ h , ∂h

(1,1)

or ∂h1 ∂h (2,0)

:= ∂h1 ∂h2 ∂h

(1,2)

, ∂h

(1,1)

:= ∂h2 ∂h

(j)

(0,2)

or ∂h1 ∂h

,

j

, . . . , and ∂ν,h v h (x) := (∂hν ) v h (x).

These symmetric formulas are invariant with respect to replacing h by −h. The problematic (2,0) ∂h,u v h (x) := ∂h1 ∂h1 v h (x) = [v h (x + 2he1 ) − 2v(x) + v h (x − 2he1 )]/(2h)2 (8.14) is avoided. It would yield in (8.16) a ﬁve-point star of double step size 2h. Therefore, (i,j) the previous higher order ∂h v h (x) in (8.13) are chosen to include only points as

566

8. Finite diﬀerence methods

Table 8.1: Stencil for the ﬁve-point star around x ∈ Ωh ⊂ R2 for −h2 Δh .

0

−uh (x + he2 )

0

−u (x − he1 ) 0

h

−u (x + he1 ) 0

h

4u (x) −uh (x − he2 )

h

abbreviated as

0

−1

0

−1 0

4 −1

−1 0

(i,j) h

close as possible to x. So we allow only points x ± νhe1 + μhe2 in ∂h that in

v (x) show

(i,j) h

v (x) with2(m − 1) ≤ i + j ≤ 2m, only x ± νhe1 + μhe2 with|ν|, |μ| ≤ m. (8.15)

∂h

For the two-dimensional Laplacian in (8.8) and Ω = (0, 1) × (0, 1) we formulate its diﬀerence approximation analogously to (8.1) and (8.7). We replace the partial (2,0) + −Δu(x) = −(∂ (2,0) + ∂ (0,2) )u(x) in (8.8) by the diﬀerence approximations −(∂h (0,2) h h 2 ∂h )u (x), and only admit x = (xi , yj ) ∈ Ω0 . Omitting the O(h ) term, the diﬀerence equation for uh0 : Ωh → R has the form of a ﬁve-star approximation

% (2,0) (0,2) uh0 (x) = − uh0 (x + he1 ) + ∂h (8.16) f (x) = Ah uh0 (x) := −Δh uh0 (x) := − ∂h & + uh0 (x − he1 ) + uh0 (x + he2 ) + uh0 (x − he2 ) − 4uh0 (x) /h2 , ∀x = (xi , yj ) ∈ Ωh0 , h = 1/k, ∀0 < i, j < k, uh0 (x) = φ(x)∀x ∈ ∂Ωh ⇔ ∀0 ≤ i, j ≤ k, i · j = 0. Sometimes we approximate the f (x) as in Proposition 8.5. Equation (8.16) is visualized in Table 8.1. We will use the same standard abbreviation, showing only the factors of the uh (x ± hei ), uh (x) in the appropriate positions, in the later tables. Equation (8.16), and the following symmetric, and unsymmetric ﬁnite diﬀerence methods, essentially maintain the order 2 and 1, and the asymptotic expansion in powers of h2 and h, in Lemma 8.2 for the diﬀerence between the exact solution, u0 , e.g. of the two-dimensional Laplacian and its discrete approximation, uh0 , e.g. of (8.16), evaluated in the grid points, cf. Theorem 8.31 in Section 8.5, and Theorem 8.43 in Section 8.8.

8.3

Discrete Sobolev spaces

Essential for our stability and consistency results is the modiﬁcation of continuous to discrete Sobolev spaces. These are spaces of grid functions only deﬁned on Ωh or G h . Correspondingly, we introduce these concepts, based upon partial diﬀerences. For the intended convergence results, we have to consider sequences of discrete Sobolev spaces, deﬁned on increasingly ﬁner grids, indicated by h ∈ H, cf. (8.10). 8.3.1

Notation and deﬁnitions

Similarly to the above, we deﬁne the higher partial diﬀerences with the usual multiindices α := (α1 , . . . , αn ) ∈ Nn0 , and the corresponding reals, and vectors in (8.17).

8.3. Discrete Sobolev spaces (i)

(i)

567

(i)

They are based upon the above ∂+ uh , ∂− uh , ∂h uh in (8.3), (8.4), generalized in α α , version, and indicate the ∂− , ∂hα forms. The (8.12)–(8.15). We formulate the ∂+ α α α notation ∂± indicates ∂+ or ∂− or their combination, applied recursively. i i With ∂+ u, ∂− u, ∂hi u

as in (8.12), (8.13) let n αn α α1 αn α 1 α1 u := ∂+ . . . ∂+ u := ∂+ · · · ∂+ u, ∇k+ u := ∂+ u |α|=k , ∂+ α 0 1 ∇≤k + u := ∂+ u |α|≤k , ∇+ u := u, ∂+ u := ∇+ u := ∇+ u,

(8.17)

i α u, ∂− u, ∇k− u, ∂− u, and ∂hi u, ∂hα u, ∇h u, ∇kh u, ∂h u, and correspondingly ∂− i α u(x), ∂+ u(x) ∈ R, ∂+ u(x) ∈ Rn , nk , Nk s.t. ∇k+ u(x) ∈ Rnk , with, e.g., u(x), ∂+ Nk ∇≤k , ϑi , ϑα := (ϑ1 )α1 · · · (ϑn )αn ∈ R, ϑ := (ϑ1 , . . . , ϑn ) ∈ Rn , + u(x) ∈ R

Θk := (ϑα )|α|=k ∈ Rnk , ϑα := ϑi := 1 for |α| = i = 0, Θ0 ∈ R, Θ := Θ1 . For the elliptic equations below, the boundary conditions, cf. (8.8), are transformed into trivial Dirichlet conditions, see (8.62). For the unsymmetric and symmetric diﬀerence methods, this is translated into vanishing uh (x) for grid points x near the α h u in the diﬀerence equations are deﬁned in boundary, cf. (8.19), (8.20). Then the ∂± the standard way for all points, in particular for those near to, and at the boundary. The symmetric diﬀerence methods, cf. Subsection 8.4.2, (8.60), ﬀ., and Section 8.7, are the motivation for choosing diﬀerent forms for the relevant grid points, cf. (8.20), compared to (8.19), e.g. [678]. For relating the strong forms in (8.7), (8.16), and (8.44) ﬀ. below to the corresponding generalized weak forms, we split the set of grid points into all relevant, interior, boundary and near boundary grid points, Ωh , Ωh0 , ∂Ωh , . . . , ∂Ωhm or Ωh \ Ωh0 , generalizing (8.11). We allow curved boundaries, such that Ω ⊂ Rn is open, bounded, nonempty, and ∂Ω is Lipschitz-continuous. (8.18) We distinguish two cases for deﬁning the relevant grid points. Often u0 ∈ H k (Ω), k < m, so it cannot be boundedly extended to an Ec u0 ∈ H m (Rn ). Or the solution u0 ∈ H k (Ω), k ≥ m with (8.18) allows the bounded extension Ec u0 ∈ H k (Rn ), cf. Theorem 4.37. The Ω-relevant grid points x ∈ G h , cf. (8.10), are deﬁned via half open cubes ch (x). In both cases, (8.19) and (8.20), the strong and weak diﬀerence equations will be formulated on Ωh0 and Ωh , respectively, and the boundary conditions on Ωh \ Ωh0 . For both cases and for the ﬁrst case, u0 ∈ H k (Ω), k < m, we deﬁne ch (x) :=

n i=1

[x − hei /2, x + hei /2), ch (Ωh ) :=

C

ch (x), and for k < m (8.19)

x∈Ωh

Ωh := {x ∈ G h : ch (x) ⊂ Ω}, ∂Ωh1 := {x ∈ G h \ Ωh : ch (x) ∩ ∂Ω = ∅}, Ωh0 := x ∈ Ωh : dist x, ∂Ωh1 ≥ mh , and ∂Ωh := ∂Ωhm := Ωh \ Ωh0 ,

568

8. Finite diﬀerence methods

of interior and boundary knots. In the second case, we deﬁne the interior knots as Ωh0 and ∂Ωh , respectively. (8.20) Ωh0 := {x ∈ G h : ch (x) ⊂ Ω}, and deﬁne boundary knots recursively: For m = 1 : ∂Ωh := ∂Ωh1 := x ∈ G h \ Ωh0 : ch (x) ∩ ∂Ω = ∅ , Ωh := Ωh1 := Ωh0 ∪ ∂Ωh1 , m ≥ 2 : Ωh := Ωhm := Ωh0 ∪ ∂Ωh1 ∪ . . . ∂Ωhm , ∂Ωh := Ωh \ Ωh0 = ∂Ωh1 ∪ . . . ∂Ωhm with ∂Ωhm := x ∈ G h \ Ωh0 ∪ ∂Ωh1 ∪ . . . ∂Ωhm−1 : ch (x) ∩ ∪x ∈Ωh0 ..∪∂Ωhm−1 ch (x ) = ∅ . The half open cubes ch (x) yield a disjoint covering of Rn , and Ω : C C Rn = ch (x), Ω ⊂ ch (x) for (8.19)), and Ω⊂

(8.21)

x∈Ωh ∪∂Ωh 1

x∈G h

C

ch (x) for (8.20)), with ch (x) ∩ ch (x ) = ∅∀x = x, x ∈ G h .

h x∈Ωh 0 ∪∂Ω1

h For Ω in (8.18), the above ∂Ω1 in (8.19) or (8.20) is equivalently deﬁned by ∂Ωh1 = x ∈ G h \ Ωh or \ Ωh0 : ch (x) ∩ Ω = ∅ . We have deﬁned the Ωh0 , Ωh , ∂Ωhm , in (8.20) diﬀerently from other authors, e.g. Temam [624]. This allows the presentation of symmetric diﬀerence methods of order 2 instead of the nonsymmetric order 1. For the consistency of these methods we need (α)

(α)

Lemma 8.2. The diﬀerence operators ∂± , ∂h satisfy the following estimates, and asymptotic expansions: With 1 := (1, . . . , 1), cα,j , cα,j ∈ R, independent of h, and cα,j = cα,j u(α+β) (x), |β| = j , cα,j = cα,j u(α+β) (x), |β| = 2j, β even , we get (α) ∂± u(x) − u(α) (x) ≤ cα,1 h for u(α) ∈ CL (Ω) for unsymmetric forms (8.22) (α) ∂ u(x) − u(α) (x) ≤ cα,1 h2 for u(α+1) ∈ CL (Ω) for symmetric forms, h (0,i)

(α=0e )

i u(x) = (u(x + ei ) + u(x − ei ))/2, and asymptotic expansions with ∂h u(x) = ∂h m (α) j ∂± u(x) − u(α) (x) − h cα,j ≤ Lhm+1 cα,m+1 for u ∈ CLα+m (Ω), (8.23)

j=1

m (α) (α+2m) 2j ∂ u(x) − u(α) (x) − h cα,j ≤ Lh2m+1 cα+2m+1 , for u ∈ CL (Ω); h j=1

u∈

CLα+m (Ω)

indicates derivatives of u only in directions ei with αi = 0 or 0ei .

8.3. Discrete Sobolev spaces

8.3.2

569

Discrete Sobolev spaces

Convergent diﬀerence methods require regularity assumptions for the boundary. For our chosen Ω ∈ CL and the deﬁnition of Ωhm the following, usually imposed conditions are satisﬁed by Theorem 1.26 and by construction. In diﬀerent papers the following conditions are imposed: 1. Assume ∂Ω smooth enough, such that the embedding W 1,p (Ω) → Lp (∂Ω) is continuous, cf. Neˇcas [510]. 2. The boundary strip Ωhm := ∪x∈Ωh ch (x) \ Ω is uniformly small. Hence there exist 0 < d, h0 ∈ R such that for all 0 < h < h0 , h ∈ H, for all x ∈ Ωhm , there exists an α with |α| ≤ m such that ∃x + dαh or ∃x − dαh ∈ Ω, cf. Raviart [545], p. 22. The continuous embedding W 1,p (Ω) → Lp (∂Ω) is a consequence of (8.9), (8.18), the uniformly small boundary strip of our construction. We consider the following grid functions, uh , and test functions v h : G h → R or Rq , always vanishing outside of Ωh . The imposed vanishing Dirichlet boundary conditions (of higher order) imply test functions, v h (x), vanishing on Ωh \ Ωh0 , uh , v h : G h → R or → Rq , with uh (x), v h (x) = 0 ∀x ∈ G h \ Ωh , and

(8.24)

m,p for test functions v h ∈ W0,+ (Ωh ) : v h (x) := 0 ∀x ∈ Ωh \ Ωh0 , cf. (8.19), (8.20).

In Section 8.6, we will discuss natural boundary conditions, and modify the previous, and following deﬁnitions. We deﬁne, for the grid functions, uh , and their diﬀerences in (8.17)–(8.20), the discrete integral, the (scaled) discrete Lebesgue, and Sobolev scalar products, their norms, seminorms, and spaces, (of grid functions of ﬁnite norms uh L2h (Ωh ) or uh W+m,p (Ωh ) ), cf. e.g. Hackbusch [387] and Zeidler [678]: uh dxh := hn uh (x), uh dxh := hn uh (x), (8.25) Ωh

Ωh 0

x∈Ωh

x∈Ωh 0

(v h , wh )L2h (Ωh ) :=

1/2

Ωh

v h wh dxh , v h L2h (Ωh ) := (v h , v h )L2 (Ωh ) and h

1/p

1 ≤ p ≤ ∞ : v h Lph (Ωh ) :=

Ωh

|v h (x)|p dxh

⎛ = ⎝hn

⎞1/p |v h (x)|p ⎠ ,

(8.26)

x∈Ωh

p h h with v h L∞ v h : v h Lph (Ωh ) < ∞ . h := max {|v (x)|}, and L (Ω ) := h h (Ω ) x∈Ωh

We turn to discrete Sobolev spaces and the dual norms for bounded linear forms. Depending upon the diﬀerent types of divided diﬀerences, mainly the ∂+ , we introduce the higher order Sobolev spaces of grid functions, denoted as W+m,p (Ωh ) or W−m,p (Ωh ), W±m,p (Ωh ), Whm,p (Ωh ). Compared to W+m,p (Ωh ) with uh of ﬁnite norms, we replace

570

8. Finite diﬀerence methods

α h in W−m,p (Ωh ), or the symmetric forms, Whm,p (Ωh ), the ∂+ v by the corresponding α h α h ∂− v , and appropriate symmetric ∂h v , cf. (8.12)–(8.17). The notation Lph (Ωh ), with h indicating the discretization parameter, is not appropriate for the following W+m,p (Ωh ), . . . , since the + , − , h have to indicate the chosen type of divided α h u (x)∀x ∈ Ωh , uh : G h → R. diﬀerences, assuming, e.g. well-deﬁned ∂+ α h α h ∂+ (uh , v h )H+m (Ωh ) := u , ∂+ v L2 (Ωh ) , and (8.27) h

|α|≤m

⎛ |v h |W k,p (Ωh ) := ⎝ +

⎞1/p

α h p ∂+ v Lp (Ωh ) ⎠

⎞1/p

, v h W+m,p (Ωh ) := ⎝

h

|α|=k

⎛

|v h |p

k,p (Ωh ) W+

0≤k≤m

⎠

,

m,p m,p m,p W0,+ (Ωh ) := uh ∈ W+m,p (Ωh ) : uh (x) = 0 ∀x ∈ Ωh0 =: W0,− (Ωh ) =: W0,h , (8.28) m (Ωh ) := W+m,2 (Ωh ), . . . , W+−m,p (Ωh ) := L W+m,p (Ωh ), R , 1/p + 1/p = 1, H+ f h , v h W −m,p (Ωh )×W m,p (Ωh ) = +

+

(8.29)

|α|≤m

Ωh

α h fαh ∂+ v dxh , 1 < p < ∞, cf. (2.107), and

⎛ f h W −m,p (Ωh ) := +

{|f h , v h |} = ⎝

sup

v h W m,p (Ωh ) =1

⎞1/p

f α p p

|α|≤m

+

Lh (Ωh )

⎠

.

(8.30)

By (8.24), (8.29), · W+m,p (Ωh ) , and · Whm,p (Ωh ) are equivalent. For systems we modify these deﬁnitions, corresponding to (8.69), as ( uh , v h )H+m (Ωh ,Rq ) :=

q

α h α h ∂+ ui , ∂+ vi

i=1 |α|≤m

L2h (Ωh )

, uh = uh1 , . . . , uhq ,

⎛

| uh |W+m,p (Ωh ,Rq ) , uh W+m,p (Ωh ,Rq )

f h W −m,p (Ωh ,Rq ) +

q α h p ∂+ := ⎝ ui p i=1 |α|≤m

q p fih −m,p := i=1

W+

1/p (Ωh )

Lh (Ωh )

(8.31)

⎞1/p ⎠

, cf. (2.107), with

, for f h = f1h , · · · , fqh .

For strong diﬀerence equations, and in Proposition 8.4, we additionally need W+k,p (Ωh ) with uh (x) = 0∀x ∈ Ωh for k = 2m or 0 ≤ k < m. Proposition 8.3. All these spaces of discrete functions with any, but ﬁxed choices α α or ∂− or ∂hα , are discrete Lebesgue and Sobolev spaces, in particular Banach of ∂+ spaces, with respect to the scalar products, and norms deﬁned in (8.25)–(8.31).

8.3. Discrete Sobolev spaces

571

As for standard Sobolev spaces, discrete Sobolev spaces satisfy (8.32), sometimes called the Sobolev inequality, cf. Yulin [673], Theorems 4, 9. The norm and seminorm in k,p (Ωh , Rq ) in (8.33) are equivalent, cf. Ladyˇzenskaja [462], Schumann and Zeidler, W0,+ [574], Lemma 4, Temam [624], a consequence of his Proposition 3.3. Its generalization to m > 1 is straightforward. Proposition 8.4. For any discrete function uh ∈ W+m,p (Ωh ), and for any given > 0 there exists a constant K(, m, q) ∀0 ≤ k < m, 2 ≤ p ≤ ∞, independent of h, and uh , such that the following inequality is valid ∀uh ∈ W+k,p (Ωh ): | uh |W k,p (Ωh ) ≤ | uh |H+m (Ωh ) + K(, m, q)| uh |L2h (Ωh ) .

(8.32)

+

k,p (Ωh ), hence there exists uh W k,p (Ωh ) and | uh |W k,p (Ωh ) are equivalent norms in W0,+ +

+

C = C(m, q) ∈ R+ , independent of h, such that k,p uh W k,p (Ωh ) ≤ C| uh |W k,p (Ωh ) ≤ C uh W k,p (Ωh ) ∀uh ∈ W0,+ (Ωh ). +

+

(8.33)

+

k,p (Ωh ). This allows a modiﬁcation of (8.32) with uh W k,p (Ωh ) , uh W+m,p (Ωh ) in W0,+ +

In the following diﬀerence equations we sometimes employ the following mean value function. We will come back to it in our later approximations. For (8.20), we assume u or f to be extended to an Ω1 ⊃ Ωh or to Rn and deﬁne: 1 f (¯ x)d¯ x. (8.34) f¯h (x) := n h ch (x) Proposition 8.5. This integral mean value f¯h (x) approximates f (x) ∈ CL1 Ωhe : 1 f (¯ x)d¯ x, |f¯h (x) − f (x)| ≤ Lh2 . (8.35) f¯h (x) := n h ch (x) For functions only deﬁned in Ω ⊂ Ωhe , extensions beyond Ω as (8.24) or in Gilbarg and Tru-dinger [346], Lemma 6.37 have to be used. 2m ,ev With |α|=2 indicating even multi-indices and α + 1, 1 := (1, . . . , 1), we obtain the asymptotic expansion for f ∈ CL2m Ωhe ⎛ ⎞ 2m ,ev α h (h/2) α f¯ (x) − ⎝f (x) + ⎠ ∂ f (x) ≤ Ch2m +1 . (8.36) (α + 1)! |α|=2

The analogous result holds for f ∈ W 2m +1,p (Ωhe ) and

2m ,ev

|α|=2 2m +1,ev

replaced by the averaged Taylor polynomial Q |f |W 2m +1,p (Ωhe ) .

f

· · · and |f |C 2m (Ωh ) L

e

in (4.33) and by

572

8. Finite diﬀerence methods

Proof. We use the Taylor expansion: f (x + tej ) = f (x) + t

R1m := R1 : = 0

−h/2

h/2

= −h/2

=

−h/2

+

⎛

···

h/2

f ⎝x +

−h/2

⎛ ⎝ ···

−h/2

···⎝

m,ev

(t1 )i

−h/2

−h/2

tj ej ⎠ dt1 · · · dtn

h/2

m

⎞

⎞

tj ej ⎠ dt1⎠ · · · dtn and with (8.37)

j=2

⎛ ⎛ ⎝f ⎝x +

−h/2

m

⎞ tj ej ⎠

j=2

m ∂xi 1 f x + j=2 tj ej i! m,ev j=2

h/2

···h

⎞

f ⎝x + t1 e1 +

and, with m = 2m and x+ := x + =

(8.37)

j=1

i=1

h/2

m

⎛ h/2

⎛ h/2

+ R1 with ∂xi j f = ∂ i f / ∂xj i and

(t)m+1 for ∂xmj f ∈ CL (Ω). m!

h/2

f (x)dx = ch (x)

i!

(t − σ)m−1 m ∂xj f (x + σ) − ∂xmj f (x) dσ with (m − 1)!

This is inserted into

∂xi j f (x)

(t)i

i=1

|R1 | ≤ L

m

f (x+ ) +

m,ev i=2

⎞

⎞

+ R1 ⎠ dt1 ⎠ · · · dtn

tj ej i i ∂x1 f (x+ )

(h/2)

(i + 1)!

+ R1

dt2 · · · dtn .

By induction we obtain the claim.

8.4 8.4.1

General elliptic problems with Dirichlet conditions, and their diﬀerence methods General elliptic problems

In Section 8.1 we have already described the unsymmetric and the symmetric types of diﬀerence methods. The previous deﬁnitions allow a generalization of these diﬀerence equations to all types of elliptic diﬀerential equations studied in Chapter 2. To avoid obscuring the basic ideas we start, similarly to the above two examples in (8.1) and (8.8) with the case that Ω has the form (8.9). In contrast to (8.1) and (8.8) we consider unsymmetric diﬀerence methods ﬁrst and generalize them to curved boundaries. This yields methods converging of order 1. We list for all cases the quartets of strong and

8.4. General elliptic equations with Dirichlet boundary conditions

573

weak diﬀerential and diﬀerence equations, e.g. As , Ahs , A, Ah . We have seen in (8.12)– (8.15) that for diﬀerence equations, replacing ∂+ uh , ∂− uh by ∂h uh does not make sense, cf. (8.14). Nevertheless, we will formulate symmetric divergence forms for all our problems. This modiﬁcation to symmetric diﬀerence methods converges of order 2. We start, summarizing for the diﬀerent elliptic problems studied here, with the deﬁnitions of ellipticity, and the conditions for the coeﬃcients, implying the boundedness, and H0m (Ω, Rq )-or W0m,p (Ω, Rq )-coercivity of the weak forms of the principal parts for m, q ≥ 1. All equations have to satisfy a strong Legendre, or in the case of higher order systems, a strong Legendre–Hadamard condition. We use the uniﬁed q (x) ∈ Rq×q for systems, notation aα,β for equations, q = 1, and Aα,β (x) = aij α,β i,j=1 q > 1, with aα,β = aij , and Aα,β = Akl for m = 1, and ∀|α|, |β| ≤ 1. We include the conditions for nonlinear operators G implying a bounded derivative G (u0 ) with a coercive principal part. Diﬀerence equations will require appropriate modiﬁcations. Summary 8.6. Let the bilinear form, a(·, ·), be induced by the (weak) linear operator A or the linearized G (u0 ). We assume the following conditions 1.–4. for boundedness, and (strong, and uniform) ellipticity with ϑ = (ϑ1 , . . . ϑn ) ∈ Rn : 1. For linear equations of second order, cf. (8.46), (8.47), (8.60), we require ∀0 ≤ i, j ≤ n, for almost every (a.e.) x ∈ Ω, cf. Chapter 2, (2.136), aij ∈ L∞ (Ω), ∃λ > 0, a.e. ∀x ∈ Ω : λ|ϑ|2n ≤

n

aij (x)ϑi ϑj ∀ϑ ∈ Rn ,

(8.38)

i,j=1

n where aij ∈ L∞ (Ω) implies a.e. i,j=1 aij (x)ϑi ϑj < Λ|ϑ|2n for Λ > λ > 0. 2. For linear equations of order 2m, m > 1, in (8.63), cf. Theorems 2.36, (2.79), (2.106), let ∀|α|, |β| ≤ m : aα,β ∈ L∞ (Ω), and for ∀|α|, |β| = m : aα,β ∈ C(Ω), (8.39) ∃λ > 0, a.e. ∀x ∈ Ω : λ|ϑ|2m aα,β (x)ϑα ϑβ ∀ϑ ∈ Rn , x ∈ Ω. n ≤ |α|=|β|=m

α β 2m ∀ϑ ∈ Rn , x ∈ Ω. This implies a.e. |α|=|β|=m aα,β (x)ϑ ϑ ≤ Λ|ϑ|n 3. For linear systems of order 2m, m ≥ 1, q > 1, in (8.69), (8.70), we combine for m = 1 the strong Legendre condition (8.40), cf. Theorems 2.89, (2.343), and for m > 1, the strong Legendre–Hadamard condition (8.41), cf. Theorems 2.104, (2.394). Assume Akl , Aα,β ∈ L∞ (Ω, Rq×q ) for m ≥ 1. For m > 1, and = (ϑ 1, . . . , ϑ n ) ∈ Rn×q , ϑ ∈ Cn , η ∈ ∀|α|, |β| = m let Aα,β ∈ C(Ω, Rq×q ). With ϑ ϑ, η Cq , we assume for (8.70) that ∃λ > 0 such that ∀ϑ,

2 ≤ for m = 1 : λ|ϑ| nq

n

l , ϑ k Akl (x)ϑ

q

2 , a.e.x ∈ Ω, ≤ Λ|ϑ| nq

(8.40)

k,l=1

for m > 1 : λ|ϑ|2m |η|2 ≤

i,j=1,...,q |α|=|β|=m

α

β aij αβ (x)ϑ ηj ϑ η i ∀x ∈ Ω.

(8.41)

574

4. 5.

6. 7.

8. Finite diﬀerence methods

i,j=1,...,q α β 2m For m > 1 this 47 implies |α|=|β|=m aij |η|2 ∀x ∈ Ω. αβ (x)ϑ ηj ϑ η i ≤ Λ|ϑ| A nonlinear problem, G, as in Subsections 8.4.5–8.4.7, is called elliptic if its derivative, G (u0 ), satisﬁes the appropriate condition 1.–3., for m, q ≥ 1. Then a(·, ·), induced by the previous linear or linearized operators, is bounded, and its principal part, ap (·, ·), is H0m (Ω, Rq )-coercive. This H0m -coercivity remains correct for bilinear forms, induced by G (u0 ) of nonlinear G, for m, q ≥ 1, and in W m,p (Ω, Rq ) for 2 ≤ p ≤ ∞, cf. Section 2.7, and Theorems 2.118, 2.119, 2.122, 2.124, 2.125, 2.126, 2.127. For a constant a0 > 0, the extended principal part, aep (u, v) := ap (u, v) + a0 ·(u, v)L2 (Ω,Rq ) , is also H m (Ω, Rq )- coercive, cf. Theorem 1.24. The strong diﬀerential operators, As or (G (u0 ))s , are bounded if some coeﬃcients aα,β , and Aα,β ,respectively, additionally are boundedly diﬀerentiable, more precisely: aα,β ∈ W |β|,∞ (Ω), and Aα,β ∈ W |β|,∞ (Ω, Rq×q ) ∀|α|, |β| ≤ m.

(8.42)

8. We consider diﬀerence equations in the weak forms (8.47), (8.63), (8.65), (8.67), (8.71)–(8.73), and in the corresponding strong forms, essentially the classical FDEs, (8.44), (8.62), (8.64), (8.66), (8.76). These diﬀerence equations are well deﬁned, if the previous L∞ (Ω), W |β|,∞ (Ω), . . . , are replaced by C(Ω), C |β| (Ω), . . . , ¯ij (x), cf. (8.34), and Remark 8.7. or by bounds on G h or aij ∈ H 1 (Ω), . . . , by a 9. The discrete principal parts, ahp (·, ·), are deﬁned as the sum of the highest order diﬀerence terms, corresponding to the coeﬃcients, e.g. in (8.38)–(8.41). The discrete forms of these linear operators, the Ah , or the linearizations of the discrete operator, Gh , for m, q ≥ 1, and the coercivity of the principal parts are essential for stability. In Subsections 8.4.2 and 8.4.3, we give the details for second order linear equations for unsymmetric and symmetric forms. The other cases we restrict to listing the equations in Subsections 8.4.4ﬀ. The domains, and ranges, e.g. the D Ahs = 2 1 H+ (Ωh ) ∩ H0,+ (Ωh ), R Ahs = L2h Ωh0 of the As , Ahs in (8.43), and the Ah in (8.47) have to be obviously modiﬁed. For nonlinear operators the domains are subsets of the corresponding Sobolev spaces in (8.43), (8.47). Conditions for the coeﬃcients are listed in Remark 8.7 and Summary 8.6. For general domains as in (8.18) with boundary grid points in (8.19) not in the boundary of Ω, speciﬁc diﬀerence methods require a separate discretization of the diﬀerential equation, and the boundary conditions. This would be necessary for fully nonlinear problems in Subsection 8.4.7, if we were to discuss the details. We indicate some other examples in Section 8.7. 8.4.2

Second order linear elliptic diﬀerence equations

We present the detailed relations between the strong and weak diﬀerential and diﬀerence equations, As , Ahs , A, Ah . We start assuming Ω in the form (8.9), and 47 For diﬀerence methods several deﬁnitions for regular elliptic systems are used in the literature. They are all more or less equivalent, cf. [151]. Here we have enforced in (8.41), towards the needs for α diﬀerence methods, the condition ϑ ∈ Rn , η ∈ Cq , ϑα for elliptic systems into ϑ ∈ Cn , η ∈ Cq , ϑ .

8.4. General elliptic equations with Dirichlet boundary conditions

575

consider the strong unsymmetric form of diﬀerence equations in divergence form. j i h , ∂+ u (x), yields one of the classical or natural diﬀerence equations. Combining the ∂− The unsymmetric methods of order 1 are modiﬁed in Subsection 8.4.3 on rectangular domains to symmetric methods of order 2. We apply partial summation (8.49) in Proposition 8.8 to diﬀerence equations in the strong form to get the weak forms. The following As is well deﬁned for u ∈ H 2 (Ω), aij ∈ 2 (Ωh ). We use (−1)j>0 = 1 for j = 0, else = −1, H 1 (Ω), cf. (2.16), and Ahs for uh ∈ H+ ¯hij (x), f¯h (x) in (8.34). and sometimes replace aij (x), f (x) by a 2 1 (Ωh ) ∩ H0,+ (Ωh ) → L2h Ωh0 , As : H 2 (Ω) ∩ H01 (Ω) → L2 (Ω), Ahs : H+ n

As u 0 =

(−1)j>0 ∂ j (aij ∂ i u0 ) = f in L2 (Ω), cf. (8.42),

(8.43)

i,j=0

Ahs uh0 (x) =

n

j i h aij (x)∂+ (−1)j>0 ∂− u0 (x) = f h (x) ∀x ∈ Ωh0 , with

(8.44)

i,j=0 1 (Ωh ). boundary conditions, cf.(8.24), (2.16), u0 ∈ H01 (Ω), uh0 (x) ∈ H0,+

(8.45)

Similarly to Table 8.1, we visualize the unsymmetric (8.44) for x ∈ Ωh0 ⊂ R2 . We use the same abbreviation as in Table 8.1, by listing the coeﬃcients of, but omitting the unknowns uh (x ± hei ). We include, here and below, only the highest order terms. i i h aii ∂+ u (x) are The three-point stars in the direction of ei , i = 1, 2, for the −h2 ∂− listed in Table 8.2. They have to be combined with the seven-point stars for the 2 1 1 2 uh (x) in Table 8.3. Summation of the numbers in Table a12 ∂+ + ∂− a21 ∂+ −h2 ∂− 8.2, i = 1, 2, and in Table 8.3 yields the seven-point star for the full (8.44) in Table 8.4. Obviously Table 8.4 reduces to Table 8.1 for a11 = a22 = 1, a12 = a21 = 0. The following dilemma applies to all our elliptic diﬀerence equations. For (8.43) we only need aij ∈ H 1 (Ω). This is certainly problematic for (8.44). Remark 8.7. 1. aij ∈ H 1 (Ω) in (8.43) is suﬃcient for analytical results: We require, for simplicity, and ﬁtting to the standard situation in diﬀerence methods, aij ∈ C(Ω), with well-deﬁned aij (x)∀x ∈ Ωh in (8.44). For interesting cases aij ∈ H 1 (Ω), . . . , ¯ij (x), as deﬁned in (8.34). This is we might replace the aij (x)∀x ∈ Ωh by a unproblematic for linear(ized) problems. With minor modiﬁcations all results remain correct. The proof for this claim is left as an exercise for the interested reader. 2. For semi-, quasi-, and fully nonlinear problems below, this averaging, necessarily deﬁned with respect to x, but for ﬁxed values of uh (x), e.g., (8.64),(8.65), would be problematic. 3. The conditions for boundedness and ellipticity of linear(ized) operators in Summary 8.6 may be appropriately modiﬁed for diﬀerence equations, and systems for higher order problems as well. In particular, generalizations to the

576

8. Finite diﬀerence methods Table 8.2: star for (8.44), x ∈ G h ⊂ R2 i Three-point h 2 i for −h ∂− aii ∂+ u (x), i = 1, 2.

−aii (x − hei )

aii (x) + aii (x − hei )

−aii (x)

2 Table star 28.3: 1Seven-point h for (8.44), x ∈ R , and 2 1 2 −h ∂− a12 ∂+ + ∂− a21 ∂+ u (x).

a21 (x − he1 ) −a21 (x − he1 ) 0

−a21 (x) (a12 + a21 )(x) −a12 (x − he2 )

0 −a12 (x) a12 (x − he2 )

Table 8.4: Seven-point star for the complete −h2 × (8.43) and x ∈ R2 .

a21 (x − he1 ) −(a11 + a21 )(x − he1 ) 0

−a22 (x) − a21 (x) a11 (x − he1 ) + a22 (x − he2 ) +(a11 + a22 + a12 + a21 )(x) −(a22 + a12 )(x − he2 )

0 −(a11 + a12 )(x) a12 (x − he2 )

W01,p (Ω), W0m,p (Ω) situation and the appropriate conditions for this and the following subsections are discussed there. The usual multiplication of As u with v ∈ H01 (Ω), integration over Ω, and partial integration yields the (standard) weak bilinear form a(·, ·), and the induced A. Both are well deﬁned for u, v ∈ H 1 (Ω) and aij ∈ L∞ (Ω), see (2.18). With Au, vH −1 (Ω)×H01 (Ω) = (As u, v)L2 (Ω) ∀u ∈ H 2 (Ω), v ∈ H01 (Ω),

(8.46)

aij ∈ H 1 (Ω)∀j > 0, n 1 aij ∂ i u0 ∂ j vdx = Au0 , vH −1 (Ω)×H01 (Ω) compute u0 ∈ H0 (Ω) : a(u0 , v) := i,j=0

Ω

= f, vH −1 (Ω)×H 1 (Ω) ∀v ∈ H01 (Ω) with f ∈ H0−1 (Ω). 0

0

Similarly to the previous strong form, Ahs , we introduce the corresponding weak operator, Ah , and weak bilinear form, ah (·, ·). Both are well deﬁned for uh , uh0 , v h ∈ 1 (Ωh ) and aij ∈ C(Ω). Choosing uh , v h as in (8.24), we determine uh0 from H+

−1 1 1 1 (Ωh ) × H0,+ (Ωh ) → R, Ah : Vbh = H0,+ (Ωh ) → H0,+ (Ωh ) = Vbh , uh0 ∈ Vbh ah : H0,+ n ah uh0 , v h := i,j=0

:

Ωh

=: Ah uh0 , v

i h j h aij ∂+ u0 ∂+ v dxh =

n

hn

i,j=0

; h

Vbh ×Vbh

i h j h aij ∂+ u0 ∂+ v (x)

x∈Ωh

1 = f h , v h V h ×V h ∀v h ∈ Vbh = H0,+ (Ωh ). (8.47) b

b

577

8.4. General elliptic equations with Dirichlet boundary conditions

Depending upon the structure of f (and of Au) we employ diﬀerent kinds of approximations. The (f, v h )L2h (Ωh ) or f, v h V h ×V h can be evaluated exactly or appropriate b b quadrature approximations may be used, cf. Section 5.4. For nonsmooth f, we may replace f (x) by f h (x) := f¯h (x) in (8.34). We denote all these approximations as f h , v h V h ×V h . For other strategies, cf., e.g. Thom´ee and Westergren [627], Heinrich b b [395], Bube et al. [151, 605] and Grossmann et al. [374–376]. Similarly to partial integration transforming diﬀerential equations in strong form into the weak form, we apply partial summation. For avoiding unnecessary repetition, we formulate Proposition 8.8 for linear and quasilinear elliptic equations of orders 2m, m ≥ 1, in (8.62), (8.66), with obvious extensions to systems. We require for v h ∈ m (Ωh ) the extension (8.24), but not for all uh considered here. H0,+ We simplify the notation for the proof, and the following results by introducing the grid points, and their function values in Ωh in the ej direction as, cf. (8.16). xi := x0 + ihej , i ∈ Z, xi ∈ Ωh0 for i = 1, . . . , k − 1, xi ∈ ∂Ωh for i ≤ 0, i ≥ k, j

ui := uh (xi ), and the Ωj := {xi , i = 1, . . . , k − 1}, Ω := {xi , i = 0, . . . , k}, h,j

j

h,j

Ωm := {xi , i = −m + 1, . . . , k + m − 1}, Ω := Ω1 , with sums, e.g.,

j h v h ∂− u

:=

k

h,j j h v h ∂− u (xi ), and obvious modiﬁcations for Ωj , Ωm , Ωh , . . . .

i=0

j

Ω

2m m (Ωh ), v h ∈ H0,+ (Ωh ), m ≥ 1, (8.48)–(8.52) are valid, Proposition 8.8. For uh ∈ H+ with obvious (v h , · · · )L2 (Ωh ) = (v h , · · · )L2h (Ωh ) , . . . in (8.49)– (8.52), hence h

for j > 0 : −

0

j h v h ∂− u =−

Ωj

⎛ ⎝

n

Ω

j h v h ∂− u =

j

Ω

⎛

j i aij ∂+ v h , (−1)j>0 ∂− u0 ⎠

=⎝

h

i,j=0

for m = 1, (8.48)

j

⎞

j h uh ∂+ v ⇒

n

⎞ j h⎠ i h aij ∂+ u0 , ∂+ v

i,j=0

2 h L2h (Ωh 0 )=Lh (Ω )

(8.49) , L2h (Ωh )

β h β h u L2 (Ωh )=L2 (Ωh ) = uh , ∂+ v L2 (Ωh ) , (8.50) for m > 1, |β| ≤ m : (−1)|β| v h , ∂− 0

h

h

β h Aβ ·, uh , ∇m ,v (−1)|β| ∂− +u ⎛ ⎝

L2h (Ωh 0 )=...

h

β h h , ∂+ v L2 (Ωh ) ⇒ = Aβ ·, uh , ∇m +u h

⎞

⎛

β α aαβ ∂+ v h , (−1)|β| ∂− u0 ⎠

=⎝

|α|,|β|≤m

h

h

L2h (Ωh 0 )=..

⎞ β h⎠ α h aαβ ∂+ u0 , ∂+ v

|α|,|β|≤m

L2h (Ωh )

(8.51)

578

8. Finite diﬀerence methods

⎛

⎝

h

⎞

β ⎠ v h , (−1)|β| ∂− Aβ ·, uh0 , · · · , ∇m + u0

|β|≤m

2 h L2h (Ωh 0 )=Lh (Ω )

⎛

=⎝

⎞

h

β h⎠ Aβ ·, uh0 , · · · , ∇m + u0 , ∂+ v

|β|≤m

.

(8.52)

L2h (Ωh )

Equations (8.48) ﬀ. remain correct if ∂+ , and ∂− are interchanged. Remark 8.9. 1. According to (8.48)–(8.52), the following weak forms of the diﬀerence methods use m m ·, · L2 (Ωh ) . The unequal previous condition uh ∈ H+ (Ωh ), m ≥ 1, v h ∈ H0,+ (Ωh ) h

for uh , v h is important, since we apply (8.48)–(8.52) in the diﬀerence methods to very diﬀerent terms uh , cf. (8.44), Proposition 8.10, and (8.56),(8.65). 2. These results remain correct, if we allow uh ∈ W+m,p (Ωh ), 2 ≤ p < ∞, and h ∈ for the linear case v h and the quasilinear case ∀uh : Aβ ·, uh , · · · , ∇m +u

−m,p W0,+ (Ωh ), 1/p + 1/p = 1, respectively

Proof. We start with the ﬁrst order result (8.48) for j > 0. Since the proofs for j h j h j h 1 u , and ∂− u are very similar, we only consider ∂− u . We use H0,+ (Ωh ) v h , ∂+ h h h h h v (x) = 0∀x ∈ G \ Ω0 , only for v , but not for u . j>0:−

j h v h ∂− u =−

Ωj

Ω

=−

j h v h ∂− u

(8.53)

j

k

vi (ui − ui−1 )/h = −

i=0 v0 =vk =vk+1 =0

=

k

k

vi u i −

(vi+1 − vi )ui /h =

vi +1 ui

/h

i =−1

i=0

i=0

k−1

j h uh ∂+ v .

j

Ω

h Replacing the previous uh by wh := aj ∂+ u we obtain with v0 = vk = vk+1 = 0 j>0:

−

Ωj

j h j h h aj ∂+ aj ∂+ v h ∂− u = u ∂+ v . Ω

(8.54)

j

For (8.49), we multiply (8.54) with hn and sum over all 0 ≤ i = , j ≤ n. The ﬁrst line of (8.50) with 1 < m ≥ |β| ≥ 1 is proved inductively. Equation (8.54) with m > 1 ≥ |α|, |β| = 1 represents the beginning of the induction. So we continue with considering the components βj ≥ 2, and the sum over the lines Ωj . In (8.53), we j h replace, for the next step of the induction, the v h , and Ωj , and Ω by, e.g. ∂+ v , and

8.4. General elliptic equations with Dirichlet boundary conditions j

579

h,j

Ω , and Ω2 , respectively. We obtain with 0 = ∂+ v−1 = ∂+ vk+1 : β −1 β −1 h βj h h j βj −1 h ∂−j ui := ∂−j u(xi ) : − ∂+ v ∂− u = − ∂+ v ∂− ∂− u Ω

j

(8.55)

h,j

Ω2

=−

k+1

β −1

j ∂+ vi ∂− ∂−j

ui

i=−1

=−

k+1

β −1 ∂+ vi ∂−j ui

−

i=−1 j 0=∂+ v−1 =∂+ vk+2

=

=

k+1

k+1

β −1 ∂+ vi ∂−j ui−1

i=−1 ∂+ vi+1

−

∂+ vi

/h

β −1 ∂−j ui

/h

i=−1

β −1 j ∂+ ∂+ v h ∂−j uh .

h,j

Ω2

This allows reducing inductively each component βj > 0 in (8.55) to 0, and proceeding m is suﬃcient for the ﬁrst line of (8.50). as above. Then v h ∈ H0,+ For the second line of (8.50), the quasilinear equations of order 2m in (8.66) below, we obtain with j β j βj βj h = ∂− ∂− , wh := (−1)|β | ∂− Aβ ·, uh (·), . . . , ∇m β j := β − ej , ∂− + u (·) :

β h v h (−1)|β| ∂− Aβ ·, uh (·), . . . , ∇m u (·) + Ω

j

= −

j j βj h (−1)|β | ∂− v h ∂− Aβ ·, uh (·), . . . , ∇m + u (·)

j

Ω

= −

j h v h ∂− w =−

Ωj

= − =

Ω

j

k

Ω

vi wi −

i=0 j h wh ∂+ v =

k i=0

k

vi (wi − wi−1 )/h

i=0

j

vi wi−1

Ω

j h v h ∂− w =−

/h

0=v0 =vk+1

=

=

k

wi (vi+1 − vi )/h

i=0

j h j βj h (−1)|β | ∂− Aβ ·, uh (·), . . . , ∇m ∂+ v . + u (·)

j

Again, we reduce |β| to 0. Now multiplying the ﬁrst and second line of (8.50) with hn , and summation over all directions yields (8.51) and (8.52). We reformulate the results in Proposition 8.8, (8.49), (8.51) now for the weak and strong diﬀerence operators, Ahs , and Ah , and extend them to systems.

580

8. Finite diﬀerence methods

Proposition 8.10. For the diﬀerence operators Ahs , Ah in (8.44), (8.47) for m = 1 m (Ωh , Rq ) : and (8.62), (8.63), (8.71) for m > 1, q ≥ 1 we obtain ∀v h ∈ Vbh = H0,+ ; : h h h; : As u , v L2 (Ωh ,Rq ) = Ahs uh , v h L2 (Ωh ,Rq ) = Ah uh , v h V h ×V h . (8.56) b b 0 Equation (8.56) remains correct for the quasilinear problems Ghs , and Gh in (8.65), (8.67), and linear and quasilinear systems (8.71) and (8.72), (8.73), cf. Remark 8.9 h h Exchanging the two diﬀerence operators ∂+ , and ∂− in the previous (8.44) (8.47), and all the following cases, allows the same convergence results. In the symmetric form below we will use the mean value of these two equations. Equation (8.44) is a classical system of diﬀerence equations for (8.43), and has to be solved. It is, possibly with slight modiﬁcations, included in the usual software packages. For the corresponding weak diﬀerence form (8.47) of (8.44), and its generalizations below, a direct stability, and convergence proof with respect to the · H+1 (Ωh ) will be given in Subsections 8.5.4 ﬀ. This implies the results for (8.44) as well. In fact, a strong solution of (8.44) satisﬁes, by (8.56), the weak (8.47) as well, and vice versa: 2 1 (Ωh ) ∩ Vbh = H0,+ (Ωh ) : Ahs uh0 , v h 2 h = (f h , v h ) 2 h ∀v h ∈ Vbh uh0 ∈ H+ Lh Ω0

f ∈ L2 (Ω) =⇒ uh0 ∈ Vbh :

:

Ah uh0 , v

; h

Vbh ×Vbh

Lh Ω0

= (f h , v h )L2h (Ωh ) ∀v h ∈ Vbh .

(8.57)

Hence we solve (8.44) by standard software packages. This solution inherits the convergence properties of the weak solution, uh0 , of (8.47), cf. Theorem 8.11. If a discrete solution uh0 (x)∀x ∈ Ωh for (8.44) or (8.47) with the boundary conditions (8.45) is computed, it is straightforward to determine its ﬁrst and second divided j i h i h 2 u0 (x), ∂+ ∂+ u0 (x), yielding the desired uh0 ∈ Vbh or uh0 ∈ H+ (Ωh ) ∩ Vbh . diﬀerences, ∂+ h h m h This remains correct for order 2m, and the solutions u0 ∈ Vb = H0,+ (Ω , Rq ), and 2m uh0 ∈ H+ (Ωh , Rq ) ∩ Vbh , m, q ≥ 1, of the weak, and strong diﬀerence problems in the following subsections. We use the same approach as in the previous chapters, related to Stetter [596]. This contrasts to, e.g. Raviart [545], Temam [621] and Schumann and Zeidler [574, 678]. They apply external approximations. ˆh0 indicate We want to outline the diﬀerences in alternative approaches. Let u ˘h0 and u ˘h0 = the solutions of the strong and the weak problem, respectively. For the strong Ahs u h f (= f ) with special A, often stability results are obtained via M-matrices, see, e.g. Hackbusch [387], Sections 4.2 ﬀ. This yields estimates of the form ˘h0 (x) = f (x)∀x ∈ Ωh0 , u ˘h0 (x) = 0∀x ∈ ∂Ωh Ahs u h h =⇒ sup u ˘0 (x) = u ˘0 L∞ (Ωh ) ≤ Cf L∞ h . h (Ω ) x∈Ωh

(8.58)

h

The approach, based upon the weak Au0 = f, starts with the coercive bilinear form, ap (·, ·), of the principal part, Ap , of the above general A. We will show that its discrete counterpart ahp (·, ·), cf. Summary 8.6, is again coercive, so stable as well. Thus the consistency, and Theorem 3.29, imply, for a boundedly invertible A, the stability of

581

8.4. General elliptic equations with Dirichlet boundary conditions

Ah , hence the existence of a weak solution u ˆh0 = u ˘h0 by (8.57), such that h ˆh0 (x) = 0 ∀ x ∈ ∂Ωh : Ahs u ˆh0 = f h ⇐⇒ (8.59) for f h ∈ L∞ h (Ω ), and for u : h h h; 1 A u ˘h0 , v h = f h , v h V h ×V h ∀v h ∈ Vbh = H0,+ ˆ0 , v V h ×V h = ah u (Ωh ), and

h u ˆ0

b

1 (Ωh ) H+

b

b

h = u ˆ0

Vbh

≤ Cf h V h b

b

h h =⇒ u ˘0 V h ≤ C f h L∞ h ≤ Cf h , V h (Ω ) b

b

is dominated by f h V h . This argument from (8.57) to (8.59) since f h L∞ h h (Ω ) b

remains correct for the following linear cases as well, if only the ˘ uh0 H+1 (Ωh ) are h replaced by ˘ u0 H+m (Ωh ,Rq ) , m, q ≥ 1. For quasilinear problems corresponding stability– consistency arguments apply. Fully nonlinear problems require another treatment. m convergence for classical (strong) diﬀerence methods: Theorem 8.11. H+ −m ˆh0 = f h ∈ H0,+ We assume a stability estimate (8.59) for the weak Ah u (Ωh , Rq ), m, q ≥ 1, in the form ˆ uh0 H+m (Ωh ,Rq ) ≤ Cf h H −m (Ωh ,Rq ) . Let A : H0m (Ω, Rq ) 0,+

→ H0−m (Ω, Rq ), and As : H 2m (Ω, Rq ) ∩ H0m (Ω, Rq ) → L2 (Ω, Rq ) ⊃ L∞ (Ω, Rq ) be for the solution boundedly invertible. This implies ˘ uh0 H+m (Ωh ,Rq ) ≤ Cf L∞ h q h (Ω ,R ) h h h ∞ h q of As u ˘0 = f ∈ Lh (Ω , R ) as well, strongly improving (8.58). is a special case of regularity results for This ˘ uh0 H+m (Ωh ,Rq ) ≤ Cf L∞ h q h (Ω ,R ) the solutions of diﬀerence equations in the style of, e.g. Bube et al. [151, 605], F¨ ossmeier [320], Geissert [339], Hackbusch [381, 383, 384, 387] and Thom´ee and Westergren [627]. Other concepts of generalized weak solutions for diﬀerence equations are studied by Jovanovich et al. [416,430–432,613], and the viscosity solutions for nonlinear problems by Crandall and Lions [217] ﬀ., cf. Subsection 8.4.7

8.4.3

Symmetric diﬀerence methods

After our excursion to linear and quasilinear equations, and systems of orders 2 and 2m, we return to linear equations of order 2, and their symmetric diﬀerence equations. The symmetric forms are particularly interesting, since the discretization error for the diﬀerence equations is O(h2 ), compared to O(h) for unsymmetric forms. The above i by ∂hi does not yield usable symmetric discussion has shown that replacing the ∂± forms. However, for the previous unsymmetric forms a simple modiﬁcation yields the desired symmetric forms. This idea is identically applicable to all the following cases. Only the boundary conditions for higher order equations have to be modiﬁed, cf. Subsection 8.4.4. We deﬁne the strong and the corresponding weak symmetric discrete operators Ahss , and Ahws , and the bilinear form ahws (·, ·) as the mean of two diﬀerence

582

8. Finite diﬀerence methods

i i equations, exchanging ∂+ , and ∂− :

Ahss uh (x) := ahws (uh , v h ) :=

n j 1 j i h i h (−1)j>0 ∂− (aij (x)∂+ u (x)) + ∂+ (aij (x)∂− u (x)) , 2 i,j=0

(8.60)

n ; : 1 i h j h i h j h aij (∂+ u ∂+ v + ∂− u ∂− v )dxh =: Ahws uh , v h V h ×V h . b b 2 i,j=0 Ωh

We turn to the boundary conditions. The error for the diﬀerence equations, of O(h2 ), only makes sense if the boundary conditions are discretized equally accurately or better. For the cuboidal domains Ω in (8.9) this is straightforward: we choose Ωh , Ωh0 , ∂Ωh as in (8.11). Then the boundary points x ∈ ∂Ωh satisfy x ∈ ∂Ω, hence, allow an exact discretization uh (x) = 0 ∀x ∈ ∂Ωh . For curved domains the techniques in Section 8.7 apply. So, based upon Proposition 8.10, and this boundary condition we determine uh0 ∈ Vbh : Ahss uh0 (x) = f (x)∀x ∈ Ωh0 ⇐⇒ : h h h; Aws u0 , v V h ×V h = f, v h V h ×V h ∀v h ∈ Vbh . b

b

b

(8.61)

b

Only the highest order divided diﬀerences in (8.60), so i, j = 1, . . . , n, are required in this special form. For the lower order terms with i · j = 0 we may choose any of the symmetric formulas, e.g. as in (8.13). In fact, they do satisfy the conditions in Proposition 8.4, thus the Vbh -coercivity of the principal part is maintained. We visualize the symmetric Ahss in (8.60), starting with the three-point stars in Table 8.5, the seven-point stars in Tables 8.6, 8.7, and the ﬁnal seven-point star in Table 8.8. Note that the factor 1/2 in front of every table represents the factor 1/2 in (8.60). Furthermore we deﬁne, in Tables 8.5–8.7, several a± ij used in Table 8.8. i h i i i u (x). Table 8.5: Three-point star for (8.60) for −h2 ∂− aii ∂+ + ∂+ aii ∂−

1/2

−a− ii := −aii (x) − aii (x − hei )

+ a− ii + aii

−a+ ii := −aii (x) − aii (x + hei )

2 1 2 1 Table 8.6: Seven-point star for (8.60), and −h2 (∂− a12 ∂+ + ∂+ a12 ∂− )uh (x).

1/2

a12 (x + he2 ) −a12 (x) 0

−a+ 12 := −a12 (x + he2 ) 2a12 := 2a12 (x) −a− 12 := −a12 (x − he2 )

0 −a12 (x) a12 (x − he2 )

1 2 1 2 Table 8.7: Seven-point star for (8.60), and −h2 (∂− a21 ∂+ + ∂+ a21 ∂− ) uh (x).

1/2

a− 21 := a21 (x − he1 ) −a21 (x − he1 ) 0

−a21 (x) 2a21 := 2a21 (x) −a21 (x)

0 −a21 (x + he1 ) a+ 21 := a21 (x + he1 )

8.4. General elliptic equations with Dirichlet boundary conditions

583

Table 8.8: Seven-point star for the complete −h2 × (8.60) and x ∈ Ωh0 ⊂ R2 .

1/2

− a+ 12 + a21 − a12 − a− 21

+ −a+ 22 − a12 − a21 − − + a11 + a+ 22 + a22 +2a12 + 2a21 − −a− 22 − a12 − a21

−a− 11

a+ 11

0

8.4.4

0 + −a+ − a 12 − a21 11 + a− 12 + a21

Linear equations of order 2m

For these equations we need the higher order Dirichlet boundary conditions mentioned m (Ωh ) above. So we assume the discrete boundary values in the form uh ∈ Vbh = H0,+ m h as in (8.28), and the same for H0,− (Ω ). Symmetric formulas only are worthwhile for cuboidal domains or for the special methods in Section 8.7. We formulate the equations of order 2m: for aαβ ∈ W |β|,∞ (Ω), f ∈ L2 (Ω), and aαβ ∈ C |β| (Ω), f ∈ C(Ω), cf. Summary 8.6, and Remark 8.7, we determine u0 , and uh0 , respectively: 2m m u0 ∈ H 2m (Ω) ∩ (Vb := H0m (Ω)), and uh0 ∈ H+ (Ωh ) ∩ Vbh := H0,+ (Ωh ) s.t. As u0 =

(−1)|β| ∂ β (aαβ ∂ α u0 ) = f ∈ L2 (Ω), and

|α|,|β|≤m

Ahs uh0 (x) =

β α h aαβ (x)∂+ (−1)|β| ∂− u0 (x) = f (x) ∀x ∈ Ωh0 .

(8.62)

|α|,|β|≤m

Similarly as above we formulate, for aαβ ∈ L∞ (Ω), the weak forms a(u, v), Au, and, for aαβ ∈ C(Ω), the ah (uh , v h ), Ah uh , cf. (8.46), (8.47), and compute u0 ∈ Vb , and uh0 ∈ Vbh for f ∈ Vb from a(u0 , v) := Au0 , vV ×Vb := b

Ω |α|,|β|≤m

uh0 ∈ Vbh : ah uh0 , v h := hn

aαβ ∂ α u0 ∂ β vdx = f, vV ×Vb ∀v ∈ Vb , b

α h β h aαβ ∂+ u0 ∂+ v (x)

x∈Ωh |α|,|β|≤m

; : =: Ah uh0 , v h V h ×V h b

b

= f h , v h V h ×V h b

= hn

b

β h fβh ∂+ v (x)∀v h ∈ Vbh ,

x∈Ωh |β|≤m

(8.63)

584

8. Finite diﬀerence methods

cf (2.107). The form for f in (8.63) is unusual in classical diﬀerence methods, cf. (2.110). However it makes sense for the weak form, and the later error estimates. Generalizations to Banach spaces with aαβ ∈ W ∞ (Ω), u, u0 , v ∈ W m,p (Ω), 2 ≤ p < ∞ or aαβ ∈ W p1 (Ω), u, u0 ∈ W m,p2 (Ω), v ∈ W m,p3 (Ω), with 1/p1 + 1/p2 + 1/p3 = 1 m,p (Ωh ) are possible, similarly to Section 2.3 and the corresponding discrete spaces W0,+ and Subsection 2.4.4, considering the corresponding coercivity problems. This allows m (Ωh ) norm. results for the discrete H+ As in (8.60) symmetric diﬀerence equations can be deﬁned, again only for cuboidal domains Ω, cf. (8.9), and ∀x ∈ Ωh0 or the methods in Section 8.7. 8.4.5

Quasilinear elliptic equations of orders 2, and 2m

According to the theoretical results we assume homogeneous Dirichlet boundary conditions of the form (8.45) or (8.62). Even more general Dirichlet systems, see (2.82), Lemma 2.26, (2.158), (2.159), and Theorem 2.50 can be transformed into the standard homogeneous Dirichlet boundary conditions. Furthermore, the ﬁrst types of nonlinear elliptic equations in Section 2.5, Subsection 2.5.3, are either special quasilinear equations, or simply the term a00 u in (8.62) has to be replaced by f (u). So we only give the diﬀerence equations corresponding to a general quasilinear form, with the boundary conditions in (8.45) or (8.62), compare (2.270), (2.306). The inhomogeneous function f is included in the nonlinear Aj and Aα . Appropriate conditions for the unique existence of solutions are listed in Subsections 2.5.4 and 2.5.6. 1,p 2,p These strong quasilinear equations are

deﬁned for u, u0 ∈ W (Ω) ∩ W0 (Ω), and 1,p (Ωh ) . Applying (8.49) to (8.64) we get the weak uh , uh0 ∈ W+2,p (Ωh ) ∩ Vbh := W0,+

form Gh , cf. (8.65), deﬁned for uh , uh0 , v h ∈ Vbh , and 1/p + 1/p = 1:

Gs u 0 =

n

(−1)j>0 ∂ j Aj (·, u0 , ∇u0 ) = 0 ∈ L1/p (Ω),

(8.64)

j=0

u0 ∈ W 2,p ∩ W01,p (Ω), Ghs uh0 (x) =

n

j (−1)j>0 ∂− Aj ·, uh0 , ∇+ uh0 (x) = 0∀x ∈ Ωh0 , and

(8.65)

j=0

:

Gh uh0 , v h

;

Vbh ×Vbh

:= hn

n

j h Aj ·, uh0 , ∇+ uh0 (x)∂+ v (x)

x∈Ωh j=0

1,p (Ωh ) with 2 ≤ p < ∞ := ah uh0 , v h = 0 ∀ v h ∈ Vbh = W0,+ uh0 ∈ Vbh , ∀uh ∈ Vbh ∩ D(Aj ) : Aj (·, uh , ∇+ uh )

−1,p ∈ Vbh = W0,+ (Ωh )

8.4. General elliptic equations with Dirichlet boundary conditions

585

for the weak form. Notice that ah (uh , v h ) here, and for the following nonlinear problems, is linear and bounded in v h , but not linear in uh . For the analysis of linearized quasilinear problems we will need general Sobolev spaces W 2m,p (Ω) ∩ W0m,p (Ω) with 1 ≤ p ≤ ∞, and their discrete counterparts. According to Theorem 2.122 this linearization essentially requires the H0m (Ω)-coercivity of the principal part, cf. the end of Subsection 8.4.4. For our discretization approach via linearization, we again restrict p from 1 < p < ∞ to 2 ≤ p < ∞ in v h ∈ Vbh = 1,p −1,p m,p (Ωh ) and Aj (·, uh , ∇+ uh ) ∈ Vbh = W0,+ (Ωh ) or v h ∈ W0,+ (Ωh ). If we employ W0,+ the monotone approach, 1 < p < ∞ is still possible, cf. Section 4.5 and Theorem 8.32. The following strong equations Ghs of order 2m are transformed, by iteratively applyand ing (8.49), into the weak form Gh . Again we consider trivial Dirichlet conditions, m,p (Ωh ) require u, u0 ∈ W 2m,p (Ω) ∩ W0m,p (Ω), and uh , uh0 ∈ W+2m,p (Ωh )∩ Vbh := W0,+ for the strong, and uh , uh0 , v h ∈ Vbh for the weak form, cf. (8.17) for ∇m + u, then (−1)|β| ∂ β Aβ (·, u0 , · · · , ∇m u0 ) = 0 ∈ L2 (Ω), (8.66) Gs u 0 = |β|≤m

Ghs uh0 (x) =

β h h (−1)|β| ∂− Aβ ·, uh0 , · · · , ∇m + u0 (x) = 0 ∀x ∈ Ω0 , and

|β|≤m

:

Gh uh0 , v

; h

Vbh ×Vbh

= hn

β h h Aβ ·, uh0 , · · · , ∇m + u0 (x)∂+ v (x)

(8.67)

x∈Ωh |β|≤m

m,p (Ωh ) with 2 ≤ p < ∞ := ah uh0 , v h := 0 ∀ v h ∈ Vbh = W0,+ −m,p h ∈ Vbh = W0,+ (Ωh ). uh0 ∈ Vbh , ∀uh ∈ Vbh ∩ D(Aj ) : Aj ·, uh , . . . , ∇m +u Schumann and Zeidler [574, 678] prove for these equations convergence of order 1 via “external approximation schemes”, so totally diﬀerently from our approach. We update Proposition 8.10 with Proposition 8.11, to the quasilinear case: Proposition 8.12. For the diﬀerence operators Ghs , and Gh in (8.65)–(8.67) we m,p (Ωh ), hence v h ∈ W 2m,p (Ωh ) as well, cf. Remark 8.9: obtain ∀v h ∈ Vbh = W0,+ ⎛ ⎞ β h ⎠ dxh (8.68) v h Ghs uh dxh = vh ⎝ (−1)|β| ∂− Aβ ·, uh (·), . . . , ∇m + u (·) Ωh 0

Ωh

=

|β|≤m

Ωh |β|≤m

β h h h Aβ ·, uh (·), . . . , ∇m + u (·) ∂+ v dx

; : = Gh uh , v h V h ×V h . b

b

For the weak form we require uh , v h ∈ W+m,p (Ωh ), 2 ≤ p < ∞, and −m,p h (Ωh ), 1/p + 1/p = 1. ∀uh : Aβ ·, uh0 , · · · , ∇m + u0 ∈ W0,+ For the symmetric diﬀerence equations we proceed as in (8.68).

586 8.4.6

8. Finite diﬀerence methods

Systems of linear and quasilinear elliptic equations

We formulate the diﬀerential and diﬀerence systems only for order 2m, cf. (2.391), (2.306). The transition from order 2m back to second order is obvious from (8.65)– (8.67). The main reason is that for systems of order 2m the proof for coercivity requires a modiﬁcation, based upon the linearized forms. Preparing the quasilinear problems, we formulate the linear problems in the Sobolev setting W+2m,p (Ω, Rq ). We maintain the notation in Chapter 2, e.g. (2.389), and formulate the necessary diﬀerence modiﬁcations compared to (8.17), e.g. as l h α h l h α h l h α h u := ∂+ u1 , · · · , ∂+ uq , ∂+ u := ∂+ u1 , · · · , ∂+ uq , (8.69) uh = uh1 , · · · , uhq , ∂+ α h α h ∇≤k uh := ∂+ u ∀|α| ≤ k , ∇k+ uh := ∂+ u ∀|α| = k , + l h α h u := ∂+ u := uh for k = l = |α| = 0, and with ∇+ uh := ∇1+ uh , ∇k+ uh := ∂+ l h u (x) ∈ Rq , ∇+ uh (x) ∈ Rq×n , ∇k+ uh (x) ∈ Rq×nk , and uh (x), ∂+ j = ϑ1 , . . . , ϑn ∈ Rn , j = 1, . . . , q, ϑ l = ϑl , . . . , ϑl ∈ Rq , l = 1, . . . , n, with ϑ j j 1 q

1, · · · , ϑ := ϑ n = (ϑ q ) ∈ Rn×q , and|ϑ| = |ϑ| nq ∈ R, 1, · · · , ϑ ϑ

α )|α|=k ∈ Rq×nk , Θ α )|α|≤k . α := ϑα , · · · , ϑα ∈ Rq , and Θ k := (ϑ ≤k := (ϑ ϑ 1 q Correspondingly we deﬁne, e.g. the ∇k− uh , ∇kh uh . Then for the strong linear diﬀerential and diﬀerence systems, and the weak diﬀerence systems we determine the solu2m,p (Ω, Rq ) ∩ W0m,p (Ω, Rq ), tions u0 , and uh0 from (8.70), (8.71). We choose h u0 ∈ hW 2m,p m,p h h q h h q u0 ∈ W+ (Ω , R ) ∩ Vb := W0,+ (Ω , R ) , v ∈ Vb , for the strong, and delete the W+2m,p (Ω, Rq ) for the weak problems. The conditions for the Nemyckii operators, cf. Subsection 2.5.5, Aαβ in (8.70), and Aβ in (8.72), and their second order form, Akl , and Ak , are discussed in Subsections 2.6.3–2.6.6, and are summarized for diﬀerence methods in Summary 8.6. (−1)|β| ∂ β (Aαβ ∂ α u0 ) ∈ Lp (Ω, Rq ), Aαβ (x) ∈ Rq×q , (8.70) f = As u0 = |β|,|α|≤m

1/p + 1/p = 1, uh0 ∈ Vbh ∩ W+2m,p (Ωh , Rq ), f h ∈ Lp+ (Ωh , Rq ), and β α h Aαβ (x) ∂+ (−1)|β| ∂− u0 (x) ∀ x ∈ Ωh0 f h (x) = Ahs uh0 (x) = |β|,|α|≤m

f h , v h V h ×V h = hn b

b

ah uh0 , v h :=

α h v f αh , ∂+

x∈Ωh |α|≤m

Ωh |β|,|α|≤m

q

(8.71)

; : (x) = Ah uh0 , v h V h ×V h :=

β h α h Aαβ ∂+ u0 , ∂+ v

b

b

q

−m,p dxh ∀ v h ∈ Vbh , f h ∈ W0,+ (Ωh , Rq ).

The quasilinear versions with f included in the Aβ are deﬁned, and the solutions, u0 , uh0 , are computed in D(G), D(Gh ), subsets of the same spaces as in (8.70), (8.71),

587

8.4. General elliptic equations with Dirichlet boundary conditions

see (2.424), (2.425), so Gs u0 :=

(−1)|β| ∂ β Aβ (·, ∇≤m u0 ) = 0 ∈ Lp (Ω, Rq ),

and

(8.72)

|α|≤m

Ghs uh0 (x) :=

β h h (−1)|β| ∂− Aβ ·, ∇≤m u 0 (x) = 0, ∀x ∈ Ω0 , +

|β|≤m

: ; m,p (Ωh , Rq ) (8.73) ah uh0 , v h := Gh uh0 , v h V h ×V h with Vbh := W0,+ b b

β h Aβ ·, ∇≤m uh0 , ∂+ v dxh = 0 ∀ v h ∈ Vbh . := + q

Ωh |β|≤m

Again we proceed as in (8.60) for symmetric diﬀerence equations. 8.4.7

Fully nonlinear elliptic equations and systems

For the fully nonlinear equation (8.74) in general no weak formulations are possible, cf. Subsections 2.5.7, 2.6.8, and Section 5.2. So, as for FEMs in (5.2) ﬀ., we formulate only the strong form of the diﬀerence methods. As for FEMs, stability is proved via the detour to the linearized operator A = G (u0 ). It requires regularity results of the form Au = f ∈ L2 (Ω) =⇒ u ∈ H 2m (Ω) =: U. These results are available for equations of order 2m for boundaries ∂Ω ∈ C 2m or ∂Ω ∈ C ∞ , cf. Theorems 2.79 ﬀ. For equations and systems of order 2 this is known for convex polygonal boundaries, cf. Kozlov et al. [452]. We consider essentially unsymmetric diﬀerence methods for curved boundaries of second order equations, cf. (8.76). For special cuboidal domains as in (8.9), symmetric methods are formulated at the end of this subsection. For G in (8.76) we ﬁnd its derivative in the form (8.43). We restrict the discussion to order 2m = 2. Crandall et al., e.g. [216, 217] discuss the concept of viscosity solutions for fully nonlinear uniformly elliptic, e.g. [216], and parabolic equations of second order using speciﬁc monotonicity properties of the operator. They extend these concepts to diﬀerence methods on equidistant grids and their convergence to the exact solutions. Kuo and Trudinger [456] consider discrete versions of maximum principles, H¨ older estimates and Harnack inequalities for linear elliptic diﬀerence operators of positive type and generalize them to fully nonlinear problems on equidistant and moderately nonequidistant grids. Relations between discrete and continuous elliptic equations, but no convergence results, are discussed. We compute u0 (x), uh0 (x) for (8.74), (8.76), cf. (8.45), Gs : D(G) ⊂ Ub := H 2m (Ω) ∩ H0m (Ω) → V := L2 (Ω),

(8.74)

Gs u0 := G(·, u0 , ∇u0 , ∇2 u0 ) = 0 ∈ L2 (Ω), with u0 (x) = 0 on ∂Ω, and Ghs

: D(Ghs ) ⊂ Ubh

:=

2m H0,+ (Ωh ) →

V := L (Ω h

2

h

), Ghs uh0 :=

(8.75)

G(., uh0 , ∇+ uh0 , ∇2+ uh0 )|Ωh0

= 0,

(8.76) uh0 (x)|∂Ωh = 0 h h h ⇐⇒ Gs u0 , v L2 (Ωh ) = 0 ∀v h ∈ V h , uh0 |∂Ωh = 0, hence u0 (x) ∈ Ub , uh0 (x) ∈ Ubh .

588

8. Finite diﬀerence methods

We do not reformulate the highly technical proof for stability, consistency and convergence for fully nonlinear problems for the ﬁnite diﬀerence form (8.76). We only indicate the necessary steps, strongly analogous to the FEM approach: For special polyhedral, e.g. cuboidal, domains in (8.9) the boundary conditions are satisﬁed exactly, otherwise they are violated. Correspondingly, (8.74) is discretized as (8.76) for (8.9). The general discretization approach in Subsection 5.2.4 is then reduced to elaborating the diﬀerence operator, our Gh , the F1h = Gh in (5.41). Consequently, we only need projectors for the G component and no extension operators for the consistency. This strongly simpliﬁes the analogue of the proof for Theorem 5.4. For the other domains we again need the full machinery of discretization errors for diﬀerential and boundary operators Gh , B h as for FEMs in Section 5.2. In the proof of stability via the linearized operator, G (u0 ), we use the equality of weak and strong discrete bilinear forms induced by G (u0 ), cf. Proposition 8.10, (8.56). We obtain the regularity of the diﬀerence solutions, as in Lemma 5.10, and Hackbusch [387], Theorem 9.2.26. The corresponding convergence results are summarized in Theorem 8.31 below. This diﬀerence method avoids the main disadvantage of the preceding FEMs, the need of C 1 FEs. These require necessarily piecewise polynomials of higher degree, e.g. ≥ 5 for n = 2 and ≥ 9 for n = 3. For symmetric diﬀerence equations, we slightly modify (8.60). The boundary conditions are formulated in (8.19), (8.20). So we require: 1 G x, uh0 , ∇+ uh0 , ∇2+ uh0 + G x, uh0 , ∇− uh0 , ∇2− uh0 (x) = 0 ∀x ∈ Ωh0 . Ghss uh0 (x) := 2 Extensions of these results to equations and systems of order 2m, and to curved boundaries ∂Ω ∈ C 2m can be formulated as in the preceding subsections. A totally diﬀerent approach is due to Crandall and Lions. They developed, in a series of papers, the theoretical tools and full convergence proofs for diﬀerence methods for quasi- and fully nonlinear elliptic and parabolic equations [217, 226, 227], and with Kocan and Swiech in [216]. The basis is a concept of generalized weak solutions, the viscosity solutions; for a user’s guide see Crandall et al. [215]. For other types of results Kuo and Trudinger combine nonlinear elliptic and parabolic diﬀerence equations of positive type with their concepts of weak solutions, local, and Schauder, Harnack and H¨ older estimates, and a discrete local maximum principle, see [456–460].

8.5

Convergence for diﬀerence methods

Here we apply our general discretization theory from Chapter 3 to the diﬀerence methods formulated in the last chapter. As mentioned, we have only studied methods of orders 1 and 2. We formulate the Richardson extrapolation and the defect correction methods, presented in Section 8.8, as an appropriate strategy for developing high order m (Ωh ) norms methods as well. The obtained convergence results with respect to the H+ is a special case of the regularity results for diﬀerence methods mentioned at the end of Subsection 8.4.2. Important in this context are exact conditions for the linearized operators, collected in Summary 8.6 and in Remark 8.7. From the many papers for

8.5. Convergence for diﬀerence methods

589

general diﬀerence equations we only mention Borz`ı et al. [134] and Vainikko with Tamme [646, 646–649]. −m,p (Ωh ), we often do Although W −m,p (Ω) = W0−m,p (Ω) and W+−m,p (Ωh ) = W0,+ −m,p not distinguish them. The restriction, e.g. of f ∈ W (Ω) yields an f |W −m,p (Ω) ∈ 0

W0−m,p (Ω). 8.5.1

Discretization concepts in discrete Sobolev spaces

We brieﬂy summarize the general discretization theory in Chapter 3, cf. Summary 4.52 as well. Our approach is much more similar to the FEM or DCGM form than to the earlier combination of strong forms of the diﬀerence equations with M -matrices. In contrast, we discuss here stability and convergence for the weak forms, cf. Proposition 8.11. These variational diﬀerence methods yield results that are usually stronger than M -matrix results. These ideas have been applied to several problems, e.g. by Samarskii [564, 565] and Temam [621]. However, we simplify the major part of the following proof for convergence. We restrict the full presentation to the case of discretely exactly reproduced boundary conditions. This is satisﬁed for the cuboidal domains Ω in (8.9), and second order elliptic problems, hence m = 1, q ≥ 1. The extension to general Ω in (8.18), and m > 1, q ≥ 1, can be proved by modifying the FE approach for fully nonlinear problems in Section 5.2, and Subsection 5.2.4. We would have to replace the diﬀerential operator A or G by the two-component operator, F, including the boundary operator, B, thus F = (A, B) or F = (G, B), e.g. F = (A, B) : H01 (Ω) → H0−1 (Ω) × H 1/2 (∂Ω). This allows verifying the consistency of the “nonconforming ﬁnite diﬀerence method” as in Subsection 5.2.4, and the stability by combining the techniques in Subsections 8.5.5 and 5.2.7. This technique will be applied to the natural boundary conditions in Section 8.6 as well. For the diﬀerence methods for curved domains in Section 8.7 totally diﬀerent techniques are necessary. These proofs will be omitted there. In the last section we have transformed the strong (classical) form of diﬀerence equations into the corresponding weak form except for fully nonlinear problems. For the case of trivial Dirichlet conditions we need lines 1, 2 of (8.76), for the strong and weak form the next two lines for their weak discretization.

U = H 2m (Ω), or W 2m,p (Ω), or H 2m (Ω, Rq ), and V = Lp (Ω), . . . ,

(8.77)

Ub = U ∩ H0m (Ω), W0m,p (Ω), Vb = Lp (Ω), for m, q ≥ 1, 1 ≤ p ≤ ∞, m U h = V h = H+ (Ωh ), W+m,p (Ωh ), V

h

=U

h

= W+−m,p (Ωh ),

m,p −m,p m Vbh = Ubh = H0,+ (Ωh ), W0,+ (Ωh ), Vbh = Ubh = W+,0 (Ωh ). m,p (Ωh ) forms. or one can use the or the W−m,p (Ωh ), . . . , or the W0,−

For nonlinear G, Gh , the domains D(G), D(Gh ) are subsets of U, U h . The A, G, will usually be deﬁned on U → V , and the Ah , Gh on U h → V h , cf. Figure 8.3. Uniqueness

590

8. Finite diﬀerence methods Ub ⊂

U

A, G

⊃ D(G)

Φ h⏐

P h⏐ U hb

⊂

U

V′

⊃

h

←→

Vb

tested by

V bh

′h

Q

A h, G h

D (G h )

tested by

V ′h

←→

Figure 8.3 Diﬀerence methods: Spaces, and operators, usually with Au, Gu ∈ Vb ∀u ∈ Ub and Ah uh , Gh uh ∈ Vbh ∀uh ∈ Ubh .

requires boundary conditions Ub , Vb and Ubh , Vbh , tested according to Au, Gu ∈ Vb and Ah uh , Gh uh ∈ Vbh . For applying the general discretization theory to A, G, we need restriction, and projection operators, P h , and Q h , relating the original and the discrete spaces.

P h : U → U h , Q h : Vb → Vbh , or only P h : Ub → Ubh .

(8.78)

Then Ah = Φh A and Gh = Φh G are deﬁned by the constructions in Section 8.4, cf. Figure 8.3. We formulate the W+k,p (Ωh ) case. The modiﬁcation to W−k,p (Ωh ) is obvious. For our diﬀerence method, applicable to the problem Au0 = f or Gu0 = 0, see Deﬁnition 3.12, we thus have the sequence of quintuples Ubh , Vbh , P h , Q h , Gh , with inf{0 < h ∈ H} = 0, dim Ubh = dim Vbh < ∞, (8.79) h∈H

and h < h0 , thus limh→0 := limh∈H,h→0 always makes sense. According to Section 3.2, cf. (3.16), (3.24), we need bounded, not necessarily linear P h , Q h in (8.78), such that P h 0 = 0, Q h 0 = 0, and

lim P h uU h = uU ∀u ∈ U, lim Q h f hV = f V ∀f ∈ Vb .

h→0

b

h→0

(8.80)

b

We employ the claim of Theorem 3.21 in Chapter 3: consistency and stability essentially imply uniquely existing discrete solutions, uh0 , and their convergence to the exact solution u0 . The (classical) consistency error, and consistency for diﬀerential operators in u, are deﬁned as, cf. Deﬁnition 3.14,

(Ah P h u − f h ) − Q h (Au − f ) ∈ Vbh , or Gh P h u − Q h Gu ∈ Vbh , and

(8.81)

lim ||(Ah P h u − f h ) − Q h (Au − f )||V h = 0 or lim ||Gh P h u − Q h Gu||V h = 0.

h→0

b

h→0

b

Gh are called stable in uh , if ∃h0 , r, S ∈ R+ ﬁxed, such that for all h ∈ H, h < h0 : (8.82) uh1 , uh2 ∈ Br (uh ) ∩ Ub ⇒ uh1 − uh2 U h ≤ S Gh uh1 − Gh uh2 V h . b

For A or A − f this reduces to (for any r, and u , but for h < h0 )

(Ah )−1 ∈ L Vbh , Ubh exists and ||(Ah )−1 ||U h ←V h ≤ S. h

h

h

h

b

b

Gh is stable if its derivative (G (u0 ))h is stable, cf. Theorem 3.23.

591

8.5. Convergence for diﬀerence methods

So we have to introduce P h , Q h , and prove consistency and stability, via coercivity or an inf-sup condition of its principal part. Alternatively the monotony arguments in Section 4.5 are applicable. 8.5.2

The operators P h , Q h

We do not repeat the basic proofs for well-deﬁned, linear and bounded P h and Q h , cf. e.g. Raviart [545], Schumann and Zeidler [574, 678] and Temam [621]. But we prove, in a slightly nonstandard way, consistency and stability. Again we consider the two cases (8.19) and (8.20), with the two diﬀerent deﬁnitions of the Ωh0 ⊂ Ωh , ∂Ωh in parallel. Either Ωh ⊂ Ω for the original u ∈ W k,p (Ω) or we extend u to u ∈ W k,p (Ω1 ) with Ωh ⊂ Ω1 . This is possible for Ω in (8.9) by Theorem 4.37. The same holds for functionals. The deﬁnition of the P h : U → U h , and Q h : V → V h is nearly the same. But the trivial boundary conditions in P h : Ub → Ubh require modiﬁcations. This is achieved by deﬁning P h u(x) either ∀x ∈ Ωh or ∀x ∈ Ωh0 . We discuss two diﬀerent possibilities. P h = P1h is studied, e.g. in Raviart [545], Schumann and Zeidler [574, 678], and P h = P2h , e.g. in Temam [621]. We mainly consider P h = P2h , and only sometimes indicate P1h . Note that in (8.84) supp u is closed, and Ω is open. Deﬁnition 8.13. For functions, u ∈ U, deﬁned in Ω, or extended to Ω1 as Ec u, and with ch (x) ⊂ Ωh ⊂ Ω or Ωh ⊂ Ω1 , for Ec , Ωh , Ωh0 cf. (8.19)–(8.20), deﬁne P h = P1h by the integral mean value u ¯h (x) of u or Ec u as 1 P h = P1h : U → U h : ∀x ∈ Ωh : P1h u(x) := u ¯h (x) := n u(¯ x)d¯ x, and h ch (x) ¯h (x) otherwise = 0. P h = P1h : Ub → Ubh : ∀x ∈ Ωh0 : P1h u(x) := u

(8.83)

For P h = P2h we modify (8.83) by replacing P1h u(x) by P2h u(x)∀x ∈ Ωh or Ωh0 : For u ∈ C(Ω) := {v ∈ C(Ω) or C(Ω1 , R) : carr v ⊂ Ω or ⊂ Ω1 } : P2h u(x) := u(x) (8.84) and uniquely extend P2h from C(Ω) to U = W k,p (Ω) deﬁning P h = P2h : U → U h . Obviously these P h and the Q h below can be trivially extended as P h : U ∪ U h → U h : P h |U = P h in (8.83) and P h |U h = I, identity on U h .

(8.85)

Parallel to the standard grid functions, uh , we employ the step functions, uhe , with χch (x) , the characteristic functions of the ch (x), cf. (8.19), ⎧ ⎫ ⎨ ⎬ uh (x)χch (x) . E h : U h → Ueh := uhe := E h uh := (8.86) ⎩ ⎭ h x∈Ω

We ﬁnd, possibly except for a subset of measure 0 on the boundaries of the ch (x), that α h h α h E u = E h ∂+ u . ∂+

(8.87)

592

8. Finite diﬀerence methods

This implies that ⎛

U h U h := uh W k,p (Ωh ) := [[E h uh ]]W k,p (Ωh ) := ⎝ +

+

⎞1/p α h h p ∂+ E u Lp (Ωh ) ⎠

. (8.88)

|α|≤k

Remark 8.14. It is important to realize that the results in this and the next subsection are measured with respect to this speciﬁc norm uh U h = uh W k,p (Ωh ) , unless stated + otherwise. This holds, in particular, for the consistency estimates. The following proposition is an obvious extension of Temam [621], Lemma 3.1 on p. 51 for Lp (Ω), to higher order derivatives, and to both our cases (8.19), (8.20). His Proposition 3.1, p. 42, shows that Proposition 8.15 had only to be proved for functions u in a dense subspace of W k,p (Ω). This remains correct for Theorem 8.21, (8.104) as well. We do not repeat the argument in [621], Proposition 3.1, since we use it in part (b) of the proof of the following Theorem 8.21. The Ec in Theorem 4.37 combined with this result yields, cf. (8.19)–(8.20): Proposition 8.15. P h , E h , cf. (8.84), (8.86), satisfy ∀u ∈ W k,p (Ω), |α| ≤ m α h h E P u − ∂ α uLph (ch (Ωh )) = 0 for (8.19): lim ∂+

(8.89)

h→0

α h h E P u − Ec ∂ α uLph (ch (Ωh )) = 0, this implies for (8.20): lim ∂+ h→0

lim P uW k,p (Ωh ) h

h→0

= lim [[E h P h u]]W k,p (Ωh )

+

h→0

+

= uW k,p (Ω) ,

cf. (8.80). Similarly, we obtain α h P u − P h (∂ α u)Lph (Ωh ) = 0 for (8.19): lim ∂+ h→0

(8.90)

α h P u − P h (∂ α Ec u)Lph (Ωh ) = 0. for (8.20): lim ∂+ h→0

Remark 8.16. A direct comparison between the original functions and grid functions is not possible. So the concept of external approximation schemes was introduced, cf. Petryshyn [528–530, 532], Temam [621] and Zeidler [677]. To this end a mapping is deﬁned for V = W m,p (Ω) as ω : V → W := Π|α|≤m Lp (Ω), ωu := (∂ α u)|α|≤m , ωuW := max ∂ α uLp (Ω) . |α|≤m

ω is bounded, since all norms in Π|α|≤m R are equivalent. Then, e.g. the ﬁrst line of (8.89) can be reformulated as α h h E P u |α|≤m − ωuW = 0. (8.91) lim ∂+ h→0

In this generalized sense, grid functions approximate functions in Sobolev spaces. By (8.30), cf. Proposition 2.34, (2.107), (2.108), for 1 < p < ∞, 1/p + 1/p = 1, and functionals, f ∈ V = W −m,p (Ω), unique fβ ∈ Lp (Ω) exist, such that,

8.5. Convergence for diﬀerence methods

f, vV ×V :=

⎛ fβ ∂ β vdx ∀ v ∈ V, with f V = ⎝

Ω |β|≤m

593

⎞1/p

fβ pLp (Ω) ⎠ , (8.92)

|β|≤m

cf. Proposition 2.34. For 1 = p or p = ∞ the fβ ∈ Lp (Ω) are no longer unique, and we obtain f V = minfβ { |β|≤m fβ Lp (Ω) } over all possible fβ .

Two diﬀerent possibilities for Q h correspond to the above P h = P1h or = P2h .

Deﬁnition 8.17. For functionals, f ∈ V = W −m,p (Ω), tested by v ∈ V = W m,p , we deﬁne the Q h : V → V h componentwise for the fβ in (8.92). Using the ﬁrst line of (8.83) or (8.84), and testing ∀ v h ∈ V h = W+m,p yields : Q h f := fβh (x) := P h fβ (x)∀x ∈ Ωh |β|≤m , fβh ∈ Lph (Ωh ), (8.93) β β h fβh ∂+ v dxh = hn ∂+ v h (x)(P h fβ )(x). =⇒ Q h f, v h V h ×V h :=

Q h : V → V

h

Ωh |β|≤m

x∈Ωh

|β|≤m

In contrast to Deﬁnition 8.13, usually P h fβ (x) = 0 for x ∈ Ωh \ Ωh0 .

Proposition 8.18. The previous P h : V → V h in (8.83), (8.84) and Q h : V → V in (8.93), are well-deﬁned linear bounded operators, satisfying (8.80), (8.89).

h

Proof. We start with the properties of the P h . Obviously the P1h are well deﬁned. By Raviart [545], pp. 22–25, and Schumann and Zeidler [574, 678], Lemma 6, P h = P1h are linear bounded operators. For P h = P2h this is a consequence of Temam [621], Proposition 3.1, and Lemma 3.1, cf. its proof. In particular, the extension of P2h from C(Ω) to U = W k,p (Ω) is possible, and yields this result. Relation (8.80) is implied, and obtained for P2h by combining [621], Proposition 3.1 with the proof of Lemma 3.1. Obviously Q h is well deﬁned and linear. The boundedness follows from (8.93), and twice the discrete H¨older inequality (1.44) with 1/p + 1/p = 1: β Q h f, v h V h ×V h = hn ∂+ v h (x)(P h fβ )(x) (8.94) |β|≤m

x∈Ωh

≤

|β|≤m

=

|β|≤m

⎛ ⎝

hn

⎞1/p ⎛ ⎞1/p

p β h ∂+ v (x)(x) ⎠ ⎝ hn (P h fβ (x))p ⎠

x∈Ωh

x∈Ωh

β h ∂+ v )Lph (Ωh ) P h fβ Lp (Ωh ) h

≤ v h Whm,p (Ωh ) Q h f W −m,p (Ωh ) . h

h

This shows that Q is bounded. A combination with (8.80) for the P h , and (8.92), (8.35) with the minimal fβ , implies (8.80).

594

8. Finite diﬀerence methods

8.5.3

Consistency for diﬀerence equations

For the previous examples consistency does not require u ∈ Ub , v h ∈ Vbh , but is valid for u ∈ U, v h ∈ V h as well. In fact, the boundary conditions are exactly satisﬁed. Since here the norms in V and Vb are equivalent, we often do not distinguish them. The previously introduced P h , Q h , Ah , Gh allow the proof of this consistency. We only consider ﬁnite diﬀerence methods with equal step sizes, h, in all directions. For special Ω, Dirichlet boundary conditions can be exactly reproduced: Ω :=

n

(ai , bi ) ⊂ Rn s.t. ∃h > 0 with ai /h, bi /h ∈ Z, i = 1, · · · , n.

(8.95)

i=1

For Ω ⊂ Rn in (8.95) or in (8.98) below we introduce step sizes, h, and grids: H := {h as in (8.95) or h ∈ R+ for (8.98)} :

(8.96)

h := {x = (x1 , . . . , xn )T ∈ Rn : xi /h ∈ Z}, G h := GD

Glh := {∀j = 1, . . . , n : x = (x1 , . . . , xn )T ∈ Rn : ∀i = j : xi /h ∈ Z, xj /h ∈ R}. For Ω in (8.95) we introduce, cf. Figures 8.4, Ωh0 := Ω ∩ G h , Ωh := Ω ∩ G h , ∂Ωh := ∂Ω ∩ G h = Ωh \ Ωh0 .

(8.97)

This has to be modiﬁed as in (8.20) for Ω ⊂ Rn is open, bounded, nonempty, and ∂Ω is Lipschitz-continuous. (8.98) Let, with the norm for U h in (8.87), P h = P1h : Ub → Ubh : ∀x ∈ Ωh0 : P1h u(x) := u ¯h (x) otherwise = 0.

: points in Ωh0 : points in ∂Ωh

Figure 8.4 Two-dimensional grid for Dirichlet boundary conditions.

(8.99)

8.5. Convergence for diﬀerence methods

595

For proving consistency of orders 1 or 2, for the diﬀerence methods, we have to approximate the u ∈ V = H m+μ (Ω). A very up-to-date choice, and very appropriate for our bounded Ω, and equidistant Ωh , are radial basis functions. Since they are not quite as well known as other approximations, we give a very short introduction, and refer for all the speciﬁc conditions to Buhmann [152] and B¨ ohmer [120]. We start with φ, deﬁned by a positive completely monotonic g F φ : R+ → R+ , φ(r) := g(r2 ), g ∈ C(R+ ), e.g. g(r2 ) = r2 + c2 (8.100) and deﬁne ψ(x) := ck φ(x − k)ck ∈ R, x ∈ Ω ⊂ Rn , k∈Zn ,|k|
with just one ψ, a ﬁnite sum, and appropriate ck . We introduce ⎧ ⎫ ⎨ ⎬ cj φ(x/h − j) , e.g. sh (x) := f (jh)ψ(x/h − j), S h := sh (x) := ⎩ ⎭ n n j∈Z

j∈Z

(8.101) the latter interpolating f in the jh, if ψ is appropriately chosen. The following proposition requires a slightly modiﬁed semi-inner product and norm in H k (Ω): (u, v)H∗k (Ω) :=

k! (∂ α u, ∂ α v)L2h (Ωh ) , uH∗k (Ω) := ((u, u)H∗k (Ω) )1/2 , (8.102) α!

|α|=k

n . and pH∗k (Ω) = 0∀p ∈ Pk−1

If Ω is not too narrow and h is small enough, Ωh contains a unisolvent subset Ωhu for n , the set of polynomials of degree ≤ k − 1, in Rn , such that Pk−1 n p ∈ Pk−1 : ∀x ∈ Ωhu : p(x) = 0

⇒

p ≡ 0.

We have mentioned that the second sh in (8.101) interpolates f in Ωh , however not uniquely in S h . This is possible in a modiﬁed way by, cf. [152], Proposition 5.2. n , for u ∈ H k (Ω), Proposition 8.19. Assume Ωh contains a unisolvent subset for Pk−1 h the function values u(x)∀x ∈ Ω are well deﬁned, cf. Theorem 1.26 or [152]. Then there exists a unique minimal norm interpolant sh ∈ S h interpolating sh (x) = u(x)∀x ∈ Ωh . Compared to any other interpolating g ∈ S h , this sh has minimal seminorm, hence sh H∗k (Ω) ≤ gH∗k (Ω) .

For these interpolants Buhmann [152] proves, cf. Theorem 5.5 for m = 0 and Wendland [662], for m ≥ 0: Theorem 8.20. Convergence for radial basis functions: For u ∈ V = W m,p (Ω), and small enough h, the conditions of Proposition 8.19 are satisﬁed. They guarantee a unique minimal norm interpolant shu ∈ S h . For μ = 1, 2 we obtain convergence and

596

8. Finite diﬀerence methods

bounds for u ∈ Z := W m+μ,p (Ω), μ = 1, 2,

=⇒ u − shu V < Chμ uZ and

(8.103)

=⇒ ωu − ωshu W < 2hμ ωCuZ with ωshu W < 2ωuV . Theorem 8.21. Consistency of linear operators with respect to · V h in (8.87):

1. With P h , Q h in (8.99), (8.84), (8.93), and the conditions for the coeﬃcients of A in Summary 8.6 and Remark 8.7, we obtain for Au = f ∈ V , u ∈ Ub = W0m,p (Ω), A ∈ L(U, V ) and f h = Q h f , consistent unsymmetric and symmetric approximations Ah , Ahsw , cf. (8.60), for linear operators, e.g.

∀v h ∈ V h , v h = 0 : lim Ah P h u − Q h Au, v h V h ×V h /||v h ||V h = 0, or h→0

(8.104)

lim ||Ah P h u − Q h Au||V h = lim ||Ahsw P h u − Q h Au||V h = 0 ∀u ∈ W m,p (Ω).

h→0

h→0

2. More precisely, we obtain, for u ∈ W m+μ,p (Ω), A ∈ L(W m+μ,p (Ω), W −m+μ,p (Ω)) consitency of order μ = 1 for unsymmetric, and μ = 2 for symmetric differences:

||Ah P h u − Q h Au||V h ≤ ChuW m+1,p (Ω) ∀u ∈ W m+1,p (Ω), or

||Ahsw P h u − Q h Au||V h ≤ Ch2 uW m+2,p (Ω) ∀u ∈ W m+2,p (Ω).

(8.105) (8.106)

3. For possible choices of f h = Q h f , limh→0 ||f h − Q h f ||V h = 0 has to be proved. Proof. We prove the result only for m ≥ q = 1; the generalization to q ≥ 1 is straightforward. The proof requires three steps:

(a) We estimate Ah P h u − Q h Au, v h V h ×V h for u ∈ CLm (Ω), and Ahsw P h u − Q h Au, v h V h ×V h for u ∈ CLm+1 (Ω), thus (8.105), (8.106), cf. (8.108) (b) The dense CLm (Ω) ⊂ W m,p (Ω) implies consisteny in (8.104) for u ∈ W m,p (Ω). (c) For the stronger forms with order 1 in (8.105), and 2 in (8.106) we need interpolation with radial basis functions with the error estimates in Theorem 8.20 on the rectangular grids in (8.95), (8.98). (a) For aαβ ∈ C(Ω) the aαβ (x) have to be replaced by P h aαβ (x). We obtain, cf. (8.47), for ∀u ∈ V = W m,p (Ω), ∀v h ∈ V h = W+m,p (Ωh ), ah : V h × V h → R :

ah (P h u, v h ) − Q h Au, v h V h ×V h

(8.107)

= Ah P h u − Q h Au, v h V h ×V h β h h α h aαβ ∂+ = P u − P h aαβ ∂ α u ∂+ v dx Ωh |α|,|β|≤m

= hn

x∈Ωh |α|,|β|≤m

= hn

x∈Ωh |α|,|β|≤m

β h α h (aαβ ∂+ P u)(x) − P h aαβ ∂ α u (x) ∂+ v (x)

β h α h aαβ (x) ∂+ P u (x) − P h ∂ α u (x) ∂+ v (x).

8.5. Convergence for diﬀerence methods

597

In (8.107), only the terms with |α| > 0 are diﬀerent. We use Lemma 8.2. For the unsymmetric and symmetric divided diﬀerence, respectively, This yields with (8.22) for u ∈ CLm (Ω) and u ∈ CLm+1 (Ω), and with (8.99), (8.84), α h ∂+ P u(x) − P h ∂ α u (x) ≤ cα,1 h ∀x ∈ Ωh for u ∈ CLm (Ω) and (8.108) α h ∂h P u(x) − P h ∂ α u (x) ≤ cα,1 h2 ∀x ∈ Ωh for u ∈ C m+1 (Ω). L (b) This part is an update of Temam’s proof of his Proposition 3.1 [621]. CLm (Ω) is a dense subset of V = W m,p (Ω). So we approximate the u ∈ W m,p (Ω) by appropriate un ∈ CLm (Ω), and employ the approximation results in (8.91), (8.108). For each n ∈ N a un ∈ CLm (Ω) exists such that u − un V <

1 n

⇒

ωu − ωun W <

ω . n

(8.109)

α h h We ﬁx this un . Since ∂+ E P u − ∂ α uLph (ch (Ωh )) → 0 by (8.89), (8.91), there exists an ηn > 0, such that for 0 < h < ηn , α h h 1 (8.110) ∂+ E P un |α|≤m − ωun W < . n We may additionally choose ηn < 1/n, thus {ηn } monotonely converge to 0. For

P+h u := P h un for ηn+1 < h < ηn we obtain that

α h h ωu − ∂+ E P+ u |α|≤m W

α h h ≤ ωu− ωun W + ωun − ∂+ E P un |α|≤m W α h h α h h + ∂+ E P un |α|≤m − ∂+ E P+ u |α|≤m W

1 + ω . n α h h This ωu − ∂+ E P+ u |α|≤m W ≤ (1 + ω)/n for h < ηn with (8.91), (8.107), (8.108), proves the claimed consistency for u ∈ W m,p (Ω). ≤

(c) A slight modiﬁcation of the proof in (b) yields the stronger results (8.105), (8.106). Only for this case do we need the approximation properties in (8.103). It implies with P h u := P h shu that α h h E P u |α|≤m We ωe Ec u − ∂+ α h h h ≤ ωe Ec u − ωe shu We + ωe shu − ∂+ E P su |α|≤m We α h h h α h h + ∂+ E P su |α|≤m − ∂+ E P u |α|≤m We ≤ h(C + ω)uZ and ≤ h2 (C + ω)uZ .

598

8. Finite diﬀerence methods

α h h This ωe Ec u − ∂+ E P u |α|≤m We ≤ hμ (C + ωe )uZ proves the claimed consistency of orders μ = 1, 2, for u ∈ W m+μ,p (Ω) for unsymmetric and symmetric diﬀerences. In Subsections 8.4.5 and 8.4.6 we introduced the diﬀerence equations for the quasilinear equations of orders 2 and 2m, cf. (8.65) and (8.67), and for linear and quasilinear systems in (8.71) and (8.72). The symmetric forms only were described analogously to the linear problems. For the diﬀerent types of quasilinear problems diﬀerent conditions are discussed for unique existence of solutions and their regularity, cf. Subsections 2.5.4, 2.5.6, 2.6.4, 2.6.6 and bounded Frechet derivatives, cf. Subsections 2.7.2, 2.7.3, 2.7.5. To simplify the following consistency result for quasilinear problems, we assume a unifying condition. It is oriented towards that for bounded Frechet derivatives, needed anyway for the proof of the coercivity, stability and convergence in Subsections 8.5.4 and 8.5.5. For minimal conditions the reader has to consult Subsections 2.5.4, 2.5.6, 2.6.4, 2.6.6, 2.7.2, 2.7.3, 2.7.5. We do not fully formulate the next condition, and the theorem for systems, since the generalization is obvious. We assume all the Aj in (8.65) and Aα in (8.67) to be equicontinuous in a “neighborhood” of u0 , more precisely, with Nm in (8.17), Aj , Aα equicontinuous in Ω × N∇m ⊂ Ω × RNm with bounded N∇m , s.t. ∀u ∈ Ub : u − u0 V ≤ ρ, ∇ Theorem 8.22. (8.87)

≤m

u(x) ∈ N∇m ∀x ∈

Ω, ∇≤m + u(x)

(8.111)

∈ N∇m ∀x ∈ Ωh0 .

Consistency of quasilinear equations and systems with · V h in

1. Choose the P h , Q h in (8.84), (8.99), (8.93), and let Aj , Aα in (8.67) and u satisfy (8.111). Choose Gh , similarly Ghsw , as in (8.65), (8.67), and (8.71), (8.72). Then we obtain for Gu ∈ W −m,p (Ω) consistent unsymmetric and symmetric h h approximations G , Gsw , hence, e.g.

∀v h ∈ V h , v h = 0 : lim Gh P h u − Q h Gu, v h V h ×V h /||v h ||V h = 0, h→0

(8.112)

thus

lim ||Gh P h u − Q h Gu||V h = lim ||Ghsw P h u − Q h Gu||V h = 0 ∀u ∈ W m,p (Ω).

h→0

h→0

2. For u ∈ W m+μ,p (Ω), Aα : W m+μ,p (Ω) → W −m+μ,p (Ω)) with Lipschitz-continuous Aj , Aα with respect to the variables in N∇m , we obtain consitency of order μ = 1 for unsymmetric, and μ = 2 for symmetric diﬀerences:

||Gh P h u − Q h Gu||V h ≤ ChuW m+1,p (Ω) ∀u ∈ W m+1,p (Ω),

(8.113)

or

||Ghsw P h u − Q h Gu||V h ≤ Ch2 uW m+2,p (Ω) ∀u ∈ W m+2,p (Ω).

(8.114)

8.5. Convergence for diﬀerence methods

599

Proof. The proof is very similar to the proof of Theorem 8.21, here restricted to m = q = 1. The generalization to m, q ≥ 1 is straightforward. From the three steps, we only formulate (a) and only indicate (b) and (c) (a) Again, we formulate this proof for u, ∇u ∈ C(Ω), Aj ∈ C(Ω, R, Rn ). Otherwise these u, ∇u, Aj have to be replaced by P h u, P h ∇u, P h Aj . We obtain, cf. (8.65), for ∀u ∈ V = W 1,p (Ω), ∀v h ∈ V h = W+1,p (Ωh ),

ah (P h u, v h ) − Q h Gu, v h V h ×V h = Gh P h u − Q h Gu, v h V h ×V h n Aj (·, P h u, ∇+ P h u) =

(8.115)

Ωh j=0

j h h v dx −P h Aj (·, u, ∇u) ∂+ = hn

n

Aj ·, P h u, ∇+ (P h u)

x∈Ωh j=0

−Aj (·, u, ∇u

j h (x)∂+ v (x).

Lemma 8.2 yields with (8.22) for u ∈ CL1 (Ω), u ∈ CL2 (Ω), but now for Aj ∈ CL (Ω, R, Rn ),

Aj ·, (P h u), (D+ (P h u)) − Aj (·, u, ∇u) (x) ≤ c1,1 h for u ∈ CL1 (Ω) (8.116)

Aj ·, (P h u), (D+ (P h u)) − Aj (·, u, ∇u) (x) ≤ c1,1 h2 for u ∈ CL2 (Ω). (b) We approximate the u ∈ W 1,p (Ω) by uk ∈ CL1 (Ω). Then a combination of the equicontinuity in (8.111), (8.109), (8.110), implies the consistency (8.112). (c) The ﬁnal consistency of orders 1 and 2 follows as in the proof of Theorem 8.21, now with the Lipschitz-continuity of the Aj . Finally, we introduced in Subsection 8.4.7 the diﬀerence equations for the fully nonlinear equations restricted to order 2, cf. (8.76) and (8.67). The symmetric form is listed at the end of Subsection 8.4.7. Again we restrict the discussion to the Hilbert space setting U := H 2 (Ω) → V := L2 (Ω). For fully nonlinear equations diﬀerent conditions for unique existence of solutions and their regularity are listed in Subsection 2.5.7 and the bounded Frechet derivatives in Subsections 2.7.3 and 2.7.5. For systems nearly nothing is known, cf. Section 5.2.8. Similarly to (8.111), we assume the fully nonlinear G to be equicontinuous in a “neighborhood” of u0 , more precisely, with Nm in (8.17), G equicontinuous in Ω × N∇2 ⊂ Ω × RN2 with bounded N∇2 , s.t.

(8.117)

∀u ∈ V : u − u0 V ≤ ρ, ∀x ∈ Ω∇≤2 u(x) ∈ N∇2 ∀x ∈ Ωh0 .∇≤2 + u(x) ∈ N∇2 . Theorem 8.23. Consistency for fully nonlinear problems with · V h = · L2h (Ωh ) :

600

8. Finite diﬀerence methods

1. Choose the P h , Q h in (8.99), (8.84), (8.93), and let G now in (8.74) satisfy (8.117). Then we obtain for G : D(G) ⊂ U := H 2m (Ω) → V := L2 (Ω) and the grid functions, U h , V h , a consistent strong approximation, Gh = Ghs , and the strong symmetric approximation, Ghss , hence we ﬁnd for u − u0 U ≤ ρ,

∀v h ∈ V h , v h = 0 : lim (Gh P h u − Q h Gu, v h )V h /||v h ||V h = 0, h→0

(8.118)

thus

lim ||Gh P h u − Q h Gu||V h = lim ||Ghss P h u − Q h Gu||V h = 0 ∀u ∈ U.

h→0

h→0

(Ω), G : D(G) ⊂ H (Ω) → H μ (Ω)) with Lipschitz-continuous G 2. For u ∈ H with respect to the variables in N∇2 , we obtain consitency of order μ = 1 for unsymmetric, and μ = 2 for symmetric diﬀerences: 2+μ

2+μ

||Gh P h u − Q h Gu||V h ≤ Chμ uH 2+μ (Ω) ∀u ∈ H 2+μ (Ω). Proof. This is an obvious modiﬁcation of the last proofs. 8.5.4

(8.119)

Vbh -coercivity for linear(ized) elliptic diﬀerence equations

In contrast to the previous consistency results, the boundary conditions are crucial for coercivity and the following stability of elliptic diﬀerence operators. Vbh -coercivity is, by Theorems 3.23, 2.12 ﬀ., and 8.29, the essential step towards stability of our diﬀerence methods. In fact, let the principal part of a linear, or linearized, elliptic diﬀerence m (Ω, Rq )-coercive, see below, and the method be consistent, operator be Vbh := H0,+ cf. Subsection 8.5.3. Assume a bounded and boundedly invertible Frechet derivative G (u0 ) of the original problem in the exact solution, u0 . Then the diﬀerence equation, Gh (uh ) = 0, similarly Ah (uh ) = 0, is stable, by Theorem 8.29. With its consistency, this implies its uniquely existing solution, uh0 , converging to u0 with respect to · H m (Ω,Rq ) . We summarize these results in Subsection 8.5.5. + We start with coercivity results for the original linear operators. In Chapter 2, Theorems 2.43, 2.89, and 2.104, we have seen that coercive bilinear forms induce boundedly invertible linear operators. For the diﬀerent cases, we have formulated in Summary 8.6 the deﬁnitions of ellipticity and the conditions for the coeﬃcients, implying H0m (Ω, Rq )-or W0m,p (Ω, Rq )-coercivity, and boundedness of the weak (and strong) forms of the principal parts of the original operators for m, q ≥ 1. As we have previously discussed, cf. Chapters 2–5, we will usually only have H0m (Ω, Rq )-coercivity, even for (nonlinear) problems deﬁned in W0m,p (Ω, Rq ). As indicated in Remark 8.7 with its pros and cons, the conditions in Summary 8.6 for the coeﬃcients of the linear diﬀerence equations, (8.38)–(8.41), could be considerably reduced towards those of diﬀerential equations, by replacing the C(Ω, Rq ), C |β| (Ω, Rq ) by L∞ (Ω, Rq ), W |β|,∞ (Ω, Rq ). The nonlinear problems in Subsections 8.4.5 and 8.4.6 are included via linearizations of their discrete operators, for m, q ≥ 1. They are bounded and have H0m (Ω, Rq )-coercive principal parts, for 2 ≤ p < ∞, under appropriate conditions, cf. (8.38)–(8.41), by Theorems 2.124, 2.125, 2.126, 2.127. Thus

601

8.5. Convergence for diﬀerence methods

m we obtain stability for 2 ≤ p < ∞, with respect to the H+ (Ωh , Rq ) norm. The case m,p h q 1 ≤ p < ∞, with respect to the W+ (Ω , R ) norm can be studied via monotone operators. We memorize the discrete Vbh -coercivity and ellipticity. We choose the discrete spaces m,p h m (Ωh , Rq ), or Vbh = W0,+ (Ωh , Rq ), m, q ≥ 1, 1 ≤ p ≤ ∞ for the homogeneous Vb = H0,+ Dirichlet boundary value problem. Let ah (uh , v h ), cf. e.g. (8.47), (8.63), (8.71) for m ≥ 1, be a bounded bilinear form, deﬁned for V h , and Ah : V h → V h the induced linear operator. Then ah (uh , v h ) are called Vbh -coercive, and Vbh -elliptic, if h-independent constants α ∈ R+ , Cc ∈ R exist such that

ah (uh , uh ) > αuh 2V h , and ≥ αuh 2V h − Cc uh 2W h ∀uh ∈ Vbh ⊂ W h , m respectively, e.g. Vbh = H0,+ (Ωh , Rq ) or

m,p W0,+ (Ωh , Rq )

(8.120)

⊂ W h = L2h (Ωh , Rq ),

m, q ≥ 1. If additionally C ∈ R+ exist such that

Ah : Vbh → Vbh is boundedly invertible with (Ah )−1 V h →V h ≤ C ∀h > 0, b

(8.121)

b

then Ah is called Vbh -stable or Vbh -regular, e.g. by Thom´ee and Westergren [627], Hackbusch [381, 383, 387] and Bube and Strikwerda [151]. The proofs for the Vb -coercivity in Chapter 2, and the Vbh -coercivity here are very similar. So we only indicate the proof for one equation of order 2, but present it for a system of order 2m. This case shows the strongest diﬀerences between the Vb -, and the Vbh -situation. 48 48 We recapitulate the discrete Fourier transform, cf. Theorem 2.105, Thom´ ee and Westergren [627], Stoer and Bulirsch [603] and Bube and Strikwerda [151], pp. 660 ﬀ., and Bube et al. [605]. In standard applications to periodic functions it yields (the coeﬃcients of) the trigonometric polynomials h = G h → R, see (8.96), e.g. via FFT, but not the Fourier interpolating discrete function values uh : Gn series. This method will be extensively studied in spectral methods, see [120]; see also any standard textbook on numerical methods, cf. [151,603]. We only summarize the results needed for the following coercivity proof in this special case. We assume Ω in (8.95) or (8.98). For (8.98) we extend Ω to a n cuboidal Ωe with uh (x) = 0∀x ∈ Ωh 0 . Then we transform it, such that Ω = (−π, π) , and

Ωh := {xν := hν} ⊆ [−π, π]n with h := π/N, N ∈ N, 1

n

n

(8.122)

j

ν = (ν , · · · , ν ) ∈ Z , ∀j = 1, · · · , n : |ν | ≤ N. m (Ωh ), to G h := {hν : ν = (ν 1 , · · · , ν n ) ∈ Zn }, We periodically extend the grid functions, uh ∈ H0,+ since uh has identical, here even vanishing, values on opposite faces of [−π, π]n . We extend scalar products, and norms, see (8.25), to uh , v h : Ωh → C with uh (x)∗ = uh (x) as

h

h

(u , v )L2 (Ωh ) := h h

n

x∈Ωh

⎡ h

∗ h

h

u (x) v (x), v H m (Ωh ) +

:= ⎣

|α|≤m

h

n

⎤1/2 α h |∂+ v (x)|2 ⎦

.

x∈Ωh

We deﬁne the discrete Fourier transform with the imaginary unit, ι, ι2 = −1, and ξ, x n =

n k=1

ξk xk ∀ x ∈ Ωh , ∀ ξ ∈ Γh := {ξ = (ξ1 , · · · , ξn ) ∈ Zn : |ξj | ≤ N }

(8.123)

602

8. Finite diﬀerence methods

For order 2m, discrete Fourier transforms are employed. They have been used in the literature to prove regularity results for the solutions of diﬀerence equations, similar to those of diﬀerential equations, and the convergence to exact solutions, cf. Agmon et al. [2, 3], Thom´ee and Westergren [627], Bube and Strikwerda [151], Strikwerda et al. [605] and Hackbusch [387]. But the author did not ﬁnd any application towards discrete coercivity as in this section. Theorem 8.25. Vbh -coercive principal parts: Assume the conditions in Summary m 2m (Ωh , Rq ), Ubh := H+ (Ωh , Rq ) ∩ Vbh , Vbh = Vbh := 8.6, and Remark 8.7, let Vbh := H0,+ 2 h q h h Lh (Ω , R ), and let f ∈ Vb be bounded so p = 2 compared to Subsections 8.5.2, 8.5.3. 1. Then the weak bilinear forms ah (uh , v h ) are bounded with respect to V h , the principal parts ahp (uh , v h ), and ah (uh , v h ) are Vbh -coercive, and Vbh -elliptic, respectively. The ahp (uh , v h ) induce a stable Ahp , which is consistent with Ap . 2. The strong diﬀerence operators, Ahs , or (G (u0 ))hs , are also bounded with respect to Vsh . Proof. (1) The claim that the strong operators, Ahs , (G (u0 ))hs and ah (uh , v h ) are bounded with respect to Vsh , and V h , respectively, is nearly obvious. (2) Since the ah (uh , v h ) are compact perturbations of the principal parts ahp (uh , v h ) the Vbh -ellipticity of the ah (uh , v h ) can be shown, via the detour as ˆh (ξ) := hn (Fh uh )(ξ) := u

m e−ι ξ,xn uh (x) = (eι ξ,·n , uh (·))L2 (Ωh ) ∀uh ∈ H0,+ (Ωh ). h

x∈Ωh

Obviously this u ˆh (ξ) is 2N periodic in each ξj for ξ ∈ Zn . The fact that uh (x) and u ˆh (ξ) are deﬁned h h on diﬀerent grids Ω and Γ or their extensions, is reﬂected in a certain asymmetry of the inverse discrete Fourier transform, in contrast to (2.397). The following properties are proved in standard textbooks, and, e.g. Bube and Strikwerda [151]. Theorem 8.24. For the discrete Forier transform its inverse is deﬁned as

ˆh (x) := eι ξ,xn u ˆh (ξ), it yields uh (x) = Fh−1 u ˆh (x), and Fh−1 u

(8.124)

ξ∈Γh

(uh , v h )L2 (Ωh ) = hn h

uh (x)∗ v h (x) =

x∈Ωh

u ˆh (ξ)∗ vˆh (ξ) = (ˆ uh , vˆh )L2 (Γh ) ∀uh , v h ∈ L2 (Ωh ), h

ξ∈Γh

β h β h u (ξ) = ∂+ u (ξ) = ξ h,β u ˆh (ξ), with the so-called Parseval formula, and Fh ∂+ ξ h,β :=

n n n (exp(ιhξ ) − 1)β h,β 2(exp(ιhξ /2)β sin(hξ /2))β ,ξ = , |ξ h,β | ≈ ± 2(ξ /2)β β β h h =1 =1 =1

for small h. The squares of the norm and seminorm, uh 2H k (Ωh ) , and |uh |2H k (Ωh ) , are related as +

2 h u

k (Ωh ) H+

=

2 2 h ∗ h ˆ (ξ) u ˆ (ξ), uh ξ h,α u

ξ∈Γh |α|≤k

k (Ωh ) H+

+

=

2 h ∗ h ˆ (ξ) u ˆ (ξ). ξ h,α u

ξ∈Γh |α|=k

8.5. Convergence for diﬀerence methods

603

ap (u, v), a(u, v) as in Chapter 2 by results similar to Theorem 1.26 and Proposition 2.24. (3) So we turn to the Vbh -coercivity of the ahp (uh , v h ), only indicating the proof for i u(x), and sum m = q = 1. In fact for (8.38) we only have to replace ϑi by ∂+ h over x ∈ Ω to get the claim. This is analogous to (8.38) where ϑi = ∂ i u(x) is combined with integrating over Ω. (4) For m, q > 1, so for systems (8.71) of order 2m, we impose the above (8.41). For equations and systems of order 2m, similarly to the situation in Theorem 2.36, we cannot repeat the above in (3) based upon (8.38). The αi i argumentation αi u = ∂+ u for αi > 1. So we apply instead reason is again the inequality ∂+ β α the discrete Fourier transform to each component ∂+ uj and ∂+ uj . So we prove: (4) With Theorem 8.24 for the discrete Fourier transform, we prove the Vbh = m (Ωh )-coercivity of the principal part. We only discuss constant coeﬃcients H0,+ Aαβ , |α|, |β| = m. The reduction of Aαβ ∈ C m (Ω) to constant coeﬃcients by the technique of frozen (constant) coeﬃcients is presented in the proof of Theorem 2.43. It can be applied to diﬀerence methods as well. We combine Theorem 8.24 with the Legendre–Hadamard condition (8.41). With the notation in (8.123), (8.124) and ξ h,β =

n n (exp(ιhξ ) − 1)β 2(exp(ιhξ /2)β sin(hξ /2))β = , β h hβ

=1

we obtain with ξ h,β = ⎛ q h h h ⎝ ap ( u , u ) =

β

n

n =1

ξh

h

, ϑ := ξh ,

=⎝ ⎛ =⎝ ⎛ =⎝

q

⎛

⎝hn ajk αβ

⎞⎞ β h α h ∂+ uk (x)∂+ uj (x)⎠⎠

x∈Ωh

β h α h ajk αβ ∂+ uk , ∂+ uj

ajk αβ

Fh

β h ∂+ uk

L2 (Ωh )

=

ajk αβ

j,k=1 |α|=|β|=m

≥λ

ξ∈Γh

n h 2 ξ =1

⎠

%

⎞

h

α , Fh ∂+ uj

j,k=1 |α|,|β|=m q

⎞

j,k=1 |α|,|β|=m q

⎞ α h ∂+ uj (x)⎠

x∈Ωh

j,k=1 |α|=|β|=m q

β h ajk αβ ∂+ uk (x)

j,k=1 |α|=|β|=m

⎛

(8.125)

=1

L2 (Γh )

⎠

& ξ h,β u ˆk (ξ) [ξ h,α u ˆj (ξ)] , by (8.41),(8.125)

ξ∈Γh

m

q h 2 u ˆj (ξ) . j=1

604

8. Finite diﬀerence methods

n h 2 m For a ﬁxed m > 1 there exists an > 0, such that > |α|=m |ξ h,α |2 . =1 ξ This is veriﬁed by the polynomial formula. So we ﬁnd with (8.124) ⎛ ⎞ q h 2 ⎝ ˆi (ξ) = λ| uh |2H m (Ωh ,Rq ) . ahp ( uh , uh ) > λ (ξ h,α )2 ⎠ u ξ∈Γh i=1

+

|α|=m

m (Ωh , Rq ), With the equivalent norms | uh |H+m (Ωh ,Rq ) and uh H+m (Ωh ,Rq ) on Vbh = H0,+ h h h h the Vb -coercivity of ap ( u , v ) is proved for constant coeﬃcients. The general case is proved as in Theorem 2.42.

8.5.5

Stability and convergence for general elliptic diﬀerence equations

In Subsection 8.5.3 we proved the consistency of our discrete operator with the original operators. Subsection 8.5.4 showed the Vb -coercivity of the principal parts of the linear(ized) operators, and hence its stability. This implies, by Theorem 8.21, the convergence of the discrete to the exact solutions of these elliptic problems for the principal part. The corresponding Lemma 8.28 is combined in Theorem 8.29 for proving the stability for boundedly invertible linear elliptic operators.

−m m (Ωh , Rq ), V h = H+ Condition 8.26. Choose Ubh = Vbh and their duals as H+ (Ωh , Rq ), cf. (8.27)–(8.31), (8.77), and P h = I h and Q h as in Subsection 8.5.2 for p = 2. The original linear and nonlinear problems and the diﬀerence methods are listed in Section 8.4. For the consistency, and its order 1 or 2, we refer to Theorems 8.21, 8.22, and 8.23; for the coercivity of the principal parts we refer to Theorem 8.25. m (Ωh , Rq ), in contrast to V = Remark 8.27. Notice that we have chosen the V h = H+ m,p q (Ω, R ). This allows our linearization approach based upon coercive principal W parts, as used in this section. The direct discrete counterpart V h = W+m,p (Ωh , Rq ) for V = W m,p (Ω, Rq ) is possible for quasilinear problems and their diﬀerence methods as well. Then we have to use the monotony technique in Section 4.5, and Theorem 4.67 remains valid for 1 < p < ∞ and e.g. cf. (4.197), (8.146) h P u0 − uh0 h ≤ hμ Lu0 W m+μ (Ω) 1/(p−1) , (8.126) U b

m and symmetric diﬀerence methods. for μ = 1 and = 2 for unsymmetric V h = H+

Even better results are obtainable by linearization techniques. They require 2 ≤ p < m (Ωh ) norm, cf. ∞ and yield stability and convergence with respect to the discrete H+ the discussion in Subsection 8.4.5, following (8.65). The following lemma enforces the consequences of Theorem 8.21. The diﬀerent errors are related with Ah uh = Q h Au as:

Q h Ah u − Ah uh = Q h Ah (u − P h u) + Ah P h u − Q h Au; Ah uh = Q h Au. < => ? => ? < => ? < var.cons.error class.cons.err. interp.error

(8.127)

8.5. Convergence for diﬀerence methods

605

Lemma 8.28. Under Condition 8.26, let Ap : Ub → V be a coercive principal part, cf. (5.278) or (5.281). Then Ap u = f ∈ W −m,p (Ω), and for small enough h0 > h ∈ H, Ahp uh = Q h f as well, have unique solutions, and uh converge to u as uh − P h uU h ≤ uU , for h < h0 , hence lim uh − P h uU h = 0. (8.128) h→0

For general A, and u ∈ Ub , the preceding consistency errors satisfy, for h < h0 ,

Q h Au − Ah P h uV h ≤ uU , hence lim Q h Au − Ah P h uV h = 0. h→0

(8.129)

Proof. We can essentially dublicate the proof of Lemma 5.76, if we replace u − uh U h there by P h u − uh U h here. Theorem 8.29. Stability inherited to invertible compact perturbations. Under Condition 8.26, let A, B ∈ L (Ub , Vb ) , Ub = W0m,p (Ω), Vb = W0−m,p (Ω), be boundedly

invertible, and B h ∈ L Ubh , Vbh

be stable, e.g. B = Ap . Let A := B + C, with C ∈

C (Ub , Vb ), the set of compact operators from Ub → Vb , and let Ah , B h , hence C h , be consistent with A, B, C. Then, for small enough h ∈ H, A−1 ∈ L(V , Ub ) =⇒ Ah is stable in P h u hence (Ah )−1 ∈ L Vbh , Ubh , (Ah )−1 U h ←V h ≤ C. b

b

m,p Remark 8.30. We can choose B = Δm . Then, for Ubh = W0,+ (Ωh ) with 2 < p ≤ ∞ m h only H0,+ (Ω )-coercivity, stability and convergence is obtained, cf. Summary 8.6, Remarks 8.7, 8.14 and Theorems 8.31, 8.32.

Proof. This proof essentially modiﬁes that for FEMs with variational crimes. We consider the appropriate extensions E h , and Eh , cf. (8.86), (8.131). (1) Again we deﬁne a kind of anticrime transformation from Ubh to Ub = W0m,p (Ω) ⊂ U, cf. Lemma 5.77. It allows a direct comparison between the elements in, m,p h Tc and Ub = W0m,p (Ω), Vb = W0−m,p (Ω). We again choose e.g. Ubh = W0,+ the boundedly invertible B : Ub → Vb , here B = Δm . We have proved B h : Ubh → Vbh to be stable and consistent with B for every u ∈ Ub , with a consistency order p > 0 for smoother u. In the following procedure we combine Proposition 8.18 and Theorem 8.21 with the results in Theorems 8.25 and 3.21: For any given uh ∈ Ubh deﬁne fˆh := B h uh ∈ V h , and fˆh := E h fˆh ∈ V ,

with fˆh V ≤ ||B h ||V h ←U h uh U h (1 + o(1)), cf. (8.30). b

(8.130)

606

8. Finite diﬀerence methods

We deﬁne the bounded Eh : Ubh → Ub : uh := u0,h := Eh uh := (B)−1 fˆh ∈ Ub = W0m,p (Ω) yielding

Eh uh U ≤ C1 uh U h , and ∀ > 0 ∃h0 = h0 uh 1 h : ∀h < h0 : H (Tc )

(8.131)

h P E h u h − u h

Ubh

< uh Ubh , hence, lim P h Eh uh − uh Ubh = 0. h→0

(2) We derive several relations and inequalities. For an arbitrary u ∈ Ub , and v := ˆ, and u ˆh , of Cu ∈ Vb , we determine the exact, and discrete solutions, u

Bu ˆ = v and B h u ˆh = Q h Cu = Q h v ∈ Vbh . By assumption they uniquely exist and the u ˆh converge to u ˆ in the sense that h ∀ > 0 ∃h0 = h0 (ˆ uU ) : ∀h < h0 : P u ˆ−u ˆh U h < ˆ uU . (8.132)

With the operators B −1 , (B h )−1 Q h , and the notation T := B −1 ∈ L (Vb , Ub ) , and

T h := (B h )−1 Q h ∈ L Vb ∪ Vbh , Ubh , Q h |V h = Id, b

we relate this result to the stability and consistency of B h , as

Bhu ˆh − B h P h u ˆ = Q h Cu − Q h B u ˆ + Q hBu ˆ − BhP hu ˆ = Q hBu ˆ − BhP hu ˆ⇒

ˆ−u ˆh U h ≤ (B h )−1 U h ←V h Q h B u ˆ − BhP hu ˆV h → 0 for h → 0. P h u b

b

b

Applying B u ˆ = Cu and the equibounded (B h )−1 to the terms in the last norm, we ﬁnd

(B h )−1 Q h Cu − P h B −1 CuU h = (T h − P h T )CuU h → 0 for h → 0 ∀ u ∈ Ub . Since C is compact, and (T h − P h T )Ubh ←Vb are equibounded, this implies h (T − P h T )C h → 0 for h → 0. (8.133) U ←U b

b

We combine uh ∈ Ubh with Eh . With the boundedly invertible A we estimate uh U h ≤ 2 Eh uh U ≤ 2 A−1 U ←V AEh uh V b = 2 A−1 U ←V B(I + T C)Eh uh V (8.134) b −1 ≤ 2 A U ←V BV ←Ub (I + T C)Eh uh U , b

hence, (I + T C)Eh uh ≥ uh h /(2 A−1 U U U

b ←V

BV ←Ub ).

(8.135)

8.5. Convergence for diﬀerence methods

607

With the stability of B h we get (I + T h C h )uh h = (B h )−1 B h (I + T h C h )uh h U U h h −1 ≤ (B ) U h ←V h B (I + T h C h )uh V h . b

This implies h B (I + T h C h )uh h ≥ (I + T h C h )uh h / (B h )−1 h . V U U ←V h

(8.136)

b

The linearity of diﬀerence methods implies Φh (A) = Ah = Φh (B + C) = B h + C h

(8.137)

= B h (I + (B h )−1 C h ) = B h (I + T h C h ) : Ubh → V h . (3) For the ﬁnal stability estimates we return to (8.136) with (B h )−1 Ah uh = (I + T h C h )uh , and use (8.137), cf. Remark 8.14. We extend Q h : V → V h as Q h : V ∪ V h → V h by deﬁning for the new component V h , the Q h |V h = IV h . This is combined with T h |V h = (B h )−1 Q h |V h = (B h )−1 , the equiboundedh h h ness of C and T , the consistency of C , the indicated error terms, and the triangle inequality estimating (B h )−1 U h ←V h ||Ah uh ||V h b

(8.138)

b

= (B h )−1 U h ←V h ||B h (I +(B h )−1 Q h C h )uh ||V h b

b

≥ (B h )−1 U h ←V h ||B h (I + T h C h )uh ||V h b b

h ≥ ||P (I + T C)Eh uh ||U − ||uh − P h Eh uh ||U h − ||T h C h uh − P h T CEh uh ||U h ≥ P h ||(I + T C)Eh uh ||U − ||uh − P h Eh uh ||U h

− ||(T h − P h T )CEh uh ||U h − ||T h (C h uh − CEh uh )||U h ≥ ||(I + T C)Eh uh ||U /2 − ||uh − P h Eh uh ||U h − ||(T h − P h T )CEh uh ||U h − ||T h (C h P h Eh uh − CEh uh )||U h

h h h h h ⎛ − ||T (C − C P Eh )u ||U h ⎜ ≥ ⎝|| (I + T C)Eh uh ||U /2 − || uh − P h Eh uh ||U h < => ? < => ? by (8.135)

diﬀ. err. (8.131)

− 2|| (T h − P h T )C ||Ubh ←Ub ||uh ||U h < => ?

⎞

by (8.133)

⎟ − ||T h (C h P h Eh uh − CEh uh ) ||U h − ||T h C h (uh − P h Eh uh ) ||U h ⎠ . < < => ? => ? class.consist. C,(8.129)

diﬀ. error (8.131)

608

8. Finite diﬀerence methods

We choose h small enough, such that all the preceding errors in (8.131), (8.133), (8.129), (8.89), (8.91) are < ||uh ||U h with

< 1/ 8 2 + ||T h ||U h ←V h + ||T h C h ||Ubh ←Ubh ||A−1 ||Ub ←V ||B||V ←Ub . b

b

This implies

−1 ||Ah uh ||V h ≥ ||uh ||U h / 8||A−1 ||Ub ←V ||B||V ←Ub /||B h ||U h ←V h b

(8.139)

≥ K ||uh ||U h , i.e. (Ah )h∈H is stable.

For diﬀerence methods, Theorem 8.29, similarly to Theorem 5.78 for FEMs, yields a criterion for the stability of discretizations of operators, which are compact perturbations of coercive operators. An important class are A ∈ L (Ub , Ub ) satisfying a so-called G˚ arding inequality. Hence Theorem 3.32 can directly be generalized to our diﬀerence methods. We combine the consistency and coercivity/stability results in Theorems 8.21–8.23, 8.29 with Condition 8.26 for obtaining the following theorem. We only formulate the convergence results of orders μ for u ∈ W m+μ,p (Ω) and μ = 1, 2. For the change from H m (Ω) to W m+μ,p (Ω), compare the remarks at the end of Subsection 8.4.4. For convergence for u ∈ W m,p (Ω) we refer directly to Theorems 8.21–8.23. We start listing the conditions for linear, quasilinear and fully nonlinear problems. 1. We use in Theorem 8.31 the following spaces, cf. (8.77), often we omit Rq and set p = p = 2

Ub = Vb = W0m,p (Ω, Rq ), Vb = W0−m,p (Ω, Rq ), Ubh = Vbh =

m,p W0,+ (Ωh , Rq ),

Vbh = Ubh =

and for strong equations Ush =

for m, q ≥ 1, 1 ≤ p ≤ ∞,

−m,p W+,0 (Ωh , Rq ),

h W+2m,p (Ωh ), Us,b

= Ush ∩

(8.140)

m,p W0,+ (Ωh ),

Vsh = Vsh = L2+ (Ωh ). 2. For μ = 1, 2, and for linear problems we assume, cf. Remark 8.7,

aα,β ∈ C(Ω) ∩ W μ,∞ (Ω), u ∈ W m+μ,p (Ω), A : W m+μ,p (Ω) → W −m+μ,p . (8.141) 3. For quasilinear problems we assume, with Nm in (8.17), Aj , Aα equicontinuous in Ω × N∇m ⊂ Ω × RNm , N∇m bounded, s.t. ≤m

∀u : u − u0 V ≤ ρ, V, ∇

u(x) ∈ N∇m ∀x ∈

Ω, ∇≤m + u(x)

(8.142)

∈ N∇m ∀x ∈ Ωh0 ,

with Lipschitz-continuous Aj , Aα with respect to the variables in N∇m . Finally let for

u ∈ W m+μ,p (Ω), Aα : D(Aα ) ∩ W m+μ,p (Ω) → W −m+μ,p (Ω))

(8.143)

8.5. Convergence for diﬀerence methods

609

4. For fully nonlinear problems, here only formulated for order 2m = 2, let G be equicontinuous in Ω × N∇2 ⊂ Ω × RN2 with bounded N∇2 , s.t.

(8.144)

h ∀u : u − u0 V ≤ ρ, V, ∇≤2 u(x) ∈ N∇2 ∀x ∈ Ω, ∇≤2 + u(x) ∈ N∇2 ∀x ∈ Ω0 ,

with Lipschitz-continuous G with respect to the variables in N∇2 . Furthermore require for u ∈ H 2+μ (Ω), G : D(G) ⊂ H 2+μ (Ω) → H μ (Ω)).

(8.145)

We obtain consitency of order 1 and 2, respectively. Theorem 8.31. Convergence for symmetric and unsymmetric ﬁnite diﬀerence methods: Let Ub = Vb ⊂ V and Ubh = Vbh be the spaces and their duals in (8.140), cf. (8.27)–(8.31), (8.77), and P h = I h and Q h the projectors in Subsection 8.5.2. For the original linear and nonlinear elliptic problems, Au = f ∈ W −m.p (Ω, Rq ) and −m.p q (Ω, R ) deﬁne the diﬀerence methods as listed in Section 8.4, and G(u) = 0 ∈ W impose, for the diﬀerent cases, the conditions (8.141)–(8.145). Finally, let A or G (u0 ) : W m,p (Ω, Rq ) → W −m,p (Ω, Rq ) be boundedly invertible. 1. Then under the conditions in Chapter 2, a (locally) unique exact solution u0 and for small enough h, a unique discrete solution uh0 for these problems exist. m (Ωh ) 2. Stability, and hence convergence, follows for 2 ≤ p < ∞ from the H0,+ h h coercivity of the principal part ap (.,.) in the · H+m (Ω ) norm. m,p 3. If for U0h = Vbh = W+ (Ωh ) with 2 ≤ p < ∞ an inf-sup condition for ahp (.,.) is h valid, the u0 converge in · W+m,p (Ωh ) . 4. For strong diﬀerence equations Theorem 8.21 can be updated. More precisely, we obtain convergence of order μ, with μ = 1, and 2, for unsymmetric and symmetric diﬀerence methods, cf. (8.105), (8.106), (8.103), for the linear and quasilinear problems. Symmetric methods, with μ = 2, are worthwhile in cuboidal domains, cf. (8.95). h P u0 − uh0 h ≤ Chμ u0 W m+μ,p (Ω) for μ = 1, 2, u0 ∈ W m+μ,p (Ω), (8.146) U with p = 2 and p > 2 for the two previous cases in 2. and 3., respectively. For fully nonlinear second order problems we ﬁnd h P u0 − uh0 h ≤ Chμ u0 H 2+μ (Ω) for u0 ∈ H 2+μ (Ω). (8.147) U Theorem 8.32. Diﬀerence methods for monotone operators, eigenvalue problems, nonlinear boundary operators, quadrature approximations: Under the conditions of Theorems 8.29 and 8.31, diﬀerence methods applied to the following problems yield unique converging solutions. For monotone operators the convergence satisﬁes (4.196)– (4.198). For the other problems (8.146) and (8.147) are valid: variational methods for eigenvalue problems, cf. Section 4.7, for nonlinear boundary operators, cf. Section 5.3, quadrature approximations, cf. Section 5.4.

610

8. Finite diﬀerence methods

Solving nonlinear equations in diﬀerence methods As for FEMs and DCGMs we use a discrete Newton’s method based upon the mesh independence principle (MIP). For the MIP we have to determine the linearized form of the nonlinear system and its discretization, cf. Subsections 8.4.6, 8.4.7. The derivatives have to be consistently diﬀerentiable and of order μ, hence, additionally to the standard consistency,

(Gh ) (P h u)P h v − Q h (G (u)v)V h → 0 for small u − u0 V and, μ = 1, 2,

(Gh ) (P h u)P h v − Q h (G (u)v)V h ≤ Chμ (1 + uUs )vUs for h → 0.

(8.148)

These vUs are usually vW m+μ,p (Ω) and the above m, μ, p. The proof of this property follows essentially the lines of FEMs and DCGMs, so we do not repeat it here, cf. Theorems 5.21, 7.46, 7.47. We transform these results to diﬀerence equations: Theorem 8.33. Discrete Newton method for diﬀerence equations Gh : Let u0 ∈ of G(u0 ) = 0, such that G (u0 ) : U → V is boundedly H s (Ω) be an isolated solution h h invertible and let G u0 = 0 indicate one of the diﬀerence equations for the nonlinear problems in Subsections 8.4.6, 8.4.7. Assume G and G (u) to be Lipschitz-continuous with respect to the variables u, for u − u0 V small enough. Let the conditions in Theorems 8.21–8.23 be satisﬁed for G and G . Then the corresponding diﬀerence equation Gh for G is consistently diﬀerentiable. Start the original Newton process with u1 ∈ W m+μ,p (Ω), μ = 1, 2, and the discrete Newton process for i = 1, . . . , with uh1 := P h u1 , for small enough u0 − u1 U and h. Then in the original Newton method and in −1 (8.149) Gh uhi . uhi+1 := uhi − Gh uhi the ui+1 and the uhi+1 in (3.84) uniquely exist and converge quadratically to u0 and to uh0 and uhi to ui+1 of order μ, such that h ui+1 − uhi h ≤ C uhi − uhi−1 2 h and h u i − P h u i

U

Uh

8.6

U

≤ Chμ ui Us , i = 1, . . . .

Natural boundary value problems of order 2

We turn to natural boundary, e.g. Neumann conditions, instead of the previous Dirichlet case. We only consider second order equations, and their symmetric diﬀerence methods on cuboidal domains, cf. (8.95). Generalizations to order 2m are possible, but require many technicalities, cf. the discussion in Subsection 2.4.2 (2.92) ﬀ. and Subsection 2.4.4, Lions and Magenes [478], p. 120, Oden and Reddy [518], p. 290 ﬀ. Other boundary conditions are discussed, e.g. by Collatz [206], Forsythe and Wasow [319] and Hackbusch [387]. Standard natural boundary conditions as in (8.152) for linear problems (8.151) are generalized to quasilinear problems in (8.160). Mixed boundary value problems can be included by generalized test functions. As in the previous sections, there is nearly as

8.6. Natural boundary value problems of order 2

611

close a relation between the weak and strong diﬀerential equations as in (8.151)–(8.152) for natural boundary conditions, cf. Proposition 8.35. This allows us to prove unique existence of discrete solutions, uh0 , and their convergence of order 1 very similarly to the previous sections. For the proof of the full order 2 of convergence of this diﬀerence method, the technique in Section 5.2 for fully nonlinear elliptic problems with FEMs would have to be employed. We omit that here. 8.6.1

Analysis for natural boundary value problems

As mentioned already, we only study second order elliptic problems in U := V := H 1 (Ω), U := H −1 (Ω), with Ub := Vb := H01 (Ω), Ub := H0−1 (Ω). (8.150) We give a short summary of the analytic results for natural boundary value problems of second order, cf. Chapter 2. They again allow the transformation from strong to weak forms, cf. (2.24), and Hackbusch [387], Section 7.4. The As , A and a(·, ·) are maintained and the boundary conditions in (8.43)–(8.45) are modiﬁed. The conditions for the coeﬃcients of A are formulated in Summary 8.6, and, for diﬀerence methods, in Remark 8.7. We solve Au0 = f ∈ U , u0 ∈ U (or for u0 ∈ H 2 (Ω) : As u0 = g ∈ L2 (Ω), Ba u0 |∂Ω ∈ L2 (∂Ω)) n a(u0 , v) = aij ∂ i u0 ∂ j vdx = Au0 , vU ×U = f, vU ×U (8.151) i,j=0

Ω

= (As u0 , v)L2 (Ω) +

(Ba u0 )vds ∂Ω

with the natural boundary operator Ba u :=

j=1,...,n

ν j aij ∂ i u cf. (2.25).

(8.152)

i=0,...,n

We do not consider the general f ∈ U , e.g. in (8.71), but restrict it in (8.151), (8.156), to the form ˆ ϕvds, ∀v ∈ U with (8.153) f (v) = f, vU ×U = f vdx + Ω

∂Ω

1/2 . fˆ ∈ L2 (Ω), ϕ ∈ H −1/2 (∂Ω), f U = fˆ2U + ϕ2H −1/2 (∂Ω) b

For c0 > 0, the extended principal part of a(u, v), in (8.151), ap,e (u, v) := ap (u, v) + c0 (u, v)L2 (Ω) : U × U → R, c0 > 0,

(8.154)

is U-coercive, cf. Theorems 2.43, 2.89, Corollary 2.44. Let A, and Ap,e : U → U , be the bounded linear operators induced by a(u, v), and ap,e (u, v). Hence Theorem 2.12 implies a unique solution for f ∈ U , ∀f ∈ U : ∃1 u1 ∈ U : ap,e (u1 , v) = f, vU ×U ∀v ∈ U,

(8.155)

612

8. Finite diﬀerence methods

and for A−1 ∈ L(H −1 (Ω), U) ⇒ ∃1 u0 ∈ U : a(u0 , v) = f, vU ×U ∀v ∈ U.

(8.156)

Ap,e is coercive and its diﬀerence form Ahp,e will turn out to be consistent with Ap,e . So Theorem 3.29 implies the stability of a boundedly invertible A, as a compact perturbation of Ap,e . The most important special case, showing some diﬃculties, is the Laplacian n ∂ i u∂ i v = (As u, v)L2 (Ω) + v∂u/∂νds As u = −Δu, Au0 = fˆ, a(u, v) = i=1

Ω

∂Ω

Neumann boundary condition Ba u = ∂u/∂ν = ϕ, fˆ ∈ L2 (Ω), ϕ ∈ L2 (∂Ω). (8.157) Obviously, for any constant c ∈ R we ﬁnd −Δc = 0, ∂c/∂ν = 0, so (8.157) is not uniquely solvable. Hence the kernel, ker F (u0 ) ∼ = R of the operator F u := (−Δu, ∂u/∂ν) is not empty. By the Fredholm alternative, cf. Theorem 2.21, a unique solution exists only for the modiﬁed problem ˆ ϕds = 0. f0 dx + ∃1 u0 ⊥ ker F (u0 ) ⇔ f ⊥ ker F (u0 ) ⇔ f, 1U ×U = Ω

∂Ω

This is a special form of conditions relevant in bifurcation. A direct approach is discussed in [387]. We will study these problems in [120]. The numerical Liapunov– Schmidt method systematically proves the convergence for these cases. For avoiding these diﬃculties, we assume here a boundedly invertible linear operator, A. Then we solve (8.156), and summarize consequences for the diﬀerential equation and the boundary condition, cf. Remark 2.6. We will partially prove again the results in Theorem 2.43 and Corollary 2.44. The transition in (8.159) from the weak Au0 − f = 0 to the strong form is only possible for special f ∈ H −1 (Ω), cf. (8.153), so we assume a Lipschitz-continuous ∂Ω, and Au0 − f = 0 ⇔ As u0 = fˆ = fˆ0 ∈ L2 (Ω), Ba u0 = ϕ ∈ H −1/2 (∂Ω).

(8.158)

Theorem 8.34. Analytical results for natural boundary conditions: 1. The ap,e (·, ·) in (8.154) is V-coercive, the original in (8.156) is V-elliptic, and satisﬁes the Fredholm alternative. Regularity results are available in Theorems 2.45 and 2.47. The latter yields u0 ∈ H 2 (Ω) for convex Ω and coeﬃcients satisfying (2.149), (2.47), and fˆ ∈ L2 (Ω), ϕ ∈ H 1/2 (∂Ω) in (8.158). 2. The solution, u0 ∈ H 1 (Ω), of (8.156), (8.158), with f in (8.153), satisﬁes Au0 = fˆ ∈ L2 (Ω), Ba u0 = ϕ ∈ H −1/2 (∂Ω) or, cf (8.158), for smooth data u0 ∈ H 2 (Ω), As u0 = fˆ ∈ L2 (Ω), Ba u0 = ϕ ∈ H 1/2 (∂Ω).

(8.159)

3. An analogous result holds for the quasilinear problems in (8.160), (8.161). Proof. We start testing (8.156) with ∀v ∈ Ub . This eliminates the boundary term, ∂Ω ϕvds = ϕ, vH −1/2 (∂Ω)×H 1/2 (∂Ω) = 0, by 0 = v|∂Ω ∈ H 1/2 (∂Ω), and yields a(u0 , v) = (fˆ, v)L2 (Ω) ∀v ∈ Ub , hence Au0 − fˆ ∈ L2 (Ω), by (8.156), (8.151), and since

8.6. Natural boundary value problems of order 2

613

L2 (Ω) is dense in V. We even get u0 ∈ H 2 (Ω), for fˆ ∈ L2 (Ω), ϕ ∈ H 1/2 (∂Ω), smooth enough ∂Ω and coeﬃcients, or convex ∂Ω, hence As u0 = fˆ cf. Theorem 2.45. Next, testing (8.156) with general ∀v ∈ U implies ∂Ω (Ba u − ϕ)vds = 0 ∀v|∂Ω ∈ H 1/2 (∂Ω). For the general quasilinear systems of order 2 we obtain Gs ( u) :=

n

(−1)k>0 ∂ k Ak (x, u, ∇ u)

(8.160)

k=0

∀ u ∈ V = H 2 (Ω, Rq ), v ∈ H 1 (Ω, Rq ) n (Ak (x, u, ∇ u), ∂ k v)q dx Gu, vV ×V = a( u, v ) = k=0

−

BA ( u, ∇ u), v q ds,

∂Ω

where BA ( u, ∇ u) :=

n

ν k Ak (·, u, ∇ u),

k=1

For these general quasilinear systems of order 2 we obtain, with the precise conditions for the Ak discussed in Chapter 2, for diﬀerence equations in Remark 8.7, (8.111), and with Ak (·, u, ∇ u)(x) ∈ Rq , BA ( u, ∇ u)(x) ∈ Rq : Gs ( u) :=

n

(−1)k>0 ∂ k Ak (x, u, ∇ u) ∈ L2 (Ω, Rq ), ∀ u ∈ H 2 (Ω, Rq ), v ∈ H 1 (Ω, Rq )

k=0

(Gs ( u), v )L2 (Ω,Rq ) +

e∈∂Ω

e

BA ( u, ∇ u), v q ds =

n

(Ak (x, u, ∇ u), ∂ k v)q dx

Ω k=0

= G( u), v H −1 (Ω,Rq )×H 1 (Ω,Rq ) , and u ∈ H 1 (Ω, Rq ). 8.6.2

(8.161)

Diﬀerence methods for natural boundary value problems

For these natural boundary value problems we are going to deﬁne the corresponding diﬀerence methods. As in Section 8.4 we could formulate the ﬁrst order unsymmetric forms for domains Ω in (8.98). This would yield similar equations as for Dirichlet conditions, however with a modiﬁed discrete natural boundary operator Bah . Or we assume (8.95) and restrict the presentation to symmetric diﬀerence approximations, cf. Hackbusch [387], Subsections 4.7.2, 4.7.3 and Section 7.4. We only present this second choice in two diﬀerent variants. We start by extending the original Ωh by a layer of exterior points. Another possibility, a shifted grid, will only be indicated. 8.4.3, we have to avoid the ∂hj uh , and instead use arithmetic means,

As in Subsection j j uh /2. This complicates the presentation. It requires modifying the previous ∂− + ∂+

614

8. Finite diﬀerence methods

Ωh in (8.96), (8.97). For Ω :=

n

(ai , bi ), Ωh := Ω ∩ G h ,

(8.162)

i=1

we add to Ωh left and right outer boundary grid points, x ∈ ∂Ωhl and x ∈ ∂Ωhr both ⊂ G h in the distance h from Ωh , more precisely ∂Ωhl := {x = (x1 , . . . , xn ) ∈ G h : ∃j : xj = aj − h, ∀i = j : ai ≤ xi ≤ bi }, ∂Ωhr

(8.163)

:= {x = (x1 , . . . , xn ) ∈ G : ∃j : xj = bj − h, ∀i = j : ai ≤ xi ≤ bi } and h

Ωhr := Ωh ∪ ∂Ωhr , Ωhl := Ωh ∪ ∂Ωhl , Ωhe := Ωhr ∪ ∂Ωhl , ∂Ωhe := ∂Ωhr ∪ ∂Ωhl . The results for unsymmetric diﬀerence equations are obtained essentially by choosing one of the unsymmetric versions used in the following formulas for the averaging. Straightforward extensions to systems with u(x) ∈ Rq are omitted here. In this section the outer normal vectors ν in boundary grid points, x ∈ ∂Ωh , play a crucial role. Sometimes even the number, #ν(x) ≥ 1, of these unit normals in x, or their parerallel shifts to ∂Ωhe , are relevant: for x ∈ ∂Ωh , x ∈ interior of a face of ∂Ω : #ν(x) = 1, otherwise #ν(x) > 1. (8.164) In particular, we ﬁnd for n = 2 two normals in the vertices, and for n = 3 two normals in the edges, and three in the vertices. For an interior point x of a face of ∂Ω, hence #ν(x) = 1, this ν = ±ej for one j. For an x in an edge or a vertex x, we need #ν(x) unit vectors ej . For every j = 1, . . . , n, there are two boundary points x ∈ ∂Ωje ⊂ ∂Ωh , cf. (8.176), with the outer normal ν = −ej in ∂Ωjl and ν = +ej in ∂Ωjr , ν j = ∓1, cf. (8.163). h j Consequently, we use in the following formulas the notation ± a,j ∂± u (· ∓ he ) = j h ν a,j ∂± u (· ∓ hν). Accordingly, we modify, for our symmetric forms, the discrete functions,uh , the Sobolev spaces, the inner product, seminorms, norms. This yields, with the Ωh , Ωh , l , cf. (8.25), (8.163), and a new boundary integral, e.g. Ωh r uh , v h : Ωhe → R, uh dxh := hn uh (x), uh dsh := hn−1 uh (x), Ωh

(uh , v h )H±1 (Ωhe ) := and

1 2

x∈Ωh

Ωh l

n

∂Ωh e

j h j h ∂+ u ∂+ v dxh +

j

x∈∂Ωh e n

Ωh r j=0

j h j h ∂− u0 ∂− v dxh ,

h 1 Ωe := uh : Ωhe → R, uh H±1 (Ωhe ) < ∞ . V h = H±

(8.165)

(8.166)

The relation between the weak and the strong linear diﬀerence operators is based upon the natural discrete boundary operator Bah uh and ∂Ωh v h Bah uh dsh , cf. (8.174). e Accordingly, we modify the previous Ah , Ah , ah (·, ·), fˆh (·), into the form for natural s

8.6. Natural boundary value problems of order 2

615

boundary conditions motivated by (8.151)–(8.152), (8.153), and Proposition 8.35. We maintain the notation Ah , Ahs , . . . , instead of Ahws , Ahss , . . . , for the weak and strong form of symmetric diﬀerence equations in Subsection 8.4.3: h h h 2 1 1 Ωe → L2 (Ωh ), Ah : V h := H± Ωe → V h , Bah : H± ∂Ωe → L2 ∂Ωhe , Ahs : H± fh : V

→ R, uh , uh0 , v h ∈ V h , cf. (8.166): n

1 j j h h h h a,j ∂+ aj ∂− (x) (−1)j>0 ∂− u + ∂+ u As u (x) := 2 h

(8.167)

j,=0

ah (uh , v h ) := Ah uh , v h V h ×V h ⎛ ⎞ n n 1⎝ h j h h j h a,j ∂+ u0 ∂+ v dxh + aj ∂− u0 ∂− v dxh ⎠, := 2 h Ωh Ω r l j,=0

(8.168)

j,=0

⎛ h h Bah uh |∂Ωhl := Ba,l u :=

1⎝ 2

j=1...,n

h

⎞

h ν j a,j ∂+ u + aj ∂− u (· + hej ) ⎠, in x ∈ ∂Ωhl

=0,...,n

Bah uh |∂Ωhr

⎛ ⎞ j=1...,n

1 h h h ⎠ in x ∈ ∂Ωhr , := Ba,r uh := ⎝ ν j (a,j ∂+ u )(· − hej ) + aj ∂− u 2

=0,...,n

v ∂Ωh e

h

Bah uh dsh

f , v := h

:=

v

h

∂Ωh l

h h h Ba,l u ds

fˆh v h dxh +

h h

Ωh

∂Ωh e

+ ∂Ωh r

h v h Ba,r uh dsh , and

v h ϕh (· − hν/2)dsh , cf.(8.153).

(8.169) (8.170)

1 n 1 and V h = H± , . . . , ∂± (Ωh , Rq ), Quasilinear diﬀerence systems satisfy, with ∇h± = ∂± (8.171) Gh uh , v h V h ×V h = ah ( uh , v h ) ⎛ ⎞ n n h h h j h h j h h 1⎝ Aj ·, u , ∇+ u ∂+ v dx + Aj ·, uh , ∇h− uh ∂− v dx ⎠ := h 2 Ωl j=0 Ωh r j=0 =

1 h j j v , (−1)j>0 ∂− Aj ·, uh , ∇h+ uh + ∂+ Aj ·, uh , ∇h− uh dxh (8.172) 2 q h Ω j=0 n

+ ∂Ωh l

h h h v h Ba,l u ds +

=: Ghs uh , v h L2 (Ωh ) +

∂Ωh e

∂Ωh r

h v h Ba,r uh dsh

v h Bah uh dsh , with

(8.173)

616

8. Finite diﬀerence methods

Bah uh |∂Ωhl

Bah uh |∂Ωhr

⎛ ⎞ n

1 h h := Ba,l u := ⎝ ν j Aj ·, uh , ∇h+ uh + Aj ·, uh , ∇h− uh (· + hej ) ⎠, 2 j=1 ⎛ ⎞ n

1 h := Ba,r uh := ⎝ ν j Aj ·, uh , ∇h+ uh (· − hej ) + Aj ·, uh , ∇h− uh ⎠ 2 j=1

Proposition 8.35. For natural boundary conditions, the linear symmetric weak and strong diﬀerence operators, Ah uh , Ahs uh , ah (uh , v h ), and the discrete boundary operators, Bah uh , in (8.167)–(8.169), are related by h h h h h h h h h v h Bah uh dsh . (8.174) a (u , v ) = A u , v V h ×V h = As u , v L2 (Ωh ) + ∂Ωh e

1 n 1 and V h = H± Quasilinear diﬀerence systems satisfy, with ∇h± = ∂± , . . . , ∂± (Ωh , Rq ), cf. (8.171)–(8.173) h h h h h h h h h v h Bah uh dsh . (8.175) G u , v V h ×V h = a ( u , v ) = Gs u , v L2 (Ωh ) + ∂Ωh e

Proof. Again, cf. (8.53), we start with the ﬁrst order result for j > 0 and the simpliﬁed notation for grid points and function values in Ωh in the ej direction: Ωj := {xi := x0 + ihej , i = 0, . . . , k},

(8.176)

Ωjr := {xi := x0 + ihej , i = 0, . . . , k + 1}, Ωjl := {xi := x0 + ihej , i = −1, . . . , k}, ⊂ G h cf. (8.163). Then we obtain for j > 0 : −

j h v h ∂− u

=−

Ωj

=−

k

vi u i −

vi +1 ui

/h

i =−1

i=0 k

k−1

(vi ui − vi+1 ui ) − v0 u−1 + vk+1 uk /h

i=0

=

k

j ui ∂+ vi + (v0 u−1 − v−1 u−1 + v−1 u−1 − vk+1 uk )/h

i=0

=

j h uh ∂+ v + (v−1 u−1 − vk+1 uk )/h

Ωjl

=−

Ωj

j h v h ∂− u

(8.177)

8.6. Natural boundary value problems of order 2

617

and −

j h v h ∂+ u =

=

j ui ∂− vi + v−1 u0 − vk uk+1 + vk+1 uk+1 − vk+1 uk+1 /h

i=0

Ωj

k

j h uh ∂− v

j h + v−1 u0 − vk+1 uk+1 /h = − v h ∂+ u .

Ωjr

(8.178)

Ωj

n 0 h h With ∂± u = uh, we insert for uh in (8.177) the wh := =0 a,j ∂+ u , and in n h n (8.178) the wh := =0 aj ∂− u , multiply with h , and sum for j ≥ 0: vh Ωh

n

j h a,j ∂+ (−1)j>0 ∂− u dxh =

n

Ωh l

j,=0

h j h a,j ∂+ u ∂+ v dxh

j,=0

⎛

vh ⎝

+ ∂Ωh l

⎞ h⎠ dsh a,j ∂+ u

=0,...,n

⎛

−

j=1...,n

(8.179)

v ⎝ h

∂Ωh r

j=1...,n

⎞ h⎠ (· a,j ∂+ u

− hej )dsh

=0,...,n

and Ωh

v

h

n

j h aj ∂− (−1)j>0 ∂+ u dxh j,=0

n

=

Ωh r j,=0

+ ∂Ωh l

−

∂Ωh r

h j h aj ∂− u ∂− v dxh

⎛

vh ⎝

j=1...,n

(8.180) ⎞

h⎠ (· + hej )dsh aj ∂− u

=0,...,n

⎛ vh ⎝

j=1...,n

⎞ h⎠ dsh . aj ∂− u

=0,...,n

We take the arithmetic mean of the last two equations and ﬁnd (8.174). Similarly, applying these results to uh = Aj (·, uh , ∇ uh ), yields (8.175).

For the second variant we consider the shifted grid, shown for n = 2 in Figure 8.5, and we modify (8.163). To the inner grid points Ωh ⊂ Ω of the shifted grid G h , we add those grid points in G h with distance ≤ h to Ωh , cf. (8.95)–(8.97): For Ωh let Ωh := Ωh ∪ {◦ in Figure 8.5} and ∂Ωh := {× in Figure 8.5}.

(8.181)

Compared to the ﬁrst variant, we mainly have to modify the boundary conditions, but do not give the details. For, e.g. Neumann conditions, choose for each x ∈ ∂Ωh , never a vertex of Ω, the next neighbor xh = x ∓ hν/2 ∈ Ωh , and the corresponding outer next neighbor xh±h = x ± hν/2 ∈ {◦ in Figure 8.5} ⊂ Gnh \ Ω. Then we require,

618

8. Finite diﬀerence methods

: points in Ω h : points in ∂Ω h

Figure 8.5 Two-dimensional grid for a square, and Neumann conditions.

cf. (8.174), ν uh (x) ≈ ϕ(x) = ∂h/2

∂ u(x)∀x ∈ ∂Ωh . ∂ν

(8.182)

We might want to avoid outer points xh±h ∈ {◦ in Figure 8.5} outside Ωh . To this end we combine, in the ﬁve-point star, the above xh = x ∓ hν/2 ∈ Ωh with the corresponding xh±h ∈ Ωh . We replace the value for the uh xh±h by a linear combination of ϕ(x), and uh (xh ), computed from (8.182). For an x ∈ ∂Ωh , next to an edge for n > 2 or a vertex of Ωh , for n ≥ 2, e.g. the point • next to (0, 0) in Figure 8.5, there are at least two directions with neigboring points xh±h for x to apply this replacement. For Neumann boundary conditions we are done. Now we have to solve ﬁrst the linear then the quasilinear problems. −1 (Ωh ), (8.183) Ahs uh0 (x) = fˆh (x) ∀x ∈ Ωh , fˆh ∈ H± : ; h h h h h h h h h h h h h u0 ∈ V : a u0 , v = A u0 , v V h ×V h = f , v , ∀v ∈ V , (8.184)

Bah uh0 (x) = ϕh (x − hν/2)∀x ∈ ∂Ωhe , or

v

∂Ωh e

h

Bah uh0 dsh

:= ∂Ωh e

v h ϕh (· − hν/2)dsh ,

ah uh0 , v h = Ahs uh0 , v h L2 (Ωh ) + h

∂Ωh e

v h Bah uh0 dsh = f h , v h h ∀v h ∈ V h . (8.185)

Either fˆ, ϕ are continuous or we use appropriate approximations as in Proposition 8.5. The original f and ϕ deﬁne the corresponding fˆh := Q h fˆ and ϕh := Q h ϕ in (8.170) and (8.191). The quasilinear diﬀerence systems has the form v h Bah uh dsh = 0. (8.186) Gh uh , v h V h ×V h = ah ( uh , v h ) = Ghs uh , v h L2 (Ωh ) + ∂Ωh e

8.6. Natural boundary value problems of order 2

619

As in Subsection 8.6.1 we have the two choices of either the weak or the strong formulation. The latter essentially yields the classical diﬀerence equations and boundary conditions. The weak form allows using our variational techniques. For this we prove stability, unique discrete solutions and their convergence for our diﬀerence method. The necessary relation between the weak Ah , the strong Ahs , the boundary condition, Bah , the coercivity and consistency are collected in Propositions 8.36–8.38. They guarantee the claim in Theorem 8.39. So we contrast the weak solution, uh0 , cf. Proposition 8.35, similarly to Theorem 8.34, deﬁned by : ; h 1 Ωe . (8.187) ah uh0 , v h = Ah uh0 , v h V h ×V h = f h , v h ∀v ∈ V h = H± The strong solution solves the diﬀerence equation and the discrete natural boundary conditions:

Ahs uh0 − fˆh (x) = 0∀x ∈ Ωh and Bah uh0 (x) − ϕh = 0∀x ∈ ∂Ωhe . (8.188) h 2 Ωe , of (8.188) satisﬁes (8.187) as well, Proposition 8.36. Every solution, uh0 ∈ H± and vice versa. An analogous result holds for quasilinear problems. Proof. We modify the proof of Theorem 8.34 according to (8.169)–(8.174). With V0h := v h ∈ V h : v h |∂Ωhe = 0 we obtain : ; ah uh0 , v h = Ah uh0 , v h V h ×V h = Ahs uh0 , v h L2 (Ωh ) = (fˆh , v h )L2h (Ωh ) ∀v h V0h , h

hence the strong diﬀerence equation in (8.188). Next we extend testing (8.169) with v h ∈ V h . By the previous equation, we are left with: for all v h ∈ V h : hν hν v h dsh = 0 ⇒ Bah uh0 (x) − ϕh x− ∀x ∈ ∂Ωhe , Bah uh0 − ϕh · − 2 2 ∂Ωh e (8.189) so a natural boundary condition. 1 Ωhe , satisfying the strong diﬀerence equation, and Vice versa, a solution uh0 ∈ H± natural boundary conditions in (8.188), satisﬁes the weak problem (8.187) as well. The same strategy applies to quasilinear problems. By an obvious modiﬁcation of part (3) of the proof of Theorem 8.25 we get Proposition 8.37. Under the condition (8.38), and with c0 > 0, the extended principal part ahp,e (uh , v h ) := ahp (uh , v h ) + c0 (uh , v h )L2 (Ωh ) : V h × V h → R

(8.190)

is V h -coercive, hence stable; the original ah (·, ·) in (8.169) is V h -elliptic. We ﬁnally need the consistency of our approximations. To that end, we use the obvious extensions of the P h , Q h in Subsection 8.5.2 to the Ωhe , ∂Ωhe in this section.

620

8. Finite diﬀerence methods

The only not totally obvious modiﬁcation concerns the boundary integral terms. We come back to the boundary term of the f h in (8.170): for v h ϕh (· − hν/2)dsh = hn−1 (v h (x)ϕh (x − hν/2), let (8.191) ϕh , v h h := ∂Ωh e

x∈∂Ωh e

Qbh : L2 (∂Ω) → V B A ⇒ Qbh ϕ, v h V

h

: Qbh ϕ(x) := ϕh (x − hν/2) := P h (Ec ϕ)(x − hν/2)∀x ∈ ∂Ωhe , := v h Qbh ϕdsh = hn−1 v h (x)(P h Ec ϕ)(x − hν/2).

h ×V h

∂Ωh e

x∈∂Ωh e

Proposition 8.38. With the extended P h , Q h , as in (8.99), (8.84), (8.93), Qbh in (8.191), and the conditions for the coeﬃcients of A in Summary 8.6 and Remark 8.7, we obtain for Au = f ∈ V , A ∈ L(V, V ), As ∈ L(H 2 (Ω), L2 (Ω)), h 1 Ωe , fˆh = Q h fˆ0 , ϕh = Qbh ϕ, u, v ∈ V = H 1 (Ω), V h = H±

(8.192) (8.193)

consistent symmetric approximations Ah , Ahs , Bah cf. (8.60), (8.194), for linear operators, hence,

lim ||Ah P h u − Q h Au||V h = lim ||Ahs P h u − Q h Au||V h = 0 ∀u ∈ H 1 (Ω),

h→0

h→0

lim ||Bah P h u − Qbh Ba u||V h = 0∀u ∈ H 2 (Ω).

h→0

(8.194)

b

More precisely, we obtain, for the weak form and u ∈ H 2 (Ω), A ∈ L(H 2 (Ω), L2 (Ω)), Ba ∈ L(H 2 (Ω), L2 (∂Ω)), consistency of order 1 for Ah , and for Bah ,

||Ah P h u − Q h Au||V h ≤ ChuH 2 (Ω) , ||Bah P h u − Qbh Ba u||V h ≤ ChuH 2 (Ω) . b

For u ∈ H (Ω) with A ∈ L(H (Ω), H (Ω)), and Ba ∈ L(H (Ω), H 1 (∂Ω)), we obtain consistency of orders 2 and 1 for Ahs , and for Bah , 3

3

1

3

||Ahs P h u − Q h Au||V h ≤ Ch2 uH 3 (Ω) , ||Bah P h u − Qbh Ba u||V h ≤ ChuH 3 (Ω) b

With the technique in Section 5.2 we obtain consistency of order 2 for Bah as well. For possible choices of f h = Q h f , and ϕh = Qbh ϕ, the limh→0 ||f h − Q h f ||V h = 0 and limh→0 ||ϕh − Qbh ϕ||V h = 0 have to be proved. b Analogous results hold for the quasilinear problems.

Proof. Except for ||Bah P h u − Qbh Ba u||V h we can argue for the strong form as in b Theorem 8.21. In ∂Ωh v h Bah uh dsh we had to shift from x ∈ ∂Ωh to ∞ − hν/2. This e reduces the consistency to order 1. Order 2 would be obtained if we could compare Ba and Bah exactly along ∂Ωh . The technique in Section 5.2, splitting into diﬀerential and boundary ooperators, allows this exact comparison along ∂Ωh , and thus yields order 2. We do not repeat that proof.

8.6. Natural boundary value problems of order 2

621

We summarize the necessary conditions for our diﬀerence methods for natural boundary conditions, essentially restricted to convergence of order 2. 1. We use the following spaces, cf. (8.77), often we omit the Rq : V = H 1 (Ω, Rq ), V = H −1 (Ω, Rq ), for q ≥ 1, 1 1 V h = H± (Ωh , Rq ), Vbh = H0,± (∂Ωh , Rq ), V

h

−1 = H± (Ωh , Rq ). (8.195)

2. For linear problems we assume, cf. Remark 8.7, aij ∈ C(Ω), and A : H 3 (Ω) → H 1 (Ω), Ba : H 3 (Ω) → H 1 (∂Ω).

(8.196)

3. For quasilinear problems we assume, with N1 in (8.17), Aj equicontinuous in Ω × N∇1 ⊂ Ω × RN1 , N∇1 bounded, s.t.

(8.197)

h ∀u ∈ V : u − u0 V ≤ ρ, ∇≤1 u(x) ∈ N∇1 ∀x ∈ Ω, ∇≤1 h u(x) ∈ N∇1 ∀x ∈ Ω ,

with Lipschitz-continuous Aj with respect to the variables in N∇1 . Finally we assume Aj : D(Aj ) ∩ H 3 (Ω) → H 1 (Ω)).

(8.198)

Theorem 8.39. Diﬀerence methods for natural boundary conditions: 1. Let A or G (u0 ) : H 1 (Ω, Rq ) → H −1 (Ω, Rq ) be boundedly invertible, for a (locally) unique exact solutions u0 for the exact equations, cf. the conditions in Chapter 2. Under the conditions (8.195)–(8.197) we extend P h = I h and Q h , from Subsection 8.5.2, to Ωhe and deﬁne Qbh as in (8.191). For the exact linear and nonlinear problems, Au = f ∈ H −1 (Ω, Rq ) and G(u) = 0 ∈ H −1 (Ω, Rq ) deﬁne the symmetric diﬀerence methods as in (8.167)–(8.175), deﬁned on cuboidal domains, cf. (8.95). 2. A unique discrete solution uh0 , simultaneously for the weak and strong discrete linear and nonlinear problems exist near u0 for small enough h. It converges of order 1. With the f in (8.153), and the corresponding f h in (8.170) with fˆh = Q h fˆ0 , ϕh = Qbh ϕ, in (8.192) we ﬁnd Ah uh0 = f h ⇒ |P h u0 − uh0 |V h ≤ Chu0 H 2 (Ω) for u0 ∈ H 2 (Ω). (8.199) 3. This uh0 satisﬁes the classical diﬀerence and a modiﬁed boundary equation:

Ah uh0 = f h ⇒ Ahs uh0 − fˆh (x) = 0∀x ∈ Ωh , and (8.200) Bah uh0 (x) = ϕh (x − hν/2)∀x ∈ ∂Ωhe . 4. If we maintain, in (8.200), (Ahs uh0 − fˆh )(x) = 0, and replace the modiﬁed boundary condition by the classical discrete boundary condition, then this implies

622

8. Finite diﬀerence methods

convergence of order 2:

Ahs uh0 − fˆh (x) = 0∀x ∈ Ωh , Bah uh0 (x) = ϕh (x)∀x ∈ ∂Ωh =⇒ |P h u0 − uh0 |V h ≤ Ch2 u0 H 3 (Ω) for u0 ∈ H 2 (Ω).

(8.201)

5. We replace the above (8.167) ﬀ., (8.174) for linear problems by (8.171) ﬀ., (8.175) for quasilinear problems. Then we obtain the corresponding results. Proof. We only prove ﬁrst convergence, and indicate second order. We have horder h −1 1 Ωe → H± Ωe , induced by ah (uh , v h ), ahp,e (uh , v h ), and shown that Ah , Ahp,e : H± A, Ap,e induced by a(u, v), ap,e (u, v), are bounded linear operators. By Proposition 8.37, the ahp,e (uh , v h ), and ap,e (u, v) are V h -, and V-coercive. This A is a compact perturbation of Ap,e with stable Ahp,e . Furthermore, the Ahp,e , Ah are consistent with the Ap,e , A, by Proposition 8.38. h 1 Ωe -stability of Ahp,e , and the consisHence for a boundedly invertible A, the H± −1 tencies in Proposition 8.38 and Theorem 3.29 imply the stability of Ah . With h A ∈ −1 −1 1 −1 h h L(H (Ω), H (Ω)), and the corresponding f ∈ H (Ω), f = Q f ∈ H± Ωe , as in (8.170), we obtain : ; ∃1 uh0 ∈ Hh1 (Ωh ) : ah uh0 , v h = Ah uh0 , v h = f h , v h ∀v h ∈ Hh1 (Ωh ), (8.202) with the convergence. By the equivalence in (8.174), we verify the strong diﬀerence equation, and the modiﬁed discrete natural boundary condition. This is shown as in the previous sections. For second order convergence, we have to replace the above modiﬁed boundary condition in (8.200) by classical discrete natural boundary conditions in (8.201), and similarly for the quasilinear case. Then the full operator is split into diﬀerential and boundary operators, (As ,Ba ). The corresponding classical pair of discrete diﬀerence and boundary operators, Ahs , Bah , is consistent of order 2. This allows the full proof by modifying the technique in Section 5.2. We omit that here.

8.7

Other diﬀerence methods on curved boundaries

In Section 8.4 we discussed nonsymmetric ﬁrst order methods for domains with curved boundaries for general elliptic problems. Here we modify these methods for the following special cases, allowing, by appropriate treatment of the boundary, methods of higher order. Points more or less near the boundary will have to be distinguished. We summarize the results for problems with Dirichlet boundary conditions, but do not prove them for several reasons: For FEMs, an extended theory essentially for all types of elliptic equations and systems is available, and is preferably used and presented in Chapter 4. For the mentioned special cases, the proofs for stability, consistency and convergence use techniques totally diﬀerent from those in Sections 8.5, 8.6. They do not seem to allow generalizing and unifying them towards the problems in Section 8.4.

8.7. Other diﬀerence methods on curved boundaries

623

The results in Subsections 8.7.1 and 8.8.1 are only known for special cases. Hackbusch [387] formulates his results only for the Laplacian, but his proofs apply (as he mentions) to most of the problems in Subsection 8.7.1. A coercivity proof as in the last sections is not known for the Shortley-Weller method. We refer, cf. (8.210), to the V h = Hh1 (Ωh )-regularity result in [387], Theorems 9.2.3, and 9.2.8 for the principal part. Our techniques imply the stability of (8.205) for a boundedly invertible operator. For the polynomial interpolation case, even asymptotic expansions for the discretization error are available, allowing higher order methods via extrapolation or defect corrections. Proving the convergence in the following subsections would probably be possible for special cases by applying the techniques in Section 5.2, splitting the discretization error into errors for the diﬀerential and the boundary operator. However this is beyond the scope of this chapter. Finally, Heinrich [395] discusses with appropriate machinery ﬁnite diﬀerence methods on irregular networks. 8.7.1

The Shortley–Weller–Collatz method for linear equations h

We assume Ω as in (8.98), and have to modify the Ωh , Ω and ∂Ωh . The original grid G h , the j-th grid line Gjh , and diﬀerent types of grid points, Ωh , Ωhi , Ωh are distinguished, cf. Figure 8.6. With ei , the i-th unit vector, i = 1, · · · , n, let G h := {x = (x1 , . . . , xn )T : xi /h ∈ Z for i = 1, · · · , n}, Gjh

(8.203)

:= {x = (x1 , . . . , xn ) ∈ R : xj ∈ R, xi /h ∈ Z for j = i = 1, · · · , n}, T

n

Ωh := {x ∈ Ω ∩ G h : x ± hei ∈ Ω for i = 1, · · · , n}, Ωhi := (Ω ∩ G h ) \ Ωh , h ∂Ωh := P ∈ ∂Ω ∩ Gjh for j = 1, · · · , n , Ω := Ωh ∪ Ωhi ∪ ∂Ωh , Ωhe := Ωh ∪ Ωhi . We denote Ωh and Ωhi , and ∂Ωh as the set of regular, Ωhi as the irregular, ∂Ωh as the h boundary, Ω as all mesh points in Ω, and Ωhe as the essential mesh points in Ω. In Figure 8.6 we have marked these points by •, by , and by

points in Ωh : far from ∂Ω points in Ωih : close to ∂Ω

Ω

points in ∂Ωh (on ∂Ω): Ωh : = Ωh ∪ Ωhi ∪ ∂Ωh Ωeh : = Ωh ∪ Ωih

Figure 8.6 Grid for curved ∂Ω.

624

8. Finite diﬀerence methods

We only consider the following special second order equations, and formulate their classical strong diﬀerence equation, cf. (8.43)–(8.47). Note that we exclude mixed derivatives, and use symmetric formulas As u(x) =

n

− aj (x)(∂ j )2 u(x) + bj (x)∂ j u(x) + c(x)u(x) = f (x),

(8.204)

j=1

2e j j j j /2, and ∂h j := ∂− and with ∂hj := ∂− + ∂+ ∂+ , cf. (8.12), (8.13), deﬁne Ahs uh (x) =

n

2e − aj (x)∂h j uh (x) + bj (x)∂hj uh (x) + c(x)uh (x) = f (x)

(8.205)

j=1 h

∀x ∈ Ωh , uh : Ω → R, with boundary conditions uh (x) = 0∀x ∈ ∂Ωh ,

(8.206)

with bounded and continuous aj , bj , j = 1, . . . , n, cf. Remark 8.7. Then Ahs uh is well deﬁned for ∀x ∈ Ωh , but not for x ∈ Ωhi . The ellipticity condition (8.38) for this As has the form, see (2.16), (2.20), n

aj (x)(ϑj )2 ≥ |ϑ|2 or aj (x) ≥ > 0, ∀x ∈ Ω.

(8.207)

j=1

There is a long history of dealing with the missing x ∈ Ωhi in (8.205). We discuss only two diﬀerent approaches. We choose an x ∈ Ωhi with at least one neighbor outside of Ω, hence − for x ∈ Ωhi choose the smallest possible 0 < s+ j , sj ≤ 1, j = 1, · · · , n, + − h − sj , sj < 1 s.t. x + s+ with min j hej , x − sj hej ∈ Ω .

(8.208)

j=1,··· ,n

2e

Shortley and Weller [588] use, instead of the symmetric diﬀerences, ∂h j uh (x), in (8.205), the unsymmetric diﬀerences with respect to x = x − s− j hej < x < x = + h x + sj hej , with u(x ) = 0 or u(x ) = 0 for x or x ∈ ∂Ω , [588] replace, cf. [387], (4.8.5), (4.8.6), u(x) − u(x ) + / sj + s− j − sj − u C[x ,x ] /3 with u (x) − (∂ j )2h u(x) ≤ h max s+ (8.209) j , sj

∂h j uh (x) by (∂ j )2h u(x) := 2h−2 2e

u(x ) − u(x) s+ j

and ∂hj uh (x) by (∂ j )h u(x) := h−1

−

u(x ) − u(x) s+ j

with |u (x) − (∂ j )h u(x)| ≤ hs+ j u C[x ,x ] . The previous discrete Sobolev spaces have to be modiﬁed accordingly. The special diﬀerences in (8.205), corresponding to (8.209), are indicated by special multi-indices, c α, so let αc := jei , j = 0, 1, 2, i = 1, . . . , n. Finite diﬀerences, ∂hα , are chosen as in c (8.205), (8.209). With these divided diﬀerences, ∂hα uh , the discrete Sobolev inner

8.7. Other diﬀerence methods on curved boundaries

625

products, norms, and spaces are deﬁned, for m = 0, 1, 2, as

c c ∂hα uh , ∂hα v h 2 h , and (8.210) (uh , v h )H m (Ωh ) := (uh , v h )L2 (Ωh ) + h h Lh (Ωe ) 0<|αc |≤m ⎞1/2 ⎛ 2 αc h |v h |2H k (Ωh ) := |v h |H k (Ωh ) ⎠ . ∂h v 2 h , v h H m (Ωh ) := ⎝ h

0<|αc |=k

Lh (Ω ))

h

0≤k≤m

h

Applying (8.209) for (8.208) instead of (8.205), we obtain the Shortley–Weller approx− < 1, usually for one j : , s imation for an x ∈ Ωhi , with minj=1,··· ,n s+ j j ' n 2 2 h h x + s+ Ahs uh (x) := aj (x)h−2 − + + (8.211) j hej + + − u (x) − u s s + s s s j j j j j j=1 + h h x + s − u u he (x) 2 j j uh x − s− − − + + bj (x) j hej sj sj + s− s+ j j h + c(x)uh (x). Reinterpreted, this is essentially Collatz’s method as well [205, 206], see below. For computations we multiply AhSW = Ahs in (8.205), (8.211) by the diagonal matrix Dh + h obtaining, cf. [387], Corollary 9.2.10, with s− j sj := 1 for x ∈ Ω , + h , and ASW Dh := (diag d(x))x∈Ωhe , d(x) := min 1, 2s− := Dh AhSW . (8.212) j sj j=1...,n

h Then instead of AhSW uh0 = f h we have to solve ASW uh0 = Dh f h . Hackbusch’s [387] Theorems 4.3.16, 4.8.4 and 9.2.8 are formulated for As u = −Δu. His proofs for consistency in (4.8.11) or Theorem 9.2.14 including his regularity in Theorems 9.2.3, 9.2.8 apply to bj = c = 0 in (8.205), (8.211). The extension to boundedly invertible operators A : H01 (Ω) → H −1 (Ω) follows with the previous consistency and regularity of the principal part by Theorem 3.29. The preceding Shortly–Weller method can be generalized to special strong quasilinear equations of the form, cf. (8.64), yielding a linearized form (8.204), G : D(G) ⊂ H 1 (Ω) → H −1 (Ω), Gh : D(Gh ) ⊂ Hh1 Ωhe → Hh−1 Ωhe ,

Gs u 0 =

n

(−1)j>0 ∂ j Aj (·, u0 , ∂ j u0 ) = 0 ∈ L2 (Ω), u0 ∈ H 2 ∩ H01 (Ω),

(8.213)

j=0

Ghs uh0 (x) =

n

j j h j j h (−1)j>0 ∂− Aj ·, uh0 , ∂+ u 0 + ∂+ Aj ·, uh0 , ∂− u0 (x)/2 = 0

j=0

∀x ∈ Ωh , and uh0 ∈ Vbh .

(8.214)

For the proof, we combine, as usual, Theorems 3.23, 8.40, 3.29, 3.21 with the above consistency. We summarize these results in Theorem 8.40. Convergence of the Shortly–Weller method:

626

8. Finite diﬀerence methods

1. Let Ω be bounded, and in C 2 or convex, and A : H01 (Ω) → H0−1 (Ω) in (8.204) h , the matrix deﬁned by the Shortley–Weller be boundedly invertible. Then ASW h method (8.205), (8.211), (8.212) as ASW : Hh1 Ωhe → Hh−1 Ωhe , cf. (8.210), is stable and consistent. The Shortley–Weller equations have, for small enough h, a unique solution uh0 , converging to the exact solution u0 of Au0 = f : h u0 − P h u0 1 h ≤ Chu0 H 2 (Ω) . (8.215) H (Ω ) h

e

2. Whenever the conditions for an M matrix are satisﬁed, cf. [387], Theorems 4.8.4, 4.8.6, then max uh0 (x) − u0 (x) ≤ Ch2 u0 C 3 (Ω) . x∈Ωh e

L

3. Whenever the conditions for well-posed quasilinear equations of the form (8.213) and for a locally unique solution u0 with boundedly invertible G (u0 ) are satisﬁed, cf. Chapter 2, the Shortly–Weller equations (8.214) have, for a small enough step size h, a unique solution uh0 , converging as in (8.215).

8.8

Asymptotic expansions, extrapolation, and defect corrections

This area of research for ordinary and partial diﬀerential equations ﬂorished between 1960 and 1990. Since the interest in these results is pretty low these days, we will only list a choice of relevant results and apply them to extrapolation methods. The following authors have contributed to this area: textbooks are, e.g. Conte and de Boor, Deuﬂhard and Hohmann, and Fox, Isaacson and Keller, and Stetter [207, 284,324,414,596]. Papers are due to, e.g. Axelsson and Layton, B¨ ohmer, with Hemker, Gross, Schmitt, R. Schwarz, and with Stetter, Chibi and Moore, Doedel with Reddien, Fox, Frank and Ueberhuber, Gragg, Hackbusch, Keller and Pereyra, Stetter, Monnet, Pereyra et al., Reinhardt, and Thoma [44,101,103,106,107,110,125,126,130,172,288– 291,321–323,325,326,358,381–384,442,496,526,527,548,549,596,597,625]. Extensions to asymptotic expansions for FEMs are due to Lin, Liu, Rannacher, Zou, [476,477,541]. In particular, Hackbusch’s regularity results applied to defect corrections [382–384] showed that defect corrections yielded the same orders of acccuracy as the extrapolation methods. The approach presented in this chapter, with convergence results with respect to the discrete Sobolev norms, · H±m (Ωh ) , is a special regularity result as well, now proved for all the previous problems. So the result [384] is valid for our methods and problems as well. Here we combine extrapolation techniques or defect corrections with asymptotic expansions for the discretization error. They are available for cuboidal and curved domains, cf (8.95) and (8.98). Similarly to extrapolation for numerical integration, cf. e.g. [414, 602, 603], they allow, by linear combinations of numerical results for step sizes, say h, h/2, h/4, higher order methods via extrapolation. Defect corrections are strongly related to discrete Newton methods. Deferred corrections are another choice, but are more diﬃcult to explain. So we omit them here, cf. the above papers.

8.8. Asymptotic expansions, extrapolation, and defect corrections

8.8.1

627

A diﬀerence method based on polynomial interpolation for linear, and semilinear equations

The methods discussed now, and the necessary tools for their proofs, ﬁt even less than in Subsection 8.7.1, to the previous approach. We include them here, since their discretization error admits an asymptotic expansion. For smooth Ω, and small enough h we ﬁnd for x ∈ Ωhi , cf. (8.203), e.g. x + hej ∈ Ω, the x + sj hej ∈ ∂Ω, 0 < sj < 1,

(8.216)

but x − (ν − 1)hej ∈ Ω, ν = 1, · · · , k ≥ 1. Again we choose the smallest 0 < sj < 1, such that x + sj hej ∈ ∂Ω and the full segment {x − t h ej } ⊂ Ω, −sj ≤ t ≤ k − 1. All the following methods essentially replace uh (x + hej ) in (8.216) by the value of a polynomial interpolating the boundary function ϕ(x + sj hej ) in x + sj hej ∈ ∂Ω, and the uh (x) at the other points in (8.216): Gerschgorin [340] replaced uh (x + hej ) by a nontrivial ϕ(x + sj hej ). Pereyra et al. [527], referring to unpublished notes of H.-O. Kreiss, use the following polynomial Pjk for k ≥ 2. It is a univariate k-th order interpolating polynomial xk := Pjk ∈ Πk interpolates: Pjk (x + sj hej ) = ϕ(x + sj hej ), Pj+

and Pjk (x − (ν − 1)hej ) = uh (x − (ν − 1)hej ), ν = 1, · · · , k.

(8.217)

Then they replace uh (x + hej ) by Pjk (x + hej ). If in (8.205), bj (x) ≡ 0, the above Shortley–Weller method is obtained for k = 2. Pereyra et al. [527] conjectured an asymptotic expansion for the discretization error of order k based upon Bramble–Hubbard techniques [139]. B¨ ohmer [106] proved this expansion u(x) − uh (x) − h2 e2 (x) − h4 e4 (x)∞ ≤ O(hk+1 ) for k ≤ 4. Since large values for k impose restrictive conditions for Ω, and h, k ≤ 4 is not a really severe restriction. [106, 139] essentially work with M -matrix arguments, not applicable to our previous technique. Diﬀerently, Collatz [205,206] replaced uh (x), not uh (x + hej ), by the value of the linear polynomial interpolating in (x + sj hej , ϕ(x + sj hej )), and (x − hej , uh (x − hej )). Then again (8.211) results. In fact, rescale (4.8.14a) and (4.8.14b) in [387], by the equibounded 2/(s + sr ), and 2/(su + so ) and then proceed to the modiﬁed (4.8.16), yielding (4.8.7). So [205, 206] is included in the last subsection. For x + hej ∈ Ω, see (8.216) [106, 527], we compute the approximate, k h k ν h cj u (x − (ν − 1)hej ) /c0j , u (x + hej ) := Pj (x + hej ) = ϕ(x + sj hej ) − ν=1

(8.218) with cνj :=

k (1 − sj ) − , ν−

ν ==0

628

8. Finite diﬀerence methods

and with our ϕ(x + sj hej ) = 0. − The situation x − hej ∈ Ω, and x − s− j hej ∈ ∂Ω, 0 < sj < 1, in (8.216)–(8.218), is handled analogously by replacing h by −h. j h u (x) by Pjk (x + hej )− Summarizing, we replace in (8.205), e.g. the ∂+ uh (x − hej ) /h and obtain for an x ∈ Ωhi , x + hej ∈ Ω f (x) = Ahs uh (x) n 5 6 2e −aj (x)∂h j uh (x) + bj (x)∂hj uh (x) + c(x)uh (x) = j =i=1

'

− aj (x)h−2

0−

+ bj (x)h

-J cνj uh (x − (ν − 1)hej )

ν=1

' −1

k

0−

k

c0j − 2uh (x) + uh (x − hej )

-J cνj uh (x

− (ν − 1)hej )

(8.219)

c0j

− u (x − hej ) /2. h

ν=1

By (8.216), the situation x ∈ Ωhi , x + hej ∈ Ω, and x − hej1 ∈ Ω for j1 = j or even x ± hej ∈ Ω for k = 1 is not excluded. Then in (8.219) the ﬁrst sum over j = i = 1, . . . , n has to be modiﬁed into i = 1, . . . , n, x ± hei ∈ Ω, and the missing i have to be collected in the last two lines of (8.219). We avoid the messy formulation. For (8.219) we need a connectedness condition for the grid Ωh . We deﬁne, for a ﬁxed x ∈ Ωh ∪ Ωhi , the set of neighbors as N (x) := x ¯ ∈ Ωh ∪ Ωhi occurring in (8.205) or (8.219) . Then we reformulate (8.205), and (8.219) with σ(x, x ¯) = 0 as x ∈ Ωh ∪ Ωhi : Ahs uh (x) − f (x) =: σ(x, x ¯)uh (¯ x) + B h (−aj , bi , f, 0 = ϕ, x). x ¯∈N (x) h

∪ Ωhi

is called connected, if for every proper subset, S h , ∀ S h ⊂⊂ Ωh ∪ Ωhi : ∪ N (x) ∩ Ωh ∪ Ωhi \ S h = ∅.

Then the grid Ω

x∈S h

(8.220)

Furthermore we assume for small enough h the existence of an Ω∗ ⊃ Ω such that ⎧ ⎫ 1/2 n ⎨ n 2 j=1 aj (x) aj (x) ⎬ , min min h ≤ min min (8.221) j=1 x∈Ω |bj (x)| ⎭ ⎩ x∈Ω |c(x)| and that

Ω ⊂ Ω∗ ⊃ x ± hej : j = 1, · · · , n, x ∈ Ωh ∪ Ωhi .

(8.222)

We summarize parts of Propositions 4.1, 4.3, and Theorems 4.4, 4.11 in [106]: Theorem 8.41. Convergence for a Collatz–Pereyra method: For Ω in (8.98) let Ωh ∪ Ωhi be connected, let As : C 2 (Ω) ∩ C(Ω) → C(Ω) in (8.204) satisfy (8.207), and

8.8. Asymptotic expansions, extrapolation, and defect corrections

629

be boundedly invertible. Let Ahk be the matrix deﬁned by the preceding method, see h

h

h ∞ ∞ (8.216)–(8.219), let and h satisfy (8.221). Then Ak : L (Ω ) → L (Ω ) is invertible −1 with bounded Ahk ≤ C, hence Ahk is stable in the sense of (8.58). If additionally ∞

for Ω∗ in (8.222) an extension u∗0 ∈ C q+α (Ω∗ ), 2 ≤ q ≤ 6, 0 < α ≤ 1 exists, then the discrete solution uh0 converges according to h u0 − u0 ∞ h ≤ Chmin{2,k+1,q−2+α} . (8.223) L (Ω ) h

In addition let −aj , bj , c, f ∈ C k,α (Ω), ϕ ∈ C k+2,α (∂Ω), ∂Ω ∈ C k+2,α , 0 ≤ k ≤ 4. Then, by Theorems 2.38 and 2.39, the exact solution satisﬁes u0 ∈ C k+2,α (Ω), and there exist f2 ∈ C k+α (Ω), f4 ∈ C k−2+α (Ω), independent of h, such that the following asymptotic expansion is valid h u0 − (u0 + h2 f2 + h4 f4 ) ∞ h ≤ Chmin{4+α,k+1} . (8.224) L (Ω ) h

In B¨ohmer [104] we have extended results in Theorem 8.41 to quasilinear equations Gs u(x) =

n

−aj x, u(x), ∂ j u(x) (∂ j )2 u(x) − f (x, u(x), ∇u(x)) = 0,

(8.225)

j=1

Ghs uh (x) =

n

2e −aj x, u(x), ∂hj u(x) ∂h j uh (x) − f (x, uh (x), ∇h uh (x)) = 0

j=1

∀x ∈ Ω with boundary conditions u(x) = 0∀x ∈ ∂Ω, and uh (x) = 0∀x ∈ ∂Ωh , h

with bounded and continuous f, aj , j = 1, . . . , n. Then Ghs is well deﬁned for ∀x ∈ Ωh , for x ∈ Ωhi the modiﬁcation as in (8.219) has to be employed. For the stability we have to consider the derivative G (u0 ) of (8.225), evaluated in the exact solution, u0 . This again has the form (8.204). We impose condition (8.226) for stability, and (8.227) for consistency, and asymptotic expansions, basic for Theorems 4.5, and 5.3 in [104]: Let u0 be the unique solution of (8.225), and 0 < a ≤ aj (·, u0 (·), ∂ u0 (·)) in Ω, ∀j = 1, . . . , n ⎛ ⎞ n ∂a ∂f j (·, u0 (·), ∂ j u0 (·))(∂ j )2 u0 (·)⎠ 0 < Q∗ ≤ ⎝ (·, u0 (·), ∇u0 (·)) − ∂u ∂u j=1 j

⎛ ×⎝

n j=1

⎞−1 aj (·, u0 (·), ∂ j u0 (·))⎠

≤ Q∗ in Ω, and as the aj , bj

(8.226)

630

8. Finite diﬀerence methods

for G (u0 ) in (8.204) we get aj (·) := aj (·, u0 (·), ∂ j u0 (·)), ∀j = 1, . . . , n bj (·) := −

∂aj ∂f (·, u0 (·), ∂ j u0 (·))(∂ j )2 u0 (·) − (·, u0 (·), ∇u0 (·)) in Ω, ∂(∂ j u) ∂(∂ j u)

and assume bj (·)/aj (·)L∞ (Ω) ≤ Pj∗ hence for our case ∂f ∂aj j j 2 (·, u (·, u (·), ∂ u (·))(∂ ) u (·) + (·), ∇u (·)) 0 0 0 0 0 ∂(∂ j u) ∂(∂ j u) −1 × aj (·, u0 (·), ∂ j u0 (·)) ≤ Pj∗ , for stability, with aj ,

∂aj ∂f ∂f ∂aj , , , (·, u0 (·), ∇u0 (·)) ∈ C(Ω), and, for asymptotics, j ∂u ∂u ∂(∂ u) ∂(∂ j u)

u0 ∈ C(Ω) ∩ C 2q+2+α (Ω), aj ∈ C 2q+α (Ω × B), f ∈ C 2q+α (Ω × Bn ), ∂Ω ∈ C 2q+2+α , with closed B ⊃ {(x, u0 (x), ∂ j u0 (x))∀x ∈ Ω, j = 1, . . . , n}

(8.227)

Bn ⊃ {(x, u0 (x), ∇u0 (x))∀x ∈ Ω}, and 0 < α < 1. Theorem 8.42. Asymptotic expansions for quasilinear problems: For Ω in (8.98) let Ωh ∪ Ωhi be connected, and Gs : D(Gs ) ⊂ C 2 (Ω) ∩ C(Ω) → C(Ω) in (8.225) satisfy (8.226). Then Gh (·) in (8.225) is stable in P h u0 . Additionally let (8.216), k − 1 ≤ min{3, 2q + α}, (8.227), and the more or less geometric type conditions for these types of problems in Agmon et al. [2], Chapter 7, be satisﬁed, implying u0 ∈ C 2q+2+α (Ω). Then we obtain the asymptotic expansion q˜ h h 2i h e2i ≤ Chk−1 uC 2q+2+α (Ω) , (8.228) u 0 − R u 0 − ∞ h i=1

L

(Ω )

with q˜ := max{m ∈ N|2m < k − 1, m ≤ q}. These e2i are independent of h, and e2i ∈ C 2q+2−2i+α (Ω), and e2i |∂Ω = 0, i = 1, . . . , q˜. 8.8.2

Asymptotic expansions for other methods

Results based upon the earlier sections in this chapter are summarized. Basic is Proposition 8.5, where we have proved the asymptotic expansion for 1 f (¯ x)d¯ x, |f¯h (x) − f (x)| ≤ Lh2 as (8.229) f¯h (x) := n h ch (x) ⎛ ⎞ 2m ,ev α h (h/2) α f¯ (x) − ⎝f (x) + ⎠ ∂ f (x) ≤ Ch2m +1 . (8.230) (α + 1)! |α|=2 In Subsection 8.8.1, we have already listed results on asymptotic expansions of the discretization errors for those special methods. Here we extend, but do not prove, these results to the methods in Sections 8.4–8.6. We obtain asymptotic expansions in powers of h and h2 for unsymmetric and symmetric diﬀerence methods for the Dirichlet

8.8. Asymptotic expansions, extrapolation, and defect corrections

631

problem on curved and cuboidal domains, respectively. The asymptotic expansions in powers of h2 are much more interesting for extrapolation methods. Theorem 8.43. Asymptotic expansion for the diﬀerence methods in Sections 8.4– 8.6: For unsymmetric and symmetric diﬀerence methods on curved and cuboidal domains, cf. (8.95), with μ = 1 and μ = 2, respectively, we assume the conditions of Theorems 8.31, 8.39. Furthermore, assume u0 ∈ W μ(r+1),p (Ω), and Lipschitzμ(r+1) with respect to the variables in N∇m . This essentially implies continuous G ∈ CL the additional technical conditions, e.g. the (μr, μ)-smooth asymptotic expansion of the local discretization error, of the Ah P h u − Q h Au, cf. (8.105), (8.106), and ohmer [103]. Gh P h u − Q h Gu, cf. (8.113), (8.114), (8.119), cf. Stetter [596] and B¨ Then for small enough h, unique discrete solutions uh0 for the linear and nonlinear problems in Sections 8.4, 8.5 exist, converge, and admit an asymptotic expansion of the discretization error. More precisely, there exist eμk ∈ W μ(r+1−k),p (Ω), 1 ≤ k ≤ r, independent of h, such that r h u0 − hμk eμk − uh0 ≤ Chμr+1 u0 W μ(r+1),p (Ω) for u0 ∈ W μ(r+1),p (Ω). P k=1

Vh

(8.231) We combine asymptotic expansions with Richardson extrapolation: a few step h = h 0 , h/2, h/4, h/8, are chosen with Ωh ⊂ Ωh1 ⊂ sizes, h = h0 > h1 > h2 > h3 , often h2 h3 hi Ω ⊂ Ω . The corresponding G uh0 i = 0, i = 0, 1, . . . , t, deﬁned on Ωhi are computed. Appropriate linear combinations of these uhi require the restriction of the uhi to Ωh . So deﬁne, for Ωh ⊂ Ωh1 ⊂ Ωh2 . . . : ui : Ωh → Rq : ui (x) := (Rh uhi )(x) := (uhi )(x)∀x ∈ Ωh ; (8.232) next deﬁne the Pi,0 := ui , i = 0, 1, . . . , t, and for i = 0, 1, . . . , t − j, j = 1, . . . , t < q, P −Pi,j−1 ﬁnally Pi,j := Pi+1,j−1 + i+1,j−1 (hi /hi+j )μ −1 . Theorem 8.44. High order diﬀerence methods by Richardson extrapolation: Let a problem G(u0 ) = 0 and its stable, consistent discretization Gh uh0 = 0 deﬁne solutions u0 and uh0 with an asymptotic expansion (8.231), and let h be small enough. Compute the ui as in (8.232). Then we obtain for u0 ∈ W μ(r+1),p (Ω) and a constant γ := hi /hi−1 > 1 : h P u0 − hμ . . . hμ (−1)j eμ(j+1) − Pi,j = O hμ hμ . . . hμ , t < q, (8.233) i i+j i i i+j Vh

μ(j+1) μ(j+1) for ﬁxed γ : P h u0 − hi+j (−1)j γ −j(j−1)/2 eμ(j+1) − Pi,j = O hi+j . Vh

Defect and deferred correction methods have strong advantages compared to Richardson extrapolation. As indicated we restrict the discussion to defect corrections, and here to elliptic problems. The basic version has to solve the nonlinear system Gh uh0 = 0 only once and only for one Ωh . A combination with mesh reﬁnement strategies is possible by computations on diﬀerent Ωhi , cf. B¨ohmer, et al. [11, 12, 101, 102, 104, 107–109, 112, 125, 126]. The basic idea of this method is

632

8. Finite diﬀerence methods

a modiﬁcation of Newton’s method. This has to be combined with the asymptotic expansions in (8.231). We start with the original discrete problem (8.234) deﬁne uh0 by Gh uh0 = f h := Q h f, or Ah uh0 = f h . For our linear diﬀerence methods, we can update this uh0 iteratively by the defect correction process. Note that (8.234) is usually a nonlinear system, whereas (8.235) is a sequence of linear problems, always with the same linear Gh uh0 . The nonlinearity is reduced to computing the nonlinear defect Q h G Ph uhi : h h h G u0 ui+1 − uhi = −Q h G Ph uhi − f , i = 1, . . . . (8.235) Here, it is important that the asymptotic expansion of the uhi is reproduced by Ph , cf. (8.231), hence with eμk,i , gμk,i , g μk,i independent of h, r h h μk h eμk,i ≤ Chμr+1 u0 W μ(r+1),p (Ω) with (8.236) P u 0 − u i − k=i+1

eμk,i ∈ W

μ(r+1−k),p

Vh

(Ω), gμk,i , g μk,i ∈ W μ(r+1−k)−2m,p (Ω), 1 ≤ k ≤ r,

has to imply r h μk h eμk,i ≤ Chμr+1 u0 W μ(r+1),p (Ω) =⇒ P h u i − u 0 − k=i+1 Vh r h hμk gμk,i ≤ Chμr+1 u0 W μ(r+1),p (Ω) Q G Ph uhi − Q h G(u0 ) − k=i+1

or

V

r h h h hμk g μk,i Gi+1 ui − Q G(u0 ) − k=i+1

h

V

h

≤ Chμr+1 u0 W μ(r+1),p (Ω) , h

where this last Ghi+1 ui indicates a diﬀerence approximation for Q h G Ph uhi of order μ(i+ 1) again with an asymptotic expansion. The Ph can be deﬁned as above, for Ghi+1 uhi see e.g. [101, 102, 104, 107]. Theorem 8.45. Defect correction method for Gh : Under the conditions of Theorem 8.44, and for our linear diﬀerence methods the defect correction method (8.235) for Gh in (8.234) yields for small enough h, high order approximations uhi of the order μ(i + 1), hence r hμk eμk,i ≤ Chμr+1 u0 W μ(r+1),p (Ω) for i = 0, . . . , r − 1, Ph uhi − u0 − k=i+1

and with eμk,i ∈ W

μ(r+1−k),p

Vh

(Ω), 1 ≤ k ≤ r independent of h.

(8.237)

8.9. Numerical experiments, v.K´arm´ an equations

8.9

633

Numerical experiments for the von K´ arm´ an equations, with C.S. Chien

The strong form of the von K´ arm´an equations is, cf. (2.403) ⎞ ⎛ ⎞ ⎛ 2 Δ u −[u, w] f1 ⎠=⎝ ⎠ Gs (u, w) := ⎝ f2 Δ2 w + 12 [u, u]

(8.238)

in Ω = [0, 1]2 ⊂ R2 , with Δ2 = ΔΔ, boundary conditions u = Δu = 0, w = Δw = 0 on ∂Ω, and the bracket operator [u, v] := uxx vyy − 2uxy vxy + uyy vxx = [v, u]. For demonstrating the power of extrapolation, we choose f1 (x, y) := π 4 [4 sin πx sin πy − 5 sin πx sin 2πx sin2 πy + 4 cos πx cos 2πx cos2 πy] and f2 (x, y) := π 4 [25 sin 2πx sin πy + sin2 πx sin2 πy − cos2 πx cos2 πy]. Then the exact solution is u(x, y) = sin πx sin πy, w(x, y) = sin 2πx sin πy. We employ the symmetric diﬀerence method for fourth order systems on the square Ω = [0, 1]2 with Ωh as in (8.95)–(8.97). With step sizes h = 1/8, 1/16, 1/32, 1/64, and 1/128, we obtain coeﬃcient matrices of order 98 × 98, 450 × 450, 1922 × 1922, 7938 × 7938, and 32258 × 32258. The nonlinear diﬀerence systems corresponding to (8.238) are solved with Newton’s method. In this academic example we have guaranteed all necessary smoothness conditions. In Table 8.9 we list the computed ehu := P h u0 − uh0 L∞ (Ωh ) , ehw := P h w0 − h w0 L∞ (Ωh ) and determine the squares of the corresponding experimental orders of h/2

h/2

convergence, (EOC)2 , from the quotients ehu /eu , ehw /ew , corresponding to the expected h2 /(h/2)2 = 4, cf. (8.231) In Table8.10 we compare the errors inthe midpoint of the square, in X := (1/2, 1/2) ehu (X) := P h u0 − uh0 (X) ehw (X) := P h w0 − w0h (X) . Again, we determine the Table 8.9: Computational errors and the corresponding experimental orders of convergence (EOC)2 of uh , wh approximations in the discrete L∞ norm.

Grid 1 2 3 4 5

h 1/8 1/16 1/32 1/64 1/128

h eu

L∞ (Ωh )

3.4731045E − 02 8.156673E − 03 2.0079870E − 03 5.0015205E − 04 1.2568853E − 04

(EOC)2 – 4.2600 4.0601 4.0148 3.9793

h ew

L∞ (Ωh )

9.6361989E − 02 2.3262711E − 02 5.7658634E − 03 1.4383812E − 03 3.5913250E − 04

(EOC)2 – 4.1423 4.0346 4.0086 4.0051

634

8. Finite diﬀerence methods Table 8.10: Computational errors and the corresponding experimental orders of convergence, (EOC)2 of the uh (X), wh (X) approximations at the midpoint X = (1/2, 1/2).

Grid 1 2 3 4 5

h

h eu (X)

(EOC)2

h ew (X)

(EOC)2

1/8 1/16 1/32 1/64 1/128

3.0417972E − 02 7.2228714E − 03 1.7835678E − 03 4.4453207E − 04 1.1192101E − 04

– 4.2113 4.0497 4.0122 3.9718

8.2382692E − 02 1.9354054E − 02 4.7722116E − 04 1.1890687E − 04 2.9399067E − 05

– 4.2566 4.0556 4.0134 4.04455

Table 8.11: Computational errors and the corresponding experimental orders of convergence, (EOC)2 of the uh (X) approximations at the midpoint X.

Grid 1 2 3 4 5

h

uh (X)

uh (X)

|uh (X) − uh (X)|

(EOC)2

1/8 1/16 1/32 1/64 1/128

1.0304179722 1.0072228714 1.0017835678 1.0004445321 1.0001119210

0.999491170 0.999970465 0.999998187 1.000001051

5.0883E − 04 2.9535E − 05 1.8130E − 06 1.0510E − 06

– 17.2280 16.2906 1.7250

squares, (EOC)2 , corresponding to the expected h2 /(h/2)2 = 4, now for the X,h/2 X,h/2 , and eX,h . eX,h u /eu w /ew In Table 8.11, we compare the errors in the midpoint X, but now for the ﬁrst extrapolated approximation. Instead of the Pi,j in (8.232), (8.233), with γ = 2, we use the notation uh (X) := (4uh/2 (X) − uh (X))/3. We list the uh (X), uh (X), |u(X) − uh (X)| and determine the squares, (EOC)2 , corresponding to the expected h4 /(h/2)4 = 16 from the |u(X) − uh (X)|/|u(X) − uh/2 (X)|. Similar results hold for wh (X), wh (X), |w(X) − wh (X)|.

9 Variational methods for wavelets, with S. Dahlke 9.1

Introduction

In the previous variational approaches we have used FEs, DCGEs, and grid functions in this book, and will use radial and spectral functions in B¨ ohmer [120]. In this chapter, the spaces of ansatz and test functions are deﬁned by wavelets. Despite the uniﬁed convergence theory, cf. Chapter 3, Petryshyn [528, 532], Stetter [596], Stummel [607– 609] Vainikko [644] and Zeidler [676,677], these diﬀerent methods require us to carefully distinguish their diﬀerent approximation characteristics. Until now, only conforming wavelet Galerkin methods have been applied. We present them here for general elliptic, saddle point, and Navier–Stokes equations. This extends some recent results for wavelet methods. Again, we employ the basic concepts of invertibility, compact perturbation, approximation and general discretization theory. First results, for nonsymmetric, noncoercive, boundedly invertible linear operators have been derived by B¨ ohmer and Dahlke [122]. Here nonlinear problems and bifurcation are indicated, however not explicitely formulated. B¨ ohmer and Dahlke [122] constitute the initial form of this chapter. Our standard result for elliptic problems speciﬁed for conforming wavelet Galerkin schemes is applied. Let B be a boundedly invertible operator with stable discretization B h and A = B + C a compact perturbation of B, such that A is boundedly invertible as well. Then Ah is also stable. In fact, the additional condition of consistent Ah , B h , C h is always satisﬁed for conforming methods. In Chapter 3, cf. [122], we have additionally shown that, under appropriate technical conditions, the stability for the nonlinear problem is guaranteed if it is correct for the linearized operator, cf. Stetter [596] and B¨ ohmer [116], for variational methods. This applies to wavelet methods, if the exact evaluation of all nonlinear terms in the operator is possible. For wavelet methods this is still a problem for future research, in particular since in the wavelet setting, exact evalutions of nonlinear operators are hard to achieve, see e.g. [69, 251] for details. Therefore we restrict the discussion to nonlinear problems, e.g. in (9.13), where the nonlinear terms can be evaluated for wavelet arguments, see Subsection 9.3.5. The linearized Navier–Stokes equations can be interpreted as compact perturbations of the Stokes equations, at least for moderate Reynolds numbers. In this sense, we obtain convergence results for wavelet applied to the Navier–Stokes equations on the same level as the ﬁnite element methods. These problems have been carefully studied by a diﬀerent approach in, e.g. Girault and Raviart [348] and Temam [622].

636

9. Variational methods for wavelets

For our conforming wavelet methods, convergence is, for linear coercive problems, an immediate consequence of Cea’s Lemma 4.48. If the solution u0 is well approximated, here inf vn ∈Sn u0 − vn L2 (Ω)→ 0 , with the stability implied by the coercivity, we obtain “optimal” convergence. During the last few years, wavelets have shown some potential for the numerical treatment of partial diﬀerential and integral equations, c.f. e.g. Barinka et al. [70], Cohen et al. [197, 198], Dahlke et al. [236, 237, 241] and Dahmen [243]. The advantages of wavelet methods can be described as follows. It turns out that a simple diagonal scaling applied to stiﬀness matrices relative to wavelet bases suﬃces to produce uniformly bounded condition numbers. Moreover, for a wide class of integral or pseudo-diﬀerential operators the stiﬀness matrix relative to wavelet bases can be shown to be suﬃciently close to sparse matrices so that eﬃcient sparse solvers can be applied and yield powerful stable and convergent Galerkin schemes. The most far-reaching results were obtained for linear self-adjoint and saddle point problems. For these problems, it has even been possible to derive optimal convergent adaptive wavelet schemes, see Cohen et al. [197] and Dahlke et al. [236,237]. Numerical tests for self-adjoint and saddle point problems, e.g. the Stokes problem, with wavelet methods are documented in Barinka et al. [70] and Dahlke et al. [237]. Adaptive wavelet techniques require highly technical proofs, but do have a lot of advantages whenever they are needed. They can be applied only to boundedly invertible (linearized) operators A. This is the general assumption in the cited papers: Barinka et al. [70], Cohen et al. [197, 198], Dahlke et al. [236] and Dahmen [243]. For a nonsymmetric, boundedly invertible A, for these techniques the use of the symmetric coercive A∗ A is unavoidable, but expensive, cf. e.g. Sections 3 and 7 in Cohen et al. [198] for further details. Here A∗ denotes the adjoint of A. This excludes bifurcation problems. There is another diﬃculty related to A∗ A: It is well-known that cond (A∗ A) = (cond(A))2 explodes for increasing cond(A). First results for nonsymmetric, boundedly invertible problems that avoid these diﬃculties and which are based on perturbation arguments have been derived by Gantumur [332]. The adaptive treatment of problems where the underlying operator is not boundedly invertible, i.e. of ill-posed problems, is a very challenging task that has recently been intensively studied. We refer to Bonesky et al. [131,132] for further details. First results for nonlinear equations have been developed by Cohen et al. [199]. This chapter is organized as follows. In Section 9.2, we brieﬂy discuss the scope of problems we shall be concerned with. Especially, we introduce the setting of elliptic problems. In Section 9.3 we present the basic concepts of wavelet analysis, i.e. the discrete wavelet transform, biorthogonal bases, approximations of functions by wavelets, construction of wavelets on domains and manifolds, and the evaluation of nonlinear functionals applied to wavelet expansions. Section 9.4 discusses the ﬁrst steps, deﬁnitions and basic tools, e.g. stability and preconditioning for wavelet methods. Section 9.5 is devoted to wavelet methods for general linear and nonlinear elliptic problems. We explain how suitable Galerkin schemes based on wavelets can be constructed and discuss their stability and consistency properties. Then, in Section 9.6, we apply these results via its linearization to the Navier–Stokes equations. Finally,

637

9.2. The scope of problems

in Section 9.7, we explain how adaptive numerical methods based upon wavelets can be designed. The following problems, only presented for FEMs, can be solved and yield the corresponding results for wavelet methods as well: general convergence theory for monotone operators, cf. Section 4.5, variational methods for eigenvalue problems, cf. Section 4.7, methods for nonlinear boundary operators which can be evaluated for wavelet functions, cf. Section 5.3, quadrature approximate methods, cf. Section 5.4. All this is limited to problems where the nonlinearities can be evaluated for wavelets, cf. Theorem 9.13. For all the nonlinear problems, considered here, the corresponding nonlinear equations can be solved with Newton’s method, satisfying the mesh independence principle. Methods for fully nonlinear elliptic problems are impossible with our technique. In this chapter we use the notation Rd , Cd , . . . , instead of the previous Rn , Cn , . . . , and restrict the presentation to a Hilbert space setting.

9.2

The scope of problems

In this section, we shall brieﬂy explain the scope of problems we shall be concerned with. The goal is to derive a stable and convergent wavelet scheme for problems with a linearization, that can be interpreted as boundedly invertible compact perturbations of an operator equation with stable discretization. Especially, we are interested in problems related to well-known elliptic and saddle point problems. The Stokes operator and, for moderate Reynolds numbers, its compact perturbation, the (linearized) Navier–Stokes operator, fall into this category. Therefore we ﬁrst recall the general setting of operator equations. Suppose that U is a Hilbert space with norm · U induced by the inner product ·, ·. Let A : U −→ U denote a linear operator into the normed dual U of U. In contrast to the other methods, we assume throughout that in this chapter U and the approximating subspaces, SΛ := {ψλ : λ ∈ Λ} ⊂ U, include the appropriate boundary conditions. We shall mainly discuss the case that A can be written as A = B + C,

(9.1)

with a boundedly invertible operator, B ∈ L(U, U ), with stable discretization, and a compact C ∈ C(U, U ). We start with a short presentation of general elliptic equations. It is well known that, given a Hilbert space U, its dual U and the dual pairing ·, ·U ×U , a general elliptic operator A : U → U induces a continuous and elliptic bilinear form, see Sections 2.3 and 2.4, a : U × U → R, a(u, v) := Au, vU ×U ∀ u, v ∈ U and a(v, v) ≥

αv2U

−

γv2WU ,

α > 0, for Hilbert spaces U → WU =

(9.2) WU

→ U ,

usually a Gelfand triple U → WU = WU → U with dense and continuous embedding U → WU . Furthermore, in A = B + C we usually assume B, inducing a coercive

638

9. Variational methods for wavelets

bilinear form b(·, ·), i.e. Bv, vU ×U = b(v, v) ≥ αv2U ,

α > 0,

for all v ∈ U.

(9.3)

It is known that, for an invertible A, in particular for a B in (9.3), the equation a(u0 , v) = f, vU ×U

for all v ∈ U

(9.4)

is uniquely solvable. Consequently, typical examples for A and B are given by general elliptic partial diﬀerential operators and their invertible special cases satisfying (9.3). For example, the operator induced by the Poisson equation 49 −'u = f, u = 0,

in Ω ⊂ Rd

(9.5)

on ∂Ω,

would play the role of B. This B = −' is a bounded and boundedly invertible mapping of H01 (Ω) onto its dual H0−1 (Ω), i.e. BuU ∼ uU . Here “a ∼ b” means that both quantities can be uniformly bounded by some constant multiple of each other. Likewise, “< ∼” indicates inequality up to constant factors. We generalize the Laplacian to elliptic operators of order 2m. The generalization to systems follows exactly Section 4.3.2, cf. (9.13). For the reader’s convenience we repeat the relevant notation and equations: for the multi-indices α, we deﬁned, see (2.73), partials, vectors and reals as ∂iu =

∂u ∂ |α| , ∇u = (∂ 1 u, . . . , ∂ d u) and ∂ α u = ∂1α1 . . . ∂dαd u = u, α1 ∂xi ∂x1 . . . ∂xd αd ϑ = (ϑ1 , . . . , ϑd ) ∈ Rd , ϑα = (ϑ1 )α1 · · · (ϑd )αd , ϑi , ∂ i u(x), ∂ α u(x) ∈ R.

A linear second order elliptic equation, see (2.160), determines the solution u0 , such that d aij ∂ i u0 ∂ j vdx = f, vU ×U u0 ∈ U : a(u0 , v) = Au0 , vU ×U = Ω i,j=0

∀ v ∈ U = H01 (Ω), with aij ∈ L∞ (Ω).

(9.6)

The linear form f and our elliptic bilinear form a(u, v) are bounded, so |f, vU ×U | ≤ C vU ,

|a(u, v)| ≤ CuU vU .

(9.7)

The principal part, ap (u, v), satisﬁes for all ϑ ∈ Rd λ|ϑ|2d ≤ λ(x)|ϑ|2d ≤

d

aij (x)ϑi ϑj ≤ Λ|ϑ|2d ∀ x ∈ Ω with λ > 0.

i,j=1 49

Note that we use in this chapter Ω ⊂ Rd instead of Ω ⊂ Rn in the previous chapters.

(9.8)

9.3. Wavelet analysis

639

By (2.22), (2.79), ap (u, v) is bounded, and U = H01 (Ω)-coercive, hence there exists λ∗ > 0, such that d aij ∂ i u∂ j udx ≥ λ∗ u2U , ∀u ∈ U, i.e. |ap (u, u)| ∼ u2U . (9.9) ap (u, u) = Ω i,j=1

Elliptic equations of order 2m require U = H0m (Ω). For linear A, determine aαβ ∂ α u0 ∂ β vdx = f, vU ×U u0 ∈ U = H0m (Ω) : a(u0 , v) :=

(9.10)

Ω |α|,|β|≤m

¯ ∀v ∈ U with aαβ ∈ L∞ (Ω), and for 1 < m = |α| = |β| : aαβ ∈ C(Ω). The previous a(u, v) is bounded and has to be elliptic, hence the principal part, ap (u, v), is U-coercive, satisfying with λ, λ∗ > 0 and ∀ϑ ∈ Rd , see (2.79), (4.132), 2m m λ|ϑ|2m aαβ (x)ϑα ϑβ ≤ Λ|ϑ|2m ∀ x ∈ Ω, (9.11) d ≤ λ(x)|ϑ|d ≤ (−1) d ∀u ∈ U : ap (u, u) =

|α|=|β|=m

Ω |α|=|β|=m

aαβ ∂ α u0 ∂ β udx ≥ λ∗ u2U , i.e. |ap (u, u)| ∼ u2U .

Quasilinear elliptic systems of order 2m extend and modify the above equations and conditions, cf. Subsection 2.6.3 ﬀ. In the introduction we have already mentioned the diﬃculties in evaluating nonlinear functionals with wavelet arguments, cf. Subsection 9.3.5. Therfore, we impose the following condition: we admit only coeﬃcients such that Aα in (9.13), Aα (x, u, . . . , ∇m u), can be evaluated for wavelet arguments. Under this condition we consider for U = determine u0 ∈ D(G) ⊂ U

H0m (Ω, Rq ), q

(9.12)

≥ 1, the weak form, and

u0 ∈ D(G) s.t.

a( u0 , v ) := G u0 , v U ×U := Ω |α|≤m (Aα (x, u0 , . . . , ∇m u0 ), ∂ α v )q dx = f , v U ×U = |α|≤m (f α ∂ α v )q dx ∀ v ∈ U = H0m (Ω, Rq ), f α ∈ L2 (Ω, Rq ), q ≥ 1,

(9.13)

cf. Proposition 2.34. The appropriate conditions are listed in Sections 2.6.4, 2.6.6. For the linearized principle part, (9.11) has to be replaced by (2.394). Applications to saddle point and Navier–Stokes equations will be discussed in Section 9.6.

9.3

Wavelet analysis

Our goal is conforming Galerkin methods for the approximate solution of Au = f or Gu = f for operators A, G as in Section 9.2. However, in contrast to conventional

640

9. Variational methods for wavelets

ﬁnite element discretizations we will work with trial spaces that not only exhibit the usual approximation properties and good localization but in addition lead to expansions of any element in the underlying Hilbert spaces in terms of multiscale or wavelet bases with certain stability properties. This fact can be used to derive, e.g. very eﬃcient preconditioning strategies as well as powerful adaptive algorithms, see Section 9.4, Theorem 9.6, and Section 9.7. To understand these relationships, we shall ﬁrst of all brieﬂy recall the basic setting of wavelet analysis as far as it is needed for our purposes. In Subsection 9.3.1, we want to collect some facts concerning the discrete wavelet transform. Then, in Subsection 9.3.2, we discuss the biorthogonal wavelet approach. It is one of the most important properties of wavelets that they form stable bases for scales of smoothness spaces such as Sobolev and Besov spaces. These aspects will be discussed in Section 9.3.3. Then, in Subsection 9.3.4, we brieﬂy recall the construction of wavelet bases on bounded domains. Finally, in Subsection 9.3.5, we discuss how nonlinear functionals applied to wavelet expansions can be evaluated. 9.3.1

The discrete wavelet transform

In general, a function ψ is called a wavelet if all its scaled, dilated, and integertranslated versions ψj,k (x) := 2j/2 ψ(2j x − k),

j, k ∈ Z,

(9.14)

form a (Riesz) basis of L2 (R). Usually, they are constructed by means of a multiresolution analysis introduced by Mallat [482]: Deﬁnition 9.1. A sequence {Vj }j∈Z of closed subspaces of L2 (R) is called a multiresolution analysis (MRA) of L2 (R) if · · · ⊂ Vj−1 ⊂ Vj ⊂ Vj+1 ⊂ . . . ; ∞ C

(9.15)

Vj = L2 (R);

(9.16)

Vj = {0};

(9.17)

j=−∞ ∞ K j=−∞

f (·) ∈ Vj ⇐⇒ f (2·) ∈ Vj+1 ; f (·) ∈ V0 ⇐⇒ f (· − k) ∈ V0

(9.18) for all

k ∈ Z.

(9.19)

Moreover, we assume that there exists a function ϕ ∈ V0 such that V0 := span{ϕ(· − k), k ∈ Z}

(9.20)

9.3. Wavelet analysis

641

and that ϕ has stable integer translates, i.e., with ⎧ ⎫ 1/2 ⎨ ⎬ c2k <∞ , 2 (Z) := c := {ck }k∈Z : c2 := ⎩ ⎭ k∈Z

C1 ||c||2 ≤ ||

ck ϕ(· − k)||L2 ≤ C2 ||c||2 ,

c := {ck }k∈Z ∈ 2 (Z). (9.21)

k∈Z

The function ϕ is called the generator of the multiresolution analysis. The properties (9.15), (9.18), (9.20), and (9.21) immediately imply that ϕ is reﬁnable, i.e. it satisﬁes a two-scale relation ak ϕ(2x − k), (9.22) ϕ(x) = k∈Z

with the mask a = {ak }k∈Z ∈ 2 (Z). A function satisfying a relation of the form (9.22) is sometimes also called a scaling function. Because the union of the spaces {Vj }j∈Z is dense, and their intersection is {0}, it is easy to see that the construction of a wavelet basis reduces to ﬁnding a function whose translates span a complement space W0 of V0 in V1 , V1 = V0 ⊕ W0 ,

W0 = span{ψ(· − k) | k ∈ Z}.

(9.23)

Indeed, if we deﬁne Wj := {f (·) ∈ L2 (R) | f (2−j ·) ∈ W0 },

(9.24)

it follows from (9.16), (9.17) and (9.18) that L2 (R) =

∞ L

Wj ,

(9.25)

j=−∞

so that ψj,k (·) = 2j/2 ψ(2j · −k),

j, k ∈ Z

(9.26)

2

forms a wavelet basis of L (R). Obviously, (9.18) and (9.20) imply that the wavelet ψ can be found by means of a functional equation of the form bk ϕ(2x − k), (9.27) ψ(x) = k∈Z

where the sequence b := {bk }k∈Z has to be judiciously chosen; see, e.g. [173, 254, 490] for details. The construction outlined above is quite general. In many applications, it is convenient to impose some more conditions, i.e. to require that functions on diﬀerent scales are orthogonal with respect to the usual L2 inner product, i.e.

ψ(2j • −k), ψ(2j • −k ) = 0,

if j = j .

(9.28)

642

9. Variational methods for wavelets

This can be achieved if the translates of ψ not only span an (algebraic) complement but the orthogonal complement, V0 ⊥ W0 ,

W0 = span{ψ(• − k) | k ∈ Z}.

(9.29)

The resulting functions are sometimes called pre-wavelets. The basic properties of scaling functions and (pre-) wavelets can be summarized as follows:

r Reproduction of polynomials. If ϕ is contained in C0r (R) := {g | g ∈

C r (R) and supp g compact}, then every monomial xα has an expansion of the form cα α ≤ r. (9.30) xα = k ϕ(x − k), k∈Z

r Oscillations. If the generator ϕ is contained in C0r (R), then the associated wavelet ψ has vanishing moments up to order r, i.e. xα ψ(x)dx = 0 for all R

0 ≤ α ≤ r.

(9.31)

r Approximation. If ϕ ∈ C0r (R) and f ∈ H r (R), then the following Jackson-type inequality holds, cf. Lemma 9.7: inf f − gL2 (R) ≤ C2−jr |f |H r .

g∈Vj

(9.32)

In practice, it is clearly desirable to work with an orthonormal wavelet basis. This can be realized as follows. Given an 2 -stable generator in the sense of (9.21), one may deﬁne another generator φ by ϕ(ξ) ˆ ˆ . φ(ξ) := ( k∈Z |ϕ(ξ ˆ + 2πk)|2 )1/2

(9.33)

It can be checked directly that the translates of φ are orthonormal and span the same space V0 . The generator φ is also reﬁnable, φ(x) = a ˆk φ(2x − k) (9.34) k∈Z

and it can be shown that the function bk φ(2x − k), ψ(x) =

bk := (−1)k a ˆ1−k

(9.35)

k∈Z

is an orthonormal wavelet with the same regularity properties as the original generator ϕ. However, this approach has a serious disadvantage. If the generator ϕ is compactly supported, this property will in general not carry over to the resulting wavelet since it gets lost during the orthonormalization procedure (9.33). Therefore the compact support will only be preserved if we can dispense with the orthonormalization procedure, i.e. if the translates of ϕ are already orthonormal. This observation was the starting point for the investigations of I. Daubechies who constructed a family φN , N ∈ N, of generators with the following properties [253, 254].

9.3. Wavelet analysis

643

Theorem 9.2. Good wavelet generators: There exists a constant β > 0 and a family φN of generators satisfying φN ∈ C βN (R), supp φN = [0, 2N − 1], and φN (·), φN (· − k) = δ0,k ,

φN (x) =

N2

ak φN (2x − k).

(9.36)

k=N1

Obviously, (9.36) and (9.35) imply that the associated wavelet ψ N is also compactly supported with the same regularity properties as φN . Given such an orthonormal wavelet basis, any function v ∈ Vj has two equivalent representations, the single-scale representation with respect to the functions φj,k (x) = 2j/2 φ(2j x − k) and the multiscale representation which is based on the functions φ0,k , ψl,m , k, m ∈ Z, 0 ≤ l < j, ψl,m := 2j/2 ψ(2j x − m). From the coeﬃcients of v in the single-scale representation, the coeﬃcients in the multiscale representation can easily be obtained by some kind of ﬁltering, and vice versa. Indeed, given j λk φj,k v= k∈Z

and using the reﬁnement equation (9.34) and the functional equation (9.35), it turns out that j j −1/2 −1/2 ¯bk−2l λ ψj−1,m . 2 a ¯k−2l λk φj−1,l + 2 (9.37) v= k l∈Z

m∈Z

k∈Z

k∈Z

From (9.37) we observe that the coeﬃcient sequence λj−1 = {λj−1 k }k∈Z which describes the information corresponding to Vj−1 can be obtained by applying the low-pass ﬁlter H induced by a to λj , λj−1 = Hλj , λj−1 = 2−1/2 a ¯k−2l λjk . (9.38) l k∈Z

The wavelet space Wj−1 describes the detailed information added to Vj−1 . From (9.37), we can conclude that this information can be obtained by applying the high-pass ﬁlter D induced by b to λj : −1/2 ¯bk−2l λj . cj−1 = Dλj , cj−1 = 2 (9.39) l k k∈Z

By iterating this decomposition method, we obtain a pyramid algorithm, the so-called fast wavelet transform: H

H

λj

H lj−2

lj−1 D

D c j−1

lj−3 D

c j−2

c j−3

644

9. Variational methods for wavelets

A reconstruction algorithm can be obtained in a similar fashion. Indeed, a straightforward computation shows j 1 j−1 φj,l λl φj,l = √ al−2k λk + bl−2n cj−1 n 2 n∈Z l∈Z l∈Z k∈Z so that λjl = 2−1/2

al−2k λj−1 + 2−1/2 k

bl−2n cj−1 n .

(9.40)

n∈Z

k∈Z

Similar decomposition and reconstruction schemes also exist for the pre-wavelet case. There are several methods to generalize this concept to higher dimensions. The simplest way is to use tensor products. Given a univariate multiresolution analysis with generator ϕ, it turns out that φ(x1 , . . . , xd ) := ϕ(x1 ) · · · ϕ(xd ) 2

(9.41)

d

generates a multiresolution analysis of L (R ). Let E denote the vertices of the unit cube in Rd . Deﬁning ψ 0 := ϕ and ψ 1 := ψ, it can be shown that the set Ψ of 2d − 1 functions e

ψ (x1 , . . . , xd ) :=

d

ψ el (xl )

e = (e1 , . . . , ed ) ∈ E\{0},

(9.42)

l=1

generates by shifts and dilates a basis of L2 (Rd ). There also exist multivariate wavelet constructions with respect to nonseparable reﬁnable functions φ satisfying ak φ(2x − k), {ak }k∈Zd ∈ 2 (Zd ), (9.43) φ(x) = k∈Zd

see, e.g. [420] for details. Analogously to the tensor product case, a family ψ i , i = 1, . . . , 2d − 1, of wavelets is needed. Each ψ i satisﬁes a functional equation similar to (9.27): bik φ(2x − k). (9.44) ψ i (x) = k∈Zd

The basic properties of wavelets and scaling functions (approximation, oscillation, etc.) carry over to the multivariate case in the usual way. 9.3.2

Biorthogonal bases

Given an orthonormal wavelet basis, the basic calculations are usually quite simple. For instance, the wavelet expansion of a function v ∈ L2 (R) can be computed as v= v, ψj,k ψj,k , ψj,k (x) := 2j/2 ψ(2j x − k). (9.45) j,k∈Z

However, requiring smoothness and orthonormality is quite restrictive, and consequently, as we have already seen above, the resulting wavelets are usually not

9.3. Wavelet analysis

645

compactly supported. It is one of the advantages of the pre-wavelet setting that the compact support property of the generator can be preserved. Moreover, since we have to deal with weaker conditions, the pre-wavelet approach provides us with much more ﬂexibility. Therefore, given a generator ϕ, many diﬀerent families of pre-wavelets adapted to a speciﬁc application can be constructed. Nevertheless, since orthonormality is lost, one is still interested in ﬁnding suitable alternatives which in some sense provide a compromise between both concepts. This can be performed by using the biorthogonal approach. For a given (univariate) wavelet basis {ψj,k , j, k ∈ Z}, ψj,k (x) := 2j/2 ψ(2j x − k), one is interested in ﬁnding a second system {ψ˜j,k , j, k ∈ Z} satisfying ψj,k (·), ψ˜j ,k (·) = δj,j δk,k ,

j, j , k, k ∈ Z.

Then all the computations are as simple as in the orthonormal case, i.e. v, ψ˜j,k ψj,k = v, ψj ,k ψ˜j ,k . v=

(9.46)

(9.47)

j ,k ∈Z

j,k∈Z

To construct such a biorthogonal system, one needs two sequences of approximation spaces {Vj }j∈Z and {V˜j }j∈Z . As for the orthonormal case, one has to ﬁnd bases ˜ 0 satisfying the biorthogonality for certain algebraic complement spaces W0 and W conditions ˜ 0, V0 ⊥ W

V˜0 ⊥ W0 ,

V0 ⊕ W0 = V1 ,

˜ 0 = V˜1 . V˜0 ⊕ W

(9.48)

This is quite easy if the two generators φ and φ˜ form a dual pair, ˜ − k) = δ0,k . φ(·), φ(·

(9.49)

Indeed, then two biorthogonal wavelets ψ and ψ˜ can be constructed as ˜ ˜ (−1)k d1−k φ(2x − k), ψ(x) = (−1)k a1−k φ(2x − k) ψ(x) = k∈Z

(9.50)

k∈Z

where φ(x) =

k∈Z

ak φ(2x − k),

˜ φ(x) =

˜ dk φ(2x − k).

(9.51)

k∈Z

Therefore, given a primal generator φ, one has to ﬁnd a smooth and compactly supported dual generator φ˜ satisfying (9.49) which is much less restrictive than the orthonormal setting. Elegant constructions can be found, e.g. in [202]. There, especially, the important case where the primal generator is a cardinal B-spline, φ = Nm , m ≥ 1, is discussed in detail. Generalizations to higher dimensions also exist [201]. For further information on wavelet analysis, the reader is referred to one of the excellent textbooks on wavelets which have appeared quite recently [173,254,433,490, 669].

646 9.3.3

9. Variational methods for wavelets

Wavelets and function spaces

It is one of the most important properties of wavelets that they give rise to stable bases in scales of function spaces. For wavelet applications, the Besov spaces are the most important smoothness spaces. Therefore in this section we introduce the Besov spaces and recall their characterization by means of wavelet expansions. For simplicity, we restrict ourselves to the biorthogonal setting and the univariate case. Similar results also hold for the multivariate and the pre-wavelet setting. We refer to [329] and [490] for details. Let us start by recalling the deﬁnition of Besov spaces. The modulus of smoothness ωr (v, t)Lp (R) of a function v ∈ Lp (R), 0 < p ≤ ∞, is deﬁned by ωr (v, t)Lp (R) := sup Δrh (v, ·)Lp (R) ,

t > 0,

|h|≤t

with Δrh the r-th diﬀerence with step h. For s > 0 and 0 < q, p ≤ ∞, the Besov space Bqs (Lp (R)) is deﬁned as the space of all functions v for which . ∞ 1/q [t−s ωr (v, t)Lp (R) ]q dt/t , 0 < q < ∞, 0 |v|Bqs (Lp (R)) := (9.52) −s q=∞, supt≥0 t ωr (v, t)Lp (R) , is ﬁnite with r := [s] + 1. Then, (9.52) is a (quasi-)seminorm for Bqs (Lp (R)). If we add vLp (R) to (9.52), we obtain a (quasi-)norm for Bqs (Lp (R)). Let P0 be the biorthogonal projector which maps L2 (R) onto V0 , i.e. ˜ − k)φ. P0 (v) := v, φ(· k∈Z

Then, P0 has an extension as a projector to Lp (R), 1 ≤ p ≤ ∞. For each v ∈ Lp (R), we have v, ψ˜j,k ψj,k . (9.53) v = P0 (v) + j≥0,k∈Z

Bqs (Lp (R))

The Besov spaces can be characterized by wavelet coeﬃcients provided the parameters s, p, q satisfy certain restrictions. For simplicity, we shall only discuss the case q = p. Then, the following characterization holds: Proposition 9.3. Let φ and φ˜ be in C r (R). If 0 < p ≤ ∞ and r > s > (1/p − 1), then a function v is in the Besov space Bps (Lp (R)), if and only if, v, ψ˜j,k ψj,k (9.54) v = P0 (v) + j≥0,k∈Z

with

⎛ P0 (v)Lp (R) + ⎝

⎞1/p 2j (s+(

1 1 2−p

))p |v, ψ˜ |p ⎠ j,k

j≥0,k∈Z

and (9.55) provides an equivalent (quasi-)norm for Bps (Lp (R)).

<∞

(9.55)

9.3. Wavelet analysis

647

In the case p ≥ 1, this is a standard result and can be found for example in Meyer [490] (Section 10 of Chapter 6). For the general case of p, this can be deduced from general results in Littlewood–Paley theory (see e.g. Section 4 of Frazier and Jawerth [329]) or proved directly. We also refer to [636] and [195] for further information. The condition that s > (1/p − 1) implies that the Besov space Bps (Lp (R)) is embedded in Lp˜(R) for some p˜ > 1 so that the wavelet decomposition of v is deﬁned. Also, with this restriction on s, the Besov space Bps (Lp (R)) is equivalent to the nonhomogeneous s deﬁned via Fourier transforms and Littlewood–Paley theory. Besov spaces Bp,p Another important scale of function spaces we shall be concerned with are the Sobolev spaces H s (R). Usually, these spaces are deﬁned by means of weak derivatives or by Fourier transforms, see, e.g. [1] for details. However, it can be shown that for p = q = 2, Sobolev and Besov spaces coincide: H s (R) = B2s (L2 (R)),

s > 0.

Therefore Proposition 9.3 immediately provides us with a characterization of Sobolev spaces. Proposition 9.4. Let φ and ψ be in C r (R), r > s. Then a function v is in the Sobolev space H s (R), if and only if, ⎛ P0 (v)Lp (R) + ⎝

⎞1/2 22js |v, ψ˜j,k |2 ⎠

<∞

(9.56)

j≥0,k∈Z

and (9.56) provides an equivalent (quasi-)norm for H s (R). 9.3.4

Wavelets on domains

Our aim is to employ wavelet bases for the design of eﬃcient numerical schemes. However, usually the operator under consideration is deﬁned on a bounded domain Ω ⊂ Rd , and therefore we are faced with the problem of designing a wavelet basis adapted to this domain. We cannot simply cut all elements of a wavelet basis on the whole Euclidean plane, for this would destroy almost all of the important properties described above. Nevertheless, it is by now well known how to construct suitable biorthogonal wavelet bases for many cases of interest, see [159, 160, 204, 250]. The ﬁrst step is always to construct nested sequences S = {Sj }j≥j0 , S˜ = {S˜j }j≥j0 whose unions are dense in L2 (Ω). Then, one has to ﬁnd suitable bases Ψj = {ψj,k : k ∈ ∇j },

˜ j = {ψ˜j,k : k ∈ ∇j }, Ψ

˜ j of S˜j in S˜j+1 , such that the for some complements Wj of Sj in Sj+1 and W biorthogonality condition ψj,k , ψ˜j ,k = δj,j δk,k

(9.57)

holds. Moreover, one has to ensure that all the convenient properties of wavelets, especially the characterization of Sobolev spaces according to Propositon 9.4, can still

648

9. Variational methods for wavelets

be saved. What turns out to matter is that C Ψ= Ψj j≥j0

forms a Riesz basis of L2 (Ω), i.e. every v ∈ L2 (Ω) has a unique expansion v=

∞

v, ψ˜j,k ψj,k

(9.58)

j=j0 k∈Jj

such that

⎛ vL2 (Ω) ∼ ⎝

∞

⎞ 12 |v, ψ˜j,k |2 ⎠ ,

v ∈ L2 (Ω),

(9.59)

j=j0 k∈∇j

and that both S and S˜ should have some approximation and regularity properties which can be stated in terms of the following pair of estimates. There exists some ρ > 0 such that the inverse estimate ns vn H s (Ω) < ∼2 vn L2 (Ω) ,

vn ∈ Sn ,

(9.60)

holds for s < ρ. Moreover, there exists some m ≥ ρ such that the direct estimate −sn inf v − vn L2 (Ω) < vH s (Ω) , ∼2

v ∈ H s (Ω),

vn ∈Sn

(9.61)

holds for s ≤ m, compare with (9.32). Such estimates are known to hold for every ﬁnite element or spline space. For instance, for piecewise linear ﬁnite elements one has ρ = 3/2, m = 2. It will be convenient to introduce the following notation. Let J := {λ = (j, k) : k ∈ ∇j , j ≥ j0 } =

∞ C

({j} × ∇j ),

j=j0

and deﬁne |λ| := j

if λ = (j, k) ∈ J, that is, k ∈ ∇j .

Then the following result holds, [242]. ˜ = {ψ˜λ : λ ∈ Theorem 9.5. Equivalent norms: Suppose that Ψ = {ψλ : λ ∈ J} and Ψ J} are biorthogonal collections in L2 (Ω) satisfying (9.59). If both S and S˜ satisfy (9.60) and (9.61) relative to some ρ, ρ > 0, ρ ≤ m, ρ ≤ m , then 12 2|λ|s 2 vH s (Ω) ∼ 2 |v, ψ˜λ | , s ∈ (−ρ , ρ), (9.62) λ∈J

∼

λ∈J

12 22|λ|s |v, ψλ |2

, s ∈ (−ρ, ρ ),

v ∈ H s (Ω).

9.3. Wavelet analysis

649

It will also be convenient to introduce the diagonal matrix (Ds )λ,λ := 2s|λ| δλ,λ .

(9.63)

By using (9.63), the norm equivalence (9.62) can be written as ˜ T (J) , vH s ∼ Ds v, Ψ 2

s ∈ (−ρ , ρ).

(9.64)

All the construction in [159, 160, 204, 250] induce isomorphisms of the form (9.62) at least for −1/2 < s < 3/2. For special domains such as rectangular domain even much sharper results are available. To explain the basic idea, let us brieﬂy recall the construction from [250]. It consists of three steps: (a) Construct dual pairs of generator bases Φj = {φj,k , : k ∈ 'j }, i.e.

˜ j = {φ˜j,k : k ∈ 'j }, Φ

˜ j := φj,k , φ˜j,l Φj , Φ

= I,

(9.65)

k,l∈j

whose elements have local support, diam(suppφj,k ) ∼ 2−j ,

diam(suppφ˜j,k ) ∼ 2−j ,

(9.66)

and such that their spans Sj = S(Φj ) := span Φj ,

˜ j ) := span Φ ˜j, S˜j = S(Φ

are nested, S(Φj ) ⊂ S(Φj+1 ),

˜ j ) ⊂ S(Φ ˜ j+1 ). S(Φ

(9.67)

˜ j ) are referred to as the primal and dual multiresolution spaces. S(Φj ), S(Φ ˘ j of some complement S(Ψ ˘ j ) of S(Φj ) in S(Φj+1 ), i.e. (b) Find a stable basis Ψ ˘ j ) ⊕ S(Φj ), S(Φj+1 ) = S(Ψ

j = j0 , . . . .

(9.68)

˘ j into a basis Ψj (c) Given such an initial decomposition project the initial basis Ψ ∞ ˜ which is perpendicular to S(Φj ). The union Ψ := Φj0 ∪ ∪j=j0 Ψj will be the ﬁnal primal wavelet basis satisfying our requirements. Note that (9.67) states that each coarse scaling function can be written as a linear combination of ﬁne scale basis functions. Viewing the bases as vectors whose components are the individual scaling functions, (9.67) is equivalent to saying that ˜ j,0 such that there must be #'j+1 × #'j matrices Mj,0 , M ΦTj = ΦTj+1 Mj,0 ,

˜ j,0 . ˜T = Φ ˜T M Φ j j+1

(9.69)

This program can be carried out for various types of domains. As an example, let us consider the interval Ω = (0, 1). The common strategy is to start with a biorthogonal multiresolution analysis on R as explained in Subsection 9.3.2. Speciﬁcally one often chooses the biorthogonal system from [202] where the primal scaling functions consist

650

9. Variational methods for wavelets

of cardinal B-splines. For j ≥ j0 where j0 is ﬁxed and suﬃciently large to disentangle endpoint eﬀects, one builds Φj by keeping those translates 2j/2 φ(2j · −k), k ∈ Z, that are fully supported in [0, 1]. These will be refered to as interior basis functions. For B-splines of order m one adds at each end of the interval m ﬁxed linear combinations of the 2j/2 φ(2j · −k) in such a way that the resulting collection Φj spans all polynomials of order m on (0, 1), compare with (9.30). One proceeds in the same way with the dual scaling functions restoring the original order of polynomial exactness while ˜ j . At this point one can verify (9.67), (9.66) and (9.69), i.e. keeping #Φj = #Φ nestedness, reﬁnability and locality. However, only the interior basis functions inherit the biorthogonality from the line whereas the boundary modiﬁcations have perturbed biorthogonality. One can show that in this spline family of dual multiresolution sequences one can always biorthogonalize [247], ending up with pairs of generator ˜ j satisfying all the properties mentionend above. In fact, there are additonal bases Φj , Φ noteworthy features: (a) Exploiting symmetry properties of the original scaling functions, one can arrange the bases to be invariant under the transformation x → 1 − x. (b) One can arrange that only a single primal and dual basis function diﬀers from zero at the endpoints of the interval. ˜ j always consist of three (c) According to the above comments, the bases Φj , Φ L I R parts signiﬁed by the index sets 'j , 'j , 'j , identifying the left boundary, interior and right boundary basis functions. Only the size of the interior sets 'Ij depends on j. The number of boundary functions always stays the same. Moreover, at each endpoint one has a ﬁxed number of scaling relations which can be computed a priori and stored. The interior basis functions satisfy the classical reﬁnement rule from the real line. ˜ j,0 are invariant under reversing the order of rows and (d) The matrices Mj,0 , M columns. (e) The reﬁnement matrices have ﬁxed upper left and lower right blocks (in the leftmost and rightmost column) with the above symmetry properties. Only the interior block (column) changes its size with growing level j, see Figure 9.1. This construction can also be modiﬁed in order to incorporate homogeneous Dirichlet boundary conditions. There are two ways that suggest themselves: (I) In view of property (b) above one can simply remove those basis functions from ˜ j that do not vanish at the endpoints of the interval. the generator bases Φj , Φ Obviously, the resulting collections are still biorthogonal and span appropriate subspaces of H01 (0, 1), see [160, 204, 250]. (II) Following [250], remove the two endpoint boundary functions from Φj that do not vanish at 0 and 1 but discard two interior basis functions from ˜ j . It can be shown that the resulting sets can again the dual collection Φ be biorthogonalized retaining all the above properties [250]. Note that now ˜ j ) still contains all polynomials up to a certain only Φj ⊂ H01 (0, 1) while S(Φ order.

9.3. Wavelet analysis

651

0

0 2

10 4 20 6 30

8 10

40

12 50 14 60 16 0

2

4 6 nz = 30

8

0

10

20 nz = 159

30

Figure 9.1 Nonzero elements (nz) of the matrix, Mj,0 , for a quadratic and cubic wavelet basis with three and six vanishing moments, respectively.

While (I) is most simple and immediate, option (II) appears to be conceptually preferable for the following reasons. Quite in line with the structure of the dual H −1 (0, 1), the functional on H01 (0, 1) should not be constrained at the endpoints. This is supported by the following observation. Biorthogonality of the wavelets combined with the fact that in (II) the dual system retains full polynomial exactness immediately ensures that the wavelets have vanishing moments of a certain order, cf. (9.31). This approach also works for more complicated domains such as polygonal or polyhedral domains in Rd . In this case, one can use a domain decomposition technique. The domain Ω of interest is devided into nonoverlapping subdomains Ωi , ¯= Ω

N C

¯ i, Ω

Ωi ∩ Ωj = ∅, i = j.

(9.70)

i=1

Here each Ωi is a smooth parametric image Ωi = κi () of the unit cube. In order to construct wavelet bases on Ω we follow the recipe from above, i.e. we have to construct ﬁrst dual pairs of biorthogonal bases on the composite domain Ω. This in turn is fairly easy be stitching together parametric liftings of generator bases on the unit cube. This is essentially due to the following facts. Firstly, the boundary properties (b) of the univariate ingredients conﬁne the gluing process to very few functions associated with the domain. Secondly, the symmetric properties (a) leaves convenient ﬂexibility

652

9. Variational methods for wavelets

concerning invariance under rotations in the parametric mappings κi . On the reference domain, we use tensor products of the scaling functions constructed for the interval, i.e. ϕ j,k := ϕj,k1 ⊕ · · · ⊕ ϕj,kj ,

k ∈ I1 × · · · × Id ,

(9.71)

see also (9.41). Then we deﬁne a generator bases on Ω by deﬁning −1 x∈Ω ϕij,k (x) := ϕ j,k κi (x) ,

(9.72)

and gluing together those scaling functions across interfaces of subdomains which do not vanish at the interfaces. The analogous construction is applied to the dual system. ˜Ω The result is a global system ΦΩ j , Φj which is globally continuous and biorthogonal to the modiﬁed inner product (v, w) :=

N

(v ◦ κi , w ◦ κi )

i=1

which is equivalent to the canonical inner product on Ω. So far, we have accomplished step (a) from above. Step (b) consists of identifying suitable bases for the complement spaces. There are several possibilities described in [250] which work for generator bases of any order. The result is the so-called composite wavelet basis. Once again, homogeneous Dirichlet boundary conditions can be incorporated. As before, we have the options (I) and (II). 9.3.5

Evaluation of nonlinear functionals

If we want to discretize nonlinear operator equations by means of wavelets, then sooner or later the problem of evaluating a nonlinear functional applied to a wavelet expansion occurs. That is, given an expansion uΛ = λ∈Λ dλ ψλ , one is interested in determining the coeﬃcients f (uΛ ), ψλ . Usually, this is a nontrivial problem and many diﬀerent solutions have been suggested. In this chapter, we shall only discuss the approach described in [251], which, in our opinion, leads to the most far-reaching results. Let u ∈ Bqs (Lτ ) ∩ L∞ , τ > (s/d + 1/p)−1 , for some q ≤ p with q ∈ (0, ∞] and suppose that uL∞ ≤ < ∞. Furthermore, let a function f ∈ C m [− , ], m > s with f (0) = 0 be given. Suppose that we have found an approximation in terms of a wavelet basis duλ ψλ with u − uΛ < . (9.73) uΛ = λ∈Λ

Then one is interested in computing f (uΛ ), ψλ with an accuracy of order . Expand˜ leads to ing f (uΛ ) with respect to the dual system Ψ f (u ) f (u ) d˜λ Λ = f (uΛ ), ψλ , d˜λ Λ ψ˜λ , f (uΛ ) = λ∈J

i.e. the quantities f (uΛ ), ψλ are just the expansion coeﬃcients of f (uΛ ) in terms of the dual wavelet bases. Let gΛˆ = λ∈Λˆ dˆλ ψ˜λ be some approximation of f (uΛ ) in the

9.4. Stable discretizations and preconditioning

653

˜ satisfying system Ψ gΛˆ − f (uΛ )L2 < ∼.

(9.74)

Using the norm equivalences as stated in Theorem 9.5 (for s = 0), we obtain ˆ ˆ − f (uΛ ), Ψ ˆ ≤ d ˆ ˆ − f (uΛ ), Ψ <g ˆ − f (uΛ )L2 <. d 2 2∼ Λ Λ Λ Λ ∼ This states that a good approximation of f (uΛ ) in terms of the dual wavelet basis automatically yields a good approximation of the desired coeﬃcients. The algorithm in [251] focuses on constructing such an approximation. Roughly speaking, this is ˜ in such a way that the corresponding space done by ﬁrst enlarging the set Λ to Λ ˜ SΛ˜ contains a function satisfying (9.74). Under the above assumptions, there exists a constant C(f, uL∞ ) such that f (u)Bqs (Lp ) ≤ C(f, uL∞ )uBqs (Lp ) ,

(9.75)

and it can be shown that this guarantees that only a ﬁnite enlargement has to be done. In a second step one now constructs the functions gΛˆ . This is done by performing a ˜ then approximating local transformation to scaling functions φ˜λ for all indices in Λ, f (u), φλ (e.g. by quadrature via point evaluations), and decomposing this in an appropriate way.

9.4

Stable discretizations and preconditioning

Our goal is to develop a suitable Galerkin scheme for approximating the solution of Au = f for an A as in (9.1) which is based on a wavelet basis as introduced in Section 9.3. Therefore we consider subspaces of the form SΛ := {ψλ : λ ∈ Λ}

⊂ U,

Λ ⊂ J.

(9.76)

We project our problem onto these spaces, i.e. the Galerkin approximation uΛ ∈ SΛ is deﬁned by AuΛ , v = f, v

for all v ∈ SΛ .

(9.77)

˜ we can associate canonical projectors With the pair of biorthogonal bases Ψ and Ψ QΛ and QΛ by v, ψ˜λ ψλ , QΛ (v) = v, ψλ ψ˜λ . (9.78) QΛ (v) := λ∈Λ

λ∈Λ

By means of these projectors, the Galerkin scheme (9.77) may very conveniently be written as QΛ AQΛ uΛ = QΛ f.

(9.79)

Obviously, (9.79) is equivalent to a linear system with the stiﬀness matrix AΛ := AΨΛ , ΨΛ T .

(9.80)

654

9. Variational methods for wavelets

The most far-reaching results were obtained for linear operators A : U → U , which are boundedly invertible, AuU ∼ uU ,

u ∈ U.

(9.81)

In this case, the equation Au = f

(9.82)

clearly has a unique solution, compare again with Section 9.2. Typical examples are second order coercive boundary value problems with Dirichlet boundary conditions on some open domain Ω ⊂ Rd such as the Poisson equation (9.5). In this case U = H01 (Ω) and U = H0−1 (Ω). Other examples are obtained by turning an exterior boundary value problem into a singular integral operator on the boundary Γ of the domain. For a formulation in terms of the single layer potential operator one obtains for instance U = H −1/2 (Γ) and U = H 1/2 (Γ). Thus U typically is a Sobolev space and U ⊂ L2 ⊂ U

or

U ⊂ L2 ⊂ U.

Unless A is a diﬀerential operator, one further property of A will matter. Whenever A is an operator with global Schwartz kernel (Av)(x) = K(x, y)v(y) dy, we will assume in addition that when (9.81) holds for U = H t then α β ∂x ∂y K(x, y) dist(x, y)− d+t+|α|+|β| .

(9.83)

This assumption covers a wide range of cases, including Calder´ on–Zygmund operators, cf. [248]. In any case, to obtain an applicable numerical algorithm, it is essential that the Galerkin scheme has some basic stability properties. By using the projectors QΛ , QΛ this requirement can be formulated as QΛ AuΛ U = QΛ AQΛ uΛ U ∼ uΛ U ,

uΛ ∈ S Λ .

(9.84)

Before we study this issue in detail, let us discuss one of its consequences. It is one of the main advantages of wavelet Galerkin schemes that they give rise to very simple preconditioning strategies. Essentially, this is a consequence of the norm equivalences stated in Theorem 9.5. Indeed, if the Galerkin scheme is stable in the sense of (9.84), then a simple diagonal preconditioning involving the matrices introduced in (9.63) gives rise to uniformly bounded condition numbers of the stiﬀness matrices (9.80). As we shall see below, (9.84) holds under natural conditions for all the problems considered in this chapter. Therefore the following result is very important: Theorem 9.6. Simple diagonal preconditioning for all problems in this chapter: Suppose that the Galerkin scheme (9.79) for A : H t → H −t is stable (9.84). Furthermore, suppose that the parameters ρ, ρ in (9.62) satisfy |t| < ρ, ρ .

(9.85)

9.4. Stable discretizations and preconditioning

655

Let DsΛ be the diagonal matrix deﬁned in (9.63). Then the matrices −t BΛ := D−t Λ AΛ DΛ

(9.86)

have uniformly bounded spectral condition numbers, = O(1), Λ ⊂ J. BΛ B−1 Λ

(9.87)

Proof. The ﬁrst step is to introduce the projectors Qj (v) := v, φ˜j,k φj,k , Qj (v) := v, φj,k φ˜j,k .

(9.88)

k∈j

k∈j

With the aid of these projections, we can deﬁne the operators Σs v :=

∞

2js (Qj − Qj−1 )v,

where

Qj0 −1 := 0,

(9.89)

j=j0

which, due to the norm equivalences (9.62), act as shifts in the Sobolev scale, Σs vL2 (Ω) ∼ vH s (Ω) ,

s ∈ (−ρ , ρ).

(9.90)

It can be shown that Σ−1 s = Σ−s ,

Σs =

∞

2js Qj − Qj−1 .

(9.91)

j=j0

Now consider any v ∈ SΛ and set w := Σt v. Thus, by (9.85), (9.90) and (9.84), we obtain wL2 = Σt vL2 ∼ vH t ∼ QΛ AQΛ vH −t . Employing the norm equivalence (9.90), now relative to the dual basis, and bearing (9.91) in mind, yields wL2 ∼ Σ−t QΛ AQΛ Σ−t wL2 . This means that the operators At,Λ := Σ−t QΛ AQΛ Σ−t : SΛ → S˜Λ are uniformly boundedly invertible, that is, At,Λ A−1 t,Λ = O(1),

#Λ → ∞.

(9.92)

It is now a matter of straightforward calculation to verify that the matrix representation of At,Λ relative to ΨΛ is −t At,Λ ΨΛ , ΨΛ T = D−t Λ AΛ DΛ ,

proving the claim.

(9.93)

656

9. Variational methods for wavelets

Sending #Λ to inﬁnity, the original equation Au = f can be viewed as an inﬁnite discrete system D−t AD−t d = D−t f ,

(9.94)

where D−t and A are the inﬁnite counterparts of D−t Λ , AΛ , respectively, and f := f, ΨT is the coeﬃcient sequence of f expanded relative to the dual basis. The sequence d then consists of the wavelet coeﬃcients of the solution, u = dT Ψ. The inﬁnite matrix D−t AD−t is, on account of Theorem 9.6, a boundedly invertible mapping from 2 (J) onto 2 (J). In the remaining part of this chapter, we shall mainly discuss Galerkin schemes based on sets of the form, cf. (9.88), Λn := {λ ∈ J : |λ| ≤ n},

i.e.

QΛn = Qn , SΛn = Sn .

(9.95)

Therefore, in this case the approximation spaces consist of the spaces of the underlying multiresolution analysis. We project our problem onto these spaces, i.e. the Galerkin approximation un := uΛn ∈ SΛn for Au0 = f , U = H0t (Ω) is deﬁned by u n = u Λn ∈ S n = S Λn :

Aun , v = f, v

for all v ∈ Sn = SΛn .

(9.96)

These methods correspond to uniform mesh reﬁnements. Later on, in Subsection 9.7, we shall also discuss adaptive numerical schemes based on wavelets, and then we will clearly dispense with these assumptions. In terms of the adjoint projectors Qn and Qn , Qn (v) = v, ψ˜λ ψλ , Qn (v) = v, ψλ ψ˜λ , (9.97) λ∈Λn

λ∈Λn

the Galerkin scheme (9.96) now reads as Qn Aun = Qn AQn un = Qn f.

(9.98)

(We sometimes use the redundant notation Qn AQn to indicate self-adjointness.) For later use, we summarize a generalized version of Propositions 4.50 for the wavelet setting which implies that the operators Qn and Qn are projection and linear approximation operators. Lemma 9.7. Qn , Qn are linear approximation and projection operators: The previous projectors Qn ∈ L(U, Sn ), Qn ∈ L(U , S˜n ), satisfy ∀v ∈ U, f ∈ U :

lim Qn v − vU = 0 and lim Qn f − f U = 0.

n→∞

n→∞

(9.99)

More precisely we obtain for U = H m (Ω), cf. (9.32), U = H m (Ω) : ∀v ∈ H m+s (Ω) : Qn v − vU C2−ns vH m+s (Ω) .

(9.100)

9.4. Stable discretizations and preconditioning

657

Proof. By using the norm equivalences (9.62), we obtain 22|λ|m |v, ψ˜λ |2 Qn v − v2H m < ∼ |λ|>n

< ∼

22|λ|(m+s) 2−2ns |v, ψ˜λ |2

|λ|>n

< 2−2ns ∼

22|λ|(m+s) |v, ψ˜λ |2

|λ|∈J

< 2−2ns v2H m+s . ∼

Let us now come back to the stability requirement as stated in (9.84). For uniform wavelet schemes, this property can be formulated as Qn Aun U ∼ un U ,

un ∈ Sn .

(9.101)

When A is positive deﬁnite and self-adjoint, then the property (9.101) holds for any trial subspace of U. The same is correct, if A induces a coercive bilinear form as B in (9.3). Moreover, in the framework of pseudo-diﬀerential operators, suﬃcient conditions have been derived by Dahmen et al. [248]. Let us brieﬂy recall the deﬁnition of pseudo-diﬀerential operators. To do this consider the Fourier inversion formula, cf. Theorem 2.105, and notice that here i ∈ C with i2 = −1, u(x) = (2π)−d eix·ξ u ˆ(ξ)dξ , u ∈ C0∞ (Rd ). (9.102) Rd

Diﬀerentiating this expression and employing here the notation ∂ j = β

−d

ξ β eix·ξ u ˆ(ξ)dξ .

D u(x) = (2π) Let P = p(x, D) = Ω ⊂ Rd . Then

∂ ∂xj

/i yields (9.103)

Rd

|β|≤k

aβ (x)Dβ be a diﬀerential operator deﬁned on a domain

(P u)(x) = (2π)−d

p(x, ξ)eix·ξ u ˆ(ξ)dξ .

(9.104)

Rd

Instead of the polynomial p(x, ξ) one can take a function σ(x, ξ) belonging to a more general class of functions. Then the operator A deﬁned by σ(x, ξ)eix·ξ u ˆ(ξ)dξ (9.105) (Au)(x) := Rd

is called a (linear) pseudo-diﬀerential operator with symbol σ(x, ξ). (For details we refer to the standard literature concerning pseudo-diﬀerential operators, e.g. Taylor [617].) It turns out that injectivity of A and a G˚ arding inequality also imply stability. More

658

9. Variational methods for wavelets

precisely, it turns out that the uniform Galerkin scheme based on the projectors Qn , Qn will be stable and convergent if the following three conditions are satisﬁed [248]: m r A is a pseudo-diﬀerential operator with symbol σ and is in the class S1,0 , the

subclass of the H¨ormander class with the property that β α Dx Dξ σ(x, ξ) ≤ cα,β (1 + |ξ|)(m−|α|) .

(9.106)

m can be split as a sum of the principal part σ0 representing an The symbols in S1,0 operator of order m and a second part σ1 which represents an operator of order m < m. r A satisﬁes a G˚ arding inequality, i.e.

σ0 (x, ξ) ≥ c|ξ|m ,

ξ ∈ Rd ,

(9.107)

holds.

r A is injective, Ker A = {0}.

(9.108)

In summary, it is by now possible to construct stable wavelet Galerkin schemes for a large class of problems. One of the aims of this chapter is to investigate to what extent these stability properties are preserved under perturbations. The relations are clariﬁed in the following theorem. This is, for our Hilbert space setting, a known result, see e.g. Zeidler [677], extended to our situation in Chapter 3, and B¨ ohmer et al. [129,567]. Nevertheless, we want to repeat the simple and straightforward proof for Theorem 9.8 formulated for wavelets. Theorem 9.8. Stability inherited to compact perturbations: Let B ⊂ L(U, U ) and suppose that the biorthogonal wavelet Galerkin scheme B n := Qn BQn is stable. Let A := B + C with C ∈ C(U, U ), the set of compact operators from U → U . Then A−1 ∈ L(U , U) =⇒ An := Qn AQn is stable. Proof. See Chapter 3, and e.g. B¨ ohmer et al. [129, 567]. We determine for an ˆ and u ˆn , of arbitrary u ∈ U and v := Cu the unique exact and discrete solutions, u ˆn = Qn v = Qn B u ˆ. Since B n is assumed to be stable, the equations B u ˆ = v and B n u the corresponding Galerkin scheme converges, hence for any u ∈ U we obtain, with the notation T = B −1 , Tn = (B n )−1 Qn , lim (B n )−1 Qn Cu − B −1 CuU = lim (Tn − T )CuU = 0. n→∞

n→∞

C is compact, so we get lim (T − Tn )C = 0.

n→∞

(9.109)

Now let un ∈ Sn . Because A is boundedly invertible, we can estimate un U ≤ A−1 Aun U ≤ A−1 Aun U ≤ A−1 B(I + B −1 C)un U ≤ A−1 B(I + T C)un U .

(9.110)

9.5. Applications to elliptic equations

659

Hence, we obtain An un U = Qn (B + C)Qn un U = B n I + (B n )−1 Qn CQn un U ≥ 1/B n −1 I + (B n )−1 Qn CQn un U = 1/B n −1 (I + Tn CQn )un U ≥ 1/B n −1 ((I + T C)Qn un U − (T − Tn )CQn un U ). By using the fact that A = B + C implies I + B −1 C = B −1 A, this reduces to An un U ≥ 1/B n −1 1/A−1 B − (T − Tn )C un U . Because of (9.109) and the stability of B n there exists a positive constant K, independent of n, such that for all n ≥ n0 the following holds: An un U ≥ Kun U

for all un ∈ Sn ,

hence An is stable.

9.5

Applications to elliptic equations

Based upon the previous results, in this section we will employ our general discretization approach for general convergence results for general linear and semilinear elliptic problems. Quasilinear problems are still excluded, since they cannot yet be evaluated for wavelet arguments. Combining the general stability theory with the conforming wavelets in Theorem 9.11, we obtain our goal, the convergence of (uniform) wavelet methods. The convergence analysis of nonuniform methods, i.e. of adaptive wavelet schemes, will be carried out in Section 9.7. The situation here is very similar to conforming FEMs in Subsection 4.3.3. In fact, only conforming wavelets are employed for our Galerkin methods. For the reader’s convenience, we update the general approach in Subsection 4.3.3 to the wavelet situation. For the following it is important that any elliptic operator A induces an elliptic bilinear form a(·, ·). This can be split into the sum of a coercive bilinear form b(·, ·) and its complement c(·, ·) such that the induced operators B, C satisfy A = B + C with a compact perturbation, C. For our U = H0m (Ω), m ≥ 1, we use for b(·, ·) the principal part ap (·, ·). Wavelet methods for linear elliptic equation of order 2m require U = H0m (Ω). For linear A, determine un0 ∈ Sn ⊂ U = H0m (Ω) such that n n n n a (u0 , v ) = Au0 , v U ×U = aαβ ∂ α un0 ∂ β v n dx = f, v n U ×U (9.111) Ω |α|,|β|≤m

¯ ∀ v n ∈ Sn with aαβ ∈ L∞ (Ω), and for 1 < m = |α| = |β| : aαβ ∈ C(Ω).

660

9. Variational methods for wavelets A, G −→

U Qn

Φ

n

tested by

←→

U

S˜n

tested by

Sn

Qn′

An, Gn −→

Sn

U′

←→

Figure 9.2 Conforming wavelet methods: Spaces and operators.

(Of course, for preconditioning reasons, we shall always employ the multiscale wavelet bases instead of the single-scale bases, compare with Theorem 9.6.) Equation (9.111) deﬁnes a bounded, elliptic a(un , v n ), with U-coercive principle part, cf. (9.11). Linear pseudo-diﬀerential operators: As already stated in Subsection 9.4, a uniform wavelet Galerkin scheme for operator equations induced by a linear pseudo-diﬀerential operator, B, in the H¨ ormander class is stable if the fundamental conditions (9.107) and (9.108) are satisﬁed. Wavelet methods for quasilinear elliptic equation and systems of order 2m: We may use again the spaces Sn of our multiresolution analysis as test and approximation spaces, i.e. we require Sn ⊂ U = H0m (Ω, Rq ), q ≥ 1. In fact, we use here and in the following Summary the same symbols for u0 ∈ U = H0m (Ω, Rq ), un0 ∈ Sn ⊂ H0m (Ω, Rq ), q ≥ 1, a (un0 , v n ) = Aun0 , v n U ×U = An un0 , v n U ×U = f, v n U ×U

(9.112)

∀ v n ∈ S n , hence, An := Qn A|S n = Qn AQn |Sn , similarly Gn := Qn G|S n . We recall and impose the condition (9.12) for the Aα in (9.113): the Aα (x, u, . . . , ∇m u) can be evaluated for wavelet arguments. Then we determine n n n n n (Aα (·, un0 , . . . , ∇m un0 ) , ∂ α v n )q dx u0 ∈ Sn s.t. a (u0 , v ) = Gu0 , v U ×U := = f, v n U ×U =

Ω |α|≤m

(fα , ∂ α v n )q dx

|α|≤m

∀ v ∈ Sn ⊂ U = H0m (Ω, Rq ), q ≥ 1. n

(9.113)

All these methods are special cases of the general discretization methods in Chapter 3. For the proof of the convergence we have to verify the conditions in Subsections 3.3–3.4 for the diﬀerent spaces and operators constituting these methods. Their relations are sketched in the diagram in Figure 9.2. We summarize the necessary conditions and deﬁnitions. As mentioned, the wavelet methods in this chapter are conforming variational methods. We discuss wavelet methods for quasilinear problems only under the condition (9.12), but do not discuss wavelet methods for fully nonlinear problems at all. There are still too many problems open in this context.

9.5. Applications to elliptic equations

661

Summary 9.9. Wavelet methods are special discretization methods: 1. The wavelet method has to be applicable to the problem Au0 = f or Gu0 = 0, cf. Deﬁnition 3.12. This requires, throughout a sequence of approximating spaces Sn for U, projectors Qn , Qn , cf. Lemma 9.7, and the discrete problems An , Gn , cf. (9.112), {Sn , Qn , Qn , Gn }n∈N , with dim Sn < ∞.

(9.114)

This is provided by the spaces of the underlying multiresolution analysis together with the associated biorthogonal projectors. The wavelet spaces Sn are approximating spaces, compare with (9.100): ∀v ∈ U : lim dist (v, Sn ) = 0, e.g. lim ||v − Qn v||U = 0. n→∞

n→∞

(9.115)

2. For semi- and quasilinear problems, we require that the Aα (x, u, . . . , ∇m u) in (9.113) can be evaluated for wavelet arguments. 3. Qn ∈ L(U, Sn ), Qn ∈ L(U , S˜n ), are linear approximation operators, such that ∀v ∈ U, f ∈ U : lim Qn vU = vU , lim Qn f U = f U . n→∞

n→∞

(9.116)

4. Wavelet methods, cf. (9.112), deﬁne a mapping Φn : D(Φn ) ⊂ (D(G) ⊂ U → U ) → (Sn → S˜n ), transforming A, f, G into An , f n , Gn as An := Φn A := Qn A|Sn , f n := Φn f := Qn f |Sn , Gn := Φn G := Qn G|S n . (9.117) 5. According to Deﬁnition 3.14, we deﬁne un , for a chosen u, as An un = Qn Au and Gn un = Qn Gu. Then Qn Au − An un = Qn A(u − un ) and Qn Gu − Gn un = Qn (Gu − Gun )

(9.118)

are the so-called (variational) consistency errors, cf. Theorem 9.11. 6. The (classical) consistency error or local discretization error in u is AQn u − Qn Au or GQn u − Qn Gu.

(9.119)

7. The G are called stable in u , and A stable, if for ﬁxed n0 , r, S ∈ R+ , n

n

n

uni ∈ Br (un ), i = 1, 2, ⇒ un1 − un2 U ≤ S Gn (un1 ) − Gn (un2 )S˜n ∀n > n0 (9.120) For An or An · −f n this reduces to (for any r and un , but for n > n0 ) (An )−1 ∈ L(S˜n , Sn ) exists and ||(An )−1 ||Sn ←S˜n ≤ S. The stability of Gn is essentially implied by the stability of its derivative (G (u0 ))n , cf. Theorems 9.8, 3.23. 8. For possible quadrature approximations, the Qn , An , Gn have to be replaced by ˜n. ˜ n , A˜n , G appropriate Q 9. All our wavelet methods are so-called linear methods in the sense Φn (B + C) = Φn B + Φn C, and Φn (G + f ) = Φn G + Φn f.

(9.121)

662

9. Variational methods for wavelets

Theorem 9.10. Existence and convergence for wavelet methods, cf. Theorem 3.21: Let the original problem Gu = 0 have the exact solution u0 ∈ U. Let its discretization Gn = Qn G |S n = Qn GQn |S n : S n → (S n ) in (9.117) satisfy 1. Gn : S n → (S n ) is deﬁned and continuous in Br (Qn u0 ), r > 0, n-independent; 2. Gn is consistent with G in Qn u0 , hence Qn Gn (Qn u0 )U → 0 for n → ∞; 3. Gn is stable for Qn u0 . Then the discrete problem Gn (un ) = 0 possesses a unique solution un0 ∈ S n near u0 for all suﬃciently large n and un0 converge to u0 as n un0 − u0 U < ∼ Qn G (Qn u0 )U → 0.

(9.122)

Hence, good convergence, essentially determined by the consistency, is usually only satisﬁed if u0 is smoother than u0 ∈ U. For our claim of a uniquely determined discrete solution, un0 , we have to verify the previous conditions 1., 2., 3. in Theorem 9.10. The continuity of G implies that of Gn . So consistency and stability have to be proved. Theorem 9.11. Variational and classical consistency errors vanish and go to 0, respectively: 1. Wavelet methods applied to linear problems, A ∈ L(U, U ), yield vanishing variational consistency errors Qn Au0 − An un0 ≡ 0. 2. For A ∈ L(U, U ), the classical consistency or local discretization error is An (Qn u) − Qn AuU = Qn A(Qn u − u)U < ∼Qn u − uU .

(9.123)

This remains correct for G if it is Lipschitz-continuous and (9.12) is satisﬁed. 3. By Lemmas 9.7, the classical consistency error tends to 0, according to Qn A(Qn u − u)U → 0 and Qn (GQn u − Gu)U → 0 for u ∈ U or for U = H m (Ω), m ≥ 1 : u ∈ H m+s (Ω) : Qn u − uU ≤ C2−ns uH m+s (Ω) . 4. The above (9.116) is a consequence of (9.115). Proof. The variational consistency errors vanish by (9.117) for the exact solution u0 . This is a simple consequence of Sn ⊂ U, e.g. (9.112) implies for A, similarly for G, Au0 − An un0 , v n U ×U = a (u0 − un0 , v n ) = f, v n − f, v n = 0 ∀ v n ∈ Sn . (9.124) For the classical consistency error we use Sn ⊂ U, and (Φn A)un = An un = Qn Aun . Then we ﬁnd An Qn u − Qn AuU ≤ CAU ←U Qn u − uU → 0, by Lemma 9.7. Finally, the stability of the nonlinear Gn is an immediate consequence of the stability of its linearized (Gn ) (un0 ) or the invertibility of G (u0 ), cf. Theorem 3.23. Theorem 9.12. Convergence for the wavelet method applied to the above cases: Let Au = f and let G(u) = 0 have (locally) unique solutions u0 . Hence A : U → U or, for the nonlinear operator G : D(G) ⊂ U → U , the linearization G (u0 ) : U → U be boundedly invertible. For nonlinar G in (9.13), (9.113), we impose the condition (9.12), so the Aα (x, u, . . . , ∇m u) can be evaluated for wavelet arguments. For the

9.5. Applications to elliptic equations

663

problems (9.111), (9.13) choose the wavelet methods according to (9.111)–(9.113). They are stable and consistent. Hence for n > n0 the wavelet equations have a (locally) unique solution un0 as well near the exact u0 . If this exact solution satisﬁes u0 ∈ H m+s (Ω), the un0 ∈ S n converge according to −sn un0 − u0 U < u0 H m+s (Ω) . ∼2

(9.125)

As already indicated in Summary 9.9 7., these results remain correct, if the exact evaluation of the integrals in (9.111), (9.113) is replaced by applying quadrature formulas. The following results are obtained by a straightforward modiﬁcation of the results in Section 5.4. ˜n (un , v n ) := quadr. formula applied to a(un , v n )|S n ×S n , A˜n un , v n |S n ×S n := a f˜n , v n |S n ×S n := quadr. formula applied to := f, v n |S n ×S n ,

(9.126)

˜ n un , v n | n n := a ˜n (un , v n ) := quadr. formula applied to a(un , v n )|S n ×S n . G S ×S We require good enough quadrature formulas, such that for smooth enough functions un , v n ∈ H m+s (Ω) and coeﬃcients, aij ∈ H m+s (Ω), terms Aα (un ) ∈ H −(m+s) (Ω) for un ∈ H m+s (Ω), and f ∈ H −(m+s) (Ω) the following quadrature errors hold: −sn |f, v n − f˜n , v n | < f H −(m+s) (Ω) · v n H m+s (Ω) , (9.127) ∼2 −sn ˜n (un , v n )| < · un H m+s (Ω) v n H m+s (Ω) . |an (un , v n ) − a ∼2 We formulate these quadrature approximate results with other generalizations:

Theorem 9.13. Wavelet methods for monotone operators, eigenvalue problems, nonlinear boundary operators, quadrature approximations: Under the conditions of Theorem 9.12, wavelet methods applied to the following problems yield unique converging solutions. For monotone operators the convergence has to be modiﬁed as in Section 4.5. For the remaining problems (9.125) is valid: variational methods for eigenvalue problems, cf. Section 4.7, and methods for nonlinear boundary operators, cf. Section 5.3. Additionally, let the quadrature approximate methods in (9.126) satisfy (9.127), cf. Section 5.4. All these general convergence properties are only valid for problems where the nonlinearities can be evaluated for wavelets. Finally, all the nonlinear equations for all problems, considered in this chapter, can be solved with Newton’s method, satisfying the mesh independence principle. We aim for a better convergence with respect to the L2 norm for second order equations, cf. Theorem 4.60. The solution, φg , of the dual problem, φg ∈ U s.t. a(w, φg ) = (g, w)∀w ∈ U can be combined with the Aubin–Nitsche Lemma 5.73 for modifying Theorem 5.74: n u0 − un0 U < sup φg − φng S n /gL2 (Ω) (9.128) ∼ u0 − u0 S n 0 =g∈L 2 (Ω) to obtain

664

9. Variational methods for wavelets

Theorem 9.14. Aubin–Nitsche result: Under the conditions of Theorem 9.10 and for an exact solution u0 ∈ H 1+s (Ω), the solution of the wavelet method equation satisﬁes −(s+1)n u0 H 1+s (Ω) . u0 − un0 L2 (Ω) < ∼C2

9.6

(9.129)

Saddle point and (Navier–)Stokes equations

We start with saddle point equations as a generalization of Stokes equations, a special case of the Navier–Stokes equations. For the reader’s convenience, we partially repeat Section 2.8. 9.6.1

Saddle point equations

Saddle point problems cf. Subsection 2.8.2 and, for the name, Theorem 2.129, require two pairs of Hilbert spaces U ⊆ WU , M ⊆ WM , with continuous bilinear forms a : U × U → R, b : U × M → R with U → WU → U , M → WM → M . (9.130) Usually, U → WU → U and M → WM → M are Gelfand triples with continuous and dense embeddings U → WU , M → WM . For given (f, g) ∈ U × M one has to determine a pair (u0 , p0 ) ∈ U × M such that a(u0 , v) + b(v, p0 ) = f, vU ×U for all v ∈ U, = g, qM ×M for all q ∈ M. b(u0 , q)

(9.131)

In general, we assume the bilinear form a(·, ·) to be coercive on the subspace U0 := {v ∈ U : b(v, q) = 0 ∀ q ∈ M}, hence, a(v, v) ≥ αv2U ∀v ∈ U0 , α > 0, (9.132) compare with (9.3). To ensure that the problem (9.131) is uniquely solvable, we also have to assume that U and M fulﬁll the inf–sup condition: inf

sup

0 =q∈M 0 =v∈U

b(v, q) ≥β vU qM

(9.133)

for some constant β > 0. For details, we refer to Subsection 2.8.2, and e.g. to Hackbusch [387]. The following equivalent formulation will be very useful in the sequel. Deﬁning the operators A : U → U , Au, vU ×U := a(u, v), v ∈ U, B : M → U , Bp, vU ×U := b(v, p), v ∈ U, B : U → M , B u, qM ×M := b(u, q), q ∈ M,

(9.134)

the problem (9.131) is equivalent to: ﬁnd (u0 , p0 ) ∈ U × M such that Au0 + Bp0 = f in U , = g in M . B u0

(9.135)

9.6. Saddle point and (Navier–)Stokes equations

If (9.131) is well posed, the operator

B :=

A B B 0

665

(9.136)

is bounded and boundedly invertible with respect to the usual graph norm, see again Subsection 2.8.2, and Hackbusch [387] for details, (u, p)U ×M ∼ B(u, p)U ×M where (u, p)2U ×M := u2U + p2M .

(9.137)

As saddle point problems are deﬁned on product spaces of the form U × M, we ˜ = {ψ˜λ : λ ∈ J U } need two pairs of biorthogonal wavelet bases Ψ = {ψλ : λ ∈ J U }, Ψ M M ˜ ˜ = {ϑμ : μ ∈ J } forming Riesz bases for WU and WM , and Θ = {ϑμ : μ ∈ J }, Θ ˜ also induces a pair of respectively. The second pair of biorthogonal bases Θ and Θ projectors in the sense of (9.79), (9.88): Pn (q) :=

n

Pn (q) :=

q, ϑ˜j,k ϑj,k ,

j=0 k∈JjM

n

q, ϑj,k ϑ˜j,k .

(9.138)

j=0 k∈JjM

(We use the notation Pn here since in our setting the projectiors Qn are related with the space U.) In our applications, U, MU and M, WM are mainly Hilbertian Sobolev spaces on suitable domains or manifolds Ω1 ⊂ Rd , Ω2 ⊂ Rd , i.e. U = H t (Ω1 ), WU = L2 (Ω1 ), where

M = H s (Ω2 ), WU = L20 (Ω2 ),

L20 (Ω) :=

q ∈ L2 (Ω) :

q(x)dx = 0 .

(9.139)

(9.140)

Ω

Then, we assume that the norm equivalences of the form (9.62) hold for both spaces, cf. Theorem 9.5, 1/2 1/2 2|λ|τ 2 2|λ|τ 2 ˜ 2 |v, ψλ | ∼ 2 |v, ψλ | , τ ∈ [−t, t], vH τ (Ω1 ) ∼ λ∈J U

qH ζ (Ω2 ) ∼

λ∈J U

1/2 2|μ|ζ

2

|q, ϑ˜μ |2

λ∈J U

∼

1/2 2|λ|τ

2

|v, ϑλ |

2

, ζ ∈ [−s, s].

λ∈J U

So we choose for (9.135), the trial spaces (UΛ , MΛ ) ⊂ (U, M) with the previous pair of index sets, (J U , J M ), Λ := (ΛU , ΛM ) ⊂ (J U , J M ).

(9.141)

It is well known [144] that stability of the discretization of (9.135) is ensured if the Ladyshenskaja–Babuska–Brezzi (LBB) condition inf

sup

qλ ∈MΛ vλ ∈UΛ

b(vλ , qλ ) ≥β vλ U qλ M

(9.142)

666

9. Variational methods for wavelets

is satisﬁed. Quite recently, explicit conditions to check (9.142) in the wavelet context have been derived by Dahlke et al. [241], cf. also Dahmen et al. [246], and the conditions for FEMs in Subsection 4.6.2. For any subset U¯ ⊆ U we will use the notation ¯ U¯⊥b := {q ∈ M : b(v, q) = 0 for all v ∈ U},

(9.143)

¯ ⊆M and similarly for M ¯ ⊥b := {v ∈ U : b(v, q) = 0 for all q ∈ M}. ¯ M

(9.144)

In terms of these sets, the fundamental result from Dahlke et al. [241] reads as follows. Theorem 9.15. Ladyshenskaja–Babuska–Brezzi (LBB) condition: The multiscale spaces UΛ , MΛ deﬁned above fulﬁll the LBB condition (9.142) provided that one of the following equivalent conditions holds (for B c.f. (9.134)): 1. MΛ ⊆ (U , UΛ )⊥b ; 2. B(MΛ ) ⊆ U˜Λ = span{ψ˜λ , λ ∈ Λ}; ˜ Λ and M ˜ Λ is deﬁned analogously to U˜Λ . 3. B (U , UΛ ) ⊆ M , M For simplicity, we treat (9.135) again by a uniform method, i.e. we consider multiscale spaces of the form V n := VJnV , JnV = {λ ∈ J V , |λ| ≤ n}, Mn := MJ M , JnM = n {μ ∈ J M , |μ| ≤ n }. Let Qn and Pn denote the associated biorthogonal projectors as deﬁned in (9.79), (9.88), and (9.138), respectively. Then the resulting Galerkin scheme for (9.135) is given by: ﬁnd a pair (un0 , pn0 )

un0 ∈ V n := VJnV , pn0 ∈ Mn := MJ M : νQn Aun0 + Qn Bpn0 = Qn f, n

Pn B un0 = Pn g. (9.145) Consequently we obtain for saddle point equations: Theorem 9.16. Stability and convergence for the saddle point wavelet methods:

1. Assume (9.133), and choose the multiscale spaces V n and Mn in such a way that one of the conditions in Theorem 9.15 is satisﬁed. 2. Then (9.145), the saddle point wavelet method, is stable and consistent for (9.131) with unique and converging solutions, cf. (9.158). Adaptive numerical schemes for saddle point problems have been derived in [237], see also Section 9.7. 9.6.2

Navier–Stokes equations

For the Stokes problem let Ω be a bounded, simply connected domain in Rd . Then, given a vector ﬁeld f ∈ H −1 (Ω, Rd ) and a function g ∈ L20 (Ω) one has to determine the velocity u0 ∈ H01 (Ω, Rd ) and the pressure p0 ∈ L20 (Ω) such that, in the strong

9.6. Saddle point and (Navier–)Stokes equations

667

formulation, −'u0 + ∇p0 = f

in Ω,

−∇ · u0 = g

in Ω.

(9.146)

In formulation this problem reads as follows: ﬁnd a solution (u0 , p0 ) ∈ the mixed U := H01 (Ω, Rd × M := L20 (Ω) for (9.131),with a(u, v) := ∇u, ∇v =

d ∂vi ∂ui (x) (x)dx for all u, v ∈ H01 (Ω, Rd ), ∂x ∂x j j Ω i,j=1

b(v, q) := −∇ · v, q = −

d

q(x)

Ω

i=1

∂ vi (x)dx for all q ∈ L20 (Ω). ∂xi

For further information concerning the theory and the numerical treatment of the Stokes equations, the reader is referred to Section 4.6, and, e.g. to Hackbusch [387] and Temam [622]. Our goals is to present a stable and convergent wavelet scheme for the stationary (nonlinear) Navier–Stokes equation. It has the form ⎞ ⎛ d i f u ∂ u + ∇ p −νΔu + i ⎠ ⎝ , G(u0 , p0 ) = in Ω, G(u, p) : = i=1 0 div u p0 dx = 0. (9.147) u0 = 0 on ∂Ω, Ω

After multiplying with test functions in the usual way, this problem ﬁts into our setting as follows: ﬁnd a pair (u0 , p0 ) ∈ H01 (Ω, Rd ) × L20 (Ω) such that νa(u0 , v) + d(u0 , u0 , v) + b(v, p0 ) = f, v = g, q b(u0 , q)

for all v ∈ H01 (Ω, Rd ), for all q ∈ L20 (Ω),

(9.148)

where d(u, v, w) :=

d i,j=1

ui (∂ i vj )wj dx.

(9.149)

Ω

For bounded Ω, and d ≤ 4, see Temam [622], Lemma 1.2, Ch. II, U 1, d(u, v, w) is a bounded trilinear form on H01 (Ω, Rd ) × H01 (Ω, Rd ) × H01 (Ω, Rd ). To discuss wavelet methods, in particular the stability, for (9.148), we consider its linearized form. We obtain the derivative G (u, p) evaluated in (u, p) and applied to (w, r) as ⎞ ⎛ d i i −νΔw + (wi ∂ u + ui ∂ w) + ∇ r ⎠ (9.150) G (u, p)(w, r) = ⎝ i=1 div w

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and, with the a(·, ·), b(·, ·), d(·, ·) in (9.148), and the pair of test functions (v, q) νa(w, v) + d(u, w, v) + d(w, u, v) + b(v, r) . G (u, p)(w, r), (v, q)2 := b(w, q) (9.151) d

In fact, the term i=1 (wi ∂ i u + ui ∂ i w) in (9.150) of lower order derivatives represents, for a moderate Reynolds number, a compact perturbation of the Stokes operator. This can immediately be seen as in Section 4.6. We shall derive a stable numerical scheme for the treatment of (9.150), (9.151). Some preparations are necessary. We want to solve the problem f1 , v = F (v, q). (9.152) G (u, p)(w, r), (v, q)2 = f2 , q For ﬁxed u, the continuous bilinear forms d(u, ·, w) and d(·, u, v) deﬁne elements in H −1 (Ω), hence they induce linear continuous operators, D1 , D2 , d1 (v) := d(u, ·, v) and d2 (v) := d(·, u, v) induce D1 , D2 ∈ L(U, U ).

(9.153)

Therefore we observe that (9.151) can be written as A(w, r) = B(w, r) + C(w, r) = F where, slightly diﬀerent from (9.136), cf. (9.134), (9.135), D1 + D2 0 νA B . , C= B= 0 0 B 0

(9.154)

(9.155)

We treat this problem again by a uniform method, i.e. we consider the same multicale spaces as in Section 9.6.1. Then the resulting Galerkin scheme for the linear (9.154) requires: ﬁnd a pair (un0 , pn ) un0 ∈ V n := VJnV ,

pn0 ∈ Mn := MJ M :

(9.156)

n

νQn Aun0 + Qn Bpn0 + Qn (D1 + D2 )un0 = Qn f1 , Pn B un0 = Pn f2 . The stationary (nonlinear) Navier-Stokes equation has the form (9.148), (9.147). So the wavelet method requires: ﬁnd a pair (un0 , pn0 ) ∈ V n × Mn such that

for all v n ∈ V n , νa (un0 , v n ) + d (un0 , un0 , v n ) + b v n , pn0 = f, v n

(9.157) = g, q n for all q n ∈ Mn . b un0 , q n Consequently we obtain for the (linearized) Navier–Stokes equations, cf. Subsection 4.6.2 and B¨ ohmer et al. [122, 129, 567].

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Theorem 9.17. Stability and convergence for the Navier–Stokes wavelet methods:

1. Choose the multiscale spaces V n and Mn such that one of the conditions in Theorem 9.15 is satisﬁed for the Stokes equations. 2. Then the linearized form of the Navier–Stokes operator A in (9.154) represents, for suﬃciently large ν, a compact perturbation of the Stokes operator B in (9.155). For boundedly invertible A, in particular for suﬃciently large ν, the Galerkin scheme (9.156) yields stable and consistent B n and An . 3. The wavelet Navier–Stokes problems (9.156) and (9.157) for the linearized and the nonlinear equations, possess, for large enough n, unique discrete solutions, un0 , pn0 , near the exact solution,u0 , p0 . For the nonlinear equations the nonlinear terms have to be evaluatible for wavelet arguments. 4. If the exact solution satisﬁes u0 ∈ H 1+s (Ω), p0 ∈ H 1+s (Ω), the un0 , pn0 converge according to, cf. [246],

−sn u0 H 1+s (Ω) + 2−sn p0 H 1+s (Ω) . (9.158) un0 − u0 U + p0 − pn0 M < ∼2 Proof. Compare Subsection 4.6.2 and B¨ ohmer et al. [122, 129, 567]. All the operators are continuously diﬀerentiable. So we show that our wavelet method for both problems satisﬁes the conditions of Theorem 3.21. The method is stable by assumption or by the compact perturbation property of the Navier–Stokes equation, by Theorem 9.8 for large enough parameters ν. Finally the conformity of our methods implies their consistency. This ﬁnishes the proof.

9.7

Adaptive wavelet methods, by T. Raasch

For the eﬃcient numerical simulation of realistic problems coming from technical applications, adaptive approximation methods with a highly nonuniform spatial discretization are of ultimate importance. The core ingredient of any adaptive algorithm is an appropriate coupling of a posteriori error estimators and adaptive space reﬁnement. By following this general strategy, reliable approximations of the unknown solution can be provided within a prescribed error tolerance and with a reasonable amount of computational work. For more than 25 years, adaptive ﬁnite element methods have been successfully used in practical applications. However, a full theoretical comprehension of their convergence and complexity properties, even for second order elliptic problems, could be acquired only recently. We refer to Morin et al. [499], Stevenson [600] and to Chapter 6 for an overview. Since the late 1990s, particular interest has also been drawn to the analysis of adaptive discretization schemes based on wavelets. Contrary to adaptive ﬁnite element methods, the convergence and complexity properties of adaptive wavelet algorithms could be analyzed already at an early stage of their development. This is in particular due to the strong analytic properties of wavelet systems. In the sequel, we will give a brief overview of the current status in adaptive wavelet methods; see also Dahmen [243] and Cohen [195]. Unless otherwise stated, we shall assume in this section that the underlying wavelet system Ψ = {ψλ }λ∈J is a Riesz basis for the solution space U. By this we mean that

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any u ∈ U can be expanded with respect to Ψ into a series u = λ∈J uλ ψλ with a unique coeﬃcient array u := (uλ )λ∈J , and with the stability condition cR u2 (J) ≤ uU ≤ CR u2 (J) ,

(9.159)

where cR , CR > 0 are suitable Riesz constants. Therefore, in contrast to the previous sections, the wavelet system is normalized in U, i.e. we have ψλ U 1 for all λ ∈ J. From the density of span Ψ in U, it follows that the variational problem a(u, v) = f, vU ×U for all v ∈ U is equivalent to the inﬁnite system Au = f , where we deﬁne

A := a(ψμ , ψλ ) λ,μ∈J ,

(9.160)

f = f, ψλ U ×U λ∈J .

The following observation is of fundamental importance to any adaptive wavelet algorithm: Theorem 9.18. Bounded bilinear form and A: If the bilinear form a : U × U → R is continuous, a(v, w) ≤ βvU wU for all v, w ∈ U and some β > 0, then A is bounded 2 . on 2 (J) with A ≤ βCR Moreover, if a = b + c decomposes into bilinear forms b and c, where b is coercive, b(v, v) ≥ αv2U , and c is bounded, c(u, v) ≤ γuU vU with γ < α, then A is boundedly invertible on 2 (J) with A−1 ≤ (α − γ)−1 c−2 R . Proof. The claim immediately follows by inserting the respective properties of the bilinear forms and (9.159) into the identity

a v ψ , w ψ μ μ λ λ μ λ sup . Av2 (J) = w2 (J) 0 =w∈2 (J) In the mathematical analysis of adaptive wavelet methods, typically the following three major issues are addressed. (i) First of all, one is interested in establishing convergence of the adaptive reﬁnement strategy. By this we mean that, given a target accuracy > 0 as input argument, the algorithm (u, ) → u under consideration terminates after ﬁnitely many steps and outputs a ﬁnitely supported coeﬃcient sequence u , such that u − u 2 (J) ≤ . Note that via the Riesz stability (9.159), such convergence estimates are equivalent to controlling the error in the U norm. (ii) As a second issue, the convergence rate of the adaptive method is usually analyzed in detail, i.e. the ratio between the number of active degrees of freedom and the corresponding discretization error. Results on attainable convergence rates of adaptive approximation schemes and on the conditions to guarantee them can be used to quantify whether, for a given operator equation, the

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additional computational work of adaptive algorithms asymptotically pays oﬀ at all, compared to nonadaptive strategies. In order to analyze convergence rates of adaptive approximation schemes in detail, certain tools from nonlinear approximation theory are necessary. We will summarize them in Subsection 9.7.1. (iii) The third issue in the mathematical analysis of an adaptive reﬁnement strategy is related with its computational complexity. In order to scale in an optimal way when applied to a realistic problem, the algorithm should work in linear complexity, i.e. the number of ﬂoating point and storage operations needed to compute the output u should stay proportional to the number of degrees of freedom #suppu . In that case, a reasonable response time of the adaptive algorithm can be expected for small target accuracies also. Implementable algorithms which feature one or more of these key properties typically have an iterative structure and make use of a posteriori error estimators. Choosing the inﬁnite dimensional system (9.160) as a starting point, the following two major strategies for the development of adaptive wavelet schemes for stationary problems have been pursued in the literature. (i) A ﬁrst natural strategy is to consider adaptive wavelet–Galerkin methods. They select certain ﬁnite subsets Λ ⊂ J and compute the solutions uΛ ∈ 2 (Λ) to the associated Galerkin systems PΛ AIΛ uΛ = PΛ f , where PΛ : 2 (J) → 2 (Λ) denotes the restriction operator and IΛ = P∗Λ : 2 (Λ) → 2 (J) is the trivial inclusion. The particular choice of the active index sets Λ is steered by the biggest coeﬃcients in the current discrete residual rΛ = r(IΛ uΛ ) = f − AIΛ uΛ ∈ 2 (J) or in an approximation thereof. This strategy leads to the adaptive wavelet– Galerkin algorithms considered in [70, 197, 203, 236, 333], and we note that it is similar to analogous residual-based reﬁnement strategies in a ﬁnite element setting [499,600]. In any of these variants, the actual computation of approximate residuals is made feasible by exploiting certain matrix compression properties of wavelet systems, discussed in Subsection 9.7.2. However, we remark that for wavelet–Galerkin methods, it is mostly assumed that the bilinear form a and hence A are symmetric. This is due to the fact that convergence is usually proven in the energy norm, see Subsection 9.7.3 for details. Only for a compact antisymmetric part of a, has convergence of adaptive wavelet–Galerkin methods recently been analyzed in [332]. (ii) A second approach, initially propagated in [198], is focused on the approximate application of well-known iterative methods for ﬁnite dimensional systems to inﬁnite dimensional systems like (9.160). For linear iterative schemes, like the damped Richardson iteration and the steepest descent method, converge can be veriﬁed also in an inﬁnite-dimensional setting [161, 198] by straightforward

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9. Variational methods for wavelets

arguments. Moreover, by replacing the evaluations of the inﬁnitely supported right-hand side f and of biinﬁnite matrix-vector products Av by inexact computable versions thereof, inexact linear iterations were derived in [161,198] and in subsequent related papers. We refer to Subsection 9.7.4 for an overview. For several reasons, adaptive descent iterations are more ﬂexible than adaptive Galerkin methods. On the one hand, linear descent iterations of the aforementioned type can be readily applied also to more general problems than those with a linear symmetric operator A. Generalizations towards nonsymmetric elliptic equations have been discussed in [198]. By using adaptive variants of the classical Uzawa iteration, also saddle point problems can be handled [237]. Moreover, the analysis of best tree approximation methods in [69, 196, 200] has enabled the development of adaptive wavelet algorithms for semilinear stationary equations, based on an adaptive Newton-type iteration [199]. We refer to Subsection 9.7.5 for details. In order to improve the linear convergence properties of iterations like the Richardson and the steepest descent method, one may also consider adaptive variants of more complicated Krylov iterations. As an example, inexact conjugate gradient methods have been analyzed in [252], although their numerical performance usually does not outperform the simpler steepest descent scheme. Another important feature is that, when choosing approximate iterative schemes, one has the possibility of replacing the wavelet basis Ψ by a redundant system, e.g. by a wavelet frame. Frames are usually easier to construct than bases, in particular on nontrivially shaped bounded domains. Moreover, the matrix compression properties of wavelet bases can be preserved when using suitable frame systems. It must be noted that in a frame discretization, the discrete system (9.160) may become singular and the coeﬃcient representation of a given function u will no longer be unique. However, these potential drawbacks can be handled in practice, e.g. for Richardson iterations [239,538,598] and for the steepest descent method [240]. Moreover, domain decomposition methods in 2 (J) can be used to improve the quantitative performance of wavelet frame discretizations considerably, see [601, 663] for details. 9.7.1

Nonlinear approximation with wavelet systems

The underlying principle of adaptive wavelet methods consists in approximating the unknown solution u in the norm of U by ﬁnite linear combinations from the ansatz system Ψ, such that the number of nonzero coeﬃcients is as small as possible. By using the Riesz stability (9.159), we may conﬁne the discussion to the sequence space regime, approximating the unknown coeﬃcient array u in 2 (J) by sparsely populated sequences. As a benchmark, the most economical approximations of a given v ∈ 2 (J) are the best N-term approximations vN , deﬁned by discarding in v all but the N largest coeﬃcients in absolute value. The error of best N -term approximation of v is then

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deﬁned as σN (v) := v − vN 2 (I) . Obviously, an upper bound for the convergence rate of an adaptive discretization scheme is given by the largest value s > 0, such that the best N -term approximation error of the unknown coeﬃcient array u decays like σN (u) N −s , as N tends to inﬁnity. All sequences with this property are usually collected in the nonlinear approximation space As := v ∈ 2 (J) : vAs := v2 (J) + sup N s σN (v) < ∞ . N ∈N

Note that · As is in general only a quasinorm. 50 It is well known that for > 0, 0 < p ≤ 2 − and s = p1 − 12 > 0, we have the continuous and dense embeddings p (J) → As → p+ (J),

(9.161)

see [286, 331] for a proof. Therefore, As is also frequently referred to as the weak p space pw (J), and As may be endowed with several equivalent quasinorms; we refer to [286] for details. In order to ensure that the unknown coeﬃcient array u is contained in As , certain nonclassical regularity assumptions on the solution u play a crucial role. Wavelet theory tells us, see [195], that if Ψ is a wavelet Riesz basis for U = H t (Ω) which is suﬃciently smooth and has polynomial order r, then for all 0 < s < (r − t)/d we have v ∈ p (J) if and only if v = vλ ψλ ∈ Bpsd+t (Lp (Ω)), (9.162) λ∈J

−1 and d is the spatial dimension of the domain Ω. Therefore, comwhere p = s + 12 bining (9.162) with the embedding (9.161), we see that whenever the unknown solution u is contained in the Besov space Bpsd+t (Lp (Ω)), then the nonlinear H t -approximation rate of u with respect to the wavelet system Ψ is at least s, uAs uBpsd+t (Lp (Ω)) . The maximum nonlinear H t -convergence rate that can be expected by only assuming appropriate smoothness conditions on the unknown solution u is (r − t)/d. Remark 9.19. 1. In order to obtain an H t -approximation rate s > 0 for linear approximation methods with uniform grid reﬁnement, it would be necessary that u belongs to the smaller space H sd+t (Ω). However, recent results from regularity theory of PDEs show that there exist several relevant classes of operator equations whose solutions are much smoother in the scale Bpsd+t (Lp (Ω)) than in the Sobolev scale, see [233, 235, 238] and the references therein. For example, corner singularities of polygonal domains in R2 have inﬁnitely high regularity in the Besov scale Bp2s+t (Lp (Ω)), whereas their Sobolev regularity is strongly limited [234]. For problems with those kinds of solutions, nonlinear approximation methods will 50

Here the triangle inequality is replaced by ∃C > 0. x + y ≤ C( x + y )∀x, y ∈ As

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therefore asymptotically outperform linear approximation strategies with uniform reﬁnement. 2. Note that we will not address the approximation of u in weaker norms than that of U. This is due to the fact that, to the best of our knowledge, a counterpart of Nitsche duality arguments is not known in the case of nonlinear approximation methods. In practical computations, where the target quantity v is only implicitly given, we will not be able to compute a best N -term approximation vN exactly. Instead, it is easier to aim at the computation of an approximate or near best N -term ˜ N , by which we mean that #supp˜ vN ≤ N and approximation v ˜ N 2 ≤ CσN (v), v − v with C ≥ 1 being a uniform constant which only depends on the particular algorithm. Due to the fact that the best N -term approximation of an element v ∈ As up to 1/s accuracy needs at most −1/s vAs entries, a convergent algorithm (u, ) → u is called asymptotically optimal if, for any u from the unit ball of As , the output u fulﬁlls #suppu −1/s , as tends to 0. The algorithm has asymptotically optimal complexity if, moreover, the number of ﬂoating point and storage operations to compute u are bounded by a constant multiple of #suppu . The algorithm is called s∗ -optimal and of s∗ -optimal complexity, respectively, if these implications hold true for all 0 < s < s∗ . In many adaptive schemes, computational optimality is ensured by inserting certain thresholding operations into the algorithm. The rationale is to thin out the current iterate in such a way that the perturbed algorithm still converges, but it additionally features an optimal work/accuracy balance. It is therefore of crucial importance to perform thresholding operations onto a given ﬁnitely supported vector v in a computationally optimal way. The naive approach of sorting all entries of v in modulus and then collecting the most important ones has the disadvantage of log-linear complexity. For this reason, sorting operations were not taken into account in the initial complexity analysis of adaptive wavelet methods [197]. However, it has been shown in [69, 598] that quasisorting is suﬃcient to compute an approximate best N -term approximation ˜ N in linear complexity. The following binning variant of hard thresholding is therefore v used in current adaptive wavelet schemes. Algorithm 9.1. COARSE[v, ] → v

q := log2 ((#suppv)1/2 v 2 / ) sort nonzero entries of v into bins: for all λ ∈ suppv do if |vλ | ≤ 2−q v 2 then put (λ, vλ ) into the bin Vq else put (λ, vλ ) into that bin Vi with 2−(i+1) v 2 < |vλ | ≤ 2−i v 2

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end if end do aggregate large entries until tolerance is met: v := 0; err := v 2 2 for i = 0 to q do for all (λ, vλ ) ∈ Vi do if err ≤ 2 then return v

(v )λ := vλ ; err := err − |vλ |2 end for end for

The computational optimality of the subroutine COARSE has been analyzed in [69, 598]. Theorem 9.20. v has few nonzero entries: Let v ∈ 2 (J) be ﬁnitely supported and > 0. Then for the output v := COARSE[v, ], it holds that v − v 2 ≤ and v has at most #suppv inf{N : σN (v) ≤ } nonzero entries. Moreover, the number of arithmetic operations and storage locations needed to compute v is bounded by a 1/s constant multiple of #suppv + max{log(−1 v2 ), 1} −1/s vAs . Moreover, ﬁnitely supported approximations to a given vector v ∈ As can be thinned out eﬃciently by an application of the subroutine COARSE, at the price of a slightly enlarged error, see [197, 598]. Theorem 9.21. Good approximation by COARSE with few nonzero entries: Let θ < 1/3 be ﬁxed and s > 0. Then, for any > 0, v ∈ As , and a ﬁnitely supported w ∈ 2 (J) ¯ := COARSE[w, (1 − θ)] fulﬁlls v − w ¯ 2 ≤ and with v − w2 ≤ θ, the output w ¯ is bounded by the number of nonzero entries in w ¯ −1/s vAs . #suppw 1/s

Consequently, there is a constant C > 0 which only depends on s > 0, such that ¯ As ≤ CvAs . w However, we note that thresholding routines like COARSE are not a necessary ingredient of any adaptive scheme. Recently, it has been proved that certain variants of adaptive wavelet–Galerkin methods attain optimal convergence rates also without intermediate coarsening of the iterands, see [333]. 9.7.2

Wavelet matrix compression

In almost all variants of adaptive wavelet methods, it is exploited that the biinﬁnite system matrix A is not arbitrarily structured but it often features certain compressibility properties. By this we mean that A can be approximated well by sparse matrices with a ﬁnite number of entries per row and column. Such compressibility properties can indeed be observed and veriﬁed, see [91, 197, 245, 571, 599], whenever the underlying operator is local or at least pseudo-local, i.e. for diﬀerential equations and certain classes of integral equations. In both cases, using

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the locality and the vanishing moment properties of wavelet systems, oﬀ-diagonal decay estimates of the following type can be shown for individual matrix entries, see [243, 571, 599]: a(ψλ , ψλ ) 2−||λ|−|λ ||σ 1 + δ(λ, λ ) −β , for all λ, λ ∈ J. (9.163) Here σ > d/2 and β > d depend on the particular wavelet system Ψ and on the operator, and δ is given by

δ(λ, λ ) := 2min{|λ|,|λ |} dist(suppψλ , suppψλ ),

for all λ, λ ∈ J.

In the special case of a diﬀerential operator, the decay parameter β > 0 in (9.163) may be chosen arbitrarily large, i.e. the oﬀ-diagonal decay is essentially exponential in the level distance ||λ| − |λ ||, a(ψλ , ψλ ) 2−||λ|−|λ ||σ , for all λ, λ ∈ J. (9.164) Moreover, it is important here that the wavelets ψλ are normalized in the solution space U = H t (Ω). Otherwise, in the case of an L2 normalization, the left-hand sides of −|λ|t . (9.163) and of (9.164) would entail also the normalization factors ψλ −1 H t (Ω) 2 For an appropriate formalization of compressibility from the viewpoint of the full system matrix, we recall the following deﬁnition from [197] and related works. A bounded operator A : 2 (J) → 2 (J) is called s∗ -compressible, when for each j ∈ N αj 2j nonzero entries per row and there exists an inﬁnite matrix Aj with ∞at most ∞ js column and j=0 αj < ∞, such that j=0 2 A − Aj < ∞ holds for any 0 < s < s∗ . Equivalently, for any 0 < s < s∗ and any N ∈ N, there exists an inﬁnite matrix with at most N nonzero entries per row and column and with distance to A of order N −s . Remark 9.22. 1. We do not assume that the nonzero entries of Aj coincide with those of A, i.e. quadrature errors are allowed. 2. Any s∗ -compressible A is bounded from As to As , 0 < s < s∗ [331]. In general, the particular compression rules mostly work by retaining in Aj only those entries Aλ,λ (or approximations thereof), for which the level diﬀerence is below the threshold ||λ| − |λ || ≤ j/d. For diﬀerential operators this is already suﬃcient, at least for a certain range of the compressibility exponent s∗ . In the case of singular integral operators, additional thresholding techniques have to be applied per column; we refer to [243, 571, 599] for details. In order to estimate the associated computational work, an s∗ -compressible matrix A is called s∗ -computable, if the computation of any column of Aj takes at most O(2j ) ﬂoating point operations. An interpretation is that the average computational cost of each individual entry of Aj should be constant. This notion of computability was introduced in [334, 335] in order to verify s∗ -optimal complexity of adaptive wavelet schemes for diﬀerential and singular integral operators, also taking the quadrature subproblems into account.

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Within an adaptive numerical scheme, the matrices Aj will be used to realize the multiplication of A with a given ﬁnitely supported vector v, up to a prescribed error tolerance . In particular, the estimate A − Aj ≤ Cj 2−js for 0 < s < s∗ can be used to construct a numerical routine APPLY for approximate matrix-vector multiplication. The initial deﬁnition of APPLY goes back to [197]. However, analogous to the COARSE Algorithm 9.1 from Subsection 9.7.1, also the APPLY subroutine has undergone a certain evolution in the literature. In Algorithm 9.2 below we present the most advanced binning variant from [598], for which also optimal complexity estimates could be established. Algorithm 9.2. APPLY[A, v, ] → w

q := log2 ((#suppv)1/2 v 2 A 2/ ) sort nonzero entries of v into bins: for all λ ∈ suppv do if |vλ | ≤ 2−q v 2 then put (λ, vλ ) into the bin Vq else put (λ, vλ ) into that bin Vi with 2−(i+1) v 2 < |vλ | ≤ 2−i v 2 end if end do regroup large nonzero entries of v into segments v[k] : k := 0; v[0] := 0; err := v 2 2 ; i := 0 while err > 2 /(4 A 2 ) do if Vi = ∅ then i := i + 1 if #suppv[k] > 2k − 2k−1 then k := k + 1; v[k] := 0 select (λ, vλ ) ∈ Vi and remove it from bin Vi (v[k] )λ := vλ ; err := err − |vλ |2 end do l := k compute approximation to Av: compute smallest j ≥ l with lk=0 Cj−k 2−(j−k)s v[k] 2 ≤ /2 l w := k=0 Aj−k v[k]

Theorem 9.23. Bounded operations, storage, nonzero entries by APPLY: Let A be s∗ –compressible and v be ﬁnitely supported. Then, for any > 0, the output w := 1/s APPLY[A, v, ] fulﬁlls Av − w 2 ≤ and w has at most #suppw −1/s vAs nonzero entries. Moreover, the number of arithmetic operations and storage locations 1/s needed to compute w is bounded by a constant multiple of −1/s vAs + #suppv. It remains to specify the concrete value of s∗ for which compressibility and computability of A can be expected, being of crucial importance to the convergence properties of an adaptive wavelet scheme. In order not to spoil the theoretically optimal value (r − t)/d, it is desirable that the system matrix A is s∗ -computable for values of s∗ strictly greater than (r − t)/d. The ﬁrst results on the compressibility of operators in wavelet coordinates can be found in [197, 243], assuming only generic properties of the wavelet system, like

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smoothness and vanishing moments, and certain properties of the operator. However, the results from [197, 243], on the actual range of s∗ were still suboptimal. It turned out shortly afterwards that the restriction to spline wavelet systems can substantially improve the compressibility properties of the system matrix. This is mostly due to the fact that spline wavelets are locally smooth, away from their lower-dimensional singular support. Therefore, in the case of spline wavelets more entries of A can be expected to have small modulus, compared to the case of generic wavelet systems. Considering the Laplace operator, compressibility of order s∗ = (γ − t)/d has been proved in [70, 237], where γ = sup{ν : ψλ ∈ H ν (Ω) for all λ ∈ J} is the critical Sobolev smoothness of Ψ. The most far-reaching compressibility results, however, have been obtained in [599]. For spline wavelet discretizations of diﬀerential operators and of integral operators with Schwartz kernel, the results from [599] show that A is indeed s∗ -compressible for s∗ > (r − t)/d. In particular, there exist compression rules A → Aj , such that A − Aj 2−js for s larger than any value for which u ∈ As can be expected. Therefore, the theoretically optimal convergence rate will not be limited by the compressibility of the system matrix. Concerning computability, in the early results on the convergence rates of adaptive wavelet schemes [197], it was assumed that the entries of A are available exactly at uniformly bounded computational cost. However, using spline wavelets, this is only possible for diﬀerential operators with constant coeﬃcients. The computational complexity of the adaptive matrix-vector multiplication APPLY in the case of general diﬀerential operators and of integral operators with Schwartz kernel was discussed recently in [335] and [334], respectively. Computing the nonzero entries of Aj by suitable quadrature rules with an error proportional to the compression error of merely dropping entries from A, the computational optimality of APPLY could be veriﬁed for a large range of compressible operators as well. 9.7.3

Adaptive wavelet–Galerkin methods

In the history of adaptive wavelet methods, convergence was ﬁrst established for certain Galerkin strategies [197, 236] Due to the fact that the respective proofs are using the energy norm, we will assume in this subsection that the bilinear form a : U × U → R is coercive and symmetric. Moreover, we denote with |||v|||:= Av, v1/2 the discrete energy norm of a given coeﬃcient sequence v ∈ 2 (J). The following norm equivalence is well known: A−1 −1/2 v2 (J) ≤ |||v||| ≤ A1/2 v2 (J) ,

for all v ∈ 2 (J).

Therefore, convergence of the coeﬃcient arrays in the discrete energy norm is equivalent to convergence in 2 (J) and, by the Riesz basis property (9.159), to U-convergence of the expansions. Adaptive Galerkin methods typically have an iterative structure. Starting with an initial Galerkin set Λ0 , they compute a sequence of nested index sets Λj , j ∈ N0 .

9.7. Adaptive wavelet methods, by T. Raasch

679

The iteration stops as soon as the associated Galerkin solution uΛj ∈ U is considered suﬃciently good by an embedded a posteriori error estimator. During the iteration, successive Galerkin approximations uΛj are chosen to realize a certain saturation property, i.e. the new iterate uΛj+1 and the corresponding coeﬃcient sequence uΛj+1 are strictly better than the previous ones: |||u − uΛj+1 ||| ≤ ρ|||u − uΛj |||,

ρ ∈ (0, 1).

(9.165)

In order to predict a new Galerkin set Λj+1 with a prescribed linear error reduction (9.165), the current discrete residual rΛj = f − AIΛj uΛj plays a key role. The following lemma points out that we have to determine the positions of the biggest entries of rΛj . The result is implicitly used in [197, 236], and the proof can be found in [331, 333]: Lemma 9.24. Let 0 < μ ≤ 1 and w ∈ 2 (J) with ﬁnite support suppw ⊂ Λ ⊂ J be such that PΛ r(w) 2 ≥ μr(w)2 (J) . (9.166) (J) Then the Galerkin solution uΛ associated with Λ fulﬁlls μ2 1/2 |||u − IΛ uΛ ||| ≤ 1 − |||u − w|||, κ(A) with κ(A) = AA−1 . In fact, if w = IΛj uΛj is the current iterate, an application of Lemma 9.24 shows that the new Galerkin set Λj+1 ⊃ Λj should fulﬁll (9.166) with 1/2 μ = κ(A)(1 − ρ2 ) . Note that due to the constraint μ ≤ 1 in Lemma 9.24, the attainable error reduction per iteration is bounded from below by 1 1/2 . ρ≥ 1− κ(A) However, it is still a nontrivial task to detect the biggest entries in the discrete residual since rΛj will almost always be an 2 sequence with inﬁnite support. Moreover, in order to end up with an optimal computational complexity, the new index set Λj+1 should ideally be the minimal superset of Λj with the prescribed saturation property. Both tasks can in fact be accomplished in practice, by the development of a suitable numerical subroutine GROW for the extension process. In [236], approximations of the discrete residual rΛj were constructed by the aforementioned adaptive matrix-vector multiplication and by inexact evaluations of the right-hand side f , ending up with a convergent Galerkin scheme. In [197], the problem of computational optimality was addressed. By interleaving the extension steps with certain thresholding operations, it was shown that the resulting sequence of Galerkin approximations shows an optimal convergence rate. Moreover, the corresponding algorithm was numerically tested and compared with ﬁnite element schemes in [70]. The experiments revealed that in many cases and for judicious parameter choice, the intermediate thresholding steps could

680

9. Variational methods for wavelets

even be avoided. This observation was analyzed and justiﬁed for a speciﬁc variant of GROW in [333]; we cite the corresponding main result. Theorem 9.25. Bounds for operations and storage by GROW: There exists an implementable algorithm GROW[w, ν¯, ] → (Λ, ν) with the following properties: 1. For ﬁnitely supported w ∈ 2 (J), ν¯ ≥ r(w)2 (J) and > 0, GROW terminates with ν ≥ r(w)2 (J) and ν min{ν, }. If u ∈ As for some s < s∗ , then the number of ﬂoating point and storage operations for the call is bounded by a constant multiple of min{ν, ν¯}−1/s (wAs + uAs + ν¯1/2 (#suppw + 1)). 2. If GROW terminates with ν > , then α−ω PΛ r(w) 2 ν ≥ (J) 1+ω and #(Λ \ suppw) ν −1/s uAs . 1/s

Here 0 < ω < α are internal parameters of GROW with (α + ω)/(1 − ω) < κ(A)−1/2 . Therefore, unless the GROW subroutine outputs a new Galerkin set with residual . error below the target accuracy , the new iterate uΛj+1 fulﬁlls (9.166) for μ = α−ω 1+ω In consequence, the error reduction rate of the global iteration is at least ρ = 1 − 1/2 . (α − ω)2 /((1 + ω)2 κ(A)) 9.7.4

Adaptive descent iterations

A conceptually diﬀerent approach to the discretization of stationary elliptic equations has been proposed in [198]. Instead of cutting out ﬁnite portions of the equivalent 2 problem (9.160) and then solving them with standard algorithms from numerical linear algebra, the strategy advocated in [198] proceeds the other way around. The idea is to directly apply well-known iterative schemes in the inﬁnite dimensional setting, preferably with ﬁxed error reduction per iteration step. Only after that, are all inﬁnite dimensional quantities replaced by ﬁnitely supported and computable ones. In particular, one has to work with an inexact right-hand side ¯f ≈ f , approximate matrixvector operations, and appropriately matched tolerances for subroutines like APPLY or COARSE. Among the relevant iterative schemes are, e.g. Richardson iteration u(n+1) = u(n) + αr(n) ,

r(n) = f − Au(n) ,

n = 0, 1, . . .

(9.167)

with ﬁxed relaxation parameter α > 0, and the closely related steepest descent method u(n+1) = u(n) + αn r(n) ,

αn =

r(n) , r(n) , Ar(n) , r(n)

n = 0, 1, . . .

(9.168)

with a posteriori choice of the descent parameter αn . Both iterative schemes can be shown to converge also in an 2 setting. For instance, convergence of the exact

9.7. Adaptive wavelet methods, by T. Raasch

681

Richardson iteration (9.167) follows from the contractivity ρ = ρ(α) = I − αA < 1, which can be achieved whenever 0 < α < 2/A. In that case, we have linear error reduction per iteration, u − u(n+1) 2 (J) ≤ ρu − u(n) 2 (J) , and the exact solution u has the series representation u = A−1 f = α

∞

(I − αA)n f .

n=0

If A is symmetric, then the optimal relaxation parameter α ˆ = arg minα ρ(α) can be computed as α ˆ=

2 , A + A−1 −1

with ρ(ˆ α) =

κ(A) − 1 . κ(A) + 1

In practical applications, however, we will almost never be able to determine α ˆ exactly. Implementations of the Richardson iteration (9.167) will therefore often suﬀer from suboptimal performance. In contrast, the steepest descent iteration (9.168) automatically adapts the descent parameter to the current iterate, and it usually outperforms the Richardson iteration exactly for this reason. We refer to [240, 252] for illustrative numerical experiments. The convergence and complexity analysis of the resulting inexact iterations follows general principles. Usually, it is exploited that the exact iteration u(n+1) = G(u(n) ),

n = 0, 1, . . .

(9.169)

has at least linear error reduction u − u(n+1) ≤ ρu − u(n) ,

n = 0, 1, . . .

(9.170)

for some 0 < ρ < 1, and · is either the 2 norm or the discrete energy norm. Many important schemes have this property: we mention Richardson iteration (9.167), the steepest descent method (9.168) and other Krylov methods, like Chebyshev iteration [331] or the conjugate gradient method [252]. In practice, the exact iteration mapping G is replaced by an inexact algorithm ITER[w, ] → v whose output v fulﬁlls G(w) − v ≤ , for all > 0. Linear convergence of the perturbed but implementable iteration v(n+1) = ITER[v(n) , n ],

n = 0, 1, . . .

with a slightly larger error reduction constant can easily be shown.

(9.171)

682

9. Variational methods for wavelets

Lemma 9.26. Assume that the exact iteration (9.169) has linear error reduction ρ − ρ)¯ ρn , the iterates (9.170) with ρ < ρ¯ < 1. Then for all tolerances n ≤ u − v(0) (¯ from (9.171) fulﬁll the error estimate ρn , u − v(n) ≤ u − v(0) ¯

n = 0, 1, . . .

Proof. The claim immediately follows from the induction u − v(n+1) ≤ u − G(v(n) ) + G(v(n) ) − ITER[v(n) , n ] ≤ ρu − v(n) + n ≤ ρu − v(0) ¯ ρn + (¯ ρ − ρ)u − v(0) ¯ ρn = u − v(0) ¯ ρn+1 .

However, with the simple iteration (9.171) it is in general impossible to verify either convergence rates or linear computational complexity of the overall algorithm. In order to end up with an optimal convergence rate, one may consider intermediate thresholding operations with the COARSE subroutine. Note that Theorems 9.20 and 9.21 have analogs when the 2 norm is replaced by the discrete energy norm. Due to the fact that coarsening of a given iterate slightly enlarges the approximation error, the tolerances have to be adjusted carefully to preserve convergence. We therefore consider the following generic SOLVE Algorithm 9.3, see also [198, 331]. Algorithm 9.3. SOLVE[v(0) , ] → v

i let θ < 1/3 be ﬁxed and K ∈ N such that ρK + (¯ ρ − ρ) K−1 ¯ i=0 ρ ≤ θ ρ n := 0 w(0,0) := v(0)

0 := A−1 f − Av(0) while n > do for i = 1 to K do % & ρ − ρ) n w(n,i) := ITER w(n,i−1) , (¯ end do % & ρ n v(n+1) := COARSE w(n,K) , (1 − θ)¯

n+1 := ρ¯ n n := n + 1 end do v := v(n)

Theorem 9.27. Estimates for errors and support in SOLVE: Assume that the exact iteration (9.169) fulﬁlls (9.170) with ρ < ρ¯ < 1. Then SOLVE[v(0) , ] → v terminates 1/s with u − v ≤ . If, moreover, u ∈ As for some s > 0, then #suppv −1/s uAs . Proof. Concerning convergence, it is suﬃcient to verify that u − v(n) ≤ n = 0 ρ¯n = A−1 f − Av(0) ¯ ρn .

(9.172)

9.7. Adaptive wavelet methods, by T. Raasch

683

For n = 0, this is trivial. Otherwise, after the K iterations in the while loop, then with an analogous argument as in Lemma 9.26, ρ − ρ)n u − w(n,K) ≤ ρu − w(n,K−1) + (¯ K−1 ≤ ρK u − v(n) + (¯ ρ − ρ)n i=0 ρi K−1 ≤ ρK + (¯ ρ − ρ) i=0 ρi n ≤ θρ¯n . By the properties of COARSE, see Theorem 9.20, we can deduce (9.172), ρn = n+1 . u − v(n+1) ≤ θρ¯n + (1 − θ)¯ As an immediate consequence of (9.172) and Theorem 9.21, we ﬁnally estimate ρn )−1/s uAs −1/s uAs . #suppv (¯ 1/s

1/s

In order to obtain an algorithm with linear computational complexity, further assumptions on the subroutine ITER are necessary. The following theorem is adapted from [331]. Theorem 9.28. Estimates for operations, storage and support in SOLVE: In the situation of Theorem 9.27, assume that w = ITER[v, ] fulﬁlls #suppw #suppv + −1/s uAs , 1/s

wAs (#suppv)1/s + uAs ,

where the number of ﬂoating point and storage operations needed to compute w 1/s is bounded by a constant multiple of −1/s uAs + #suppv + 1. Then the output 1/s v = SOLVE[v(0) , ] can be computed with at most an absolute multiple of −1/s uAs ﬂoating point and storage operations, as tends to zero. Remark 9.29. The generic algorithm SOLVE requires knowledge of the error reduction constant ρ < 1 or at least an approximation thereof. For a particular iterative scheme like Richardson iteration or the steepest descent method, such estimates are available whenever the spectral bounds A and A−1 can be approximated. 9.7.5

Nonlinear stationary problems

The paradigm which has been outlined in the previous subsection, namely the treatment of the original operator equation in the sequence space regime by inﬁnitedimensional iterations and by a subsequent discretization of the single iteration step, may be readily applied to problems with nonlinearities also. Starting out from [199], the adaptive discretization of certain nonlinear operator equations has been discussed recently in the literature. Assume that we are looking for a solution u ∈ U to F (u) = 0,

(9.173)

684

9. Variational methods for wavelets

where F : U → U has a continuous Fr´echet derivative F : U → U which is invertible in a neighborhood of u, F (v)wU wU ,

for all v ≈ u, w ∈ U.

Using a wavelet Riesz basis Ψ = {ψλ }λ∈J for U, (9.173) has an equivalent reformulation as the inﬁnite nonlinear system F(u) = 0, where

F(v) = F (v), ψλ U ×U λ∈J ,

v=

vλ ψ λ .

λ∈J

Such systems of equations can be handled by iterative schemes of the form u(n+1) = u(n) − Rn F(u(n) ),

n = 0, 1, . . .

(9.174)

Here the Rn : U → U are suitable bounded linear operators. As typical examples, we mention Richardson iteration with stationary choice Rn = αI, or the Newton method with nonstationary Rn = F (u(n) )−1 . Note that the discrete mapping F has the Fr´echet derivative F (v) = F (v)ψλ , ψμ U ×U μ,λ∈J . The convergence properties of the exact method (9.174) have been analyzed in [199], for several classes of nonlinearities and iterative strategies. In case we are dealing with semilinear operator equations Au + g(u) = f,

(9.175)

where A : U → U is boundedly invertible and g : U → U is monotone, v − w, g(v) − g(w)U ×U ≥ 0,

for all v, w ∈ U,

global convergence of the exact discrete Richardson iteration u(n+1) = u(n) − α Au(n) + g(u(n) ) − f , n = 0, 1, . . . can be veriﬁed for suﬃciently small values of α > 0. For general nonlinear problems like (9.173), the simpliﬁed Newton iteration Rn = F (u(0) )−1 has been discussed in [199]. As a ﬁrst-order scheme, it at least shows linear error reduction (9.170) with rate ρ < 1 in the vicinity of the exact solution u. When it comes to concrete numerical implementations, the question arises of how to eﬃciently realize the application of the nonlinearity in (9.174). As already mentioned in Subsection 9.3.5, tree approximation techniques are an appropriate tool for this kind of problem. By imposing a tree structure on their output arguments, numerical routines for thresholding and for the adaptive application of nonlinear mappings to ﬁnitely supported sequences have been developed in [199], based on the ﬁndings from [251]. We cite the corresponding main results in the following theorems. Thresholding of ﬁnitely supported vectors can be performed with a subroutine TCOARSE, i.e. tree coarsening.

9.7. Adaptive wavelet methods, by T. Raasch

685

Theorem 9.30. Quasi-optimal tree in TCOARSE: There exists an implementable algorithm TCOARSE[v, ] → v with the following properties: 1. For ﬁnitely supported v ∈ 2 (J) and > 0, TCOARSE computes a ﬁnitely supported v ∈ 2 (J), such that v − v 2 (J) ≤ and suppv is a quasioptimal tree, i.e. suppv is a tree with #suppv ≤ C inf #T : T ⊂ J is a tree and v − IT PT v2 (J) ≤ . 2. The number of ﬂoating point and storage operations to compute the output v stays proportional to inf #T : T is a tree with suppv ⊂ T ⊂ J . Due to its similarity with the COARSE subroutine from Subsection 9.7.1, results like Theorems 9.20 and 9.21 carry over to the case of tree coarsening with TCOARSE. The inexact evaluation of the nonlinearity is done with a subroutine EVAL. As outlined in Subsection 9.3.5, the algorithmic ﬂow of EVAL decomposes into two steps. The ﬁrst step is the recovery step, where an estimate for the support of the output is computed. The concrete entries indexed by the recovery set are computed in a second step. The basic properties of EVAL read as follows, see [199]. Theorem 9.31. Approximation of nonlinear F(v) by w : There exists an implementable algorithm EVAL[v, ] → w with the following properties: 1. For ﬁnitely supported v ∈ 2 (J) and > 0, EVAL computes a ﬁnitely supported w ∈ 2 (J), such that F(v) − w 2 (J) ≤ and suppw is a tree with #suppw ≤ C inf #suppw : suppw is a tree and F(v) − w2 (J) ≤ . 2. The number of ﬂoating point and storage operations to compute the output w stays proportional to #suppw + #supp v + 1 By combining the subroutines TCOARSE, EVAL and APPLY, an implementale adaptive iteration has been designed in [199]. At least in the case of semilinear problems (9.175), the overall algorithm converges at an optimal rate and with linear computational complexity. In [69, 72], the recovery algorithm is explained in more detail, with an explicit analysis of all computational ingredients and with extensive numerical experiments for nonlinearities depending on function values only. As an interesting variant, we mention that the recovery algorithm can also be applied to adaptive quadrature schemes for the stiﬀness matrix A of linear problems; see [71] for details. Moreover, in [435], it has been shown that also in the case of redundant ansatz systems, tree approximation techniques can be used to obtain adaptive recovery algorithms with optimal computational complexity.

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Index (−1)j>0 := 1 for j = 0 else := −1, 40 α = (α1 , . . . , αn ) ∈ Nn 0 , 54 ∇0 u = u, 55 ∇u = ∇1 u, 54 ϑ = (ϑ1 , . . . , ϑn ) ∈ Rn , 54 (Ai , Bi ), 68 (·, ·)n , 37 (·, ·)n Euclidean scalar product, 456 (u, v)H 1 (Ω) , 36 (u, v)H 2 (Ω) , 36 (u, v)H m (Ω,Rq ) , 119 u C k (Ω) , 58 u C k,γ (Ω) , 58 u H i (Ω) , 36 u W m,p (Ω,Rq ) , 119 v H i (Ω) , i = 0, 1, . . ., 36 v W m,p (Ω,Rq ) , 119 A, 37 A∗ , 47 Ad , 47 Ac , 37 Ap , 40 As , 37, 40 B, 41 Ba , 41 C 2m+s,γ (Ω), 58 D≤k , 55 Dk , 55 h, 293 H 1 (Ω), 36 H 2 (Ω), 36 H m (Ω, Rq ), 119 Hk−1 (Tch ), 351 H −m (Ω) =??H0−m (Ω)???, 39 P h , Q h , 591 W 2m+s,p (Ω), 58 W m,p (Ω, Rq ), 119 W s,p (∂Ω) Sobolev space, 25 % Euclidean product, 43 C: complex numbers, 36 Δu, 36, 119 N: natural numbers {1, 2, . . .}, 36 N0 : {0} ∪ N = {0, 1, 2, . . .}, 36 Ω, 36, 119 Ω0 ∈ C 0,1 , 75 Ω0 ⊂⊂ Ω1 for open Ω0 , Ω1 means Ω¯0 ⊂ Ω1 , 75 Q+ : positive rational numbers, 36 Q: rational numbers, 36

R+,0 = R+ ∪ {0}, 140 R: real numbers, 36 RNk , 119 Rnk , 119 R+ : positive real numbers, 36 Z: integer numbers, 36 α, 55 ¯ 19 Ω, L(U , V ), 46 N (A), 46 N L Ub , Vb , 184 O(xα ), 5 O(hp ), 229 R(A), 46 UT h = W k,p (T h ), 231 V = H0m (Ω, Rq )−coercivity and ellipticity, 121 Q set of all vertices T , 235 δij = 1fori = j, else= 0, 213 , 233 v with few nonzero entries, 675 div w, 114 index(A), 60 ∇k , Θk ∈ Rnk , 54 ∇≤k , Θ≤k ∈ RNk , 55 ∂u, 54 ∂Ω, 36, 119 curved, 395 polygonal, 395 ∂ α u, 54 ∂ i ∂ j u, 36, 119 ∂ i u, 36, 119 ϑ = (ϑ1 , . . . , ϑn )T ∈ Rn , 40 ϑα , 54 ϑi , 54 | · |n , 37 |ϑ|n = |ϑ|, 56 |α|, 54 | · |n Euclidean norm, 456 a< ∼b, 638 a ∼ b, 638 a(·, ·), 37 a := b: a is deﬁned by b, 36 a =: b: b is deﬁned by a, 36 a ∼ b, 423, 638 ad (u, v), 47 ap (·, ·), 40 as (·, ·), 37, 40 ei , i-th unit vector, 30 f ∈ W0−m,p (Ω), 63 −1 f ∈ Hk (Ω), 351

f ∈ W0−m,p (Ω), 63

734

Index

f ∈ W0−1,p (Ω, Rq ), 119 f, v V ×V , 119 o(x), 5 o(hp ), 229 v h ∈ C(Ω) ⇒ v h ∈ H 1 (Ω), 372 vl , vr , [v], {v}, 462 X → Y, 21 h.o.t., 5 T h non–degenerate, 229 a posteriori error estimator, 669 a priori error bound, 424 a.e. =almost everywhere, 20, 40 adaptive approximation method, 669 convergence, 670 convergence rate, 670 adaptive Finite Element Method, 420 adaptive wavelet-technique, 636 advantage wavelet methods, 636 algorithm APPLY, 677 COARSE, 674 SOLVE, 682 almost everywhere (a.e.), 20, 40 analysis natural boundary conditions, 611, 612 anti-crime transformation, 190, 527 approximating spaces admissible Galerkin, 272 bi-dual pair, 181 approximation cubature, 346 edge singularities,boundary layers, 234 external, 180 ﬁve star, 566 good by COARSE, few nonzero entries, 675 internal, 180 nonlinear F(v) by w , 685 quadrature, 305, 346 saturated data approximation, 439 approximation projector, P h , Q h , 184 approximation scheme external, 592 asymptotic expansion extrapolation, defect correction, 626 asymptotic optimality, 674 atlas, 23, 25 Aubin–Nitsche result, 664 Banach space approximate, 334 reﬂexive, V = V , 15, 52 basis biorthogonal, 644 interpolation, 223, 367 nodal, 213 stable and local, 241 stable local, 249

basis functions interior, 650 Besov space, 646, 673 best N -term approximation, 672 bifurcation multiple, 11 bilinear form, 157 Vbh -coercive, 601 Vbh -elliptic, 601 approximate, 346 bounded, 45 coercive, 45, 50, 70, 121, 321, 637 coercivity implies invertibility, 49 coinciding strong and weak, 338 continuous, 45 diﬀerence weak, 576 dual, 45 elliptic, 50, 70, 121, 637 induced linear operator, 45 strong, 39, 66, 159, 160, 320, 337 strong and weak form, 36 symmetric, 46 weak, 40, 66, 160, 320, 321, 337, 576 biorthogonality of wavelets + full polynomial exactness imply vanishing moments, 651 blue reﬁnement, 436 bound upper a priori, 430 lower a priori, 431 a preori, 423 boundary curved, 250 isoparametric, 253 polynomial interpolation, 252 parametrization, 24 boundary derivative complementary tangential, 75 boundary condition, 5, 8, 588 (2.177) - (2.179), 83 Dirichlet, 42, 56, 70, 101, 132, 145, 388, 397, 403, 414, 531, 546, 564, 594 Dirichlet and/or Neumann, 531 excluded tangential derivative, 42 general, 56, 70, 75 homogeneous Dirichlet, 258 included, 637 incorporated, 47 modiﬁed, 75 natural, 8, 41, 42, 71, 74, 388, 403, 614, 617 natural, Neumann, 610 Neumann, 8, 56, 70, 612 results for 2m = 2 and Bi in (2.185), (2.186), 83 transformation into homogeneous, 57 vanishing Dirichlet, 569 boundary layers, 423 boundary operator Dirichlet BD , 56 natural, 41

Index natural discrete, 614 principal part, 75 boundary value problem, 5, 8, 459 Dirichlet, 42, 56, 66, 86 existence, uniqueness, regularity, 67 ﬁrst, 66 generalized Neumann, 66 generalized Robin, 66 linear, 472 natural, 44, 66 normal, 56, 59, 61 second, 67 semilinear, 474 third, 66 bounded operations, storage, nonzero entries by APPLY, 677 bounded bilinear form and A, 670 boundedness, 498 for the Laplacian, 495 SIPG, NIPG, IIPG, GIPG, 501 bounds for operations, storage by GROW, 680 bubble edge, 432 element, 432 carr(σ), 25 carr u = supp u, 591 case μ, 400 σ, 400 Cea’s Lemma, 424 center of gravity, 450 Chladny sound ﬁgures, 10, 290 chunkiness parameter, 230, 428 co-ordinate system, 25 coercivity, 49, 94, 99, 387 H01 (Ω), 41 H m (Ω), 71 H0m (Ω), 71 H0m (Ω) ⇒ W0m,p (Ω), p < 2, 155, 191 H0m (Ω) ⇒ W0m,p (Ω), 2 < p, 155, 191 W0m,p (Ω), 155, 191 V, 121 V h quasi-linear, 502 V h for the Laplacian, 495 Vbh −, 603 discrete, 416 discrete, relations, 387 in V h general linear, semi-linear case, 499 isoparametric, 416 modiﬁed, 105 NIPG, SIPG, IIPGMs, 497, 500 nonlinear, 103, 143 coercivity implies invertibility, 49 collar Dirichlet BVP, 137 estimate, 137 norm, 137 compact, 14 pre-, 14

735

relatively, 14 complement generalized orthogonal, 16 complexity of MAKECONFORM, 443 complexity of SOLVE, 446 compressibility, 675 computational complexity, 671 condition uniform Legendre, 160 boundary and quadrature, 396 boundedness, 94 Brezzi-Babuska, 47 Carath´eodory, 98, 103, 131, 141, 142, 144 Carath´eodory type, 128 Carath´eodory, growth, 145 complementary, 59, 82 controllable growth, 128, 478 ellipticity, 94 exterior sphere, 110 for linearization, 151 growth, 98, 103, 110, 141–144 controllable, 129 ﬁrst, 94 fourth, 94 natural, 129 second, 94 third, 94 H, 304,307,311,317,341 Hm, 335,340 inf-sup-, 48 Ladyshenskaja–Babuska–Brezzi, 166, 283, 477 Ladyshenskaja–Babuska–Brezzi (LBB), 665 Legendre, 120 Legendre–Hadamard, 121, 128 modiﬁed strong Legendre, 136, 339 modiﬁed Brezzi–Babuska, 166 modiﬁed ellipticity, 157 monotone strict, 103, 143 monotone strict uniform, 103 monotone uniform, 103, 143 natural boundary, 66 natural growth, 128, 478 stability, consistency for DCGMs, 529 strong Legendre, 128, 131, 144 strong Legendre–Hadamard, 122, 133, 266 uniform Legendre–Hadamard, 161 violated boundary, 378 violated continuity, 378 condition numbers uniformly bounded, 636 conforming ﬁnite elements, 428 conjecture, smooth splitted splines, 250 connected, 628 conservative ﬂux, 466 consistency, 174 Jhσ , 491 h , 505 h , 506

736

Index

consistency (cont.) ah for general linear, 511 ah for the Laplacian, 507 ch and bh , 503 classical, 185, 187, 590 diﬀerence methods natural boundary condition, 620 diﬀerentiable quasilinear DCGMs, 536 diﬀerentiable, semilinear DCGMs, 537 for violated boundary conditions, 392 for violated Dirichlet conditions, 393 fully nonlinear, 599 general linear ah ,, 513 linear operator, 596 linearized quasilinear, 532 of order p in u, 187 quadrature for quasilinear problems, 363 quasilinear ah in [409], 523 quasilinear equations, systems, 598 quasilinear system ah , 518 quasilinear systems, 526 semilinear ah , 513 variational, 185, 188 consistency error variational, 392 consistent ﬂux-, 466 consistent diﬀerentiability, 416, 486, 610 constant depending upon parameters, 212 Poincar´e’s, 21, 430 shape regularity, 239 continuation embedding method, 206 continuity H¨ older, 20 Lipschitz, 20 modiﬁed Lipschitz, 514 uniform, 387 controllable and natural growth conditions, 128 convergence, 190, 268 adaptive FEM, 440 order p, 190, 268 radial basis function, 595 symmetric, unsymmetric ﬁnite diﬀerence method, 609 weak, 15 convergence theory general for monotone operators, 277, 279 general for quasilinear operators, 277, 279 Navier–Stokes wavelet method, 669 saddle point wavelet method, 666 solution for monotone operators, 279 solution for quasilinear operators, 279 corner point, 421 corner singularity, 421 corner stone of modern analysis, 37

covering open, 14 cubature formula bivariate, 390 cuboid, 214 data approximation error, 439 DCGM hp, 539, 540 hp−, convergence, 546 hp−variants, 538 coercive linearized principal part, 494 consistency, 503, 507, 511, 531 convergence, 527, 531 convergence of hp, 542 convergence via monotony, 531 estimate, inverse and interpolation error, 487 general form, 468, 473 generalization to systems, 482 geometry of the mesh, 486 IIPG form, 480 incomplete form, 468, 473 inverse estimate, 486 linearized consistency, 534 linearized IIPG form, 533 NIPG or SIPG or GIPG, 480 non symmetric form, 468, 473 nonstandard, 471, 530 numerical experiences, 546 standard, 471, 477, 529 symmetric form, 468, 473 derivative, 32 directional, 28 Frech´et, 28 Gateaux, 28 inverse function, 31 normal, 10 partial, 36, 54, 230, 264, 638 relation:Frech´et/Gateaux-, 29 weak, 20 diagram, 267 diﬀeomorphism C 1 , 31 local C 1 , 31 diﬀerence backward, 562, 564 forward, 562, 564 nonsymmetric, 562 normal, 564, 565 operator, 563, 568 partial, 566 symmetric, 563, 565 diﬀerence approximation symmetric, 613 diﬀerence equation, 574 classical form, 575 coercivity for elliptic, 600 consistency, 593 divergence form, 574, 575

Index elliptic 2. order, 574 natural form, 575 partial summation, 575, 578 symmetric, 581, 615 symmetric form, 560, 573 unsymmetric form, 561 diﬀerence method asymptotic expansion, 631 convergence, 588, 609 symmetric, unsymmetric method, 609 strong (classical) form, 609 curved boundaries, 622 elliptic nonlinear order 2m, 584 elliptic linear order 2 Dirichlet, 583 fully nonlinear order 2, 587 natural boundary conditions, 611, 613, 621 nonlinear order 2m, 586 polynomial interpolation, 626 Shortley–Weller–Collatz, 623 simple examples, 562 solving nonlinear equation, 610 strong form, 609 symmetric, 581 unsymmetric/symmetric, 567 variational, 589 diﬀerentiability partial, 30 diﬀerentiable Frech´et, 28 Gateaux, 28 consistent, 207, 344, 532 partially, 30 diﬀerential operator linear linear of order 2m, 56, 62 linear system order 2m, 133 order 2m, 70 dilemma the G, Gh , 196 Dirac measure, 449 Dirichlet system, 75 discontinuous Galerkin methods (DCGMs), 455 Discrete Brezzi–Babuska condition:, 272 discrete solution quasilinear, unique, convergence, 275 unique existence,convergence, 269 discrete space Lebesgue, 569 discretization DCGM, 460, 469 quasilinear system, 478 discretization and preconditioning, 653 discretization errors diﬀerence, 189 discretization method linear, 269, 661 wavelets, 661

737

discretization schemes inner and outer, 185 distance A, B, 254 domain curved, 461 Lipschitz, 23 partition of, 461 polygonal, 251 polyhedral, 459, 461 reference, 213 star-shaped, 229, 487 dual pair, 645 dual problem, 452 dualgenerator, 645 duality argument, 449 edge jump, 430 edge residual, 430 eigen function, 290 space, 290 value problem linear operator, 289 eigenvalue, 11, 51, 60, 64, 73, 81, 83, 89, 169, 290 variational method, xxiii eigenvalues, 109 element C r , 223 reference, 213 element residual, 430 elliptic, 94, 96 H m (Ω), 71 H0m (Ω), 71 V, 121 Vbh , 601 linear system order 2, 117 strongly, 56 strongly 2m−, 136 uniformly, 56, 89 elliptic equation quasilinear, nonlinear coercive, 106 elliptic diﬀerence equation stability,convergence, 604 elliptic diﬀerential operator order 2, 35 elliptic equation existence quasilinear, 91 fully nonlinear order 2, 108 linear, order 2m, 639 linear, order 2, 638 nonlinear, 77 nonuniformly, 91 quasilinear divergent, 88, 100 non divergent, 88 order 2, 88 special quasilinear, 149 special semilinear, 149 uniformly, 91 weak, order 2m, 69

738

Index

elliptic operator characteristic polynomial, 56 deﬁnition nonlinear, 79 fully nonlinear, 81 linear Fredholm alternative, 54 order 2, 64 order 2m, 58 regular solution, 54 nonlinear, linearization, 147 order 2m, m, 80 quasilinear, 80 semilinear, 78, 80 symbol, 56 elliptic problem general linear DCGM, 472 general, Dirichlet condition, 572 semilinear and quasilinear, 474 elliptic system, 114 characteristic polynomial, 115 divergent quasilinear order 2m, 137 fully nonlinear order 2m, 146 linear, nonlinear, 112 local solvability, nonlinear, 147 nonlinear order 2m, 586 order, 115 quasilinear, 477 quasilinear of order 2m, 639 quasilinear, variational method, 125 symbol, 115 ellipticity Vbh −, 602 fully nonlinear, 112 embedding, 21 compact, 21, 50 continuous, 21, 50 dense, 50 equation Bratu, 78, 87 diﬀerential, 5 elliptic divergent quasilinear, 273 fully nonlinear form, 306 general quasilinear, 306 fully nonlinear, xxiii, 151, 156 K´ arm´ an, 137 Lam´ e, 117 linear order 2m, 83 minimal surface, 12, 103 Monge-Amp`ere, 13, 109, 307 regular solution, 111 monotone convergence, xx Navier–Stokes, 13, 163, 211 Navier–Stokes, steady compressible, 554 non coercive, 148 non divergent, xx non divergent quasilinear, 156 nonuniformly elliptic, 35, 57, 90 order 2m, 336 quasilinear divergent, 151

reaction-diﬀusion, 12 saddle point, 664 Stokes, 114 surface with Gauss curvature, 13, 110, 307 uniformly elliptic, 35, 89 von K´ arm´ an, 116 von K´ arm´ an, numerical experiments, 632 equilibrium, 4 error (classical) consistency, 187, 268, 270, 590 (classical) consistency, nonconf. FEMs, 270 (global) discretization, 190 (variational) consistency, 268 boundary, 468 classical consistency, 179, 189, 296, 373, 389, 401, 485, 662 classical consistency=local discretization, 275 classical discretization, 412 conforming variational, 188, 189 consistency, 187, 207, 417 FE interpolation global estimates, 232 local estimates, 232 tensor FEs, 232 interpolation, 189, 229, 241, 424 local discretization, 187, 268, 270, 661, 662 local discretization, nonconf. FEMs, 270 nonconforming variational, 189 nonconforming, variational,classical consistency, 270 quantities of interest, 449 variational consistency, 179, 186, 188, 269, 296, 389, 393, 401, 661, 662 variational discretization, 412 variational,classical consistency,conforming FEM, 269 error bound a priori, 423 a preori, 423 error estimate a posteriori, 433 other types, 447 quantities of interest, 449 error representation formula, 431 errors wavelet consistency, vanish, 662 estimate hpinverse and error estimate, 540 computable a posteriori, 438 consistency, 414 errors and support in SOLVE, 682 interior, 65, 136 inverse, 229, 232, 237, 240, 257 inverse, nonquasiuniform triangulation, 233 lower a posteriori error, 432 operations, storage, support in SOLVE, 683 point error, 449 quantities of interest, 449 upper a posteriori error, 430 estimator

Index other type, 447 eststimation quantities of interest, 449 Euclidean product in Rn , 43 Euclidean scalar product and norm in Rn , 37 Euler system strong form, 127 weak form, 127 examples nonlinear PDEs, 10 excluded tangential derivative, 43, 75 existence, 145 interior, 65 expansion, 640 extension bilinear form, 373 energy, 37 Friedrichs, 36 from Ω to Ωh 0 , 311 extremal problem, 49 FE, 213 n (T h ), 223 Pd−1 aﬃne equivalent, 221 approximating spaces, 222 approximation theory, 212 Argyris, 218 Bell, 218 bilinear, biquadratic, 219 condition interpolation points on edges, 400 conforming, 218, 258 cubic Hermite, 227 discontinuous Crouzeix–Raviart, 177 equivalent, 221 aﬃne, 221 example, 221, 258 Hermite, 215 interpolation, 221 interpolation error, 256 isoparametric, 251, 254 isoparametric polynomial, 253 Langrange, 215 linear Lagrange, 226 Mortar, 234 non–degenerate, 229 nonconforming, 189, 218, 368 nonconforming, conditions, 390 of degree, 223 order 2m, 271 polynomial, 214 at least degree, 223 degree or order, 223 quadrilateral, hexahedral, 235 rectangular K, 214 simple computation, 219 smooth curved domain, 238 polyhedral domain, 238

739

space, 221 spaces, updated, 237 standard case, 390 curved boundary, 396 discontinuous FEs, 400 subdivision, 213 tensor product, 232 triangular, 215 triangular K, 214 uniformly star-shaped, 229 violated boundary condition, 371, 392, 397 violated continuity, 399 violating continuity condition, 372 violating continuity, consistency, 403 FEM hp, 448 adaptive, 434 bubble function, 284 conforming, 209, 210, 257, 274, 275, 282 conforming generalized Petrov–Galerkin, 199 conforming,basic idea, 258 consistency, 336 convergence, 336 u0 − uh 0 L2 (Ω) , 272 linear, quasilinear problem, 271 convergence of adaptive, 439 convergence, adaptive, 438 convergence, quadrature with crimes, 411 crimes linear problems, 380 quasilinear problems, 380 crimes, quasilinear, quadrature, 413 discrete Newton method, MIP, 276 equation order 2m, 332 general convergence theory, 266 general linear equations, 264 general linear systems, 264 impacts of variational crimes, 373 isoparametric, 414, 417 quasilinear, 418 linear, 276 linear + quadratic, 284 linear elliptic system order 2, 266 order 2m, 266 linear elliptic equation order 2, 264 order 2m, 265 linear, sparse equation, 342 main idea, 258 mixed, 211, 282 Mixed for Navier–Stokes operator, 286 mixed, nonlinear Navier–Stokes,convergence, 288 nonconforming, 210, 283, 297, 371 nonlinear boundary conditions, 345 equation, 345 system, 345

740

Index

FEM (cont.) notation, requirement, 428 optimal mesh, 451 quadrature second order linear, 350 second order nonlinear, 357 systems order 2m, 361 quadrature for nonconforming, 412 reducing costs, 262 second order, violating boundary condition and/or continuity, 411 semi-conforming, 308 semi-conforming, fully nonlinear, 309 solution of nonlinear equations, 413 sparse linear equations, 342 sparse nonlinear system, 342 stability, 336 stability nonconforming, 406 stable linear, quasilinear problem, 271 stable reﬁnement, triangulations in R2 , 436 summary isoparametric second order problems, 415 system fully nonlinear, 297 general quasilinear, 297 order 2m, 332 variational crimes, 297, 368 consistency, 368 convergence, 368 example, 370 stability, 368 violating boundary conditions, 371 Fichera corner, 423 ﬁnite element, xv ﬂux inviscid (Euler), 556 numerical, 465 viscous, 556 ﬂux term, 459 form bilinear, 41, 45 bounded bilinear, 41 bounded linear, bilinear, 14 continuous, 63 divergence, 63, 66, 86 linear special representation, 63 strong, 36 strong quasilinear, 80 strong quasilinear diﬀerence, 625 weak, 37, 76, 86, 101, 132, 583 weak diﬀerence, 584, 585 formula fundamental Green’s, 37, 101 generalized Green’s, 41 Green’s, 37, 77, 459 Frech´et derivative, 27 Fredholm alternative, 51, 52, 59, 66, 67, 73, 134, 151, 155, 157, 290

for order 2m, 137 divergence form, 64 Fredholm-index, 18 fully nonlinear problem existence, uniqueness, 110 order 2 and 2m, 108 function cut-oﬀ, 230 Dirac delta, 213 mesh-size, 425 shape, 213 step, 591 functional regular integral, 126 Galerkin approximation, 424 Galerkin method, 659 adaptive, 671 Gelfand triple, 50, 637 general approach, 451 general discretization method DCGM, 482 geometric problem quasilinear, fully nonlinear, 12 geometrically stable, 436 geometry of the mesh, 487 GIPG, 468 global Schwartz kernel, 654 Gram determinant, 235 green reﬁnement, 436 Green’s function approximate, 449 G˚ arding inequality, 658 hanging node, 435 Hilbert space setting, 458 hyper plane non–degenerate, 215 IIPG, 457, 468, 529 inclusion, 21 index, 52 index A, 82 inequality Cauchy–Schwarz, 19 discrete Chebyshev, 18 discrete H¨ older, 19 G˚ arding, 50, 70, 411, 608 H¨ older, 19, 231, 349, 392 modiﬁed Poincar´e–Friedrich, 397 multivariate Markov, 240 Sobolev, 571 inf–sup condition (BB), 664 integral discrete, 569, 601 integral mean value, 571, 591 interior node property, 438 interior regularity, 124 interpolant

Index FE, simple inductive computation, 219 interpolation error, 232 interpolation operator, 424, 429 invariance property, 563 inverse estimate, 432 invertibility, equibounded, xvi isomorphism, 16 iteration adaptive descent, 680 iterative method, 671 Jacobian, 235 jumping coeﬃcients, 423 Ladyshenskaja–Babuska–Brezzi (LBB) condition, 666 Lagrange function, 451 Laplacian, 10 LDC, 477 Lebesgue-/Sobolev space discrete, 570 lemma Aubin–Nitsche, 272 Bramble-Hilbert, 230, 349 Cea, 259 generalized Aubin–Nitsche, 404 nonconforming Aubin–Nitsche, 405 Strang, 378 Liapunov-Schmidt method numerical, 612 linear approximation method, 673 linear complexity, 674 linear conforming ﬁnite elements, 429 linear form approximate, 346 special representation, 63, 260 linear operator Vbh -regular, 601 Vbh -stable, 601 linearization bounded, 155, 191 fully nonlinear systems, 161 linearized problem, 32 local error indicator, 433 local solvability and regularity, 112 locality, 239 longest node bisection, 437 lower bound a posteriori, 431 mapping isoparametric, 254 mapping properties, 235 mask, 641 matrix M, 179 mass, 257, 262 real symmetric, 109

741

stiﬀness, 257, 262 symmetric, 81 weakly diagonal dominant, 179 matrix compression, 676 mean value symmetric, 563 measure positive, 71 mesh anisotropically graded, 236 optimal, 425 mesh function, 237 mesh independence principle, xviii, 205, 561 mesh parameter, 234 mesh reﬁnement necessary nonuniform, 425 optimal, 425 uniform, 656 mesh sequence shape-regular, 234 method (linear) discretization, 183 adaptive wavelet, 678 optimal FEM, 442, 451 hp FEM, 448 all discretization quasilinear equations+systems, monotone operators, 279 applicable discretization, 184, 316 applicable Galerkin, 272 approximation conforming, 180 Galerkin, 180 generalized Petrov–Galerkin, 180 nonconforming, 180 Petrov–Galerkin, 180 conforming wavelet Galerkin, 635 DCGMs, 538 defect correction, 631, 632 deferred correction, 631 diﬀerence, xxiv diﬀerence equations, 610 diﬀerence for eigenvalue problems, 609 diﬀerence for monotone operators, 609 diﬀerence for nonlinear boundary operators, 609 diﬀerence for quadrature approximations, 609 discontinuous Galerkin (DCGM), xvii, xxiv discontinuous Galerkin (DCGMs), 212, 297, 455 discrete Newton, 207 eigenvalue problems, 531, 663 existence,convergence, 662 ﬁnite diﬀerence, xvii ﬁnite element, xvii approximation theory, 212 general interior penalty Galerkin (GIPG), 468 generalized gradient, 280

742

Index

method (cont.) incomplete interior penalty Galerkin (IIPG), 468 linear discretization, 184 local discontinuous Galerkin, 477 mesh free or radial basis, xvii monotone operators, 531, 663 Newton–Kantorovich, 207 non symmetric interior penalty Galerkin (NIPG), 468, 529 nonconforming approximation Galerkin, 311 nonlinear boundary operators, 531, 663 other, nonlinear boundary conditions, 345 Petrov–Galerkin generalized, 211 quadrature approximations, 531, 663 quasilinear problems, 660 Runge–Kutta, 185 space discretization, xxi, xxv spectral, xvii symmetric, 587 symmetric interior penalty Galerkin (SIPG), 468 wavelet, xvii, xxiv minimal surface equation, 153 minimum potential, 4 MIP, 205 mixed formulation, 667 model nonlinear, 3 model problem for DCGMs, 459 model problem, 421 modulus of smoothness, 646 Monge -Ampere operator G(uh ) deﬁned and continuous, 197 monotone, 99 strictly, 99 uniform, 103, 143, 274 multi-indices, 54, 230, 264, 638 multiresolution analysis, 640 multiresolution spaces dual, 649 primal, 649 multiscale, 640 Navier–Stokes operator linearization, 286 near best N -term approximation, 674 Nemyckii operator, 586 newest node bisection, 436 NIPG, 468, 529 coercivity linear, semilinear, 499, 500 node hanging, 436 nondegenerate triangulations, 328 nonuniqueness interior, 65 norm, 69 broken Sobolev, 462

discrete dual, 569 discrete Sobolev, 569 discrete Sobolev, uh H 1 (Ωh ) , 614 ±

e

dual, 15 dual Au, 48 energy, 41, 46 eqivalent to Sobolev, 21 equivalent, 41, 46 Lebesgue discrete, 569 penalty, 468, 469, 484, 493 scaled Sobolev, 367 semi, 230 Sobolev, 36, 55, 119, 230 Sobolev–Slobodeckij, 23 norms equivalent, 648 notation often used, 259 number of unknowns, N (h), 425 operator P h , Q h , 591 Qn , Qn linear approximation, 656 ν−Stokes, 167, 169 adjoint, 17, 47 approximate, 346 bijective, 16 bounded, 97, 278 bounded linear, 13 Calder´ on–Zygmund, 654 classical, strong and weak form, 36 coercive, 97, 278 compact, 13, 14 continuous, 97, 278 continuous linear, 13 diﬀerence weak, 576 dual, 16, 47 elliptic, 40, 56, 81, 388 elliptic nonlinear, 80 estimates for nonlinear, 99, 141 existence for the quasilinear, 91 extension, 16, 22 extension on Ω, 246 Fredholm, 18, 51 Fredholm-index, 18 global interpolation, 222 hemi continuous, 97, 278 induced, 46 induced boundary, 42, 66 induced linear, 45 injective, 16 interpolation, 424 isoparametric interpolation, 254, 256 kernel, 14 linear, 41 linear, continuous, bounded, 14, 41 linearization, quasilinear, order 2m, coercive, 158 linearized Navier–Stokes, 167, 286

Index linearized Navier-Stokes, 114, 167 Lipschitz-continuous, 97, 278 local interpolation, 213, 222 monotone, 34, 97, 278 natural boundary, 42, 66 Navier-Stokes, 114, 167 Nemyckii, 96, 98, 141 Nemyckii for systems, 140 nonlinear, 96 nonlinear system, 140 nonlinearcoercive, 97, 278 norm, 14 null space, 14 properties for nonlinear, 98, 140 properties of Nemyckii, 100 properties of the trace, 26 properties,compact, 15 pseudo–diﬀerential, 657 quasilinear, 81, 150 H0m − coercive linearization, 158 quasilinear order 2, 85 range, 14 saddle point problems, 163 self-adjoint, 17 semilinear, 81, 105 semilinear in nonlinear spaces, 86, 150 semilinear non autonomous, 150 semilinear Order 2m, 149 special linear, 14 stable, 97, 278 stationary Navi´ er–Stokes, 281 Stokes, 116, 163 strictly monotone, 97, 278 strongly continuous, 97, 278 strongly elliptic, 40, 56, 81 strongly monotone, 97, 278 surjective, 16 symmetric, 17 trace or restriction, 26 uniformly elliptic, 56, 81, 109, 307, 320 uniformly monotone, 97, 278 weak, 160 optimal complexity, 443 optimal mesh for 2D corner singularity, 427 optimality, 442 order, 115, 133 orthogonality Galerkin, 440 outﬂow boundary point, 423 paralellotope, 214 parameter chunkiness, 230 parametrization of ∂Ω, 25 partition of unity, 25 PDE divergence form, 34

743

divergent, 34 penalty interior, 457 penalty function interior discontinuities, 468 penalty norm, 491 plate buckling, 11 pollution error, 434 polynomial P d = Pd1 , 215 Pdn , 215 Pd = Pd2 , 215 characteristic, 40 average Taylor, 230, 243 bilinear,biquadratic, 219 degree d, 214 positive, 99 asymptotic, 99 pre–wavelets, 642 primal generator, 645 principal part, 40, 64, 67, 73, 76, 80, 81, 106, 122, 134, 151, 158, 160 Vbh −coercive, 602 discrete, 574 elliptic operator, 56, 70, 133 symbol, 115 system, 115, 120, 133, 266 principal symbol system, 120 principle mesh independence, 486 problem and Stokes, 282 convection-diﬀusion, 458 dual, 272, 404, 663 existence and uniqueness for semilinear, 106 extremal, 49 linearization, 151 nonlinear stationary, 683 reaction-diﬀusion, 458 saddle point, 163, 165, 166, 282, 664 unique solution, 166 scope of, 637 unique discrete solution++, 283 variational, 49, 64, 67, 126 product tensor, 219 projector Q h = Qh , 260, 270, 385, 401 approximation, P h , Q h = Qh , 184 orthogonal, 17 property k-linear algebraic, 342 Galerkin orthogonality, 472 proposition useful, 366

744

Index

quadratic eigenvalue problem, 423 quadrature formula Gauss, 390 Gauss–Lobatto, 390 Gauss–Radau, 390 Gauss–type, 390 high order, 390 quasi uniform, 229, 233 quasi-interpolant, 241 quasi-norm, 673 quasi-optimal tree in TCOARSE, 684 quasilinear problem violated boundary conditions, 399 quasilinear equation, 629 special DCGM, 481 quasilinear operator special, 150 quasilinear system order 2, 2m, 157 quasiuniform locally, 234 radial basis functions, 595 red reﬁnement, 436 redundant system, 672 reﬁnable, 641 reﬁnement, red-green, 436 blue, 436 newest node bisection, 436 longest node bisection, 437 regular discretisation, 428 regularity, 146 ¯ 65 in Ω, bootstrap arguments, 84 in Morrey spaces, 125 inherited, 84 inherited to perturbations, 85 results of Ladyˇzenskaja/Uralceva, 93 solutions of FEMs, 327 regularity for order 2m, 137 relation two–scale, 641 residual error estimator, Poisson problem, 429 residuals, 433 Reynolds number, 164, 281 Richardson extrapolation high order diﬀerence, 631 Richardson iteration, 680 Riesz basis, 648, 669 rod axial load, 7 axial loaded, 15 bending, 3 clamped end, 3 loaded, 27 perp load, 3

simply supported end, 7 stationary position, minimum, 8 saturation property, 679 scalar product, 69 discrete Sobolev, 569, 614 Lebesgue discrete, 569 Sobolev, 36, 55, 119 scale representation multi, 643 single, 643 scaling function, 641 Schwartz kernel, 678 semi norm, 55, 231 broken Sobolev, 462 discrete, 569, 614 Sobolev, 36, 119 sequence approximating spaces, 267, 661 set admissible, 126 dense, compact, 14 set of marked triangles Ah , 435 shape factor, 428 Shortley–Weller approximation, 625 Shortly–Weller Method convergence, 625 singular solutions general structure, 422 singularities corner, 421 edge, 423 SIPG, 468, 529 smooth manifold, 25 Sobolev are Banach spaces, 20 Sobolev space, 20, 422, 647 broken, 231 discrete, 566, 568 discretization in discrete, 589 fractional order, 23 trace, fractional order, 27 solution Ω convex, 74 hp-DCG approximate, 540 classical, 38, 459, 475 discrete approximate, 471, 474 existence, 106, 110 existence and uniqueness, 86, 87 existence and uniqueness semilinear, 84 existence/regularity, 96 for convex domain, 74, 124 for fully nonlinear, 111 for quasilinear, 110 generalized, 104, 121, 134, 143 weak Euler system, 129 interior regularity, 75 linear and semilinear, 88 regular, 60, 65, 73, 74, 95, 112, 123, 130, 139, 146

Index singular, 421 smooth data imply smooth solution, 170 spurious, 87 strong, 38, 42 strong and weak, 133 unique, 93 uniqueness, 106, 110 viscosity, 296, 299, 587 weak, 38, 42, 121, 134, 460, 475 weak discrete, 374 weak=strong, 42 space admissible approximating, 180 approximating, 484 bi–dual, 180 conforming, 180 Galerkin, 180 generalized Petrov–Galerkin, 180 nonconforming, 180 Petrov–Galerkin, 180 Banach, 14 Banach, L(X , Y), 18 Banach, dense, 14 broken Sobolev, 462 discrete Sobolev, 569, 624 1 (Ωh ), 614 discrete Sobolev,H± e dual, 14 H¨ older, 58 H¨ older/Sobolev, 58 Morrey, 125 nonconforming approximating, 311 reﬂexive Banach, 47, 49 Sobolev, 36, 69, 119 surplus, 441 space reﬁnement adaptive, 669 uniform, 673 spectrum bilinear form, 291 operator, 51, 52 spline super spline subspace, 247 splitting stable, 250 stability, 174, 190, 191, 239, 268, 485, 661 bound, 191 compactly perturbed discretization inherits, 408 discrete W m,p (Ω) norm, 155 elliptic operators, 406 in uh , 191, 268, 485, 590 proofs, 194 threshold, 191 stability and consistency yield convergence, 189 stability implies existing A−1 , 204 stability inherited to compact perturbations, 658 star of a vertex, 240 star-shaped, 230

state vector, 556 steepest descent method, 680 step size, 562 maximal, 229 Stokes problem unique discrete solutions,convergence, 285 unique, solution, regularity, 166 strategy ﬁxed energy fraction, 435 maximum, 435 subdivision, 229, 231 cf. triangulation, 214 isoparametric, 254 supp(σ), 25 symbol, 657 elliptic operator, 70 symmetric formula, 563 system bounded nonlinearity, 145 Dirichlet, 56, 62, 132, 145 elliptic, 115 Euler, 34, 126 fully nonlinear, 158 quasilinear divergent, 140 linear elliptic order 2m, 132 nonlinear order m, 112, 147 order 2m, 115 nonlinear elliptic, 115 normal, 56 order 2m, 336 quasilinear, 144 divergent, 158 quasilinear elliptic, 157 strongly elliptic, 120 uniform Legendre condition, 382 uniformly elliptic, 120, 266, 382 weak Euler, 129 tensor products, 644 term penalty, 464 tetrahedra/bricks, 233 The Stokes problem, 666 theorem (local) inverse function, 31 Aubin–Nitsche, 272 Banach, open mapping, 16 closed range, 18 extension of linear forms, operators, 16 Gauss integral, 27 G˚ arding, 71 Hahn–Banach Extension, 16 implicit function, 31 mean value, 30 modiﬁed Sobolev–Stein extension, 246 nonconforming Aubin–Nitsche, 405

745

746

Index

theorem (cont.) Riesz–representation, 17 Riesz–Schauder, 51 Sobolev embedding, 22, 350 Sobolev–Stein extension, 22 trace, 26 transformation, 367 theory Riesz–Schauder, 51 trace from Ω to ∂Ω, 75 transform discrete Fourier, 603 Fourier, 134 inverse discrete Fourier, 602, 603 transformation anti crime, xvi, xxiii, 293, 408, 605 tree approximation, 684 triangles/quadrilaterals curvilinear, 233 triangulation, 214, 251, 254, 428, 461 admissible, 214 locally quasi-uniform, 540 non–degenerate, 229, 231 quasi-uniform, 229, 239 shape-regular, 540 Tychonov regularization, 57 unique existence, 169 uniqueness in the small, 95 interior, 65 uniqueness results, 91 unisolvent, 213 unit ball, 23 upper bound a posteriori, 430 variables nodal, 213

variational crimes, 175 variational formulation, 462 discretization, 475 variational problem regular, 126 vertex interior, 247 vertices, 214 wavelet, 640 analysis, 639 bases, 647 biorthogonal, 647 composite, 652 dual, 652 evaluation of nonlinear functionals, 652 generator, 643 good, 643 nonlinear approximation, 672 orthonormal, 642 sdiagonal preconditioning for all problems, 654 wavelet method conforming, 660 convergence, 662 linear elliptic equation order 2m, 659 not for fully nonlinear problems, 660 wavelet transform discrete, 640 fast, 643 wavelets, 669 conforming, 659 elliptic equations, 659 function spaces, 645 on domains, 647 p weak w (J) space, 673 weak form, 423 welldeﬁned G, Gh for the Monge-Ampere operator, 196