hp-Finite Element Methods for Singular Perturbations

Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1796 3 Berlin Heid...

Author: Jens M. Melenk

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Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1796

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Jens M. Melenk

hp-Finite Element Methods for Singular Perturbations

13

Author Jens M. Melenk Max Planck Institute for Mathematics in the Sciences Inselstr. 22 04103 Leipzig Germany e-mail: [email protected]

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Melenk, Jens M.: hp-finite element methods for singular perturbations / Jens M. Melenk. Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; Vol. 1796) ISBN 3-540-44201-4

Mathematics Subject Classification (2000): 65N30, 65N35, 58J37, 35J25 ISSN 0075-8434 ISBN 3-540-44201-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10891013

41/3142/ du - 543210 - Printed on acid-free paper

Preface

Many partial diﬀerential equations arising in practice are parameter-dependent problems and are of singularly perturbed type for small values of this parameter. These include various plate and shell models for small thickness in solid mechanics, the convection-diﬀusion equation, the Oseen equation, and NavierStokes equations in ﬂuid ﬂow problems where the ﬂuid is assumed to have small viscosity, and ﬁnally equations arising in semi-conductor device modelling. Analysis of such equations by numerical methods such as the ﬁnite element method is an important task in today’s computational practice. A signiﬁcant design aspect of numerical methods for such parameter-dependent problems is robustness, that is, that the performance of the numerical method is independent of, or at least fairly insensitive to, the parameter. Numerous methods have been proposed and analyzed both theoretically and computationally for a variety of singularly perturbed problems—we merely refer at this point to the three recent monographs [97, 99, 108] and their extensive bibliographies. Most numerical methods employed in the study of singularly perturbed problems are low order methods. In contrast, the present work is devoted to a complete analysis of a high order ﬁnite element method, the hp-version of the Finite Element Method (FEM), for a class of singularly perturbed problems on curvilinear polygons. To the knowledge of the author, this work represents the ﬁrst robust exponential convergence result for a class of singularly perturbed problems under realistic assumptions on the input data, that is, piecewise analyticity of the coeﬃcients of the diﬀerential equation and the geometry of the domain. This work is at the intersection of several active research areas that have their own distinct approaches and techniques: numerical methods for singular perturbation problems, high order numerical methods for elliptic problems in nonsmooth domains, regularity theory for singularly perturbed problems in terms of asymptotic expansions, and regularity theory for elliptic problems in curvilinearpolygons. Although, naturally, the present work draws on techniques employed in all of these ﬁelds, new tools and regularity results for the solutions had to be developed for a rigorous robust exponential convergence proof. This book comprises research undertaken during my years at ETH Z¨ urich. I take this opportunity to thank Prof. Dr. C. Schwab for many stimulating discussions on the topics of this book and for his support and encouragement over the years. Leipzig, June 2002

J.M. Melenk

Contents

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem class and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Principal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Review of existing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.1 Elliptic problems in non-smooth domains . . . . . . . . . . . . . . . 7 1.4.2 Regularity in terms of asymptotic expansions . . . . . . . . . . . 9 1.4.3 hp ﬁnite element methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.4 Numerical methods for singular perturbations . . . . . . . . . . . 16 1.5 Outline of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Part I. Finite Element Approximation 2.

hp-FEM for Reaction Diﬀusion Problems: Principal Results . 2.1 Setting and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Prelude: the one-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Regularity in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 hp-FEM in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Regularity: the two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 hp-FEM approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 hp-meshes and spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The minimal hp-mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 hp-FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The classical L-shaped domain . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Robustness with respect to mesh distortion . . . . . . . . . . . . . 2.5.3 Examples with singular right-hand side . . . . . . . . . . . . . . . . 2.6 h-FEM approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Approximation on Shishkin meshes in one dimension . . . . . 2.6.2 h-FEM meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 h-FEM boundary layer meshes . . . . . . . . . . . . . . . . . . . . . . . .

23 23 24 24 25 30 31 38 38 39 43 45 46 49 50 54 55 57 62

VIII

3.

Contents

hp Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1 Motivation and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1.1 General overview of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1.2 Outline of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1.3 Robust exponential convergence: key ingredients of proof . 76 3.2 Polynomial approximation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.2.1 Notation and properties of polynomials . . . . . . . . . . . . . . . . 87 3.2.2 Approximation of analytic functions: intervals and squares 90 3.2.3 Approximation of analytic functions on triangles . . . . . . . . 93 3.2.4 The projector Πp∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.2.5 Anisotropic projection operators: Πp1,∞ . . . . . . . . . . . . . . . . 105 3.2.6 An optimal error estimate for an H 1 -projector . . . . . . . . . . 109 3.3 Admissible boundary layer meshes and ﬁnite element spaces . . . . 111 3.3.1 hp-meshes for the approximation of boundary and corner layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.3.2 Patchwise structured meshes . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.3.3 The p-version boundary layer and corner layer patches . . . 118 3.3.4 Boundary layer mesh generation via mesh patches . . . . . . . 120 3.3.5 Properties of the pull-backs to the patches . . . . . . . . . . . . . . 122 3.4 hp Approximation on minimal meshes . . . . . . . . . . . . . . . . . . . . . . . . 123 3.4.1 Regularity on the reference element . . . . . . . . . . . . . . . . . . . . 123 3.4.2 Approximation on minimal meshes . . . . . . . . . . . . . . . . . . . . 127

Part II. Regularity in Countably Normed Spaces 4.

l The Countably Normed Spaces Bβ,ε ......................... 4.1 Motivation and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Outline of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector . . . . . . . . . . . . . . . . . . . . . . . .

141 141 141 146 146

m,l 4.2.1 Properties of the spaces Hβ,ε (Ω) . . . . . . . . . . . . . . . . . . . . . . 150 l . . . . . . . . . 154 4.2.2 Properties of the countably normed spaces Bβ,ε 4.3 Local changes of variables for analytic functions . . . . . . . . . . . . . . . 165

5.

Regularity Theory in Countably Normed Spaces . . . . . . . . . . . . . 5.1 Motivation and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Outline of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Analytic regularity results of Babuˇska and Guo . . . . . . . . . . . . . . . . 5.3 Analytic regularity: Dirichlet problems . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Analytic regularity in sectors . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Regularity in curvilinear polygons . . . . . . . . . . . . . . . . . . . . . 5.4 Neumann and transmission problems . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Neumann and Robin corners . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 169 172 173 176 177 184 188 188

Contents

5.4.2 5.4.3 5.5 Local 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5

Mixed corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries: local H 2 -regularity . . . . . . . . . . . . . . . . . . . . . . Interior regularity: Proof of Proposition 5.5.1 . . . . . . . . . . . Regularity at the boundary: Proof of Proposition 5.5.2 . . . Regularity of transmission problems: Proof of Prop. 5.5.4 . Regularity of Neumann problems: Proof of Prop. 5.5.3 . . .

IX

195 196 197 200 202 208 215 224

Part III. Regularity in Terms of Asymptotic Expansions 6.

Exponentially Weighted Countably Normed Spaces . . . . . . . . . . 6.1 Motivation and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Outline of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m,l l 6.2 The exponentially weighted spaces Hβ,ε,α and Bβ,ε,α in sectors . . 6.2.1 Properties of the exponentially weighted spaces . . . . . . . . . 6.3 Change of variables: from polar to Cartesian coordinates . . . . . . . 6.4 Analytic regularity in exponentially weighted spaces . . . . . . . . . . . 6.4.1 Transmission problem: problem formulation . . . . . . . . . . . . . 6.4.2 Transmission problem in exponentially weighted spaces . . . 6.4.3 Analytic regularity in exponentially weighted spaces . . . . . 6.4.4 Analytic regularity for a special transmission problem . . . .

227 227 227 230 231 231 235 238 238 243 246 248

7.

Regularity through Asymptotic Expansions . . . . . . . . . . . . . . . . . . 7.1 Motivation and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Outline of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Regularity of the outer expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Regularity of the boundary layer expansion . . . . . . . . . . . . . . . . . . . 7.3.1 Deﬁnition and properties of the boundary layer expansion 7.3.2 Proof of Theorem 7.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Regularity through asymptotic expansions . . . . . . . . . . . . . . . . . . . . 7.4.1 Notation and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Proof of Theorem 7.4.5: smooth and boundary layer parts 7.4.3 Proof of Theorem 7.4.5: corner layer and remainder . . . . . .

255 255 255 262 263 267 267 270 283 283 288 290

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Some technical lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Transformations of elliptic equations . . . . . . . . . . . . . . . . . . . A.1.2 Leibniz formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.3 Hardy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Kondrat’ev’s theory for a special transmission problem . . . . . . . . . A.2.1 Problem formulation and notation . . . . . . . . . . . . . . . . . . . . . A.2.2 Proof of Proposition A.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297 297 297 298 301 302 302 303

X

Contents

A.3 Stability properties of the Gauss-Lobatto interpolant . . . . . . . . . . . 309 A.4 L∞ projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Notation General notation N

The set of positive integers, {1, 2, . . . , }.

N0

The set of non-negative integers N ∪ {0}. The set of integers N0 ∪ −N.

Z R, R

+

, R+ 0

The real, the positive real, and the non-negative real numbers.

C

The complex numbers.

i

The imaginary unit with i2 = −1.

Γ (·)

The Gamma function with Γ (j + 1) = j! for j ∈ N0 . The Kronecker symbol: δij = 0 for i = j and δii = 1.

δij

C, C , γ, γ , K, b

Generic constants independent of critical parameters such as ε, the diﬀerentiation order, the polynomial degree, etc. These constants may be diﬀerent in diﬀerent instances.

·

x = max {n ∈ Z | n ≤ x}.

[·] ⊂⊂

In Section 5.5: [p] = max {1, p} for p ∈ Z (see p. 198). In all other sections [ · ] denotes the jump operator. Compact embedding.

|K|

for K ⊂ Rn represents the Lebesgue measure (volume) of K.

EΩ

The characteristic function of the set Ω.

Matrices Mn

The set of (real) n × n matrices.

Sn

Sn ⊂ Mn are the symmetric matrices.

Sn>

The set of symmetric positive deﬁnite matrices. For matrices A, B, we set A : B = i,j Aij Bij .

:

Sets, balls, sectors, neighborhoods Br (x)

The ball of radius r around the point x.

+ , BR , BR − BR

Ball and half balls with radius R; see (5.5.1).

Uκ (K)

The κ-neighborhood of the set K, i.e., ∪x∈K Bκ (x).

S

A generic sector, Deﬁnition 4.2.1, p. 146.

SR (ω)

A sector with opening angle ω, see (4.1.2).

XII

Notation

0,δ SR (ω), ω,δ SR (ω) Ix , Iy

SX , SX (δ)

Conical neighborhood of the lateral parts Γ0 , Γω of the sector SR (ω), see (5.4.30). Intervals on R; Iy is the form Iy = [0, b] for a b > 0; see outset of Section 7.3.1. Complex neighborhoods of interval Ix , see (7.3.11).

Norms, diﬀerential operators, standard function spaces L2 (Ω) k

H (Ω) H01 (Ω) 1 (Ω) H0,ε

H 1/2 (Ω) 1/2

The space of square integrable functions. Sobolev space H k of L2 -functions whose distributional derivatives of order up to k are also in L2 ; cf. [1]. Sobolev space of H 1 -functions with vanishing trace on ∂Ω; cf. [1]. The Sobolev space of H 1 functions with vanishing trace on ∂Ω equipped with the energy norm · L2 (Ω) + ε∇ · L2 (Ω) ; cf. p. 184. The usual Sobolev space H 1/2 ; cf. [1]. 1/2

H00 (Ω)

The usual Sobolev space H00 ; cf. [1].

· ε

Energy norm uε ∼ uL2 (Ω) + ε∇uL2 (Ω) ; cf. p. 3.

· ε,α

Exponentially weighted energy norm, cf. p. 243.

α

D u ∇p u(x)

For multi-indices α = (α1 , . . . , αn ) ∈ Nn0 and (smooth) functions u deﬁned on an open subset of Rn : Dα u = ∂xα11 ∂xα22 · · · ∂xαnn u. n |∇p u(x)|2 = |∂α1 ∂α2 · · · ∂αp u(x)|2 , where, for α1 ,...,αp =1

tensor-valued functions u = (ui )N i=1 and shorthand ∂α u for ∂xα u N |∂α1 ∂α2 · · · ∂αp ui (x)|2 . |∂α1 ∂α2 · · · ∂αp u(x)|2 = i=1

A(G)

For domains G ⊂ Rn (or Cn ) A(G) denotes the set of functions analytic on G. For closed sets G, f ∈ A(G) is understood to imply the existence of an open neighborhood of G on which f is analytic; see also (1.2.2).

A(G, Rn )

The set of vector-valued functions that are (componentwise) analytic on G.

A(G, Sn> )

The set of functions from G to the symmetric positive deﬁnite matrices Sn> that are (componentwise) analytic on G.

Lε

The diﬀerential operator, (1.2.1).

[·]

The jump operator across a curve. Only in Sec. 5.5: [p] = max {1, p} for p ∈ Z; p. 198. The co-normal derivative operator nT ∇·.

∂nA

Notation

XIII

Weight functions and weighted spaces ˆp,β,ε Φ

Weight function in a sector, p. 147.

Φp,β,ε Ψˆp,β,ε,α

Weight function in a curvilinear polygon, p. 184.

Ψp,β,ε,α

Exponentially weighted weight function in Ω; see (7.4.7).

m,l Hβ,ε l Bβ,ε m,l Hβ,ε,α l Bβ,ε,α

Weighted Sobolev space, p. 149.

E

Smallest characteristic length scale, p. 177.

Exponentially weighted weight function in a sector, p. 231.

Countably normed space, p. 149. Exponentially weighted Sobolev space, p. 232. Exponentially weighted countably normed space, p. 232.

Semi norms for controlling high order derivatives NR,p (u), NR,p,q (u), ,± NR,p,q (u)

Bounds on higher derivatives of u, p. 198.

MR,p (f )

Bounds on higher derivatives of f , p. 202.

(f ), MR,p ˜ MR,p (f ), NR,p (u)

Bounds on higher derivatives of f and u, p. 208.

,± NR,p (u), HR,p (u), ,± (f ) MR,p

Bounds on higher derivatives of u and f , p. 217.

Description of the boundary and corner layer ψj

Boundary ﬁtted coordinates (x, y) = ψj (ρj , θj ) in neighborhood of arc Γj , where ρj measures the distance of the point (x, y) to Γj ; see Notation 2.3.3

Aj

Vertex of the curvilinear domain Ω, Section 1.2.

Γj Sj , Sj+ , Sj−

Analytic arc being part of the boundary of the curvilinear polygon Ω, Section 1.2. Sectors near Aj for the deﬁnition of corner layer, (7.4.2), (7.4.3).

Ωj , χBL , χCL

Subdomains of Ω and cut-oﬀ functions associated with arcs Γj and the vertices Aj ; see Notation 2.3.3 and outset of Section 7.4.1.

Ωj , χBL j , CL χj

Subdomains of Ω and cut-oﬀ functions associated with arcs Γj and the vertices Aj ; see Notation 2.3.3.

sκ , s˜j,κ

Anisotropic and anisotropic stretching maps; see Notation 2.4.3.

XIV

Notation

Polynomials, approximation, and projections I, S, T

The reference interval I = (0, 1), square S = I × I, and triangle T = {(x, y) | 0 < x < 1, 0 < y < 1 − x}, p. 87.

S, T

The references square and triangle in Section 3.2.3; see (3.2.19).

Pp (T ), Qp (S), Πp (K)

Spaces of polynomials, p. 87.

(α,β)

Pp

Jacobi polynomials, see [124].

˜p Lp , L

Lp = Pp

ψp,q

Orthogonal polynomials on the triangle, (3.2.23).

GL

Gauss-Lobatto points, p. 87.

ip , jp

1D and 2D Gauss-Lobatto interpolation operators, p. 87.

ip,Γ

Gauss-Lobatto interpolation operator on an edge Γ , p. 88.

E

Polynomial extension operator from the boundary, p. 89.

Πp∞

Polynomial projector deﬁned in Theorem 3.2.20, p. 103.

Πp1,∞

Polynomial projector deﬁned in Theorem 3.2.24, p. 108.

2

ΠpL

(0,0)

˜ p (x) = Lp (2x − 1). is the usual Legendre polynomial; L

The L2 projector into the space Pp .

Meshes and ﬁnite element approximation T

Triangulation, p. 39. p

S (T ), S0p (T ) ∞ Πp,T

Spaces of piecewise mapped polynomials of degree p on the mesh T , p. 113. Elementwise application of Πp∞ on a mesh T , p. 113.

1. Introduction

1.1 Introduction 1.2 Problem class and assumptions This work presents numerical analysis and regularity results for singularly perturbed equations of the form (1.2.1). Such equations are ubiquitous, appearing, for example in convection-dominated ﬂuid ﬂow, in semi-conductor device modelling, and solid mechanics (where their analysis is crucial for an understanding of the layer structure of Reissner-Mindlin plate models, [9, 10]). We consider the following class of singularly perturbed equations: Lε uε := −ε2 ∇ · (A(x)∇uε ) + b(x) · ∇uε + c(x)uε = f on Ω, (1.2.1a) uε = g on ∂Ω. (1.2.1b) The bounded Lipschitz domain Ω ⊂ R2 is assumed to be a curvilinear polygon as depicted in Fig. 1.2.1. The boundary ∂Ω is assumed to consist of ﬁnitely (1)

Γ3 (1)

A3

(1)

(1) ω3

(2)

A1 (1)

(2)

AJ1 −1 (1)

(1)

(1) ωJ1

ωJ1 −1

(2) (1)

(2) A2

(2)

AJ2

A2

Γ2

Γ1 (1)

ΓJ1 = Γ0

(1) ω2

Γ2

(2)

ΓJ2

(1)

(1) ω1

A1

(1)

Γ1 (1) AJ1

=

(1) A0

Fig. 1.2.1. A curvilinear polygon. (i) many curves Γ (i) , i.e., ∂Ω = ∪N , each of which consists of ﬁnitely many i=1 Γ (i) analytic arcs Γj : (i)

i Γj . Γ (i) = ∪Ji=1

J.M. Melenk: LNM 1796, pp. 1–20, 2002. c Springer-Verlag Berlin Heidelberg 2002

2

1. Introduction (i)

The arcs Γj

are parametrized by (i)

Γj (i)

(i)

(i)

= {(xj (θ), yj (θ) | θ ∈ (0, 1)},

(i)

where the functions xJ , yj are analytic on a neighborhood of the interval [0, 1]. We assume that d (i) 2 d (i) 2 x + y > 0 on [0, 1] for all i, j. dθ j dθ j (i)

The curves Γj

are oriented such that the domain Ω is “on the left”; that is, the (i)

(i)

normal vector (− dθ yj (θ), dθ xj (θ)) points into Ω (cf. Fig. 1.2.1). The endpoints (i)

of the arc Γj

(i)

(i)

(i)

(i)

(i)

(i)

are the vertices Aj−1 = (xj (0), yj (0)), Aj = (xj (1), yj (1)),

(i)

(i)

(i)

and we set A0 := AJi . The internal angle at vertex Aj

(i)

is denoted ωj , and

(i)

we exclude cusps by stipulating 0 < ωj < 2π. In order to simplify the notation in this work, we assume without loss of generality that N = 1 and drop the (1) (1) superscript (i); i.e., we write J = J1 , Γj = Γj , Aj = Aj , etc. It is also convenient to write Γ0 = ΓJ . The remaining data appearing in (1.2.1) are assumed to be analytic: We suppose that c ∈ A(Ω), b ∈ A(Ω, R2 ), and A ∈ A(Ω, S2> ); i.e., we stipulate the existence of CA , Cb , Cc , and γA , γb , γc > 0 such that p ∇p AL∞ (Ω) ≤ CA γA p! ∇p bL∞ (Ω) ≤ Cb γbp p!

∇p cL∞ (Ω) ≤ Cc γcp p!

∀p ∈ N0 , ∀p ∈ N0 , ∀p ∈ N0 .

(1.2.2a) (1.2.2b) (1.2.2c)

Furthermore, the matrix A(x) is symmetric positive deﬁnite for each x ∈ Ω and there exists λmin > 0 such that A(x) ≥ λmin

∀x ∈ Ω.

(1.2.2d)

We require the existence of µ > 0 such that 1 − (∇ · b)(x) + c(x) ≥ µ > 0 2

∀x ∈ Ω.

(1.2.2e)

The right-hand side f in (1.2.1) satisﬁes f ∈ A(Ω), i.e., there are Cf , γf > 0 such that ∇p f L∞ (Ω) ≤ Cf γfp p! ∀p ∈ N0 . (1.2.3) Finally, the boundary data g ∈ C(∂Ω) are assumed to be analytic on the arcs Γj : For each j, the function g(xj , yj ) is analytic on [0, 1], i.e., there are Cg , γg > 0 such that Dp g(xj (·), yj (·))L∞ ((0,1)) ≤ Cg γgp p! ∀p ∈ N0 . (1.2.4) For most of our analysis, the singular perturbation parameter ε ∈ (0, 1] is assumed to be small, i.e., ε << 1.

1.2 Problem class and assumptions

3

Remark 1.2.1 The assumption of analyticity of the data on Ω can be relaxed. In fact, in most of the subsequent analysis, only piecewise analyticity of the data A, b, c, f , and g needs to be assumed. Solutions of (1.2.1) are understood in the weak sense; i.e., uε is the solution of the following problem: Find uε ∈ H 1 (Ω) s.t. uε |∂Ω = g and Bε (uε , v) = F (v)

∀v ∈ H01 (Ω). (1.2.5)

Here, the bilinear form Bε and the linear form F are deﬁned by 2 (A(x)∇u) · ∇v + b(x) · ∇uv + c(x)uv dx, Bε (u, v) := ε Ω F (v) := f (x)v dx.

(1.2.6a) (1.2.6b)

Ω

This bilinear form Bε is closely connected with the energy norm u2ε := ε2 ∇u2L2 (Ω) + u2L2 (Ω)

(1.2.7)

as will become apparent in the subsequent Lemma 1.2.2. The bilinear form Bε is coercive on the space H01 (Ω), and the variational formulation (1.2.5) has a unique solution even under the weaker assumptions f ∈ L2 (Ω) and g ∈ H 1/2 (∂Ω): Lemma 1.2.2. Let the coeﬃcients A, b, c satisfy (1.2.2), ε ∈ (0, 1], and let f ∈ L2 (Ω), g ∈ H 1/2 (∂Ω). Then there exists a unique solution uε of (1.2.5) and a constant C > 0 independent of ε, f , and g such that ε∇uε L2 (Ω) + uε L2 (Ω) ≤ C f L2 (Ω) + gH 1/2 (Ω) . Moreover, the bilinear form Bε is coercive on the space H01 (Ω), and there holds Bε (u, u) ≥ ε2 λmin ∇u2L2 (Ω) + µu2L2 (Ω)

∀u ∈ H01 (Ω).

(1.2.8)

The bilinear form Bε is also continuous on the space H 1 (Ω): There is C > 0 independent of A, b, c, and ε such that for all u, v ∈ H 1 (Ω) we have: |Bε (u, v)| ≤ C AL∞ (Ω) + cL∞ (Ω) + bL∞ (Ω) ε−1 uε vε . (1.2.9) Proof: (1.2.9) follows immediately from the Cauchy-Schwarz inequality. As a ﬁrst step, we show (1.2.8). We start by noting that for u ∈ H01 (Ω), an integration by parts gives (b · ∇u)u dx = − (∇ · b)u2 + u(b · ∇u) dx. Ω

Therefore,

Ω

1 (b · ∇u)u dx = − 2 Ω

(∇ · b)u2 dx. Ω

4

1. Introduction

Combining this with assumption (1.2.2e) implies the coercivity of the bilinear form Bε on the space H01 (Ω): 2 Bε (u, u) = ε (A(x)∇u) · ∇u + (b(x) · ∇u)u + c(x)u2 dx Ω 1 2 =ε (A(x)∇u) · ∇u + c(x) − ∇ · b(x) u2 dx 2 Ω ≥ ε2 λmin ∇u2L2 (Ω) + µu2L2 (Ω) . This coercivity gives uniqueness of the solution of (1.2.5). In order to see existence of a solution, let G ∈ H 1 (Ω) be an extension of g into Ω satisfying G|∂Ω = g,

GH 1 (Ω) ≤ CgH 1/2 (∂Ω)

for some C > 0 depending only on Ω. The diﬀerence u ˜ := uε − G must be the solution of the problem: Find u ˜ ∈ H01 (Ω) s.t. Bε (˜ u, v) = F (v) − Bε (G, v)

∀v ∈ H01 (Ω).

(1.2.10)

We see that for all v ∈ H 1 (Ω) |F (v) − Bε (G, v)| ≤ f L2 (Ω) vL2 (Ω)

+ C GH 1 (Ω) ε2 ∇vL2 (Ω) + GH 1 (Ω) vL2 (Ω) + GL2 (Ω) vL2 (Ω)

≤ C f L2 (Ω) + GH 1 (Ω) vε , where we assumed ε ≤ 1. Therefore, by the classical Lax-Milgram Lemma, [36, 82], (1.2.10) indeed has a unique solution u ˜ satisfying ˜ uε ≤ C f L2 (Ω) + GH 1 (Ω) . Using ε ≤ 1, we see that uε := G + u ˜ satisﬁes the desired bounds.

2

The greater part of our analysis will be done for the special case b ≡ 0; i.e., we consider the following singularly perturbed problem of elliptic-elliptic type: −ε2 ∇ · (A(x)∇uε ) + c(x)uε = f uε = g

on Ω,

(1.2.11a)

on ∂Ω,

(1.2.11b)

where assumption (1.2.2e) implies that c ≥ µ > 0 on Ω.

1.3 Principal results The main result of the present work is the robust exponential convergence result Theorem 2.4.8 for high order ﬁnite element methods applied to (1.2.11). It is shown that with the proper choice of conforming subspaces VN of dimension

1.3 Principal results

5

N ∈ N, the ﬁnite element method, i.e., Galerkin projection, yields approximants uN ε to the exact solution uε that satisfy −bN uε − uN ε ε ≤ Ce

1/3

.

(1.3.1)

Here, the constants C, b > 0 are independent of ε; in fact, in our numerical experiments in Section 2.5 b ≈ 1 and likewise C = O(1). The ﬁnite element spaces VN are given explicitly in Section 2.4. They consist of the usual piecewise polynomial spaces of degree p deﬁned on meshes that are adapted to the length scale ε of the problem. Speciﬁcally, for the approximation with polynomial of degree p, these meshes are designed according to three principles: 1. near the edges of the domain, long, thin needle elements of width O(pε) are employed in order to capture boundary layer phenomena; 2. in an O(pε) neighborhood of the vertices a geometric mesh reﬁnement is used in order to resolve corner singularities; 3. in the interior of the domain a standard coarse mesh is utilized for the resolution of smooth solution components. It is worth stressing that the only information required for an application of these mesh design principles is the length scale ε of the problem, which is typically known in practice. Let us compare our robust exponential convergence result with previous convergence analyses. Thus far only algebraic robust convergence results have been available, typical of low order methods. Robust algebraically convergent methods deliver approximate solutions uN ε from spaces VN of dimension N ∈ N that satisfy error bounds of the form −α uε − uN . ε ε ≤ CN

(1.3.2)

Here, C, α > 0 are independent of ε. Even for optimally chosen meshes, α ≤ 2 is typical for two-dimensional problems. A good measure for comparing approximation results (1.3.1) and (1.3.2) is the alphanumerical work W required to compute the approximate solution uN ε . In the case of low order methods, an eﬃcient iterative solver such as multigrid, [65], is essential for acceptable solution times. Such an optimal solution algorithm would solve the resulting linear system with linear complexity, i.e., W = O(N ). The best rate of convergence of these low order methods in terms of work W is therefore −α uε − uN . ε ε ≤ CW

This work estimate, however, is based upon two strong assumptions. First, in order for α to be reasonable, e.g., α ≈ 1, the mesh has to be carefully designed so as to capture the relevant features of the solution. In particular, it has to contain highly anisotropic elements in the boundary layer. However, most state-of-theart adaptive strategies do not allow for such elements: Their use of shape-regular elements precludes robustness, and the convergence rates visible in practice are

6

1. Introduction

low, i.e., α is small. The second strong assumption made is the existence of multigrid methods (or, more generally, preconditioned iterative solvers) with linear complexity for meshes that contain anisotropic elements. Their construction is non-trivial, and, in fact, few results in this direction are available to date. Let us turn to work estimates for high order approximations. Standard Gaussian elimination allows us to solve a linear system with work W = 13 N 3 + O(N 2 ). Hence, if the linear system obtained in our high order method is so solved, our hp-approximation result takes the following form in the “error versus work” perspective: −bW 1/9 uε − uN . (1.3.3) ε ε ≤ Ce Thus, even in terms of work, our exponential convergence result will (asymptotically) outperform methods with algebraic convergence rates. It should be pointed out that (1.3.3) is a rather crude estimate in that the estimate W = 13 N 3 +O(N 2 ) for the solution of the linear system does not make any sparsity assumptions on the matrix. However, even in high order methods, the resulting system matrices have structure and are sparse, which can be exploited by sophisticated modern direct solvers, [44, 52]. In practice, the work estimate (1.3.3) is therefore pessimistic. Our robust exponential convergence result depends strongly on detailed regularity assertions for the solution of (1.2.1). A large portion of this work (Parts II, III) is therefore devoted to the derivation of new regularity results for (1.2.1). While our regularity assertions are interesting in their own right, they are derived in order to enable us to obtain a priori error bounds for piecewise polynomial approximation. This intended main application determines the type of the results and shapes their form. For our application it is essential to have bounds on higher order derivatives of the solution (or solution components) at any given point of the domain that are explicit in critical parameters such as the singular perturbation parameter, the distance to the vertices, and the distance to the boundary of the domain. For exponential convergence results it is furthermore necessary to have bounds on the derivatives of the solution that are explicit in the diﬀerentiation order. Finally, it is also important that these bounds depend on the given input data only, i.e., the coeﬃcients of the diﬀerential equation, the geometry of the domain, the right-hand side, and the boundary data. This last requirement is closely connected with our goal to place assumptions on the input data that are typically met in practice and that can be checked explicitly. In contrast, existing convergence analyses are often done under the unrealistic assumption that certain – typically uncheckable – compatibility conditions are satisﬁed by the input data; we will elaborate this point in Sections 1.4.2 and 1.4.4. Regularity assertions that meet all the above requirements are proved in the present work. The main regularity assertions proved here are the following: 1. A shift theorem in weighted spaces (Theorem 5.3.8), where the solution uε of (1.2.1) is shown to be in an appropriately weighted H 2 space.

1.4 Review of existing results

7

2. A shift theorem in countably normed spaces (Theorem 5.3.10) where the growth of the derivatives of the solution at a point is controlled in terms of ε and the distance to the nearest vertex. 3. Complete asymptotics with error bounds for the solutions of (1.2.11) on curvilinear polygons. Analytic regularity results for all terms arising in the asymptotics—in particular the so-called corner layers—are provided. Since these results are spread over Parts II, III, we collect the main results in Section 2.3 in Theorems 2.3.1, 2.3.4 in a form that enables us to prove robust exponential convergence of the hp-FEM.

1.4 Review of existing results Numerical analysis and regularity theory are closely connected since the ability to characterize solution behavior precisely is essential for the design and analysis of eﬃcient numerical methods. For this reason the present work is placed at the intersection of several ﬁelds, namely, regularity theory on non-smooth domains, asymptotic expansions methods, numerical methods for singular perturbation problems, and high order ﬁnite element methods. In the present section, we brieﬂy review existing results in these areas. Sections 1.4.1 and 1.4.2 discuss two kinds of regularity results. Section 1.4.1 is concerned with regularity theory of elliptic problems on non-smooth domains. The classical theory discussed in that area does not address the issue of singular perturbation problems. Section 1.4.2 presents the approach of describing solution behavior through asymptotic expansions. There, the classical work is not concerned with problems posed on non-smooth domains. Finally, Sections 1.4.3 and 1.4.4 discuss high order methods for problems on non-smooth domains and numerical methods for singularly perturbed problems. 1.4.1 Elliptic problems in non-smooth domains The main points of regularity theory for elliptic problems in non-smooth domains can already be understood for the following model problem: −∆u = f

on Ω ⊂ R2 ,

u=0

on ∂Ω.

(1.4.1)

If ∂Ω is smooth, then the classical elliptic shift theorem, [2–4], holds, i.e., f ∈ H k−1 (Ω), k ≥ 0, implies u ∈ H k+1 (Ω). This shift theorem breaks down if the boundary ∂Ω fails to have suﬃcient regularity. For piecewise smooth boundaries, e.g., if Ω is a polygon, the shift theorem holds only for k ∈ [0, k0 ) where k0 depends on the domain. Speciﬁcally, for (1.4.1) and polygonal domains Ω with interior angles ωj ∈ (0, 2π), k0 = π/ maxj ωj . It turns out that a “modiﬁed” shift theorem holds: For k beyond k0 , the classical shift theorem holds provided that a certain number of singular functions is subtracted. Let us associate with each vertex Aj of the polygon Ω polar coordinates (rj , ϕj ) (such that the lines

8

1. Introduction

ϕj = 0 and ϕj = ωj coincide with the edges meeting at Aj ) and deﬁne singular functions Slj by  rlπ/ωj sin lπ ϕj if lπ/ωj ∈ N, j ωj Slj (rj , ϕj ) = lπ/ω j lπ lπ r ln rj sin ωj ϕj + ϕj cos ωj ϕj if lπ/ωj ∈ N. j With the aid of these singular functions, we can formulate the modiﬁed shift theorem: Proposition 1.4.1. Let Ω be a polygon with vertices Aj , j = 1, . . . , J, and interior angles ωj ∈ (0, 2π). If f ∈ H k (Ω), then the solution u of (1.4.1) can be decomposed as J alj (f )Slj + u0 (1.4.2) u= j=1

l∈N lπ/ωj
for some alj (f ) ∈ R and u0 ∈ H k+1 (Ω). The coeﬃcients alj (·) are in fact linear functionals, and for each k there exists Ck > 0 independent of f such that J j=1

|alj (f )| + u0 H k+1 (Ω) ≤ Ck f H k−1 (Ω) .

(1.4.3)

l∈N lπ/ωj
Hence, up to ﬁnitely many singular components, the solution u of (1.4.1) satisﬁes a shift theorem. Forerunners of regularity assertions similar to Proposition 1.4.1 are [83,128]. The seminal paper, however, is [76,77]. This kind of regularity theory is by now very well developed, in that corresponding results for equations with variable coeﬃcients, diﬀerent kinds of boundary conditions, and elliptic systems are available. The monographs [39, 62, 63, 79] are classical in this area. The ﬁrst decomposition of the type presented in Proposition 1.4.1 for the system of Lam´e equations in polyhedral domains was presented in [126]. General results for ADNelliptic systems can be found in [37]. We also mention the newer monographs [78, 91, 100] and the important papers [89, 90]. Proposition 1.4.1 is essentially concerned with ﬁnite regularity in the sense that for right-hand side f from a Sobolev space H k−1 (Ω), the solution is ascertained to be, up to ﬁnitely many singular components, in the Sobolev space H k+1 (Ω). Many problems of practical interest, however, have input data that are piecewise C ∞ or even piecewise analytic. Then, the solution u of (1.4.1) is piecewise C ∞ or piecewise analytic on Ω, [70, 98]. In particular, therefore, all derivatives of u exist. Decompositions of the form (1.4.2) are not an adequate means for control of all the derivatives as the constant Ck in (1.4.3) depends on k in an unspeciﬁed way. Especially in the context of high order approximation, it is important to be able to control the growth of the derivatives of the solutions to elliptic problems (cf. Section 1.4.3 below). These considerations motivated [15] to introduce socalled countably normed spaces Bβl . On a polygon Ω, the spaces Bβl are deﬁned as follows. With each vertex Aj , j = 1, . . . , J, we associate a number βj ∈ [0, 1). We then set β = (β1 , . . . , βJ ) and introduce for each p ∈ N0 the weight function

1.4 Review of existing results

Φp,β (x) :=

J

p+βj

(dist(x, Aj ))

9

.

j=1

A function u that is analytic on Ω is in Bβl (Ω, Cu , γu ), if for Cu , γu > 0 and l ∈ N0 there holds uH l−1 (Ω) ≤ Cu , Φp,β ∇p+l uL2 (Ω) ≤ Cu γup p!

∀p ∈ N0 .

We note that such growth conditions on the derivatives impose conditions on the analytic functions. In particular, a function from a space Bβl is analytic up to the boundary of the polygon Ω with the exception of the vertices Aj ; i.e., for each x ∈ ∂Ω \ ∪Jj=1 Aj there is a neighborhood Br (x) to which u has an analytic extension. It can be checked that the singular functions Slj introduced above are in the space Bβ2 if βj ≥ 0 is chosen such that additionally βj ∈ (1 − π/ωj , 1). In the framework of these countably normed spaces, [15, 16] proved the following shift theorem for analytic functions: Proposition 1.4.2. Let βj > 0 satisfy βj ∈ (1 − π/ωj , 1). Then for a righthand side f ∈ Bβ0 (Ω, Cf , γf ) with Cf , γf > 0, the solution u of (1.4.1) is in the countably normed space Bβ2 (Ω, Cu , γu ) for some Cu , γu > 0. Proposition 1.4.2 thus characterizes the solution of (1.4.1) for analytic right-hand sides. The solutions are analytic on Ω, and the derivatives become singular at the vertices of the domain in a controlled way. This characterization is of essential importance in the proof of exponential convergence of hp-FEM on geometric meshes as will be discussed in greater detail below. As stated, Proposition 1.4.2 is only a typical representative of regularity results in countably normed spaces. Analogous results hold for strongly elliptic systems, [15, 16] and the Lam´e equations, [17]. [15] considered analytic input data (here: the right-hand side f ) as they typically arise in computational practice. However, other kinds of regularity theory for C ∞ data have been considered in the literature. One such class of functions are Gevrey classes, in which less stringent conditions are placed on the growth of the derivatives. Shift theorems akin to Proposition 1.4.2 can also be formulated in such classes, [30]. The regularity theory in countably-normed spaces of [15] exempliﬁed in Proposition 1.4.2 is not directly suited for an application to singular perturbation problems in polygonal domains because the spaces Bβl have no explicit means of controlling the dependence of the solution on the perturbation parameter. The need for such explicit control motivates in the present work us to introduce l parameter-dependent countably normed spaces Bβ,ε . 1.4.2 Regularity in terms of asymptotic expansions In the preceding Section 1.4.1, we presented various approaches to the description of the behavior of solutions of elliptic problems in polygonal domains. For

10

1. Introduction

singularly perturbed problems, a diﬀerent kind of regularity theory is prevalent, namely, the use of asymptotic expansions. This approach has a long history, dating back at least to the middle of the 19th century. We mention here [46,47,85] and [71,81] in which the method of matched asymptotics has been applied to a variety of singularly perturbed problems. Note that [71] is mostly concerned with these techniques for problems posed over smooth domains or rectangular domains, due to the fact that asymptotic expansions typically require smoothness of the input data in order to be deﬁned. This point is best illustrated by the ensuing example, taken from [95]. An example of asymptotic expansions. We consider the problem −ε2 ∆uε + uε = f

on Ω ⊂ R2 ,

uε = g on ∂Ω.

(1.4.4)

Here, ε ∈ (0, 1] is a small parameter and the data f , g, and ∂Ω are assumed to be C ∞ . The solution uε exhibits boundary layers that are classically described with the aid of asymptotic expansions. These can be created by the classical method of matched asymptotic expansions, [71]. For the deﬁnition of the asymptotic expansions, we introduce boundary-ﬁtted coordinates: Let L > 0 be the length of ∂Ω and let (X(θ), Y (θ)), θ ∈ [0, L), be the smooth, L-periodic parametrization of ∂Ω by arc length such that the normal vector (−Y (θ), X (θ)) always points into the domain Ω. Introduce the notation κ(θ) for the curvature of the boundary curve and denote by TL the one-dimensional torus of length L, i.e., R/LZ, endowed with the usual topology. The functions X, Y and hence also κ are smooth on TL by the smoothness of ∂Ω. For ρ0 > 0 suﬃciently small, the mapping ψ : [0, ρ0 ] × TL → Ω (ρ, θ) → (X(θ) − ρY (θ), Y (θ) + ρX (θ))

(1.4.5)

is smooth on (a neighborhood of) [0, ρ0 ]×TL . The function ψ maps the rectangle (0, ρ0 )×TL onto a half-tubular neighborhood Ω0 of ∂Ω. Furthermore the inverse ψ −1 exists and is also smooth on (a neighborhood of) the closed set Ω 0 . The ﬁrst step in the method of matched asymptotics is to deﬁne the outer expansion w, which can be viewed as an approximation to a particular solution of (1.4.4). Here, it is obtained by making the formal ansatz w(x, y) ∼

∞

εi wi (x, y)

i=0

and then inserting this ansatz into (1.4.4) to get a recurrence relation for the unknown functions wi . For the present problem, we obtain w(x, y) ∼

∞

ε2i ∆(i) f = f + ε2 ∆f + ε4 ∆∆f + · · · .

i=0

For every M ∈ N0 the outer expansion of order 2M + 1 is deﬁned by

(1.4.6)

1.4 Review of existing results

w2M +1 :=

M

ε2i ∆(i) f.

11

(1.4.7)

i=0

The function uε − wM then satisﬁes Lε (uε − w2M +1 ) = f − Lε wM = ε2M +2 ∆(M +1) f.

(1.4.8)

Hence, asymptotically as ε tends to zero, the functions wM satisfy the diﬀerential equation in Ω. However, the functions wM do not satisfy the given boundary conditions g. We therefore introduce a boundary layer correction uBL of wM , which will lead to the inner expansion. For each M the correction uBL is deﬁned as the solution of Lε uBL = 0 uBL

in Ω, M =g− ε2i ∆(i) f i=0

on ∂Ω. ∂Ω

The inner expansion is an asymptotic expansion for this correction function uBL . In order to deﬁne this expansion, we need to rewrite the diﬀerential operator Lε in boundary-ﬁtted coordinates (ρ, θ). With the curvature κ(θ) of ∂Ω and the function 1 (1.4.9) σ(ρ, θ) = 1 − κ(θ)ρ we have (see, for example, [10]) ∆u(ρ, θ) = ∂ρ2 u − κ(θ)σ(ρ, θ)∂ρ u + σ 2 (ρ, θ)∂θ2 u + ρκ (θ)σ 3 (ρ, θ)∂θ u. Introducing now the stretched variable notation ρ = ρ/ε, we have Lε = −∂ρ2 + Id + εκ(θ)σ(ε ρ, θ)∂ρ − ε2 σ 2 (ε ρ, θ)∂θ2 − ε3 ρκ (θ)σ 3 (ε ρ, θ)∂θ . Expanding in power series in ε, we can write the operator Lε formally as Lε =

∞

εi Li ,

(1.4.10)

i=0

where the operators Li have the form L0 = −∂ρ2 + Id,

Li = − ρi−1 a1i−1 ∂ρ − ρi−2 a2i−2 ∂θ2 − ρi−2 ai−3 3 ∂θ ,

i ≥ 1,

and the coeﬃcients aij are given by ai1 = −[κ(θ)]i+1 , ai2 = (i + 1)[κ(θ)]i , ai3 = ai1 = ai2 = ai3 = 0

(i + 1)(i + 2) [κ(θ)]i κ (θ), i ∈ N0 , 2

for i < 0.

Now, in order to deﬁne the inner expansion, we make the formal ansatz

12

1. Introduction

uBL =

∞

i ( εi U ρ, θ),

(1.4.11)

i=0

i are to be determined. Setting Lε uBL = 0 in (1.4.10) where the functions U yields i ∞ i−j = 0. εi Lj U i=0

j=0

Hence, upon setting the coeﬃcients of this formal power series in ε to zero, we i : obtain a recurrence relation for the desired functions U i = Fi = F1 + F2 + F3 , i + U −∂ρ2ˆ U i i i Fi1 =

i−1

i−1−j , Fi2 = ρj aj1 ∂ρˆ U

j=0

i−2

i = 0, 1, . . . ,

i−2−j , Fi3 = ρj aj2 ∂θ2 U

j=0

i−3

i−3−j , ρj+1 aj3 ∂θ U

j=0

where we use the tacit convention that empty sums take the value zero. As we expect the boundary layer function uBL to decay away from the boundary ∂Ω and we want to satisfy the boundary conditions, we supplement these ordinary i with the boundary conditions diﬀerential equations for the U i → 0 U i ] [U ∂Ω

as ρ → ∞,   g − [f ]∂Ω = Gi := −[∆(i/2) f ]∂Ω   0

(1.4.12) if i = 0 if i ∈ N is even if i ∈ N is odd.

(1.4.13)

The inner expansion of order 2M + 1 is obtained by truncating the (formal) sum ∞ i ρ, θ) after 2M + 1 terms and transforming back to (x, y)-coordinates i=0 ε Ui ( with the map ψ: 2M +1 2M +1 BL i −1 i u2M +1 (ρ, θ) := ε Ui ( ρ, θ) ◦ ψ = ε Ui (ρ/ε, θ) ◦ ψ −1 . (1.4.14) i=0

i=0

i do decay exponentially away It is not too diﬃcult to see that the functions U from ∂Ω. They have the form   i   i ( ρ, θ) = Θj (θ) ρj e−ρ , (1.4.15) U   j=0

where the functions Θj (θ) are smooth functions of θ. We note that uBL 2M +1 + w2M +1 = g

on ∂Ω for all M .

It can be shown that for each M there exists CM (depending on the data f and ∂Ω) such that

1.4 Review of existing results 2M +2 |Lε uBL 2M +1 | ≤ CM ε

13

in a neighborhood of ∂Ω.

This allows us to obtain error estimates for the diﬀerence between the exact solution uε and the expansion uBL 2M +1 + w2M +1 . Letting χ be an appropriate cut-oﬀ function supported by a neighborhood of ∂Ω that is identically one in a (smaller) neighborhood of ∂Ω, we can deﬁne the remainder r2M +1 via the following equation uε = wM + χuBL 2M +1 + r2M +1 . By construction, r2M +1 satisﬁes Lε r2M +1 = R2M +1

on Ω,

r2M +1 = 0

on ∂Ω,

where the residual R2M +1 satisﬁes (for some CM > 0 depending on the data f , ∂Ω) R2M +1 L∞ (Ω) ≤ CM ε2M +2 . Thus, by standard energy estimates, we get r2M +1 ε = ε∇r2M +1 L2 (Ω) + r2M +1 L2 (Ω) ≤ CM ε2M +2 .

(1.4.16)

Discussion of the example. The above example illustrates several points that are typically encountered when trying to describe solution behavior with the aid of asymptotic expansions. The ﬁrst point to note is the formal nature of the expansions: Asymptotic expansions such as (1.4.6), (1.4.11) are formal sums only and do in general not converge for a ﬁxed ε. To give meaning to such approximations, the error (in our example: r2M +1 ) has therefore to be estimated. The typical procedure is to derive an equation for the remainder and then to use a priori bounds for the solution operator of this equation. This was our procedure in the example and we arrived at (1.4.16), which justiﬁes the approximation by asymptotic expansions since for ﬁxed M and ε → 0, r2M +1 → 0. In practice, however, ε is given; then the question of the size of the bound CM ε2M +2 in (1.4.16) arises. In general, this bound diverges to ∞ as M → ∞ because asymptotic expansions are typically divergent sums. The implications of this fact are twofold: 1. In general, one has to expect that, for ﬁxed ε, the remainder r2M +1 ε can be bounded from below for all M . Thus, asymptotic expansions are only useful up to a certain error level, because the remainder r2M +1 cannot be made arbitrarily small. An implication is that expansion-based regularity assertions are not adequate for the description of the asymptotic behavior of a numerical method (such as the FEM), that is, they cannot describe its convergence behavior for ﬁxed ε and a large number of unknowns (i.e., small mesh size h or large polynomial degree p). In this asymptotic regime, other forms of regularity assertions are required. 2. Given that the remainder r2M +1 cannot be made arbitrary small for ﬁxed ε, the question arises of determining the “optimal” expansion order 2M + 1 for which the remainder is minimal. As the constant CM depends in an almost

14

1. Introduction

intractable way on higher order derivatives of the data f , ∂Ω, this optimal choice is a formidable task in general. For certain classes of data, however, nearly optimal choices of M can be computed. It was shown in [95] that for analytic data f and ∂Ω, the dependence of CM on M can be made explicit: CM = C(2M + 2)!γ 2M +2 for some C, γ > 0 independent of ε and M . For analytic data f , ∂Ω, the choice M ∼ ε−1 then minimizes the error bound CM ε2M +2 , which takes the form e−γ/ε for some γ > 0 independent of ε. The next point to note about the use of asymptotic expansions is that they require diﬀerentiability of the data. For example, from the representation of the outer expansions w2M +1 , we see that diﬀerentiability of order 2M is required of the right-hand side f . Likewise, the ability to deﬁne the inner expansion depends on diﬀerentiability properties of the parametrization of the boundary. This approach therefore fails for domains with non-smooth boundaries (such as polygons). The boundary layer expansions can of course be deﬁned wherever the boundary is smooth, that is, for each edge of the polygon separately. Near a vertex, however, two boundary layer expansions meet in an incompatible way. To remove these incompatibilities, new functions, called corner layers, have to be introduced. These corner layers are solutions of appropriate auxiliary problems on cones. Being solutions of elliptic problems over cones, the corner layers are not smooth, as we just saw in Section 1.4.1, and thus they are not easily handled in the framework of asymptotic expansions, which heavily depends on diﬀerentiability. These diﬃculties highlight the reason why asymptotic expansions for singular perturbation problems have typically been restricted to domains with smooth boundaries or particularly simple geometries such as the square. Regularity results for problems of the type (1.4.4) on the square can be found in [32, 67, 116]. The restriction to the square is largely due to the very special nature of the Laplace operator on that domain: The linear functionals alj of Proposition 1.4.1 are local in the sense that the functional alj depends on the behavior of the datum f at the vertex Aj only, i.e., on f (Aj ) and possibly higher derivatives Dα f (Aj ). This coincidence allows for formulating explicit compatibility conditions on the data that suppress certain singularities. In the context of regularity theory for singular perturbations, this is a common occurrence. Another diﬃculty encountered when trying to apply regularity results that include corner layers arises from the fact that these corner layers are typically deﬁned implicitly as the solutions of auxiliary problems. While such a deﬁnition is mathematically convenient and removes some of the technical diﬃculties in deﬁning expansions, it often has the disadvantage of not giving precise enough information about these corner layers for use in numerical schemes. Results for problems of the type (1.4.4) on general polygons are scarce. The most notable one for the application to ﬁnite element methods is in [74, 75]. However, even these results are not quite sharp enough for an application in low-order FEM as is discussed in [7, Sec. 25.1]. Furthermore, the results of [74,75] are cast in the classical framework of ﬁnite Sobolev regularity. The results are therefore

1.4 Review of existing results

15

not directly applicable to the hp-FEM, which requires control of all derivatives of the solution. 1.4.3 hp ﬁnite element methods Most methods for the numerical treatment of partial diﬀerential equations are based on approximating the sought solution by piecewise polynomials. In Finite Element Methods (FEM) and Finite Volume Methods, the piecewise polynomial approximation is done explicitly, while in ﬁnite diﬀerence schemes it is done implicitly. Many methods employed today are so-called low order methods, in which the underlying polynomial degree p is ﬁxed (typically p ≤ 2) and convergence is achieved by decreasing the mesh size h. These methods are also called h-methods and have algebraic rates of convergence at best; i.e., error bounds of the form O(N −α ) for some α > 0 are typical (here, N stands for the number of unknowns of the resulting linear system and is some measure for the work required to compute the numerical solution). On the other hand, high order methods such as the p- and hp-version ﬁnite element methods in solid mechanics (see, e.g., [25, 112] and the references there) and spectral methods/spectral element methods in ﬂuid mechanics (see, e.g., [33, 73] with their bibliographies) emerged in the early 1980s. In these methods, the polynomial degree p is increased while still keeping the option to perform mesh reﬁnement. These high order methods are most appreciated for their high accuracy, for their rates of convergence are often very high and can in certain cases even be exponential. Let us discuss the claim of high accuracy for high order methods and in particular high order FEM in more detail. The ability of (piecewise) polynomials to approximate a function that is analytic on an open neighborhood of the computational domain at an exponential rate has been known for a long time (see [40] for a nice exposition of this fact). However, most elliptic problems do not in practice have solutions that are analytic on such open neighborhoods. A large class of practical problems is posed on piecewise analytic domains in which case the solution becomes singular at certain boundary points only, as we saw in Section 1.4.1. In the context of the h-version on quasi-uniform meshes, the presence of these singularities downgrades the convergence rate. However, on so-called radical meshes, which are suitably reﬁned toward the singularities, it is possible to recover the optimal algebraic rate of convergence achievable by ﬁxed-order polynomial approximation: See the classical [21,106] for two-dimensional results and the recent [7] for the corresponding results in three dimensions. We point out that regularity results of the type presented in Proposition 1.4.1 contain suﬃcient information to design these radical meshes and are essential for the proof of the optimal rate of convergence. Corresponding algebraic convergence results hold for the p-version where the mesh is ﬁxed and the approximation order, i.e., the polynomial degree p, is increased. While the convergence rate of the p-version is still algebraic, it is twice that of the h-version (on quasi-uniform meshes) for problems with piecewise smooth input data, [22]. While both the h- and p-version achieve only algebraic rates of convergence in the presence of typical singularities, appropriate combinations of the two

16

1. Introduction

leads to exponential rates of convergence. In the hp-version of the ﬁnite element method, this combination occurs via mesh reﬁnement toward the singularities in conjunction with an increase of the polynomial degree in regions where the solution is smooth. The ﬁrst such exponential convergence result was obtained in [19, 20], where it was shown that functions from the countably normed space Bβ2 can be approximated at an exponential rate on so-called geometrically reﬁned meshes. The regularity assertion Proposition 1.4.2 shows that this situation is met in practice. Although the high accuracy of hp-ﬁnite element methods is their most striking feature, there are additional reasons for their increasing popularity, especially in solid mechanics. We mention here the issue of numerical locking in parameterdependent problems, where high order methods tend to be more robust than their low-order counterparts, [24]. The hardest problems in linear ﬁnite element analysis where locking problems are rampant are shell problems. Numerical evidence of [54, 66, 104] for cylindrical shells shows that high order methods are indeed much better suited for dealing with these parameter-dependent problems than low order methods. Similar observations about the eﬃciency of high order methods were made for plate problems in [69, 132]. 1.4.4 Numerical methods for singular perturbations As mentioned previously, singular perturbation problems arise in many practical applications. However, the problems vary greatly in structure and a correspondingly large number of diﬀerent numerical schemes has been devised—by far too large to be treated here in a comprehensive way, and the reader is therefore referred to the monographs [42,97,99,108] for an up-to-date overview. The presence of layers is a trademark of singularly perturbed problems. Numerical methods for such problems have therefore to address the issue of approximating such layers. Layer approximation in one dimension. One of the most notable features of singularly perturbed problems is the presence of layers. In the example (1.4.4) they appear as boundary layers. These boundary layers are given by (1.4.15) and can be written in the form S(ρ, θ)e−ρ/ε , where S is a smooth function. This structure has motivated many authors to approximate such functions on meshes that have tensor product structure and to obtain mesh design principles from the analysis of the simpler one-dimensional case, that is, the approximation of the function e−x/ε by piecewise polynomials on the interval (0, 1). It is clear that the mesh must depend on ε and that mesh points have to be condensed in a neighborhood of x = 0 in order to resolve the boundary layer function. The ﬁrst such adapted grid was proposed in [26], where essentially half the grid points are concentrated in an O(ε| ln ε|) neighborhood of the point x = 0 and the remaining half form a uniform grid on rest of (0, 1). We refer to [108, Sec. 2.4.2] for a nice exposition of these meshes and their many variations such as the Bakhvalov-type meshes of [127] and meshes of Gartland type, [51]. Closer to the type of meshes that we use for high order methods are so-called Shishkin meshes, [118], which are piecewise uniform meshes and again cluster half the grid points in a small

1.4 Review of existing results

17

neighborhood of the point x = 0. Speciﬁcally, for N ∈ N and a parameter τ > 0, these meshes are piecewise uniform meshes deﬁned by the set of nodes xi as follows: σ := min {1/2, τ ε ln N }, i xi := σ, i = 0, . . . , N, N i xN +1+i := σ + (1 − σ), i = 0, . . . , N. N The transition point σ, at which the mesh switches abruptly from a very ﬁne one to a coarse one is chosen such that the boundary layer function e−x/ε is small on the coarse grid. In fact, it is not diﬃcult to see that the diﬀerence between u = e−x/ε and the piecewise linear interpolant Iu on the Shishkin grid satisﬁes bounds of form u − IuL∞ (0,1) ≤ Cτ N −1 ln N for each ﬁxed τ ≥ 1. The constant Cτ depends on τ but is independent of ε and N . Similar approximation results hold, of course, for other norms as well. The analog of this Shishkin mesh in the p-version FEM is the so-called “twoelement” mesh of [113, 114]. For polynomial degree p, it is deﬁned by the three nodes x0 = 0, x1 = min {1/2, τ pε}, x2 = 1, where τ > 0 is a ﬁxed parameter of size O(1). Approximation of the boundary layer function u = e−x/ε by piecewise polynomials of degree p on this mesh can be achieved at a robust exponential rate; in other words, the rate is exponential (in p) and all constants are independent of ε, [113]. To illustrate this robust exponential approximability result, we point to the fact that it is not diﬃcult to construct, for each 0 < τ < e−1 , a continuous, piecewise polynomial approximation Iu satisfying u − IuL∞ ((0,1)) ≤ Cτ [(τ e)p + e−τ p ], where Cτ > 0 depends on τ but is independent of ε and p. As in the case of Shishkin meshes, similar results can be obtained for the approximation in other norms as well. The design of meshes and grids for the approximation of boundary layer functions in higher dimensions is guided by these ideas obtained from analyzing one-dimensional examples. Typically, the meshes are obtained through tensor product constructions using highly reﬁned meshes of Bakhvalov or Shishkin type in the direction normal to the boundary and quasi-uniform meshes in the tangential direction. We refer to [7, 107] for good surveys on recent results concerning the use of such layer-adapted anisotropic grids. Numerical methods for (1.2.11). The key properties of a numerical method are stability and approximability. The design of stable methods for non-symmetric problems (arising, for example, in convection-dominated ﬂuid ﬂow problems) is non-trivial and currently an active and extensive research area. In the present subsection, we restrict our discussion to numerical methods for symmetric problems of type (1.2.11). Typical FEM or ﬁnite diﬀerence discretizations are stable due to the symmetric positive-deﬁnite structure of the problem, and numerical analysis of this problem class is thus reduced to an approximability problem.

18

1. Introduction

In the literature, equations (1.2.11) are typically simpliﬁed further, and the following model problem is analyzed: −ε2 ∆uε + uε = f

on Ω,

uε = 0

on ∂Ω.

(1.4.17)

We now review existing work for the numerical solution of problems of type (1.4.17). An early low order ﬁnite element analysis was performed by [109] on quasi-uniform meshes. In the practical regime of mesh sizes h >> ε, the boundary layer eﬀects near ∂Ω cannot be captured, and the authors therefore focussed on local error estimates away from the boundary. Methods devised to capture the boundary layer eﬀects were presented in [117, 119] in a ﬁnite diﬀerence context on Shishkin-type meshes and in [8, 29, 84] in a ﬁrst order ﬁnite-element context on Bakhvalov-type and Shishkin type meshes. Typically, these results were obtained on a special domain, the square, because their convergence proofs hinge on decompositions of the solution into smooth parts, boundary layer parts, and possibly corner layer parts. However, due to a lack of suﬃciently precise knowledge of the regularity of the corner layers, their approximation is often avoided by stipulating compatibility conditions on the datum f so as to suppress corner layers or to show that they are suﬃciently small. The ability to formulate such compatibility conditions rests on the fact that for the very special case of a square, the linear functionals describing the corner singularities are local. [7] gives results that are not restricted to a square by basing its convergence proof on some regularity assumptions. As pointed out in [7, Sec. 25.1], these assumptions are not rigorously proved. The present work provides a complete regularity analysis of the solutions of (1.4.17) that is suited for polynomial approximation results. Although we apply our regularity results to high order approximations, they could also be employed to rigorously establish these low order approximation results. We present such an approach in Sec. 2.6. While a fair body of literature is available for lower order methods on meshes that can capture the boundary layer behavior of the solution, the situation is not so well developed for spectral and hp-methods. The ﬁrst result in one dimension appears to be [113,114]. There, a two-element mesh was designed that allowed for the robust exponential approximation of boundary layer functions. This result, however, applies to the case of a constant-coeﬃcients diﬀerential equation only. The ﬁrst robust exponential approximation result in one dimension for general analytic input data (coeﬃcients of the diﬀerential equation and right-hand side) is [92]. The ﬁrst extension to two dimensions was done in [131, 133, 135]. On smooth domains, boundary-ﬁtted tensor product meshes were employed and robust convergence of arbitrary order was proved for smooth input data. [131, 134] also considered the case of a square with simple right-hand side f for which, again due to the special nature of the domain and the data, robust convergence of arbitrary order was obtained. The ﬁrst robust exponential convergence result for analytic input data f and ∂Ω was obtained in [94,95]. A further improvement of [94] over existing high order approximation results is that the meshes do not have to be boundary-ﬁtted tensor product meshes. This is important in practice since in the presence of curved elements, boundary-ﬁtted tensor product meshes

1.5 Outline of the book

19

are not easily generated. The present work extends these results to the case of general curvilinear polygons and arbitrary analytic right-hand side.

1.5 Outline of the book The book comprises three parts, each contributing to the main result of the work, the proof of robust exponential convergence of a properly designed hp-FEM for (1.2.11). While Part I focuses on hp-approximation on anisotropic meshes, Parts II and III provide two diﬀerent types of regularity assertions for the solution that are required for proving robust exponential convergence. Chapter 2 serves as an overview chapter to make the main results of this work more accessible. The main regularity assertions for solutions uε of (1.2.11), rigorously established in Parts II, III, are collected in Section 2.3: Theorem 2.3.1 extracts from Part II the assertion that uε is an element of a countably normed 2 ; Theorem 2.3.4 consolidates the analysis of Part III concerning the space Bβ,ε regularity of the description of uε by means of asymptotic expansions. In this overview chapter, we include the hp-FEM approximation result Theorem 2.4.8 together with numerical examples in Section 2.5 illustrating the robust exponential convergence result. In order to motivate the regularity assertions Theorems 2.3.1, 2.3.4, we begin Chapter 2 with a discussion of the one-dimensional case. Also key features of hpapproximation on meshes that are suitable for the resolution of layer phenomena are examined in this one-dimensional setting and may serve as a motivation for the two-dimensional boundary layer meshes of Deﬁnition 2.4.4. Part I: Finite element approximation. The main result of Part I is the robust exponential convergence result Theorem 2.4.8 for the boundary value problem (1.2.11). This result follows from the approximation result Theorem 3.4.8, where an approximant is explicitly constructed. Chapter 3 embeds Theorem 3.4.8 in a general discussion of hp-approximation on non-aﬃne, anisotropic meshes. An important point to note is that the meshes for which robust exponential convergence is proved, the admissible boundary layer meshes of Deﬁnition 2.4.4, are essentially the minimal meshes with that property. The application of the regularity assertions Theorems 2.3.1, 2.3.4 is not limited to the hp-FEM; we show in Section 2.6 that, based on these regularity assertions, robust algebraic convergence can be established for piecewise linear ansatz functions on Shishkin meshes. Part II: Regularity in countably normed spaces. Part II is concerned with regularity assertions for solutions of (1.2.1) in countably normed spaces. The key feature is that the dependence on the parameter ε is made explicit. l,m l In Chapter 4 the weighted spaces Hβ,ε and the countably normed spaces Bβ,ε of analytic functions are introduced and their main properties are proved. The weights are functions of the perturbation parameter ε, the distance to the nearest vertex, and the diﬀerentiation order. The main result of Chapter 4 is Theol rem 4.2.20, which states that membership in the countably normed spaces Bβ,ε

20

1. Introduction

is invariant under analytic changes of variables. This property proves very useful in inferring regularity results for domains with curved boundaries. The spaces l Bβ,ε are deﬁned through an L2 -based control of all derivatives. Pointwise control l of all derivatives for functions from the spaces Bβ,ε is given in Theorem 4.2.23. In Chapter 5 two kinds of regularity assertions for the solutions of (1.2.1) are proved. The ﬁrst regularity statement is Theorem 5.3.8, which is a shift theorem in weighted spaces for (1.2.1). When applied to (1.2.11), it yields that for 0,0 2,2 right-hand sides from the spaces Hβ,ε , the solution is in Hβ,ε . Theorem 5.3.10 represents a second kind of regularity result, where for right-hand sides from countably normed spaces, the solution of (1.2.1) is asserted to be in a countably normed space as well. For example, for (1.2.11), membership of the right-hand 0 2 side in Bβ,ε implies that the solution is an element of Bβ,ε . Section 5.3 discusses the case of Dirichlet boundary conditions. The techniques employed are emenable to other kinds of boundary conditions: We analyze Neumann and Robin boundary conditions in Section 5.4 and additionally discuss a transmission problem. Part III: Regularity in terms of asymptotic expansions. In Part III, we describe the regularity of the solutions u of (1.2.11) by means of asymptotic expansions. The deﬁnition of the so-called corner layer is done with the aid of a transmission problem with exponentially decaying data. Analytic regularity results for such transmission problems are therefore provided in Chapter 6. These regularity assertions are utilized in Chapter 7 to obtain analytic regularity results for the corner layers in the asymptotic expansions. Theorem 6.4.13 and Corollary 6.4.14 synthesize the results of Chapter 6: Theorem 6.4.13 shows that the solutions of the transmission problems considered are in exponentially weighted countably normed spaces, and Corollary 6.4.14 gives pointwise estimates for the growth of the derivatives of these solutions.

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

This part of the book is devoted to the ﬁnite element approximation to solutions of (1.2.11). The principal aim of the present Chapter 2 is the robust exponential convergence result Theorem 2.4.8, which is illustrated by numerical examples in Section 2.5. Essential for this robust exponential convergence result are detailed regularity assertions for the solution. For the convenience of the reader, the present chapter collects from Parts II, III the regularity results that are required for the proof of Theorem 2.4.8. The proofs of both the approximation result and the regularity assertions are very technical and therefore not included in this chapter. In order to motivate the two-dimensional results of this chapter, we present the analogous results in the one-dimensional setting in Section 2.2. Technically, this setting is considerably simpler than the two-dimensional case, yet it exhibits many features that are relevant for the two-dimensional case. We conclude this chapter with a discussion of a low-order method in Section 2.6, since the regularity assertions of Section 2.3 can also be employed to prove robust convergence of the h-FEM on Shishkin meshes.

2.1 Setting and Introduction We consider the singularly perturbed problem (1.2.11) with b ≡ 0, i.e., −∇ · (A(x)∇u) + c(x)u = f u=g

in Ω, on ∂Ω.

(2.1.1a) (2.1.1b)

The coeﬃcients A ∈ A(Ω, S2> ), c ∈ A(Ω), and the right-hand side f are assumed to satisfy (1.2.2), (1.2.3), i.e., p ∇p AL∞ (Ω) ≤ CA γA p! ∇p cL∞ (Ω) ≤ Cc γcp p!

∀p ∈ N0 , ∀p ∈ N0 ,

A(x) ≥ λmin > 0 ∀x ∈ Ω, c(x) ≥ µ > 0 ∀x ∈ Ω, ∇p f L∞ (Ω) ≤ Cf γfp p!

∀p ∈ N0 .

(2.1.2a) (2.1.2b) (2.1.2c) (2.1.2d) (2.1.2e)

The boundary ∂Ω is assumed piecewise analytic as described in Section 1.2 and the Dirichlet data g are also piecewise analytic in the sense of (1.2.4). Solutions uε are understood in the weak sense, i.e., uε ∈ H 1 (Ω) solves (2.1.1) if

J.M. Melenk: LNM 1796, pp. 23–72, 2002. c Springer-Verlag Berlin Heidelberg 2002

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

24

uε |∂Ω = g where

and

Bε (uε , v) = F (v) ∀v ∈ H01 (Ω),

(2.1.3)

(A(x)∇u) · ∇v + c(x)uv dx,

Bε (u, v) = ε2

F (v) =

Ω

f (x)v dx. (2.1.4) Ω

The energy norm · ε is deﬁned as in (1.2.7), i.e., u2ε = ε2 ∇u2L2 (Ω) + u2L2 (Ω) .

(2.1.5)

Due to the assumptions on the coeﬃcients A, c, we have min {λmin , µ}u2ε ≤ Bε (u, u) ≤ max {AL∞ (Ω) , cL∞ (Ω) }u2ε

∀u ∈ H 1 (Ω).

Lemma 1.2.2 gives the existence and uniqueness of the solution uε to (2.1.1). As described in Section 1.1, the solution uε is analytic on Ω but has boundary layers near the boundary ∂Ω and corner singularities/corner layers near the vertices of the curvilinear polygon Ω. Theorems 2.3.1, 2.3.4 below collect the main regularity properties of the solution uε , which will enable us to prove robust exponential convergence of the hp-version of the ﬁnite element method applied to problems of type (2.1.1).

2.2 Prelude: the one-dimensional case Many features of both regularity theory and high order approximation can be seen more clearly in one dimension. The present section is therefore devoted to an overview of the results in this considerably simpler setting. 2.2.1 Regularity in one dimension To illustrate the main points of the regularity results of Theorems 2.3.1, 2.3.4, we recall a one-dimensional result. Proposition 2.2.1. Let Ω = (−1, 1) and let c, f be analytic 1 on Ω with c ≥ c > 0 on Ω. Then there exist C, γ, α > 0 depending only on c, f such that for every ε ∈ (0, 1] the solution uε to −ε2 uε (x) + c(x)uε (x) = f (x)

in Ω,

uε (±1) = 0

(2.2.1)

∀n ∈ N0 .

(2.2.2)

satisﬁes: 1. uε is analytic on Ω and −1 n u(n) } ε L2 (Ω) ≤ C max {n + 1, ε 1

A function g : Ω → C is said to be analytic on the closed set Ω if there exists an → C with G| = g. of Ω and an analytic function G : Ω open neighborhood Ω Ω

2.2 Prelude: the one-dimensional case

25

2. For each ε ∈ (0, 1], the solution uε can be decomposed as + rε , uε = wε + uBL ε

(2.2.3)

where, upon setting ρ(x) := dist(x, ±1), there holds for all n ∈ N0 wε(n) L∞ (Ω) ≤ Cγ n n!, BL (n) (x) ≤ Cγ n max {n + 1, ε−1 }n exp(−αρ(x)) uε

∀x ∈ Ω,

rε H 1 (Ω) ≤ C exp(−α/ε). Furthermore, rε (±1) = 0, rε is smooth, and for each k ∈ N0 , there exist Ck , αk > 0 such that rε(k) L∞ (Ω) ≤ Ck exp(−αk /ε),

k = 0, 1, . . . . 2

Proof: See [92] and the proof of Lemma 7.1.1.

The decomposition (2.2.3) is obtained with the aid of the classical asymptotic expansions. It captures the main solution components: a smooth part wε , which is even analytic, and a boundary layer part uBL that decays very fast away from ε the boundary points ±1. The remainder rε is ascertained to be small. For the application of the FEM, we will design the mesh such that the smooth part wε and the boundary layer part uBL can be approximated well. The small remainε der rε is simply approximated by zero. However, for ﬁxed ε, the remainder rε is small but ﬁnite. Thus, for FEM approximations with a required accuracy below O(exp(−α/ε)), the decomposition (2.2.3) cannot be employed; instead, the assertion (2.2.2) concerning the growth of the derivatives is employed. In this regime of accuracy, the bounds (2.2.2), although strongly ε-dependent, are suﬃcient. We will illustrate the interplay of these two types of regularity assertions in Section 2.2.2 below for the one-dimensional case; an analogous interplay takes place in the two-dimensional situation. Remark 2.2.2 For the case of convection-diﬀusion equations, i.e., solutions to the equation −ε2 u (x) + b(x)u (x) + c(x)u(x) = f (x), the analog of Proposition 2.2.1 is available in [96]. 2.2.2 hp-FEM in one dimension We illustrate the hp-version of the FEM in the one-dimensional setting ﬁrst, because some important features of the hp-FEM, in particular for the approximation of boundary layers, can already be seen in one dimension. Abstract FEM in one dimension. We consider the approximation of the solution to (2.2.1). Upon writing Ω = (−1, 1), the weak formulation is: Find uε ∈ H01 (Ω) s.t. Bε (uε , v) = F (v) ∀v ∈ H01 (Ω),

(2.2.4)

where the bilinear form Bε and the right-hand side functional F are given by

26

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

Bε (u, v) =

ε2 u v + c(x)uv dx,

F (v) =

Ω

f (x)v dx. Ω

For arbitrary subspace VN ⊂ H01 (Ω), dim VN = N < ∞, the FEM reads: Find uN ∈ VN s.t. Bε (uN , v) = F (v) ∀v ∈ VN .

(2.2.5)

By C´ea’s Lemma, uN exists and is the best-approximant in the energy norm · ε , i.e., uε − uN ε = inf uε − vε . v∈VN

Here, the energy norm is given by u2ε = Bε (u, u) ∼ εu 2L2 (Ω) + u2L2 (Ω) . hp-FEM in one dimension. In the hp-version of the FEM in one dimension, the spaces VN are spaces of piecewise polynomials. Let −1 = x0 < x1 < · · · < xL = 1 be mesh points and deﬁne the intervals Ii := (xi−1 , xi ), i = 1, . . . , L. The mesh T is then deﬁned as T = {Ii | i = 1, . . . , L}. For a mesh and a polynomial degree p ∈ N, we deﬁne the hp-FEM spaces S p (T ), S0p (T ) as S p (T ) := {u ∈ H 1 (Ω) | u|Ii is a polynomial of degree p for each i = 1, . . . , L}, S0p (T ) := S p (T ) ∩ H01 (Ω). It is easy to check that dim S0p (T ) = pL − 1. Remark 2.2.3 Analogously to the procedure in two dimensions, one could deﬁne the space S p (T ) as S p (T ) = {u ∈ H 1 (Ω) | u ◦ Mi ∈ Pp ,

i = 1, . . . , L},

where Pp is the space of polynomials of degree p and the element maps Mi are given by Mi : I → ii ξ → xi−1 + hi ξ,

hi = xi − xi−1

for the reference interval I = (0, 1). We now focus on the question of how to choose the mesh T for the approximation of the solution uε of (2.2.1). Our aim is to ﬁnd meshes T with as few elements as possible. For problems with boundary layers such as (2.2.1), [92, 113, 131] have proposed and analyzed the so-called boundary layer mesh, which yields robust exponential convergence with meshes consisting of three, judiciously chosen elements. We deﬁne boundary layer meshes as follows. Deﬁnition 2.2.4 (boundary layer mesh). For κ > 0 the boundary layer mesh Tκ is deﬁned by the points x0 = −1,

x1 = −1 + min {0.5, κ},

x2 = 1 − min {0.5, κ},

x3 = 1.

2.2 Prelude: the one-dimensional case

27

Boundary layer meshes are suitable for the approximation of problems with boundary layers as is shown in the following proposition. Proposition 2.2.5. Let c, f satisfy the assumptions of Proposition 2.2.1. Denote by uε the solution to (2.2.1). Then there exist C, b, λ0 > 0 independent of ε such that for every λ ∈ (0, λ0 ) and p ∈ N0 inf

v∈S0p (Tλpε )

uε − vL∞ (Ω) + λpε(uε − v) L∞ (Ω) ≤ Ce−bλp .

In particular, therefore, if uN denotes the hp-FEM solution of (2.2.5) based on VN = S0p (Tλpε ), then there exist for each λ ∈ (0, λ0 ) constants C, b > 0 independent of ε, p such that

uε − uN ε ≤ Ce−b N ,

N = dim S0p (Tλpε ).

Proof: This result was ﬁrst obtained in [92]. We will, however, prove it here because the ideas will be encountered again in the proof of the technically more involved two-dimensional case, Theorem 2.4.8. The required approximants to uε are constructed with the aid of the piecewise Gauss-Lobatto interpolation operator, ip,T . Speciﬁcally, for a mesh T = {Ii | i = 1, . . . , L} we associate a polynomial degree vector p := (pi )L i=1 ⊂ {1, . . . , p} and set ip,T : C(Ω) → S p (T ) u → ip,T u where

ip,T u|Ii = (ipi (u ◦ Mi )) ◦ Mi−1

∀Ii ∈ T ;

here, Mi : I → Ii is the element map of element Ii and ipi denotes the GaussLobatto interpolation operator ipi : C(I) → Ppi on the reference element. The essential properties of the Gauss-Lobatto operator ipi can be found in Lemma 3.2.1 (stability) and Lemma 3.2.6 (approximation of analytic functions). The proof employs both regularity assertions of Proposition 2.2.1. Step 1: We start with the case λpε ≥ 1/2. In this case, the boundary layer mesh Tλpε consists of the three elements I1 = (−1, −0.5), I2 = (−0.5, 0.5), I3 = (0.5, 1). From the regularity assertion (2.2.2) the pull-backs u ˆi := uε |Ii ◦ Mi satisfy for some C, γ > 0 independent of ε ˆ ui L2 (I) ≤ Cγ n max {n + 1, ε−1 }n (n)

∀n ∈ N0 .

Hence, employing the Sobolev embedding L∞ (I) ⊂ H 1 (I) ˆ ui L∞ (I) ≤ Cγ n max {n + 1, ε−1 }n+1 (n)

∀n ∈ N0 ,

(2.2.6)

for some suitably chosen C, γ > 0. The term max {n + 1, ε−1 }n+1 is brought to a more familiar form by noting max {n + 1, ε−1 }n+1 = max {(n + 1)n+1 , ε−(n+1) } (n + 1)n+1 ε−(n+1) , = (n + 1)! max (n + 1)! (n + 1)! = (n + 1)! max en+1 , e1/ε ≤ C (n + 1)! e1/ε ,

(2.2.7)

28

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

where we used ε ∈ (0, 1] and Stirling’s formula in the form nn ≤ n!en . Inserting this result in (2.2.6) we obtain after appropriately adjusting C, γ > 0 (n)

ˆ ui L∞ (I) ≤ Ce1/ε γ n n!

∀n ∈ N0 .

From Lemma 3.2.6 (setting Cu = Ce1/ε ), we then obtain

ˆ ui − ip u ˆi L∞ (I) + (ˆ ui − ip u ˆi ) L∞ (I) ≤ Ce1/ε e−bp for some C, b > 0 independent of C, p. Mapping back to the elements Ii , we obtain using λpε ≥ 0.5 and p = (p, p, p)

uε −ip,T uε L∞ (Ω) + (uε − ip,T uε ) L∞ (Ω) ≤ Ce1/ε e−bp ≤ Ce(2λ−b)p ≤ Ce−(b/2)p provided that λ < λ0 ≤ b/4. Step 2: For the case λpε < 0.5, we employ the decomposition (2.2.3). Denoting the pull-back of the smooth part wε to the reference element I by w ˆi := wε |Ii ◦ Mi , we have (n)

w ˆi L∞ (I) ≤ C(γhi )n n!

∀n ∈ N0 ,

where hi = |Ii | is the length of Ii . Employing Lemma 3.2.6 and mapping back to the physical elements Ii , we get

wε − ip,T wε L∞ (Ω) + (wε − ip,T wε ) L∞ (Ω) ≤ Ce−bp for some C, b > 0 independent of ε and p, where we set p = (p, p, p). Thus, the smooth part wε can be approximated in the desired fashion. We also note that, since the endpoints are sampling points for the Gauss-Lobatto interpolation operator, wε (±1) = ip,T wε (±1). We now turn to the approximation of the boundary layer part uBL by ip,T uBL ε ε , BL ◦ where we choose p = (p, 1, p). Again, we introduce the pull-backs u ˆi := uBL ε Mi . Using the results of Proposition 2.2.1, we have on I1 = (−1, −1 + λpε) BL (n) u ˆ1 L∞ (I) ≤ Cγ n max {n + 1, ε−1 }n (λpε)n

∀n ∈ N0 .

In the same way as in (2.2.7), we can estimate max {n + 1, ε−1 }n (λpε)n ≤ e n!

e n 2

eλp .

Thus, Lemma 3.2.6 (with Cu = Ceλp ) implies the existence of C, b > 0 independent of ε, p such that BL − ip u ˆ uBL ˆBL ˆ1 − ip u ˆBL L∞ (I) ≤ Ceλp e−bp . 1 1 L∞ (I) + u 1 Again, assuming λ < λ0 ≤ b/2, we obtain after mapping back to I1 : BL uBL − ip,T uBL − ip,T uBL L∞ (I1 ) ≤ Ce−(b/2)p . ε ε L∞ (I1 ) + λpε uε ε

2.2 Prelude: the one-dimensional case

29

By symmetry, the same estimate holds on I3 = (1−λpε, 1). It remains to consider is small. I2 = (−1 + λpε, 1 − λpε). On I2 , we merely exploit the fact that uBL ε BL Speciﬁcally, since i1,T uBL reduces to the linear interpolant of u on the interval ε ε I2 and the length of I2 is bounded by 1 ≤ |I2 | < 2, we have BL i1,T uBL L∞ (I2 ) ε L∞ (I2 ) + i1,T uε BL BL ≤ C uε (−1 + λpε) + uε (1 − λpε) ≤ Ce−αλp . Hence we can bound BL BL uBL − i1,T uBL ε ε L∞ (I2 ) ≤ uε L∞ (I2 ) + i1,T uε L∞ (I2 )

BL

λpε uBL − i1,T uε ε

L∞ (I2 )

≤ Ce−αλp , ≤ λpε uBL L∞ (I2 ) + λpε i1,T uBL L∞ (I2 ) ε ε ≤ Cλpe−αλp .

BL By construction, we have uBL ε (±1) = ip,T uε (±1). It remains to approximate rε . We simply approximate rε by zero, since

rε L∞ (Ω) + rε L∞ (Ω) ≤ Ce−α/ε by Proposition 2.2.1. Our approximant coincides with uε in the endpoints ±1 and has the desired approximation properties. 2 Remark 2.2.6 From an application point of view, the draw-back of Proposition 2.2.5 is that λ0 , while independent of ε, is essentially left unspeciﬁed. In actual calculations, when good choices of λ are not available, one might use the following two ideas: 1. Since the dependence on λ is made explicit in the approximation result Proposition 2.2.5, it is possible to choose λ as a function of p; choosing for example λ = 1/ ln p in the deﬁnition of the boundary layer mesh gives for the hp-FEM approximation uε − uN ε ≤ Ce−bp/ ln p , with C, b independent of ε, p. 2. In view of C´ea Lemma, one could consider the approximation from spaces VN ⊃ S0p (Tλpε ). Good choices are meshes that are geometrically reﬁned towards the endpoints ±1. Speciﬁcally, for a grading factor q ∈ (0, 1) and a number of layers L ∈ N, we deﬁne a geometric mesh TLgeo by the points {−1, 1} ∪ {−1 + q i | i = 1, . . . , L} ∪ {1 − q i | i = 1, . . . , L}. If L is chosen such that q L ≈ ε, then it is not diﬃcult to see that inf

v∈S0p (TLgeo )

u − vε ≤ Ce−bp

30

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

for some C, b > 0 independent of ε and p. The condition q L ≈ ε implies L = O(| ln ε|) and, since dim VN = (2L + 1)p − 1, we obtain therefore in terms of the number of degrees of freedom N = dim VN inf

v∈S0p (TLgeo )

u − vε ≤ Ce−bp/| ln ε| .

While the hp-FEM on such a geometrically reﬁned mesh with O(| ln ε|) elements is not robust in a strict sense, the ε-dependence is very weak. Note that a geometric mesh is shape-regular in the usual sense that neighboring elements are comparable in size, in contrast to the boundary layer mesh.

2.2.3 Numerical examples Proposition 2.2.5 shows that the robust exponential convergence in the energy norm can be achieved on boundary layer meshes. The following numerical example shows that a) this robust exponential convergence is achieved in computational practice and that b) the small elements of size O(pε) in the layer are necessary to resolve the layers. Example 2.2.7 We consider the approximation to the solution of −ε2 uε (x) + uε (x) =

1 2 − x2

on Ω = (−1, 1),

uε (±1) = 0.

(2.2.8)

We apply the hp-FEM (2.2.5) based on the boundary layer mesh of Proposition 2.2.5. In the deﬁnition of the boundary layer mesh Tλpε we take λ = 0.71, remarking in passing that a more careful analysis in [92, 113] shows that in the present case, λ0 = 4/e. We have dim S0p (Tλpε ) = 3p − 1. For ε = 10−2 , ε = 10−4 , ε = 10−6 , ε = 10−8 , the left graph in Fig. 2.2.1 shows the error in energy (i.e., the square of the energy norm (2.1.5)) vs. dim S0p (Tλpε ) = 3p − 1. In this linearlogarithmic plot, we observe robust exponential convergence, as the curves are almost straight lines and very close to each other, in spite of the very wide range of parameter values ε. To show that the small elements of size O(pε) are necessary to achieve robust exponential convergence, we consider the performance of the p-version FEM on a single element in the right graph in Fig. 2.2.1. The p-version FEM is deﬁned by (2.2.5) with VNp-FEM = S0p (T p-FEM ),

T p-FEM = {Ω}.

We note that dim VNp−F EM = p − 1. Again, calculations are performed for ε = 10−2 , ε = 10−4 , ε = 10−6 , ε = 10−8 and shown in the right part of Fig. 2.2.1. We observe very poor convergence for small ε (the error curves corresponding to ε = 10−4 , ε = 10−6 , ε = 10−8 are practically on top of each other). In fact, in the regime of polynomial degrees shown (p ≤ 40), robust exponential convergence

2.3 Regularity: the two-dimensional case

31

√ −1

is not visible but only robust convergence O(p 1 + ln p), [113]. We therefore conclude that the small elements of size O(pε) in the layer are necessary for good performance of the hp-FEM. Example 2.2.8 In the preceding example, we saw that the small elements of size O(pε) in the layer are necessary for robust exponential convergence. In this example, we show that they may not be chosen too small. In fact, Proposition 2.2.5 suggests a deterioration of the approximation if λ → 0. We now show numerically that such a deterioration arises indeed in computations. We consider the equation −ε2 uε + uε = 1

on (0, 1),

uε (0) = 0,

uε (1) = 0

(2.2.9)

with solution uε (y) is given by uε (x) = 1 −

cosh ((1 − x)/ε) . cosh(1/ε)

We consider the p-version FEM based on a two-element mesh determined by the points 0 = y0 < y1 = aε < y2 = 1. The performance for ε = 10−3 and various choices of the parameter a is reported in Fig. 2.2.2. For ﬁxed a, we note an initial exponential convergence which deteriorates if p becomes large. In fact, the exponential rate of convergence is visible until p ≈ a. For p > a, the large element (which is aε away from the boundary point y = 0) dominates the overall possible error reduction. This can be seen as follows. As the boundary layer function in this particular case is essentially e−y/ε , the function to be approximated on the large element is e−a e−(y−aε)/ε . For small ε polynomial approximation of e−(y−aε)/ε on the element (aε, 1) is quite poor as we have seen in Example 2.2.7, and the factor e−a is comparatively large if a is small (relative to p). However, if a is large (a ≥ p, say), then the boundary layer function e−y/ε on the large element is exponentially small (in p), and thus the contribution of the large element to the total error as well. We conclude therefore that for ﬁxed a, the error on the large element is negligible for p < a, and the global error reduction is controlled by the error on the small element. In the regime p > a, the error on the large element dominates the global error. The choice of a variable mesh, i.e., taking a = p balances the two errors; we see in Fig. 2.2.2 that this choice allows us to obtain exponential convergence.

2.3 Regularity: the two-dimensional case Our regularity assertions in the two-dimensional setting take a form similar to that of Proposition 2.2.1; that is, the regularity is described both in terms of bounds on the growth of the derivatives and in terms of decompositions that capture the main features of the solution. In the one-dimensional case of Proposition 2.2.1, the data c and f are smooth. In the two-dimensional case of (2.1.1),

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

32

HP−VERSIONS (3 elements), λ = 0.71

−2

10

ε =10^(−2) ε =10^(−4) ε =10^(−6) ε =10^(−8)

−4

10

−6

−1

10

−2

10

10

Relative Error in Energy

Relative Error in Energy

P−VERSION (1 element)

0

10

−8

10

−10

10

−3

10

−4

10

−5

10

−6

10

ε =10^(−2) ε =10^(−4) ε =10^(−6) ε =10^(−8)

−7

10 −12

10

−8

10 −14

10

5

−9

10

15

20

25 30 Degrees of Freedom

35

40

45

50

10

0

5

10

15 20 25 Degrees of Freedom

30

35

40

Fig. 2.2.1. Example 2.2.7: hp-FEM (left) and p-FEM (right) for a problem with boundary layers. p versions (2 elements);

−3

10

ε =10^(−3)

−4

10

−5

Rel. Error in Energy

10

−6

10

−7

10

−8

10

a=2 a=4

−9

10

a=6 a=8

−10

10

a=p

−11

10

0

5

10

15 20 25 Degrees of Freedom

30

35

40

Fig. 2.2.2. Example 2.2.8 p-version for 1D example and various values of a; ε = 10−3 .

part of the data, namely, the boundary ∂Ω is only piecewise analytic. This introduces singularities into the solution. Thus, boundary layers as well as corner singularities have to be captured. The analog of the assertion on the growth of the derivatives of Proposition 2.2.1 is therefore replaced with estimates in weighted L2 -spaces. The analog of the decomposition of Proposition 2.2.1 includes an additional term that captures the corner singularities. In order to be able to formulate the regularity assertions of Theorem 2.3.1, we need to introduce some notation. First we introduce the weight functions Φp,β,ε as follows. With each vertex Aj , j = 1, . . . , J, we associate a number βj ∈ [0, 1), set β = (β1 , . . . , βJ ), and write for p ∈ N0 ˆp,β ,ε (x) := Φ j

min

1,

dist(x, Aj ) min {1, ε(p + 1)}

p+βj .

2.3 Regularity: the two-dimensional case

33

The weight function Φp,β,ε is then deﬁned as Φp,β,ε (x) :=

J

ˆp,β ,ε (x). Φ j

j=1

Using the weight functions Φp,β,ε we can formulate the two-dimensional analog of (2.2.2): Theorem 2.3.1 (Regularity in countably normed spaces). Let Ω be a curvilinear polygon and A, c, f , g satisfy (2.1.2), (1.2.4). Then there exist C, K > 0, and a vector β ∈ [0, 1)J independent of ε such that the solution uε of 2 (2.1.1) satisﬁes uε ∈ Bβ,ε (Ω, C, K), i.e., Φp,β,ε ∇

p+2

uε ε ≤ C uε L2 (Ω) ≤ CK p max {p + 1, ε−1 }p+2

∀p ∈ N0 .

In particular, for ﬁxed neighborhoods Uj of the vertices Aj there holds for all p ∈ N0 and all x ∈ Uj ∩ Ω r 1−βj rj p+1 j |∇p (u(x) − u(Aj )) | ≤ CK p ε−1 min 1, rj−p max p + 1, , ε ε where rj := dist(x, Aj ). In the interior Ω \ ∪Jj=1 Uj there holds |∇p u(x)| ≤ CK p max {(p + 1), ε−1 }p+2

∀p ∈ N0

∀x ∈ Ω \ ∪Jj=1 Uj .

Proof: The energy estimate uε ε follows from Lemma 1.2.2. The L2 -based bound for higher derivatives of uε is proved as Theorem 5.3.14, which asserts 2 in particular that uε ∈ Bβ,ε . The pointwise bounds are then obtained from Theorem 4.2.23. An outline for the key ingredients of the proof can be found in Section 5.1. 2 Remark 2.3.2 1. Theorem 2.3.1 merely asserts the existence of β ∈ (0, 1)J . The values of βj , however, are available from the proof. For A = Id, for example, one may choose any βj ∈ (0, 1) ∩ (1 − π/ωj , 1), where ωj is the interior angle of the curvilinear polygon Ω at vertex Aj . 2. The assumption that f is analytic up to the boundary ∂Ω can be relaxed. 0 Theorem 2.3.1 still holds true if f ∈ Bβ,ε , i.e., Φp,β,ε ∇p f L2 (Ω) ≤ Cf γfp p! for all p ∈ N0 and suitable constants Cf , γf > 0. 3. The assumption that c, f are real-valued is not essential; see Remark 5.3.5. 4. Theorem 2.3.1 can also be formulated for solutions to the diﬀerential equation (1.2.1) with b = 0. We refer to Theorem 5.3.14, where this case of covered.

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

34

5. The coeﬃcients A, c, and the right-hand side f are assumed analytic on Ω. This is done for simplicity of notation. The results of this work can be extended to the case of piecewise analytic functions A, c, f . 6. In the case of analytic data c, f , and ∂Ω, the assertion of Theorem 5.3.14 holds with the weight function Φp,β,ε replaced with Φp,β,ε ≡ 1, [95]. 7. The boundary value problem (1.2.11) is a Dirichlet problem. Regularity assertions analogous to that of Theorem 2.3.1 hold for other boundary conditions also as we show in Section 5.4 for Neumann, Robin boundary conditions, and a transmission problem.

The regularity assertion of Theorem 2.3.1 does not capture the boundary layer character of the solution. This is done classically with the aid of asymptotic expansions. Since additionally corner singularities are present in the solution uε , we present next a decomposition of the solution uε into a smooth part wε , a CL boundary layer part uBL ε , a corner layer part uε , and a small remainder rε . In order to be able to formulate this decomposition, we need to introduce some notation.

x

Ωj

Γ˜j+1

Ωj

ρj

Γj

Aj

Ωj+1 Γj+1

Ωj+1

Aj

θj

Γj

Γ˜j

Fig. 2.3.1. Boundary ﬁtted coordinates ψj : (ρj , θj ) → x.

A5

Γ5 Ω5

A4

Ω6 = Ω0 Γ6 = Γ0 Ω1 A6 = A0 Γ1

Ω4

A1 Γ2

Γ4

Ω2

A2

Ω3 Γ3

A3

Γj 111111111111111111111111111 000000000000000000000000000 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 BL 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 supp χ 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 supp χCL 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 B 000000000000000000000000000 111111111111111111111111111 j111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 00000000000000000 11111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 00000000000000000 11111111111111111 000000000000 111111111111 Aj 111111111111 00000000000000000 11111111111111111 Γj 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 Γj+1 111111111111 000000000000 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111

Fig. 2.3.2. Scheme of the supports of χBL and χCL .

2.3 Regularity: the two-dimensional case

35

Notation 2.3.3 The notation introduced here is illustrated in Figs. 2.3.1, 2.3.2. 1. (boundary ﬁtted coordinates/ψj ) For each boundary arc Γj we introduce boundary-ﬁtted coordinates (ρj , θj ) as follows: Recalling that Γj Γj = {xj (θ), yj (θ) | θ ∈ (0, 1)} is parametrized such that the normal vector (−y (θ), x (θ)) points into Ω, we deﬁne ψj : R × (0, 1) → Ω xj (θ) (ρ, θ) → + yj (θ)

ρ

x (θ)2 + y (θ)2 j

j

−yj (θ) xj (θ)

.

The maps ψj are real analytic and in fact invertible in a neighborhood of {0} × (0, 1). Without loss of generality, we may assume that ρ0 and Θ > 0 are chosen so small that the analytic continuation of ψj (again denoted ψj ) is real analytic and invertible on (−2ρ0 , 2ρ0 ) × (−2Θ, 1 + 2Θ) for every j ∈ {1, . . . , J}. The analytic continuation of the arc Γj is the arc Γ˜j := {(xj (θ), yj (θ)) | θ ∈ (−Θ, 1 + Θ)}. The inverse functions ψj−1 deﬁne boundary-ﬁtted coordinates (ρj , θj ) in a neighborhood of Γ˜j via (ρj , θj ) = (ρj (x), θj (x)) = ψj−1 (x). A geometric interpretation of ρj is ρj (x) = dist(x, Γ˜j ) (cf. Fig. 2.3.1). 2. (arcs Γj , subdomains Ωj ) For each j ∈ {0, . . . , J − 1} choose an analytic arc Γj passing through vertex Aj such that the angles ∠(Γj , Γj ) and ∠(Γj+1 , Γj ) at Aj are strictly less than π/2. Set ΓJ := Γ0 and deﬁne Ωj as the subset of ψj ((0, ρ0 ) × (−Θ, 1 + Θ)) that is bordered by Γj−1 , Γj , Γj (see Fig. 2.3.2). BL CL 3. (cut-oﬀ functions χ , χ ) Choose smooth cut-oﬀ functions χCL , χBL and numbers ρ1 < ρ0 , r2 < r1 with the following properties: supp χBL ⊂ Uρ0 (∂Ω), χBL ≡ 1 on Uρ1 (∂Ω), χCL ≡ 1 on supp χBL ∩ Γj , CL

⊂

CL

≡

supp χ χ

j = 1, . . . , J,

∪Jj=1 Br1 (Aj ), 1 on ∪Jj=1 Br2 (Aj ).

Finally, choose r3 < r2 so small that j := Br (Aj ) ⊂ {x ∈ R2 | χBL (x) = 1}. B 3 These notations enable us to formulate the following regularity results, in which the solution uε is decomposed into a smooth part wε , a boundary layer part CL uBL ε , a corner layer part uε , and a small remainder rε . Theorem 2.3.4 (Regularity through asymptotic expansions). Fix sets Ωj , arcs Γj , and cut-oﬀ functions χBL , χCL as in Notation 2.3.3. Then there

36

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

exist C, K, α > 0, and β ∈ [0, 1)J such that for each ε ∈ (0, 1] the solution uε of (2.1.1) can be decomposed as uε = wε + χBL uBL + χCL uCL + rε ε ε

(2.3.1)

with the following properties: (i) The smooth part wε is analytic on Ω and ∇p wε L∞ (Ω) ≤ Cγ p p!

∀p ∈ N0 .

2 (Ω) satisﬁes (ii) The remainder rε ∈ H01 (Ω) ∩ Hloc

rε L∞ (Ω) + rε H 1 (Ω) + Φ0,β,ε ∇2 rε L2 (Ω) ≤ Ce−α/ε . (iii) The boundary layer uBL satisﬁes on each Ωj ε sup θj ∈(−Θ,1+Θ)

|(∂ρnj ∂θmj (uBL ◦ ψj ))(ρj , θj )| ≤ Cε−n γ n+m m! e−αρj /ε , ρj ≥ 0. ε

j ) ∪ (Ωj+1 ∩ B j ) for each (iv) The corner layer uCL is analytic on Uj := (Ωj ∩ B ε j = 0, . . . , J − 1 and satisﬁes for all p ∈ N0 αrj /ε eαrj /ε uCL ∇uCL j ) + εe j ) ≤ Cε, ε L2 (U ε L2 (U p −1 p+2 } eαrj /ε Φp,β,ε ∇p+2 uCL j ) ≤ Cεγ max {p + 1, ε ε L2 (U

together with the pointwise bounds (we abbreviate rj = dist(x, Aj )) p βj −1 rj |∇p uCL ε (x)| ≤ Cγ p! ε

1−p−βj

e−αrj /ε

∀x ∈ Uj

∀p ∈ N0 .

Additionally, for Uj := (Ωj ∩ Br1 (Aj )) ∪ (Ωj+1 ∩ Br1 (Aj )) we have αrj /ε eαrj /ε uCL ∇uCL ε L2 (Uj ) + εe ε L2 (Uj ) ≤ Cε.

Proof: This result can be inferred from Corollary 7.4.6. An outline of the key steps of the construction of the decomposition an be found in Section 7.1. 2 Remark 2.3.5 1. In the language of Part III, Theorem 2.3.4 asserts for the corner layer uCL membership in an exponentially weighted countably 2 (Uj , Cε, γ) for each j ∈ {1, . . . , J}. normed space, namely, uCL ∈ Bβ,ε,α 2. Theorem 2.3.4 asserts diﬀerentiabilty properties for the corner layer on the sets Uj . However, uCL is also deﬁned on Uj , it is smooth on Uj , and inspection of the proof in Section 7.4 reveals that for each k ∈ N0 there exists Ck > 0 (independent of ε) such that |∇k uCL (x)| ≤ Ck εβj −1 rj

1−k−βj

e−αrj /ε

We emphasize that ∪Jj=1 Uj ⊃ supp χCL \ ∪Jj=1 Γj .

∀x ∈ Uj .

2.3 Regularity: the two-dimensional case

37

2 3. Theorem 2.3.4 merely asserts rε ∈ Hloc (Ω). However, inspection of the procedure in Section 7.4 reveals that rε is obtained as the solution of an elliptic equations with piecewise smooth data; in can be shown that for each k ∈ N0 there exists Ck > 0 such that J

Φk,β,ε ∇k+2 rε L2 (Ωj ) + Φk,β,ε ∇k+2 rε L2 (Ω\∪Jj=1 Ωj ) ≤ Ck ε−k e−α/ε .

j=1

The decomposition whose existence is ascertained in Theorem 2.3.4 is constructed using the classical method of asymptotic expansions: First, the outer expansion is constructed that gives a particular solution to the diﬀerential equation. In the second step, the boundary conditions are corrected using the boundary layer functions. These are constructed on each of the subdomains Ωj separately. may have jumps across the In particular, the piecewise deﬁned function χBL uBL ε arcs Γj . These jumps are corrected with the corner layer function uCL ε , which are solutions to transmission problems where the jump and the jump of the CL CL co-normal derivative across Γj is prescribed such that χBL uBL uε is C 1 ε +χ across the arcs Γj . We formulated these observations in the following theorem for future reference: Theorem 2.3.6 (Regularity through asymptotic expansions). Assume the hypotheses of Theorem 2.3.4. Denote by Lε the diﬀerential operator Lε v := −∇ · (A(x)∇v) + c(x)v. With the constants C, α > 0 of Theorem 2.3.4, the CL terms wε , uBL of the decomposition in Theorem 2.3.4 have the following ε , uε additional properties: (i) Lε wε − f L∞ (Ω) ≤ Ce−α/ε . + wε = g on ∂Ω and for each j ∈ {1, . . . , J} there holds (ii) χBL uBL ε −α/ε Lε (χBL uBL . ε )L∞ (Ωj ) ≤ Ce = 0 on ∂Ω. (iii) χCL uCL ε + χCL uCL is in C 1 (Ω \ ∪Jj=1 {Aj }) and (iv) The function uC := χBL uBL ε ε Lε uC L∞ (Ω\∪Jj=1 Γj ) ≤ Ce−α/ε . Proof: The asserted properties follow from the construction of the decomposition in Section 7.4. Assertion (i) is shown in (7.4.12). The assertion (ii) follows from Lemma 7.4.7. The last assertion follows from the construction, viz., the deﬁnition of uCL in (7.4.33). 2 ε Remark 2.3.7 The constants C, K in Theorems 2.3.4, 2.3.6, depend on the choice of the analytic arcs Γj . In the proof of exponential convergence of the hp-FEM, the mesh has to be chosen such that at each vertex Aj , a mesh line can be chosen as an arc Γj with the properties stipulated in Notation 2.3.3.

38

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

We conclude our discussion of the regularity properties of the solution uε with remarks on two special cases, namely, convex corners and smooth domains. In general, the boundary layer function uBL (and therefore also the corner layer ε function uCL and the remainder r ) is only piecewise smooth; near convex verε ε tices, a special construction is possible that aﬀords a smooth boundary layer uBL ε . This special case is formulated in the following theorem: Theorem 2.3.8 (Regularity near convex vertices). Assume the hypotheses of Theorem 2.3.4. Let Aj be convex vertex, i.e., ∠(Γj , Γj+1 ) < π/2. Then uBL ε and uCL can be constructed such that, in addition to the properties listed in ε Theorem 2.3.4, the following holds: (i) On Bρ0 (Aj ) the function uBL is analytic and can be written as uBL = ε ε BL BL BL BL BL uj,ε + uj+1,ε , where the functions u ˜BL := u ◦ ψ , u ˜ := u ◦ ψ j j+1 j j,ε j+1 j+1,ε satisfy −n n+m |(∂ρn ∂θm u ˜BL γ m! exp(−αρ/ε), j (ρ, θ)| ≤ Cε

ρj ≥ 0,

−n n+m |(∂ρn ∂θm u ˜BL γ m! exp(−αρ/ε), j+1 (ρ, θ)| ≤ Cε

ρ ≥ 0.

sup θ∈(−Θ,1+Θ)

sup θ∈(−Θ,1+Θ)

j and satisﬁes for all p ∈ N0 and (ii) The corner layer uCL is analytic on B ε j x∈B CL p βj −1 1−p−βj |∇p uCL rj exp(−αrj /ε), ε (x) − uε (Aj ) | ≤ Cγ p! ε + χBL uε = g on ∂Ω. where rj = dist(x, Aj ). Additionally, χCL uCL ε Remark 2.3.9 In the construction of the corner layer uCL in the case of convex ε corners Aj , the parameter βj can be characterized more precisely. For example, in the case A = Id, we may choose any βj ∈ (0, 1). Remark 2.3.10 In the case of analytic boundary curves ∂Ω, a simpliﬁed expansion holds: The boundary layer function uBL is analytic on a neighborhood ε CL of ∂Ω, where uCL may be chosen as u = 0 and the weight function Φk,β,ε in ε ε the estimates for the remainder rε may be replaced with Φk,β,1 ≡ 1. We refer to [95] for the details.

2.4 hp-FEM approximation 2.4.1 hp-meshes and spaces The reference square S and the reference triangle T are deﬁned as S = (0, 1) × (0, 1),

T = {(x, y) | 0 < x < 1 | 0 < y < x}.

We start by deﬁning regular triangulations without “hanging nodes”.

2.4 hp-FEM approximation

39

ˆ i ) | i ∈ I(T )} is said Deﬁnition 2.4.1. A collection of triples T = {(Ki , Mi , K to be a triangulation of a domain Ω if the subsets Ki ⊂ Ω, the element maps ˆ i ∈ {S, T }, and the index set I(T ) ⊂ N ˆ i → Ki , the reference elements K Mi : K satisfy the following conditions (M1)–(M4): (M1) The elements Ki partition the domain Ω, i.e., Ω = ∪i Ki . (M2) For i = j, Ki ∩ Kj is either empty, or a vertex or an entire edge (vertices and edges are the images of the vertices and edges of the reference elements under the maps Mi ). ˆ i → Ki are analytic diﬀeomorphisms. (M3) The element maps Mi : K (M4) The common edge of two neighboring elements has the same parametrization “from both sides”: For two neighboring elements Ki , Kj , let γij = Ki ∩ Kj be the common edge with endpoints P1 , P2 . Then for any point P ∈ γij , we have dist(Mi−1 (P ), Mi−1 (Pl ))/Li = dist(Mj−1 (P ), Mj−1 (Pl ))/Lj , l = 1, 2 where Li and Lj denote the lengths of the edges corresponding to γij in the reference elements. ˆ i = S then Ki is said to be a (curvilinear) quadrilateral and if K ˆ i = T then If K Ki is a (curvilinear) triangle. Once a triangulation T is chosen, one can deﬁne ﬁnite-element spaces S p (T ) based on this triangulation. ˆ i) | i ∈ Deﬁnition 2.4.2 (FE-spaces). Given a triangulation T = {(Ki , Mi , K p 1 p I(T )}, the H -conforming ﬁnite element spaces S (T ), S0 (T ) of piecewise mapped polynomials are deﬁned as S p (T ) := {u ∈ H 1 (Ω) | u|Ki = ϕp ◦ Mi−1 S0p (T ) := S p (T ) ∩ H01 (Ω),

ˆ i )}, for some ϕp ∈ Πp (K

(2.4.1) (2.4.2)

ˆ i ) are deﬁned as where the polynomial spaces Πp (K ! ˆ ˆ i ) = Qp (S) if Ki = S Πp (K ˆi = T , Pp (T ) if K Qp (S) := span{xi y j | 0 ≤ i, j ≤ p}, Pp (T ) := span{xi y j | i + j ≤ p}. 2.4.2 The minimal hp-mesh The regularity assertion of Theorem 2.3.4 shows that we have to deal with two types of phenomena: boundary layers in the vicinity of the boundary curves Γj and corner singularities in neighborhoods of the vertices Aj . Meshes with special properties are required to resolve these two phenomena; namely, the meshes should contain needle elements near the boundary curves Γj to capture the boundary layers and they should include geometric reﬁnement near the vertices Aj to catch the corner singularities. The (essentially) minimal mesh family that

40

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

incorporates these two mesh design principles is characterized by our notion of admissible boundary layer meshes in Deﬁnition 2.4.4. An example of such an admissible mesh is presented in Fig. 2.4.1: The rectangles at the boundary are boundary layer elements of width O(κ), the elements in the shaded regions are corner layer elements, and the remaining elements are interior elements. The parameter κ, which characterizes the width of the needle elements, will be chosen κ = O(pε), where p is the approximation order. Our hp-FEM convergence results ahead rests on the decomposition of the exact solution into a smooth part, boundary layers, and corner layers given in Theorem 2.3.4. Our notion of admissible boundary layer meshes in Deﬁnition 2.4.4 reﬂects this by aligning mesh lines with the subdomains Ωj introduced in Notation 2.3.3. Additionally, we employ the cut-oﬀ functions χBL , χCL , and the maps ψj providing the transfer from Cartesian coordinates to the boundary-ﬁtted coordinates (ρj , θj ) of Notation 2.3.3. For the deﬁnition of admissible meshes, it is convenient to introduce the following two stretching maps: Notation 2.4.3 For κ > 0 and each j ∈ {1, . . . , J} introduce + sκ : R+ 0 × R → R0 × R (ρ, θ) → (κρ, θ),

s˜j,κ : R2 → R2 (x, y) → κ ((x, y) − Aj ) + Aj . The value κ0 > 0 is such that on a κ0 -neighborhood of each arc Γj the cut-oﬀ function χBL is identically 1 and such that on a κ0 -neighborhood of the vertices Aj , the cut-oﬀ functions χCL are identically 1: Uκ0 (Γj ) ∩ Ωj ⊂ {x ∈ Ω | χBL (x) = 1}, Bκ0 (Aj ) ∩ Ω ⊂ {x ∈ Ω | χCL (x) = 1},

j = 1, . . . , J, j = 1, . . . , J.

(2.4.3a) (2.4.3b)

where we set Uκ0 (Γj ) := {x ∈ R2 | dist(x, Γj ) < κ0 }. The following deﬁnition introduces admissible boundary layer meshes, which includes needle elements of width O(κ). The key requirement for the element "i := sκ−1 ◦ψj ◦Mi maps Mi of such needle elements is that the “stretched” maps M satisﬁes the standard assumptions on element maps. We note that the anisotropic stretching sκ−1 ◦ψj corresponds to a stretching “normal” to the boundary, which correctly reﬂects the boundary layer behavior of the solution. Deﬁnition 2.4.4 (admissible boundary layer mesh). The two-parameter ˆ i )}κ,L , (κ, L) ∈ (0, κ0 ] × N0 of meshes satisfying family T (κ, L) = {(Ki , Mi , K (M1)–(M4) is said to be admissible if there are C, γ, ci > 0, i = 1, . . . , 4, σ ∈ (0, 1) and sets Ωj , j = 1, . . . , J, of the form described in Notation 2.3.3 ˆ i ) ∈ T (κ, L) falls into exactly one of the following such that each triple (Ki , Mi , K three categories:

2.4 hp-FEM approximation

41

(C1) Ki is a boundary layer element, i.e., for some j ∈ {1, . . . , J} there holds Ki ⊂ Uκ (Γj ) ∩ Ωj \ (Bc1 κ (Aj−1 ) ∪ Bc1 κ (Aj )), and the element maps Mi satisﬁes −1

(Mi )

C , κ ≤ Cγ |α| α!

L∞ (Kˆ i ) ≤

−1 Dα s−1 ˆ i) κ ◦ ψj ◦ Mi L∞ (K

∀α ∈ N20 ;

(C2) Ki is a corner layer element, i.e., for some j ∈ {0, . . . , J − 1} either Ki ⊂ Bκ (Aj ) ∩ Ωj or Ki ⊂ Bκ (Aj ) ∩ Ωj+1 . Additionally, denoting hi = diam Ki , the element map Mi satisﬁes |α| α! ∀α ∈ N20 , Dα s˜−1 ˆ i ) ≤ Cγ j,hi ◦ Mi L∞ (K −1 (˜ s−1 L∞ (Kˆ i ) ≤ C; j,hi ◦ Mi ) furthermore, exactly one of the following situations is satisﬁed: either Aj ∈ Ki and hi ≤ c4 κσ L or Aj ∈ Ki and c3 hi ≤ dist(Aj , Ki ) ≤ c4 hi . (C3) Ki is an interior element, i.e., Ki ⊂ Ω \ Uc2 κ (∂Ω), and the element map satisﬁes Dα Mi L∞ (Kˆ i ) ≤ Cγ |α| α! −1

(Mi )

L∞ (Kˆ i ) ≤

∀α ∈ N20 ,

C . κ

The parameter κ in the deﬁnition of admissible meshes controls the width of the needle elements required to capture the boundary layer phenomena; in the hp-FEM approximation result Theorem 2.4.8, we will choose κ = O(pε), where p is the approximation order. The parameter L in admissible meshes represents the number of layers of geometric reﬁnement towards the vertices in an O(κ) neighborhood of the vertices. We refer to Fig. 2.4.1 for examples of admissible meshes. Remark 2.4.5 Admissible boundary layer mesh families are essentially the minimal meshes that lead to robust exponential convergence for an hp FEM (see Theorem 2.4.8). In particular, highly distorted elements violating the maximal angle condition, [12], are admitted in our notion of admissible meshes: Minimal and maximal angles are allowed to be of sizes O(κ) and O(π − κ), cf. Fig. 2.4.2.

Remark 2.4.6 The notion of admissible meshes of Deﬁnition 2.4.4 includes the standard p-FEM and hp-FEM meshes as special cases. For ﬁxed κ and L, the

42

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

ˇ ˇ

κ

κ

Fig. 2.4.1. Example of an admissible mesh T (κ, L) with L = 3: the domain (top) and zoom-ins at convex and concave vertices.

O(1) O(κ)

O(1)

Fig. 2.4.2. Elements near the boundary may be distorted.

2.4 hp-FEM approximation

43

mesh T (κ, L) reduces to a standard p-version mesh with non-distorted elements. For ﬁxed κ but variable L, the mesh T (κ, L) is a standard p-version outside a O(κ) neighborhood of the vertices Aj . In the O(κ) neighborhood of the vertices, a geometric mesh with L + 1 layers is used. Finally, for ﬁxed κ and ﬁxed η > 0, the meshes T (κ, ηp) are standard hp-FEM meshes with large elements in the interior of Ω and geometric mesh reﬁnement with O(p) layers toward the vertices as proposed in, e.g., [14, 112, 123]. Remark 2.4.7 As mentioned above, the elements in admissible meshes are allowed to be quite distorted in the sense that minimal and maximal angles may be close to 0 or π. For implementational reasons, it is advisable to be able to control the distortion of the elements, in particular the maximal angles. The introduction of regular admissible meshes in Deﬁnition 3.3.10 allows for this control, and we refer to Section 3.3 for a more detailed discussion of this issue. 2.4.3 hp-FEM With the hp-spaces S p (T ) and the notion of admissible boundary layer meshes of Deﬁnition 2.4.4 in hand, we can proceed to the robust exponential approximation result for the approximation of solutions to (2.2.1). The starting point of the hp-FEM for (2.1.1) is the weak formulation (2.1.3). In order to incorporate the Dirichlet boundary condition uε |∂Ω = g, let uD ∈ H 1 (Ω) be an arbitrary function with uD |∂Ω = g. Then uε can be sought in the form uε = uD + u0 where u0 is the solution to: Find u0 ∈ H01 (Ω) s.t. Bε (u0 , v) = F (v) − Bε (uD , v)

∀v ∈ H01 (Ω).

Here, the bilinear form Bε and the right-hand side F are deﬁned in (2.1.4). In the hp-FEM, the inﬁnite dimensional space H01 (Ω) is replaced with the ﬁnite dimensional space S0p (T ) and the function uD is replaced with a suitable element of S p (T ). For the sake of deﬁniteness, we choose one particular form of enforcing the Dirichlet boundary conditions, namely, that of sampling the Dirichlet data g elementwise in the Gauss-Lobatto points: Let uD,p ∈ S p (T ) be any element of S p (T ) satisfying uD,p = ip,Γ uε

∀ element edges Γ with Γ ⊂ ∂Ω.

(See p. 88 for a precise deﬁnition of ip,Γ .) Such an interpolant is easily constructed In fact, only the elements abutting on the boundary ∂Ω are aﬀected. The ﬁnite element approximant uN is then given by uN := uD,p + u0,p ,

(2.4.4)

where the functions u0,p ∈ S0p (T ) is the solution of the following problem: Find u0,p ∈ S0p (T ) s.t. Bε (u0,p , v) = F (v) − Bε (uD,p , v) ∀v ∈ S0p (T ).

(2.4.5)

The approximation error uε − uN can be controlled with C´ea’s Lemma, [35, 36] in the standard way: Noting that uε − uD,p satisﬁes the variational problem

44

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

Bε (uε − uD,p , v) = F (v) − Bε (uD,p , v)

∀v ∈ H01 (Ω),

C´ea’s Lemma allows us to estimate uε − uN ε = (uε − uD,p ) − u0,p ε =

inf

πp ∈S0p (T )

(uε − uD,p ) − πp ε =

inf{uε − vε | v ∈ S p (T ) s.t. ip,Γ uε = v for all edges Γ of T with Γ ⊂ ∂Ω}. In Theorem 3.4.8, we construct a speciﬁc approximant to uε that leads to an upper bound in the last inﬁmum. We then arrive at the following robust exponential convergence result: Theorem 2.4.8. Let Ω be a curvilinear polygon and T (κ, L) be a two-parameter family of admissible meshes in the sense of Deﬁnition 2.4.4. Let uε be the solution of (2.1.1) where the piecewise analytic Dirichlet data g satisfy (1.2.4) and the analytic function f satisﬁes (1.2.3). Then there are b, λ0 > 0 independent of ε and p with the following property: For each λ ∈ (0, λ0 ) there exists C > 0 such that the ﬁnite element solution uN ∈ S p (T (min {λpε, 1}, L)) as deﬁned by (2.4.4), (2.4.5) satisﬁes u − uN L2 (Ω) + ε∇(u − uN )L2 (Ω) ≤ Cp2 (1 + ln p) e−bλp + εp3 e−b L . Furthermore, ip,Γ uε = uN |Γ

∀ edges Γ of T (min{1, λpε}, L) with Γ ⊂ ∂Ω.

In particular, if the number of elements |T (min {λpε, 1}, L)| ∼ L ∼ p, then N = dim S p (T (min {λpε, 1}, L)) ∼ p3 and therefore

u − uN ε ≤ Ce−b N

1/3

,

where the constants C, b are independent of ε (albeit dependent on λ). Proof: The result follows by combining C´ea’s Lemma and the approximation result Theorem 3.4.8. A rigorous proof of Theorem 3.4.8 is involved. The main ideas, however, are similar to the one-dimensional case of Proposition 2.2.5. We outline in Subsection 3.1.3 the key steps of the proof. 2 We have |T (min {λpε, 1}, L)| ∼ L in admissible boundary layer meshes that are typically chosen (cf. Fig. 2.4.1 and the meshes of the numerical examples in Section 2.5). Remark 2.4.9 The p-dependence in the approximation result Theorem 2.4.8 is likely to be suboptimal. This is due to the our choice of the particular approximant of Theorem 3.4.8.

2.5 Numerical Examples

45

2.5 Numerical Examples The aim of the numerical examples of the present section is to illustrate and highlight several properties of the hp-FEM applied to (2.1.1). We discuss the following aspects: 1. Robust exponential convergence is indeed observed numerically in the practical range of values of polynomial degrees p. 2. Our numerical experiments conﬁrm that the corner layer components of the solution of (1.2.11) are weak. Theorem 2.4.8 reﬂects this as, neglecting algebraic terms in p, we have uε − uN ε ≤ C e−bλp + εe−b L . The ﬁrst term, e−bλp , reﬂects the approximation of the smooth and the boundary layer parts whereas the second term, εe−b L , is due to the approximation of corner layers. Balancing these two error contributions, we see that for small ε, the number L of layers of geometric reﬁnement may be chosen small compared to p; that is, few layer of geometric reﬁnement are suﬃcient for adequate resolution of the corner layers. 3. As noted in Remark 2.4.5 the ﬁnite element mesh may contain highly distorted elements with minimal angles of size O(pε) and maximal angles of size π − O(pε). In agreement with the predictions of Theorem 2.4.8, our numerical examples show robustness of the FEM with respect to mesh distortion as the presence of highly distorted elements has practically no eﬀect on the energy convergence. Several of these points have already been observed in previous numerical studies [94, 115, 131, 133, 134]. These studies considered the model problem −ε2 ∆uε + uε = f

on Ω,

uε |∂Ω = g.

(2.5.1)

For the case of smooth boundary curves, [131, 133] obtained robust algebraic rates of convergence of arbitrary order if the hp-FEM is based on boundary-ﬁtted tensor product meshes with needle elements of width O(pε) in the layer. Robust exponential convergence was observed numerically and conjectured. In [94] this conjecture was rigorously established for the model problem (2.5.1) under the assumption of analyticity of the input data f , g, and ∂Ω. [94] also removed the restriction to boundary-ﬁtted tensor product meshes and allowed highly distorted elements as discussed in Remark 2.4.5. Robust algebraic convergence of arbitrary order for (2.5.1) for the special case of Ω being square were obtained in [131, 134]. It was also observed numerically that corner layers are weak in the sense that few layer of geometric reﬁnement are required for good resolution in the energy norm. As we pointed out above, Theorem 2.4.8 rigorously establishes this observation for arbitrary curvilinear polygons, and our numerical examples below corroborate this claim for the classical L-shaped domain.

46

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

2.5.1 The classical L-shaped domain The numerical experiments that we show in this section illustrate that the hpFEM achieves robust exponential convergence for (1.2.11) on polygonal domains as well. Additionally, we show that the corner layer is weak and that thus few layers of geometric reﬁnement are suﬃcient for good convergence in the energy norm. We consider the problem −ε2 ∆uε + uε = f (x, y)

on Ω,

uε = 0

on ∂Ω.

(2.5.2)

Here, the domain Ω is the classical L-shaped domain, Ω = (−1, 1)2 \ (−1, 0) × (0, 1). For our calculations, we chose f (x, y) = exp(x+y) so that the assumptions on the data of Theorem 2.4.8 are satisﬁed. The computations were done with the code Concepts 1.4, [80]. The ﬁnite element mesh T (κ, L)

(2.5.3)

employed is depicted in Fig. 2.5.1 where the boundary layer elements are of width κ = min {0.5, pε} and L + 1 layers of geometric reﬁnement with grading factor σ = 0.5 are employed in the vicinity of the vertices. The actual ﬁnite element spaces employed in our numerical examples were slightly diﬀerent from those analyzed so far: We employed meshes with “hanging nodes” (cf. Fig. 2.5.1) and used the ﬁnite element space S˜0p (T ) based on the “trunk spaces” Qp (S) (cf., e.g., [123]). The precise deﬁnition of S˜0p (T ) is S˜0p (T ) := {u ∈ H01 (Ω) | u ◦ Mi ∈ Qp (S)}, Qp (S) := span {xp y, xy p , xi y j | 0 ≤ i + j ≤ p}. The ﬁnite element spaces S˜0p (T ) are thus H01 (Ω)-conforming ﬁnite element spaces, and this conformity constraint on meshes with hanging nodes is properly treated during the assembly procedure. It can be shown that an approximation result analogous to that of Theorem 2.4.8 could be formulated for the spaces S˜0p (T ) on meshes with hanging nodes as well. Our numerical results are presented in Figs. 2.5.2, 2.5.3, 2.5.4 for ε = 1, ε = 10−3 , ε = 10−6 , where the energy error uε − uN 2ε is graphed versus the number of degrees of freedom N = dim S˜0p (T ). In each ﬁgure, lines correspond to meshes with a ﬁxed number of L + 1 layers of geometric reﬁnement and polynomial degree p increasing from 1 to 15. Comparing Figs. 2.5.2–2.5.4 we ﬁrst observe robustness of the FEM based on admissible meshes as the energy errors do not depend on ε. We now discuss the numerical results of Figs. 2.5.2–2.5.4 in more detail and start with the case ε = 1 in Fig. 2.5.2. Each line in Fig. 2.5.2 corresponds to the p-version on a ﬁxed mesh with L + 1 layers of geometric reﬁnement toward the vertices. Fig. 2.5.2 exhibits the typical convergence pattern of the p-version FEM: On ﬁxed meshes, there is an initial phase of exponential convergence (visible for larger values of L in Fig. 2.5.2) followed by the asymptotic algebraic convergence. The asymptotic convergence rate is p−4/3 in the energy norm (cf. [22]), i.e., a

2.5 Numerical Examples

47

convergence of N −4/3 in the energy. Indeed, the asymptotic slopes in Fig. 2.5.2 are very close to the predicted value 4/3. “True” exponential convergence can only be observed if the number of layers of geometric reﬁnement is taken proportional to the polynomial degree p, i.e., if L = κp for some ﬁxed κ > 0. We now illustrate that the corner layers are indeed rather weak. This weakness manifests itself numerically in the fact that for ﬁxed L, the pre-asymptotic exponential convergence is visible for a large range of values of p for small ε. Put diﬀerently, for ﬁxed L, the onset of the asymptotic algebraic convergence occurs at larger values of p as ε decreases. We see this pattern by comparing lines corresponding to the same value of L in Figs. 2.5.2, 2.5.3, and 2.5.4. In particular, in Fig. 2.5.4, the asymptotic algebraic convergence occurs below machine accuracy (16 digits). This extension of the pre-asymptotic exponential convergence behavior can be explained with the aid of Theorem 2.4.8. Theorem 2.4.8 states that the FEMerror behaves like (ignoring the algebraic factors involving powers of p)

e−bλp + εe−b L

(2.5.4)

for some b, b > 0 independent of ε, p, L. The proof of Theorem 3.4.8 shows that the factor εe−b L is due to the approximation of the corner layers. For small ε and ﬁxed L, (2.5.4) suggests an exponential decrease (in p) for the ﬁnite element error. This exponential decay is visible until the two terms are equilibrated, i.e., until e−bλp ≈ εe−b L . This observation explains why the pre-asymptotic exponential convergence phase of the FEM increases as ε becomes small. (2.5.4) suggests another characteristic of the FEM applied to (2.1.1): For small ε the FEM is quite insensitive to an increase of the number of layers L of geometric reﬁnement near the vertices. This is indeed visible in Fig. 2.5.3 for p = 1, 2, 3, and in Fig. 2.5.4 for p ranging from 1 to 9 since the energy error is eﬀectively constant for all values of L.

Fig. 2.5.1. The mesh and the local reﬁnement near the reentrant corner (L = 3).

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results Energy convergence on L−shaped domain:, σ = 0.5, ε = 1

0

10

L=0 L=2 L=4 L=6 L=8 L=10

−1

10

−2

10

rel. error in energy

−3

10

−4

10

−5

10

−6

10

−7

10

−8

10

1

10

2

10

3

10 degrees of freedom

4

5

10

10

Fig. 2.5.2. Rel. energy error vs. DOF on mesh of Fig. 2.5.1; ε = 1. Energy convergence on L−shaped domain:, σ = 0.5, ε = 0.001

−2

10

L=0 L=2 L=4 L=6 L=8 L=10

−3

10

−4

10

−5

10 rel. error in energy

48

−6

10

−7

10

−8

10

−9

10

−10

10

−11

10

−12

10

1

10

2

10

3

10 degrees of freedom

4

10

5

10

Fig. 2.5.3. Rel. energy error vs. DOF on mesh of Fig. 2.5.1; ε = 10−3 .

2.5 Numerical Examples

49

Energy convergence on L−shaped domain:, σ = 0.5, ε = 1e−06

−2

10

L=0 L=2 L=4 L=6 L=8 L=10

−4

10

−6

rel. error in energy

10

−8

10

−10

10

−12

10

−14

10

1

2

10

3

10

10

4

10

degrees of freedom

Fig. 2.5.4. Rel. energy error vs. DOF on mesh of Fig. 2.5.1; ε = 10−6 .

2.5.2 Robustness with respect to mesh distortion Our next numerical example illustrates robustness of the FEM on admissible boundary layer meshes with respect to mesh distortion. Theorem 2.4.8 states that robust exponential convergence is achieved on admissible boundary layer meshes, which may be highly distorted as discussed in Remark 2.4.5. To show that the presence of highly distorted elements does not aﬀect the energy convergence, we consider the following quasi one-dimensional model problem. −ε2 ∆uε + uε = 1 uε = 0 ∂n u ε = 0

on Ω := (0, 1)2 ,

(2.5.5a)

on ΓD := {(x, y) ∈ ∂Ω | y = 0},

(2.5.5b)

on ΓN := ∂Ω \ ΓD .

(2.5.5c)

The solution of this problem is given by uε (x, y) = 1 −

cosh((1 − y)/ε) . cosh(1/ε)

(2.5.6)

We note that uε has a boundary layer at ΓD and no corner singularities although Ω is a square. We therefore do not need any geometric reﬁnement near the vertices of Ω, and needle elements are required only near ΓD . For ε = 10−3 our numerical calculations were performed with the commercial code STRESS CHECK, [48], a p-version code with highest polynomial degree pmax = 8. The meshes consisted of straight triangles and quadrilaterals

50

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

(see Figs. 2.5.5– 2.5.7), where the parameter b determines the distortion of the meshes. On a ﬁxed quadrilateral mesh as depicted in Fig. 2.5.5 the tensor product spaces Qp (S) with p ranging from 1 to pmax were used. The relative error in energy uε − uN 2ε versus the square root of the number of degrees of freedom (DOF) is reported in Fig. 2.5.5. In the case b = 0.5 all quadrilaterals satisfy a maximum and minimum angle condition (even as ε tends to zero). For the case b = 0.25 the maximum angle is π − O(ε) and the minimum angle is O(ε), i.e., the mesh is highly distorted. Nevertheless, the error curves in Fig. 2.5.5 are practically on top of each other showing the robustness with respect to mesh distortion of the approximation properties of admissible meshes. The situation is completely analogous for triangular meshes: the error graph in Fig. 2.5.6 shows the performance of the p version on the triangular mesh also shown in Fig. 2.5.6; again, the convergence is not visibly aﬀected by the presence of highly distorted elements in the boundary layer. According to Theorem 2.4.8, the needle elements in the boundary layer should have width O(pε), i.e., the mesh should depend on ε as well as on p. However, for practical purposes, it is more convenient to ﬁx a mesh and to increase p. The question arises then what the appropriate width of the needle elements is. If only one layer of needle elements is used, we advocate the use of needle elements of width O(pmax ε). The following numerical example supports this choice. In Fig. 2.5.7, we show the relative error in energy versus the number of degrees of freedom for the mesh also shown in Fig. 2.5.7. While robustness with respect to mesh distortion is again clearly visible as the choice of the parameter b has practically no eﬀect, we note that the error curves in Fig. 2.5.7 level oﬀ at an error of about 10−7 corresponding to p = 6. Actually, already for p = 5, some deterioration of the rate of convergence is visible. This is due to the fact that the width of the needle elements is ﬁxed at 4ε instead of 8ε = pmax ε (In the meshes of Figs. 2.5.5, 2.5.6, the width was at least 10ε, which exceeds pmax ε). As demonstrated in the one-dimensional setting in Example 2.2.8, the large elements are too close to ΓD and dominate the global error reduction. 2.5.3 Examples with singular right-hand side So far, the right-hand side f was chosen analytic in Ω. This implies that, as ε → 0, the “limit solution” u0 (x) := limε→0 uε (x) = f (x)/c(x) is smooth as well and does not exhibit corner singularities. For many singularly perturbed problems such as the Reissner-Mindlin plate model on polygonal domains, the limit solution has corner singularities as well. From an approximation point of view the mesh should be designed to capture the boundary and corner layers as well as the behavior of the limit solution. A good mesh design strategy is therefore to combine two types of meshes, namely, a) the meshes presented above, i.e., meshes capable of resolving boundary layer and corner singularities with length scale O(ε), and b) meshes with classical geometric reﬁnement towards the corners, which allows for resolving corner singularities present in the limit solution.

2.5 Numerical Examples

51

6 elements; eps=10^(−3); both middle lines slightly skewed

−3

10

b=0.5 b=0.25

−4

10

−5

10

y −6

Rel. Error in Energy

1

12ε

10

−7

10

−8

10

−9

10

10ε

−10

10

6ε

−11

4ε

10

−12

b

1/2

1

10

x

2

4

6

8

10

12 14 SQRT(DOF)

16

18

20

22

Fig. 2.5.5. Left: mesh with parameter b determining distortion (not drawn to scale); right: p-version on that mesh, ε = 10−3 . hp FEM; triangles; eps=10^(−3)

−2

10

b=0.5

−3

10

b=0.25 b=0.125

y

−4

10 Rel. Error in Energy

1

−5

10

−6

10

−7

10

10 ε

−8

10

−9

b

1/2

1

x

10

2

4

6

8

10 SQRT(DOF)

12

14

16

18

Fig. 2.5.6. Left: mesh with parameter b determining distortion (not drawn to scale); right: p-version on that mesh, ε = 10−3 .

Example 2.5.1 This example shows two points. First, combining geometric mesh reﬁnement toward singularities with boundary layer meshes is very effective. Second, as proposed in Remark 2.2.6, geometric mesh reﬁnement can also be used to replace boundary layer meshes, albeit at the expense of losing robustness in a strict sense. We consider −ε2 uε (x) + uε (x) = f (x) := (1 + x)−0.45

on (−1, 1),

uε (±1) = 0.

The mesh employed consists of the points {−1, 1} ∪ {xi = −1 + q L+1−i | i = 1, . . . , L} ∪ {1 − λpε},

q = 0.15,

λ = 0.71,

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

52

4 elements; eps=10^(−3); middle line slightly skewed

−2

10

b=0.5 b=0.25 −3

10

y −4

Rel. Error in Energy

1

10

−5

10

−6

10

8ε −7

10

4ε

−8

b

1/2

1

x

10

2

4

6

8

10 SQRT (DOF)

12

14

16

18

Fig. 2.5.7. left: mesh with parameter b determining distortion (not drawn to scale); right: p-version on that mesh, ε = 10−3 .

where the number of layers L of geometric reﬁnement is a parameter of our numerical investigations. The boundary layer at the right endpoint x = 1 is resolved with the aid of the small element of size λpε (cf. Example 2.2.7). Fig. 2.5.8 show the performance of this choice of mesh for ε = 10−4 , ε = 10−6 and diﬀerent values of L. While the mesh is designed to resolve the small-scale features at the right endpoint, we have to choose L such that q L ≈ ε in order to resolve the features at the left endpoint x = −1. For our choices of ε, this happens for the moderate values L = 2 for the case ε = 10−4 and L = 4 in the case ε = 10−6 . This shows that, as already discussed in Remark 2.2.6, geometric mesh reﬁnement toward an endpoint is a very eﬀective means of generating elements that reach size O(ε) very quickly. Indeed, the numerical experiments in Figs. 2.5.8 conﬁrm this observation. In an O(ε)-neighborhood of the endpoint x = −1, the solution uε has a singularity; in a fact, a calculation reveals uε (x) = −

1 ε−2 (x + 1)2−0.45 + o(1). (1 − 0.45)(2 − 0.45)

Singular functions of this type can be approximated well in the context of the hp-FEM by geometric meshes of exactly the type considered here. Outside a small neighborhood of x = −1, we expect the solution uε to be close to u0 (x) := f (x). Again, the geometric mesh of the type considered here is very suitable for the approximation of the function u0 . On such geometric meshes, exponential convergence can be achieved if the number of layers L is chosen proportional to p, the approximation order. For ﬁxed L, asymptotically, only algebraic convergence can be expected, which is visible in Fig. 2.5.8. Also visible is a preasymptotic exponential convergence, and we note that for fairly moderate values of L, the asymptotic algebraic convergence sets in at very small error levels.

2.5 Numerical Examples

53

Example 2.5.2 On the L-shaped domain Ωt = (−1, 1)2 \ (−1, 0)2 , we consider the boundary value problem −ε2 ∆uε + uε = f (x) = #

ex+y

in Ω,

x2 + y 2

uε = 0

on ∂Ω.

The meshes employed are depicted in Fig. 2.5.9. They consist of a combination of the meshes T (min {0.5, pε}, L2 ) of (2.5.3) combined with a classical geometric mesh T geo (L1 , σ1 ) with L1 layers of reﬁnement towards the origin and grading factor σ1 ∈ (0, 1). We refer to Fig. 2.5.9 for an example with L1 = 2; the inset ﬁgure shows the reﬁnement with σ2 = 0.5 and L2 = 2 layers of reﬁnement in the small O(pε)neighborhood of the origin. The computations are performed with σ1 = 0.05 and ε = 10−4 . We compare the case L1 = 0, i.e., simply the mesh T (min {0.5, pε}, L2 ) with the case L1 = 2, which represents a mesh that is suitable for the approximation of u0 . The polynomial degree varies from p = 1 to p = 15. In Fig. 2.5.10 we note that for the case L1 = 2, a polynomial degree p ≈ 4 leads to the same accuracy as the case L1 = 0 with p = 15. Remark 2.5.3 To combine geometric mesh reﬁnement toward the corners starting at distance O(1) with boundary layer meshes has also been proposed and successfully employed for Reissner-Mindlin plate calculations in [132]. Remark 2.5.4 For elliptic systems, the additional problem of stable discretizations arises, especially on meshes with highly anisotropic elements. We mention here [5, 110, 111, 125] for Stokes’ equations. An additional issue is locking; we refer to [112, Sec. 6.3] and the references therein for a discussion of this issue in the context of p- and hp-FEM.

bdy layer mesh at right endpt, fixed geom. mesh, ε =10^(−4)

0

10

bdy layer mesh at right endpt, fixed geom. mesh, ε =10^(−6)

0

10

−2

10

−4

10 Rel. Error in Energy

Rel. Error in Energy

−5

10

−10

−6

10

−8

10

2 layers

10

2 layers

4 layers

−10

10

6 layers

4 layers

8 layers

6 layers

−12

10

8 layers −15

10

10 layers

−14

0

10

1

2

10

10 Degrees of Freedom

3

10

10

0

10

1

2

10

10 Degrees of Freedom

Fig. 2.5.8. Example 2.5.1, ε = 10−4 (left) and ε = 10−6 (right).

3

10

54

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

11 00 00 11

11 00 00 11

11 0 01 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 01 1 01 0 1 0 1 0 1 0 1 0 1 0 1 0 01 1 01 0 1 0 1 0 1 0 1 0 1 0 1 0 01 1 01 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 000 111 0 1 0 1 000 111 0 1 0 1 000 111

11 00 00 11

000 111 000 111

00 11 11 00

Fig. 2.5.9. Example 2.5.2: meshes. σ1= 0.05, σ2 = 0.5, ε= 0.0001

0

rel. error in energy

10

−1

10

−2

10

no geom. ref. geom. ref. (L1=2, L2=2) geom. ref. (L =2, L =4) 1 2 geom. ref. (L =2, L =8) 1 2 −3

10

1

10

2

10

3

10 degrees of freedom

4

10

Fig. 2.5.10. Example 2.5.2: Convergence behavior.

2.6 h-FEM approximation The regularity assertions of Section 2.3 can also be utilized to obtain a priori estimates in the context of low order methods. Robust algebraic convergence can be achieved on suitably designed meshes. We present these ideas in the context of Shishkin meshes. As in the case of the hp-FEM, we start with the one-dimensional case in Section 2.6.1. We then move to the two-dimensional case

2.6 h-FEM approximation

55

in Section 2.6.3, where we introduce a class of Shishkin-type meshes and analyze the performance of the h-FEM on these meshes. 2.6.1 Approximation on Shishkin meshes in one dimension The boundary layer mesh of Deﬁnition 2.2.4 can be viewed as the hp-version analog of the Shishkin mesh, [118]. In the context of the approximation of the solution uε to (2.2.1), the Shishkin meshes may be deﬁned as follows: Deﬁnition 2.6.1 (Shishkin mesh in one dimension). For N ∈ N \ {1} and a transition point κ ∈ (0, 1/2) deﬁne Ω1 := (−1, −1 + min {κ, 0.5}), Ω2 := (−1 + min {κ, 0.5}, 1 − min {κ, 0.5}), Ω3 := (1 − min {κ, 0.5}, 1). Shishkin The Shishkin mesh Tκ,N is then given by the piecewise uniform mesh obtained by placing N elements of equal size min {κ, 0.5}/N in Ω1 , Ω3 and N elements of equal size (2 − min {κ, 0.5})/N in Ω3 .

The h-FEM on a Shishkin mesh is then given by (2.2.5) with ansatz space Shishkin VN = S01 (Tκ,N ). The analog of Proposition 2.2.1 reads Proposition 2.6.2. Let c, f satisfy the assumptions of Proposition 2.2.1, and let uε be the solution to (2.2.1) for ε ∈ (0, 1]. Then there exist C, λ0 > 0 such that for every λ > λ0 , N ∈ N, and transition point κ := min {1/2, λε ln N } inf

Shishkin ) v∈S01 (Tκ,N

uε − vL2 (Ω) + ε(uε − v) L2 (Ω) ≤ C ε1/2 N −1 (λ ln N )3/2 + N −2 (λ ln N )2 .

Shishkin Furthermore, dim S01 (Tκ,N ) = 3N − 1.

Proof: Results of the type presented in Proposition 2.6.2 are by now classical. Proofs for the case c ≡ 1 can be found, e.g., in the monographs [97, 108]. In order to stress the fact that the same regularity results as in the proof of Proposition 2.2.5 are really key to the proof, we sketch the main procedure. Shishkin The statement dim S01 (Tκ,N ) = 3N − 1 follows immediately from the fact Shishkin that the mesh Tκ,N has 3N − 1 internal nodes. Shishkin 1. Step: The case λε ln N ≥ 0.5. The mesh Tκ,N consists of a quasi-uniform mesh with mesh size h ∼ 1/N . Proposition 2.2.1 asserts uε L2 (Ω) ≤ Cε−2 . Thus, standard estimates for the piecewise linear interpolant I give

uε − Iuε L2 (Ω) ≤ Ch2 uε L2 (Ω) ≤ CN −2 ε−2 ≤ CN −2 (λ ln N )2 , ε(uε − Iuε ) L2 (Ω) ≤ Cεhuε L2 (Ω) ≤ Cε1/2 N −1 ε−3/2 ≤ Cε1/2 N −1 (λ ln N )3/2 , which is the desired bound.

56

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

2. Step: The case λε ln N < 0.5. We employ the decomposition uε = wε +uBL ε +rε of (2.2.3). We choose 2 (2.6.7) λ0 := α where α > 0, which measures how fast the boundary layer part decays away from the boundary, is given in Proposition 2.2.1. We approximate each of the terms wε , uBL ε , rε in turn. The standard estimates for the piecewise linear interpolant allows us to infer for the smooth part wε wε − Iwε L2 (Ω) ≤ h2 wε L2 (Ω) ≤ CN −2 , (wε − Iwε ) L2 (Ω) ≤ hwε L2 (Ω) ≤ CN −1 , where we exploited h = maxi hi ≤ CN −1 . Thus, for a C > 0 (which depends on λ0 but is independent of ε, N ) we have for any λ ≥ λ0 : wε − Iwε L2 (Ω) + ε(wε − Iwε ) L2 (Ω) ≤ C ελ3/2 N −1 + λ2 N −2 . For the remainder rε we note that the assumption λε ln N < 0.5 implies α/ε ≥ 2λα ln N. Since rε H 1 (Ω) ≤ Ce−α/ε , we obtain for λ ≥ λ0 in view of (2.6.7) rε L2 (Ω) + rε L2 (Ω) ≤ Ce−2λα ln N ≤ Ce−2λ0 α ln N = CN −4 . We ﬁnally approximate the boundary layer term uBL ε . To that end, we consider the approximation on the subdomains Ω1 , Ω2 , and Ω3 separately. Noting that the mesh size on Ω1 , Ω3 is h = λε ln N/N , we get # BL uBL − IuBL L2 (Ω1 ) ≤ Chε−1 |Ω1 |, ε ε L2 (Ω1 ) ≤ Ch uε # − IuBL L2 (Ω1 ) ≤ Cεh uBL L2 (Ω1 ) ≤ Chε−1 |Ω1 | ε uBL ε ε ε and therefore

BL uBL − IuBL − IuBL L2 (Ω1 ) ε ε L2 (Ω1 ) + ε uε ε # √ ≤ Chε−1 |Ω1 | = Chε−1 λε ln N = Cε1/2 N −1 (λ ln N )3/2 .

By symmetry, a completely analogous estimate holds for Ω3 . On the large subdomain Ω2 , we note that h ∼ N −1 and that dist(Ω2 , ∂Ω) ≥ λε ln N . Hence, using again (2.6.7) and a standard inverse estimate for piecewise linear functions: BL BL uBL − IuBL ε ε L2 (Ω2 ) ≤ C uε L∞ (Ω2 ) + Iuε L∞ (Ω2 )

BL

ε uBL − Iuε ε

L2 (Ω2 )

≤ Ce−αλ ln N ≤ CN −αλ0 = CN −2 , ε ≤ ε uBL L∞ (Ω2 ) + IuBL ε ε L2 (Ω2 ) h ≤ Ce−αλ ln N [1 + εN ] ≤ CN −2 [1 + εN ].

Combining the estimates for wε − Iwε , uBL − IuBL ε ε , and rε gives the desired result. 2

2.6 h-FEM approximation

57

2.6.2 h-FEM meshes In any h-FEM, approximation is achieved by increasing the number of elements, i.e., by reducing the size of the elements. To that end, the notion of “element size” needs to be properly introduced, which is the purpose of the following Deﬁnition 2.6.3. ˆ → M (K) ˆ ⊂ Deﬁnition 2.6.3 (normalizable). An invertible C 2 -map M : K R2 is said to be (CM , κ, h)-normalizable if it can be factored as M = G ◦ S, where the aﬃne stretching map S satisﬁes h0 S = , 0h

(2.6.8)

and the C 2 -map G satisﬁes ∇p Gi L∞ (R) ≤ CM , CM . (G )−1 L∞ (R) ≤ κ

p ∈ {0, 1, 2},

(2.6.9a) (2.6.9b)

ˆ under the aﬃne map S, Here, the set R is the image of the reference element K ˆ i.e., R = S(K).

M

K

1

S

ˆ K = M (K) G

h ˆ R = S(K)

Fig. 2.6.11. Factorization of a normalizable element map M as M = G ◦ S.

Several comments concerning Deﬁnition 2.6.3 are in order. Remark 2.6.4 1. Since in this section we are interested in the approximation from S 1 (T ), analyticity of the element maps is not required; Deﬁnition 2.6.3 therefore restricts the regularity requirement to C 2 . 2. Deﬁnition 2.6.3 allows us to introduce the notion of “element size” for ˆ i ) where, for some non-aﬃne elements: For triangulations T = (Ki , Mi , K ﬁxed CT , each element map Mi is (CT , 1, hi )-normalizable for some suitably chosen hi > 0, it is meaningful to speak of hi as the element size, since hi ∼ diam Ki .

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

58

3. The chain rule allows us to infer the existence of C depending only on CM such that the element map M of a (CM , κ, h)-normalizable map satisﬁes M L∞ (K) ˆ ≤C ,

(M )−1 L∞ (K) ˆ ≤C

1 . κh

(2.6.10)

Singularities of the type arising in solutions to elliptic boundary value problems posed on domains with piecewise smooth boundary can be eﬀectively treated numerically with meshes that are reﬁned towards the singularities. The appropriate reﬁnement strategy is as follows: A mesh T is radically reﬁned towards a point A with reﬁnement exponent µ ∈ [0, 1), if the elements Ki , which are of size hi , satisfy the following dichotomy: Either A ∈ Ki (i.e., Ki abuts on the point A) and hi ∼ h1/(1−µ) µ or A∈ Ki , in which case the element satisﬁes hi ∼ h (dist(Ki , A)) . A more formal deﬁnition of such radical meshes is given in Deﬁnition 2.6.5 below. It diﬀers slightly from the standard notion of radical meshes because it is formulated in view of a later application in the context of Shishkin meshes. There, mesh reﬁnement is required in small neighborhoods of the vertices only; therefore, our notion of radical meshes contains a parameter κ > 0, which controls the size of the region in which mesh reﬁnement takes place. The Shishkin meshes that are the ﬁnal goal of this section are meshes whose elements have very diﬀerent character depending on the location of the element. It is therefore of interest to introduce the notion of a collection of elements: A ˆ i )i∈I , where I is some index set, is said to be a collection of triples (Ki , Mi , K collection of elements, if conditions (M2)–(M4) of Deﬁnition 2.4.1 are satisﬁed. In the following deﬁnition, the reader will recognize the “standard” notion of radical meshes for the special case κ = 1; the parameter µ controls the strength of the reﬁnement near A: ˆ i )i∈I(h) } be a oneDeﬁnition 2.6.5 (radical mesh). Let T (h) = {(Ki , Mi , K parameter family (parametrized by h ∈ (0, 1)) of collections of elements, i.e., ˆ i )i∈I(h) } satisfy (M2)–(M4) of Deﬁnifor each ﬁxed h the triples {(Ki , Mi , K 2 tion 2.4.1. Let A ∈ R , CT , crad > 0, κ ∈ (0, 1], µ ∈ [0, 1) be given. Then T (h) is said to be a (CT , crad , κ, µ)-radically reﬁned, if for each h the elements satisfy the following conditions: 1. Mi is (CT , 1, hi )-normalizable for some suitably chosen hi > 0; hi is called the size of element Ki . 2. Ki ⊂ Bcrad κ (A) for all i ∈ I(h), i.e., the elements Ki are in a neighborhood of A. 3. The following dichotomy holds: Either A ∈ Ki or

and

1/(1−µ) c−1 ≤ rad h

hi ≤ crad h1/(1−µ) κ

(2.6.11a)

2.6 h-FEM approximation

A ∈ Ki

and

hi $ $µ,κ (x), c−1 ≤ crad h inf Φ rad h sup Φµ,κ (x) ≤ x∈Ki κ x∈Ki

where $µ,κ (x) = min Φ

1,

dist(x, A) κ

59

(2.6.11b)

µ .

The parameter µ controls the reﬁnement near the point A. Choosing µ = 0 corresponds to no reﬁnement near A. One way to construct radical meshes is illustrated in the following example: Example 2.6.6 Radical meshes are meshes that are reﬁned toward a boundary point can be constructed by mapping uniform meshes. This is illustrated in Fig. 2.6.12 where a radical mesh on (0, 1)2 is obtained as the image of a uniform −1+1/(1−µ) mesh under the map x → xx∞ ; the exponent µ is chosen as µ = 2/3 in Fig. 2.6.12.

Fig. 2.6.12. (see Example 2.6.6) Radical meshes obtained by mapping uniform meshes: points of uniform mesh (left) are mapped under x → xx2∞ (right).

Essential properties of radical meshes are collected in the following lemma: Lemma 2.6.7. Let the collection T (h) of elements be (CT , crad , µ, h)-radically reﬁned in the sense of Deﬁnition 2.6.5. Then there exists a constant C > 0 depending only on CT , crad , µ, such that for each h ∈ (0, 1]: max diam Ki ≤ Cκh,

i∈I(h)

|T (h)| ≤ Ch−2 , dist(Ki , A) ≥ C −1 κh1/(1−µ)

if A ∈ Ki .

Here, |T (h)| denote the number of elements in T (h). Proof: The third estimate follows immediately from Deﬁnition 2.6.5. The second estimate follows from idea of [21]. 2 An important tool for the approximation from the space S 1 (T ) is the piecewise linear interpolant, which we deﬁne here for completeness’ sake:

60

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

Deﬁnition 2.6.8 (linear interpolation operator I). Let T be a mesh on a domain Ω. The linear operator I : C(Ω) → S 1 (T ),

u → Iu,

is uniquely deﬁned by the condition that u(V ) = (Iu)(V ) for all vertices V of the mesh (vertices of the mesh are the images of the vertices of the reference elements under the element maps). The approximation properties of radical meshes is well understood (see, e.g., [21]). Nevertheless, we formulate an approximation result here in a form that will be convenient for our convergence analysis on Shishkin meshes in Section 2.6.3. Proposition 2.6.9. Let the collection T (h) of elements be (CT , crad , µ, h)radically reﬁned in the sense of Deﬁnition 2.6.5. Let β ∈ [0, 1) and set Ωh := ∪i∈I(h) Ki . Then the piecewise linear interpolant Iu of a function u satisﬁes $β,κ ∇2 uL2 (Ω ) + C(κh)2 uH 1 (Ω ) , u − IuL2 (Ωh ) ≤ Cκ2 h1+δ Φ h h δ $ 2 ∇(u − Iu)L2 (Ω ) ≤ Cκh Φβ,κ ∇ uL2 (Ω ) + C(κh)uH 1 (Ω ) , h

h

h

provided the function u is such that the right-hand sides are ﬁnite. Here, 1−β δ = min 1, . 1−µ Proof: For ﬁxed h, we decompose the set I(h) as I(h) = I int ∪ IA , where I int = {i ∈ I(h) | A ∈ Ki },

IA = {i ∈ I(h) | A ∈ Ki } = I(h) \ I int .

Next, we observe that Deﬁnition 2.6.5 implies that all elements Ki satisfy Ki ⊂ Bcrad κ (A) and $β,κ (x) ∼ Φ

dist(x, A) κ

β ∀x ∈ Bcrad κ (A),

(2.6.12)

with implied constants independent of x and κ. The assumption that the element maps Mi are (CT , 1, hi ) normalizable implies ˆi → that each Mi can be factored as Mi = Gi ◦ Si , where the aﬃne map Si : K 2 ˆ Si (Ki ) satisﬁes Si = hi and the map Gi is a C -diﬀeomorphism between Ri := ˆ i ) and Ki with implied constants depending only on CT . We set ui := Si (K u ◦ Gi . For simplicity of notation, we assume that the aﬃne maps Si are of the form Si (x) = hi x; note that this implies that the origin 0 is a vertex of Ri . Additionally, we require that A ∈ Ki implies Gi (0) = A. The fact that the maps Gi are C 2 -diﬀeomorphisms gives the existence of C > 0 (depending only on CT and µ) such that: ∇2 ui L2 (Ri ) ≤ C ∇2 uL2 (Ki ) + uH 1 (Ki ) ∀ i ∈ I int , (2.6.13a) $β,κ ∇2 uL2 (K ) + hβ uH 1 (K ) ∀i ∈ IA ; (2.6.13b) rβ ∇2 ui L2 (Ri ) ≤ C κβ Φ i i i

2.6 h-FEM approximation

61

here, we employed the shorthand r = r(x) = |x|. ˆ i ) is square or a triangle of diameter O(hi ), standard interpoSince Ri = Si (K lation estimates (see, e.g., Proposition 3.2.21 for the case p = 1 together with a scaling argument) yield for the error of the linear interpolant ! , i ∈ I int h2 ∇2 ui L2 (R ) ui − Iui L2 (Ri ) +hi ∇(ui − Iui )L2 (Ri ) ≤ C i2−β β 2 i hi r ∇ ui L2 (Ri ) , i ∈ IA . Exploiting the fact that the maps Gi are C 1 -diﬀeomorphisms gives us: ∇(u − Iu)2L2 (Ωh ) ≤ C ∇(ui − Iui )2L2 (Ri ) + C ∇(ui − Iui )2L2 (Ri ) . i∈I int

i∈IA

We treat the two sums separately. For the second sum, we get using (2.6.12) 2(1−β) ∇(ui − Iui )2L2 (Ri ) ≤ C hi rβ ∇2 ui 2L2 (Ri ) i∈IA

≤C

i∈IA 2(1−β) $ κ2β hi Φβ,κ ∇2 u2L2 (Ki )

+C

i∈IA

h2i u2H 1 (Ki ) .

i∈IA

Since the assumptions on the mesh imply hi ≤ Cκh1/(1−µ) ≤ Cκh for i ∈ IA , we get ∇(ui − Iui )2L2 (Ri ) i∈IA

2 2 2 $β,κ ∇2 u2 2 ≤ C κ2 h2(1−β)/(1−µ) Φ + κ h u 1 L (Ωh ) H (Ωh ) $β,κ ∇2 uL2 (Ω ) + Cκ2 h2 u2 1 ≤ Cκ2 h2δ Φ H (Ωh ) . h

For

i∈I int

∇(ui − Iui )2L2 (Ri ) we bound

∇(ui − Iui )2L2 (Ri ) ≤ C

i∈I int

≤C

h2i ∇2 ui 2L2 (Ri )

i∈I int

h2i ∇2 u2L2 (Ki ) + u2H 1 (Ki )

i∈I int

≤C

h2i ∇2 u2L2 (Ki ) + Cκ2 h2 u2H 1 (Ωh )

i∈I int

≤C

i∈I int

inf x∈Ki

h2i 2 2 2 $ ∇2 u2 2 Φ L (Ki ) + Cκ h uH 1 (Ω) . $β,κ (x)|2 β,κ |Φ

In view of (2.6.12) and Lemma 2.6.7, we can estimate sup x∈Ki

$µ,κ (x) Φ ≤ Ch−1+δ . $ Φβ,κ (x)

62

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

Thus, we get

  2 h h2i i  sup ≤ $β,κ (x)|2 $µ,κ (x)|2 x∈Ki inf |Φ inf |Φ

x∈Ki

x∈Ki

$µ,κ (x) Φ $β,κ (x) Φ

2 ≤ crad h2 h−2+2δ = crad h2δ .

Combining the above estimate yields the bound for ∇(u − Iu)L2 (Ωh ) . Analogous reasoning yields $β,κ ∇2 uL2 (Ω ) + (κh)2 uH 1 (Ω ) . u − IuL2 (Ωh ) ≤ C κ2 h1+δ Φ h h 2 2.6.3 h-FEM boundary layer meshes Our aim is the introduction of meshes T such that robust approximation of solution of (2.1.1) from the h-FEM spaces S 1 (T ) is possible. Such meshes T need to provide the capability to resolve boundary layers and corner singularities. As in the case of admissible meshes in Deﬁnition 2.4.4, the approximation of boundary layers is made possible with anisotropic boundary layer elements, whose aspect ratio is controlled by a parameter κ; the resolution of corner singularities is achieved with radical meshes, where the reﬁnement is near a vertex Aj is controlled by a parameter µj ∈ [0, 1). Deﬁnition 2.6.10 (h-FEM boundary layer mesh). Consider a three-paraˆ i )}, where the three parameters satisfy meter family T (κ, h, µ) = {(Ki , Mi , K κ > 0, h > 0, and µ ∈ [0, 1)J . This family T is said to be of boundary layer type if there are ci , i = 1, . . . , 4, σ ∈ (0, 1), CM > 0, and sets Ωj , j = 1, . . . , J, of the form given in Notation 2.3.3 such that the elements Ki of the meshes fall into exactly one of the following three categories: (C1) Ki is a boundary layer element, i.e., for some j ∈ {1, . . . , J} we have the inclusion Ki ⊂ Uκ (Γj ) ∩ Ωj \ (Bc1 κ (Aj−1 ) ∪ Bc1 κ (Aj )); additionally the map −1 ˜ i := s−1 G κ ◦ ψ j ◦ Mi

is (CM , 1, h)-normalizable in the sense of Deﬁnition 2.6.3. (C2) Ki is a corner layer element, i.e., for some j ∈ {1, . . . , J}, the element Ki satisﬁes Ki ⊂ Bκ (Aj ) ∩ Ωj or Ki ⊂ Bκ (Aj ) ∩ Ωj+1 , and Mi is (CM , 1, hi )normalizable in the sense of Deﬁnition 2.6.3, where the parameter hi = diam Ki > 0 is the diameter of the element Ki . Additionally, the following dichotomy holds: Either hi Aj ∈ Ki and ≤ c4 h1/(1−µj ) κ or Aj ∈ Ki

and

c3 h sup Φ0,µ,κ (x) ≤ x∈Ki

hi ≤ c4 h sup Φ0,µ,κ (x). κ x∈Ki

2.6 h-FEM approximation

63

(C3) Ki is an interior element, i.e., Ki ⊂ Ω \ Uc2 κ (∂Ω) and Mi is (CM , κ, h)normalizable in the sense of Deﬁnition 2.6.3. The conditions placed on the diﬀerent types of elements in Deﬁnition 2.6.10 are best understood by considering the following examples of meshes. Example 2.6.11 For ﬁxed κ0 > 0 and µ0 ∈ [0, 1)J , a one-parameter family T (κ0 , h, µ0 ) of meshes coincides with a “standard” radical mesh for the approximation of solutions of elliptic boundary value problems posed on domains with piecewise smooth boundary in the following sense: The mesh family T (κ0 , h, µ0 ) can be viewed as a generalization to the context of non-aﬃne element maps of standard radical meshes of [21,106], which consist of shape-regular elements and the reﬁnement toward the vertices of the domain is such that the element size hi satisﬁes (2.6.11). Example 2.6.12 The mesh shown in the left part of Fig. 2.6.13 is the tensorproduct of the one-dimensional Shishkin mesh of Deﬁnition 2.6.1. It is an h-FEM boundary layer mesh in the sense of Deﬁnition 2.6.10 for the special case µj = 0, j = 1, . . . , 4. This special choice of reﬁnement parameters µj corresponds to not performing mesh grading in the neighborhood of the vertices. For the present case of convex corners and an intended approximation by piecewise linear/bilinear functions, this is completely adequate as will be shown in Theorem 2.6.15 and the discussion following Theorem 2.6.15. Example 2.6.13 The right part of Fig. 2.6.13 shows an example of a boundary layer mesh for an L-shaped domain. In the shaded regions near the re-entrant corner, reﬁned meshes are inserted that can be constructed as illustrated in Example 2.6.6 by mapping a uniform mesh with mesh size h on (0, 1)2 under −1+1/(1−µ) the map x → xx∞ and afterwards inserting scaled (by κ) and rotated versions into the three shaded regions.

κ

h = 1/N

κ

h = 1/N

h = κ/N

Fig. 2.6.13. Left: Shishkin-type mesh on a square with µj = 0, j = 1, . . . , 4. Right: Shishkin-type mesh on L-shaped domain.

64

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

From the regularity result Theorem 2.3.4, we can extract the following simpliﬁed version that is suitable for approximating with piecewise linear functions: Proposition 2.6.14. Let the subdomains Ωj , j = 1, . . . , J, and the boundary ﬁtted coordinates (ρj , θj ) be as in Theorem 2.3.4. Write r(x) = min dist(x, Aj ). j=1,...,J

Let β ∈ [0, 1)J be given by the statement of Theorem 2.3.4 and write 1 − β for the vector (1 − β1 , . . . , 1 − βJ ). Then there exist constants C, α > 0 such that for each ε ∈ (0, 1] the solution uε of (2.1.1) can be decomposed as uε = wε + u ˜BL +u ˜CL + rε , ε ε with the following properties: (i) wε ∈ C 2 (Ω) and wε L∞ (Ω) + ∇wε L∞ (Ω) + ∇2 wε L∞ (Ω) ≤ C. (ii) On Ωj the function u ˜BL satisﬁes ε |(∂ρkj ∂θmj (˜ uBL ◦ ψj ))(ρj , θj )| ≤ Cε−k e−αρj /ε , ε

k, m ∈ {0, 1, 2}.

˜CL satisﬁes (iii) On Ωj , j = 1, . . . , J, the function u ε −k |˜ uCL Φ0,1−β,ε (x) e−αr/ε , ε (x)| ≤ Cr

k ∈ {0, 1, 2}.

(iv) uC := u ˜BL +u ˜CL is in C 1 (Ω \ ∪Jj=1 {Aj }) and for all x ∈ Ω: ε ε |∇k uC (x)| ≤ C ε−k e−α dist(x,∂Ω)/ε + r−k Φ0,1−β,ε (x) e−αr/ε ,

k ∈ {0, 1}.

2 Additionally, uC ∈ Hloc (Ω) and in neighborhoods of the vertices Aj , we have for κ > 0 and j ∈ {1, . . . , J}: √ uC L2 (Ω∩Bκ (Aj )) + ε∇uC L2 (Ω∩Bκ (Aj )) ≤ C κε + ε , √ κε−3/2 + ε−1 . Φ0,β,ε ∇2 uC L2 (Ω∩Bκ (Aj )) ≤ C 2 (v) rε ∈ Hloc (Ω) ∩ H01 (Ω) and

rε ε + Φ0,β,ε ∇2 rε L2 (Ω) ≤ Ce−α/ε . Proof: The proposition is a corollary to Theorem 2.3.4. Let wε , rε be the functions given in Theorem 2.3.4. Assertions (i), (v) follow immediately from Theorem 2.3.4. Next, we deﬁne u ˜BL := χBL uBL ε ε ,

u ˜CL := χCL uCL ε ε ,

2.6 h-FEM approximation

65

CL where the cut-oﬀ functions χBL , χCL and the functions uBL are deﬁned ε , uε CL asin Theorem 2.3.4. The result then follows from the properties of uBL ε , uε C certained in Theorem 2.3.4. For example, for the bound on u L2 (Bκ (Aj )) we compute

uC L2 (Bκ (Aj )) ≤ ˜ uBL uCL uBL ε L2 (Bκ (Aj )) + ˜ ε L2 (Bκ (Aj )) ≤ ˜ ε L2 (Bκ (Aj )) + Cε, where we employed Theorem 2.3.4 to bound ˜ uCL ε L2 (Bκ (Aj )) ≤ Cε. Next, using again Theorem 2.3.4, we estimate κ κ 2 ˜ uBL ≤ C e−αρ/ε dρ ≤ Cκε, 2 ε L (Bκ (Aj )) 0

0

which leads to the desired bound.

2

On boundary layer meshes we can formulate the following approximation result, which is the h-FEM analog of Theorem 2.4.8: Theorem 2.6.15. Let T (κ, h, µ) be a family of boundary layer meshes in the sense of Deﬁnition 2.6.10. Let uε be the solution to (2.1.1) and let β ∈ [0, 1)J be given by Proposition 2.6.14. Then there exist λ0 > 0 and C > 0 independent of ε and h ∈ (0, 1/2) with the following properties: Setting, for each λ ≥ λ0 and µ ∈ [0, 1)J , κ := min {1, λε| ln h|}, 1 − βj δ := min 1, j=1,...,J 1 − µj there exists v ∈ S 1 (T (κ, h, µ)) with v = Iuε on ∂Ω and uε − vL2 (Ω) + κh∇(uε − v)L2 (Ω) ≤ C εh1+δ |λ ln h|3 + ε1/2 h2 |λ ln h|5/2 + h2 |λ ln h| . Proof: As in the one-dimensional case, Proposition 2.6.2, we distinguish the asymptotic case λε| ln h| > 1 and the preasymptotic case λε| ln h| ≤ 1. The case λε| ln h| > 1: For ﬁxed µ ∈ [0, 1)J , we consider boundary layer meshes T (1, h, µ), which is a one-parameter family of meshes as described in Example 2.6.11. Proposition 2.6.9 then yields for the piecewise linear interpolant Iuε of uε : uε − Iuε L2 (Ω) ≤ C h1+δ Φ0,β,1 ∇2 uε L2 (Ω) + h2 uε H 1 (Ω) , ∇(uε − Iuε )L2 (Ω) ≤ C hδ Φ0,β,1 ∇2 uε L2 (Ω) + huε H 1 (Ω) . Noting that ελ| ln h| > 1 implies κ = 1, we get using Φ0,β,1 (x) ≤ Φ0,β,ε (x) and the estimate of Theorem 2.3.1

66

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

uε − Iuε L2 (Ω) + κh∇(uε − Iuε )L2 (Ω) ≤ C h1+δ Φ0,β,ε ∇2 uε L2 (Ω) + h2 uε H 1 (Ω) ≤ C h1+δ ε−2 + h2 ε−1 ≤ C εh1+δ ε−3 + h2 ε−1 ≤ C εh1+δ |λ ln h|3 + h2 |λ ln h| , where, in the last step we used the assumption λ| ln h| ≥ ε−1 . The case λε| ln h| ≤ 1: In this case, we employ the decomposition given by Proposition 2.6.14, i.e., uε = wε + u ˜BL +u ˜CL + rε . ε ε

(2.6.14)

We approximate the function rε by zero; in view of Proposition 2.6.14 we obtain with κ = λε| ln h| rε L2 (Ω) + κh∇rε L2 (Ω) ≤ (1 + hλ| ln h|)rε ε ≤ C(1 + hλ| ln h|)e−α/ε ≤ C(1 + hλ| ln h|)e−αλ| ln h| ≤ C(1 + hλ| ln h|)hαλ ≤ Ch2 for λ ≥ λ0 suﬃciently large. It remains to approximate the function wε + u ˜BL ˜CL ε +u ε . To that end, we denote uC := u ˜BL +u ˜CL ε ε and consider the error of the piecewise linear interpolant, e := (wε + uC ) − I(wε + uC ) = (wε − Iwε ) + (uC − IuC ) uBL − Iu ˜BL uCL − Iu ˜CL = (wε − Iwε ) + (˜ ε ε ) + (˜ ε ε ).

(2.6.15) (2.6.16)

We will use both representations of the error e. To that end, we consider the approximation on each the three types of elements, namely, boundary layer elements, corner layer elements, and interior elements, separately. For notational convenience, we write the set IT of element indices i as the pairwise disjoint union IT = I int ∪ I BL ∪ I CL of indices corresponding to interior elements, boundary layer elements, and corner layer elements, respectively. 1. step: boundary layer elements. Let Ki be a boundary layer element with Ki ⊂ Ωj . Our aim is to show wε − Iwε L∞ (Ki ) + κh∇(wε − Iwε )L∞ (Ki ) ≤ Ch2 , ˜ uBL ε ˜ uCL ε

− Iu ˜BL ε L∞ (Ki ) − Iu ˜CL ε L∞ (Ki )

+ κh∇(˜ uBL ε + κh∇(˜ uCL ε

− Iu ˜BL ε )L∞ (Ki ) − Iu ˜CL ε )L∞ (Ki )

(2.6.17a)

≤ Ch |λ ln h| , (2.6.17b) 2

≤ Ch2 .

2

(2.6.17c)

−1 ◦ Mi By assumption the element map Mi is such that the maps s−1 κ ◦ ψj −1 −1 is (CT , 1, h)-normalizable, i.e., sκ ◦ ψj ◦ Mi = Gi ◦ Si , where Gi is a C 2 diﬀeomorphism and Si = hi . The formula Mi = (ψj ◦ sκ ◦ Gi ) ◦ Si shows that Mi is (C , κ, h)-normalizable with a constant C > 0 that is independent of κ and i.

2.6 h-FEM approximation

67

We start with proving (2.6.17a). Using the fact that Mi is (C , κ, h) -normalizable and that wε ∈ C 2 (Ω) (cf. Proposition 2.6.14), we get wε ◦ Mi − I(wε ◦ Mi )W 1,∞ (Kˆ i ) ≤ Ch2 .

(2.6.18)

Returning to Ki , we get in view of the fact that Mi is (C , κ, h)-normalizable, wε − Iwε L∞ (Ki ) + κh∇(wε − Iwε )L∞ (Ki ) ≤ Ch2 ,

(2.6.19)

which is (2.6.17a). ˆ i , we have We now turn to the proof of (2.6.17b). On the reference element K −1 u ˜BL ◦ Mi = (˜ uBL ◦ ψj ◦ sκ ) ◦ (s−1 ˜i ◦ (Gi ◦ Si ). ε ε κ ◦ ψj ◦ Mi ) =: u

Proposition 2.6.14 implies k

∇k u ˜i L∞ (R˜ i ) ≤ C{1 + (κ/ε) },

k ∈ {0, 1, 2},

(2.6.20)

˜ i = (Gi ◦ Si )(K ˆ i ). Since Gi is a C 2 -diﬀeomorphism, we get where R 2 uBL ◦ Mi )L∞ (K) ∇2 (˜ ˆ ≤ Ch ε

2

k

{1 + (κ/ε) }.

k=0

Inserting κ = λε| ln h| gives for λ ≥ λ0 2 2 ∇2 (˜ uBL ◦ Mi )L∞ (K) ˆ ≤ Ch (λ ln h) . ε

ˆi Therefore for the interpolation error on the reference element K ˜ uBL ◦ Mi − I(˜ uBL ◦ Mi )W 1,∞ (Kˆ i ) ≤ Ch2 (λ ln h)2 ; ε ε

(2.6.21)

thus, on Ki we get, again due to the fact that Mi is (C , κ, h)-normalizable, 2 2 ˜ uBL − Iu ˜BL uBL − Iu ˜BL ε ε L∞ (Ki ) + κh∇(˜ ε ε )L∞ (Ki ) ≤ Ch (λ ln h) ,

which is (2.6.17b). We ﬁnally turn to (2.6.17c), where we simply exploit the fact that u ˜CL is small ε away from the vertices. We start with the observation that the fact that Mi is (C , κ, h)-normalizable and that a standard inverse estimate for linear functions on the reference element gives ∇I u ˜CL ε L∞ (Ki ) ≤ C

1 ˜ uCL L∞ (Ki ) . κh ε

This bound allows us to estimate ˜ uCL − Iu ˜CL uCL ε ε L∞ (Ki ) ≤ 2˜ ε L∞ (Ki ) , ∇(˜ uCL − Iu ˜CL uCL ε ε )L∞ (Ki ) ≤ ∇˜ ε L∞ (Ki ) + C

1 ˜ uCL L∞ (Ki ) . κh ε

68

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

Proposition 2.6.14 together with dist(Ki , Aj ) ≥ cκ for all j ∈ {1, . . . , J} imply

−α κ/ε ˜ uCL − Iu ˜CL uCL − Iu ˜CL ε ε L∞ (Ki ) + κh∇(˜ ε ε )L∞ (Ki ) ≤ Ce

for some C, α > 0 independent of ε, h. Taking λ0 > 0 suﬃciently large, we get 2 ˜ uCL − Iu ˜CL uCL − Iu ˜CL ε ε L∞ (Ki ) + κh∇(˜ ε ε )L∞ (Ki ) ≤ Ch ,

which is (2.6.17c). Since all boundary layer elements are contained in an O(κ)-neighborhood of ∂Ω, we get from the L∞ -estimates of (2.6.17) for the error e of (2.6.16) by summing over all elements: 1/2 √ √ 2 2 2 eL2 (Ki ) + (κh) ∇eL2 (Ki ) ≤ C κh2 (λ ln h)2 ≤ C εh2 |λ ln h|5/2 . i∈I BL

This is the desired bound on boundary layer elements. 2. step: corner layer elements. On corner layer elements, we use ProposiC tion 2.6.9. For wε + (˜ uBL +u ˜CL ε ε ) =: wε + u , we note that Proposition 2.6.14 yields # wε + uC H 1 (Bκ (Aj )∩Ω) ≤ C κ/ε + 1 , √ √ Φ0,β,ε ∇2 (wε + uC )L2 (Bκ (Aj )∩Ω) ≤ C κε−3/2 + ε−1 + κ ≤ C κε−3/2 + ε−1 where we used the hypotheses κ ≤ 1 and ε ≤ 1. Proposition 2.6.9 then allows us to estimate for the error e 1/2 √ 2 2 2 eL2 (Ki ) + (κh) ∇eL2 (Ki ) ≤ κ2 h1+δ κε−3/2 + ε−1 i∈I CL

≤ Cεh1+δ |λ ln h|5/2 , where we inserted κ = λε| ln h| and used the assumptions λ ≥ λ0 , h ≤ 1/2. 3. step: interior elements. Let Ki be an interior element. Reasoning exactly as in the proof of (2.6.17a), we get wε − Iwε L∞ (Ki ) + κh∇(wε − Iwε )L∞ (Ki ) ≤ Ch2 . Likewise, reasoning as in the proof of (2.6.17c), we can estimate for uC in view of the fact that dist(Ki , ∂Ω) ≥ cκ: uC − IuC L∞ (Ki ) + κh∇(uC − IuC )L∞ (Ki ) ≤ Ch2

(2.6.22)

for suitably chosen λ0 > 0. Squaring and summing these two estimates gives

i∈I int

1/2 e2L2 (Ki )

2

+ (κh)

∇e2L2 (Ki )

≤ Ch2 .

2.6 h-FEM approximation

69

Combining the estimates for boundary layer elements, corner layer elements, and interior element proves the theorem. 2 A few comments concerning Theorem 2.6.15 are in order. Remark 2.6.16 1. The corner singularities are weak: The regularity assertion of Proposition 2.6.14 shows that the corner singularities are restricted to an O(ε) neighborhood of the vertices. The factor ε in front of the term h1+δ reﬂects this. Note that if no reﬁnement in the O(κ)-neighborhoods is performed, i.e., a quasi-uniform mesh with mesh size hκ is used, this corresponds to µ = 0, i.e., δ < 1. Nevertheless, the factor ε in the term εh1+δ mitigates this neglect of reﬁnement. 2. Results analogous to Theorem 2.6.15 have been obtained in [8] and [7, Chap. 5], where also numerical examples can be found. We now show that this estimate can be improved if further assumptions on the mesh are made. We note that the interior elements of h-FEM boundary layer meshes of Deﬁnition 2.6.10 may be quite distorted. An inspection of the proof of Theorem 2.6.15 reveals that additional assumptions on the element maps for interior elements allow us to improve the approximation of the smooth part wε . This observation is formulated in the Corollary 2.6.19 below. The key observation is that the notion of normalizable element maps of Deﬁnition 2.6.3 can be generalized to anisotropic elements in the following form: ˆ → M (K) ˆ ⊂ R2 is said to be Deﬁnition 2.6.17. An invertible C 2 -map M : K anisotropically (CM , hx , hy )-normalizable if it can be factored as M = G ◦ S, where the aﬃne stretching map S satisﬁes hx 0 S = , 0 hy and the C 2 -map G satisﬁes ∇p Gi L∞ (R) ≤ CM , −1

(G )

p ∈ {0, 1, 2},

L∞ (R) ≤ CM

ˆ under the aﬃne map S, Here, the set R is the image of the reference element K ˆ i.e., R = S(K). For the approximation on anisotropic elements in the interior, we will need the following lemma concerning the interpolation error on anisotropic elements: Lemma 2.6.18. Let hx , hy > 0 and let R be either the rectangle (0, hx )×(0, hy ) or the triangle with vertices (0, 0), (hx , 0), (0, hy ). Then there exists C > 0 independent of hx , hy such that for every u ∈ C 2 (R) the linear/bilinear interpolation Iu in the vertices of R leads to the following errors:

70

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

u − IuL∞ (R) ≤ Ch2 ∇2 uL∞ (R) , ∂x (u − Iu)L∞ (R) + ∂y (u − Iu)L∞ (R) ≤ Ch∇2 uL∞ (R) , where h = max {hx , hy }. Proof: Approximation results of this type can be found in [7, Chap. 2]. For the sake of completeness, however, we include a proof of the present simple case. We deﬁne the function u ˆ by u ˆ = u◦S, where S is the anisotropic stretching given ˆ := S −1 (R), which is the reference square (0, 1)2 by S : (x, y) → (xhx , yhy ). Set K if R is a rectangle and the reference triangle {(x, y) | 0 < x < 1, 0 < y < 1 − x} if R is a triangle. We claim the following interpolation error for u ˆ: 2 ˆ u − Iu ˆL∞ (K) ˆL∞ (K) ˆ ≤ C∇ u ˆ , 2 ∂x (ˆ , u − I y ◦ I xu ˆ)L∞ (K) ˆL∞ (K) ˆL∞ (K) ˆ ≤ C ∂x u ˆ + ∂x ∂y u ˆ

(2.6.24) (2.6.25)

and an analogous estimate for ∂y (ˆ u − I y ◦ I xu ˆ)L∞ (K) ˆ . The desired bounds follows from (2.6.24), (2.6.25) from the fact u ˆ = u ◦ S and Iu = (I u ˆ) ◦ S −1 . It remains to show (2.6.24), (2.6.25). The bound (2.6.24) is standard. For the ˆ being the reference square and K ˆ bound (2.6.24), we consider the cases of K being the reference triangle separately. ˆ is the reference square: We note the following bounds for univariate functions: K w − IwL∞ ((0,1)) ≤ Cw L∞ ((0,1))

∀w ∈ C 1 ([0, 1]), (2.6.26a)

w − IwL∞ ((0,1)) ≤ Cw L∞ ((0,1))

∀w ∈ C 2 ([0, 1]), (2.6.26b)

(w − Iw) L∞ ((0,1)) ≤ Cw L∞ ((0,1))

∀w ∈ C 2 ([0, 1]). (2.6.26c)

The interpolation operator I has the form I = I y ◦ I x , where I x , I y denote the one-dimensional linear interpolation operator acting on the x- and the yvariable, respectively. We then get with the triangle inequality and the fact that the operators I y , ∂x commute: y ∂x (ˆ u − I y ◦ I xu ˆ)L∞ (K) ˆ − I y ∂x u ˆL∞ (K) u − I xu ˆ))L∞ (K) ˆ ≤ ∂x u ˆ + I (∂x (ˆ ˆ .

Next, exploiting the one-dimension estimates (2.6.26) allows us to bound ∂x u ˆ − I y ∂x u ˆL∞ (K) ˆ = sup

sup |∂x u ˆ(x, y) − I y ∂x u ˆ(x, y)|

x∈(0,1) y∈(0,1)

≤ C sup ∂y ∂x u ˆ(x, ·)L∞ ((0,1)) , x∈(0,1)

I (∂x (ˆ u−I u ˆ))L∞ (K) u(·, y) − I x u ˆ(·, y))L∞ ((0,1)) ˆ ≤ sup ∂x (ˆ y

x

y∈(0,1)

≤ C sup ∂x ∂x u ˆ(·, y)L∞ ((0,1)) , y∈(0,1)

which is (2.6.25).

2.6 h-FEM approximation

71

ˆ is the reference triangle: The derivative ∂x I u ˆ is a constant, which is computed K in (3.2.49) as 1 ∂x I u ˆ= ∂x u ˆ(x, 0) dx. 0

From the mean value theorem, we conclude that there exists ξ ∈ (0, 1) such that ˆ the mean value theorem applied to ∂x I u ˆ = ∂x u ˆ(ξ, 0). For arbitrary (x, y) ∈ K, ˆ the function t → ∂x u ˆ(tx + (1 − t)ξ, ty) then gives in view of the convexity of K ∗ the existence of t ∈ (0, 1) such that |∂x u ˆ(x, y) − ∂x I u ˆ| = |∂x u ˆ(x, y) − ∂x u ˆ(ξ, 0)| ≤ ∂x2 u ˆL∞ (K) ˆL∞ (K) ˆ + ∂y ∂x u ˆ . ˆ gives (2.6.25). Taking the supremum over all (x, y) ∈ K

2

We now come to the improvement of Theorem 2.6.15 when the element maps of interior elements are anisotropically (CM , h, hκ)-normalizable: Corollary 2.6.19. Assume the hypotheses and notations of Theorem 2.6.15. Assume additionally that the mesh family T (κ, h, µ) satisﬁes the following condition: each interior element is anisotropically (CM , h, hκ)-normalizable in the sense of Deﬁnition 2.6.17. Then there exist C, λ0 > 0 independent of ε and h ∈ (0, 1/2) such that for every λ ≥ λ0 and κ = min {1, λε| ln h|} there exists v ∈ T (κ, h, µ) with v = Iuε on ∂Ω and uε − vL2 (Ω) + κ∇(uε − v)L2 (Ω) ≤ C εhδ |λ ln h|3 + ε1/2 h|λ ln h|5/2 + h2 . Proof: We ﬁrst note that the assumption that interior elements be anisotropically (CM , h, hκ)-normalizable implies that they are (CM , κ, h)-normalizable in the sense of Deﬁnition 2.6.3 since κ ≤ 1. Thus, most of the arguments in the proof of Theorem 2.6.15 can be employed in the present situation as well; we will therefore merely highlight the main diﬀerences. The case ελ| ln h| > 1: Inspection of the arguments in the proof of Theorem 2.6.15 leads to uε − Iuε L2 (Ω) + κ∇(uε − Iuε )L2 (Ω) ≤ C hδ ε−2 + hε−1 + h1+δ ε−2 + h2 ε−2 ≤ Cεhδ ε−3 ≤ Cεhδ |λ ln h|3 . The case ελ| ln h| ≤ 1: Arguing as in the proof of Theorem 2.6.15 allows us to bound for suitably chosen λ0 : rε L2 (Ω) + κ∇rε L2 (Ω) ≤ Ch2 . Concerning the boundary layer elements, we see that (2.6.17) holds; therefore arguing as in the proof of Theorem 2.6.15, we obtain for the error e given by (2.6.16)

2. hp-FEM for Reaction Diﬀusion Problems: Principal Results

72

1/2 e2L2 (Ki ) + κ2 ∇e2L2 (Ki )

√ √ ≤ C κh(λ ln h)2 ≤ C εh|λ ln h|5/2 .

i∈I BL

For the corner layer elements, we check the proof of Theorem 2.6.15 to see that

1/2 e2L2 (Ki )

+κ

2

∇e2L2 (Ki )

≤ Cεhδ |λ ln h|5/2 .

i∈I CL

We ﬁnally turn to the interior elements Ki . Since the corresponding element map Mi is anisotropically (CM , h, hκ)-normalizable, it can be factored as Mi = Gi ◦Si , where Gi is a C 2 -diﬀeomorphism and Si is aﬃne with h 0 . Si = 0 hκ ˆ i ). If K ˆ i is the unit square, then Ri is congruent to the We deﬁne Ri := Si (K ˆ i is the reference triangle, rectangle with vertices (0, 0), (h, 0), (0, hκ), (h, hκ); if K then Ri is congruent to the triangle with vertices (0, 0), (h, 0), (0, hκ). Since the function wi := wε ◦ Gi satisﬁes ∇2 wi L∞ (Ri ) ≤ C for a constant C > 0 independent of ε and the element Ki , we conclude with Lemma 2.6.18 for the interpolation error wi − Iwi L∞ (Ri ) + h∇(wi − Iwi )L∞ (Ri ) ≤ Ch2 ; transforming this result to the element Ki gives, since Gi is a C 2 -diﬀeomorphism wε − Iwε L∞ (Ki ) + h∇(wε − Iwε )L∞ (Ki ) ≤ Ch2 .

(2.6.27)

Inspection of the arguments leading to (2.6.22) reveals that λ0 can be chosen such that the factor h2 can be replaced with h3 (in fact, arbitrary powers of h can be obtained), i.e., uC − IuC L∞ (Ki ) + κh∇(uC − IuC )L∞ (Ki ) ≤ Ch3 . We therefore obtain for the error e 1/2 e2L2 (Ki ) + κ2 ∇e2L2 (Ki ) ≤ C h2 + κh ≤ C h2 + εh|λ ln h| , i∈I int

which can be bounded in the desired fashion.

2

3. hp Approximation

3.1 Motivation and outline 3.1.1 General overview of Chapter 3 The aim of the present chapter is Theorem 3.4.8, where an hp-approximant to solutions of (1.2.11) on the minimal meshes of Deﬁnition 2.4.4 is constructed. The hp-FEM approximation result Theorem 2.4.8 then follows from Theorem 3.4.8. The actual proof of the approximation result Theorem 3.4.8 is very technical; in Section 3.1.3, we therefore sketch the key steps of the proof. The scope of the present chapter, however, goes beyond proving the approximation result Theorem 3.4.8 on admissible boundary layer meshes. This chapter addresses several issues pertinent to the design of hp-FEMs in general and to hp-FEMs for singularly perturbed problems in particular. Two lines of ideas are developed in this chapter in parallel: 1. The main line of ideas is concerned with the design of minimal meshes and the analysis of the hp-FEM for our model problem (1.2.11) on these meshes. It can be traced through the introduction of the polynomial projection op∞ erator Πp∞ in Theorem 3.2.20, the projector Πp,T in (3.3.3), the notion of admissible boundary layer meshes in Deﬁnition 2.4.4, the analysis of the ap∞ proximation properties of the operator Πp,T on admissible boundary layer meshes in Section 3.4, and ﬁnally its application to the FEM in Section 2.4.3. 2. In the second line of ideas, the construction and analysis of anisotropic meshes that have more structure than the minimal meshes is explored. The development of these ideas can be seen in the deﬁnition of the projection operator Πp1,∞ and the notion of regular admissible boundary layer meshes in Deﬁnition 3.3.10. This line of thought culminates in the introduction of mesh patches in Sections 3.3.2–3.3.4. Let us discuss the ﬁrst line of ideas in more detail. Central is the notion of admissible boundary layer meshes already introduced in Deﬁnition 2.4.4. These meshes are motivated by the description of the solution behavior in Section 2.3 (cf. Theorems 2.3.1, Theorem 2.3.4). In particular, in Theorem 2.3.4 we characterized the solution behavior in terms of asymptotic expansions. This decomposition suggests the main features of the hp-FEM to be used: thin needle elements near the boundary to capture the boundary layer behavior and geometric reﬁnement

J.M. Melenk: LNM 1796, pp. 73–138, 2002. c Springer-Verlag Berlin Heidelberg 2002

74

3. hp Approximation

near the vertices to resolve the corner layers. The admissible meshes of Deﬁnition 2.4.4 reﬂect these requirements by containing needle elements of width O(κ) near the boundary and geometric mesh reﬁnement with L+1 layers near the vertices; in the robust exponential approximation result Theorem 3.4.8 (and hence, also in the robust exponential convergence result Theorem 2.4.8), this parameter κ is then chosen as κ = O(pε), where p is the polynomial degree. Admissible meshes can be quite distorted, i.e., minimal angles can be very small (they may be of size O(κ)) and maximal angles may be very large (they may be π − O(κ)). Due to the distortion of the elements, the standard polynomial approximation results, which are essentially based on H 1 -projectors on the reference elements, do not lead to robust bounds. We therefore base our approximation theory on an operator that is essentially an L∞ projector. The main advantage of this approach is the invariance of the L∞ norm under changes of variables, which allows us to avoid some of the diﬃculties associated with distorted meshes. Our technical tool is the operator Πp∞ deﬁned on the reference element (see Theo∞ deﬁned on the triangulation rem 3.2.20) and the corresponding operator Πp,T T as the element-by-element application of Πp∞ (see (3.3.3)). The operators ∞ Πp∞ and Πp,T are constructed so as to interpolate the given function in the Gauss-Lobatto points of the edges of the triangulations. This is done mostly for convenience’s sake as this permits constructing H 1 -conforming approximations in a truly element-by-element fashion (see Remark 3.3.9). Section 3.4 is then ∞ devoted to the analysis of the error uε − Πp,T uε on admissible meshes. A direct consequence of this approximation result is the robust exponential convergence result Theorem 2.4.8 for the hp-FEM. Let us now turn to a discussion of the second line of ideas in this chapter. The minimal meshes of Deﬁnition 2.4.4 may be very distorted and have little structure. From implementational considerations, minimal meshes have the following disadvantages: 1. The presence of distorted elements may increase the sensitivity of the FEM to quadrature error. 2. The presence of thin needle elements may aﬀect the conditioning of the resulting stiﬀness matrix. If the mesh has some structure, one may selectively condense out degrees of freedom locally to improve the conditioning of the matrix and/or devise preconditioners that can handle these needle elements. The ability to control mesh distortion is captured with the notion of regular admissible meshes in Deﬁnition 3.3.10. In essence, Deﬁnition 3.3.10 stipulates that needle elements be the images of reference needle rectangles (or triangles). Thus, maximal angles of elements cannot degenerate to π; we note that we have used a similar idea in the context of the h-FEM in Deﬁnition 2.6.3. A step further in the direction of structured meshes is taken with the notion of mesh patches. This idea is closely related to domain decomposition and substructuring. Meshes are created in two steps: In a ﬁrst step, the computational domain is covered by a ﬁxed coarse mesh, the “patches”. In a second step, the ﬁnal mesh is constructed by mapping reference conﬁgurations (see Section 3.3.3 for some reference conﬁgurations relevant for the resolution of boundary layer and corner

3.1 Motivation and outline

75

layer phenomena) on the reference elements to physical space with the patch maps. The reference conﬁgurations can be chosen to reﬂect the solution behavior, for example, boundary layers and corner singularities. The main advantage of this approach is that practically only few typical conﬁgurations can occur and that the resulting mesh has considerable structure. These mesh patches represent natural divisions for parallel implementations and domain decomposition techniques. The distinction between admissible and regular admissible meshes can be embedded in a larger context, namely, deﬁning element size for curved anisotropic elements. We present two approaches to this issue with the notions of (CM , γM )regular triangulations in Deﬁnition 3.3.1 and (CM , γM )-normalizable triangulations in Deﬁnition 3.3.3. In both approaches, the element size (isotropic or anisotropic) is encoded in an aﬃne stretching map Ai . In an (CM , γM )-regular triangulation, an aﬃne maps Ai is associated with each element Ki such that the concatenations A−1 i ◦Mi can be controlled uniformly in i. (CM , γM )-normalizable triangulations are more restrictive than (CM , γM )-regular triangulations because uniform control of Mi ◦ Ai for appropriate aﬃne stretching maps Ai is required. Conceptually, admissible meshes are regular meshes while regular admissible meshes and meshes generated with mesh patches are normalizable meshes. Regular admissible meshes (and hence also meshes generated with mesh patches) are also admissible meshes (Proposition 3.3.11). Hence, the approximation theory developed for admissible meshes applies to these meshes as well. However, regular admissible meshes have more structure and therefore sharper polynomial approximation results could be obtained. In the context of the h-FEM in Section 2.6.3, such an additional structure allowed us to improve Theorem 2.6.15 to Corollary 2.6.19. While we do not develop a complete theory for polynomial approximation on normalizable meshes, we do provide two essential tools for doing so, namely, a) results in Section 3.3.5 that show how regularity results on the physical domain can be transferred to the reference conﬁguration; and b) the operator Πp1,∞ of Theorem 3.2.24 together with the approximation result Proposition 3.2.25 which show how the regularity results on the reference patch obtained with the tools of Section 3.3.5 can be used for obtaining sharp polynomial approximation on the reference patch. 3.1.2 Outline of Chapter 3 We begin this chapter with a sketch of the key steps of the proof of Theorem 3.4.8 in Section 3.1.3. As mentioned at the beginning of this chapter, this chapter develops in parallel two lines of thought on polynomial approximation. We will outline their development separately. We start with the line of ideas connected with minimal meshes. In Section 3.2, we provide polynomial approximation results on the reference element, i.e., on the reference square S and the reference triangle T . Our approximation results on minimal meshes are based on the projector Πp∞ deﬁned in Section 3.2.4. The

76

3. hp Approximation

essential result about the projector Πp∞ is formulated in Proposition 3.2.21. After these approximation results on the reference square and reference triangle, Section 3.4 is then devoted to the proof of polynomial approximation results on minimal meshes, culminating in Theorem 3.4.8, where robust exponential approximability of solutions to (1.2.11) is shown on minimal meshes. For a rigorous proof of Theorem 3.4.8, both forms of regularity results for the solution uε of (1.2.11) are required: In the “pre-asymptotic” range, in which the polynomial degree p is small compared with ε−1 , we employ the asymptotic expansions of Theorem 2.3.4. However, as can be seen in Theorem 2.3.4, such an asymptotic expansions can describe the solution uε only up to a certain error level, since the remainder rε is not arbitrarily small. Hence, in the “asymptotic” range in 2 which the polynomial degree p is large compared with ε−1 , we resort to the Bβ,ε regularity results of Theorem 2.3.1. The second line of ideas can be traced through the following sections. We introduce in Section 3.2.5 the projector Πp1,∞ that is suitable for approximation on anisotropic meshes and normalizable triangulations in particular. We recall that regular admissible meshes and meshes generated via mesh patches fall in this category. The essential approximation properties of Πp1,∞ are collected in Proposition 3.2.25. The approximation of analytic functions based on the projector Πp1,∞ relies on polynomial approximation results obtained in Section 3.2.2. The main ideas concerning mesh patches are developed in Section 3.3.2. Patches that are required for the resolution of boundary and corner layers are collected in Section 3.3.3, and a formal deﬁnition of meshes generated by mesh patches is given in Section 3.3.4. Of more general nature is Section 3.3.5, in which results are obtained concerning the regularity of functions pulled back to the reference conﬁguration with the patch map. 3.1.3 Robust exponential convergence: key ingredients of proof In this subsection, we brieﬂy present the key ingredients of the proof of the robust exponential convergence result, Theorems 2.4.8 and 3.4.8. Structurally, the proof is similar to that of Proposition 2.2.5 in the one-dimensional situation in that the distinction λpε ≥ 1 and λpε < 1 is made. For the “asymptotic case” λpε ≥ 1 the regularity assertions of Theorem 2.3.1 are employed; for the “preasymptotic case” λpε < 1 the decomposition result of Theorem 2.3.4 is used. We start with the preasymptotic case λpε < 1. hp-approximation in the pre-asymptotic regime λpε < 1. For piecewise polynomial approximation by polynomials of degree p, the meshes T (min {λpε, 1}, L) have two characteristic features: 1. thin needle elements of width O(pε) are employed in the boundary layer to capture the solution’s boundary layer components; 2. a geometric mesh reﬁnement in an O(pε) neighborhood of the vertices is used for corner singularity resolution. The aim of the present section is to illustrate the main mechanisms at work in the piecewise polynomial approximation of solutions to (1.2.11) using meshes

3.1 Motivation and outline

77

that contain thin needle elements of width O(pε) in the boundary layers and geometric reﬁnement in an O(pε) neighborhood of the vertices as stated above. In Theorem 2.3.4 we decomposed uε into a smooth part wε , a boundary layer CL part uBL ε , a corner layer part uε , and a (small) remainder rε . The ﬁnite element mesh on which piecewise polynomial approximation of uε is performed has to be designed such that each of these components can be approximated well. The smooth part wε is easily approximated by (piecewise) polynomials. The remainder rε , being exponentially small in ε, may be neglected. Approximability of the boundary and corner layer contributions therefore dictate the ﬁnite element mesh design. In the following, we illustrate for simple model situations the mesh design principles that allow for robust exponential approximability of these two solution components. Boundary layer approximation. We start with the boundary layer approximation. We consider the domain Ω := (0, 1)2 = {(ρ, θ) | 0 < ρ < 1, 0 < θ < 1}. We say that the function u = u(ρ, θ) is of boundary layer type with length scale ε ∈ (0, 1] if there are constants α, C, γ > 0 such that |∂ρs ∂θt u(ρ, θ)| ≤ Cu γus+t t! max {s + 1, ε−1 }s e−αρ/ε

∀(s, t) ∈ N20 .

(3.1.1)

We easily recognize this to be a generalization of the regularity assertions about uBL in Theorem 2.3.4. We now wish to approximate such functions of boundary ε layer type on Ω by piecewise polynomials. This can be done very eﬃciently on a two-element mesh, i.e., a mesh containing a long thin needle element in the layer and one large element away from the layer. Speciﬁcally, for κ > 0 we deﬁne the “two-element” mesh Tκ as (cf. Fig. 3.1.1) ˆ (K2 , M2 , K)}, ˆ Tκ := {(K1 , M1 , K), where K1 = (0, min {κ, 0.5}) × (0, 1), K2 = (min {κ, 0.5}, 1) × (0, 1)}, ˆ = (0, 1)2 onto and the element maps Mi are simply aﬃne maps mapping K Ki . We now show that the choice κ = O(pε) allows for robust exponential

min{0.5, κ} Fig. 3.1.1. Two-element mesh for boundary layer resolution.

approximability of function of boundary layer type from the space S p (Tκ ):

3. hp Approximation

78

Lemma 3.1.1. Let u satisfy (3.1.1) on Ω = (0, 1)2 . Then there are C, b > 0 and λ0 > 0 depending only on γu and α such that for all λ ∈ (0, λ0 ) and all p∈N inf

λpε∇(u − πp )L∞ (Ω) + u − πp L∞ (Ω) ≤ C(1 + ln p)2 (1 + εp2 )e−bλp .

πp ∈S p (Tλpε )

Proof: A slightly sharper version of this lemma is proved in [55]; a similar result is used implicitly in [94]. Lemma 3.1.1 is not formulated in its sharpest form as its aim is to expose the basic mechanics of hp-approximation of functions of boundary layer type. We construct the approximant πp explicitly for each element K1 , K2 . 1. Step: Approximation on K1 . Let us consider the needle element K1 . For κ = λpε, the element map M1 may be assumed to be of the form M1 (x, y) = ˆ (min {λpε, 0.5}x, y). Thus, the pull-back u ˆ of u|K1 to the reference element K, ˆ i.e., u ˆ := u ◦ M1 , satisﬁes on K s s+t ∂xs ∂yt u ˆ(x, y)L∞ (K) max {s + 1, ε−1 }s t! ∀(s, t) ∈ N20 . ˆ ≤ Cu min {0.5, λpε} γu

We now claim that there are C , γ > 0 depending only on γu such that 2λp s+t ˆ(x, y)L∞ (K) γ s!t! ∂xs ∂yt u ˆ ≤ C Cu e

∀(s, t) ∈ N20 .

(3.1.2)

In order to see this, we consider the case min {0.5, λpε} = λpε and the converse case min {0.5, λpε} = 0.5 separately. Consider ﬁrst the case min {0.5, λpε} = λpε. min {0.5, λpε}s max {s + 1, ε−1 }s = (λpε)s max {s + 1, ε−1 }s (λp)s s!} = max {(λpε)s (s + 1)s , (λp)s } ≤ max {0.5s (s + 1)s , s! ≤ max {0.5s (s + 1)s , eλp s!} ≤ Cγ s eλp s! (3.1.3) for some appropriate C, γ > 0 independent of λ, p, and ε. Let us now consider the second case, min {0.5, λpε} = 0.5. Then we bound by similar reasoning min {0.5, λpε}s max {s + 1, ε−1 }s = 0.5s max {s + 1, ε−1 }s (2λp)s s!} ≤ 0.5s max {(s + 1)s , (2λp)s } ≤ 0.5s max {(s + 1)s , s! ≤ Cγ s e2λp s!. (3.1.4) Combining (3.1.3), (3.1.4) then gives (3.1.2). We now apply polynomial approxˆ which are proved in detail in Section 3.2.2. For example, imation results on K, the two-dimensional Gauss-Lobatto interpolation operator jp (cf. Section 3.2.1) applied to the function u ˆ yields the existence of C, b > 0 depending only on γ and C of (3.1.2) such that 2λp −bp ˆ u − jp u ˆL∞ (K) u − jp u ˆ)L∞ (K) e . ˆ + ∇(ˆ ˆ ≤ CCu e

3.1 Motivation and outline

79

(A rigorous proof of this approximation result is obtained by combining Theorem 3.2.19 with Theorem 3.2.20 and noting that the operator Πp∞ of Theorem 3.2.20 coincides with jp in the present case of approximation on the reference square.) Observe the presence of the factor e2λp . This factor is already present in the bounds on u ˆ and reappears in this bound due to the linearity of the Gauss-Lobatto interpolation operator jp . As the constant b is independent of λ, p, we choose λ0 := b/4 to get −(b/2)p ˆL∞ (K) u − jp u ˆ)L∞ (K) . ˆ u − jp u ˆ + ∇(ˆ ˆ ≤ CCu e

Mapping back to K1 gives the desired bound on K1 . 2. Step: Approximation on K2 . We note that dist(K2 , {ρ = 0}) ≥ κ = min {0.5, λpε}. The assumptions on u therefore imply the existence of C, b > 0 (depending only on γ and α) such that uL∞ (K2 ) ≤ CCu e−bλp ,

∇uL∞ (K2 ) ≤ CCu ε−1 e−bλp .

(3.1.5)

Thus, u is already exponentially small (in p) and fairly crude polynomial approximations suﬃce. For deﬁniteness’ sake, we will use again the Gauss-Lobatto interpolant jp u. We start by noting that the element map M2 satisﬁes Mi L∞ (K) ˆ ≤ 1,

−1

(Mi )

L∞ (K) ˆ ≤ 2.

From this, it is easy to see that the pull-back u ˆ := u|K2 ◦ M2 satisﬁes bounds analogous to (3.1.5): −bλp ˆ uL∞ (K) , ˆ ≤ CCu e

−1 −bλp ∇ˆ uL∞ (K) e , ˆ ≤ CCu ε

where, in fact, only the constant C may have changed by a factor 2. From basic properties of the Gauss-Lobatto interpolation operator (Lemma 3.2.1) and inverse estimates for polynomials (Lemma 3.2.2), we have 2 ˆL∞ (K) uL∞ (K) ˆL∞ (K) uL∞ (K) ˆ u − jp u ˆ ≤ ˆ ˆ + jp u ˆ ≤ C(1 + ln p) ˆ ˆ ,

∇(ˆ u − jp u ˆ)L∞ (K) uL∞ (K) ˆL∞ (K) ˆ ≤ ∇ˆ ˆ + ∇jp u ˆ , 2 ≤ ∇ˆ uL∞ (K) ˆL∞ (K) ˆ + Cp jp u ˆ 2 2 ≤ ∇ˆ uL∞ (K) uL∞ (K) ˆ + Cp (1 + ln p) ˆ ˆ .

These estimates imply after an adjustment of the constant b: 2 −bλp ˆL∞ (K) , ˆ u − jp u ˆ ≤ C(1 + ln p) e −bλp 2 2 λpε∇(ˆ u − jp u ˆ)L∞ (K) . ˆ ≤ C 1 + εp (1 + ln p) e

Mapping this last estimate back to K2 gives the desired bound.

2

A few comments concerning the proof of Lemma 3.1.1 are in order. First, the algebraic powers of p in front of the term e−bλp are suboptimal. They were accepted in the proof of Lemma 3.1.1 in order to be able to concentrate on the essential mechanisms in the proof:

80

3. hp Approximation

1. On the element K2 that is O(pε) away from the line ρ = 0, the function u to be approximated is exponentially small (in p) and thus crude approximations (e.g., by the zero-function) are suﬃcient. 2. The main mechanism for the approximation on the element K1 is better seen by writing the proof in a slightly diﬀerent way as follows. Introducing the anisotropic stretching map sκ : (ρ, θ) → (κρ, θ), we can write the pull-back u ˆ := u ◦ M1 to the reference element as u ˆ = ◦ M ) where κ = min {0.5, λpε}. Next, we deﬁne the auxiliary (u ◦ sκ ) ◦ (s−1 1 κ $ := s−1 (K1 ). Paralleling the arguments in the proof of Lemma 3.1.1, set K κ ˜ we obtain for u ◦ sκ on K: λp n ∇n (u ◦ sκ )L∞ (K) ˜ ≤ Ce γ n!

∀n ∈ N0 ,

(3.1.6)

where C, γ are independent of ε and p. In order to get bounds on the derivatives of u ˆ, we require control of the map s−1 κ ◦ M1 . We observe that there exists C > 0 independent of ε such that ∇n s−1 n ∈ {0, 1}, (3.1.7a) ˆ ≤ C, κ ◦ M1 L∞ (K) −1 −1 (sκ ◦ M1 ) L∞ (K) (3.1.7b) ˆ ≤ C. As s−1 κ ◦ M1 is aﬃne, (3.1.7a) in fact holds trivially for all n ∈ N0 . As we will see later on, it suﬃces to stipulate that the map s−1 κ ◦ M1 be an analytic diﬀeomorphism whose constants of analyticity C, γ can be bounded independently of ε: n ∇n s−1 ∀n ∈ N0 , (3.1.8a) ˆ ≤ Cγ n! κ ◦ M1 L∞ (K) −1 −1 (sκ ◦ M1 ) L∞ (K) (3.1.8b) ˆ ≤ C. Combining (3.1.6) and (3.1.7) gives (3.1.2). Key to the present approach is the existence of anisotropic stretching maps sκ such that both u ◦ sκ and s−1 κ ◦ M1 can can be controlled uniformly in ε as ascertained in (3.1.6), (3.1.8). Corner layer approximation. We now turn to the presentation of the main points for the approximation of corner layers. For the purpose of this discussion, we will say that a function u(x, y) deﬁned on Ω = (0, 1)2 is of corner layer type, if there are Cu , γu , α > 0, β ∈ [0, 1) such that eα|x|/ε uL2 (Ω) + εeα|x|/ε ∇uL2 (Ω) ≤ εCu ) ) ! * β ) ) |x| 2 ) α|x|/ε 2 ) ∇ u) min 1, ≤ εCu , ε )e ) ) 2 ε L (Ω) 1−β |x| |∇n u(x)| ≤ Cu γun |x|−n e−α|x|/ε ∀n ∈ N0 . ε

(3.1.9a) (3.1.9b)

(3.1.9c)

3.1 Motivation and outline

81

σ L min {0.5, λpε}

min {0.5, λpε}

Fig. 3.1.2. Geometrically reﬁned mesh with L + 1 (here: L = 3) layers in O(pε) neighborhood of origin for corner layer resolution.

We recognize these bounds to be typical of the corner layer functions uCL in ε Theorem 2.3.4. We now discuss meshes T on Ω that are appropriate for the approximation of such functions of corner layer type. We note that (3.1.9c) shows that u is exponentially small (in p) outside an O(pε) neighborhood of the origin x = 0. Thus, just as in the proof of Lemma 3.1.1, fairly crude polynomial approximations suﬃce outside this neighborhood. It remains to consider the approximation in this O(pε) neighborhood of the origin. There, we propose the use of a geometrically reﬁned mesh to resolve the singularity as is standard in hp-FEM. For illustration purposes, we consider the approximation on meshes ˆ as depicted in Fig. 3.1.2. Note that the corresponding element T = {(Ki , Mi , K)} maps Mi are bilinear functions, i.e., ∇ 3 Mi = 0

∀i.

(3.1.10)

It is also convenient to introduce the element size hi as hi = diam Ki . The essential features of the meshes T as depicted in Fig. 3.1.2 are: 1. Outside an O(pε) neighborhood of the origin, few elements are employed (here: 2). 2. In an O(pε) neighborhood of the origin, a geometrically reﬁned mesh is employed with grading factor σ ∈ (0, 1) and L + 1 ∈ N layers of geometric reﬁnement. This means: a) For all elements Ki with dist(Ki , 0) > 0 there holds the two-sided bound C −1 hi ≤ dist(Ki , 0) ≤ Chi . b) If dist(Ki , 0) = 0, then hi ≤ Cλpεσ L . This list decomposes the elements into three categories; correspondingly, we can decompose the set of indices of the elements of such an mesh T into three sets: Iorg := {i | 0 ∈ K i }, −1

Igeom := {i | C hi ≤ dist(Ki , 0) ≤ Chi }, Iout := {i | i ∈ Iorg and i ∈ Igeom }.

(3.1.11a) (3.1.11b) (3.1.11c)

82

3. hp Approximation

We note self-similarity properties of the elements in the geometric reﬁnement in Fig. 3.1.2 (i.e., the elements Ki with i ∈ Igeom ), which entail certain uniformity properties of the element maps. In order to describe those, it is convenient to introduce for κ > 0 the following aﬃne stretching maps: sκ (x) := κx

∀x ∈ R2 .

By self-similarity of most elements, it is not diﬃcult to see that the maps s−1 hi ◦Mi given by ˆ →K $ i := s−1 (Ki ) K hi −1 x → shi ◦ Mi (x) ˆ and the stretched elements K $ i and are analytic diﬀeomorphisms between K satisfy in fact for some C, γ > 0 n ∇n (s−1 ˆ ≤ Cγ n! hi ◦ Mi )L∞ (K) −1 (s−1 L∞ (K) ˆ ≤ C. hi ◦ Mi )

∀n ∈ N0 ,

(3.1.12a) (3.1.12b)

Note that (3.1.10) in fact implies that ∇n (s−1 hi ◦ Mi ) = 0 for n ≥ 3 so that (3.1.12) is not diﬃcult to check. An additional important property of the elements of the geometric reﬁnement is that for each δ > 0, there exists Cδ,σ (depending on δ and the grading factor σ) such that 2δ h2δ (3.1.13) i ≤ Cδ,σ (λpε) . i∈Igeom

This estimate follows easily from the fact that the elements of the geometric mesh are self-similar and that hence the element size decays in a geometric progression as the elements approach the origin; a formal and more general proof can be found in Lemma 3.4.6. After these preparatory considerations, we formulate the following approximation result for functions of corner layer type: Lemma 3.1.2. Let u satisfy (3.1.9) and let T be the mesh with L + 1 layers of geometric reﬁnement and grading factor σ ∈ (0, 1) in an O(pε) neighborhood as in Fig. 3.1.2. Then there exist C, b > 0 independent of ε and p such that inf

πp ∈S p (T )

u − πp L2 (Ω) + ε∇(u − πp )L2 (Ω) ≤ C(1 + ln p)2 (1 + εp2 )e−bλp + εp5 σ (1−β)L .

Proof: As in the proof of Lemma 3.1.1, the approximant πp is constructed as the piecewise Gauss-Lobatto interpolant. The main advantage of this choice is that it automatically takes care of the correct interelement continuity requirement and yields elements of S p (T ) (cf. Lemma 3.3.7 for the details).

3.1 Motivation and outline

83

We distinguish the three kinds of elements corresponding to the three sets Iout , Igeom , Iorg of (3.1.11): The two elements Ki outside the O(pε) neighborhood of the origin where u is exponentially small, the elements Ki in the O(pε) neighborhood with dist(Ki , 0) > 0, and ﬁnally the element with dist(Ki , 0) = 0. The proof is therefore divided into three steps. 1. Step: Elements outside O(pε) neighborhood of origin. From Fig. 3.1.2, these are the elements Ki with dist(Ki , 0) ≥ min {0.5, λpε} ≥ λpε. From (3.1.9c) and the fact that we can bound for suitable Cα,β > 0 x1−β e−αx ≤ Cα,β e−(α/2)x

∀x > 0,

we obtain for appropriate C, b > 0 independent of ε, p, and λ: ∇n uL∞ (Ki ) ≤ C(λpε)−n e−bλp ,

n ∈ {0, 1}.

We observe that these bounds are the same as those on boundary layer functions on the element K2 in the proof of Lemma 3.1.1. We may therefore conclude in the same manner that the approximation πp |Ki deﬁned by πp |Ki = (jp (u ◦ Mi ))◦ Mi−1 satisﬁes u − πp L∞ (Ki ) + λpε∇(u − πp )L∞ (Ki ) ≤ C(1 + ln p)2 (1 + εp2 )e−bλp for some appropriate C, b > 0 independent of ε, p, and λ. 2. Step: Elements in O(pε) neighborhood of origin with dist(Ki , 0) > 0. We consider polynomial approximation of the pull-back u ˆ := u|Ki ◦ Mi on the ˆ It is not diﬃcult to see from (3.1.9c) and the property reference element K. $ i := s−1 (Ki ) the C −1 hi ≤ dist(Ki , 0) ≤ Chi that on the stretched element K hi function u ◦ shi satisﬁes for some appropriate C, γ > 0: 1−β hi n n n γ n n! ∀n ∈ N0 . ∇ (u ◦ shi )L∞ (K i ) ≤ Chi ∇ uL∞ (Ki ) ≤ C ε Writing u ˆ = (u ◦ shi ) ◦ (s−1 hi ◦ Mi ) and using (3.1.12), we infer from Lemma 4.3.4 that there are C, γ > 0 independent of ε and i such that 1−β hi n ∇ u ˆL∞ (K) γ n n! ∀n ∈ N0 . ˆ ≤C ε From the approximation results of Section 3.2.2, we can then infer that the Gauss-Lobatto interpolant jp u ˆ satisﬁes for some C, b > 0: 1−β hi ˆL∞ (K) u − jp u ˆ)L∞ (K) e−bp , p = 1, 2, . . . . ˆ u − jp u ˆ + ∇(ˆ ˆ ≤C ε Mapping this approximation result back to the physical element Ki , we obtain by setting πp |Ki = (jp u ˆ) ◦ Mi−1 : 1−β hi e−bp , p = 1, 2, . . . . u − πp L∞ (Ki ) + hi ∇(u − πp )L∞ (Ki ) ≤ C ε

84

3. hp Approximation

Integrating over Ki and summing on i ∈ Igeom gives 2+2(1−β) u − πp 2L2 (Ki ) ≤ Ce−2bp ε−2(1−β) hi , i∈Igeom

i∈Igeom

∇(u − πp )2L2 (Ki ) ≤ Ce−2bp ε−2(1−β)

i∈Igeom

2(1−β)

hi

.

i∈Igeom

The bound (3.1.13) gives u − πp 2L2 (Ki ) ≤ Ce−2bp ε−2(1−β) (λpε)2+2(1−β) , i∈Igeom

∇(u − πp )2L2 (Ki ) ≤ Ce−2bp ε−2(1−β) (λpε)2(1−β) ,

i∈Igeom

which in turn implies u − πp 2L2 (Ki ) + (λpε)2 ∇(u − πp )2L2 (Ki ) ≤ Ce−2bp ε2 (λp)2+2(1−β) . i∈Igeom

Absorbing the λ-dependence into the constant C gives an expression that can be bounded in the desired fashion. 3. Step: The element Ki abutting on the origin. Polynomial approximation on the element abutting on the origin just exploits that the element size is exponentially small in the number L of layers of geometric reﬁnement, i.e., hi ≤ Cλpεσ L . Fairly crude polynomial approximation therefore suﬃces. Again, polynomial apˆ and we consider u proximation is done on the reference element K ˆ = u|Ki ◦ Mi . We exploit the following fact about the linear interpolant j1 u ˆ (see Lemma 3.2.4) β 2 ˆL∞ (K) u − j1 u ˆ)L2 (K) ˆL2 (K) ˆ u − j1 u ˆ + ∇(ˆ ˆ ≤ C |x| ∇ u ˆ .

(3.1.14)

In order to estimate |x|β ∇2 uL2 (K) ˆ , let us assume for notational convenience that Mi (0) = 0. Observing that for our mesh of Fig. 3.1.2 the element map Mi is aﬃne and has the form Mi (x) = hi x, scaling yields 1−β |x|β ∇2 u ˆL2 (K) |x|β ∇2 uL2 (Ki ) ˆ = hi

= h1−β εβ (|x|/ε)β ∇2 uL2 (Ki ) . i To continue the estimates, we use |x| ≤ hi to get |x| |x| hi |x| |x| max 1, ≤ min 1, 1+ = min 1, ε ε ε ε ε

(3.1.15)

(3.1.16)

and then employ (3.1.9b) to arrive at )

hi ) )min 1, (|x|/ε)β ∇2 u) 2 1+ ≤ L (Ki ) ε 1−β hi hi hi β−1 ≤ C . 1 + 1 + ≤ Ch1−β ε i ε ε ε

h1−β εβ (|x|/ε)β ∇2 uL2 (Ki ) i

h1−β εβ i

3.1 Motivation and outline

85

As we want to approximate u ˆ by jp u ˆ rather than j1 u ˆ, we use stability properties of the Gauss-Lobatto interpolation operator jp (Lemma 3.2.1), inverse estimates (Lemma 3.2.2), and the fact that jp (j1 u ˆ) = j1 u ˆ to arrive at ˆ u − jp u ˆH 1 (K) u − j1 u ˆ − jp (ˆ u − j1 u ˆ) H 1 (K) ˆ = ˆ ˆ 2 ≤ ˆ u − j1 u ˆH 1 (K) u − j1 u ˆ)L2 (K) ˆ + Cp jp (ˆ ˆ 2 2 ≤ ˆ u − j1 u ˆH 1 (K) u − j1 u ˆL∞ (K) (3.1.17) ˆ + Cp (1 + ln p) ˆ ˆ 1−β hi hi . ≤ Cp2 (1 + ln p)2 1+ ε ε

Exploiting now the assumption hi ≤ Cλpεσ L , we ﬁnally get 4 2 (1−β)L ˆ u − jp u ˆH 1 (K) . ˆ ≤ Cp (1 + ln p) σ

Upon setting πp |Ki = (jp u ˆ) ◦ Mi−1 we obtain by mapping back to the physical element Ki u − πp L2 (Ki ) ≤ Chi p4 (1 + ln p)2 σ (1−β)L ≤ Cεp5 (1 + ln p)2 σ (1−β)L , λpε∇(u − πp )L2 (Ki ) ≤ Cεp5 (1 + ln p)2 σ (1−β)L . 2 hp-approximation in the asymptotic regime λpε ≥ 1. The key key point is that the assumption ε−1 ≤ λp allows us to replace negative powers of ε with powers of the polynomial degree p. We illustrate the main mechanisms with the following lemma, where we consider the approximation of a function u with the regularity properties given by Theorem 2.3.1 on geometrically reﬁned meshes: Lemma 3.1.3. Let Ω = (0, 1)2 and let T be a mesh on Ω with L layers of reﬁnement and grading factor σ ∈ (0, 1) as depicted in Fig. 3.1.2. Assume that u is analytic on Ω and satisﬁes for some C, K > 0, β ∈ [0, 1) and all n ∈ N0 |∇n u(x)| ≤ CK n ε−1 |x|−n (|x|/ε)

1−β

max {n + 1, ε−1 }n+1 ,

min {1, |x|/ε} ∇2 uL2 (Ω) ≤ Cε−2 . β

Then there exist C, b, λ0 > 0 independent of ε ∈ (0, 1] such that for every λ ∈ (0, λ0 ) and every p satisfying λpε ≥ 1 there holds infp

πp ∈S (T )

(3.1.18)

u − πp H 1 (Ω) ≤ Cp2 (1 + ln p)2 (λp)4 e−bp + εσ L(1−β) .

86

3. hp Approximation

Before we prove the lemma, we point out that the regularity assumptions on u are slightly weaker than those ascertained in Theorem 2.3.1 for the solutions uε of (1.2.1). We remark that the powers of p in the statement of Lemma 3.1.3 are not optimal and chosen so as to keep the exposition simple. Proof: The approximant is taken as the piecewise Gauss-Lobatto interpolant. We distinguish the elements Ki that are away from the origin and the (single) element Ki that touches the origin. The proof is very similar to that of Lemma 3.1.2, and we will merely sketch the main diﬀerences. We introduce the two sets of indices Igeom := {i | 0 ∈ K i },

Iorg := {i | 0 ∈ K i },

where in view of Fig. 3.1.2 the set Iorg consists of a single element. 1. Step: Elements Ki with dist(Ki , 0) > 0: We parallel the second step of the $ i := s−1 (Ki ), the function proof of Lemma 3.1.2. On the stretched element K hi u ◦ shi satisﬁes ∇n (u◦shi )L∞ (K i ) ≤ Chni ∇n uL∞ (Ki ) ≤ Cγ n ε−2+β h1−β max {n+1, hi /ε}n+1 . i Simplifying further max {n + 1, hi /ε}n+1 = max {(n + 1)n+1 , (hi /ε)n+1 } ≤ max {(n + 1)n+1 , (n + 1)!ehi /ε } ≤ Cγ n n!ehi /ε for some appropriate C, γ > 0 independent of ε, we get ∇n (u ◦ shi )L∞ (K i ) ≤ Cε−2 h1−β γ n n!e1/ε i for some suitably chosen constants C, γ > 0 independent of ε. Using Lemma 4.3.4, we can then conclude for the function u ˜ = (u ◦ shi ) ◦ (s−1 hi ◦ Mi ) 1−β −2 n ˜L∞ (K) ε γ n!e1/ε ∇n u ˆ ≤ Chi

∀n ∈ N0

for some suitably chosen C, γ > 0. Proceeding as in the in second step of the proof of Lemma 3.1.2, we conclude that the function πp = (jp u ˜) ◦ Mi−1 satisﬁes u − πp L∞ (Ki ) + hi ∇(u − πp )L∞ (Ki ) ≤ Ch1−β ε−2 e1/ε e−bp i

(3.1.19)

for some C, b > 0 independent of ε. Paralleling the arguments leading to (3.1.13), we get 2(1−β) 2(1−β) hi ≤ C (diam Ω) ≤C i∈Igeom

for some suitable constant C > 0. Hence, integrating the estimates (3.1.19) over Ki and then summing over i ∈ Igeom gives i∈Igeom

u − πp 2H 1 (Ki )

1/2

≤ Cε−2 e1/ε e−bp .

(3.1.20)

3.2 Polynomial approximation results

87

2. Step: The element Ki abutting on 0: The element Ki abutting on 0 is treated as in Lemma 3.1.2. Combining the estimates (3.1.14), (3.1.17), (3.1.16), (3.1.15), we conclude that πp |Ki = (jp (u ◦ Mi )) ◦ Mi−1 satisﬁes u − πp L∞ (Ki ) + ∇(u − πp )L2 (Ki ) ≤ Cp2 (1 + ln p)2 h1−β εβ (1 + hi /ε) min {1, |x|/ε}β ∇2 uL2 (Ki ) i ≤ Cp2 (1 + ln p)2 σ (1−β)L ε−3 .

(3.1.21)

Combining (3.1.20) and (3.1.21) allows us to conclude that u − πp H 1 (Ω) ≤ Cp2 (1 + ln p)2 ε−4 e1/ε e−bp + εσ L(1−β) ≤ Cp2 (1 + ln p)2 (λp)4 e(λ−b)p + εσ L(1−β) , where we employed in the last step the assumption λpε ≥ 1. Choosing now λ0 so small that λ − b ≤ λ0 − b ≤ −b/2 allows us to conclude the proof. 2

3.2 Polynomial approximation results 3.2.1 Notation and properties of polynomials We start with some notation: I = (0, 1), S = I × I = (0, 1)2 , T = {(x, y) | x ∈ I, 0 < y < 1 − x}. On the interval I, the reference square S, and the reference triangle T , we introduce the space of polynomials Pp (I), the tensor product space Qp (S), and the space of polynomials Pp (T ) by Pp (I) := span {xi | i = 0, . . . , p}, Qp (S) := Pp (I) ⊗ Pp (I) = span {xi y j | 0 ≤ i, j ≤ p}, Pp (T ) := Pp (S) := span {xi y j | i, j ∈ N0 , i + j ≤ p}.

(3.2.1) (3.2.2) (3.2.3)

We introduce the following shorthand for the spaces Pp (T ) and Qp (S): ! Qp (S) if K = S, (3.2.4) Πp (K) := Pp (T ) if K = T . We will be interested in two types of polynomial approximation operators: The Gauss-Lobatto interpolant and the L2 projection. As usual, Lp denotes the Legendre polynomial of degree p normalized to satisfy Lp (1) = 1 (cf., e.g., [61]). As we work on I = (0, 1) rather than (−1, 1), we introduce the scaled polynomials ˜ p (x) := Lp (2x − 1). Next, for given p, let L GLp := {xi | i = 0, . . . , p}

(3.2.5)

˜ (x). It is a well-known fact (see, be the zeros of the polynomial x → x(1 − x)L p e.g., [27]) that this polynomial has p + 1 distinct zeros lying in [0, 1]. Clearly,

88

3. hp Approximation

{0, 1} ⊂ GLp and by symmetry properties of the Legendre polynomials the set GLp is symmetric with respect to the midpoint 1/2 of the interval I. Using the Gauss-Lobatto points GLp = {xi | i = 0, . . . , p} we can then deﬁne the GaussLobatto interpolation operator ip : C(I) → Pp (I) by interpolation in the (p + 1) Gauss-Lobatto nodes GLp , i.e., ip : C(I) → Pp (I) f → (ip f )(x) :=

p

(p)

f (xi )li (x),

i=0 (p)

where the Lagrange polynomials li

(p) li (x)

∈ Pp (I) are deﬁned as

p x − xj = . x − xj j=0 i j=i

Similarly, we deﬁne the two dimensional Gauss-Lobatto interpolation operator jp : C(S) → Qp (S) by interpolation in the (p + 1)2 nodes obtained by taking the tensor product of the one-dimensional nodes, i.e., jp = ixp ◦ iyp = iyp ◦ ixp , where we denoted by ixp , iyp the one-dimensional Gauss-Lobatto interpolation operators with respect to the x and y variable. Finally, for edges Γ of S or T we introduce the one-dimensional Gauss-Lobatto interpolant operators ip,Γ by identifying the edge with the interval I and using ip . More speciﬁcally, if P1 , P2 are the two endpoints of Γ , then γ : I → Γ given by γ(t) := tP1 + (1 − t)P2 parametrizes Γ , and for a function u deﬁned on Γ we can set ip,Γ u := ip (u ◦ γ) ◦ γ −1 . We note that for a square S there holds for all four edges Γ ip,Γ u|Γ = (jp u)|Γ . For future reference we state the following stability result. Lemma 3.2.1. There is C > 0 independent of p such that ip f L∞ (I) ≤ C(1 + ln p)f L∞ (I)

∀f ∈ C(I),

jp f L∞ (S) ≤ C(1 + ln p)2 f L∞ (S)

(ip f ) L∞ (I) ≤ Cpf L∞ (I)

∀f ∈ W

∀f ∈ C(S), 1,∞

(I).

(3.2.6) (3.2.7) (3.2.8)

Proof: The ﬁrst two estimates are due to [121, 122]. For the last estimate, we ﬁrst note an inverse estimate, [105] (cf. also [112, Thm. 3.92]), f L∞ (I) ≤ 4pf L2 (I)

∀f ∈ Pp .

This bound is employed in the following way: (ip f ) L∞ (I) ≤ 4p(ip f ) L2 (I) ≤ Cpip f H 1 (I) ≤ Cpf H 1 (I) ,

(3.2.9)

where we employed the H 1 -stability of the Gauss-Lobatto interpolation operator that is asserted in Theorem A.3.1. Applying this last estimate to the function f − c with c ∈ R arbitrary yields

3.2 Polynomial approximation results

89

(ip f ) L∞ (I) ≤ Cp inf f − cH 1 (I) ≤ Cpf L2 (I) ≤ Cpf L∞ (I) . c∈R

2 Next, we recall the following inverse estimates of Markov type (see, e.g., [112]). Lemma 3.2.2. There is C > 0 such that for all p ∈ N πp L∞ (I) ≤ 2p2 πp L∞ (I)

∀πp ∈ Pp (I),

(3.2.10)

∇πp L∞ (S) ≤ Cp πp L∞ (S)

∀πp ∈ Qp (S),

(3.2.11)

∇πp L∞ (T ) ≤ Cp πp L∞ (T )

∀πp ∈ Pp (T ).

(3.2.12)

2 2

Lemma 3.2.3. Let K = T or K = S. Then there is C > 0 such that the following holds. For each f ∈ C(∂K) that is a polynomial of degree p on each edge of K, there exists F ∈ Πp (K) such that F L∞ (K) + p

−2

F |∂K = f, ∇F L∞ (K) ≤ Cf L∞ (∂K) .

Moreover, the extension map E : f → F is in fact a bounded linear operator. Proof: Since the inverse estimate of Lemma 3.2.2 implies p−2 ∇F L∞ (K) ≤ CF L∞ (K) , it suﬃces to prove the bound F L∞ (K) ≤ Cf L∞ (∂K) . For this pointwise estimates of F , we proceed as follows. After subtracting a linear (if K = T ) or a bilinear (if K = S) interpolant, we may assume that f vanishes at the vertices. Next, we may assume without loss of generality that f vanishes on all sides but one. This side may be taken to be Γ := {(x, 0) | x ∈ I}. For K = S, we set F (x, y) = (1 − y)f (x). The pointwise estimate follows immediately. For K = T , we observe that, since f is assumed to vanish at the vertices of K, it can be written in the form f (x) = x(1 − x)f$(x) for some suitable f$ ∈ Pp−2 . We then set F (x, y) := x(1 − x − y)f$(x) ∈ Pp and bound sup |F (x, y)| = sup (x,y)∈T

sup

x(1 − x − y)|f$(x)|

x∈(0,1) y∈(0,1−x)

= sup

sup

x∈(0,1) y∈(0,1−x)

x(1 − x)|f$(x)| = sup |f (x)|. x∈(0,1)

2 We ﬁnally conclude this section with a result concerning the approximation properties of the linear/bilinear interpolant. For reasons that will become clear in the following subsection, the linear/bilinear interpolation operator is denoted by Π1∞ .

90

3. hp Approximation

Lemma 3.2.4. Let K = T or K = S and let A be one of the vertices of K. Let β ∈ [0, 1). Then there is C > 0 depending only on β such that the diﬀerence u − Π1∞ u between u and its linear/bilinear interpolant Π1∞ u satisﬁes u − Π1∞ uL∞ (K) ≤ C u − Π1∞ uH 1 (K) + |x − A|β ∇2 (u − Π1∞ u)L2 (K) ≤ C |x − A|β ∇2 uL2 (K) . Proof: The set K is a sector with apex A. From Lemma 4.2.9, we get u − Π1∞ uL∞ (K) ≤ Cu − Π1∞ uH 2,2 (K) , which is the ﬁrst inequality. The second β,1

2

bound is taken directly from [112, Lemmata 4.16, 4.25].

3.2.2 Approximation of analytic functions: intervals and squares Approximation on the interval I. For ρ > 1 we denote by Eρ the ellipse (in the complex plane) with foci ±1 and sum of semi-axes ρ, i.e., Eρ = {z ∈ C | |z − 1| + |z + 1| < ρ + ρ−1 }.

(3.2.13)

Remark 3.2.5 An important property of the ellipses Eρ is the fact that ∂Eρ is the image of ∂Bρ (0) ⊂ C under the conformal map z → w = 12 (z + z −1 ). We start with the following one-dimensional approximation result. Lemma 3.2.6. Let u be analytic on I and satisfy for some Cu , γ > 0 and h ∈ (0, 1] Dn uL∞ (I) ≤ Cu (γh)n n! ∀n ∈ N. (3.2.14) Then there are C, σ > 0 depending only on γ such that the Gauss-Lobatto interpolant ip u satisﬁes

u − ip uL∞ (I) + (u − ip u) L∞ (I) ≤ CCu

h h+σ

p+1 ∀p ∈ N.

Proof: The proof proceeds in three steps. + 1. step: Let u ∈ R be the average of u, i.e., u = I u dx. By the mean value theorem, there is ξ ∈ I with u = u(ξ). Thus, u $(x) := u(x) − u satisﬁes Dn u $L∞ (I) ≤ max {1, 2γ}Cu (γh)n n!

∀n ∈ N0 .

Next, we observe that these bounds on the derivatives of u $ imply the existence of σ, C > 0 depending only on γ such that u $ is holomorphic on Eρ with ρ ≥ 1+σ/h; additionally, it satisﬁes on Eρ $ uL∞ (Eρ ) ≤ CCu . 2. step: From [40, Thm. 12.4.7], we get the existence of C such that (after appropriately adjusting σ)

3.2 Polynomial approximation results

u $(x) =

∞

˜ i (x) ui L

91

uniformly on I,

i=0

|ui | ≤ CCu (1 + σ/h)−i

∀i ∈ N0 ,

˜ i (x) := Li (2x − 1). We where for the standard polynomials Li we set L pLegendre ˜ now deﬁne up (x) := i=0 ui Li (x) − u ∈ Pp (I) and bound with (3.2.10) (u − up ) L∞ (I) ≤ ($ u− ≤ CCu

p

˜ i ) L∞ (I) ≤ ui L

i=0 ∞

∞

˜ i L∞ (I) |ui | L

i=p+1 2

−i

i (1 + σ/h)

i=p+1

≤ CCu

h h + σ

p+1

for some σ < σ and C > 0. An analogous result holds for u − up L∞ (I) . Thus, we have proved p+1 h u − up L∞ (I) + (u − up ) L∞ (I) ≤ CCu . (3.2.15) h+σ 3. step: We employ the stability result Lemma 3.2.1 in order to obtain bounds for u − ip u: u − ip uL∞ (I) ≤ u − up L∞ (I) + up − ip uL∞ (I) ≤ u − up L∞ (I) + ip (up − u)L∞ (I) ≤ C(1 + ln p)u − up L∞ (I) , (u − ip ) L∞ (I) ≤ (u − up ) L∞ (I) + (up − ip u) L∞ (I) ≤ (u − up ) L∞ (I) + p2 up − ip uL∞ (I) ≤ (u − up ) L∞ (I) + Cp2 (1 + ln p)u − up L∞ (I) . Inserting (3.2.15) gives the desired bounds on u − ip u after appropriately adjusting the constant σ. 2 Approximation on the square S. We now turn to the two-dimensional case. For the case of a square, a natural approach is to proceed by tensorizing the 1D arguments. This leads to approximants in Qp (S). However, since we want to utilize the approximation results of this section for the approximation on the triangle below, it is more convenient to construct approximant that lie in Pp (T ). The approximants are obtained as truncated Legendre series. To do so, 2 we introduce the L2 -projector ΠpL : L2 (S) → Pp (T ) by expanding u into a ˜ j (y) (convergence being understood ˜ i (x)L Legendre series u(x, y) = i,j∈N0 uij L 2 in L (S)) and then setting 2 ˜ j (y) ∈ Pp (T ). ˜ i (x)L uij L (3.2.16) ΠpL u(x, y) := i,j∈N0 i+j≤p

92

3. hp Approximation

Lemma 3.2.7. Let u be analytic on S and satisfy for some Cu , γ > 0, hx , hy ∈ (0, 1] 1 α2 |α| Dα uL∞ (S) ≤ Cu hα α! x hy γ

∀α = (α1 , α2 ) ∈ N20 \ (0, 0).

(3.2.17)

Then u can be expanded in a Legendre series on S, and there are C, σ > 0 depending only on γ such that u(x, y) =

∞

˜ j (y) ˜ i (x)L uij L

uniformly on S,

i,j=0 −i

|uij | ≤ Cu C (1 + σ/hx )

−j

(1 + σ/hy )

(i, j) = (0, 0).

,

Proof: The proof is very similar to the one-dimensional one in Lemma 3.2.6. + Deﬁning u = S u(x, y) dxdy one considers u $(x, y) := u(x, y) − u. From the mean value theorem one has the existence of (x , y ) ∈ S such that u = u(x , y ) and hence one gets √ 1 α2 |α| Dα u $L∞ (S) ≤ max {1, 2 2γ}Cu hα α! ∀α ∈ N20 . x hy γ We note that u $ can be extended holomorphically to Eρx × Eρy ⊂ C × C for ρx = 1 + σ/hx , ρy = 1 + σ/hy , where σ > 0 depends only on γ. Furthermore there is C > 0 (again depending only on γ) such that the extended function (again denoted u $) satisﬁes $ uL∞ (Eρx ×Eρy ) ≤ CCu . The remainder of the proof is now a straightforward extension to 2 dimensions of the argument given in [40, Thm. 12.4.7]. 2 The decay properties of the Legendre coeﬃcients uij of in Lemma 3.2.7 allow us to obtain exponential rates of convergence for the L2 -projection operator ΠpL2 : Proposition 3.2.8. Under the hypotheses of Lemma 3.2.7 there exist constants C, σ > 0 depending only on γ > 0 such that , p+1 p+1 2 h h x y u − ΠpL uL∞ (S) ≤ CCu , + hx + σ hy + σ . p p / 2 hx hx hy ∂x (u − ΠpL u)L∞ (S) ≤ CCu , + hx + σ hx + σ hy + σ . p p / hx hy hy L2 ∂y (u − Πp u)L∞ (S) ≤ CCu . + hy + σ hx + σ hy + σ 2

Proof: We will only prove the bound for ∂x (u − ΠpL u)L∞ (S) as the other ones are proved similarly. We have to bound 2 ˜ (x)L ˜ j (y)L∞ (S) . ∂ (u − Π L u)L∞ (S) ≤ |uij | L x

i

p

i≥1, j≥0 i+j≥p+1

3.2 Polynomial approximation results

93

˜ L∞ ((0,1)) ≤ 2i2 L ˜ i L∞ ((0,1)) . Thus, By Markov’s inequality (3.2.10) we have L i upon setting qx = hx /(hx + σ), qy = hy /(hy + σ) we get from Lemma 3.2.7 (after appropriately decreasing σ in order to absorb the factor i2 ) C

qxi qyj = C

= C(1 − qy )−1 ≤ Cqx

∞

qxi qyj + C

i=1 j=p+1−i

i≥1, j≥0 i+j≥p+1

,

p+1

p+1

qxi qyj

i=p+2 j=0

qxi qyp+1−i + C(1 − qy )−1 qxp+2

i=1 p

∞ ∞

-

,

qxi qyp−i + qxp+1 = Cqx qxp + qyp +

i=0

p−1

qxi qyp−i + qxp+1 .

i=1

Next, Young’s inequality gives qxi qyp−i ≤

i p p−i p q + q , p x p y

i = 1, . . . , p − 1,

and we obtain therefore 2 ∂x (u − ΠpL u)L∞ (S) ≤ Cqx p qxp + qyp . Decreasing again the value of σ in order to absorb the factor p, we get the desired result. 2

3.2.3 Approximation of analytic functions on triangles The main result of this section is Theorem 3.2.19 where we show that an approximation result analogous to Proposition 3.2.8 holds on triangles as well. The approximation result Proposition 3.2.8 relied on truncted Legendre expansions and estimates for the decay of the coeﬃcients in this expansion. This could conveniently be done since tensor products of Legendre polynomials are orthogonal polynomials on the square S. The general approach in the case of approximation on triangles is similar: The polynomials ψp,q of (3.2.23) are orthogonal polynomials for the triangle, and our main aim in this section is to estimate the decay of the coeﬃcients up,q in the expansion u = p,q up,q ψp,q . For the approximation on triangles, it will be convenient to employ the Ja(α,β) , which have the following orthogonality properties in cobi polynomials Pp weighted spaces (see, e.g., [124, eq. (4.3.3)], [61, eq. 7.391]):

1

−1

(1 − t)α (1 + t)β Pp(α,β) (t)Pq(α,β) (t) dt = δpq

Γ (α + p + 1)Γ (β + p + 1) 2α+β+1 . α + β + 1 + 2p p!Γ (α + β + p + 1)

(3.2.18)

94

3. hp Approximation

Due to the fact that the Jacobi polynomials are deﬁned on the reference interval (−1, 1) instead of (0, 1) we adopt in the present section the following convention concerning the reference square S and the reference triangle T : T = {(x, y) | − 1 < x < 1, −1 < y < −x}.

S = (−1, 1)2 ,

(3.2.19)

The following transformation (sometimes referred to as the Duﬀy transformation, [45]) maps S onto T : D:S→T (η1 , η2 ) → (ξ1 , ξ2 ) =

(1+η1 )(1−η2 ) 2

− 1, η2

(3.2.20)

with inverse map D−1 : T → S

1+ξ1 . (ξ1 , ξ2 ) → (η1 , η2 ) = 2 1−ξ − 1, ξ 2 2

(3.2.21)

In terms of the Jacobi polynomials, we deﬁne the following polynomials on S: p 1 − η2 ψ˜p,q (η1 , η2 ) := Pp(0,0) (η1 ) Pq(2p+1,0) (η2 ), p, q ∈ N0 . (3.2.22) 2 The functions

ψp,q := ψ˜p,q ◦ D−1

(3.2.23)

deﬁned on T , are orthogonal polynomials of degree p + q as ascertained in the following lemma, which is due to Dubiner, [43]: Lemma 3.2.9 (orthogonal polynomials on the triangle). The functions ψp,q of (3.2.23) satisfy ψp,q ∈ Pp+q (T ), they are L2 (T )-orthogonal, and 2 2 ψp,q (ξ1 , ξ2 )ψp ,q (ξ1 , ξ2 ) dξ1 dξ2 = δpp δqq . 2p + 1 2p + 2q + 2 T Proof: We start with the assertion that ψp,q is a polynomial of degree p + q. With D−1 deﬁned in (3.2.21), we get p 1 − ξ2 1+ξ1 1+ξ1 (0,0) ˜ ψp,q (ξ1 , ξ2 ) = ψp,q (2 1−ξ2 − 1, ξ2 ) = Pp (2 1−ξ2 − 1) Pq(2p+1,0) (ξ2 ). 2 (0,0)

Expanding the Legendre polynomial P (0,0) as Pp ψp,q (ξ1 , ξ2 ) =

p

(x − 1) =

p

k k=0 ck x ,

we get

ck 2k−p (1 + ξ1 )k (1 − ξ2 )p−k Pq(2p+1,0) (ξ2 );

k=0 (2p+1,0)

is a polynomial of degree q, we get ψp,q ∈ Pp+q (T ). since Pq We next demonstrate the orthogonality. By transforming to S we get using (3.2.18) twice

3.2 Polynomial approximation results

95

1 − η2 dη1 dη2 ψ˜p,q (η1 , η2 )ψ˜p ,q (η1 , η2 ) 2 T S p+p +1 1 1 1 − η2 (0,0) (2p +1,0) (0,0) = Pp (η1 )Pp (η1 ) Pq(2p+1,0) (η2 )Pq (η2 ) dη1 dη2 2 −1 −1 1 2 (2p+1,0) δpp 2−(2p+1) = (1 − η2 )2p+1 Pq(2p+1,0) (η2 )Pq (η2 )dη2 2p + 1 −1 2 2 = δpp δqq . 2p + 1 2p + 2q + 2 ψp,q (ξ1 , ξ2 )ψp ,q (ξ1 , ξ2 ) dξ1 dξ2 =

2 We are interested in estimating the decay of the coeﬃcients up,q when expanding an L2 (T )-function in terms of the orthogonal basis (ψp,q )p,q∈N0 of L2 (T ). To do that, it will be important to expand the function t → 1/(w − t) in Jacobi polynomials, which is done in the following lemma: Lemma 3.2.10. Let α, β > −1. Then for every q ∈ N0 the function ˜ (α,β) (w) = w → Q q

(α,β)

1

−1

(1 − t)α (1 − t)β

is holomorphic on C \ [−1, 1]. Writing w = ˜ (α,β) (w) can be written as Q q ˜ (α,β) (w) = Q q

∞

Pq (t) dt w−t

1 2 (z

(3.2.24)

+ z −1 ) with z ∈ C, |z| > 1,

σm,q,α,β z −(m+1) ,

m=q

where the coeﬃcients σm,q,α,β ∈ R are independent of z and satisfy √ |σm,q,α,β | ≤ 2α+β+2 m + 1 0 0 Γ (α + 1)Γ (β + 1) Γ (α + q + 1)Γ (β + q + 1) × . (α + β + 1)Γ (α + β + 1) (α + β + 1 + 2q)q!Γ (α + β + q + 1) For the case β = 0, we have the particular bounds √ m+1 α+2 √ |σm,q,α,0 | ≤ 2 , √ 2q + α + 1 α + 1 |σm,q,0,0 | ≤ 2π. Proof: The second kind Chebyshev polynomials Um ∈ Pm are deﬁned on the interval [−1, 1] by the relation Um (cos θ) =

sin(m + 1)θ , sin θ

θ ∈ [0, π].

With w = 12 (z + z −1 ) we can then write in view of [61, eq. 8.945]

96

3. hp Approximation ∞ 1 1 2 1 2 = Um (t)z −m . = = w−t (z + z −1 )/2 − t z 1 − 2tz −1 + z −2 z m=0 (α,β)

˜q Inserting this into the deﬁnition of Q and exploiting Um ∈ Pm together with (α,β) the orthogonality properties of the Jacobi polynomials Pm , we get 1 (α,β) Pq (t) dt (1 − t)α (1 + t)β w − t −1 1 ∞ =2 z −(m+1) (1 − t)α (1 + t)β Pq(α,β) (t)Um (t) dt =2

m=0 ∞

z −(m+1)

m=q

−1 1

−1

(1 − t)α (1 + t)β Pq(α,β) (t)Um (t) dt.

We then get the desired representation by setting 1 σm,q,α,β := 2 (1 − t)α (1 + t)β Pq(α,β) (t)Um (t) dt. −1

For the bound on σm,q,α,β , we employ the Cauchy-Schwarz inequality to get 1 |σm,q,α,β | = 2 (1 − t)α (1 + t)β Pq(α,β) (t)Um (t) dt ≤ 2Um L∞ ((−1,1)) −1 1 1/2 1 2 1/2 × (1 − t)α (1 + t)β dt (1 − t)α (1 + t)β Pq(α,β) (t) dt . −1

−1

(α,β) P0

The observation = 1 and the formula (3.2.18) then allows us to obtain the desired bound, since elementary considerations reveal Um (t)L∞ ((−1,1)) ≤ (m + 1). The ﬁrst bound for the particular case β = 0 follows immediately from the general case by setting β = 0. For the special case α = β = 0, we employ the fact that |P (0,0) (t)| ≤ 1 for all t ∈ [−1, 1] to bound 1 π sin(m + 1)θ (0,0) (0,0) |σm,q,0,0 | = 2 Pq (t)Um (t) dt = 2 Pq (cos θ) sin θ dθ sin θ −1 0 ≤ 2π. 2 In the following, we will merely require the special case β = 0 in Lemma 3.2.10: ˜ (α,0) of (3.2.24) satisfy Corollary 3.2.11. For ρ > 1, the functions Q q 2π ˜ (0,0) ρ−(q+1) ∀w ∈ ∂Eρ , Qq (w) ≤ 1 − 1/ρ 2α+2 (q + 2) −(q+1) ˜ (α,0) ρ ∀w ∈ ∂Eρ . Qq (w) ≤ α + 1 (1 − 1/ρ)2

3.2 Polynomial approximation results

97

Proof: We will only show the second estimate. By Remark 3.2.5 we can write w ∈ ∂Eρ as w = 12 (z + z −1 ) for a z ∈ C with |z| = ρ. Lemma 3.2.10 then implies ∞ ∞ 2α+2 √ ˜ (α,0) −(m+1) |σα,0,q,m |ρ ≤ m + 1ρ−(m+1) Qq (w) ≤ α + 1 m=q m=q

≤

∞ 2α+2 2α+2 (q + 2) −(q+1) (m + 2)ρ−(m+1) ≤ ρ , α + 1 m=q α + 1 (1 − 1/ρ)2

where the last step follows from the fact that the power series can evaluated in closed form.

∞

m+1 m=q (m+2)x

2

We seek to approximate functions u that are analytic on the closure of T . The following lemma analyzes the domain of analyticity of the function u ˜ = u ◦ D. The key observation is that the transformation D is degenerate in the sense that the line {(η1 , 1) | η1 ∈ R} is mapped to the single point (−1, 1); this implies that ˜(η1 , η2 ) is very large for η2 the domain of holomorphy of the function η1 → u close to 1. Lemma 3.2.12. Let D be deﬁned in (3.2.20) and let u be analytic on the closure of T . Then there exist C, δ > 0, ρ > 1 depending only on u such that: 1. The function u ˜ := u ◦ D is analytic on the closure of S and can be extended ˜ there holds holomorphically to Eρ × Eρ . Denoting this extension again by u ˜ uL∞ (Eρ ×Eρ ) ≤ C.

(3.2.25)

˜(η1 , η2 ) is holomorphic on 2. For each η2 ∈ (−1, 1) the function η1 → u E1+δ/(1−η2 ) with sup η2 ∈(−1,1)

˜ u(·, η2 )L∞ (E1+δ/(1−η2 ) ) ≤ C.

For each p ∈ N0 the function η2 → Up (η2 ) := following properties:

+1 −1

(0,0)

u ˜(η1 , η2 )Pp

(3.2.26) (η1 ) dη1 has the

1. Up is holomorphic on Eρ and has a zero of multiplicity p at η2 = 1; 2. |Up (ζ2 )| ≤ Cρ−(p+1) ∀ζ2 ∈ Eρ . Proof: Since u is analytic on the closure of T , there exists a complex neighborhood T ⊂ C2 of the closure cl T of T such that u is holomorphic on T . We may additionally assume that T is chosen such that u is holomorphic on a neighborhood of the closure of T . This assumption implies uL∞ (T ) < ∞. Next, we can ﬁnd a (complex) neighborhood S of S such that D(S ) ⊂ T . By the continuity of D, we can ﬁnd ρ > 1 such that Eρ × Eρ ⊂ S ; hence the ﬁrst claim concerning the domain of holomorphy of u ˜ is proved. Setting for δ > 0 Gδ := {(ζ1 , ζ2 ) | ζ2 ∈ (−1, 1), ζ1 ∈ E1+δ/(1−ζ2 ) },

98

3. hp Approximation

a direct calculation shows that for δ > 0 suﬃciently small we have D(Gδ ) ⊂ T . ˜(η1 , η2 ) Thus, the claim about the domain of holomorphy of the function η1 → u is proved. Since D(Gδ ) ⊂ T , the bound (3.2.26) follows also. We now turn to the statements concerning Up . Since there exists ρ > 1 such that u ˜ is holomorphic on Eρ × Eρ , standard results (see, e.g., [103, Chap. 2, Thm. 1.1]) give that Up is holomorphic on Eρ . In order to show that Up has a zero of multiplicity p at η2 = 1, it suﬃces to show |Up (η2 )| ≤ C(1 − η2 )p for η2 ∈ (−1, 1). To that end,we employ Cauchy’s integral representation formula, the fact that πρ2 ≤ length(∂Eρ2 ) ≤ 4ρ2 , and Corollary 3.2.11, to get with the abbreviation ρ2 := 1 + δ/(1 − η2 ) 1 1 u ˜(ζ1 , η2 ) (0,0) |Up (η2 )| = Pp (η1 ) dζ1 dη1 −1 2π i ζ1 ∈∂Eρ ζ1 − η1 2 1 ˜ (0,0) (ζ1 )dζ1 = u ˜(ζ1 , η2 )Q p 2π i ζ1 ∈∂Eρ 2

length(∂Eρ2 ) ˜ (0,0) L∞ (E ) ≤ ˜ uL∞ (Gδ ) Q p ρ2 2π p 4ρ2 1 − η2 −(p+1) ≤ ˜ uL∞ (Gδ ) ρ2 ≤C , 1 − 1/ρ2 δ + (1 − η2 ) For the last bound, we proceed similarly. The function u ˜ is holomorphic on Eρ ×Eρ and bounded there. We get with Cauchy’s integral representation 1 (0,0) 1 Pp (η1 ) u ˜(ζ1 , ζ2 ) dη1 dζ1 |Up (ζ2 )| ≤ 2π i ζ1 ∈∂Eρ ζ − η 1 1 −1 ≤

length(∂Eρ ) ˜ (0,0) L∞ (E ) ≤ Cρ−p . ˜ uL∞ (Eρ ×Eρ ) Q p ρ 2π 2

Remark 3.2.13 The proof of Lemma 3.2.12 shows that ρ depends only on the domain of holomorphy of u. Proposition 3.2.14. Let T and the polynomials ψp,q be deﬁned in (3.2.19), (3.2.23). Let u be analytic on the closure of T . Then there exist C, b > 0 such that u can be expanded as up,q ψp,q in L2 (T ), u= p,q∈N0

where the coeﬃcients up,q ∈ R satisfy |up,q | ≤ Ce−b(p+q) . Furthermore, b > 0 depends only on the domain of holomorphy of (the holomorphic extension) of u.

3.2 Polynomial approximation results

99

Proof: The polynomials ψp,q are L2 (T )-orthogonal; hence, the coeﬃcients up,q are given by 1 u(ξ1 , ξ2 )ψp,q (ξ1 , ξ2 ) dξ1 dξ2 up,q = ψp,q L2 (T ) T √ √ 2p + 1 2p + 2q + 2 = u(ξ1 , ξ2 )ψp,q (ξ1 , ξ2 ) dξ1 dξ2 . 2 T Writing u ˜ = u ◦ D, we have to estimate u(ξ1 , ξ2 )ψp,q (ξ1 , ξ2 ) dξ1 dξ2 u ˜p,q := T

1

1

= −1

−1

Pp(0,0) (η1 )

1 − η2 2

p+1 Pq(2p+1,0) (η2 )˜ u(η1 , η2 ) dη1 dη2 .

To that end, we proceed in several steps. 1. step: We deﬁne the function 1 u ˜(η1 , η2 )Pp(0,0) (η1 ) dη1 . Up (η2 ) :=

(3.2.27)

−1

By Lemma 3.2.12 there exist C, ρ > 1 independent of p such that sup |Up (η2 )| ≤ Cρ−p .

η2 ∈Eρ

This bound together with (3.2.18) implies p+1 1 1 − η2 |˜ up,q | = Up (η2 ) Pq (η2 )dη2 −1 2 ! *1/2 1 1/2 1 2p+1 2 1 − η2 1 − η2 2 (2p+1,0) ≤ dη2 Pq |Up (η2 )| (η2 ) dη2 2 2 −1 −1 ≤ Cρ−p

2 ≤ Cρ−p . 2p + 2q + 2

(3.2.28)

We require a second estimate for u ˜p,q that decays exponentially in q. To that end, we note that Lemma 3.2.12 states that Up is holomorphic on Ep and has a zero of multiplicity p at η2 = 1. Hence, we may apply Cauchy’s integral theorem to the holomorphic function η2 → Up (η2 )/(1 − η2 )p to arrive at u ˜p,q =

1

−1

1 = 2π i

Up (η2 ) ζ2 ∈∂Eρ

1 − η2 2

p+1 Pq(2p+1,0) (η2 ) dη2

Up (ζ2 ) ((1 − ζ2 )/2)p

1

−1

1 − η2 2

2p+1

(2p+1,0)

Pq

(η2 ) dη2 dζ2 . ζ2 − η2

100

3. hp Approximation

Corollary 3.2.11 and Lemma 3.2.12 then imply the existence of C, ρ˜ ∈ (1, ρ) independent of p such that −p

|˜ up,q | ≤ Cρ−p (dist(∂Eρ , 1))

˜ (2p+1,0) L∞ (E ) ≤ Cγ p ρ˜−q . 2−p Q q ρ

(3.2.29)

2. step: We now combine the estimates (3.2.28), (3.2.29) in order to get the desired result. To that end, we will consider the cases p ≤ λq and p > λq separately for a λ ∈ (0, 1) to be chosen shortly. For p ≤ λq, we have, assuming without loss of generality γ > 1 in (3.2.29), |˜ up,q | ≤ Cγ p ρ˜−q ≤ Cγ λq ρ˜−q ≤ C ρˆ−q ,

(3.2.30)

for some ρˆ ∈ (1, ρ˜), if we choose λ suﬃciently small in dependence on γ, ρ˜. Since λ < 1 and λp ≤ q, we can estimate q = 12 q + 12 q ≥ 12 q + λ2 p ≥ λ(p + q)/2. Inserting this in (3.2.30) gives |˜ up,q | ≤ C ρ˜−λ(p+q)/2

∀(p, q) ∈ N0

such that p ≤ λq.

In the converse case p > λq we reason in the same way to get p ≥ λ2 (p + q) and |˜ up,q | ≤ Cρ−p ≤ Cρ−(p+q)λ/2

∀(p, q) ∈ N20

such that p > λq.

(3.2.31)

Since ρˆ < ρ˜ < ρ, combining (3.2.29) with (3.2.31) gives the desired bound by setting b = (λ/2) ln ρˆ. The statement that b depends only on the domain of holomorphy of u stems from the observation that b is determined by ρ of Lemma 3.2.12, which in turn depends only on the domain of holomorphy of u by Remark 3.2.13. 2 Proposition 3.2.14 shows that exponential convergence can be achieved for the L2 (T )-projection onto spaces of polynomials. In order to obtain exponential convergence of the L2 (T )-projection in stronger norms such as the L∞ (T )-norm, we need a result that allows us to control the orthogonal polynomials ψp,q on (complex) neighborhoods of the closure of T : Lemma 3.2.15. Let ψp,q be the orthogonal polynomials polynomials deﬁned in (3.2.23). Then for every ρ > 1 there exist C and a complex neighborhood T ⊂ C2 of the closure cl T of T such that ψp,q L∞ (T ) ≤ Cρp+q

∀p, q ∈ N0 .

Proof: We start by recalling the Bernstein estimate for univariate polynomials of degree p: ∀ρ > 1

∀u ∈ Pp

uL∞ (Eρ ) ≤ ρp uL∞ ((−1,1))

(3.2.32)

(see, e.g., [41, Chap. 4, Thm. 2.2]). By tensor product arguments, it is easy to see that for bivariate polynomials u ∈ Qp (S) we have uL∞ (Eρ ×Eρ ) ≤ ρ2p uL∞ (S) .

3.2 Polynomial approximation results

101

Put diﬀerently, for any ρ > 1, there exists a complex neighborhood S ⊂ C2 of the closure cl S of the square S such that uL∞ (S ) ≤ ρp uL∞ (S)

∀u ∈ Qp (S).

With the aid of aﬃne changes of variables, it follows then that for every closed parallelogram P and every ρ > 1 there exists a complex neighborhood P of P such that uL∞ (P ) ≤ ρp uL∞ (P ) ∀u ∈ Pp (T ). Since the triangle T can be covered by ﬁnitely many parallelograms, we conclude that for every ρ > 1 there exists a complex neighborhood T of the triangle T such that (3.2.34) uL∞ (T ) ≤ ρp uL∞ (T ) ∀u ∈ Pp (T ). In order to conclude the proof, we require ψp,q L∞ (T ) . By a standard inverse estimate (see, e.g., [112, Thm. 4.76]) and Lemma 3.2.9, we can bound with some C > 0 independent of p, q ψp,q L∞ (T ) ≤ C(1 + p + q)2 ψp,q L2 (T ) (3.2.35) 1/2 2 2 ≤ C(1 + p + q)3/2 . = C(1 + p + q)2 2p + 1 2p + 2q + 2 The claim of the lemma now follows by combining (3.2.35) with (3.2.34).

2

Proposition 3.2.16. Let u be analytic on the closure cl T of T . Then there exist C, b > 0 and a complex neighborhood T of cl T with the following properties: u can extended holomorphically to T (the extension being again denoted u) and inf v∈Pp (T )

u − vL∞ (T ) ≤ Ce−bp

∀p ∈ N0 .

Moreover, the constant b > 0 depends only on the domain of holomorphy of u. Proof: By the analyticity of u on cl T there exists a complex neighborhood T ⊂ C2 of cl T such that u can be extended holomorphically to T . Next, Proposition 3.2.14 gives u= up,q ψp,q in L2 (T ), (3.2.36) p,q∈N0

where the coeﬃcients up,q ∈ R satisfy for some C, b > 0 |up,q | ≤ Ce−b(p+q) .

(3.2.37)

Choosing ρ > 1 so small that ln ρ − b ≤ −b/2, we get from Lemma 3.2.15 the existence of C > 0 and a closed complex neighborhood T ⊂ T of cl T such that ψp,q L∞ (T ) ≤ Cρp+q ≤ Ce−(b/2)(p+q) .

(3.2.38)

102

3. hp Approximation

Hence, the bounds (3.2.37), (3.2.38) imply that the series in (3.2.36) convergences by holomorphy to u. The desired bound is easily ascertained for in L∞ (T ) and the choice v = i,j:i+j≤p ui,j ψi,j ∈ Pp (T ). 2 Corollary 3.2.17. Let u be analytic on the closure of T and let k ∈ N0 . Then there exist C, b > 0 such that inf v∈Pp (T )

u − vW k,∞ (T ) ≤ Ce−bp

∀p ∈ N0 .

Proof: The result follows from the Cauchy integral representation of derivatives and Proposition 3.2.16. 2 Remark 3.2.18 Exponential polynomial approximability on triangles is in principle known. Previous literature has usually made the assumption that the function to be approximated can be extended analytically to a square containing the triangle—Proposition 3.2.16 removes this restriction and makes minimal assumptions on the domain of analyticity of the function u to be approximated. The following result is the analog of Proposition 3.2.8 for the case of triangles. Theorem 3.2.19 (approximation on triangles and squares). Let K be the reference square or the reference triangle. Let u be analytic on K and satisfy for some Cu , γu > 0 and hx , hy ∈ (0, 1] 1 α2 Dα uL∞ (K) ≤ Cu γu|α| α!hα x hy

∀α ∈ N20 \ (0, 0).

Then there exist C, b > 0 depending only on γu , and a sequence vp ∈ Pp (T ) of polynomials such that , p+1 p+1 hx hy u − vp L∞ (K) ≤ CCu , + hx + b hy + b . p p / hx hx hy ∞ ∂x (u − vp )L (K) ≤ CCu , + hx + b hx + b hy + b . p p / hx hy hy ∂y (u − vp )L∞ (K) ≤ CCu . + hy + b hx + b hy + b Proof: For K = S, the result is proved in Proposition 3.2.8. The case K = T can be seen as follows. There exists h0 < 1 depending only on γu such that if hx < h0 or hy < h0 then u can be extended analytically to S. Let us assume that hx < h0 . The analytic extension of u to S (again denoted u) then satisﬁes 1 n1 n2 $un1 +n2 (n1 + n2 )!h−n hx hy ∂xn1 ∂yn2 uL∞ (S) ≤ CCu γ 0

∀(n1 , n2 ) ∈ N20 \ (0, 0)

for some C, γ $u depending only on γu . The result now follows from the analysis for squares of Proposition 3.2.8. Thus, Theorem 3.2.19 is proved if hx < h0 or hy < h0 . It remains to consider the case where simultaneously hx ≥ h0 and hy ≥ h0 . Then, the result follows immediately from Proposition 3.2.16. 2

3.2 Polynomial approximation results

103

3.2.4 The projector Πp∞ Our ultimate goal is the construction of polynomial approximants in an elementby-element fashion. To that end, it is convenient to construct approximants that coincide with the Gauss-Lobatto interpolant on the edges. On squares S, it is therefore natural to consider the Gauss-Lobatto interpolation operator jp . On triangles T , the construction of an interpolation operator that coincides with the Gauss-Lobatto interpolants on the edges is not as straight forward. This is accomplished in the ensuing theorem. Theorem 3.2.20 (Deﬁnition & Properties of Πp∞ ). Let K = S or K = T . There is C > 0 such that for every p ≥ 1 there is a bounded linear operator Πp∞ : C(K) → Πp (K) with the following properties: ip,Γ u = Πp∞ u|Γ u −

Πp∞ uL∞ (K)

≤ CCK (p)

Πp∞ v = v

∀ edges Γ of K, inf v∈Πp (K)

u − vL∞ (K) ,

for all v ∈ Πp (K).

Here, the constant CK (p) is given by ! (1 + ln p)2 CK (p) := p(1 + ln p)

if K = S, if K = T .

(3.2.39) (3.2.40) (3.2.41)

(3.2.42)

In particular, for p = 1, the operator Π1∞ coincides with the linear/bilinear interpolation operator. For functions u ∈ H 1 (K) ∩ C(K) there holds ∇(u − Πp∞ u)L2 (K) ≤

inf ∇(u − v)L2 (K) + Cp2 CK (p)u − vL∞ (K) ,

(3.2.43)

v∈Πp (K)

and for functions u ∈ W 1,∞ (K) we have ∇(u − Πp∞ u)L∞ (K) ≤

inf ∇(u − v)L∞ (K) + Cp2 CK (p)u − vL∞ (K) .

(3.2.44)

v∈Πp (K)

Proof: We proceed in two steps. First, we construct the bounded linear operator Πp∞ : C(K) → Πp (K) satisfying (3.2.39), (3.2.41), and Πp∞ C(K)→L∞ ≤ CCK (p)

(3.2.45)

for some C > 0 independent of p; CK (p) is given in the statement of the theorem. In a second step, the estimates (3.2.40) and (3.2.43), (3.2.44) are obtained. For the case K = S, we set Πp∞ := jp and note that Lemma 3.2.1 gives readily (3.2.45). It is easy to see that jp also satisﬁes (3.2.39), (3.2.41). We now turn to the deﬁnition of the operator Πp∞ on triangles T . For u ∈ C(T ) let ip,∂T u be

104

3. hp Approximation

the (edgewise) Gauss-Lobatto polynomial interpolant of u. With the extension operator E of Lemma 3.2.3 we then obtain with the stability properties of the Gauss-Lobatto interpolation operator (Lemma 3.2.1) E(ip,∂T u)L∞ (T ) ≤ Cip,∂T uL∞ (∂T ) ≤ C(1 + ln p)uL∞ (∂T ) . Next, we introduce the closed subspace V := {u ∈ Pp (T ) | u = 0 on ∂T } and let Π ∞ be the bounded linear operator from C(T ) onto V given by Corollary A.4.2. The desired operator Πp∞ is then taken as Πp∞ u := E(ip,∂T u) + Π ∞ (u − E(ip,∂T u)). By the bound Π ∞ L∞ →L∞ ≤ (p + 1), it is clear that Πp∞ C(K)→L∞ ≤ Cp(1 + ln p) = CCK (p) with CK (p) given in the statement of the theorem for K = T . By construction, (3.2.39) holds, and it is easy to see that Πp∞ satisﬁes (3.2.41). In the second step, we obtain (3.2.40). As Πp∞ reduces to the identity on Πp (K), we have for all v ∈ Πp (K) u − Πp∞ uL∞ (K) = (u − v) − Πp∞ (u − v)L∞ (K) ≤ CCK (p)u − vL∞ (K) . Taking the inﬁmum over all v ∈ Πp (K) gives (3.2.40). (3.2.43) and (3.2.44) are proved in a similar way, and we therefore only show (3.2.43). For every v ∈ Πp (K), we have ∇(u − Πp∞ u)L2 (K) = ∇((u − v) − Πp∞ (u − v))L2 (K) ≤ ∇(u − v)L2 (K) + ∇(Πp∞ u − v)L2 (K) ≤ ∇(u − v)L2 (K) + Cp2 Πp∞ u − vL2 (K) ≤ ∇(u − v)L2 (K) + Cp2 u − Πp∞ uL∞ (K) + u − vL2 (K) ≤ ∇(u − v)L2 (K) + Cp2 CK (p)u − vL∞ (K) , where in the last step, we employed (3.2.40). Taking the inﬁmum over all v ∈ 2 Πp (K) gives the desired bound. For future reference, we collect some approximation results for the projector Πp∞ in the following proposition. Proposition 3.2.21. Let K = S or K = T and let A be one vertex of K. Let β ∈ [0, 1). Then there holds u − Πp∞ uL∞ (K) + u − Πp∞ uL2 (K) + p−2 ∇(u − Πp∞ u)L2 (K) ≤ Cp(1 + ln p) |x − A|β ∇2 uL2 (K) .

(3.2.46)

For functions u that are analytic on K and satisfy for some Cu , γu > 0 ∇p uL∞ (K) ≤ Cu γup p!

∀p ∈ N,

there are C, b > 0 depending only on γu such that u − Πp∞ uL∞ (K) + ∇(u − Πp∞ u)L∞ (K) ≤ CCu e−bp .

(3.2.47)

3.2 Polynomial approximation results

105

Proof: The ﬁrst estimate follows from inserting the result of Lemma 3.2.4 into the bounds of Theorem 3.2.20. Inserting the results of Theorem 3.2.19 into the statement of Theorem 3.2.20 gives the second assertion. 2 Remark 3.2.22 For analytic functions u, Proposition 3.2.21 shows that the projector Πp∞ yields exponential convergence. In the case of ﬁnite regularity, i.e., u ∈ H k for some k > 1, estimating (3.2.43) requires simultaneous approximation in diﬀerent norms, which is provided in [6]. The resulting estimates, however, are not p-optimal due to our using inverse estimates. An alternative projector leading to p-optimal estimates is presented in Section 3.2.6. 3.2.5 Anisotropic projection operators: Πp1,∞ In order to motivate the introduction of the second interpolation operator Πp1,∞ , let us consider the approximation of a (smooth) function u on a rectangle R = (0, hx ) × (0, hy ) whose aspect ratio may be very large. On the reference square S, one has to approximate the function u ˆ := u(hx x, hy y). The operator Πp∞ ∞ is essentially an L projector on the reference square. Because the L∞ -norm is invariant under bijective mappings, it also yields good approximation results for u on R in the L∞ norm. However, the operator Πp∞ does not allow for good anisotropic gradient estimates. The aim of the present section is the construction of an operator Πp1,∞ that shares with Πp∞ the property that it reduces to a Gauss-Lobatto interpolation operator on the edges of the domain (square S or triangle T ) and at the same time reﬂects anisotropy of the gradient of the function to be interpolated. The anisotropic behavior of the gradient behavior will be captured by the anisotropic norm |·|. For given hx , hy > 0 we introduce on the space W 1,∞ (K) the following weighted norm −1 |u| := uL∞ (K) + h−1 x ∂x uL∞ (K) + hy ∂y uL∞ (K) .

(3.2.48)

In order to construct the interpolation operator Πp1,∞ that is stable uniformly in hx , hy we need an lemma: Lemma 3.2.23. Let K = S or K = T . Then there is C > 0 such that the following holds. For every hx , hy > 0 and every p ∈ N there is a bounded linear operator E 1,∞ : W 1,∞ (K) → Πp (K) with the following properties: ∀ edges Γ of K, E 1,∞ u|Γ = ip,Γ u|Γ 1,∞ |E u| ≤ CCK (p)|u|, where CK (p) is given by

! p(1 + ln p) CK (p) := p

if K = S if K = T.

Furthermore, in the case K = S, the operator E 1,∞ may be taken as jp .

106

3. hp Approximation

Proof: We start with the case K = S. Then we set E 1,∞ := jp , the tensorproduct Gauss-Lobatto interpolation operator. We have from Lemma 3.2.1 E 1,∞ uL∞ (K) ≤ C(1 + ln p)2 uL∞ (K) ≤ Cp(1 + ln p)uL∞ (K) . For the derivatives, we recall jp = iyp ◦ ixp and estimate, using Lemma 3.2.1 twice, ∂x (iyp ◦ ixp u)L∞ (S) = iyp ∂x ixp uL∞ (S) ≤ C(1 + ln p)∂x ixp uL∞ (S) ≤ Cp(1 + ln p)∂x uL∞ (S) . An analogous estimate holds for ∂y jp u, which proves the lemma if K = S. We now turn to the case K = T . We ﬁrst show that for the case p = 1, we may take the linear interpolant given by Iu(x, y) = yu(0, 1) + xu(1, 0) + (1 − x − y)u(0, 0). Obviously IuL∞ (T ) ≤ uL∞ (T ) . For the derivatives, we compute

1

∂x Iu = u(1, 0) − u(0, 0) =

∂x u(ξ, 0) dξ,

(3.2.49)

0

and therefore ∂x IuL∞ (T ) ≤ ∂x uL∞ (T ) . Since an analogous estimate holds for ∂y Iu, we conclude |Iu| ≤ |u|. For p > 1, we may therefore construct the extension operator E 1,∞ for functions u that vanish at the vertices of T . Furthermore, without loss of generality we may assume hy ≤ hx . (3.2.50) Let u ∈ W 1,∞ vanish at the vertices of T . We construct E 1,∞ u by considering the edge Γ1 = {(0, y) | y ∈ (0, 1)} ﬁrst and then the two remaining edges. Set πp1 (x, y) :=

1−x−y h1 (y), 1−y

h1 (y) := iyp u(0, y).

Since u(0, 1) = 0, the function h1 ∈ Pp (I) vanishes at y = 1 and thus πp1 ∈ Pp (T ). We note furthermore that on Γ1 the polynomial π 1 coincides with the GaussLobatto interpolant of u, and π 1 vanishes on the two remaining edges of T . For the auxiliary function h1 , we obtain with the aid of Lemma 3.2.1 |h1 (y)| ≤ C(1 + ln p)uL∞ (T ) , h1 (y) h1 (y) − h1 (1) = ≤ h1 L∞ ((0,1)) ≤ Cp∂y uL∞ (T ) 1 − y 1−y for all y ∈ (0, 1). We compute the derivatives h1 (y) , 1−y x h1 (y) 1 − x − y + h1 (y). ∂y πp1 (x, y) = − 1−y 1−y 1−y

∂x πp1 (x, y) = −

(3.2.51) (3.2.52)

3.2 Polynomial approximation results

The bounds

x 1 − y ≤ 1,

1 − x − y 1−y ≤1

∀(x, y) ∈ T

107

(3.2.53)

together with (3.2.51), (3.2.52), and the assumption (3.2.50) allow us to estimate πp1 L∞ (T ) ≤ h1 L∞ ((0,1)) ≤ C(1 + ln p)uL∞ (T ) ≤ Cp|u|, 1 −1 −1 h−1 x ∂x πp L∞ (T ) ≤ Chx p∂y uL∞ (T ) ≤ Cphy ∂y uL∞ (T ) ≤ Cp|u|, 1 −1 h−1 y ∂y πp L∞ (T ) ≤ Cphy ∂y uL∞ (T ) ≤ Cp|u|.

It remains to correct the other two edges. We introduce the one dimensional interpolants h1 (x) = ixp u(x, 0), h2 (x) = ixp u(x, 1 − x) and deﬁne

y (h2 (x) − h1 (x)) . 1−x Because h1 (1) = h2 (1) = 0, we have πp ∈ Pp (T ). We note again that πp vanishes on the edge x = 0 and equals ixp,Γ u for the remaining two edges Γ of T . Analogously to the bounds (3.2.51), (3.2.52), we have for all x ∈ (0, 1): πp2 (x, y) = h1 (x) +

|h1 (x)| + |h2 (x)| ≤ C(1 + ln p)uL∞ (T ) ,

h1 L∞ ((0,1)) ≤ Cp∂x uL∞ (T ) , h2 L∞ ((0,1)) ≤ Cp ∂x uL∞ (T ) + ∂y uL∞ (T ) .

(3.2.54) (3.2.55) (3.2.56)

In order to bound (h2 (x) − h1 (x))/(1 − x), we note that, since h2 (1) = h1 (1), we have (h2 (x) − h1 (x))/(1 − x) ∈ Pp−1 . Since this polynomial coincides in the Gauss-Lobatto points with the function (u(x, 0) − u(x, 1 − x))/(1 − x), we conclude h2 (x) − h1 (x) u(x, 0) − u(x, 1 − x) = ixp . 1−x 1−x Estimating |u(x, 0) − u(x, 1 − x)| =

0

1−x

∂y u(x, t) dt ≤ (1 − x)∂y uL∞ (T ) ,

we obtain with the stability of the Gauss-Lobatto interpolation operator h2 (x) − h1 (x) ≤ C(1 + ln p) sup u(x, 0) − u(x, 1 − x) sup 1−x 1−x x∈(0,1) x∈(0,1) ≤ C(1 + ln p)∂y uL∞ (T ) .

(3.2.57)

A calculation reveals 1−x−y y y h1 (x) + h2 (x) + (h2 (x) − h1 (x)), 1−x 1−x 1−x h2 (x) − h1 (x) ∂y πp2 (x, y) = . 1−x

∂x πp2 (x, y) =

108

3. hp Approximation

Hence, we can bound using (3.2.54)–(3.2.57), (3.2.53), and (3.2.50) πp2 L∞ (T ) ≤ C(1 + ln p)uL∞ (T ) ≤ C(1 + ln p)|u|, 2 −1 h−1 ∂x uL∞ (T ) + ∂y uL∞ (T ) ≤ Cp|u|, x ∂x πp L∞ (T ) ≤ Cphx 2 −1 h−1 y ∂y πp L∞ (T ) ≤ C(1 + ln p)hy ∂y uL∞ (T ) ≤ C(1 + ln p)|u|.

2

The claim of the lemma now follows. We are now in position to deﬁne the projection operator Πp1,∞ .

Theorem 3.2.24 (Deﬁnition & Properties of Πp1,∞ ). Let K = S or K = T and hx , hy > 0. Let | · | be given by (3.2.48). Then there exists C > 0 independent of hx , hy , and p, and there exists a bounded linear operator Πp1,∞ : W 1,∞ (K) → Πp (K) with the following properties: ip,Γ u = Πp1,∞ u|Γ |u −

Πp1,∞ u|

≤ CCK (p)

Πp1,∞ v = v

∀ edges Γ of K,

(3.2.58)

|u − v|,

(3.2.59)

inf v∈Πp (K)

for all v ∈ Πp (K).

Here, the constant CK (p) is given by ! p(1 + ln p) CK (p) := p2

if K = S, if K = T .

(3.2.60)

(3.2.61)

Proof: For K = S, we choose Πp1,∞ = jp and the statement follows from Lemma 3.2.23. It remains to deﬁne Πp1,∞ for triangles. We proceed as in the proof of Theorem 3.2.20. We set V := {u ∈ Pp (T ) | u = 0 on ∂T }. Let E 1,∞ be the bounded linear operator of Lemma 3.2.23 and let Π 1,∞ be the bounded linear operator of Corollary A.4.3 mapping W 1,∞ (K) onto V . We then set Πp1,∞ u := E 1,∞ u + Π 1,∞ u − E 1,∞ u . Clearly, Πp1,∞ is linear, it satisﬁes (3.2.58), and it is bounded: |Πp1,∞ u| ≤ Cp2 |u|

∀u ∈ C 1 (T ).

Furthermore, Πp1,∞ reduces to the identity operator on the space Pp (T ). The desired ﬁnal bound (3.2.59) now follows as in the proof of Theorem 3.2.20. 2 We can therefore get the following approximation result for analytic functions with anisotropic bounds on the derivatives. Proposition 3.2.25. Let K = S or K = T . Assume that a function u is analytic on K and satisﬁes, for some Cu , γu > 0, hx , hy ∈ (0, 1] n+m ∂xn ∂ym uL∞ (K) ≤ Cu hnx hm n!m! y γu

∀(p, q) ∈ N20 \ (0, 0).

3.2 Polynomial approximation results

109

Then there exist C, σ > 0 depending only on γu such that the diﬀerence e := u − Πp1,∞ u satisﬁes with h := max {hx , hy } h h eL∞ (K) + ∂x eL∞ (K) + ∂ eL∞ (K) ≤ CCu hx hy y

h h+σ

p+1 .

Proof: The result follows by choosing in the inﬁmum in Theorem 3.2.24 the approximant of Theorem 3.2.19. 2 Remark 3.2.26 The growth conditions on the derivatives of the analytic function u in the statement of Proposition 3.2.25 appear quite naturally if so-called normalizable triangulations are considered, cf. Deﬁnition 3.3.3 ahead. 3.2.6 An optimal error estimate for an H 1 -projector For analytic functions u, the projectors Πp∞ , Πp1,∞ yield approximants that are exponentially close to u. The situation is diﬀerent if u has ﬁnite regularity, e.g., u ∈ H k (K) for some k > 1. Then the projectors Πp∞ , Πp1,∞ do not yield the expected optimal rate p−(k−1) when measuring the error in the H 1 -norm (cf., e.g., (3.2.43)). While the projectors Πp∞ , Πp1,∞ are suﬃcient for the purposes of 1 this work, it is interesting to note that it is possible to construct a projector Π H that realizes the optimal rate of convergence and at the same time is constrained to coincide with the Gauss-Lobatto interpolation operator on the boundary. Proposition 3.2.27. Let K = S or K = T , k > 3/2. Then there exist a 1 constant C(k) > 0 and a projector Π H : H k (K) → Πp (K) such that: 1

∀edges Γ of K,

ip,Γ u = Π H u|∂K 1

ΠH u = u u − Π

H

1

(3.2.62)

∀u ∈ Πp (K),

uH 1 (K) ≤ C(k)p

−(k−1)

(3.2.63)

|u|H k (K) .

(3.2.64)

Proof: By [22] there exists a bounded linear operator Π : H k (K) → Πp (K) with the following properties: u − ΠuH t (K) ≤ Cp−(k−t) uH k (K) , t ∈ {0, 1}, Πu(A) = u(A) ∀ vertices A of K, u − ΠuH t (Γ ) ≤ Cp−(k−1/2−t) uH k (K)

∀ edges Γ of K,

t ∈ {0, 1}.

We conclude by interpolation u − ΠuH 1/2 (Γ ) ≤ Cp−(k−1) uH k (K) 00

∀edges Γ of K.

Next, by Lemma 3.2.28 below and the trace theorem, we have for each edge Γ of K the bound

110

3. hp Approximation

u − ip,Γ uH 1/2 (Γ ) ≤ Cp−(k−1) uH k (K) . 00

By construction Πu|∂K − ip,∂K u vanishes at the vertices of K, it is a polynomial of degree p on each edge, and ip,∂K − ΠuH 1/2 (Γ ) ≤ Cp−(k−1) uH k (K) 00

∀ edges Γ of K.

(3.2.65)

By [13] there exists a linear map E : {f ∈ C(∂K) | f |Γ ∈ Pp for each edge Γ } → Πp (K) with the property that Ef H 1 (K) ≤ Cf H 1/2 (∂K) for some C > 0 independent of p. We conclude with (3.2.65) ip,Γ u − ΠuH 1/2 (Γ ) ≤ Cp−(k−1) uH k (K) . E ip,∂K u − Πu H 1 (K) ≤ C 00

Γ ⊂∂K

Hence, the map

Π : u → Πu + E(ip,∂K u − Πu)

satisﬁes (3.2.62) and u − Π uH 1 (K) ≤ Cp−(k−1) uH k (K) . Next, we adjust Π such that (3.2.63) is satisﬁed. To that end, we deﬁne the best approximation operator Π : H 1 (K) → Πp (K) ∩ H01 (K) by u − Π uH 1 (K) = min{u − qH 1 (K) | q ∈ Πp (K) ∩ H01 (K)}. Π is a bounded linear map with Π uH 1 (K) ≤ uH 1 (K) and Π u = u for all u ∈ Πp (K) ∩ H01 (K). We then deﬁne 1

Π H := Π + Π (Id −Π ). It is easy to see that this operator satisﬁes (3.2.62) (3.2.63) and 1

u − Π H uH 1 (K) ≤ Cp−(k−1) uH 1 (K) . 1

Since Π H u = u for all u ∈ Πp (K), we may replace the full H k -norm on the right-hand side by the H k -semi norm in the standard way. 2 Lemma 3.2.28. Let I = (0, 1), k ≥ 1. Then there exists C(k) > 0 such that the Gauss-Lobatto interpolation operator ip satisﬁes u − ip uH 1 (I) ≤ Cp−(k−1) uH k (I) , −k

u − ip uL2 (I) ≤ Cp

uH k (I) .

(3.2.66) (3.2.67)

By interpolation therefore u − ip uH 1/2 (I) ≤ Cp−(k−1/2) uH k (I) . 00

(3.2.68)

3.3 Admissible boundary layer meshes and ﬁnite element spaces

111

Proof: The estimate (3.2.66) follows directly from the H 1 -stability of the GaussLobatto interpolation, Theorem A.3.1. Likewise, the L2 -bound (3.2.67) follow from a stability estimate of Theorem A.3.1: u − ip uL2 (I) ≤ C inf u − qL2 (I) + p−1 u − qH 1 (I) ≤ Cp−k uH k (I) . q∈Pp

Since u and ip u coincide at the endpoints of I, the operator Id −ip maps in fact into H01 (I); that is, we have Id −ip : H k (I) → H01 (I) with norm bounded by Cp−(k−1) and Id −ip : H k (I) → L2 (I) with norm bounded by Cp−k . Interpolation then gives the desired bound (3.2.68). 2 1

Remark 3.2.29 The projector Π H of Proposition 3.2.27 is such that the approximant coincides with the function to be approximated in the Gauss-Lobatto points on the boundary. For a priori estimates in the p-version FEM, this allows for the construction of approximants in an element-by-element fashion. Diﬀerent approaches, which are also able to handle the case k ∈ (1, 3/2), have been taken in [18, 22, 23].

3.3 Admissible boundary layer meshes and ﬁnite element spaces We recall that the reference square S and the reference triangle T are deﬁned as S = (0, 1) × (0, 1) and T = {(x, y) | 0 < x < 1 | 0 < y < x}. We also refer the reader to Deﬁnition 2.4.1 for the precise notion of triangulations T . We now consider the problem of introducing the notion of element size for curved elements. We will not develop a general theory for characterizing the size of an element but in the next two deﬁnitions, we present two diﬀerent approaches to this issue. Deﬁnition 3.3.1 ((CM , γM )-regular triangulation). A triangulation T = ˆ i )} of a domain Ω is called (CM , γM )-regular if for each element Ki {(Ki , Mi , K there is an aﬃne map Ai with Ai = hi , hi := diamKi , such that −1 L∞ (Kˆ i ) ≤ CM , (A−1 i ◦ Mi ) p ∇p (A−1 ˆ i ) ≤ CM γM p! i ◦ Mi )L∞ (K

∀p ∈ N0 .

Remark 3.3.2 As stated, Deﬁnition 3.3.1 allows only for isotropic elements of size hi . The deﬁnition could be extended to introduce the notion of anisotropic elements—this would require that a Cartesian coordinate (possibly diﬀerent for each element) be chosen with respect to which anisotropic stretching is done. The reader will recognize that this approach is taken implicitly in the deﬁnition of boundary layer elements in Deﬁnition 2.4.4.

112

3. hp Approximation

In the approach of Deﬁnition 3.3.1, the notion of element size is introduce by stipulating the ability to control A−1 i ◦Mi where Ai is some appropriate stretching map. Clearly, one could also think of stipulating the ability to control Mi ◦ Ai for some stretching map Ai . This approach is taken in the notion of normalizable triangulations that we introduce now. Deﬁnition 3.3.3 ((CM , γM )-normalizable triangulation). A triangulation ˆ i ) is called (CM , γM )-normalizable if for each element Ki there T = (Ki , Mi , K are hx,i , hy,i ∈ (0, 1] such that the element map Mi can be written in the form M i = G i ◦ Ai , where Ai is an aﬃne map satisfying hx,i 0 , Ai = 0 hy,i and the analytic map Gi satisﬁes p p! ∇p Gi L∞ (Ri ) ≤ CM γM

(Gi )−1 L∞ (Ri )

∀p ∈ N0 ,

≤ CM .

ˆ i under the aﬃne Here, the sets Ri are the images of the reference element K ˆ maps Ai , i.e., Ri = Ai (Ki ). Remark 3.3.4 Some comments concerning Deﬁnition 3.3.3 are in order. Firstly, the notion of normalizable triangulations introduces in a very natural way anisotropic elements, and we employed a similar notion in the context of Shishkin meshes in Deﬁnition 2.6.17. Secondly, the requirement for a triangulation to be (CM , γM )-normalizable is stronger than that to be (CM , γM )-regular (see Proposition 3.3.5 and the closely related Proposition 3.3.11). We point out that the stronger notion of normalizable triangulations lends itself more easily to error bounds in approximation theory; in fact, Propositions 3.2.25, 3.2.8 are formulated so as to ﬁt into the framework of normalizable triangulations. ˆ i )} be a (CM , γM )-normalizable trianProposition 3.3.5. Let T = {(Ki , Mi , K gulation and assume that for each element Ki there holds hx,i = hy,i = hi = diamKi . Then there are C, γ > 0 depending only on CM , γM such that T is a (C, γ)-regular triangulation in the sense of Deﬁnition 3.3.1 Proof: Consider an element Ki . By assumption, the element map Mi has the form Mi = Gi ◦ Ai . The aﬃne map whose existence is stipulated in Deﬁnition 3.3.1 is now taken as A˜i x := Ai (x − Mi (0)). The assumptions on Gi then easily imply the desired result. 2 Once a triangulation T is chosen, one can deﬁned ﬁnite-element spaces S p (T ) based on this triangulation.

3.3 Admissible boundary layer meshes and ﬁnite element spaces

1

M

ˆ K

O(h)

M

ˆ K

1

113

O(h) G = A−1 ◦ M

A ˆ K = M (K)

A

ˆ K = M (K) h

G = M ◦ A−1 ˆ R = A(K)

ˆ R = G(K) O(1) Fig. 3.3.1. Two diﬀerent approaches to the concept of element size: via Def. 3.3.1 (left) and Def. 3.3.3 (right).

ˆ i) | i ∈ Deﬁnition 3.3.6 (FE-spaces). Given a triangulation T = {(Ki , Mi , K p 1 p I(T )}, the H -conforming ﬁnite element spaces S (T ), S0 (T ) of piecewise mapped polynomials are deﬁned as ˆ i )}, S p (T ) := {u ∈ H 1 (Ω) | u|Ki = ϕp ◦ Mi−1 for some ϕp ∈ Πp (K p S0 (T ) := S p (T ) ∩ H01 (Ω)

(3.3.1) (3.3.2)

ˆ i ) deﬁned in (3.2.4). with spaces Πp (K In Section 3.2.4, we introduced the interpolation operator Πp∞ on the reference square S and the reference triangle T . For a mesh T we can then deﬁne the ∞ operator Πp,T by an elementwise application of Πp∞ : ∞ u|Ki := Πp∞ (u ◦ Mi ) ◦ Mi−1 ∀ elements Ki . (3.3.3) Πp,T For a mesh T in the sense of Deﬁnition 2.4.1 we can naturally speak about vertices and edges as the images of the vertices and edges of the references elements. Hence, we can also introduce the edge-wise Gauss-Lobatto interpolation operator ip,Γ as follows: Let Γ be an edge of an element Ki , let Γˆ := Mi−1 (Γ ), and set ip,Γ u := ip,Γˆ (u ◦ Mi ) ◦ Mi−1 . (3.3.4) We note that the operator ip,Γ is well-deﬁned: An edge Γ of a triangulation T is shared (in general) by two elements Ki , Kj . However, assumption (M4) of Deﬁnition 2.4.1 guarantees that the parametrization of Γ induced by both Mi and Mj coincides, thus making the deﬁnition of ip,Γ in (3.3.4) well-deﬁned. We ∞ now show that the operator Πp,T in fact is an operator from C(Ω) into S p (T ): ˆ i ) | i ∈ I(T )} be a triangulation of a domain Lemma 3.3.7. Let T = {(Ki , Mi , K Ω in the sense of Deﬁnition 2.4.1. Let u ∈ C(Ω). For each element Ki let ˆ i ) satisfy πp,i ∈ Πp (K ip,Γˆ (u ◦ Mi ) = πp,i |Γˆ

ˆ i. ∀ edges Γˆ of K

(3.3.5)

114

3. hp Approximation

Then π deﬁned on Ω by π|Ki := πp,i ◦Mi−1 is an element of S p (T ). Furthermore, if u = 0 on ∂Ω, then π ∈ S0p (T ). Proof: One has to check that π deﬁned elementwise is in fact continuous across the edges. This follows from a) assumption (M4) of Deﬁnition 2.4.1 that shared edges of the triangulation have the same parametrization; b) that the GaussLobatto interpolation points are distributed symmetrically; and c) the uniqueness of the Gauss-Lobatto interpolation process. 2 Of particular use to us will the following result. ˆ i ) | i ∈ I(T )} be a triangulation of a Corollary 3.3.8. Let T = {(Ki , Mi , K ∞ domain Ω in the sense of Deﬁnition 2.4.1. Then the operator Πp,T : C(Ω) → p S (T ) deﬁned in (3.3.3) is a linear operator. Furthermore, if u = 0 on ∂Ω, then ∞ Πp,T u ∈ S0p (T ). The same result holds if the operator Πp∞ in (3.3.3) is replaced with Πp1,∞ . ∞ Remark 3.3.9 The operator Πp,T constructs approximations from S p (T ) in a strictly elementwise fashion. This procedure is diﬀerent from the classical approach of Babuˇska & Suri, [22]. There, approximants are constructed in two steps: In the ﬁrst step, a discontinuous approximant is constructed in an elementby-element fashion. In the second step, the interelement jumps across edges are removed with a polynomial trace-lifting. This procedure is not convenient for our purposes as we expressly want to consider meshes which may be distorted and in which thin needle elements may share an edge with large elements, i.e., where two adjacent elements may be structurally very diﬀerent. Our elementwise ap∞ proach with the operator Πp,T avoids having to consider two adjacent elements at the same time. It should be mentioned, however, that the construction of in∞ terpolants using Πp,T does not lead to optimal p-version approximation results in the case of ﬁnite Sobolev regularity, i.e., if the function u to be approximated is only known to be in a Sobolev space H k (Ω) for some k ≥ 1. The projector 1 Π H constructed in Section 3.2.6 overcomes this diﬃculty.

3.3.1 hp-meshes for the approximation of boundary and corner layers The regularity assertion of Theorem 2.3.4 shows that we have to deal with two types of phenomena: boundary layers in the vicinity of the boundary curves Γj and corner singularities in neighborhoods of the vertices Aj . In order to resolve these two phenomena, one needs meshes that contain needle elements near the boundary curves Γj and geometric meshes near the vertices Aj . The (essentially) minimal mesh family is one with needle elements near the boundary and geometric mesh reﬁnement near the vertices. An example of such an admissible mesh is presented in Fig. 2.4.2: The rectangles at the boundary are boundary layer elements of width O(κ), the elements in the shaded regions are corner layer elements, and the remaining elements are interior elements. The precise requirements for these three element types were presented in Deﬁnition 2.4.4.

3.3 Admissible boundary layer meshes and ﬁnite element spaces

115

Admissible boundary layer meshes as deﬁned in Deﬁnition 2.4.4 are essentially the minimal meshes that can lead to robust exponential convergence for the approximation of solutions to (2.1.1). As discussed in Remark 2.4.5 and shown in Fig. 2.4.1, the meshes may be severely distorted in the sense that minimal and maximal angles can be close to 0 or π. For implementational reasons, it is advisable to be able to control the distortion of the elements, in particular the maximal angles. The introduction of regular admissible meshes allows for this control. In essence, a regular admissible mesh is an admissible mesh family that is also a normalizable mesh family. The mesh of Fig. 2.4.1 is in fact an example of a regular admissible mesh. Deﬁnition 3.3.10 (regular admissible boundary layer mesh). Consider ˆ i )}κ,L , (κ, L) ∈ (0, κ0 ] × N0 , of a two-parameter family T (κ, L) = {(Ki , Mi , K (CM , γM )-normalizable meshes (in the sense of Deﬁnition 3.3.3). Denote by hx,i , hy,i the anisotropic scaling parameters of the element map Mi given by Deﬁnition 3.3.3. This family T (κ, L) is said to be regular admissible if there are ci , i = 1, . . . , 4, σ ∈ (0, 1), and sets Ωj , j = 1, . . . , J, of the form given in Notation 2.3.3 such that each element Ki of the mesh T (κ, L) falls into exactly one of the following three categories: (C1) Ki is a boundary layer element, i.e., for some j ∈ {1, . . . , J} there holds Ki ⊂ Uκ (Γj ) ∩ Ωj \ (Bc1 κ (Aj−1 ) ∪ Bc1 κ (Aj )) and Mi (E) ⊂ Γj , E = {(x, 0) | 0 < x < 1}, hx,i = 1, hy,i = κ. (C2) Ki is a corner layer element, i.e., for some j ∈ {1, . . . , J}, the element Ki satisﬁes Ki ⊂ Bκ (Aj ) ∩ Ωj or Ki ⊂ Bκ (Aj ) ∩ Ωj+1 . Additionally, denoting hi = diam Ki , the factors hx,i , hy,i satisfy hx,i = hy,i = hi . Furthermore, for the element Ki exactly one of the following situations is satisﬁed: either Aj ∈ Ki together with hi ≤ c4 κσ L or Aj ∈ Ki together with c3 hi ≤ dist(Aj , Ki ) ≤ c4 hi . (C3) Ki is an interior element, i.e., Ki ⊂ Ω \ Uc2 κ (∂Ω), and the factors hx,i , hy,i satisfy hy,i = κ. hx,i = 1, Analogous to Proposition 3.3.5, we have that regular admissible boundary layer mesh are also admissible boundary layer meshes. Proposition 3.3.11. Let T (κ, L) be a regular admissible mesh family in the sense of Deﬁnition 3.3.10. Then it is an admissible boundary layer mesh family in the sense of Deﬁnition 2.4.4. Proof: We start with the most interesting case, that of boundary layer elements. Let Ki be a boundary layer element abutting on Γj . As the triangulation T is

116

3. hp Approximation

assumed to be a normalizable triangulation, the element map Mi can be factored as Mi = Gi ◦ Ai for some aﬃne map with hx,i 0 , Ai = 0 hy,i and the mapping Gi satisﬁes for some C, γ > 0 ∇p Gi L∞ (Ri ) ≤ Cγ p p!

∀p ∈ N0 ,

(3.3.6)

where the set Ri is Ri = (0, hx,i ) × (0, hy,i ) in the case of a quadrilateral and Ri = {(x, y) | 0 < x < hx,i , 0 < y < hy,i (1 − x/hx,i )} in the case of a triangle. From the analyticity of Gi and ψj , we readily see that for some C, γ > 0 independent of κ, L there holds ∇p ψj−1 ◦ Gi L∞ (Ri ) ≤ Cγ p p! ∀p ∈ N0 . Next, including the assumption that the edge E = {(x, 0) | 0 < x < 1} is mapped into Γj , i.e., Mi (E) ⊂ Γj , we have that ψj−1 ◦ Gi has the form ψj−1 ◦ Gi (x, y) = (yRi (x, y), Θi (x, y))

∀(x, y) ∈ Ri ,

where the functions Ri , Θi are analytic on Ri with ∇p Ri L∞ (Ri ) + ∇p Θi L∞ (Ri ) ≤ Cγ p p!

∀p ∈ N0 .

Using now the assumption hx,i = 1, hy,i = κ, we get the representation s−1 κ ◦ −1 ψj−1 ◦ Mi (x, y) = s−1 ◦ ψ ◦ G ◦ A (x, y) = (yR (x, h y), Θ (x, h y)) from i i i y,i i y,i κ j which the desired bounds follow. It is easy to see that the conditions on interior elements are satisﬁed as well. For corner layer elements, we exploit the fact that Ai is aﬃne to get ∇p (Gi ◦ Ai ) L∞ (S) ≤ C∇p Gi L∞ (Ri ) hpi . −1 Combining this with ˜ s−1 hi ≤ Chi , we readily get the desired bounds for p ≥ −1 1. It remains to show that ˜ sj,hi ◦ Mi ≤ C. This follows from the fact that diamRi = Ai (S) is bounded by Chi and that hence (3.3.6) implies diamGi (Ri ) ≤ Chi . 2

We now turn to the question of constructing admissible boundary layer meshes. A general framework in which this can be accomplished is provided by the notion of patchwise structured meshes, which we describe in the ensuing subsection. The meshes constructed in this manner are in fact regular admissible boundary layer meshes. 3.3.2 Patchwise structured meshes The general idea is to start from a quasi-uniform mesh (the “macro-triangulation” whose elements are called “patches”) that resolves the geometry. The actual features such as boundary layer elements and corner layer elements are then deﬁned

3.3 Admissible boundary layer meshes and ﬁnite element spaces

117

on the reference element for the appropriate elements and “transported” by the corresponding element map to the computational domain. Rather than starting with a formal deﬁnition of this idea, we begin with an example to illustrate $j , M "j , K ˆ j ) | j = 1, . . . , N } be a ﬁxed triangulathe basic idea. Let T$ = {(K tion of Ω with N elements. This triangulation is called the macro-triangulation. ˆ 1, . . . , K ˆN . Next, let T1 , . . . , TN be N triangulations of the reference elements K ˆ These triangulations are supposed to be of the form Tj = {(Kjk , Mjk , Kjk ) | k = "j (Kjk ) | k ∈ I(j), j = 1, . . . , N } 1, . . . , Nj }, j = 1, . . . , N . Then the collection {M forms a partition of Ω satisfying (M1). The corresponding element maps are ˆ jk → M "j (Kjk ). We are interested in the case when the col"j ◦Mjk : K given by M "j ◦ Mjk , K "j (Kjk ), M ˆ jk )} actually forms a triangulation of Ω in lection T := {(M the sense of Deﬁnition 2.4.1. In that case, we say that T$ is a macro-triangulation for the triangulation T , and we have the following result. $j , M "j , K ˆ j ), j ∈ I(T$ )} be a triangulation of Ω. Lemma 3.3.12. Let T$ = {(K $ ˆ jk ) | k ∈ I(Tj )} of the For each j ∈ I(T ), let triangulations Tj := {(Kjk , Mjk , K ˆ reference elements Kj be given with bilinear/linear element maps, i.e., Mjk is ˆ jk = T and Mjk is a bilinear map if K ˆ jk = S. Deﬁne the an aﬃne map if K " ˆ " collection of triples T := {(Mj (Kjk ), Mj ◦ Mjk , Kjk | k ∈ I(Tj ), j ∈ I(T$ )}. Then T satisﬁes (M1)–(M5) (i.e., it is a triangulation of Ω) if it satisﬁes (M2). Proof: It is clear that T already satisﬁes (M1) and (M5). It is therefore enough to see that (M4) follows from (M2). This follows readily from the fact that the restrictions of the element maps Mjk to the edges of the elements Kjk are linear "j satisfy (M4). 2 and that the macro elements maps M Lemma 3.3.12 allows us to construct in a very convenient way triangulations of Ω: Starting from a ﬁxed, coarse macro-triangulation T$ , actual triangulations T ˆ j . Given the can be generated from triangulations Tj of the reference elements K ˆ fact that the reference elements Kj are only triangles and squares, one would naturally use triangulations of the reference elements that consist of straight triangles and straight quadrilaterals only. Hence, by Lemma 3.3.12, one merely has to check that the resulting collection T does not have any “hanging nodes”. In what follows, this condition is easily checked as we will allow a very limited number of types of subtriangulations. This idea of prescribing a macro-triangulation and a list of possible reference patches is formalized in the following deﬁnition: Deﬁnition 3.3.13. Let Tˆ S , Tˆ T be two collections of triangulations of the reference square S and the reference triangle T respectively. Set Tˆ ref := Tˆ T ∪ Tˆ S . $j , M "j , K ˆ j ) | j = 1, . . . , N } and T be two triangulations of Ω. We Let T$ = {(K say that T is of type Tˆ ref with respect to the macro-triangulation T$ if there ˆ jk )} ∈ Tˆ ref , j = 1, . . . , N , such that exist triangulations Tj = {(Ωjk , Mjk , K "j ◦ Mjk , K ˆ jk ) | k ∈ I(Tj ), j = 1, . . . , N }. "j (Ωjk ), M T = {(M Remark 3.3.14 The concept of a limited number of types of reference patches can have many advantages, both from an implementational point of view and

118

3. hp Approximation

from an analysis point of view. We do not dwell on the implementational ramiﬁcations here. From an analysis point of view the major advantage is that only very few canonical situations can arise which are easy to handle individually. The simplest example of a collection of reference patches Tˆ ref is provided by the trivial reference patch, i.e., T S := {(S, Id, S)}, T T := {(T, Id, T )}, and therefore T ref = {T S , T T }. Obviously, in that case, only T = T$ is possible. Another simple reference patch is given by the uniform reference square T S (M ), i.e., for a given M ∈ N, the unit square is uniformly subdivided into M 2 squares (naturally, a similar construction can be done for the reference triangle). One could then choose T ref := T S (M ) or even T ref := ∪M ∈N T S (M ). More interesting examples are provided by mesh patches that are reﬁned geometrically towards one of the corners. In our applications below, we only consider (CM , γM )-regular macro triangulations T$ ; the elements of the reference patches, however, may be highly distorted in order to be able to resolve the boundary and corner layers. 3.3.3 The p-version boundary layer and corner layer patches We now deﬁne the references patches that are the basic building blocks for the patchwise structured boundary layer meshes. For simplicity of notation, we ˆj = assume that the macro-triangulation T$ consists of quadrilaterals only, i.e., K S for all j. This is a purely notational simpliﬁcation in order to reduce the number of reference patches. Given κ ∈ (0, 1/2], σ ∈ (0, 1) and L ∈ N0 a reference patch can be only one of the following four types: 1. The trivial patch: Tˇ = {(S, Id, S)}. 2. The hp boundary layer reference patch is of the form Tˇ = {(F1 (S), F1 , S), (F2 (S), F2 , S)} with maps F1 , F2 given by (cf. Fig. 3.3.2, left) F1 : S → S (ξ, η) → (ξ, κη),

F2 : S → S (ξ, η) → (ξ, κ + (1 − κ)η).

3. The hp tensor product corner layer reference patch with grading factor σ and L + 1 layers is given by the simplest triangulation of S consisting of rectangles and triangles that contains the points (cf. Fig. 3.3.2, right) (0, 0), (0, 1), (1, 0), (1, 1), (κ, 0), (0, κ), (1, κ), (κ, 1), √ √ (σ l κ, 0), (0, σ l κ), (σ l κ 2, σ l κ 2), l = 0, . . . , L. 4. The hp mixed corner layer reference patch with grading factor σ and L + 1 layers is given by the simplest triangulation of S consisting of rectangles and triangles that contains the points (cf. Fig. 3.3.3, left) (0, 0), (0, 1), (1, 0), (1, 1), (κ, 0), (0, κ), (1, κ), √ √ (σ l κ, 0), (0, σ l κ), (σ l κ 2, σ l κ 2), l = 0, . . . , L.

3.3 Admissible boundary layer meshes and ﬁnite element spaces

119

5. The hp mixed corner layer that is the mirror image with respect to the mid-line x = 1/2 of the preceding mixed corner layer patch. 6. The hp geometric corner layer reference patch with grading factor σ and L+1 layers is given by the simplest triangulation of S consisting of rectangles and triangles that contains the points (cf. Fig. 3.3.3, right) (0, 0), (0, 1), (1, 0), (1, 1), (κ, 0), (0, κ), (1, κ), √ √ (σ l κ, 0), (0, σ l κ), (σ l κ 2, σ l κ 2), l = 0, . . . , L.

η 1

η 1

κ κ 1ξ 1

L=2

ξ

Fig. 3.3.2. Ref. boundary layer patch (left); ref. tensor product patch (right).

η 1

η 1

κ

κ

1ξ

1ξ

L=2

L=2

Fig. 3.3.3. Reference mixed patch (left); reference geometric patch (right).

The six types of patches just introduced represent the minimal number of patches required for hp-approximation of our model problem. For practical applications of the notion of mesh patches, one would introduce further types. For problems with layers, patches that feature anisotropic geometric reﬁnement towards one (or two) edges of the reference square are useful. We therefore deﬁne them at this point: 1. The anisotropically geometrically reﬁned patch with L+1 layers and grading factor σ ∈ (0, 1) is given by the simplest triangulation of S consisting of rectangles that contains the points (cf. Fig. 3.3.4, left) (0, σ i ), (1, σ i ),

i = 0, 1, . . . , L.

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3. hp Approximation

2. The tensor product anisotropically geometrically reﬁned patch with L + 1 layers and grading factor σ ∈ (0, 1) is given by the simplest triangulation of S consisting of rectangles that contains the points (cf. Fig. 3.3.4, right) (0, σ i ), (σ i , 0), (σ i , σ j ), η

i, j = 0, 1, . . . , L.

η

ξ

ξ

Fig. 3.3.4. Anisotropic and geometric reﬁnement towards one edge (left) and two edges (right).

3.3.4 Boundary layer mesh generation via mesh patches We are now in position to deﬁne meshes via mesh patches. We start by deﬁning "j , K ˆ j )}. It should satisfy the following $j , M the macro-triangulation T$ = {(K conditions (cf. Fig. 3.3.5, left): ˆ j = S for all j. (MT1) All elements are quadrilaterals, i.e., K "j , S) ∈ T$ exactly one of the following situations can $j , M (MT2) For every (K occur: $ j ∩ ∂Ω = ∅; a) ∂ K "j (0, 0); $ b) ∂ Kj ∩ ∂Ω = M "j (E) where E = {(x, y) ∈ S | y = 0}; $ j ∩ ∂Ω = M c) ∂ K "j (E) where E = {(x, y) ∈ S | y = 0 or x = 0}. $ d) ∂ Kj ∩ ∂Ω = M (MT3) If Am is a vertex of Ω at a strictly convex corner then there exists a "j , S) ∈ T$ of type (MT2).(a) with M "j (0, 0) = Am ; this element $j , M unique (K is said to be a tensor-product patch. (MT4) If Am is a vertex of Ω at a concave corner then there are exactly three "j , S), (Ω $k , M "k , S), (Ω $l , M "l , S) ∈ T$ satisfying M "j (0, 0) = $j , M elements (K " $ "j , S) is " Mk (0, 0) = Ml (1, 0) = Am and the following extra conditions: (Kj , M of type (MT2).(b) and is said to be a geometric patch; the other two elements are of type (MT2).(c) and said to be mixed patches. (MT5) Any element of type (MT2).(c) that does not fall into the category $ j ∩ ∪m Am = ∅) is said (MT4) (i.e., elements of type (MT2).(c) with ∂ K to be a boundary layer patch.

3.3 Admissible boundary layer meshes and ﬁnite element spaces T

B

B

B

M

B

B

B T

121

T B

G B M T

B

B B

Fig. 3.3.5. Left: Example of a macro-triangulation. Right: B, T,M, G indicate boundary layer, tensor product, mixed, and geometric patches, respectively.

(MT6) Any element of type (MT2).(a) is called a trivial patch. We are now in position to deﬁne the meshes T (κ, L) that we wish to use. Let T$ be a ﬁxed macro-triangulation satisfying (MT1)–(MT6) above. For convenience’s sake, we also assume that the macro-triangulation T$ is chosen such that the macro-elements abutting on the boundary are suﬃciently small so that the decompositions based on asymptotic expansions of Theorem 2.3.4 are valid on whole patches near ∂Ω. Additionally, as the corner layers are only piecewise smooth, we assume that the diagonal of the geometric patch is mapped under "i to a curve Γ that can be taken as a curve Γ of the macro-element map M j Notation 2.3.3 This is formalized in the following two additional assumptions. $ j abutting on ∂Ω are (MT7) Let κ0 be as in (2.4.3). Then the macro-elements K $ i ⊂ Uκ (∂Ω). $ in the κ0 -neighborhood of ∂Ω: K i ∩ ∂Ω = ∅ implies K 0 "j , S) at a vertex Am , we assume that the $j , M (MT8) For geometric patches (K "j (D) of the diagonal D := {(x, y) ∈ S | x = y} angles between the image M and the two boundary components Γm−1 , Γm is strictly less than π, i.e., "j (T1 ), M "j (T2 ) are completely upon setting T1 = T , T2 = S \ T1 the set M contained in the subdomain Ωm or Ωm+1 . For (κ, L) ∈ (0, 1/2] × N0 and a ﬁxed macro-triangulation T$ satisfying (MT1)– (MT6) above, the triangulation T (κ, L) is a triangulation of Ω for which T$ is a macro-triangulation where the reference patches are chosen as follows: For elements of type (MT2).(a), the trivial reference patch is chosen, for elements of type (MT5), the boundary layer reference patch is chosen, for elements of type (MT4) the reference tensor product patch is chosen, and in the situation (MT3), the reference geometric patch is chosen for the geometric patch and reference mixed patches are chosen for the two mixed patches (in the notation of (MT3), $k , M "k , S) and the mirror the reference mixed patch is chosen for the element (Ω " $ image of the reference mixed patch for the element (Ωl , Ml , S)). One should note "j guarantee that our choices that the conditions (MT2) on the element maps M of the reference patches lead to a triangulation of Ω.

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3. hp Approximation

Proposition 3.3.15. Let T macro be a ﬁxed macro triangulation satisfying the assumptions (MT1)–(MT8) and let T (κ, L) be a two-parameter family of triangulations generated by mesh patches as described above. Then T (κ, L) is a regular admissible boundary layer mesh family in the sense of Deﬁnition 3.3.10. Proof: The result is obvious as one only has to check the properties for the four types of reference patches (boundary layer, tensor product, mixed, and geometric) on the reference square. 2

3.3.5 Properties of the pull-backs to the patches In the present section, we illustrate that analytic regularity results on the physical domain Ω imply corresponding analytic regularity results on the reference ˆ if the concept of mesh patches is employed. Hence, in this context, domain K one can perform approximation theory on simple triangulations of the reference ˆ and then transport those results back to the physical domain with domains K the patch maps. We will not pursue this approach to obtaining approximation theoretical results in the present work as we concentrate on the approximation properties on admissible meshes, which have notably less structure than regular admissible meshes or meshes generated with mesh patches. Nevertheless, we collect some results that show how regularity results on the physical domain can be transferred to the reference domains. Polynomial approximation results on mesh patches can be fairly easily obtained for the operators Πp∞ and Πp1,∞ (e.g., Proposition 3.2.25) once regularity assertions on the physical domain Ω have been transferred to the reference elements via the patch maps. $j , M "j , K ˆ j )} be a (CM , γM )-regular trianguProposition 3.3.16. Let T$ = {(K lation in the sense of Deﬁnition 3.3.1. Then there exists C > 0 depending only $ j there holds with hj = diamK $j on the constants CM , γM such that for all K " and uj := u ◦ Mj : uL∞ (K j ) −1 −1 C hj ∇uj L∞ (Kˆ j ) 1/2 C −1 hj uj L2 (Kˆ j ) C −1 ∇uj L2 (Kˆ j )

= uj L∞ (Kˆ j )

≤ ∇uL∞ (K j ) ≤ ≤ uL2 (K j ) ≤

$ j ), ∀u ∈ L∞ (K −1 Chj ∇uj L∞ (Kˆ j ) 1/2 Chj uj L2 (Kˆ j )

≤ ∇uL2 (K j ) ≤ C∇uj L2 (Kˆ j )

$ j ), ∀u ∈ W 1,∞ (K $ j ), ∀u ∈ L2 (K

$ j ). ∀u ∈ H 1 (K

"j . Proof: Follows from the properties of the mapping functions M

2

"j provide analytic diﬀeomorphisms The next lemma shows that, as the maps M ˆ $ j , regularity rebetween the reference elements Kj and the physical element K $ ˆ sults on Kj are transferred to the reference elements Kj . The reader will recognize that in Lemma 3.3.17, (ii), the maps s, sˆ could be taken as appropriate stretching maps as in the deﬁnition of admissible meshes in Deﬁnition 2.4.4.

3.4 hp Approximation on minimal meshes

123

$j , M "j , K ˆ j )} be a (CM , γM )-regular triangulation. Lemma 3.3.17. Let T$ = {(K "j . Then: $ j and set uj := u ◦ M Let u be a function deﬁned on K "j (ξ). If for some C(x), γ > 0 $ j and ξ ∈ K ˆ j ; set x = M (i) Let u be analytic on K |∇p u(x)| ≤ C(x)γ p p!

∀p ∈ N0 ,

then there are C , K > 0 depending only on γ and γM , CM such that |∇p uj (ξ)| ≤ C C(x)K p p!

∀p ∈ N0 .

ˆ → sˆ(G) ˆ ⊂K ˆ j , be $ j , sˆ : G ˆ ⊂ R2 be two domains, s : G → s(G) ⊂ K (ii) Let G, G two analytic maps with analytic inverses. Assume that there are C, γ > 0 such that ∀p ∈ N0 ,

"j ◦ sˆ) ∞ ˆ ≤ Cγ p! ◦M L (G)

∀p ∈ N0 .

−1

∇ (s p

∇p (u ◦ s)L∞ (G) ≤ Cγ p p! p

Then there are C , K > 0 depending only on C, γ > 0 such that p ∇p (uj ◦ sˆ)L∞ (G) ˆ ≤ C K p!

∀p ∈ N0 .

$ j and u ∈ H 2 (G) ∩ H 1 (G). Assume that Φ∇2 uL2 (G) < ∞ for (iii) Let G ⊂ K loc ˆ := M "−1 (G), Φ ˆ := Φ ◦ M "j . Then some weight function Φ ∈ L∞ (G). Set G j there is C > 0 depending only on CM , γM such that ˆ 2 uj 2 ˆ ≤ C hj Φ∇2 uL2 (G) + Φ∇uL2 (G) . Φ∇ L (G) Proof: (i) follows immediately from Lemma 4.3.4 and (ii) is proved similarly. (iii) is a direct consequence of the chain rule. 2 Finally, in this connection, it is useful to note Theorems 4.2.20, 6.2.6, which state that for analytic changes of variables (e.g., as provided by the patch maps), a 2 2 2 2 function u ∈ Bβ,ε (or Bβ,ε,α ) is mapped again to a function in Bβ,ε (or Bβ,ε,α for some appropriate α < α).

3.4 hp Approximation on minimal meshes 3.4.1 Regularity on the reference element In Section 3.2, we presented polynomial approximation results on the reference triangle and square. The assumptions on the function to be approximated were formulated as regularity requirements on the reference element. Therefore, in order to apply these approximation results, we need to know how the element maps transform regularity asserts on Ω to regularity assertions on the reference elements. Analyzing this is the purpose of the present subsection.

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3. hp Approximation

We employ notation concerning weighted spaces that will be introduced in more generality in Sections 4.2 and 6.2.1. For the reader’s convenience, we brieﬂy ˆp,β,ε are deﬁned (see (4.2.2)) compile the relevant notions. The weight functions Φ as follows: p+β |x| ˆ Φp,β,ε (x) = min 1, , p ∈ Z, β ∈ [0, 1]. min {1, ε(|p| + 1)} For a sector S with apex 0 (see Deﬁnition 4.2.1 for the precise deﬁnition of a curvilinar sector) the norm · H 2,2 (S) is then deﬁned by β,ε

ˆ0,β,ε ∇2 u2 2 + ε∇u2 2 + u2 2 . u2H 2,2 (S) = ε2 Φ L (S) L (S) L (S) β,ε

2 For constants Cu , γu > 0, the countably normed space Bβ,ε (S, Cu , γu ) is deﬁned by (4.2.15), viz.,

ˆp,β,ε ∇p+2 uL2 (S) ≤ Cu γ p max {p + 1, ε−1 }p+2 uH 2,2 (S) ≤ Cu and Φ u β,ε

∀p ∈ N.

Finally, exponentially weighted versions of these spaces exist (cf. (6.2.11), (6.2.12)): For α ≥ 0, we can introduce the weight functions ˆp,β,ε (x). Ψˆp,β,ε,α (x) := eα|x|/ε Φ The norm ·H 2,2

2 and the countably normed space Bβ,ε,α (S, Cu , γu ) are then ˆp,β,ε replaced with Ψˆp,β,ε,α . We note in deﬁned analogously with the weights Φ 2 2 particular that · H 2,2 (S) = · H 2,2 (S) and Bβ,ε,0 (S, Cu , γu ) = Bβ,ε (S, Cu , γu ). β,ε,α (S)

β,ε,0

β,ε

Lemma 3.4.1. Let S be a sector with apex 0. Let K ⊂ S with h := diam K. ˆ → K ⊂ S be an Let A be an aﬃne mapping with A = h. Let ﬁnally M : K analytic, invertible mapping satisfying for some CM , γM , cgeo > 0 p ∇p A−1 ◦ M L∞ (K) ∀p ∈ N0 , ˆ ≤ CM γM p! −1 −1 (A ◦ M ) L∞ (K) ˆ ≤ CM , c−1 geo h ≤ dist(K, 0) ≤ cgeo h. 2 Let u ∈ Bβ,ε,α (S , Cu , γu ) for some Cu , γu > 0, ε ∈ (0, 1], α ≥ 0. Then there are C, c, γ > 0 depending only on CM , γM , α, cgeo , β such that the mapped ˆ for all p ∈ N function u ◦ M satisﬁes on K −1 ∇p (u ◦ M )L∞ (K) h/ε + (h/ε)1−β ech/ε γ p p!. (3.4.1) ˆ ≤ CCu ε

Remark 3.4.2 The assumptions on the mapping M are those that are typically met for the element maps in geometric meshes. ˜ := A−1 (K). Then the assumption c−1 h ≤ dist(K, 0) ≤ cgeo h Proof: We set K geo implies that

3.4 hp Approximation on minimal meshes

h ≤ cgeo |x|

∀x ∈ K.

125

(3.4.2)

Combining Theorem 4.2.23 for the case α = 0 with Theorem 6.2.7 for α > 0, we get the existence of C, c, γ > 0 independent of ε such that Ψˆp−1,β,ε,cα ∇p uL∞ (K) ≤ Cγ p Cu max {p + 1, ε−1 }p+1 max {1, h/ε}

∀p ∈ N.

Rewriting this last statement using Lemma 4.2.2, we obtain (using the shorthand r = |x| and max {1, h/ε} ≤ 1 + h/ε) |∇p u(x)| ≤ CCu ε−2 (1+h/ε)

max {(p + 1)/r, ε−1 }p−1 −cαr/ε p γ , e ˆ0,β,ε (x) Φ

p ∈ N, x ∈ K.

˜ Hence, we get for the “stretched” function u ˜(ξ) := u ◦ A on the set K 1 + h/ε max {(p + 1)h/r, h/ε}p−1 γ p e−cαr/ε ˆ Φ0,β,ε (x) 1 + h/ε ≤ CCu ε−2 h ∀p ∈ N, max {p + 1, h/ε}p−1 γ p e−cαh/ε ˆ Φ0,β,ε (x)

˜(ξ)| ≤ CCu ε−2 h |∇p u

where we used (3.4.2) and appropriately adjusted the constants C, γ, c. Furthermore, since (h/ε) p! ≤ ep p! max {1, eh/ε } ≤ ep p!eh/ε , max {p, h/ε}p−1 ≤ ep max p!, p! the function u ˜ satisﬁes for appropriate c > 0 1 + h/ε −cαh/ε e ˆ Φ0,β,ε (x) ε β h ch/ε h 1+ e 1+ ≤ CCu ε−1 p!γ p ε h ε

˜(ξ)| ≤ CCu hε−2 eh/ε p!γ p |∇p u

∀p ∈ N,

˜ ξ ∈ K.

Exploiting now the fact that u ˆ := u ◦ M = u ˜ ◦ (A−1 ◦ M ) and the assumptions −1 ˆ on the map A ◦ M , we get from Corollary 4.3.5 on the reference element K 1−β h h ch/ε p h −1 p e 1+ ˆL∞ (K) p!γ ∀p ∈ N. ∇ u + ˆ ≤ CCu ε ε ε ε Absorbing now the factor (1 + h/ε) into the exponential factor ech/ε by appropriately adjusting c proves the lemma. 2

3. hp Approximation

126

For the elements abutting on the vertices, we need a diﬀerent result: Lemma 3.4.3. Let S ⊂ R2 be a sector with apex 0 and K ⊂ S with h := ˆ = T or K ˆ = S. Let M : K ˆ → K be a C 2 -diﬀeomorphism diam K < ∞. Let K satisfying for some aﬃne map A with A = h and some constant CM > 0 −1 −1 A−1 ◦ M C 2 (K) ◦ M ) L∞ (K) ˆ + (A ˆ ≤ CM .

(3.4.3)

Then there is C > 0 depending only on CM and β ∈ (0, 1) such that the following 2,2 is true for all u ∈ Hβ,ε (K): If d := dist(K, 0) > 0, then , ∇ (u ◦ M )L2 (K) ˆ ≤ Cε 2

−1

h + ε

β 1−β 1−β h h h+d U, + d ε ε

(3.4.4)

where ˆ0,β,ε ∇2 uL2 (K) + εΦ ˆ0,β−1,ε ∇uL2 (K) . U := ε2 Φ If d = 0 and M (0) = 0, then , |x| ∇ (u ◦ M )L2 (K) ˆ ≤ Cε β

2

−1

h + ε

1−β h U. ε

(3.4.5)

Proof: We start with (3.4.4). From the chain rule and the bounds in (3.4.3), we ˜ := (A−1 ◦ M )(K) ˆ have upon writing u ˆ = u ◦ M, u ˜ = u ◦ A, K 2 ∇2 u = C h∇2 uL2 (K) + ∇uL2 (K) . ˆL2 (K) ˜L2 (K) uL2 (K) ˜ + ∇˜ ˜ ˆ ≤ C ∇ u Inserting 1≤

ˆ0,β,ε (x) Φ ˆ0,β,ε (x) max {1, d/ε}−β ≤Φ ˆ0,β,ε (z) inf z∈K Φ

1≤

ˆ0,β−1,ε (x) Φ ˆ0,β−1,ε (x) min {1, (h + d)/ε}1−β ≤Φ ˆ0,β−1,ε (z) inf z∈K Φ

∀x ∈ K, ∀x ∈ K,

we arrive at −1 ˆL2 (K) ∇2 u ˆ ≤ Cε

h ε

ˆ0,β,ε ∇2 uL2 (K) max {1, d/ε}−β ε2 Φ

ˆ0,β−1,ε ∇uL2 (K) ; + min {1, (h + d)/ε}1−β εΦ elementary manipulations and the deﬁnition of U then imply the bound (3.4.4). We now turn to the proof of (3.4.5). We may assume A(0) = 0 and hence ˆ0,β,ε/h = Φ ˆ0,β,ε ◦ A. Φ Thus

(3.4.6)

3.4 hp Approximation on minimal meshes 2 ˆ0,β,ε/h ∇2 u ˆ Φ ˜L2 (K) ˜ = hΦ0,β,ε ∇ uL2 (K) .

127

(3.4.7)

Hence, the chain rule and the facts that u ˆ=u ˜ ◦(A−1 ◦M ) and A−1 ◦M C 2 (K) ˆ ≤ C imply 2 ˆ ˆ |x|β ∇2 u ˆL2 (K) ˜L2 (K) uL2 (K) ˜ + CΦ0,β,1 ∇˜ ˜ ˆ ≤ CΦ0,β,1 ∇ u ˆ0,β,ε/h ∇2 u ˆ ≤ C max {1, ε/h}β Φ ˜L2 (K) uL2 (K) ˜ + Φ0,β,ε/h ∇˜ ˜ ˆ0,β,ε ∇2 uL2 (K) + Φ ˆ0,β,ε ∇uL2 (K) ≤ C max {1, ε/h}β hΦ . / h 2 ˆ h ˆ 2 ε Φ0,β,ε ∇2 uL2 (K) + εΦ ≤ C max {1, ε/h}β ε−1 ∇u 0,β−1,ε L (K) , ε ε

where we used the assumption that M (0) = 0, appealed to Lemma 4.2.3 in the second step, and used (3.4.7) in the third one; for the last step, we wrote ˆ0,β,ε = Φ ˆ0,β−1,ε Φ ˆ0,1,ε (x) = min {1, |x|/ε} ≤ Ch/ε for all x ∈ K. ˆ0,1,ε and used Φ Φ From this last estimate, the claim of the lemma follows. 2 Lemma 3.4.4. Let u be an analytic function satisfying on a domain G ⊂ R2 ∇p uL∞ (G) ≤ Cu γup max {p, ε−1 }p

∀p ∈ N0

for some Cu , γu > 0, and ε ∈ (0, 1]. Let M : G → G be an analytic map satisfying for some aﬃne map A with A = h, h ∈ (0, 1], p ∇p (A−1 ◦ M )L∞ (G ) ≤ CM γM p!

∀p ∈ N0 .

Then there are constants C, γ > 0 independent of h, ε such that the function u ˆ := u ◦ M satisﬁes h ∇p u ˆL∞ (G ) ≤ C eh/ε γ p p! ε

∀p ∈ N.

Proof: We set u ˜ := u ◦ A and observe that u ˆ=u ˜ ◦ (A−1 ◦ M ). Next, we bound for p ≥ 1 and h ≤ 1 h ∇p u ˜L∞ ≤ Cu γup hp max {p, ε−1 }p ≤ Cu γup max {p, h/ε}p−1 ε h p p h ≤ Cu γu e max {p!, p! (h/ε)p−1 /(p − 1)!} ≤ Cu (eγu )p eh/ε p!. ε ε Appealing now to Corollary 4.3.5 concludes the argument.

2

3.4.2 Approximation on minimal meshes Our approximation results are based on the operator Πp∞ deﬁned on the reference ˆ Theorem 3.2.20 gives approximation results on the reference element element K. ˆ In order to “transport” results on the reference element K ˆ to the physical K. elements, we use the following lemma.

128

3. hp Approximation

ˆ → K be an invertible map satisfying, for some CM , Lemma 3.4.5. Let M : K κ > 0, CM . (M )−1 L∞ (K) M L∞ (K) ˆ ≤ CM , ˆ ≤ κ Then there holds for u := u ˆ ◦ M −1 ˆ ∀ˆ u ∈ L∞ (K),

uL∞ (K) = ˆ uL∞ (K) ˆ

CM ∇ˆ uL∞ (K) ˆ κ ≤ CM ∇uL∞ (K)

∇uL∞ (K) ≤

ˆ ∀ˆ u ∈ W 1,∞ (K),

∇ˆ uL∞ (K) ˆ

ˆ ∀ˆ u ∈ W 1,∞ (K).

If the map M satisﬁes for some aﬃne map A with A = h −1 −1 (A−1 ◦ M ) L∞ (K) ◦ M ) L∞ (K) ˆ + (A ˆ ≤ CM , then we have for some C depending only on CM uL∞ (K) = ˆ uL∞ (K) ˆ

ˆ ∀ˆ u ∈ L∞ (K),

−1 C −1 h−1 ∇ˆ uL∞ (K) ∇ˆ uL∞ (K) ˆ ≤ ∇uL∞ (K) ≤ Ch ˆ

C −1 hˆ uL2 (K) uL2 (K) ˆ ≤ uL2 (K) ≤ Chˆ ˆ C −1 ∇ˆ uL2 (K) uL2 (K) ˆ ≤ ∇uL2 (K) ≤ C∇ˆ ˆ

ˆ ∀ˆ u ∈ W 1,∞ (K),

ˆ ∀ˆ u ∈ L2 (K), ˆ ∀ˆ u ∈ H 1 (K).

uL2 (K) . Deﬁne the set Proof: We only show the bound involving uL2 (K) and ˆ −1 ˜ K := A (K) and deﬁne the function u ˜ := u◦A. We note that u ˆ=u ˜ ◦(A−1 ◦M ). −1 From our assumptions on A ◦ M , we have uL2 (K) uL2 (K) ∼ ˜ ˜ = huL2 (K) ˆ ; here, the constants hidden in ∼ depend only on CM . The result follows.

2

We also need the following result for meshes that are reﬁned geometrically toward a point. It expresses the fact that the element size decreases in geometric progression as the elements approach the point of reﬁnement: ˆ i ) be a (CM , γM )-regular triangulation of a Lemma 3.4.6. Let T = (Ki , Mi , K $ $ ∪ ∂Ω $ in domain Ω. Let the mesh be geometrically reﬁned toward a point A ∈ Ω the sense that for a constant cgeo > 0 there holds for the choice hi = diam Ki of the parameters appearing in Deﬁnition 3.3.1 c−1 geo hi ≤ dist(Ki , A) ≤ cgeo hi

∀Ki

with A ∈ K i .

Let δ > 0 be given. Then there exists C > 0 depending only on CM , γM , cgeo , and δ such that $ 2δ h2δ i ≤ C(diam Ω) . i:A∈K i

3.4 hp Approximation on minimal meshes

129

Proof: The (CM , γM )-regularity of the mesh implies the existence of C > 0 depending only on CM , γM such that for all elements Ki : C −1 h2i ≤ |Ki | ≤ Ch2i . Next, we observe that for all i with A ∈ K i , we have c−1 geo hi ≤ dist(Ki , A) = inf dist(x, A) x∈Ki

≤ sup dist(x, A) ≤ dist(Ki , A) + hi ≤ (1 + cgeo )hi . x∈Ki

Hence, we can estimate 2(δ−1) h2δ ≤ C hi i i

≤C

i

1 dx dy ≤ C

Ki

(dist(x, A))

2(δ−1)

(dist(x, A))

dx dy

Ki

i 2(δ−1)

Ω

dx dy ≤ C

diam Ω

$ 2δ , r2(δ−1) r dr ≤ C(diam Ω)

0

where the summation is over all i such that A ∈ K i .

2

ˆ i) Lemma 3.4.7. Let S be a curvilinear sector with apex 0 and let T = (Ki , Mi , K be a (CM , γM )-regular triangulation of Ω0 ⊂ S. Assume that the element maps Mi satisfy p ∇p s˜−1 ∀p ∈ N0 , (3.4.8a) ˆ i ) ≤ CM γM p! hi ◦ Mi L∞ (K −1 −1 L∞ (Kˆ i ) ≤ CM , (3.4.8b) (˜ shi ◦ Mi ) c−1 geo hi ≤ dist(Ki , 0) ≤ cgeo hi

if 0 ∈ K i ,

(3.4.8c)

where hi = diam Ki and s˜h is the stretching map x → hx. Set H := diam Ω0 ,

h0 := max hi i:0∈K i

2 and assume H > ε. For a function u ∈ Bβ,ε (S, Cu , γu ) and the interpolation ∞ of (3.3.3), we obtain for the interpolation error operator Πp,T ∞ ei = (u − Πp,T u)|Ki

the following bounds:  1/2 , 1−β   C h h0 u 0 , p4 ei 2L∞ (Ki ) + ∇ei 2L2 (Ki ) ≤C p3 (1 + ln p) +   ε ε ε i:0∈K i

  

i:0∈K i

1/2 , 1−β  Cu H H 4 2 2 ecH/ε e−bp p ei L∞ (Ki ) + ∇ei L2 (Ki ) ≤C +  ε ε ε

for some C, b, c > 0 that depend only on CM , γM , γu , β, and the sector S.

3. hp Approximation

130

Proof: We start by noting that |Ki | ∼ h2i ,

(3.4.9)

where the constants hidden in the ∼ depend only on CM . Next, by combining Proposition 3.2.21 with Lemmata 3.4.1, 3.4.3, 3.4.5, we obtain the following error bounds: ei L∞ (Ki ) + hi ∇ei L∞ (Ki ) ≤ CCu Fi e−bp echi /ε ei L∞ (Ki ) + p

−2

∇ei L2 (Ki ) ≤ Cp(1 + ln p)Fi Ui

where

, Fi = ε

−1

hi + ε

hi ε

if 0 ∈ K i , if 0 ∈ K i ,

1−β ,

ˆ0,β,ε ∇2 uL2 (K ) + εΦ ˆ0,β−1,ε ∇uL2 (K ) , Ui = ε2 Φ i i and the constants C, c, b > 0 are independent of ε and p. Lemma 3.4.6 implies (3.4.11) Fi2 ≤ Cε−2 (H/ε)2 + (H/ε)2(1−β) . i:0∈K i

From Corollary 4.2.11 we get ˆ0,β,ε ∇2 u2 2 + ε2 Φ ˆ0,β−1,ε ∇u2 2 Ui2 ≤ ε4 Φ L (S) L (S) i:0∈Ki

≤ Cu2H 2,2 (S) ≤ CCu2 .

(3.4.12)

β,ε

Bounding ∇ei L2 (Ki ) ≤ Chi ∇ei L∞ (Ki ) , we obtain using (3.4.11) and (3.4.12)

∇ei 2L2 (Ki ) ≤ CCu2 e2cH/ε e−2bp

i:0∈K i

Fi2

i:0∈K i

(H/ε)2 + (H/ε)2(1−β) , ≤ p6 (1 + ln p)2 max Fi2 Ui2 ≤ Cε

∇ei 2L2 (Ki )

−2

Cu2 e2cH/ε e−2bp

i:0∈K i

i:0∈K i

≤

CCu2 ε−2 p6 (1

i:0∈K i

+ ln p) (h0 /ε)2 + (h0 /ε)2(1−β) , 2

which can be brought to the desired form. The L∞ -bounds and i:0∈K i ei 2L∞ (Ki ) are obtained similarly.

i:0∈K i

ei 2L∞ (Ki ) 2

3.4 hp Approximation on minimal meshes

131

Theorem 3.4.8 (hp-interpolant on minimal meshes). Let Ω be a curvilinear polygon and T (κ, L) be a two-parameter family of admissible meshes in the sense of Deﬁnition 2.4.4. Let u be the solution of (1.2.11) with piecewise analytic Dirichlet data g satisfying (1.2.4) and analytic right-hand side f satisfying (1.2.3). Then there are λ0 , C, b > 0 independent of ε such that for each p ∈ N, λ ∈ (0, λ0 ) there holds on the mesh T (min {κ0 , λpε}, L): ∞ (3.4.13a) uL∞ (Ω) ≤ Cp(1 + ln p) e−bλp + p2 e−bL , u − Πp,T −bλp ∞ 3 −bL u − Πp,T uL2 (Ω) ≤ Cp(1 + ln p) e , (3.4.13b) + εp e ∞ λpε∇(u − Πp,T (3.4.13c) u)L2 (Ω) ≤ Cp3 (1 + ln p) e−bλp + εp3 e−bL . ∞ is deﬁned in (3.3.3). In particular, The interpolation operator Πp,T ∞ u ∈ S p (T (min {κ0 , λpε}, L)). Πp,T

Furthermore, on each edge of the mesh T (min{κ0 , λpε}, L), the approximant ∞ Πp,T u coincides with the Gauss-Lobatto interpolant of u, i.e., ∞ u)|Γ ip,Γ u = (Πp,T

∀ edges Γ of T (min{1, λpε}, L).

(3.4.14)

Proof: For convenience of notation, we introduce the abbreviation Cp := p(1 + ln p).

(3.4.15)

The approximation properties of the operator Πp∞ that we utilize are collected in Proposition 3.2.21. These approximation results are formulated on the reference ˆ i , and we therefore frequently appeal to Lemma 3.4.5 to obtain results element K on the physical elements Ki . In Section 2.3, we characterized the behavior of the solution u in diﬀerent ways: Theorem 2.3.1 employs countably normed spaces and Theorem 2.3.4 asymptotic expansions. We use the regularity results in countably normed spaces for the “asymptotic case” λpε ≥ κ0 and the characterization through asymptotic expansions for the “preasymptotic” case λpε < κ0 . The asymptotic case λpε ≥ κ0 . The key ingredient for obtaining ε-independent bounds is to note that the assumption λpε ≥ κ0 implies 1 1 λp, (3.4.16) ≤ ε κ0 thus allowing us to replace powers of ε−1 by powers of p. We will employ Theorem 2.3.1, which asserts the existence of C, γ > 0, and β ∈ [0, 1)J independent 2 (Ω, C, γ). of ε such that the solution u ∈ Bβ,ε By Remark 2.4.6 the mesh T (κ0 , L) is a mesh with large elements outside an O(κ0 ) neighborhood of the vertices and a geometric mesh with L + 1 layers near the vertices. Thus, in the neighborhoods of the vertices Aj , j = 1, . . . , J, the element maps satisfy the hypotheses (3.4.8) of Lemma 3.4.7. To ﬁx the notation, we concentrate on the approximation on a sector S with apex Aj .

3. hp Approximation

132

∞ The estimates of Lemma 3.4.7 give for the error ei = u − Πp,T u p4 ei 2L∞ (Ki ) + ∇ei 2L2 (Ki ) ≤ Cε−2 (H/ε)2 + (H/ε)2(1−βj ) ecH/ε e−bp i

+ Cε−2 p6 (1 + ln p)2 (σ L /ε)2 + (σ L /ε)2(1−βj ) ,

where we inserted the fact that the element Ki with Aj ∈ K i satisfy hi ≤ Cσ L and wrote H = diam S . In order to remove the factor ecH/ε , we employ (3.4.16) to bound e−bp ecH/ε ≤ e−bp e(c diam Ω)λp/κ0 ≤ e−(b/2)p if

bκ0 . 2c diam Ω Hence, for these choices of λ, we obtain p4 ei 2L∞ (Ki ) + ∇ei 2L2 (Ki ) ≤ Cε−4 e−(b/2)p + ε−4 p6 (1 + ln p)2 σ 2(1−βj )L λ ≤ λ0 ≤

i

≤ Cp4 (1 + ln p)2 e−(b/2)p + p6 σ 2(1−βj )L ,

where in the last step, we used (3.4.16) to exchange negative powers of ε for powers of p. The bounds (3.4.13a), (3.4.13c) then follow easily. Combining (3.4.13a) with (3.4.16) yields (3.4.13b). The preasymptotic case λpε < κ0 . In this regime (which is of course the regime of practical interest), we employ the decomposition of u based on asymptotic expansions. Speciﬁcally, we use the decomposition u = wε + χBL uBL + χCL uCL + rε (3.4.17) ε ε of (2.3.1) and the regularity assertions of Theorem 2.3.4. We approximate each of the four terms separately. Before proving approximation results for each of the four terms in (3.4.17), we recall from the deﬁnition admissible meshes in Deﬁnition 2.4.4 that there are three types of elements: interior elements (collected in the set T int ), boundary layer elements (gather in the set T BL ), and corner layer elements (combined into the set T CL ). The element maps Mi for the three types of elements have the following regularity properties: interior and boundary layer elements satisfy * ∇p Mi L∞ (Kˆ i ) ≤ Cγ p p! ∀p ∈ N0 ∀Ki ∈ T int ∪ T BL , (3.4.18) −1 C (Mi ) L∞ (Kˆ i ) ≤ λpε and for corner layer elements Ki in the vicinity of vertex Aj we have with hi = diam Ki  p  ∇p s˜−1 ◦ M ≤ C γ p! ∀p ∈ N , ∞ ˆ i M 0 M j,hi L ( Ki ) −1 ∀Ki ∈ T CL ; (3.4.19a) −1  (˜ sj,hi ◦ Mi ) L∞ (Kˆ i ) ≤ CM ,

3.4 hp Approximation on minimal meshes

133

additionallly, the elements Ki not abutting the vertex Aj satisfy c−1 geo hi ≤ dist(Ki , Aj ) ≤ cgeo hi .

(3.4.19b)

1. step: approximation of wε . We claim that ∞ ∞ wε − Πp,T wε L∞ (Ω) + λpε∇(wε − Πp,T wε )L∞ (Ω) ≤ Ce−bp

(3.4.20)

for some C, b > 0 independent of ε. We prove this by showing ∞ ∞ wε −Πp,T wε L∞ (Ki ) +λpε∇(wε −Πp,T wε )L∞ (Ki ) ≤ Ce−bp

∀Ki . (3.4.21)

To that end, we deﬁne for each element Ki the pull-back w ˆi := wε ◦ Mi and consider the cases of Ki ∈ T int ∩ T BL and Ki ∈ T CL separately. We recall from Theorem 2.3.4 that wε is analytic on Ω, i.e., ∇n wε L∞ (Ω) ≤ Cγ n n! ∀n ∈ N0 . For elements Ki ∈ T int ∪ T BL we obtain from (3.4.18) and Lemma 3.4.4 (choosing ε = 1 and A = Id in the statement of Lemma 3.4.4) ∇n w ˆi L∞ (Kˆ i ) ≤ Cγ n n!

∀n ∈ N0

for some C, γ > 0 independent of ε. Proposition 3.2.21 therefore implies w ˆi − Πp∞ w ˆi L∞ (Kˆ i ) + ∇(w ˆi − Πp∞ w ˆi )L∞ (Kˆ i ) ≤ Ce−bp for some C, b > 0 independent of ε. In view of Lemma 3.4.5 and (3.4.18), this bound implies (3.4.21) for all elements Ki ∈ T int ∪ T BL . It remains to see (3.4.21) for Ki ∈ T CL . Lemma 3.4.4 (choosing ε = 1 in the statement of Lemma 3.4.4) and the regularity property (3.4.19a) yield ∇n w ˆi L∞ (Kˆ i ) ≤ Chi γ n n!

∀n ∈ N.

Proposition 3.2.21 and Lemma 3.4.5 (using (3.4.19a)) then give ∞ ∞ −bp h−1 , i wε − Πp,T wε L∞ (Ki ) + ∇(wε − Πp,T wε )L∞ (Ki ) ≤ Ce

(3.4.22)

which shows (3.4.21), since λpε ≤ κ0 . 2. step: approximation of rε . We show ∞ rε − Πp,T rε L∞ (Ω) ≤ CCp e−bλp ,

λpε∇(rε −

∞ Πp,T rε )L2 (Ω)

2 −bλp

≤ CCp p e

(3.4.23a) .

(3.4.23b)

From Theorem 2.3.4, we have rε L∞ (Ω) + rε H 2,2 (Ω) ≤ Ce−α/ε β,ε

(3.4.24)

134

3. hp Approximation

for some suitable C, α > 0. The bounds (3.4.23) follow from (3.4.24), the assumption λpε ≤ κ0 , and the following three claims:

∞ rε L∞ (Ki ) ≤ Cp rε L∞ (Ki ) ∀Ki , (3.4.25) rε − Πp,T ∞ ∇(rε − Πp,T rε )L2 (Ki ) ≤ ∇rε L2 (Ki ) (3.4.26) # 2 |Ki |p Cp + rε L∞ (Ki ) ∀Ki ∈ T int ∪ T BL , λpε 2 ∞ ∇(rε − Πp,T rε )2L2 (Ki ) ≤ C Cp p2 ε−2 rε H 2,2 (Ω) . (3.4.27) β,ε

i:Ki ∈T CL

In order to see these claims, we note that taking v = 0 in the inﬁmum in Theorem 3.2.20 readily implies (3.4.25). For (3.4.26), we use the triangle inequality, the assumptions (3.4.18) on the map (Mi )−1 , and an inverse estimate for polynomials (Lemma 3.2.2) on the reference element to get # ∞ ∞ rε )L2 (Ki ) ≤ ∇rε L2 (Ki ) + |Ki | ∇Πp,T rε L∞ (Ki ) ∇(rε − Πp,T 2 # p ≤ ∇rε L2 (Ki ) + C |Ki | Π ∞ rε L∞ (Ki ) . λpε p,T ∞ rε L∞ (Ki ) gives (3.4.26). Combining this bound with that of (3.4.25) for Πp,T CL that are in the vicinity of the For (3.4.27), we consider the elements Ki ∈ T vertex Aj . We abbreviate

Ri := ε2 Φ0,β,ε ∇2 rε L2 (Ki ) + εΦ0,β−1,ε ∇rε L2 (Ki ) and use the fact that hi ∼ dist(Aj , Ki )

and hi ≤ cλpε hi ≤ cλpε σ L

if Aj ∈ K i , if Aj ∈ K i ,

to obtain with Lemma 3.4.3 that the pull-back rˆε := rε ◦ Mi satisﬁes ∇2 rˆε L2 (Kˆ i ) ≤ Cε−1 λp + (λp)1−βj Ri if Aj ∈ K i , βj 2 −1 L L 1−βj |x| ∇ rˆε L2 (Kˆ i ) ≤ Cε λp σ + (λp σ ) Ri if Aj ∈ K i . Upon simplifying σ L ≤ 1, σ L(1−βj ) ≤ 1 and estimating λp + (λp)1−βj ≤ ε−1 λpε + εβj (λpε)1−βj ≤ Cε−1 we get from Proposition 3.2.21 ∇(ˆ rε − Πp∞ rˆε )L2 (Kˆ i ) ≤ Cp2 Cp ε−2 Ri . This estimate in turn implies by mapping back to Ki (using Lemma 3.4.5) ∇(ˆ rε − Πp∞ rˆε )L2 (Ki ) ≤ Cp2 Cp ε−2 Ri

∀Ki ∈ T CL .

3.4 hp Approximation on minimal meshes

Since Corollary 4.2.11 implies

i

135

Ri2 ≤ Crε 2H 2,2 (Ω) , we get the desired bound β,ε

(3.4.27) by squaring and summing on i. 3. step: approximation of χBL uBL ε . We claim the following estimates: ∞ −bλp χBL uBL − Πp,T (χBL uBL ε ε )L∞ (Ki ) ≤ CCp e

λpε∇(χBL uBL ε

−

∞ Πp,T

(χBL uBL ε ))L∞ (Ki )

2 −bλp

≤ CCp p e

∀Ki ∈ T , (3.4.28a) ∀Ki ∈ T . (3.4.28b)

∞ BL − Πp,T uε on each of the element We consider the approximation error uBL ε ˆBL := uBL ◦ Mi . types, T BL , T CL , T int separately. We deﬁne u ε i We start with Ki ∈ T BL . In this case, we note that χBL ≡ 1 on Ki . Theorem 2.3.4 allows us to bound

∂ρr ∂θs (uBL ◦ ψj ◦ sκ )L∞ ≤ C(λp)r γ r+s s! ≤ Ceλp γ r+s r!s! ∀(r, s) ∈ N20 , ε r

λp where we used the bound (λp)r = (λp) r! r! ≤ e /r!. In view of the assumptions −1 −1 placed on the map sκ ◦ ψj ◦ Mi (see Deﬁnition 2.4.4) and the fact that u ˆBL = i −1 BL BL −1 uε ◦ Mi = (uε ◦ ψ ◦ sκ ) ◦ (sκ ◦ ψj ◦ Mi ), Lemma 3.4.4 implies λp n ˆBL ∇n u ˆ i ) ≤ Ce γ n! i L∞ (K

∀n ∈ N0 .

Thus, Proposition 3.2.21 implies the existence of C, b > 0 independent of ε, p such that λp −bp ˆ uBL − Πp∞ u ˆBL uBL − Πp∞ u ˆBL . ˆ i ) + ∇(ˆ ˆ i ) ≤ Ce e ε ε L∞ (K ε ε )L∞ (K

We note that the choice λ0 ≤ b/2 implies eλp e−bp ≤ e−(b/2)p . In view of the property (3.4.18) and Lemma 3.4.5, we arrive at ∞ BL ∞ BL uBL − Πp,T uε L∞ (Ki ) + λpε∇(uBL − Πp,T uε )L∞ (Ki ) ≤ e−(b/2)p . ε ε

Next, we consider corner layer elements Ki ∈ T CL . Again, we note that χCL ≡ 1 on Ki . To obtain bounds on the derivatives of the pull-back u ˆBL i , we appeal to Lemma 3.4.4 to get ∇n u ˆBL ˆ i) ≤ C i L∞ (K

hi hi /ε n hi γ n! ≤ C ecλp γ n n! ∀n ∈ N, e ε ε

where we employed the fact hi ≤ cλpε in the second estimate. Hence, using the approximation results of Proposition 3.2.21, we get the existence of C, b > 0 independent of ε, p such that ˆ uBL − Πp∞ u ˆBL uBL − Πp∞ u ˆBL ˆ i ) + ∇(ˆ ˆ i) ≤ C i i L∞ (K i i )L∞ (K

hi cλp −bp e e . ε

Again, the term ecλp e−bp can be replaced with e−(b/2)p for λ0 suﬃciently small. Combining the properties (3.4.18) with Lemma 3.4.5, we obtain (3.4.28).

136

3. hp Approximation

Finally, for interior elements Ki ∈ T int , we exploit the fact that χBL uBL is ε exponentially small due to dist(Ki , ∂Ω) ≥ cλpε, viz., BL BL χBL uBL uε )L∞ (Ki ) ≤ Ce−bλp . ε L∞ (Ki ) + ε∇(χ

Reasoning as in the proof of (3.4.25) then allows us to bound ∞ BL BL χBL uBL − Πp,T (χBL uBL uε L∞ (Ki ) ≤ CCp e−bλp , ε ε )L∞ (Ki ) ≤ CCp χ

and ∞ ∇(χBL uBL − Πp,T (χBL uBL ε ε ))L∞ (Ki ) ≤ ∞ BL BL ∇(χBL uBL uε ))L∞ (Ki ) ≤ C ε )L∞ (Ki ) + ∇(Πp,T (χ

p2 Cp −bλp . e λpε

4. step: approximation of χCL uCL ε . We claim the following bounds: ∞ −bλp χCL uCL − Πp,T (χCL uCL ε ε )L∞ (Ki ) ≤ CCp e ∞ χCL uCL − Πp,T (χCL uCL ε ε )L∞ (Ki ) CL CL ∞ CL CL λpε∇(χ uε − Πp,T (χ uε ))L2 (Ω)

if Aj ∈ K i (3.4.29a)

−bL

≤ CCp pe if Aj ∈ K i (3.4.29b) −bλp 2 ≤ CCp p e + p2 εe−bL .(3.4.29c)

We distinguish again the two cases Ki ∈ T int ∪ T BL and Ki ∈ T CL . For Ki ∈ T int ∪ T BL , we reason as in the proof of (3.4.28) to exploit that is small away from the vertices and get χCL uCL ε ∞ −bλp χCL uCL − Πp,T (χCL uCL , ε ε )L∞ (Ki ) ≤ CCp e

λpε∇(χCL uCL ε

−

∞ Πp,T

(χCL uCL ε ))L∞ (Ki )

2 −bλp

≤ CCp p e

(3.4.30) .

(3.4.31)

We now turn our attention to elements Ki ∈ T CL . For simplicity of notation, we restrict our attention to the neighborhood of a single vertex, Aj , say. The line Γj divides the set Ω ∩ Bcλpε (Aj ) into two sectors; our assumptions on admissible boundary layer meshes in Deﬁnition 2.4.4 are such that each element Ki ∈ T CL is completely contained in one of these two sectors. For simplicity of notation, we ﬁx one of these two sectors and denote it by S and consider only elements Ki ∈ T CL with Ki ⊂ S . 2 Theorem 2.3.4 implies that uCL ∈ Bβ,ε,α (S , Cε, γ); in particular, therefore, ε 2 uCL ∈ Bβ,ε (S , Cε, γ). ε

(3.4.32)

Since the element maps Mi of corner layer elements satisfy the hypotheses (3.4.8), Lemma 3.4.7 is applicable, and we obtain for the errors ∞ CL ei = (uCL − Πp,T uε )|Ki , ε

from Lemma 3.4.7 (with Cu ≤ Cε, h0 ≤ Cσ L λpε, H ≤ Cλpε)

3.4 hp Approximation on minimal meshes

p4 ei 2L∞ (Ki ) + ∇ei 2L2 (Ki )

i:Aj ∈K i

137

≤ Cp4 Cp2 (σ L λp)2 + (σ L λp)2(1−βj ) ,

p4 ei 2L∞ (Ki ) + ∇ei 2L2 (Ki ) ≤ C (λp)2 + (λp)2(1−βj ) ecλp e−bp

i:Aj ∈K i

Choosing λ0 suﬃciently small allows us to estimate with suitable C, b > 0 p4 ei 2L∞ (Ki ) + ∇ei 2L2 (Ki ) ≤ Cp6 Cp2 σ 2(1−βj )L , (3.4.33a) i:Aj ∈K i

p4 ei 2L∞ (Ki ) + ∇ei 2L2 (Ki ) ≤ Ce−bp ,

(3.4.33b)

i:Aj ∈K i

The bound (3.4.33a) implies (3.4.29b); the estimate (3.4.33b) together with (3.4.30) gives (3.4.29a); combining (3.4.33) with (3.4.31) gives (3.4.29c). 5. step: Conclusion of the proof in the preasymptotic case. In order to complete the proof of the theorem, we combine (3.4.20), (3.4.23a), (3.4.28a), and (3.4.29a), (3.4.29b), to get ∞ u − Πp,T uL∞ (Ω) ≤ CCp e−bλp + pe−bL , from which the pointwise estimate (3.4.13a) follows. For the L2 -bound (3.4.13b), we proceed similarly. The only diﬀerence is that we exploit the fact that for elements Ki abutting a vertex we have hi ≤ cλpε; thus, the bound (3.4.29b) yields ∞ 2 −bL χCL uCL − Πp,T (χCL uCL ε ε )L2 (Ki ) ≤ CCp p εe

if Aj ∈ K i ,

from which we infer (3.4.13b). For the H 1 -norm bound (3.4.13c), we combine (3.4.20), (3.4.23b), (3.4.28b), and (3.4.29c), to arrive at ∞ λpε∇(u − Πp,T u)L2 (Ω) ≤ CCp p2 e−bλp + pεe−bL . 2 Remark 3.4.9 The various powers of p in front of the terms p−bλp and e−bL ∞ are largely due to our choice of the convenient elementwise interpolant Πp,T . These factors are likely to be suboptimal. A diﬀerent construction would reduce the powers of p; cf. also Remark 3.3.9. Remark 3.4.10 For simplicity of exposition (and proof), we assumed that a uniform polynomial degree p is utilized. The arguments presented, however, could be modiﬁed to accommodate the use of a reduced polynomial degree in the L + 1 layers of geometric reﬁnement near the vertices.

138

3. hp Approximation

Corollary 3.4.11. Let T (κ, L) be a two-parameter family of regular admissible boundary layer meshes (see Deﬁnition 3.3.10) or a two-parameter family of meshes generated by mesh patches as in Section 3.3.4. Then the statement of Theorem 3.4.8 holds true. Proof: The corollary follows from Theorem 3.4.8 and the fact that by Proposition 3.3.11 regular admissible boundary layer meshes in the sense of Deﬁnition 3.3.10 are admissible boundary layer meshes. 2

l 4. The Countably Normed Spaces Bβ,ε

4.1 Motivation and outline 4.1.1 Motivation Chapter 4 and the following Chapter 5 are closely connected. These two chapters describe the regularity properties of the solutions of (1.2.1) in terms of weighted Sobolev spaces. Two kinds of regularity results are presented: Proposition 5.3.2 and Theorem 5.3.8 show that the solution uε of (1.2.1) lies in the 2,2 weighted Sobolev Hβ,ε , the space of H 1 functions whose second derivatives are in 2,2 a weighted L2 -space. This Hβ,ε regularity result is then used for a boot-strapping argument to control all derivatives of the solution uε under the assumption of (piecewise) analyticity of the input data. This control of the growth of the derival tives is cast in the framework of countably normed space Bβ,ε and can be found in Theorem 5.3.8 and Theorem 5.3.10. The present chapter is preparatory in nature in that the weighted Sobolev space l,l l Hβ,ε (Ω) and the countably normed spaces Bβ,ε are deﬁned and their essential l,l properties are proved. The weighted Sobolev spaces Hβ,ε and the countably l normed spaces Bβ,ε are introduced in such a way that for the case ε = 1 the l,l l and Bβ,1 coincide with classical weighted Sobolev spaces employed spaces Hβ,1 to describe corner singularities for elliptic problems on polygonal domains (see, e.g., [79]) and the countably normed spaces Bβl introduced by Babuˇska & Guo in [15, 16]. Our spaces are thus an extension of existing spaces that allow for precise control in terms of the singular perturbation parameter ε. l,l In order to motivate our weighted Sobolev spaces Hβ,ε , we consider the following model equation:

−ε2 ∆uε + uε = f

on Ω,

uε |∂Ω = 0.

(4.1.1)

It is more convenient for our purposes to write it as −∆uε = ε−2 [f − uε ] ,

uε |∂Ω = 0.

If Ω has a smooth boundary, then bythe classical shift theorem the solution uε is in H 2 (Ω) and ∇2 uε L2 (Ω) ≤ Cε−2 f L2 (Ω) + uε L2 (Ω) . From this classical shift theorem, we therefore expect ∇2 u to be of size O(ε−2 ). Let us now consider

J.M. Melenk: LNM 1796, pp. 141–168, 2002. c Springer-Verlag Berlin Heidelberg 2002

142

l 4. The Countably Normed Spaces Bβ,ε

polygonal domains Ω. We will restrict our attention ﬁrst to a single vertex, i.e., we consider a sector Ω = SR (ω) := {(r cos ϕ, r sin ϕ) | 0 < r < R,

0 < ϕ < ω}

(4.1.2)

2 and ﬁx R < R. Then classical local regularity results give that u ∈ Hloc (SR (ω)). In fact these classical local regularity results (cf. Lemma 5.5.11) allow for the following sharper result: For δ, r > 0 we set Ωδ,r := Sr (ω) \ Bδ (0) and have the existence of C > 0 depending only on ω and R such that

∇2 uε L2 (Ω2δ,R )

≤ Cε−2 f − uε L2 (Ωδ,R ) + C δ −1 ∇uε L2 (Ωδ,R ) + δ −2 uε L2 (Ωδ,R ) .

Thus, upon choosing δ = ε, we obtain ∇2 uε L2 (Ω2ε,R ) ≤ Cε−2 f L2 (Ωε,R ) + ε∇uε L2 (Ωε,R ) + uε L2 (Ωε,R ) . (4.1.3) We remark the similarity of (4.1.3) with the corresponding bound for smooth domains Ω and conclude that the solution uε has a weak singularity at the origin in the sense that outside the ball Bε (0), the H 2 -norm of uε can be controlled in the same way as for smooth domains. For the behavior of uε in the vicinity of the origin, Proposition 5.3.2 below states the following: For β ∈ {β ∈ [0, 1) | β > 1 − π/ω}, there is C > 0 such that ) ) ) ) β )(|x|/ε) ∇2 uε ) 2 L (Ω∩B2ε (0)) .) / ) ) β ) ≤ Cε−2 )(|x|/ε) f ) + ε∇uε L2 (Ω) + uε L2 (Ω) . (4.1.4) L2 (Ω∩B4ε (0))

Combining (4.1.3), (4.1.4) we see that ∇2 uε is in the following weighted L2 -space ˆ0,β,ε ∇2 uε L2 (Ω∩B (0)) ≤ Cε−2 Φ ˆ0,β,ε f L2 (Ω) + ε∇uε L2 (Ω) + uε L2 (Ω) , Φ R ˆ0,β,ε behaves like (|x|/ε)β ˆ0,β,ε is given by (4.2.2). Essentially, Φ where the weight Φ in an O(ε) neighborhood of the origin and reduces to 1 outside an O(ε) neighborhood of the origin. Multiplying the last estimate by ε2 , we have thus obtained the following estimate: uε H 2,2 (S β,ε

R (ω))

ˆ0,β,ε ∇2 uε L2 (S (ω)) + ε∇uε L2 (S (ω)) + uε L2 (S (ω)) := ε2 Φ R R R ˆ0,β,ε f L2 (Ω) + ε∇uε L2 (Ω) + uε L2 (Ω) , ≤ C Φ

where C > 0 is independent of ε and f . The term ε∇uε L2 (Ω) + uε L2 (Ω) can also be bounded in terms of f with the aid of a Hardy inequality as shown in Theorem 5.3.8 (cf. also the energy estimate (4.1.11) in the proof of Lemma 4.1.1 of the present introduction): ˆ0,β,ε f L2 (Ω) , ε∇uε L2 (Ω) + uε L2 (Ω) ≤ CΦ

4.1 Motivation and outline

143

where again C > 0 is independent of ε and f . Thus, we arrive at the desired shift theorem in weighted spaces for sectors: uε H 2,2 (S β,ε

R (ω))

ˆ0,β,ε f L2 (Ω) . ≤ CΦ

(4.1.5)

Here, the constant C > 0 is independent of ε. This shift theorem is the motivation l,l l,m (and, more generally, Hβ,ε ) in this chapter. for introducing the spaces Hβ,ε These weighted shift theorems for sectors have natural extensions to polygonal domains. There, one chooses βj ∈ [0, 1) for each vertex Aj , j = 1 . . . , J, of the polygon Ω, sets β = (β1 , . . . , βJ ), and deﬁnes the weight function Φ0,β,ε by Φ0,β,ε (x) :=

J

ˆ0,β ,ε (x − Aj ). Φ j

(4.1.6)

j=1

With this deﬁnition of a weight function Φ0,β,ε , the shift theorem for a single sector of (4.1.5) extends to the case of polygons Ω: uε H 2,2 (Ω) := ε2 Φ0,β,ε ∇2 uε L2 (Ω) + ε∇uε L2 (Ω) + uε L2 (Ω) β,ε

≤ CΦ0,β,ε f L2 (Ω) ,

(4.1.7)

provided the components of the vector β ∈ [0, 1)J satisfy βj > 1 − π/ωj , j = 1, . . . , J. If the data of an elliptic equation are suﬃciently smooth, then higher order derivatives can be bounded as well. This is done essentially by differentiating the equation and using an elliptic shift theorem. In particular, if the data are analytic, then the solution is analytic as well, [98]. Using the techniques of [98], it was shown in [95] that, if ∂Ω is a closed analytic curve and f is analytic on Ω, then the solution uε of (4.1.1) satisﬁes ∇p+2 uε L2 (Ω) ≤ CK p+2 max {p + 2, ε−2 }p+2

∀p ∈ N0 ,

where the constants C, K are independent of ε but depend on the analytic right-hand side f . In a sector (or, more generally, in a polygon) higher order derivatives of uε will of course be in weighted L2 -spaces as we just ascertained for ∇2 uε , i.e., we expect a result of the following form Φp,β,ε ∇p+2 uε L2 (Ω) ≤ CK p+2 max {p + 2, ε−1 }p+2

∀p ∈ N0 ,

where the weight function Φp,β,ε depends on the order of the derivative and ε. It is reasonable to expect Φp,β,ε to be structurally similar to our deﬁnition of Φ0,β,ε in (4.1.6), i.e., to expect that it can be written as product weight functions ˆp,β ,ε associated with the vertices Aj of the domain Ω: Φ j Φp,β,ε (x) =

J

ˆp,β ,ε (x − Aj ). Φ j

j=1

ˆp,β ,ε for each vertex has the This is indeed the case. The weight function Φ j ˆp,β ,ε in (4.2.2) on the form (4.2.2). In order to see that the dependence of Φ j

144

l 4. The Countably Normed Spaces Bβ,ε

parameters p, ε, and the location x is a good choice, it is instructive to consider a one-dimensional example with singular right-hand side. While this example exhibit the essential features of two-dimensional problems in polygonal domains, it avoids many of the technical diﬃculties. It is therefore mostly the proof of the following result that is of interest: Lemma 4.1.1. Let Ω = (0, 1) and deﬁne for p ∈ N0 , β ∈ [0, 1), ε ∈ (0, 1] the ˆp,β,ε by weight functions Φ ˆp,β,ε (x) := Φ

min

1,

x min {1, ε(p + 1)}

p+β .

Let f be analytic on Ω and satisfy for some Cf , γf > 0, β ∈ (0, 1] ˆp,β,ε f (p) L2 (Ω) ≤ Cf γ p max {p + 1, ε−1 }p Φ f

∀p ∈ N0 .

Let uε be the solution to −ε2 uε + uε = f

on Ω,

uε (0) = uε (1) = 0.

(4.1.8)

Then there exist constants Cu , γu > 0 depending only on Cf , γf such that ˆp,β,ε u(p+2) L2 (Ω) ≤ Cγ p max {p + 2, ε−1 }p+2 Φ ε

∀p ∈ N0 .

(4.1.9)

Proof: We start with an energy estimate. Deﬁning the energy norm · ε as in (1.2.7) by u2ε = ε2 u 2L2 (Ω) + u2L2 (Ω) we get from the weak formulation of (4.1.8) 2 ˆ0,β,ε f L2 (Ω) Φ ˆ−1 uε L2 (Ω) . uε ε = f uε dx ≤ Φ (4.1.10) 0,β,ε Ω

Next using β ∈ [0, 1) and uε (0) = 0 we have from + ε [68, Thm. 327] for a constant CH > 0 independent of ε ∈ (0, 1] the bound 0 |x−1 uε |2 dx ≤ CH uε 2H 1 (Ω) . This allows us to estimate ε 1 −1 2 2 2 ˆ−1 uε 2 2 ˆ−1 uε 2 2 x uε dx + Φ ≤ Φ = ε |uε | dx 0,1,ε L (Ω) L (Ω) 0,β,ε 0

≤ CH ε

2

uε 2H 1 (Ω)

+

CH uε 2L2 (Ω)

≤

ε 2 CH uε 2ε ,

Inserting this bound into (4.1.10), we obtain the following energy estimate: ˆ0,β,ε f L2 (Ω) ≤ CH Cf . uε ≤ CH Φ The diﬀerential equation now gives ˆ0,β,ε (f − uε )L2 (Ω) ˆ0,β,ε uε L2 (Ω) ≤ ε−2 Φ Φ ˆ0,β,ε f L2 (Ω) + ε−2 Φ ˆ0,β,ε uε L2 (Ω) ≤ ε−2 Φ ≤ Cf (1 + CH ) max {1, ε−1 }2

(4.1.11)

4.1 Motivation and outline

145

from the assumptions on f and the energy estimate (4.1.11). Similarly, we get by diﬀerentiating (4.1.8) once ˆ1,β,ε (f − u ) L2 (Ω) ≤ Cf (γf + CH ) max {3, ε−1 }3 . ˆ1,β,ε u(3) L2 (Ω) ≤ ε−2 Φ Φ ε ε We show (4.1.9) by induction on p. Choosing Cu := Cf (1 + CH ) and γu := 2 + γf + CH , we have just proved (4.1.9) for p = 0 and p = 1. Let us assume that (4.1.9) holds for all 0 ≤ p < p. Diﬀerentiating (4.1.8) p ≥ 2 times, multiplying ˆp,β,ε and integrating over Ω, we arrive at by Φ ˆp,β,ε u(p+2) L2 (Ω) = ε−2 Φ ˆp,β,ε f (p) − u(p) L2 (Ω) Φ ε ε ˆp,β,ε f (p) L2 (Ω) + ε−2 Φ ˆp,β,ε u(p) L2 (Ω) . ≤ ε−2 Φ ε

(4.1.12)

ˆp,β,ε (x) is We now claim that for ﬁxed x ∈ Ω, ε ∈ (0, 1] the function p → Φ decreasing. First, the function p → min {1, ε(p + 1)} is monotonically increasing and thus x φ : p → min 1, min {1, (p + 1)ε} is monotonically decreasing. As φ ≤ 1 for all p, we conclude that p+β

p → [φ(p)]

ˆp,β,ε we get that the p → is also decreasing. Recognizing this last function as Φ ˆp,β,ε (x) is indeed decreasing. Thus, we may bound (4.1.12) further by using the Φ induction hypothesis (4.1.9) ˆp,β,ε f (p) L2 (Ω) + ε−2 Φ ˆp−2,β,ε u(p) L2 (Ω) ˆp,β,ε u(p+2) L2 (Ω) ≤ ε−2 Φ Φ ε ε ≤ ε−2 Cf γfp max {p + 1, ε−1 }p + ε−2 Cu γup max {p, ε−1 }p . p / γf p+2 −1 p+2 −2 Cf ≤ Cu γu max {p + 2, ε } γu +1 . Cu γu Our choice γu = 2 + γf + CH > 2 and Cu = 1 + Cf + CH implies . p / γf −2 Cf +1 ≤1 γu Cu γu so that we indeed obtain the desired bound ˆp,β,ε u(p+2) L2 (Ω) ≤ Cu γ p+2 max {p + 2, ε−1 }p+2 . Φ ε u 2 In the notation of this work, Lemma 4.1.1 represents a shift theorem in countably 0 normed spaces: For right-hand sides f ∈ Bβ,ε , the solution uε of (4.1.8) is in the 2 countably normed space Bβ,ε . We will see in Chapter 5 that the analogous result holds in curvilinear polygons.

146

l 4. The Countably Normed Spaces Bβ,ε

4.1.2 Outline of Chapter 4 m,l The outline of Chapter 4 is as follows. In Chapter 4 we ﬁrst deﬁne the spaces Hβ,ε l and the countably normed spaces Bβ,ε . For simplicity of exposition, these spaces are deﬁned on sectors in the present chapter. The subscript β is therefore a scalar rather than a vector as suggested above for polygonal domains. In Lemma 4.2.2, ˆp,β,ε . Of particular we collect some basic properties of the weight functions Φ relevance for the understanding of the methods of proof employed in this work is the last result of Lemma 4.2.2, (4.2.8). It states that for balls BcR (x) with ˆp,β,ε can be bounded above and below |x| = R, c ∈ (0, 1), the weight function Φ p ˆp,β,ε (x), i.e., by the value (up to a factor K with K independent of ε and x) by Φ ˆp,β,ε at the center of the ball. This feature will frequently of the weight function Φ allow us in local analyses to replace weighted norms by standard Sobolev norms. m,l Section 4.2.1 collects some properties of the spaces Hβ,ε , notably two embedding theorems. The ﬁrst embedding result, Lemma 4.2.9, shows that the functions 2,2 from the space Hβ,ε are continuous up to the boundary. The second embedding result, Lemma 4.2.10, is the key result of Section 4.2.1 where Hardy-type estimates in weighted Sobolev spaces are proved. Lemma 4.2.10 is an important technical tool for the proof of the main result of the ensuing Chapter 5, Theorem 5.3.10 and its variants Propositions 5.4.5, 5.4.8, 5.4.7. l Section 4.2.2 collects properties of the countably normed spaces Bβ,ε . The major l result of Section 4.2.2 is Theorem 4.2.20. This result shows that the spaces Bβ,ε are invariant under analytic changes of variables. This result will prove useful in Chapter 5 in our treatment of curved boundaries: Theorem 4.2.20 allows us to infer regularity results for domains with curved boundaries from those with straight boundaries by mapping arguments. The main idea of the proof of Theorem 4.2.20 is to consider in a ﬁrst step the change of variables locally and then combine in a second step these local results to a global estimate with the aid of a covering argument. The technical tool for inferring membership in a countl ably normed space Bβ,ε from local estimates is provided in Lemma 4.2.17. Local results for changes of variables need to track two parameters: the perturbation parameter ε and the distance to the apex of the sector. Such results are again technically involved and therefore provided in the separate Lemma 4.3.1. l The spaces Bβ,ε are L2 -based function spaces. It is also of interest to characterize l the pointwise behavior of Bβ,ε functions. This is achieved in Theorem 4.2.23.

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

Deﬁnition 4.2.1 (sector). A bounded Lipschitz domain S ⊂ R2 is said to be a sector with apex 0 (or simply: a sector) if 0 ∈ ∂S. A sector S is called a C 2 curvilinear sector if there are three, mutually disjoint C 2 -arcs Γi (i ∈ {1, 2, 3}) such that ∂S = ∪3i=1 Γi and 0 = Γ1 ∩ Γ2 . A C 2 -curvilinear sector is called an analytic curvilinear sector (or simply curvilinear sector) if the three arcs Γi are analytic arcs.

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

147

For R > 0 and ω ∈ (0, 2π) we deﬁne straight sectors SR (ω) as the sets SR (ω) = {(r cos ϕ, r sin ϕ) | 0 < r < R,

0 < ϕ < ω}.

(4.2.1)

ˆp,β,ε by For p ∈ Z and β ∈ R we deﬁne the weight function Φ ˆp,β,ε (x) = Φ

min

|x| 1, min {1, ε(|p| + 1)}

p+β .

(4.2.2)

ˆp,β,ε has the following properties: One can check that the weight functions Φ ˆp,β,ε ). Let S ∈ R2 be a sector. There holds for Lemma 4.2.2 (Properties of Φ all p ∈ N0 , ε ∈ (0, 1], β ∈ [0, 1], l ∈ N, and x ∈ S: ˆp,β,ε (x) ∼ Φ ˆp,0,ε (x)Φ ˆ0,β,ε (x), Φ ˆp,0,ε (x)Φ ˆ−l,β,ε (x), ˆp−l,β,ε (x) ∼ Φ Φ

(4.2.3)

ˆp,β−l,ε (x), ˆp−l,β,ε (x) ∼ Φ Φ p+β 1 min {1, ε(p + 1)} , ∼ 1+ ˆp,β,ε (x) |x| Φ

(4.2.5)

1 max {p + 1, ε−1 }p ∼ max {(p + 1)/|x|, ε−1 }p . ˆ Φp,0,ε (x)

(4.2.4)

(4.2.6) (4.2.7)

Here, the relationship a ∼ b means that there exist C, K > 0 independent of p ∈ N0 , ε ∈ (0, 1], and x ∈ S such that C −1 K −p a ≤ b ≤ CK p a. Furthermore, let c ∈ (0, 1) be given. Then for all balls Bc|x| (x) with x ∈ S min z∈Bc|x| (x)

ˆp,β,ε (x) ∼ ˆp,β,ε (z) ∼ Φ Φ

max z∈Bc|x| (x)

ˆp,β,ε (z). Φ

(4.2.8)

Here, the constants C, K in the deﬁnition of ∼ depend additionally on the constant c ∈ (0, 1) but are independent of p, ε, x ∈ S. Proof: We will only show (4.2.7) to demonstrate the general procedure. If ˆp,β,ε (x) = 1 diam(S) ≥ |x| ≥ min {1, ε(p + 1)}, then (4.2.7) follows easily as Φ and p+1 p+1 1 ≤ ≤ max {p + 1, ε−1 }. ≤ (p + 1) max 1, diam(S) |x| ε(p + 1) We therefore consider the case |x| ≤ min {1, ε(p + 1)}. We note that this implies ε−1 ≤

p+1 . |x|

(4.2.9)

From the deﬁnition of the symbol ∼, the bound (4.2.7) is proved if we can show the existence of C > 0 independent of ε such that for all p ∈ N0

l 4. The Countably Normed Spaces Bβ,ε

148

C −1

min {1, ε(p + 1)} max {p + 1, ε−1 } ≤ max {(p + 1)/|x|, ε−1 } |x| min {1, ε(p + 1)} max {p + 1, ε−1 }. ≤C |x|

First, we note that ∀z > 0

min {1, z} max {1, z −1 } = 1.

(4.2.10)

This implies readily that min {1, (p + 1)ε} 1 p+1 −1 max {p + 1, ε } = min {1, (p + 1)ε} max 1, |x| |x| (p + 1)ε p+1 p + 1 −1 = . = max ,ε |x| |x| 2

where the last equality follows from (4.2.9).

Of interest in the following will be monotonicity properties of the weight funcˆp,β,ε in the arguments β, p, and ε. We have tions Φ ˆp,β,ε ). Let S be a sector. Lemma 4.2.3 (Monotonicity properties of Φ ˆp,β,ε (x) is monotonically 1. For all ﬁxed p ∈ N, ε > 0, x ∈ S the function β → Φ decreasing on R+ . 0 ˆp,β,ε (x) is monoton2. For all ﬁxed p ∈ N, β ∈ [0, 1), x ∈ S the function ε → Φ + ically decreasing on R0 . ˆp,β,ε (x) is monotonically 3. For all ﬁxed ε > 0, β > 0, x ∈ S the function p → Φ decreasing on N0 . 4. For all β ∈ [0, 1] there are C, γ > 0 independent of ε, ε ∈ (0, 1], and p ∈ N0 such that ˆp,β,ε (x) ˆp,β,ε (x) Φ Φ p β p ≤ Cγ max {1, ε /ε} max {1, ε/ε } . max {p + 1, ε−1 }p max {p + 1, (ε )−1 }p Proof: The ﬁrst and second assertions of the lemma follow immediately from (4.2.2). The third assertion was already proved in the course of the proof of Lemma 4.1.1. For the fourth assertion, we start by ﬁrst considering the case p = 0. We have for β ≥ 0 (writing r = |x|) ˆ0,β,ε (x) = min {1, r/ε}β ≤ min {1, r/ε }β max {1, ε /ε}β Φ ˆ0,β,ε (x) max {1, ε /ε}β . ≤Φ

(4.2.11)

ˆp,0,ε we have Next, let us consider the case β = 0. By the deﬁnition of Φ p ˆp,0,ε (x) min {1, (r/m)} Φ = , max {p + 1, ε−1 } max {p + 1, ε−1 }p

m = min {1, (p + 1)ε}. (4.2.12)

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

149

Employing the formula ab ab ≤ min {a, b} ≤ 2 a+b a+b

∀a, b > 0,

(4.2.13)

we can bound the expression in parentheses in (4.2.12) as follows: min {1, (r/m)} 2r r ≤ ≤ . (p + 1) + r max {p + 1, ε−1 } max {p + 1, ε−1 } (p + 1) + r max {p + 1, ε−1 } Next, as max {p + 1, ε−1 } = max {p + 1, (ε )−1 (ε/ε )−1 } ≥ max {p + 1, (ε )−1 } min {1, (ε/ε )−1 } we arrive at 2r min {1, (r/m)} ≤ max {p + 1, ε−1 } (p + 1) + r max {p + 1, (ε )−1 } min {1, (ε/ε )−1 } r 2 ≤ · . −1 (p + 1) + r max {p + 1, (ε ) } min {1, (ε/ε )−1 } Raising both sides to the power p we get ˆp,0,ε (x) ˆp,0,ε (x) Φ Φ ≤ max {1, ε/ε }p 2p , max {p + 1, ε−1 }p max {p + 1, (ε )−1 }p which is the desired bound for the case β = 0. Appealing now to (4.2.3) of Lemma 4.2.2 allows us to conclude the claim in the desired generality. 2 m,l l We are now in position to introduce the spaces Hβ,ε (S) and Bβ,ε (S). On a sector m,l S, we deﬁne for m, l ∈ N0 , m ≥ l, and β ∈ [0, 1], ε ∈ (0, 1], the spaces Hβ,ε (S) ∞ as the completion of the space C (S) under the norms

u2H m,l (Ω) := β,ε

u2H m,0 (Ω) := β,ε

l−1 k=0 m

ε2k ∇k u2L2 (Ω) + ε2l

m

ˆk−l,β,ε ∇k u2 2 , Φ L (Ω)

l > 0, (4.2.14)

k=l

ˆk,β,ε ∇k u2 2 . Φ L (Ω)

k=0

l (S, Cu , γu ) are For a given sector S and constants Cu , γu > 0, the spaces Bβ,ε deﬁned as l,l l (S, Cu , γu ) = u ∈ Hβ,ε (S) | uH l,l (S) ≤ Cu and (4.2.15) Bβ,ε β,ε

ˆk,β,ε ∇k+l uL2 (S) ≤ Cu γuk max {k + 1, ε−1 }k+l ∀k ∈ N0 . Φ

For simplicity of notation, the dependence on the domain S and the constants Cu , γu is dropped when no confusion can arise.

150

l 4. The Countably Normed Spaces Bβ,ε

2,2 Remark 4.2.4 For ﬁxed β ∈ [0, 1), the spaces Hβ,ε (S) are all isomorphic: There holds (cf. Lemma 4.2.7 below) 2,2 ∀u ∈ Hβ,1 (S) ∀ε ∈ (0, 1]. (4.2.16)

ε2 uH 2,2 (S) ≤ uH 2,2 (S) ≤ uH 2,2 (S) β,1

β,ε

β,1

l Similarly, the spaces ∪Cu >0,γu >0 Bβ,ε (S, Cu , γu ) are algebraically identical for all m,l l (S) and Bβ,1 coincide with ε ∈ (0, 1]. Furthermore, we note that the spaces Hβ,1

the spaces Hβm,l , Bβl introduced by Babuˇska and Guo in [15,16]. We will therefore frequently make use of results by them for this special case. Lemma 4.2.5. For all ε ∈ (0, 1] and β ∈ [0, 1), there holds the embedding 2,2 Hβ,ε (S) ⊂ C(S). Moreover, there is C > 0 depending only on S and β such that uC(S) ≤ Cε−2 uH 2,2 (S)

2,2 ∀u ∈ Hβ,ε (S).

β,ε

Proof: From [21] we have uC(S) ≤ CuH 2,2 (S) . The result now follows from β,1

2

(4.2.16).

m,l l We are now interested in analyzing the behavior of the spaces Hβ,ε and Bβ,ε under smooth changes of variables. For the purposes of this work, we will limit our attention to the cases m = l ∈ {0, 2}. m,l 4.2.1 Properties of the spaces Hβ,ε (Ω)

ˆ S) be a C 2 diﬀeoˆ S ⊂ R2 be two sectors. Let g ∈ C 2 (S, Lemma 4.2.6. Let S, morphism satisfying additionally g(0) = 0. Then there exists C > 0 depending only on Sˆ and g such that for every ε ∈ (0, 1] C −1 uH l,l (S) ≤ u ◦ gH l,l (S) ˆ ≤ CuH l,l (S) β,ε

β,ε

β,ε

l,l ∀u ∈ Hβ,ε (S),

l ∈ {0, 1, 2}.

Proof: As g is a C 2 diﬀeomorphism, there are c1 , c2 > 0 such that c1 |x| ≤ |g(x)| ≤ c2 |x|. Furthermore, gC 2 (S) ˆ < ∞. The upper bounds now follow readily from the chain rule. As g is a C 2 diﬀeomorphism, the inverse g −1 is also a C 2 diﬀeomorphism and hence the lower bounds can be obtained from the upper bounds by replacing g by g −1 (and exchanging Sˆ and S). 2 ˆ0,β,ε of Lemma 4.2.3 imply The monotonicity properties of the weight function Φ properties of the norms · H l,l : β,ε

Lemma 4.2.7. Let S be a sector. Then, for ﬁxed β ∈ [0, 1) and l ∈ N0 ∀ 0 < ε ≤ ε ≤ 1

uH l,l (S) ≤ uH l,l

(S) β,ε

β,ε

uH 0,0 (S) ≤ (ε /ε) uH 0,0 (S) β

β,ε

β,ε

l,l ∀u ∈ Hβ,1 (S),

∀ 0 < ε ≤ ε ≤ 1

l > 0,

0,0 ∀u ∈ Hβ,1 (S).

For ﬁxed ε ∈ (0, 1] there holds uH l,l

β ,ε

(S)

≤ uH l,l (S) β,ε

∀ 0 ≤ β ≤ β ≤ 1

l,l ∀u ∈ Hβ,1 (S).

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

151

Proof: The estimates follow from the monotonicity properties of the weight ˆ0,β,ε of Lemma 4.2.3. In particular, the case of ﬁxed β ∈ [0, 1) and function Φ l = 0 follows from (4.2.11). 2 2,2 Lemma 4.2.8 (Hβ,ε compactly embedded in H 1 ). Let S be a sector, β ∈ 2,2 [0, 1), ε ∈ (0, 1]. Then Hβ,ε (S) is compactly embedded in H 1 (S). Furthermore, for every δ > 0, there exists C(δ) > 0 independent of ε ∈ (0, 1] such that

ˆ0,β,ε ∇2 uL2 (S) + C(δ)uL2 (S) uH 1 (Ω) ≤ δΦ

2,2 ∀u ∈ Hβ,ε (S).

Proof: The compactness of the embedding is essentially proved in, e.g., [112] (only the case of a straight polygon is considered there but Lemma 4.2.6 allows us 2,2 to infer the general case readily). For ε = 1, the compact embedding Hβ,1 (S) ⊂⊂ 1 H (S) implies by a standard argument (“Ehrling’s Lemma”, cf. [129, Thm. 7.3]) that every for every δ > 0 there exists C(δ) > 0 such that ˆ0,β,1 ∇2 uL2 (S) + C(δ)uL2 (S) . uH 1 (S) ≤ δΦ The desired result now follows from the observation that for ε ∈ (0, 1] there ˆ0,β,ε ≥ Φ ˆ0,β,1 on S. holds Φ 2 We furthermore need a result concerning L∞ bounds: Lemma 4.2.9. Let S be a sector and β ∈ [0, 1). There exists C > 0 depending 2,2 (S) only on β and S such that for all u ∈ Hβ,ε ˆ0,β,ε ∇2 uL2 (S) + uL2 (S) . uL∞ (S) ≤ C Φ Proof: By [21], we have the embedding uL∞ (S) ≤ CuH 2,2 (S) . Using the β,1 preceding lemma, we therefore have ˆ0,β,1 ∇2 uL2 (S) + uL2 (S) . uL∞ (S) ≤ C Φ ˆ0,β,ε on S for ε ∈ (0, 1] gives the desired result. ˆ0,β,1 ≤ Φ Using again Φ

2

2,2 (S) akin Next, we need embedding theorems in the weighted Sobolev spaces Hβ,ε to those studied in [21]:

Lemma 4.2.10 (embedding in weighted spaces). Let S be a C 2 -curvilinear sector, β ∈ (0, 1), l ∈ {1, 2}. Then there is C > 0 depending only on S, β, and l l,l (S) there is u ∈ R such that: such that for every ε ∈ (0, 1] and every u ∈ Hβ,ε (i) if l = 2 the constant u may be taken as u = u(0) and there holds ˆ0,β−2,ε (u − u) L2 (S∩B (0)) ≤ εΦ ˆ0,β−1,ε ∇uL2 (S∩B (0)) Φ 2ε 2ε ≤ CuH 2,2 (S∩B2ε (0)) , β,ε

uL∞ (S∩B2ε (0)) ≤ Cε

−1

uH 2,2 (S∩B2ε (0)) , β,ε

ˆ0,β−1,ε ∇uL2 (S\B (0)) ≤ Cu 2,2 ; ˆ0,β−2,ε uL2 (S\B (0)) + εΦ Φ ε ε H (S) β,ε

152

l 4. The Countably Normed Spaces Bβ,ε

(ii) if l = 1: ˆ0,β−1,ε uL2 (S∩B (0)) ≤ Cu 1,1 Φ 2ε H (S∩B2ε (0)) , β,ε

ˆ0,β−1,ε uL2 (S) ≤ Cu 1,1 , Φ H (S) β,ε

ˆ0,β,ε ∇uL2 (S∩B (0)) ≤ Cu 1,1 . ˆ0,β−1,ε (u − u) L2 (S∩B (0)) ≤ CεΦ Φ 2ε 2ε H (S) β,ε

ˆ0,β−1,ε = 1 on S \ Bε (0) ˆ0,β−2,ε = Φ Proof: We start with the proof of (i). As Φ the third assertion of (i) is trivial. We can therefore restrict our attention to the neighborhood V = B2ε (0) ∩ S. We start with the following Assertion: For τ > 0 denote Tτ := {(x, y) | 0 < x < τ, 0 < y < τ − x}. Then for any s ∈ (0, 1) there is C > 0 independent of τ such that for all functions u ∈ C(Tτ ) with |x|s ∇2 uL2 (Tτ ) < ∞ there holds |x|s−2 (u − u(0)) L2 (Tτ ) + |x|s−1 ∇uL2 (Tτ ) ≤ C |x|s ∇2 uL2 (Tτ ) + τ s−2 uL2 (Tτ ) . Proof of the Assertion: By homogeneity, it suﬃces to consider the case τ = 1. From [21, Lemma 4.4] there exists C > 0 such that for the linear interpolant p of u there holds |x|s−2 (u − p)L2 (T1 ) + |x|s−1 ∇(u − p)L2 (T1 ) ≤ C |x|s ∇2 uL2 (T1 ) . (4.2.17) By Sobolev’s embedding on T1 \ T1/2 we have the existence of C > 0 such that (4.2.18) uL∞ (T1 \T1/2 ) ≤ C |x|s ∇2 uL2 (T1 \T1/2 ) + uL2 (T1 \T1/2 ) . Next, we write p(x) = u(0) + l(x) where l is a linear function with l(0) = 0 and lL∞ (T1 ) + ∇lL∞ (T1 ) ≤ CuL∞ (T1 \T1/2 ) . An application of the reverse triangle inequality in (4.2.17) and (4.2.18) concludes the proof of the assertion. Let us now consider the neighborhood V = S ∩ B2ε (0). As S is a C 2 -curvilinear sector, there are two C 2 curves Γ1 , Γ2 comprising the boundary of S near the origin 0. Introduce another smooth curve Γ through 0 (independent of ε) that divides V into two domains V , V each having a convex corner at 0. By the smoothness of the curves Γ1 , Γ2 , Γ there is a C 2 map F : Tε → V such that F (0, 0) = 0 and F , (F −1 ) , and F can be bounded independently of ε (e.g., the blending map from Tε to V , [58–60]). From the Assertion and the fact that F (0) = 0 we get that the function u ˆ = u ◦ F satisﬁes |x|s−2 (ˆ u−u ˆ(0)) L2 (Tε ) + |x|s−1 ∇ˆ uL2 (Tε ) s 2 s−2 ≤ C |x| ∇ u ˆL2 (Tε ) + ε ˆ uL2 (Tε ) ≤ C |x|s (∇2 u) ◦ F L2 (Tε ) +

CF L∞ (Tε ) |x|s (∇u) ◦ F L2 (Tε ) + εs−2 u ◦ F L2 (Tε ) .

Employing now the fact that F , (F −1 ) , and F can be bounded independently of ε, we obtain by transforming this last estimate back to V :

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

153

rβ−2 (u − u(0)) L2 (V ) + rβ−1 ∇uL2 (V ) ≤ C rβ ∇2 uL2 (V ) + rβ ∇uL2 (V ) + εβ−2 uL2 (V ) , where r(x) = |x|. Similarly, we obtain the corresponding bound for V . Dividing by εβ−2 and using that rβ ≤ εβ on V we therefore obtain ˆ0,β−1,ε ∇uL2 (V ) ≤ ˆ0,β−2,ε (u − u(0)) L2 (V ) + εΦ Φ Cε2 Φ0,β,ε ∇2 uL2 (V ) + Cε∇uL2 (V ) + CuL2 (V ) . ˆ0,β−2,ε (u−u)L2 (S∩B (0)) ≤ Cu 2,2 This last estimate proves Φ 2ε Hβ,ε (S∩B2ε (0)) and ˆ0,β,ε ∇uL2 (S∩B (0)) ≤ Cu 2,2 . We will not explicitly prove the εΦ 2ε

Hβ,ε (S∩B2ε (0))

ﬁrst claim of (i) as its proof is very similar to that of the last claim of (ii), which we prove below. We now turn to the L∞ -bound. From the embedding theorem Lemma 4.2.9 applied to the triangle T1 we obtain with scaling arguments εβ−1 uL∞ (Tε ) ≤ C |x|β ∇2 uL2 (Tε ) + εβ−2 uL2 (Tε ) . Reasoning as above, we conclude ˆ0,β,ε ∇2 uL2 (V ) + ε2 Φ ˆ0,β,ε ∇uL2 (V ) + uL2 (V ) εuL∞ (V ) ≤ C ε2 Φ ≤ CuH 2,2 (V ) . β,ε

Part (ii) of the lemma is proved using the same ideas. For the ﬁrst part estimate of Part (ii), we employ Lemma A.1.7 and a scaling argument to conclude for the triangles Tτ : |x|β−1 uL2 (Tτ ) ≤ C |x|β ∇uL2 (Tτ ) + τ β−1 uL2 (Tτ ) . This implies with arguments as above rβ−1 uL2 (V ) ≤ C rβ ∇uL2 (V ) + εβ−1 uL2 (V ) . Dividing by εβ−1 then gives the result. The second estimate follows by combining the ﬁrst estimate of Part (ii) with the observation Φ0,β,ε (x) = 1 for x ∈ S \ Bε (0). The last estimate in Part (ii) is seen as follows: We note that by [21, Lemma 4.3] there is u ∈ R such that the function u ˆ = u ◦ F satisﬁes u − u)L2 (Tε ) ≤ C |x|β ∇ˆ uL2 (Tε ) . |x|β−1 (ˆ Transforming back to to V and dividing by εβ−1 yields ε−(β−1) |x|β−1 (u − u)L2 (Tε ) ≤ Cεε−β |x|β ∇uL2 (Tε ) . This proves (ii) of the lemma.

2

In the following corollary, we strengthen slightly the assertion of the ﬁrst part of Lemma 4.2.10.

154

l 4. The Countably Normed Spaces Bβ,ε

Corollary 4.2.11. Under the assumptions of Lemma 4.2.10 there exists C > 0 2,2 (S) independent of ε such that for all u ∈ Hβ,ε εΦ0,β−1,ε ∇uL2 (S∩B2ε (0)) ≤ C ε2 Φ0,β,ε ∇2 uL2 (S∩B2ε (0)) + ε∇uL2 (S∩B2ε (0)) . Proof: Follows from an application of the ﬁrst estimate of (ii), Lemma 4.2.10 to ∇u. 2 We also need the follow variant of Hardy’s inequality: Lemma 4.2.12. Let S := SR (ω) be a straight sector, u ∈ H 1 (S) and u = 0 on at least Γ1 or Γ2 . Then there is C > 0 depending only on ω such that for all R ∈ (0, R] there holds

1 uL2 (SR (ω)) ≤ ∇uL2 (SR (ω)) . |x|

Proof: By a scaling argument, it suﬃces to show the result for R = 1. The case R = 1 is a variant of a standard result, see, e.g., [62, Thm. 1.4.4.3]. 2

l 4.2.2 Properties of the countably normed spaces Bβ,ε l First, we show that the spaces Bβ,ε are embedded in each other.

Proposition 4.2.13. Let S be a sector. Then the following holds: 1. For each l ∈ N0 , ε > 0 membership u ∈ Bβl 0 ,ε (S, Cu , γu ) for β0 ∈ [0, 1) l implies u ∈ Bβ,ε (S, Cu , γu ) for all β ∈ [β0 , 1). l+1 2. Let β ∈ [0, 1), ε > 0, Cu > 0, γu ≥ 1, l ∈ N0 . Then u ∈ Bβ,ε (S, Cu , γu ) l (S, Cu , γu ). implies u ∈ Bβ,ε l 3. Let c > 0, ε0 ∈ (0, 1], β ∈ [0, 1), l ∈ N0 be given. Then u ∈ Bβ,ε (S, Cu , γu ) implies the existence of C, γ > 0 independent of ε ∈ (0, ε0 ] such that for all ε ∈ (0, 1] with ε /ε ≤ c there holds ˆp,β,ε ∇p+l uL2 (S) ≤ CCu γ p Φ

l−β ε max {p + 1, (ε )−1 }p+l ε

∀p ∈ N0 .

4. Under the same assumption as in the preceding statement, we obtain for the l special case l ≥ 1 that u ∈ Bβ,ε (S, CCu , γ) with C, γ > 0 independent of ε. Proof: The ﬁrst assertion follows immediately from the ﬁrst assertion of Lemma 4.2.3. For the second one, we observe that uH l,l (S) ≤ uH l+1,l+1 (S) ≤ Cu , β,ε

β,ε

−1 l ˆ0,β,ε ∇l uL2 (S) ≤ ∇l uL2 (S) ≤ ε−l u l+1,l+1 }. Φ H (S) ≤ Cu max {1, ε β,ε

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

155

ˆp,β,ε (x) ≤ It remains to consider derivatives of order p ≥ 1. For p ≥ 1 we have Φ ˆp−1,β,ε (x) by Lemma 4.2.3. Hence, we can write Φ ˆp,β,ε ∇p+l uL2 (S) = Φ ˆp,β,ε ∇(p−1)+(l+1) uL2 (S) ≤ Cu γ p−1 max {p, ε−1 }p+l Φ u ≤ Cu γup−1 max {p + 1, ε−1 }p+l , which is the desired bound. We now turn to the third assertion. From Lemma 4.2.3 we obtain the existence of C, γ > 0 independent of ε, ε such that for all p ∈ N0 ˆp,β,ε ∇p+l uL2 (S) ≤ Φ Cγ p max {p + 1, ε−1 }l max {1, ε/ε }β max {1, ε /ε}p max {p + 1, (ε )−1 }p . Observing ε /ε ≤ c allows us to simplify (by adjusting C, γ): ˆp,β,ε ∇p+l uL2 (S) ≤ Cγ p max {p+1, ε−1 }l (ε/ε )β max {p+1, (ε )−1 }p ∀p ∈ N0 . Φ Next, in order to treat the term max {p + 1, ε−1 } we bound max {p + 1, ε−1 } = ε−1 max {(p + 1)ε, 1} ≤ ε−1 (p + 1)(1 + ε0 ) ≤ (1 + ε0 )(p + 1)ε−1 max {1, (p + 1)ε } ≤ (1 + ε0 )(p + 1)(ε /ε) max {(p + 1), (ε )−1 }. Hence, we obtain (by again enlarging C, γ) that there holds for all p ∈ N0 : ˆp,β,ε ∇p+l uL2 (S) ≤ Cγ p max {p + 1, (ε )−1 }l (ε/ε )β−l max {p + 1, (ε )−1 }p . Φ For the last assertion, it suﬃces to note that β ∈ [0, 1) and l ≥ 1. Hence, the factor (ε /ε)l−β is bounded. 2 l , l ∈ {0, 1, 2}, Our main goal of this subsection is to show that the spaces Bβ,ε are invariant under analytic changes of variables. In order to prove that, we start with a variant of Besicovitch’s covering theorem:

Lemma 4.2.14 (Besicovitch’s covering theorem). Let S be a sector, c ∈ (0, 1). Then there N ∈ N and a family of balls B = {Bi = Bri (xi ) | i ∈ N} with the following properties: 1. S ⊂ ∪i Bi , 2. ri = c|xi |, 3. ∀x ∈ S there holds card{i ∈ N | x ∈ Bi } ≤ N . Proof: Consider the (uncountable) family B = ∪x∈S Bc|x| (x) of balls. [136, Thm. 1.3.5] is formulated for families of closed balls. Inspection of the proof, however, shows that it also holds for collections of open balls. Hence, by [136, Thm. 1.3.5] there is N ∈ N and N countable subfamilies of B of the form Bi = {Bi,j | j ∈ N}, i = 1, . . . , N , with the property that a) S ⊂ ∪N i=1 ∪j∈N Bij and b) for each i the balls of the subfamily Bi are (mutually) disjoint. The lemma follows now by taking as the desired family the union ∪N 2 i=1 Bi .

156

l 4. The Countably Normed Spaces Bβ,ε

Remark 4.2.15 For our purposes, it is actually not important to apply [136, Thm. 1.3.5] with open rather than closed balls. Starting from B = ∪x∈S Bc|x| (xi ) an application of [136, Thm. 1.3.5] as stated, yields N ∈ N and disjointed collections Bi = {Bij | j ∈ N} of closed balls with the desired properties. The collection B = {Bij | i = 1, . . . , N, j ∈ N} now covers S \ A where A = ∪N i=1 ∪j∈N ∂Bij is a set of Lebesgue measure zero. As this covering lemma will be used in order to obtain L2 -bounds on S using L2 -bounds on the balls Bi , the set A of measure zero is irrelevant. l For a sector S, functions belonging to a countably normed space Bβ,ε (S, C, γ) are analytic on S and can be extended analytically across ∂S \ {0}: l Lemma 4.2.16. Let S be a sector, u ∈ Bβ,ε (S, Cu , γ) for some l ∈ N0 , β ∈ $ (0, 1), Cu , γ > 0. Then there exists a sector S with S \ {0} ⊂ S$ depending only $ on S and γ such that u is analytic on S.

Proof: We start by stressing that the lemma does not allow for an explicit control of the growth of the derivatives of u—merely the analyticity is claimed. It is easy l to see that for given ε > 0 there are Cε , Kε such that u ∈ Bβ,1 (S, Cε , Kε ). The result now follows from the arguments presented in [14]. 2 The following lemma plays a key rˆ ole in the main result of this section, Theorem 4.2.20. In essence, it characterizes the functions from a countably normed l spaces Bβ,ε in terms of their local behavior. Lemma 4.2.17 (local characterization of countably normed spaces). Let S be a sector, l ∈ N0 , β ∈ (0, 1), ε > 0. Let B = {Bi | i ∈ N} be a collection of balls Bi = Bri (xi ) with the following properties: 1. there is c ∈ (0, 1) with ri = c|xi | for all i ∈ N; 2. there is N ∈ N such that ∀x ∈ S there holds card {i ∈ N | x ∈ Bi } ≤ N . l Let f ∈ Bβ,ε (S, Cf , γf ). Then there are C, K > 0 independent of ε such that for all p ∈ N0 , i ∈ N

ˆp,β,ε (xi )∇p+l f L2 (S∩B ) ≤ CC(i)K p max {p, ε−1 }p+l , Φ i ∞ 4 C 2 (i) ≤ Cf2 N < ∞, 3 i=1

(4.2.19) (4.2.20)

where the constants C(i) are given by C 2 (i) :=

∞

1 1 ˆp,β,ε ∇p+l f 2 2 Φ L (S∩Bi ) 2p −1 2(p+l) (2γ ) max {p + 1, ε } f p=0

≤ Cf2

4 < ∞. 3

(4.2.21)

$ γ > 0, and C(i) Conversely, let f be analytic on S and assume that there are C, such that f satisﬁes on the balls Bi :

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

ˆp,β,ε (xi )∇p+l f L2 (S∩B ) ≤ C(i)γ p max {p + 1, ε−1 }p+l Φ i ∞ $2 . C 2 (i) ≤ C

157

∀p ∈ N0 , (4.2.22a) (4.2.22b)

i=1

Then there exist constants C, K > 0 depending only on γ > 0 and the overlap constant N such that ˆp,β,ε ∇p+l f L2 (S) ≤ C CK $ p max {p + 1, ε−1 }p+l Φ

∀p ∈ N0 .

(4.2.23)

Proof: The bound on C 2 (i) in (4.2.21) follows from C 2 (i) = ≤

∞

1 1 ˆp,β,ε ∇p+l f 2 2 Φ L (S∩Bi ) −1 }2(p+l) (2γ )2p max {p + 1, ε f p=0

∞

∞

1 1 1 ˆp,β,ε ∇p+l f 2 2 Φ Cf2 2p L (S) ≤ 2p −1 2(p+l) (2γ ) 2 max {p + 1, ε } f p=0 p=0

4 = Cf2 . 3 The bound (4.2.20) is proved similarly using additionally the overlap properties of the sets Bi : ∞

C 2 (i) =

i=1

≤ ≤

∞ 1 1 ˆp,β,ε ∇p+l f 2 2 Φ L (S∩Bi ) −1 }2(p+l) (2γ )2p max {p + 1, ε f p=0 i=1

∞

∞

1 1 ˆp,β,ε ∇p+l f 2 2 N Φ L (S) 2p −1 2(p+l) (2γ ) max {p + 1, ε } f p=0

∞ p=0

Cf2

1 4 N ≤ Cf2 N . 22p 3

(4.2.21) implies additionally −1 p+l ˆp,β,ε ∇p+l f 2 2 Φ } (2γf )p . L (S∩Bi ) ≤ C(i) max {p + 1, ε

ˆp,β,ε (x) ≥ From Lemma 4.2.2 and our assumptions on the balls Bi , we have Φ −1 −p ˆ C K Φp,β,ε (xi ) for all x ∈ Bi for some C, K > 0 independent of ε. Hence, we get ˆp,β,ε (xi )∇p+l f L2 (S∩B ) ≤ CK p Φ ˆp,β,ε ∇p+l f L2 (S∩B ) Φ i i ≤ CK p C(i) max {p, ε−1 }p+l . The converse statement, i.e., that (4.2.22) implies (4.2.23) follows from (4.2.8) of Lemma 4.2.2 and a summation over the balls Bi . 2 For l > 0, only derivatives of order greater than or equal to l appear explicitly in Lemma 4.2.17. Local control of derivatives up to order l − 1 is possible, however. We illustrate this in the following corollary for the case l = 1.

l 4. The Countably Normed Spaces Bβ,ε

158

Corollary 4.2.18. Let S be a sector, l ∈ N0 , β ∈ (0, 1), ε > 0. Let B = {Bi | i ∈ N} be a collection of balls Bi = Bri (xi ) with the following properties: 1. there is c ∈ (0, 1) with ri = c|xi | for all i ∈ N; 2. there is N ∈ N such that ∀x ∈ S there holds card {i ∈ N | x ∈ Bi } ≤ N . 1 (S, Cf , γf ). Then there are C, K > 0 independent of ε such that for Let f ∈ Bβ,ε all p ∈ N0 , i ∈ N

ˆp−1,β,ε (xi )∇p f L2 (S∩B ) ≤ CC(i)K p max {p, ε−1 }p , Φ i ∞ C 2 (i) ≤ CCf2 < ∞,

(4.2.24) (4.2.25)

i=1

where the constants C(i) are given by ˆ−1,β,ε f 2 2 C 2 (i) := Φ L (S∩Bi ) +

∞

1 1 ˆp−1,β,ε ∇p f 2 2 Φ L (S∩Bi ) . −1 }2p (2γ )2(p−1) max {p + 1, ε f p=1

Proof: The bound (4.2.24) follows immediately from the deﬁnition of the constants C(i). It remains to see (4.2.25). We express C(i) as C(i) := C1 (i) + C2 (i), where C2 (i) is given by the inﬁnite sum in the deﬁnition of C 2 (i) and C1 (i) is deﬁned as ˆ−1,β,ε f 2 2 C1 (i) := Φ L (S∩Bi ) . From Lemma 4.2.17 we then have 4 C22 (i) ≤ N Cf2 < ∞. 3 i∈N

2 To ascertain the ﬁniteness of i∈N C1 (i), we employ a Hardy inequality of Lemma 4.2.10, (ii): 2 2 2 ˆ−1,β,ε f 2 2 ˆ Φ L (S∩Bi ) ≤ N Φ0,β−1,ε f L2 (S) ≤ N Cf H 1,1 (S) ≤ CN Cf . i∈N

β,ε

2 Lemma 4.2.19. Let S be a sector, c0 > 0, β ∈ (0, 1), l ∈ {0, 1, 2}, ε ∈ (0, 1]. Let l u ∈ Bβ,ε (S, Cu , γu ) for some ε > 0, and Cu , γu > 0. Then there are constants C, γ > 0 independent of ε and Cu and there exists a constant u ∈ R such that ˆp−l,β,ε ∇p (u − u)L2 (S∩B (0)) ≤ CCu γ p max {p + 1, ε−1 }p Φ c0 ε

∀p ∈ N0 ,

ˆp−l,β,ε ∇p uL2 (S\B (0)) ≤ CCu γ p max {p + 1, ε−1 }p Φ c0 ε

∀p ∈ N0 .

Moreover, in the case l = 2, the constant u may be taken as u = u(0) and for l = 0 we can take u = 0.

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

159

l Proof: The deﬁnition of Bβ,ε (S, Cu , γu ) implies for p ≥ l:

ˆp−l,β,ε ∇p uL2 (S) ≤ Cu γ p−l max {p − l + 1, ε−1 }p Φ u

∀p ≥ l.

As u = 0 for l = 0, this gives the desired estimates for p ≥ l. It remains to check the ﬁnitely many cases 0 ≤ p < l. On S \ Bc0 ε (0), we use the deﬁnition (4.2.2) to ﬁnd (recall: 0 ≤ |p − l| ≤ l) ˆp−l,β,ε (x) ∼ 1 Φ

∀p ∈ {0, . . . , l}

∀x ∈ S \ Bc0 ε (0).

(4.2.26)

Hence, ˆp−l,β,ε ∇p uL2 (S\B (0)) ≤ CCu , Φ c0 ε

p = 0, . . . , l − 1.

Let u be given by Lemma 4.2.10. Then (with u = u(0) in the case l = 2) ˆ−2,β,ε (u − u)L2 (S∩B (0)) + εΦ ˆ−1,β,ε ∇uL2 (S∩B (0)) ≤ CCu Φ 2ε 2ε ˆ Φ−1,β,ε (u − u)L2 (S∩B2ε (0)) ≤ CCu

if l = 2, if l = 1,

where we exploited that by deﬁnition uH l,l (S∩B2ε (0)) ≤ Cu . If c0 ≤ 2, then β,ε these estimates imply the desired result. If c0 > 2, then we note that these ˆ−l,β,ε ∼ 1 on S \ Bε (0) (cf. also (4.2.26)) estimates together with the fact Φ ˆ−l,β,ε (u − u)L2 (S∩B (0)) + uL2 (S) |u|ε ∼ uL2 (S∩B2ε (0)\Bε (0)) ≤ C Φ 2ε ≤ CuH l,l (S) ≤ CCu . β,ε

ˆ−l,β,ε ∼ 1 on S \ B2ε (0), we get from this bound and the Exploiting again Φ triangle inequality ˆ−l,β,ε (u − u)L2 (S∩B (0)\B (0)) ≤ CCu Φ c0 ε 2ε

for l ∈ {1, 2}. 2

We can now state the main result of this section, namely, the invariance of the l countably normed spaces Bβ,ε under analytic changes of variables: l Theorem 4.2.20 (Invariance of Bβ,ε under changes of variables). Let S 2 be a C -curvilinear sector and g : S → g(S) ⊂ R2 be analytic on S, g(0) = 0, and assume that g −1 is analytic on g(S). Let Cu , γu > 0, β ∈ (0, 1). Then there exist constants C, γ > 0 depending only on g, S, γu , and β, (in particular, they are independent of ε) such that for l ∈ {0, 1, 2} and ε ∈ (0, 1] l l u ∈ Bβ,ε (g(S), Cu , γu ) =⇒ u ◦ g ∈ Bβ,ε (S, CCu , γ).

Proof: We start by noting that Lemma 4.2.6 gives u ◦ gH l,l (S) ≤ CCu , β,ε

l ∈ {0, 1, 2}.

160

l 4. The Countably Normed Spaces Bβ,ε

We may therefore concentrate on the bounds of higher derivatives. To that end, we proceed in two steps and prove the following two estimates separately ˆp,β,ε ∇p+l (u ◦ g)L2 (S ) ≤ CCu γ p max {p, ε−1 }p+l Φ ε ˆp,β,ε ∇p+l (u ◦ g)L2 (S\S ) ≤ CCu γ p max {p, ε−1 }p+l Φ ε

∀p ∈ N0 ,

(4.2.27)

∀p ∈ N0 ,

(4.2.28)

where Sε := g −1 (g(S) ∩ Bε (0)). Let B = {Bi = Bri (xi ) | i ∈ N} be a covering of g(S) by balls Bi as given by Lemma 4.2.14. Note that we can choose c is so small that Bi ∩ Bε (0) = ∅ implies Bi ⊂ B2ε (0). Next, we deﬁne the index set Iε := {i ∈ N | Bi ∩ Bε (0) = ∅}. We are now in position to prove (4.2.27). From Lemma 4.2.19, we have the existence of u ∈ R such that ˆp−l,β,ε ∇p (u − u)L2 (g(S)∩B (0)) ≤ Cγ p max {p + 1, ε−1 }p Φ 2ε

∀p ∈ N0 .

We introduce the shorthand u ˜ := u − u on g(S) ∩ B2ε (0). Moreover, the constant u = 0 for l = 0. It suﬃces therefore to show that similar estimates holds for u ˜ ◦ g. For indices i ∈ Iε we deﬁne C 2 (i) :=

p∈N0

(2γ)2p

1 ˆp−l,β,ε ∇p u Φ ˜2L2 (Bi ∩g(S)) . max {p, ε−1 }2p

We note that

C 2 (i) =: C Cu < ∞,

(4.2.29)

i∈Iε

where C depends only on the covering B (and is independent of ε). This deﬁnition of C(i) implies that for each i there holds ˆp−l,β,ε ∇p u Φ ˜L2 (Bi ∩g(S)) ≤ CCu C(i)(2γ)p max {p, ε−1 }p

∀p ∈ N0 .

Let us now consider a ﬁxed i. Abbreviating r = |xi |, we get from Lemma 4.2.2 the existence of C, K > 0 independent of p, ε, and i such that C −1 K −p

max

x∈Bi ∩g(S)

ˆp−l,β,ε (xi ) ˆp−l,β,ε (x) ≤ Φ Φ ≤ CK p

min

x∈Bi ∩g(S)

(4.2.30) ˆp−l,β,ε (x). Φ

Hence, we have using (4.2.4), (4.2.7) for some C, γ > 0 independent of p, ε, i: ˆp−l,β,ε (xi )∇p u ˆ−1 (xi ) · Φ ˆ−l,β,ε (xi )∇p u ˜L2 (Bi ∩g(S)) ≤ Cγ p Φ ˜L2 (Bi ∩g(S)) Φ p,0,ε −1 p p! . ≤ CCu C(i)γ p + max p, ε rp From Lemma 4.3.1 with f2 = u ˜, f1 = 1 we get that there are C, K > 0 independent of p, ε, and i such that −1 p p! ˆ−l,β,ε (xi )∇p (˜ Φ , (4.2.31) u ◦ g) L2 (Gi ) ≤ CCu C(i)K p + max p, ε rp

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

161

where we deﬁned Gi := g −1 (Bi ∩ g(S)). Since by (4.2.7) of Lemma 4.2.2 there ˆ−1 (xi ) max {p, ε−1 }p , we get holds p!r−p + max {p, ε−1 }p ≤ CK p Φ p,0,ε ˆp−l,β,ε (xi )∇p (˜ Φ u ◦ g)L2 (Gi ) ≤ CCu C(i)γ p max {p, ε−1 }p

(4.2.32)

for some C, γ > 0. Next, by the assumptions on g, there exists C > 0 such that C −1 |x| ≤ |g −1 (x)| ≤ C|x|

∀x ∈ g(S).

ˆp,β,ε , there exist C, K > 0 independent of ε, p ∈ N0 , Hence, by the deﬁnition of Φ x ∈ g(S) such that ˆp−l,β,ε ◦ g −1 )(x) ≤ CK p Φ ˆp−l,β,ε (x) ≤ (Φ ˆp−l,β,ε (x) ∀x ∈ g(S). C −1 K −p Φ By combining this estimate with (4.2.30), we get ˆp−l,β,ε (x) = max Φ

x∈Gi

max

ˆp−l,β,ε ◦ g −1 )(x) ≤ CK p (Φ

x∈Bi ∩g(S)

max

x∈Bi ∩g(S)

ˆp−l,β,ε (x) Φ

ˆp−l,β,ε (xi ). ≤ CK p Φ Inserting this in (4.2.32), we get for suitable C, γ > 0 ˆp−l,β,ε ∇p (˜ Φ u ◦ g)L2 (Gi ) ≤ CCu C(i)γ p max {p, ε−1 }p

(4.2.33)

As ∪i Gi ⊃ g −1 (g(S)∩Bε (0)) = Sε , we get the desired result (4.2.27) by squaring (4.2.33), summing on i ∈ Iε and using (4.2.29). The estimate (4.2.28) is proved completely analogously: We consider the index set I˜ε := {i ∈ N | Bi ∩ g(S) \ Bε (0) = ∅}; since B is a covering of g(S), we get ∪i∈I˜ε g −1 (Bi ∩ g(S)) ⊃ S \ Sε . Furthermore, we may assue that the parameter c ∈ (0, 1) of the covering may be chosen so small that Bi ⊂ g(S) \ Bε/2 (0) for all i ∈ I˜ε . Additionally, we get from Lemma 4.2.19 the existence of C, γ > 0 such that ˆp−l,β,ε ∇p uL2 (g(S)\B (0)) ≤ Cγ p max {p + 1, ε−1 }p Φ ε/2

∀p ∈ N0 .

We may then prove (4.2.28) in the same way as (4.2.27) by taking u ˜ = u.

2

Let us consider l = 2 in Theorem 4.2.20. In the above proof, we essentially assumed that uL2 (g(S)) and ε∇uL2 (g(S)) are of the same size. We will see later on that this is not always the case. Inspection of the proof of Theorem 4.2.20 allows us to reﬁne the results as follows.

162

l 4. The Countably Normed Spaces Bβ,ε

Corollary 4.2.21. Let the sector S and the map g satisfy the same hypothe2 (g(S)) satisﬁes the following ses as in Theorem 4.2.20. Assume that u ∈ Bβ,ε estimates: uL2 (g(S)) ≤ C0 , ε∇uL2 (g(S)) ≤ C1 , ˆp,β,ε ∇p+2 uL2 (g(S)) ≤ C1 γ p max {p + 1, ε−1 }p+2 Φ

∀p ∈ N0 .

Then there are C, K > 0 independent of C0 , C1 , and ε ∈ (0, 1] such that the function u ◦ g satisﬁes u ◦ gL2 (S) ≤ CC0 , ε∇(u ◦ g)L2 (S) ≤ CC1 , p+2 ˆ Φp,β,ε ∇ (u ◦ g)L2 (S) ≤ CC1 K p max {p + 1, ε−1 }p+2

(4.2.34) ∀p ∈ N0 .

(4.2.35)

Proof: (4.2.34) follows easily from the chain rule. In order to see (4.2.35), it is more convenient to consider ∇u. We ﬁrst claim that there are C, K > 0 depending only on g and γ such that ˆp−1,β,ε ∇p ∇uL2 (g(S)) ≤ CC1 γ p ε−1 max {p + 1, ε−1 }p Φ

∀p ∈ N0 . (4.2.36)

For p ∈ N this follows directly from the assumptions on u. For p = 0, we note ˆ−1,β,ε = 1; it suﬃces therefore to show that that on g(S) \ Bε (0) we have Φ ˆ−1,β,ε ∇uL2 (g(S)∩B (0)) ≤ CC1 ε−1 , Φ ε which follows from Corollary 4.2.11. With the formula ∇(u ◦ g) = (g · (∇u) ◦ g) we see that we can proceed verbatim as in the proof of Theorem 4.2.20 for the case l = 1 (using Lemma 4.3.1 with f1 = ∇u, f2 = g ) to conclude that ˆp−1,β,ε ∇p ∇(u ◦ g)L2 (S) ≤ CC1 γ p ε−1 max {p + 1, ε−1 }p Φ This last estimate is readily brought to the desired form.

∀p ∈ N0 . 2

In the following, we will be particularly interested in the case l = 2. For elements 2 of the spaces Bβ,ε we have the following pointwise characterization. Proposition 4.2.22. Let S = SR (ω) be a straight sector, l ∈ {0, 1, 2}, ε ∈ (0, 1], l and let u ∈ Bβ,ε (S, Cu , γu ) for some Cu , γu > 0, and β ∈ (0, 1). Then for every R ∈ (0, R) there are C, γ > 0 independent of ε such that ) ) ) Φ ) ) ˆp−l+1,β,ε p ) ∇ u) ≤ Cγ p max {p + 1, ε−1 }p+1 ∀p ∈ N. ) ) max {1, |x|/ε} ) ∞ L

(SR (ω))

This estimate is also valid for p = 0 if either l = 0 or l = 2 together with u(0) = 0.

m,l l 4.2 The spaces Hβ,ε and Bβ,ε in a Sector

163

Proof: For given R , let ρ0 < 1 be such that R ρ0 < R. We cover SR (ω) by balls Bi = Bri (xi ) with the following properties: ri = ρ0 |xi | and Bi ∩ SR (ω) is either a full ball or a half ball, i.e., either Bi is completely contained in SR (ω) or xi is on one of the two straight sides of SR (ω). For each i ∈ N let Fi be the to Bi ∩ SR (ω), where B = B1 (0) if aﬃne map from the reference domain B Bi ∩ SR (ω) is a full ball and B = B1 (0) ∩ {y > 0} if Bi ∩ SR (ω) is a half-ball. From ρ0 < 1 we have that for every Bi there holds Bi ∩ S ⊂ S1 := S(1+ρ0 )ε or Bi ∩ S ⊂ S2 := S \ S(1−ρ0 )ε . By Lemma 4.2.19 there is u ∈ R such that the functions u $1 := u − u and the function u $2 := u satisfy on S1 , S2 : ˆp−l,β,ε ∇p u $1 L2 (S1 ) ≤ CCu γ p max {p + 1, ε−1 }p Φ ˆp−l,β,ε ∇p u $2 L2 (S2 ) ≤ CCu γ p max {p + 1, ε−1 }p Φ

∀p ∈ N0 , ∀p ∈ N0

For each i ∈ N consider the function u := u $ ◦ Fi . Then there holds for some C, γ > 0 independent of ε: p −1 p −1+p ˆp−l,β,ε (xi )∇p u Φ L2 (B) } ri ≤ Cγ max {p + 1, ε

∀p ∈ N0 .

⊂ H 2 (B) and the norm equivalence Next, from the Sobolev embedding L∞ (B) 2 · H 2 (B) ∼ · + ∇ · , we infer 2 2 L (B) L (B) p −1 p −1+p ˆp−l,β,ε (xi )∇p u 1 + ri2 max {p + 1, ε−1 }2 . L∞ (B) } ri Φ ≤ Cγ max {p+1, ε Mapping back to Bi ∩ SR (ω) yields ˆp−l,β,ε (xi )∇p u $L∞ (Bi ∩S) Φ

≤ Cγ p max {p + 1, ε−1 }p ri−1 1 + ri2 max {p + 1, ε−1 }2 . / 1 p −1 p+1 −1 ≤ C(γ ) max {p + 1, ε } + 1 + ri max {1, ε } ri max {1, ε−1 } ε ri . ≤ C(γ )p max {p + 1, ε−1 }p+1 + ri ε

Next, we estimate ri max {1, ri /ε} ε ≤ 2 max {1, ri /ε} max {1, ε/ri } = 2 + ε ri min {1, ri /ε} max {1, ri /ε} ≤C ˆ1,0,ε (xi ) Φ and therefore conclude ˆp−l+1,β,ε Φ $L∞ (Si ) ≤ Cγ p max {p+1, ε−1 }p+1 ∇p u max {1, |x|/ε}

∀p ∈ N0 ,

i ∈ {1, 2}.

For l = 0 we have u = 0 so that u $ = u and this implies the desired result. For l ∈ {1, 2} we note that u $ and u diﬀer by a constant so that we have the result for p ≥ 1. It remains to analyze p = 0 and l = 2. We then have u = u(0) = 0 by Lemma 4.2.19 so that we have again u $ = u and hence the result. 2

164

l 4. The Countably Normed Spaces Bβ,ε

l Theorem 4.2.23 (pointwise estimates for Bβ,ε -functions). Let S be an l analytic sector, l ∈ {0, 1, 2}, u ∈ Bβ,ε (S, Cu , γu ) for some Cu , γu > 0 and β ∈ (0, 1), ε > 0. Then for every neighborhood U of Γ3 there are there are C, γ > 0 independent of ε such that on S := S \ U ) ) ) Φ ) ) ˆp−l+1,β,ε p ) ∇ u) ≤ Cγ p max {(p + 1), ε−1 }p+1 ∀p ∈ N. ) ) max {1, |x|/ε} ) ∞ L

(S )

This result is also valid for p = 0 if either l = 0 or l = 2 together with u(0) = 0. Proof: Theorem 4.2.23 follows easily from Proposition 4.2.22 as follows. For simplicity, we assume that there are R, ω > 0, and an analytic mapping Λ : S → SR (ω) that is analytic on the closure S and whose inverse is also analytic SR (ω) (such a mapping can, for example, be constructed using blending maps, [58–60]). Otherwise, the argument below has to be carried out in a piecewise fashion. By l Theorem 4.2.20, the function u ◦ Λ is in the space Bβ,ε (SR (ω), C, γ) for some C, γ > 0 independent of ε. Taking R appropriately ensures that SR (ω) ⊃ Λ(S \U). Next, we employ Proposition 4.2.22 to conclude that u ◦ Λ satisﬁes for all p ∈ N ) ) ) ) Φ ) ) ˆp−l+1,β,ε p ≤ Cγ p max {p + 1, ε−1 }p+1 . (4.2.37) ∇ (u ◦ Λ)) ) ) ∞ ) max {1, |x|/ε} L

(SR (ω))

Using Lemma 4.2.2, we get that u ◦ Λ satisﬁes equivalently |∇p ∇(u ◦ Λ)(x)| ≤ Cγ p

1 max {1, |x|/ε}ε−1 max {(p + 1)/|x|, ε−1 }p+1 ˆ Φ1−l,β,ε (x)

for all x ∈ SR (ω) and p ∈ N0 . Hence, applying Lemma 4.3.3 to (∇u) ◦ Λ · Λ we conclude that for all x ∈ S \ U and p ∈ N0 |∇p ∇u(x)| ≤ Cγ p max {1, |x|/ε}

1 ε−1 max {(p + 1)/|x|, ε−1 }p+1 . ˆ Φ1−l,β,ε (x)

Using again Lemma 4.2.2, we get for all x ∈ S \ U and p ∈ N0 |∇p+1 u(x)| ≤ Cγ p max {1, |x|/ε} from which we conclude ) ) ) Φ ) ) ˆp−l+1,β,ε p ) ∇ u) ) ) max {1, |x|/ε} )

max {p + 1, ε−1 }p+1 , ˆ1−l,β,ε (x) ˆp+1,0,ε (x) Φ Φ 1

≤ Cγ p max {(p + 1), ε−1 }p+1

∀p ∈ N.

L∞ (S\U )

Since S = S \ U, this is the desired estimate for p ∈ N. We note that (4.2.37) is also valid for p = 0 if l = 0 or if l = 2 together with u(0) = 0. The theorem is now proved in full generality. 2

4.3 Local changes of variables for analytic functions

165

4.3 Local changes of variables for analytic functions Lemma 4.3.1. Let G, G1 ⊂ R2 be bounded open sets. Assume that g = (g1 , g2 ) : G1 → R2 is analytic and injective on G1 , det g = 0 and that it satisﬁes g(G1 ) ⊂ G. Let f1 : G1 → C, f2 : G → C be analytic and assume that f2 satisﬁes for some ε, Cf , γf > 0, r ≤ 1, . / p! ∇p f2 L2 (G) ≤ Cf γ p p + max {p, ε−1 }p . r Then there are constants C, K > 0 depending only on γ and g, f1 such that . / p p p! −1 p + max {p, ε } . ∇ (f1 · (f2 ◦ g))L2 (G1 ) ≤ Cf CK rp Proof: First, it is more convenient to formulate the assumption on f2 as ∇p f2 L2 (G) ≤ Cf γ p

p! r max {1, }p rp (p + 1)ε

∀p ∈ N0 .

Here, the constants Cf , γ > 0 may be diﬀerent from those in the statement of the lemma but are independent of r, ε. The growth conditions on the derivatives of f2 imply that f2 can be extended ˜ ⊂ C × C with G ⊂ G ˜ and G ˜ to a holomorphic function (also denoted f2 ) on G ˜ independent of ε > 0 (G depends only on the ratio γ/r). First, we claim that there are δ0 , γ , and C > 0 depending only on f1 , γ and Cf such that f2 (· + z1 (·), · + z2 (·))L2 (G) ≤ Ceγ

δr/ε

(4.3.38)

for all continuous functions z1 , z2 : G → C with zi L∞ (G) ≤ δr ≤ rδ0 , i = 1, 2. ˜ there is δ0 > 0 such that for all (x, y) ∈ G the As f2 is holomorphic on G, power series expansion of f2 about (x, y) converges on a ball of radius 2δ0 r. For functions z1 , z2 with zi L∞ (G) ≤ δr ≤ δ0 r we obtain: 1 α α D f (x, y)(z1 , z2 ) |f2 (x + z1 (x, y), y + z2 (x, y))| = α! 2 α∈N0

1 ≤ |Dα f2 (x, y)| (rδ)|α| . α! 2 α∈N0

Therefore we get

l 4. The Countably Normed Spaces Bβ,ε

166

f2 (· + z1 (·), · + z2 (·))L2 (G) ≤ ≤

∞

1 Dα f2 L2 (G) (rδ)|α| α! 2

α∈N0

(p!)1/2 (α!)−1/2 Dα f2 L2 (G)

(α!)−1/2 p!−1/2 (rδ)p

p=0 |α|=p ∞ 1/2 1 1 = ∇p f2 L2 (G) 2p/2 (rδ)p (rδ)2p α!p! p! p=0 p=0 |α|=p √ p 1 p! p! γ p 2p/2 δ p ≤ Cf 2γε−1 rδ + Cf p p! (p + 1) p! 0≤p≤r/ε p>r/ε p √ √ √ √ 1 √ e 2γδ ≤ Cf e 2γδ/ε + ≤ Cf e 2γδ/ε + C ≤ Ce 2γδ/ε , 1 − 2γδ0 p>r/ε

≤

∞

∇p f2 L2 (G)

√ where we made made the tacit assumption that δ0 is so small that e 2γδ0 < 1 for the second sum to be ﬁnite. This proves (4.3.38). Since g is analytic on G1 it has a holomorphic extension (also denoted g) to ˜ 1 ⊂ C × C. Thus, there are η, δ > 0 such that for i ∈ {1, 2} and all (x, y) ∈ G1 G 0 |gi (x + z1 , y + z2 ) − gi (x, y)| ≤ ηδ

∀z1 , z2 ∈ C s.t. |z1 |, |z2 | ≤ δ ≤ δ0 . (4.3.39)

Furthermore, since f1 is assumed analytic on G1 , we may suppose that the set ˜ 1 is such that f1 is analytic on G ˜ 1 and satisﬁes f1 ∞ ˜ ≤ C. For any G L (G 1 ) 0 < δ ≤ min (δ0 , δ0 /η) we obtain by Cauchy’s integral theorem for derivatives for every (x, y) ∈ G1 and every α = (α1 , α2 ) ∈ N20 (note that r ≤ 1) (f1 · (f2 ◦ g))(x + z1 , y + z2 ) −α! Dα (f1 · (f2 ◦ g)) (x, y) = dz1 dz2 . 4π 2 |z1 |=δr |z2 |=δr (−z1 )α1 +1 (−z2 )α2 +1 Hence, we can bound α!2 2 × |Dα (f1 · (f2 ◦ g)) (x, y)| ≤ 4π 2 (δr)2|α|+2 f g1 (x + z1 , y + z2 ), g2 (x + z1 , y + z2 ) 2 |dz1 | |dz2 |. |z1 |=δr

|z2 |=δr

By (4.3.39), we can write g1 (x + z1 , y + z2 ) = g1 (x, y) + ζ1 ,

g2 (x + z1 , y + z2 ) = g2 (x, y) + ζ2

where ζ1 , ζ2 are smooth functions of x, y, z1 , z2 , and |ζi | ≤ ηδ, i = 1, 2. Integrating over G1 , we obtain after the smooth change of variables g(x, y) = (x , y ) and denoting by ζ1 , ζ2 the functions corresponding to ζ1 , ζ2 after this change of variables 2

|Dα (f1 · (f2 ◦ g))(x, y)|L2 (G1 ) ≤ (α!)2 2 c2 2 |f (x + ζ1 , y + ζ2 )| dx dy |dz1 | |dz2 |. 2|α|+2 4π (δr) |z1 |=δr |z2 |=δr G

4.3 Local changes of variables for analytic functions

167

As |ζ1 |, |ζ2 | ≤ ηδr uniformly in (x , y ) ∈ G, |z1 |, |z2 | ≤ δr, the estimate (4.3.38) yields Dα (f1 · (f2 ◦ g))L2 (G1 ) ≤ C

α! eγ ηδr/ε |α| (rδ)

∀0 < δ ≤ min (δ0 , δ0 /η).

In order to extract from this estimate the claim of the lemma, we choose (p + 1)ε ,δ , δ := min δ := min (δ0 , δ0 /η). r This choice of δ implies with |α| = p: p! r −1 max {δ , }p eγ ηr min {(p+1)ε,δ}/ε p r (p + 1)ε p p! r ≤ p γ˜ p max 1, eγ η(p+1) r (p + 1)ε

Dα (f1 · (f2 ◦ g))L2 (G1 ) ≤ C

for some appropriate γ˜ > 0 independent of ε, r. This completes the proof.

2

Remark 4.3.2 The conditions on f1 be relaxed in the following way: It suﬃces that f1 satisﬁes a bound of the form |∇p f1 (x)| ≤ C(γ/r)p p! for all p ∈ N0 . Lemma 4.3.3. Let G ⊂ R2 and G ⊂ Rn (n ∈ N) be two bounded domains. Let Λ : G → G, v : G → C be analytic on G and u : G → C be analytic on G and satisfy |∇p u(x)| ≤ C(x)K p max {(p + 1)/r, ε−1 }p

∀p ∈ N0 ,

x∈G

for some K, r, ε > 0 and a function C : G → R+ . Then there exist C , γ > 0 depending only on K, Λ, and v such that |∇p (v · (u ◦ Λ)) (z)| ≤ C γ p max {(p+1)/r, ε−1 }p C(Λ(z))

∀p ∈ N0 ,

z ∈ G .

Proof: The proof is very similar to that of Lemma 4.3.1. It is worth pointing out, however, that the mapping Λ is not required to be invertible–a condition that was necessary in the proof of Lemma 4.3.1. 2 We now address the special case r = 1, ε = 1. There, the dependence of the various constants can be tracked more easily. Lemma 4.3.4. Let G ⊂ R2 and G ⊂ Rn (n ∈ N) be two bounded domains. Let Λ : G → G, v : G → C be analytic on G and u : G → C be analytic on G and satisfy for some γu and a function C : G → R+ |∇p u(x)| ≤ C(x)γup p!

∀p ∈ N0 ,

x ∈ G.

Then there are C , γ > 0 depending only on v and Λ such that |∇p (v · (u ◦ Λ)) (z)| ≤ C (γ(1 + γu )) p!(Λ(z)) p

∀p ∈ N0 ,

z ∈ G .

168

l 4. The Countably Normed Spaces Bβ,ε

Proof: The proof is very similar to that of Lemma 4.3.1. There only diﬀerence 2 is that one has to track the dependence on the constant γu . As another special case of Lemma 4.3.3, we have the following corollary, in which we stipulate only control over ∇u. Corollary 4.3.5. Let G ⊂ R2 and G ⊂ Rn (n ∈ N) be two bounded domains. Let Λ : G → G, and u : G → C be analytic on G and satisfy p ∇p ΛL∞ (G ) ≤ CΛ γΛ p!

∀p ∈ N0 ,

|∇ u(x)| ≤ C(x)K max {(p + 1)/r, ε−1 }p p

p

∀p ∈ N,

x∈G

for some CΛ , γΛ , K, r, ε > 0, and a function C : G → R+ . Then there are C , γ > 0 depending only on K and the constants CΛ , γΛ such that |∇p (u ◦ Λ)) (z)| ≤ C γ p max {(p + 1)/r, ε−1 }p C(Λ(z))

∀p ∈ N,

z ∈ G .

Proof: It is clear that in Lemma 4.3.3, the dependence of all constants on the mapping functions Λ is in fact through the constants CΛ , γΛ that determine its growth of the derivatives. In order to apply Lemma 4.3.3, we note that the function u ˜ := ∇u satisﬁes ˜ p max {(p+1)/r, ε−1 }p |∇p u ˜(x)| ≤ C max {r−1 , ε−1 } C(x)K

∀p ∈ N0 ,

x∈G

˜ Next, we observe that ∇(u ◦ Λ) = Λ · (∇u) ◦ Λ = for some appropriate K. Λ · (˜ u ◦ Λ). Hence, applying Lemma 4.3.3 we get |∇p+1 (u ◦ Λ)(z)| ≤ Cγ p C(Λ(z)) max {r−1 , ε−1 } max {(p + 1)/r, ε−1 }p for all p ∈ N0 and z ∈ G , which is the desired bound.

2

In the above results involving analytic changes of variables, we always assumed that the transformation is analytic up to the boundary. The important case of a change of variables from polar coordinates to Cartesian coordinates is therefore not covered here, but we refer to Section 6.3 where this issue is addressed.

5. Regularity Theory in Countably Normed Spaces

5.1 Motivation and outline 5.1.1 Motivation In this chapter we prove shift theorems in countably normed spaces. The most important example is Theorem 5.3.10 where we consider the Dirichlet problem (1.2.1). Analogous results for other kinds of boundary conditions, i.e., Neumann problems and transmission problems, are proved in Propositions 5.4.5, 5.4.8, 5.4.7. Such shift theorems have the following structure: If the right-hand side 0 2 f ∈ Bβ,E , then the solution uε of (1.2.1) is in the countably normed space Bβ,E . J Here, β = (β1 , . . . , βJ ) ∈ [0, 1) is a vector of numbers associated with the vertices Aj of the curvilinear polygon and E is the smallest characteristic length scale of solution uε . In the ensuing two subsections, we will motivate our notion of smallest characteristic length scale E and then outline the key steps of the proof of this shift theorem in countably normed spaces. Smallest characteristic length scale. The characteristic length scale depends on the size of ε relative to the coeﬃcients A, b, and c in (1.2.1). To see this, let us consider a one-dimensional example: −ε2 u + bu + cu = f

on (0, 1),

u(0) = u(1) = 0.

For constant b ∈ R, c ≥ 0, the solution of this problem is given by u = upart + α1 eλ1 x + α2 eλ2 x , where upart is a particular solution of the equation and λ1 , λ2 are given by 0 2 b b c λ1,2 = 2 ± + 4 2; 2 ε ε ε here, α1 , α2 are suitable constants determined by the choice of upart and the boundary conditions. Ignoring for the moment the size of the coeﬃcients α1 , α2 , we see that the growth of the derivatives of α1 eλ1 x + α2 eλ2 x is controlled by the size of λ1 and λ2 . Deﬁning now E by √ c |b| −1 E := 2 + + 1, (5.1.1) ε ε it is not diﬃcult to see that

J.M. Melenk: LNM 1796, pp. 169–224, 2002. c Springer-Verlag Berlin Heidelberg 2002

170

5. Regularity Theory in Countably Normed Spaces

E −1 ≤ max {|λ1 |, |λ2 |} + 1 ≤ 2E −1 . Thus, we expect the p-th derivatives of u to be of size E −p . This qualitative consideration neglects of course the contribution of the constants α1 , α2 , and the particular solution upart . In general, we expect—in analogy to the result of Lemma 4.1.1—that the derivatives of the solution u can be controlled by expressions of the form Cγ p max {p + 2, E −1 }p+2 .

(5.1.2)

We note that in the example of Lemma 4.1.1, i.e., b ≡ 0, c ≡ 1, we have E = ε, and (5.1.2) is indeed the estimate obtained in Lemma 4.1.1. The other case of interest is b = 1 together with c = O(1). Then, E = ε2 . Thus, for c = O(1) the cases b = 0 and b = O(1) have two diﬀerent length scales. We remark in passing that our heuristic claims on being able to control the growth of the derivatives of solutions of these one-dimensional singularly perturbed problems by expressions of the form (5.1.2) were rigorously established in [92, 96]. In the present chapter, we show that for solutions of (1.2.1) in curvilinear polygons, the growth of the derivatives can indeed be characterized by the characteristic length scale E deﬁned in (5.3.8), which is the analog of (5.1.1) in two-dimensional problems with variable coeﬃcients. Outline of the proof of Theorem 5.3.10. At the heart of the proof is Proposition 5.3.4, where a result similar to that of Theorem 5.3.10 is shown for a single straight sector, SR (ω). The general case of curved boundaries then follows easily by a mapping argument and Theorem 4.2.20. Let us outline the main ideas of the proof of Proposition 5.3.4 for the simple model equation already discussed in the introduction to Chapter 4: −ε2 ∆uε + uε = f uε = 0

on SR (ω),

(5.1.3a)

on the two sides Γ1 , Γ2 of SR (ω).

(5.1.3b)

We note that the smallest characteristic length scale is E ∼ ε. Let R < R be ﬁxed and β ∈ {β ∈ (0, 1) | β > 1−π/ω}. Then Proposition 5.3.4 states that, if the 0 right-hand side f ∈ Bβ,ε (SR (ω), Cf , γf ), then for any solution uε of (5.1.3) there 2 are Cu , γu > 0 such that uε ∈ Bβ,ε (SR (ω), Cu , γu ); furthermore, the constant Cu has the from Cu = Cgeo Cf + ε∇uε L2 (SR (ω)) + uε L2 (SR (ω)) , where the constant Cf depends only on the right-hand side f and is independent of ε and uε ; the constant Cgeo depends only on R, R , ω, and β. The proof of this result proceeds in several steps. 1. The ﬁrst ingredient of the proof are local regularity results on balls Br or half-balls Gr (near the boundary) as in [98]. Let uε solve −ε2 ∆uε + uε = f on a ball Br , where the right-hand side f satisﬁes ∇p f L2 (Br ) ≤ Cf γ p max {(p + 1)/r, ε−1 }p

∀p ∈ N0

(5.1.4)

5.1 Motivation and outline

171

for some Cf , γ > 0. The local regularity result of Proposition 5.5.1 then states that there exist C, γu > 0 depending only on γ such that rp+2 ∇p+2 uε L2 (Br/2 ) ≤ Cγup max {p + 2, r/ε}p+2 Loc(uε ), where Loc(uε ) := min {1, r/ε}ε∇uε L2 (Br ) + min {1, r/ε}2 uε L2 (Br ) + Cf . A completely analogous result, Proposition 5.5.2, holds on half-balls Gr if homogeneous boundary conditions are imposed on uε on the straight part of ∂Gr . For notational convenience in this outline of the proof of Proposition 5.3.4, however, we will ignore the technical complications introduced at the boundary of SR (ω). 2. The domain SR (ω) is covered by balls Bri /2 (xi ) that have the following key properties: The balls Bri (xi ) have ﬁnite overlap and the radii ri are (essentially) proportional to |xi |, i.e., ri = c|xi | for some ﬁxed c ∈ (0, 1). In order to avoid the regularity study near Γ3 = {(R cos ϕ, R sin ϕ) | 0 < ϕ < ω}, we assume that for some R ∈ (R , R) the balls Bri (xi ) are all contained in BR (0). 0 The characterization of functions f from the space Bβ,ε (SR (ω), Cf , γf ) that we provided in Lemma 4.2.17 allows us to bound Cf , γ > 0 in (5.1.4): The constant γ of (5.1.4) depends only on c and γf for each ball Bri (xi ), and the constant Cf can be bounded by Cf ≤

C(i) ˆ0,β,ε (xi ) Φ

for some C(i) > 0 satisfying ∞

C 2 (i) = Cf < ∞;

i=1

the constant Cf > 0 depends only on Cf , γf , and the covering by balls. 3. As the balls Bri /2 cover SR (ω), we merely sum up the local regularity results on balls (or half-balls near the boundary). Here, it is important to note that, ˆp,β,ε satisfy (4.2.8). Using due to our choice ri ∼ |xi |, the weight functions Φ additional properties of the weight function from Lemma 4.2.2, we obtain p ˆp,β,ε ∇p+2 uε 2 2 Φ L (SR (ω)) ≤ CK

∞

ˆ2p,β,ε (xi )∇p+2 uε 2 2 Φ L (Br

i /2

(xi ))

i=1

≤ CK p max {p + 2, ε−1 }2(p+2) × ∞ 2 ˆ−1,β,ε ∇uε 2 2 ˆ ε2 Φ L (Br (xi )) + Φ0,β,ε uε L2 (Br i

(xi )) i

+ C 2 (i) .

i=1

4. Next, we employ the ﬁnite overlap property of the balls Bri (xi ) to bound the inﬁnite sum by

172

5. Regularity Theory in Countably Normed Spaces ∞ ˆ−1,β,ε ∇uε 2 2 ε2 Φ L (Br

(xi )) i

ˆ0,β,ε uε 2 2 + Φ L (Br

+ C 2 (i)

(xi )) i

i=1 2 ˆ ˆ−1,β,ε ∇uε 2 2 ≤ εΦ L (SR (ω)) + Φ0,β,ε uε L2 (SR (ω)) + Cf .

ˆ−1,β,ε ∇uε 2 2 Using Lemma 4.2.10 we can control the term Φ L (S

R (ω))

to get

ˆp,β,ε ∇p+2 uε L2 (S (ω)) ≤ CK p max {p + 2, ε−1 }(p+2) × Φ R uε H 2,2 (S (ω)) + Cf . β,ε

5. It remains to replace the term uε H 2,2 (S β,ε

R (ω))

R

by expressions involving the

right-hand side f and the energy ε∇uε + uε L2 (SR (ω)) . This is done with the aid of the shift theorem in Proposition 5.3.2. It states that a solution uε of (5.1.3) satisﬁes L2 (SR (ω))

uε H 2,2 (S (ω)) ≤ R β,ε ˆ0,β,ε f L2 (S (ω)) + ε∇uε L2 (S (ω)) + uε L2 (S (ω)) C Φ R R R provided that β > 0 satisﬁes additionally β ∈ (1 − π/ω, 1). Inserting this bound in the previous one gives the desired result. 5.1.2 Outline of Chapter 5 The outline of this chapter is as follows. We start with a brief review of the analytic regularity results of Babuˇska & Guo for the case ε = 1 in Section 5.2. These results are taken from [15] and are phrased in our notation. In [15], the case of Laplace’s equation in a straight sector was analyzed. The extension to problems with variable coeﬃcients was done with a perturbation argument. The main result of this chapter is Theorem 5.3.10, which states a shift theorem in countably normed spaces for equation (1.2.1). Two cases of interest are discussed separately in Corollary 5.3.12 for the case b ≡ 0 and in Corollary 5.3.13 for the case |b| > 0 on Ω. The reason for discussing them separately is that the smallest characteristic length scale is diﬀerent in these two cases: For the case b ≡ 0 in Corollary 5.3.12 the length scale is O(ε) whereas for the case |b| > 0 in Corollary 5.3.13, the length scale is O(ε2 ). We remark at this point that Theorem 5.3.10 represents a slight improvement over the results of Babuˇska & Guo in [14] in the context of curved boundaries: There, for the case ε = 1 solutions u1 of (1.2.1) were shown to be in the space Cβ2 rather than in Bβ2 (see the discussion following Proposition 5.2.4). The key to our improvement is the l invariance of Bβ,ε functions under analytic changes of variables ascertained in Theorem 4.2.20. Theorem 5.3.10 is proved using analytic regularity results on sectors which are then combined to obtain results on (curvilinear) polygons. The analysis on sectors is done in Section 5.3.1. This is done in two steps. The ﬁrst step consists

5.2 Analytic regularity results of Babuˇska and Guo

173

in Proposition 5.3.2, which shows that the solution of (1.2.1) satisﬁes a shift 0,0 , then the theorem in weighted Sobolev spaces: If the right-hand side f is in Hβ,E 2,2 solution u of (1.2.1) is in the space Hβ,E . In the second step this regularity result 0 , then the solution u is is extended to countably normed spaces: If f is in Bβ,E 2 in Bβ,E (Proposition 5.3.4). These results are formulated for straight sectors but can readily be extended to curvilinear sectors with the aid of Theorem 4.2.20. Theorem 5.3.10 is formulated for a Dirichlet problem. In Propositions 5.4.5, 5.4.8, 5.4.7 we formulate analogous results for a variety of other boundary conditions. The main technical tool for obtaining the analytic regularity results in sectors (Proposition 5.3.4) are local regularity results for the solutions of (1.2.1). These local regularity results are provided in Section 5.5. These results are obtained with the techniques of [98]. The novel feature here is that the dependence on the parameter ε and the diameter of the ball R is explicit. We prove four types of local regularity results: interior regularity on balls of radius R, regularity on halfballs of radius R with homogeneous Dirichlet conditions, regularity results for the Neumann problem, and ﬁnally regularity results for transmission problems. These last local regularity results could easily be used to extend Theorem 5.3.10 to problems with piecewise analytic data A, b, c, and f . In fact, our procedure in Chapter 6 shows how this can be done.

5.2 Analytic regularity results of Babuˇ ska and Guo Many elliptic problems arising in practice have piecewise analytic input data such as the coeﬃcients of the diﬀerential equations, the right-hand side, and the geometry of the domain. Thus, the solution is in general piecewise analytic as well, [86,98]. From an approximation point of view this suggests that exponential rates of convergence could be possible with spectral methods. This exponential convergence of spectral methods is not obvious for problems of the form (1.2.1) as analyticity of the solution is not given up to the boundary: The solution has singularities at the vertices Aj as we discussed in Section 1.4.1. Nevertheless, as pointed out in Section 1.4.3, exponential convergence of the hp-FEM is possible in this situation, if the increase of the polynomial degree is combined with an appropriate mesh reﬁnement toward the singularities. For a rigorous proof of the exponential rate of convergence of this scheme, it is essential to control all derivatives of the solution with bounds that are explicit in their dependence on the location in the domain. A framework for controlling such bounds was developed by Babuˇska and Guo with the notion of countably normed spaces, l which coincide with the spaces Bβ,1 introduced in Chapter 4 for the special case ε = 1. In this framework, Babuˇska and Guo proved that solutions of (linear) 2 elliptic problems of second order are elements of countably normed spaces Bβ,1 as we ascertained in Proposition 1.4.2. The original proof of Proposition 1.4.2 was accomplished by Babuˇska and Guo by induction on the order of the deriva2,2 tive. The start of the induction argument was an Hβ,1 regularity result. Such regularity results in weighted Sobolev spaces are intimately linked to Proposi-

174

5. Regularity Theory in Countably Normed Spaces

tion 1.4.1 and go back to a seminal paper by Kondrat’ev, [76, 77]. In the present section, we brieﬂy highlight the key results from [14–17]. We will formulate their results for the Dirichlet problem although analogous results were obtained for the Neumann problem and problems with mixed boundary conditions (see also Sections 5.4). 2,2 We start with the Hβ,1 regularity result for the Laplacian in a straight sector: Proposition 5.2.1. Let SR (ω) be a sector, ω ∈ (0, 2π), and let R < R. Then for β ∈ [0, 1) ∩ (1 − π/ω, 1) there exists C > 0 depending only on ω, β, R, R 0,0 such that the solution u ∈ H01 (SR (ω)) of −∆u = f with f ∈ Hβ,1 (SR (ω)) is in 2 (SR (ω)) and satisﬁes Hloc uH 1 (SR (ω)) + uH 2,2 (S β,1

R (ω))

≤ Cf H 0,0 (SR (ω)) .

(5.2.1)

β,1

Perturbation arguments allow for an extension to the case of variable coeﬃcients: Proposition 5.2.2. Let 0 < R < R, ω ∈ (0, 2π), and A ∈ C 1 (SR (ω), S2> ) with 0 < λmin ≤ A on SR (ω) be given. Then there are β ∈ [0, 1), which depends only 0,0 on ω and A(0), and C > 0 such that for all f ∈ Hβ,1 (SR (ω)) the solution u of −∇ · (A∇u) = f

on SR (ω),

u=0

on ∂SR (ω),

2 is in Hloc (SR (ω)) and satisﬁes

uH 2,2 (S β,1

R (ω))

≤ Cf H 0,0 (SR (ω)) . β,1

0,0 Proof: First, from Lemma 5.3.7 (with ε = 1 there), we get Hβ,1 (SR (ω)) ⊂ H −1 (SR (ω)) so that for every β ∈ [0, 1) there exists Cβ > 0 such that uH 1 (SR (ω)) ≤ Cf H 0,0 (SR (ω)) .

(5.2.2)

β,1

2 Next, by standard elliptic regularity theory, the solution u ∈ Hloc (SR (ω)). We then ﬁx R ∈ (R , R) and claim the existence of C > 0 independent of f such that |x|∇2 uL2 (SR (ω)) ≤ C |x|f L2 (SR (ω)) + ∇uL2 (SR (ω)) . (5.2.3)

To see (5.2.3), we employ elliptic regularity locally in the following way. We choose a covering (see, e.g., Lemma 5.3.1)) of SR (ω) by balls B = {Bi | i ∈ N} with the following properties: 1. 2. 3. 4.

Bi = Bri (xi ) with ri = c|xi | for suitable c ∈ (0, 1); Bi ∩ SR (ω) is either completely contained in SR (ω) or a half ball; card{i | x ∈ Bi } ≤ N for all x ∈ SR (ω) for some ﬁxed N ∈ N; i := Br /2 (xi ) also form a covering of SR (ω). the “stretched” balls B i

5.2 Analytic regularity results of Babuˇska and Guo

175

By elliptic regularity (cf. (5.5.22) of Lemma 5.5.12 for the case of balls Bi ⊂ SR (ω) and (5.5.27) of Lemma 5.5.15 for the case of Bi ∩ SR (ω) being a half ball) we have for a constant C > 0 independent of i ri2 ∇2 uL2 (Bi ∩S (ω)) ≤ C ri2 f L2 (Bi ∩SR (ω)) + ri ∇uL2 (Bi ∩SR (ω)) . R

Dividing by ri and using ri ≤ C inf x∈Bi |x| ≤ C supx∈Bi |x| ≤ Cri , we get |x|∇2 uL2 (Bi ∩SR (ω)) ≤ C |x|f L2 (Bi ∩SR (ω)) + ∇uL2 (Bi ∩SR (ω)) . Squaring this last estimate and summing on i ∈ N gives with the overlap properties of the balls |x|∇2 uL2 (SR (ω)) ≤ C |x|f L2 (SR (ω)) + ∇uL2 (SR (ω)) , which is (5.2.3). Next, it is easy to see with aﬃne changes of variables that for every ﬁxed A˜ ∈ S2> , 0,0 there exist β ∈ [0, 1) and C > 0 such that for every f˜ ∈ Hβ,1 (SR (ω)), the ˜ = f˜ satisﬁes H 1 (SR (ω))-solution to −∇ · A∇u 0

uH 2,2 (SR (ω)) ≤ Cf˜H 0,0 (SR (ω)) . β,1

(5.2.4)

β,1

This puts us in position to prove Proposition 5.2.2 by a perturbation argument. Denoting A˜ := A(0), we calculate ˜ −∇ · A∇u = f˜ a.e. on SR (ω), (5.2.5) where

f˜ = f − ∇ · (A˜ − A) ∇u − (A˜ − A) : ∇2 u.

˜ ≤ C|x|. Hence, we get In view of A ∈ C 1 (SR (ω)) there is C > 0 with |A(x) − A| by combining (5.2.4) with (5.2.3) and exploiting β < 1 uH 2,2 (S β,1

R (ω))

≤ Cf˜H 0,0 (S (ω)) R β,1 ≤ C f H 0,0 (SR (ω)) + ∇uL2 (SR (ω)) + |x|∇2 uL2 (SR (ω)) β,1

≤ Cf H 0,0 (SR (ω)) . β,1

2 Remark 5.2.3 Note that in the proof of Proposition 5.2.2, the assumption that A be Lipschitz is exploited in an essential way—jumping coeﬃcients need a diﬀerent treatment. In that case, we need the analog of Proposition 5.2.1 for the case of a diﬀerential operator with piecewise constant coeﬃcients. Such a result is provided in Proposition A.2.1. Most of the details for this case are worked out in Chapter 6 below.

176

5. Regularity Theory in Countably Normed Spaces

By induction on the order of the derivatives, akin to the way Morrey proceeds in [98] (see also Section 5.5 ahead), one can obtain bounds on all derivatives of the solutions of elliptic equations with analytic coeﬃcients. Prototypical is the following result (a proof can be found below in Theorem 5.3.10). Proposition 5.2.4. Let SR (ω) be a sector, R < R, A ∈ A(SR (ω), S2> ) with 0 < λmin ≤ A on SR (ω) for some ﬁxed λmin . Then there exists β ∈ [0, 1), 0 which depends only on ω, A(0), such that for f ∈ Bβ,1 (SR (ω), Cf , γf ) (with Cf , 1 γf > 0) the solution u ∈ H0 (SR (ω)) of −∇ · (A∇u) = f on SR (ω) satisﬁes 2 (SR (ω), Cu , γu ) for some Cu , γu . u ∈ Bβ,1 The case of curvilinear polygons was addressed by Babuˇska and Guo in [14]. In order to characterize the solutions in curvilinear sectors S, they introduced the countably normed spaces 2,2 ˆp−1,β,1 ∇p uL∞ (S) ≤ Cp!γ p Cβ2 (S, C, γ) := {u ∈ Hβ,1 | Φ

∀p ∈ N}.

In [14], the following regularity result was shown: Proposition 5.2.5. Let S be a curvilinear sector, A ∈ A(S, S2> ) with 0 < λmin ≤ A on SR (ω). Then there exists β ∈ [0, 1) with the following property: 0 (S, Cf , γf ) (with Cf , γf > 0), the For any neighborhood U of Γ3 and f ∈ Bβ,1 1 solution u ∈ H0 (S) of −∇ · (A∇u) = f on S satisﬁes u ∈ Cβ2 (S \ U, C, γ) for some C, γ > 0. 2 2 In [14] it is also shown that Bβ,1 ⊂ Cβ2 ⊂ Bβ+δ,1 for all δ > 0. Hence, Propo2 sition 5.2.5 implies that the solution u ∈ Bβ+δ,1 for all δ > 0. This leads the 2 authors to raise in [14] the question whether the solution u is in fact in Bβ,1 for curvilinear polygons as well. Theorem 5.3.10 below answers this question in the aﬃrmative. The key ingredient for this assertion is Theorem 4.2.20, which l states that the spaces Bβ,ε are invariant under analytic changes of variables. This invariance allows us to restrict our attention to the case of straight sectors and then infer the case of curvilinear polygons by a mapping argument.

Remark 5.2.6 Proposition 5.2.4 is a form of an elliptic shift theorem in countl ably normed spaces Bβ,1 , which is a class of analytic functions. Other forms of elliptic shift theorems exist in the literature for classes of functions larger than those of analytic functions. We mention here in particular the notion of Gevrey regularity, for which we refer the reader to [30] and the reference there.

5.3 Analytic regularity: Dirichlet problems For the sake of deﬁniteness, we consider in this section the case of homogeneous Dirichlet boundary conditions. The similar cases of of homogeneous Neumann boundary conditions or transmission conditions are treated in Section 5.4.

5.3 Analytic regularity: Dirichlet problems

177

5.3.1 Analytic regularity in sectors In this subsection, we prove analytic regularity results on straight sectors. We start with the following covering lemma. Lemma 5.3.1 (Covering). Let 0 < R < R, ω ∈ (0, 2π). Then there exist constants c ∈ (0, 1), N ∈ N and a covering B = {Bi | i ∈ N} of SR (ω) by balls Bi with the following properties: 1. B covers SR (ω), i.e., SR (ω) ⊂ ∪i Bi ; 2. the balls have the form Bi = Bri (xi ) with ri = c|xi |; 3. the balls Bi satisfy a ﬁnite overlap condition, i.e., there exists N ∈ N such that card{i ∈ N | x ∈ Bi } ≤ N ∀x ∈ SR (ω); 4. the sets Di := Bi ∩ SR (ω) satisfy the following dichotomy: either Di is a ball (i.e., Di = Bi ) or Di is a half-ball (i.e., the center xi is on one of the straight parts of ∂SR (ω) and |xi | + ri < R); i := Bcr /2 (xi ) also form a covering of SR (ω). 5. the “stretched” balls B i Proof: The existence of such coverings is geometrically clear. In order to ensure that the sets Di indeed satisfy the above dichotomy, one has to choose the constant c suﬃciently small in dependence on the ratio of R to R . 2 On sectors SR (ω), we are interested in the regularity of solutions u to the following equation −ε2 ∇ · (A(x)∇u) + b(x) · ∇u + c(x)u = f (x) on SR (ω), u = 0 on ∂SR (ω) \ ∂BR (0).

(5.3.6a) (5.3.6b)

We assume that ε ∈ (0, 1] and that the coeﬃcients of (5.3.6) are analytic; that is, A ∈ A(SR (ω), S2> ), b ∈ A(SR (ω)), c ∈ A(SR (ω)) satisfy, for some Cb , γb , Cc , γc ≥ 0, p ∇p AL∞ (SR (ω)) ≤ CA γA p! ∀p ∈ N0 , 0 < λmin ≤ A on SR (ω), ∇p bL∞ (SR (ω)) ≤ Cb γbp p! ∀p ∈ N0 , p p ∇ cL∞ (SR (ω)) ≤ Cc γc p! ∀p ∈ N0 .

Next, we deﬁne for ε > 0 the relative diﬀusivity E as √ Cc Cb E −1 := 2 + + 1. ε ε

(5.3.7a) (5.3.7b) (5.3.7c) (5.3.7d)

(5.3.8)

Note that ε and E satisfy trivially the relationships Cb ≤ E −1 , ε2

Cc ≤ E −2 . ε2

(5.3.9)

2,2 H 1 solutions u of (5.3.6) are in fact in the space Hβ,E (SR (ω)) for some β ∈ (0, 1):

178

5. Regularity Theory in Countably Normed Spaces

Proposition 5.3.2 (Shift theorem). Assume that the coeﬃcients A, b, c satisfy (5.3.7). Then there exists β ∈ [0, 1), which depends only on ω and A(0), and there exists C > 0 such that each solution u ∈ H 1 (SR (ω)) of (5.3.6) with 0,0 right-hand side f ∈ Hβ,E (SR (ω)) satisﬁes ˆ0,β,E ∇2 uL2 (S (ω)) ≤ Φ R 2 −2 2 ˆ (E/ε) Φ0,β,E f L2 (SR (ω)) + E∇uL2 (SR (ω)) + Cc (E/ε) uL2 (SR (ω)) . CE Remark 5.3.3 If A(0) = Id, then we may choose β ∈ (1 − π/ω, 1) ∩ [0, 1). Proof: In this proof, we will use the shorthand Sr for the sectors Sr (ω). Let B = {Bi | i ∈ N} be the covering of SR given by Lemma 5.3.1. Recall that the i have radii balls Bi have radii c|xi |, that the corresponding “stretched” balls B c/2|xi |, and that the sets Bi ∩ SR are either full balls or half-balls. Deﬁne the index set IE := {i ∈ N | Bi ∩ SE = ∅}. Elementary geometric considerations show i ∈ IE =⇒ |xi | < (1 − c)−1 E,

i ∈ N \ IE =⇒ |xi | ≥ (1 − c)−1 E.

(5.3.10)

i satisfy It readily follows that the stretched balls B 1 1 i ⊂ S(1+c/2)(1−c)−1 E , i ⊂ R2 \ S(1−c/2)(1−c)−1 E . B B i∈IE

i∈N\IE

i ⊃ SR that Setting h := min {R , (1 + c/2)(1 − c)−1 E}, we infer from ∪i∈N B 1 i ⊃ SR \ S(1+c/2)(1−c)−1 E =: SR \ Sh , (5.3.11) B i∈N\IE

Introducing H := min {R, 2h}, we readily ascertain the existence of C > 0 independent of E such that ˆ0,β,E (x) ≤ 1 C −1 ≤ min Φ x∈Bi

∀i ∈ N \ IE ,

C −1 E ≤ |xi |

∀i ∈ N \ IE , |x| ≤ CE ∀x ∈ SH , β β |x| |x| −1 ˆ C ≤ Φ0,β,E (x) ≤ C E E

(5.3.12) (5.3.13) (5.3.14)

∀x ∈ SH .

(5.3.15)

ˆ0,β,E ∇2 u in two steps by ﬁrst estimating the L2 We estimate the L2 norm of Φ i . In order to estimate the norm over Sh and then over SR \ Sh ⊂ ∪i∈N\IE B L2 norm over Sh , let χ be a smooth cut-oﬀ function satisfying χ ≡ 1 on Bh (0), χ ≡ 0 on R2 \BH (0), and ∇j χL∞ (R2 ) ≤ CE −j , j ∈ {0, 1, 2}. Next, we calculate that uχ satisﬁes on SR −∇ · (A∇(uχ)) = χε−2 [f − b · ∇u − cu] − 2∇χ · (A∇u) − u∇ · (A∇χ), uχ = 0 on ∂SR ,

5.3 Analytic regularity: Dirichlet problems

179

where we used the fact that u satisﬁes (5.3.6). Applying Proposition 5.2.2 yields the existence of β ∈ [0, 1), C > 0 such that rβ ∇2 (uχ)L2 (SR ) ≤ Crβ ∇ · (A∇(χu))L2 (SR ) . Dividing both sides by E β , using (5.3.15), and the support properties of χ yields ˆ0,β,E ∇2 uL2 (S ) ≤ CΦ ˆ0,β,E ∇ · (A∇(χu))L2 (S ) . Φ H h Expanding the right hand-side and observing (5.3.9), (5.3.14) together with ˆ0,β,E ≤ 1 gives Φ ˆ0,β,E ∇2 uL2 (S ) ≤ C ε−2 Φ ˆ0,β,E f L2 (S ) + Cb ε−2 ∇uL2 (S ) Φ H H h + Cc ε−2 uL2 (SH ) + E −1 ∇uL2 (SH ) + E −2 uL2 (SH ) ˆ0,β,E f L2 (S ) + E −1 ∇uL2 (S ) ≤ C ε−2 Φ (5.3.16) H H + Cc (E/ε)2 E −2 uL2 (SH ) + E −2 uL2 (SH ) . Bounding further |x| ≤ CE for x ∈ SH , we get in view of Lemma 4.2.12 E −2 uL2 (SH ) ≤ CE −1

1 uL2 (SH ) ≤ CE −1 ∇uL2 (SH ) . |x|

Inserting this in (5.3.16) ﬁnally yields ˆ0,β,E ∇2 uL2 (S ) ≤ Φ (5.3.17) h ˆ0,β,E f L2 (S ) + E −1 ∇uL2 (S ) + E −2 Cc (E/ε)2 uL2 (S ) . C ε−2 Φ H H H We now turn to estimating Φ0,β,E ∇u2 L2 (SR \Sh ) . For each ball Bi , u satisﬁes −∇·(A∇u) = ε−2 [f − b∇u − cu] on Bi ∩SR . As in the proof of Proposition 5.2.2, we appeal to elliptic regularity, viz., we use (5.5.22) of Lemma 5.5.12 for the case of balls Bi completely contained in SR (ω) and (5.5.27) of Lemma 5.5.15 for the case of Bi ∩ SR (ω) being a half ball to bound with a C > 0 independent of i ri2 ∇2 uL2 (Bi ∩SR ) ≤ C ri2 ε−2 f − b∇u − cuL2 (Bi ∩SR ) + ri ∇uL2 (Bi ∩SR ) . For i ∈ N \ IE , we may employ (5.3.12), (5.3.13) and additionally use (5.3.9) to get for all i ∈ N \ IE : ˆ0,β,E ∇2 u 2 (5.3.18) Φ L (Bi ∩SR ) ≤ −2 ˆ0,β,E f L2 (B ∩S ) + E −1 ∇uL2 (B ∩S ) + E −2 Cc (E/ε)2 uL2 (B ∩S ) . C ε Φ i i i R R R Squaring this last estimate and summing on i ∈ N \ IE gives in view of (5.3.11) and the fact that the sets Bi satisfy an overlap condition:

180

5. Regularity Theory in Countably Normed Spaces

ˆ0,β,E ∇2 uL2 (S \S ) ≤ Φ h R −2 ˆ C ε Φ0,β,E f L2 (SR ) + E −1 ∇uL2 (SR ) + E −2 Cc (E/ε)2 uL2 (SR ) . (5.3.19) Combining (5.3.17), (5.3.19) and observing that ε−2 = (E/ε)2 E −2 concludes the proof. 2 We can now formulate an analytic regularity result for the solution u of (5.3.6) 0 for right-hand sides f from the countably normed space Bβ,E : Proposition 5.3.4. Let R > 0, ω ∈ (0, 2π), and coeﬃcients A, b, c satisfying (5.3.7) be given. Then there exists β ∈ [0, 1) with the following properties: For ε ∈ (0, 1], let E be given by (5.3.8). Let u ∈ H 1 (SR (ω)) be a solution of (5.3.6) 0 (SR (ω), Cf , γf ) for some Cf , γf . Then for every for a right-hand side f ∈ Bβ,E R ∈ (0, R) there are C, K > 0 independent of ε and Cf such that for all p ∈ N0 ˆp,β,E ∇p+2 uL2 (S (ω)) ≤ Φ R CK p max {p, E −1 }p+2 (E/ε)2 Cf + E∇uL2 (SR (ω)) + Cc (E/ε)2 uL2 (SR (ω)) . Remark 5.3.5 We note that the coercivity condition (1.2.2e) is not explicitly required in Propositions 5.3.2, 5.3.4. These propositions are regularity assertions for H 1 -solution, whose existence is part of the assumptions. Closely connected with this fact is the observation that the data A, b, c, f , and ε need not be real-valued: The data b, c, f , and ε may be complex-valued in which case Propositions 5.3.2, 5.3.4 still hold provided that ε in the deﬁnition of E and in the statement of the propositions is replaced with |ε|. If A(0) = Id, then any β ∈ (0, 1) ∩ (1 − π/ω, 1) may be chosen. We excluded the case β = 0 for technical convenience: Inspection of the proof shows that the β = 0 requires control of |x|−1 ∇uL2 (SR (ω)∩BE (0)) by additional arguments. Proof: Let R ∈ (R , R). We use the shorthand notation S = SR (ω), S = SR (ω). Let B = {Bi | i ∈ N} be the covering of S by balls Bi = Bri (xi ), ri = i the “stretched balls” B i = Br /2 (xi ) c|xi |, given by Lemma 5.3.1. Denote by B i as introduced in Lemma 5.3.1. An application of Lemma 4.2.17 with the covering B yields the existence of C, K > 0 independent of ε and i such that with the numbers C(i) as given by Lemma 4.2.17 we have C 2 (i) ≤ CCf2 (5.3.20) i∈N

together with

5.3 Analytic regularity: Dirichlet problems

1 ˆp,β,E (xi )∇p f L2 (S∩B ) Φ i ˆ Φp,β,E (xi ) 1 CK p C(i) max {p + 1, E −1 }p ≤ ˆ Φp,β,E (xi )

∇p f L2 (S∩Bi ) =

181

(5.3.21)

1 1 CK p C(i) max {p + 1, E −1 }p ˆ ˆ Φ0,β,E (xi ) Φp,0,E (xi ) 1 ≤ CK p C(i) max {(p + 1)/|xi |, E −1 }p , ˆ0,β,E (xi ) Φ ≤

where, in the last estimate, we appealed to (4.2.6) of Lemma 4.2.2. We note that for all i, the set Bi ∩ S is either a full ball or a half ball in which case u vanishes on the straight part of ∂(Bi ∩ S). Hence Propositions 5.5.1, 5.5.2 yield the existence of C, K > 0 independent of ε such that r p+2 i

2

∇p+2 uL2 (Bi ∩S) ≤ Cu (i)K p+2 max {(p + 3), ri /E}p+2

∀p ∈ N0 ,

where we abbreviate Cu (i) := min{1, ri /E}E∇uL2 (Bi ∩S) , 2

2

+ min{1, ri /E} (E/ε)

Cc uL2 (Bi ∩S) +

C(i) ˆ0,β,E (xi ) Φ

.

We bound with (4.2.8) of Lemma 4.2.2 ˆp,β,E ∇p+2 2 2 ≤ CK p Φ L (S )

∞

ˆ2p,β,E (xi )∇p+2 u2 2 Φ L (B

i ∩S

)

i=1

≤ CK 2(p+2)

∞

−2(p+2)

Φ2p,β,E (xi )ri

max {p + 3, ri /E}2(p+2) Cu2 (i).

i=1

We estimate further using (4.2.7) of Lemma 4.2.2 −2(p+2)

Φ2p,β,E (xi )ri

max {p + 3, ri /E}2(p+2) ˆ2 (xi ) max {(p + 1)/ri , E −1 }2p max {(p + 1)/ri , E −1 }4 (xi )Φ

ˆ20,β,E ≤ CK 2(p+2) Φ

p,0,E

ˆ20,β,E (xi ) max {(p + 1)/ri , E −1 }4 , } Φ ˆ20,β,E (xi ) max {1/ri , E −1 }4 . ≤ K 2p max {p + 1, E −1 }2p Φ

≤K

2p

max {p + 1, E

−1 2p

Hence, we obtain ˆp,β,E ∇p+2 u2 2 ≤ Φ L (S ) CK 2p max {p + 1, E −1 }2p

∞ i=1

ˆ20,β,E (xi ) max {1/ri , E −1 }4 Cu2 (i). Φ

(5.3.22)

182

5. Regularity Theory in Countably Normed Spaces

ˆ1,0,E (xi ) and using Φ ˆ2,β,E ≤ Φ ˆ1,β,E , Φ ˆ2,0,E ≤ Recognizing min {1, ri /E} ≤ C Φ 2 ˆ1,0,E ) , we can estimate (Φ ˆ1,β,E (xi )E∇uL2 (B ∩S) + Cc (E/ε)2 Φ ˆ1,β,E (xi )uL2 (B ∩S) ˆ0,β,E (xi )Cu (i) ≤ C Φ Φ i i ˆ2 (xi ) . + (E/ε)2 C(i)Φ 1,0,E Upon setting ˆ1,β,E (xi ), Wi,β := max {1/ri , E −1 }2 Φ ˆ21,0,E (xi ), Wi,0 := max {1/ri , E −1 }2 Φ we see that we have ∞ ∞ 2 ˆ20,β,E (xi ) max {1/ri , E −1 }4 Cu2 (i) ≤ E 2 Wi,β ∇u2L2 (Bi ∩S) Φ i=1

i=1

+ Cc2 (E/ε)4

∞

2 Wi,β u2L2 (Bi ∩S) + (E/ε)4

i=1

∞

2 Wi,0 C 2 (i).

i=1

Estimating these three sums with the aid of Lemma 5.3.6, and inserting the result into (5.3.22) gives ˆp,β,E ∇p+2 u2 2 ≤ CK 2p max {p + 1, E −1 }2p × Φ L (S ) ˆ Φ0,β,E ∇2 u2L2 (S) + E −2 ∇u2L2 (S)

+ Cc2 (E/ε)4 E −2 ∇u2L2 (S) + Cc2 (E/ε)4 E −4 u2L2 (S) + Cf2 (E/ε)4 E −4 .

Upon simplifying Cc (E/ε)2 ≤ 1 and using Proposition 5.3.2 to bound the term ˆ0,β,E ∇2 uL2 (S) (note that S = SR (ω) with R < R) we arrive at Φ ˆp,β,E ∇p+2 u2 2 ≤ CK 2p max {p + 1, E −1 }2p E −4 × Φ L (S ) Cf2 (E/ε)2 + E 2 ∇u2L2 (S) + Cc2 (E/ε)2 u2L2 (S) . Bounding max{p + 1, E −1 }2p E −4 ≤ max{p + 1, E −1 }2(p+2) ﬁnishes the proof. 2 Lemma 5.3.6. Let R > R > 0, ω ∈ (0, 2π), β ∈ (0, 1). Let B = {Bi | i ∈ N}, Bi = Bc|xi | (xi ), be the covering of SR (ω) given by Lemma 5.3.1. Then there 2,2 exists C > 0 such that for every E ∈ (0, 1] and every u ∈ Hβ,E (SR (ω)) 2 Wi,β ∇u2L2 (Bi ∩SR (ω)) ≤ i∈N

i∈N

i∈N

−4 ˆ ˆ0,β,E ∇2 u2 2 Φ0,β,E ∇u2L2 (SR (ω)) , E −2 Φ L (SR (ω)) + E 2 ˆ2 ˆ0,β,E ∇u2 2 Wi,β Φ1,0,E (xi )∇u2L2 (Bi ∩SR (ω)) ≤ CE −4 Φ L (SR (ω)) ,

2 −4 2 ˆ0,β,E ∇u2 2 Wi,β u2L2 (Bi ∩SR (ω)) ≤ C E −2 Φ + E u 2 L (SR (ω)) L (SR (ω)) ,

5.3 Analytic regularity: Dirichlet problems

183

where we set

ˆ1,β,E (xi ). Wi,β := max {1/|xi |, E −1 }2 Φ Additionally, if numbers C(i) satisfy i∈N C 2 (i) ≤ Cf2 < ∞, then, upon setting ˆ2 (xi ), Wi,0 := max {1/|xi |, E −1 }2 Φ 1,0,E

2 Wi,0 C 2 (i) ≤ CE −4 Cf2 .

i∈N

Proof: We abbreviate S := SR (ω). With Lemma 4.2.2, we get ˆ1,β,E (xi ) ≤ CE −2 Wi,β = max {1/|xi |, E −1 }2 Φ

ˆ1,β,E (xi ) Φ ˆ−1,β,E (xi ). ≤ CE −2 Φ ˆ2,β,E (xi ) Φ

Hence, we get exploiting (4.2.8) of Lemma 4.2.2 2 ˆ−1,β,E ∇u2 2 Wi,β ∇u2L2 (Bi ∩S) ≤ CE −4 Φ L (Bi ∩S) .

On summing over i, we obtain using the overlap properties of the covering B 2 ˆ−1,β,E ∇u2 2 . Wi,β ∇u2L2 (Bi ∩S) ≤ CE −4 Φ L (S) i∈N 2 2 ˆ ˆ ˆ−1,β,E ∇u2 2 Writing Φ L (S) = Φ−1,β,E ∇uL2 (S∩B2E (0)) +Φ−1,β,E ∇uL2 (S\B2E (0)) ˆ−1,β,E (x) = 1 for and using Corollary 4.2.11 for the ﬁrst integral and the fact Φ x ∈ S \ B2E (0) we get 2 −4 ˆ0,β,E ∇2 u2 2 Wi,β ∇u2L2 (Bi ∩S) ≤ CE −2 Φ ∇u2L2 (S) , L (S∩B2E (0)) + CE i∈N

from which the desired bound for the sum of gradients follows. The second ˆ1,0,E (xi ) ≤ CE −2 Φ ˆ0,β,E (xi ). estimate follows easily since Wi,β Φ For the third estimate, we proceed completely analogously to arrive at 2 ˆ−1,β,E u2 2 . Wi,β u2L2 (Bi ∩S) ≤ CE −4 Φ L (S) i∈N 2 2 ˆ ˆ ˆ−1,β,E u2 2 Writing Φ L (S) = Φ−1,β,E uL2 (S∩B2E (0)) + Φ−1,β,E uL2 (S\B2E (0)) and ˆ−1,β,E (x) = 1 using Part (ii) of Lemma 4.2.10 for the ﬁrst integral and the fact Φ for x ∈ S \ B2E (0) we get 2 −4 ˆ0,β,E ∇u2 2 Wi,β u2L2 (Bi ∩S) ≤ CE −2 Φ u2L2 (S) , L (S∩B2E (0)) + CE i∈N

which is the desired bound. The last claim of the lemma follows from the fact that by Lemma 4.2.2 ˆ4 (xi ) ≤ CE −4 . max {1/|xi |, E −1 }4 Φ 2 1,0,E

184

5. Regularity Theory in Countably Normed Spaces

5.3.2 Regularity in curvilinear polygons In the present subsection Ω denotes a ﬁxed curvilinear polygon as deﬁned in l,m Section 1.2. For β ∈ [0, 1)J and ε > 0 we can deﬁne the spaces Hβ,ε (Ω) and the l countably normed spaces Bβ,ε (Ω, C, γ) analogously to the way they are deﬁned on sectors as the completions of the space C ∞ (Ω) under weighted norms with ˆp,β,ε , where weight functions Φp,β,ε instead of Φ J Φp,β,ε (x) := Πj=1 Φp,βj ,ε (x − Aj ).

(5.3.23)

1 Our variational setting is based on the energy spaces H0,ε (Ω): For ε > 0, let 1 1 H0,ε (Ω) be the space H0 (Ω) equipped with the energy norm · ε given by (1.2.7), i.e., u2ε := ε2 ∇u2L2 (Ω) + u2L2 (Ω) . We have

Lemma 5.3.7. Let Ω be a curvilinear Then for all ε > 0 and all 1 polygon. 0,0 β ∈ [0, 1]J there holds Hβ,ε (Ω) ⊂ H0,ε (Ω) . Furthermore, there exists C > 0 independent of ε ∈ (0, 1] such that f (H 1

0,ε (Ω)

0,0 (Ω) ) ≤ Cf H1,ε

0,0 ∀f ∈ H1,ε (Ω).

Proof: From (4.2.6) of Lemma 4.2.2 we get the existence of C > 0 independent of ε such that ε ε ≤ C 1 + , r = dist(x, ∂Ω). Φ−1 (x) ≤ C 1 + 0,1,ε dist(x, {Aj | 1 ≤ j ≤ J}) r 1 We then calculate for v ∈ H0,ε (Ω) f v dx ≤ Φ0,1,ε f L2 (Ω) Φ−1 vL2 (Ω) 0,1,ε Ω 1 ≤ CΦ0,1,ε f L2 (Ω) vL2 (Ω) + ε vL2 (Ω) r ≤ CΦ0,1,ε f L2 (Ω) vε ,

where, in the last step we employed the embedding in weighted Sobolev spaces 1r vL2 (Ω) ≤ C∇uL2 (Ω) of Lemma 4.2.12. 2 We now consider the following boundary value problem: −ε2 ∇ · (A(x)∇u) + b(x) · ∇u + cu = f u=0

on Ω, on ∂Ω.

(5.3.24a) (5.3.24b)

The coeﬃcients A, b, c are assumed to satisfy: A ∈ W 1,∞ (Ω, S2> ), b ∈ C 0 (Ω) and b piecewise in W 1,∞ , c ∈ L∞ (Ω). Furthermore, we assume that 0 < λmin ≤ A(x)

on Ω,

bL∞ (Ω) ≤ Cb , cL∞ (Ω) ≤ Cc , 1 0<µ ≤ c− ∇·b 2

(5.3.25a) (5.3.25b) (5.3.25c)

a.e. on Ω.

(5.3.25d)

5.3 Analytic regularity: Dirichlet problems

185

As in the case of solutions on sectors, is convenient to introduce the notion of the relative diﬀusivity E via E −1 :=

Cb Cc + 1. + ε2 ε

The coercivity statement of Lemma 1.2.2 implies that the linear form Bε is 1 1 (Ω) × H0,ε (Ω). Hence, the variational coercive and continuous on the spaces H0,ε formulation 1 Find uε ∈ H0,ε (Ω) s.t. Bε (uε , v) = F (v)

1 ∀v ∈ H0,ε (Ω)

1 has a unique solution for any right-hand side F ∈ H0,ε (Ω) . In particular, as in the proof of Lemma 1.2.2, we get the existence of C > 0 independent of ε such that uε ≤ CF (H 1 (Ω)) . (5.3.26) 0,ε

This observation allows us to formulate the following theorem. Theorem 5.3.8 (shift theorem in curvilinear polygons). Let Ω be a curvilinear polygon, A, b, c satisfy (5.3.25). Let E be deﬁned by (5.3.8). Then 0,0 (Ω) the there exists C > 0 independent of ε ∈ (0, 1] such that for each f ∈ H1,ε (weak) solution u of (5.3.24) satisﬁes uε ≤ Cf H 0,0 (Ω) . 1,ε

Furthermore, if the data A, b, c satisfy additionally (1.2.2a), (1.2.2b), (1.2.2c), 0,0 then there exists β ∈ [0, 1)J such that for f ∈ Hβ,E (Ω) the solution u satisﬁes 2 u ∈ Hloc (Ω), and there exists C > 0 independent of ε ∈ (0, 1] and f such that Φ0,β,E ∇2 uL2 (Ω) ≤ 2 CE −2 (E/ε) f H 0,0 (Ω) + (E/ε) ε∇uL2 (Ω) + Cc (E/ε)2 uL2 (Ω) . β,E

Proof: The ﬁrst assertion of the theorem follows readily from the coercivity statement of Lemma 1.2.2 and (5.3.26). For polygons, the second assertion follows from Proposition 5.3.2. For the general case of curvilinear polygons, we require additionally a mapping argument. These arguments are provided by Corollary 4.2.21. Strictly speaking, Corollary 4.2.21 is formulated for analytic functions; inspection of the proof, however, shows that an analogous result holds 2,2 in the present Hβ,E -setting. 2 Remark 5.3.9 In typical applications, one has E ≤ Cε (cf. the deﬁnition of E in (5.3.8)). Lemma 4.2.3 then implies the existence of C > 0 independent of ε and β ∈ [0, 1]J such that Φ0,1,ε ≤ Φ0,β,ε ≤ CΦ0,β,E .

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5. Regularity Theory in Countably Normed Spaces

0,0 Theorem 5.3.8 then implies that for f ∈ Hβ,E (Ω) we have

Φ0,β,E ∇2 uL2 (Ω) ≤ CE −2 (E/ε)f H 0,0 (Ω) . β,E

If the data A, b, c, f are analytic on Ω, then the variational solution u of (5.3.24) is analytic on Ω as well. We assume that the data A ∈ A(Ω, S2> ), b ∈ A(Ω, R2 ), c ∈ A(Ω), satisfy (5.3.25) and (1.2.2). Then we can then formulate Theorem 5.3.10 (analytic regularity in curvilinear polygons). Let Ω be a curvilinear polygon, and let A, b, c satisfy (1.2.2). Then there exists β ∈ [0, 1)J depending only on Ω and A such that the following holds: For each f ∈ 0 Bβ,E (Ω, Cf , γf ) the solution u of (5.3.24) is analytic on Ω, and there are C, K > 0 depending only on γf , Ω, and the constants of (1.2.2) such that ∀p ∈ N0 uε ≤ CCf , Φp,β,E ∇p+2 uL2 (Ω) ≤ CK p+2 max {p, E −1 }p+2 × Cf (E/ε)2 + (E/ε)ε∇uL2 (Ω) + Cc (E/ε)2 uL2 (Ω) . Proof: The proof follows readily by combining Theorem 5.3.8, Proposition 5.3.4, and mapping arguments provided by Corollary 4.2.21. 2 Remark 5.3.11 Several comment concerning Theorem 5.3.10 are in order. Firstly, piecewise analyticity of the data A, b, c, f is suﬃcient. Then, of course, the solution u is piecewise analytic and the assertion concerning the growth of the derivatives of u has to be understood in a piecewise sense. Secondly, as observed in Remark 5.3.5, it is not necessary to require the coercivity condition (1.2.2e) and that the coeﬃcients b, c, and ε be real-valued for the weighted H 2 bound in Theorem 5.3.8 and the second bound in the statement of Theorem 5.3.10 to hold. These bounds also hold if b, c, f , and ε are complex-valued provided that ε is replaced with |ε| in these statements. Theorem 5.3.10 covers two particular cases of interest: the case of a reactiondiﬀusion equation where Cb = 0 and Cc = O(1) on the one hand and the case of a convection-diﬀusion equation where Cb = O(1). In the former case, E = O(ε) whereas in the latter case E = O(ε2 ). For future reference, we collect this observation in the following two corollaries: Corollary 5.3.12 (reaction-diﬀusion). Let Ω be a curvilinear polygon, A, b, c satisfy (1.2.2) and assume that b ≡ 0, g ≡ 0. Then there exists β ∈ [0, 1)J 0 depending only on Ω and A such that for every f ∈ Bβ,ε (Ω, Cf , γf ) there exist C, K > 0 depending only on Ω, γf , and the constants of (1.2.2) such that the 2 (Ω, CCf , K). solution u of (1.2.11) is in Bβ,ε

5.3 Analytic regularity: Dirichlet problems

187

Corollary 5.3.13 (convection-diﬀusion). Let Ω be a curvilinear polygon, A, b, c satisfy (1.2.2), g ≡ 0, and assume that Cb ≥ c > 0. Then there exists 0 β ∈ [0, 1)J depending only on Ω and A such that for every f ∈ Bβ,ε 2 (Ω, Cf , γf ) there exist C, K > 0 depending only on Ω, γf , c, and the constants of (1.2.2) such that the solution u of (1.2.1) satisﬁes: uL2 (Ω) + ε∇uL2 (Ω) ≤ CCf , Φp,β,ε2 ∇p+2 uL2 (Ω) ≤ CCf K p+2 max {p, ε−2 }p+2 ε

∀p ∈ N0 .

So far, we assumed homogeneous boundary conditions. The regularity theory can be extended to piecewise analytic (Dirichlet) boundary data. Theorem 5.3.14 (non-homogeneous Dirichlet conditions). Let Ω be a curvilinear polygon, let the data A, b, c satisfy (1.2.2), and let g ∈ C 0 (∂Ω) be piecewise analytic, i.e., g satisﬁes (1.2.4). Let E be deﬁned by (5.3.8). Then there 0 exists β ∈ [0, 1)J such that for each f ∈ Bβ,E (Ω, Cf , γf ) the solution u of (1.2.1) is analytic on Ω and there exist C, K > 0 depending only on Ω, f , g, and the constants of (1.2.2) such that uε ≤ C, ˆp,β,E ∇p+2 uL2 (Ω) ≤ CK p+2 max {p + 1, E −1 }p+2 (E/ε) + (E/ε)2 . Φ Proof: Introduce the auxiliary function u0 as the solution of the Dirichlet problem: −∇ · (A(x)∇u) = 0 on Ω and u0 = g on ∂Ω. Then, for some β ∈ [0, 1)J we 2 have from [15, 16] that u0 ∈ Bβ,1 (Ω, Cg , γg ) with Cg , γg > 0 depending only on 2 Ω and g. From Proposition 4.2.13, we immediately get that u ∈ Bβ,E (Ω, C, γ) for some C, γ > 0 independent of ε. Additionally Proposition 4.2.13 implies Φp,β,E ∇p+2 u0 L2 (Ω) ≤ Cγ p max {p + 1, E −1 }p+2 E 2−βmax

∀p ∈ N0 ,

where βmax = maxj=1,...,J βj < 1. Hence, u0 satisﬁes the desired estimates since ε ≤ 1. It remains to see that the function u $ := u − u0 satisﬁes them as well. The function u $ solves −ε2 ∇ · (A(x)∇$ u) + b(x) · ∇$ u + c(x)$ u = f + b(x) · ∇u0 + c(x)u0 on Ω, u $ = 0 on ∂Ω. 0 It suﬃces to check that f$ := b(x) · ∇u0 + c(x)u0 ∈ Bβ,E (Ω, C, γ) for some C, γ independent of ε. As u0 ∈ B2 , we get that f$ ∈ B1 . From the last assertion of β,1

β,1

1 Proposition 4.2.13, we infer that f$ ∈ Bβ,E (Ω, C, γ) for some constants C, γ > 0 independent of ε. Appealing now to the second statement of Proposition 4.2.13 0 allows us to conclude f$ ∈ Bβ,E (Ω, C, γ) for some constants C, γ > 0 independent of ε. The result now follows from Theorem 5.3.10. 2

Remark 5.3.15 The Dirichlet data g are piecewise analytic. This is, of course, not the weakest possible assumption. The appropriate setting for the boundary 3/2 data are trace spaces Bβ,ε , deﬁned as the traces of the elements of the spaces 3/2

2 Bβ,ε . The question of an “intrinsic” characterization of the spaces Bβ,ε then arises. For the case ε = 1, such intrinsic characterizations are given in [16].

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5. Regularity Theory in Countably Normed Spaces

5.4 Neumann and transmission problems We discussed in the preceding section the speciﬁc case of Dirichlet boundary conditions. The techniques employed, however, are also amenable to treating other kinds of boundary conditions. In the present section, we state the regularity assertions for Neumann boundary conditions and for transmission conditions. 5.4.1 Neumann and Robin corners On sectors SR (ω), we are interested in the regularity of solutions u to the following equation −ε2 ∇ · (A(x)∇u) + b(x) · ∇u + c(x)u = f (x) on SR (ω), ε2 ∂nA u = ε(G1,0 + G2,0 u) on Γ0 , 2

ε ∂nA u = ε(G1,ω + G2,ω u)

(5.4.27a) (5.4.27b)

on Γω , (5.4.27c)

where we deﬁned the lateral boundary Γ0 ∪ Γω by Γ0 = {(r, 0) | 0 < r < R},

Γω = {(r cos ω, r sin ω) | 0 < r < R}.

(5.4.28)

In some cases, it will be convenient to combine Γ0 and Γω into the lateral boundary Γlat given by Γlat := Γ0 ∪ Γω = ∂SR (ω) \ ∂BR (0). (5.4.29) In order to formulate conditions on the functions G1,0 , G2,0 , G1,ω , G2,ω , it will 0,δ ω,δ (ω), SR (ω) of Γ0 , Γω by be convenient to introduce conical neighborhoods SR 0,δ SR (ω) := {(r cos ϕ, r sin ϕ) | 0 < r < R,

0 < ϕ < δ}

(5.4.30a)

ω,δ SR (ω)

ω − δ < ϕ < ω}.

(5.4.30b)

:= {(r cos ϕ, r sin ϕ) | 0 < r < R,

We note that these conical neighborhoods are sectors in the sense of Deﬁnition 4.2.1. The minimal assumptions on the functions G1,0 , G2,0 , G1,ω , G2,ω , that we consider are that G2,0 , G2,ω are Lipschitz functions on the lines Γ0 , Γω , respectively, and that, for some δ > 0, the functions G1,0 , G1,ω are traces on 1,1 0,δ (SR (ω)), the lines Γ0 , Γω of functions (again denoted G1,0 , G1,ω ) G1,0 ∈ Hβ,E 1,1 ω,δ G1,ω ∈ Hβ,E (SR (ω)). We note the following lemma, which states that the functions G1,0 , G1,ω , G2,0 , G2,ω may be viewed as the traces of functions deﬁned on SR (ω):

Lemma 5.4.1. Let R > 0, ω ∈ (0, 2π), β ∈ (0, 1), δ > 0. Let χ ∈ C ∞ (R) with χ(0) = 1 and supp χ ⊂ Bω/2 (0) ∩ Bδ (0). Then there exists C > 0 such that for 0,δ every E ∈ (0, 1] and for functions G2,0 , G2,ω ∈ W 1,∞ ((0, R)) and G1,0 ∈ SR (ω), ω,δ G1,ω ∈ SR (ω), the functions G1 , G2 deﬁned in polar coordinates by G1 (r cos ϕ, r sin ϕ) := G1,0 (r cos ϕ, r sin ϕ)χ(ϕ) + G1,ω (r cos ϕ, r sin ϕ)χ(ϕ − ω), G2 (r cos ϕ, r sin ϕ) := G2,0 (r)χ(ϕ) + G2,ω (r)χ(ϕ − ω)

5.4 Neumann and transmission problems

189

satisfy G2 L∞ (SR (ω)) + |x|∇G2 L∞ (SR (ω)) ≤ C

G1 H 1,1 (SR (ω)) ≤ C β,E

G2,k W 1,∞ ((0,R)) ,

k∈{0,ω}

G1,k H 1,1 (S k,δ (ω)) . β,E

R

k∈{0,ω}

Proof: The estimate for G2 is straight forward. The estimate for G1 follows from Part (ii) of Lemma 4.2.10. 2 We start with a lemma that is analogous to case of the Dirichlet problem, treated in Proposition 5.2.2: Lemma 5.4.2. Let R > R > 0, ω ∈ (0, 2π), and A ∈ C 1 (SR (ω), S2> ) with 0 < λmin ≤ A on SR (ω) be given. Then there exist β ∈ [0, 1), which depends only on ω and A(0), and a constant C > 0 with the following properties: For 0,0 1,1 each f ∈ Hβ,1 (SR (ω)), G ∈ Hβ,1 (SR (ω)), the solutions ui , i ∈ {1, 2}, of the problems −∇ · (A∇ui ) = f on SR (ω) with u1 = u2 = 0 on ∂SR (ω) \ Γlat and lateral boundary conditions ∂nA u1 = G ∂nA u2 = G

on Γ0 ∪ Γω , on Γ0 and u2 = 0 on Γω ,

2 satisfy ui ∈ Hloc (SR (ω)) and

ui H 2,2 (S β,1

R (ω))

≤ C f H 0,0 (SR (ω)) + GH 1,1 (SR (ω)) . β,1

β,1

Proof: The case A = Id is proved in [15]. The case of variable A is handled by the same type of perturbation argument as in the proof of Proposition 5.2.2. We will therefore merely outline the main ingredients and use the notation of Proposition 5.2.2 concerning the covering B by balls. For deﬁniteness’ sake, we will consider the Neumann problem, i.e., the regularity of u1 . In the remainder of the proof, we will simply write u instead of u1 . By [15, Lemma 3.1], for every every β ∈ [0, 1) there exists Cβ > 0 such that uH 1 (SR (ω)) ≤ C f H 0,0 (SR (ω)) + GH 1,1 (SR (ω)) . β,1

β,1

By elliptic regularity, viz., (5.5.22) of Lemma 5.5.12 for the case of balls Bi completely contained in SR (ω) and Lemma 5.5.26 for the case of Bi ∩ SR (ω) being a half-ball yields ri2 ∇2 uL2 (Bi ∩S (ω)) ≤ C ri2 f L2 (Bi ∩SR (ω)) + ri ∇uL2 (Bi ∩SR (ω)) R + ri GL2 (Bi ∩SR (ω)) + ri2 ∇GL2 (Bi ∩SR (ω)) .

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5. Regularity Theory in Countably Normed Spaces

Hence, by squaring and summing over all i ∈ N and using the overlap properties of the covering, we get |x|∇2 uL2 (SR (ω)) ≤ C |x|f L2 (SR (ω)) + GL2 (SR (ω)) + |x|∇GL2 (SR (ω)) + ∇uL2 (SR (ω)) . Denoting A˜ := A(0), we see that u satisﬁes the following problem: ˜ −∇ · A∇u = f˜ a.e. on SR (ω) ˜ ∂nA˜ u = G where

on Γ0 ∪ Γω ,

f˜ = f − ∇ · (A˜ − A) · ∇u − (A˜ − A) : ∇2 u, ˜ = G−n ˜ · (A˜ − A)∇u. G

˜ is best expressed in polar coordinates as The function n ˜ (r, ϕ) = (ϕ/ω)nω + (1 − ϕ/ω)n0 , n where n0 and nω are the outer normal vectors on lines Γ0 , Γω . Since A ∈ C 1 (SR (ω)), we have |˜ n · (A˜ − A)| ≤ C|x|,

|∇(˜ n · (A˜ − A))| ≤ C

∀x ∈ SR (ω).

These estimates allow us to conclude with our bounds on ∇uL2 (SR (ω)) and |x|∇2 uL2 (SR (ω)) that ˜ 1,1 0,0 1,1 . f˜H 0,0 (S (ω)) + G ≤ C f + G H H H (S (ω)) (S (ω)) (S (ω)) R R R β,1

β,1

R

β,1

β,1

The argument is completed by observing that [15] ascertains the existence of β ∈ [0, 1) such that ˜ 1,1 1 (S (ω)) . uH 2,2 (S (ω)) ≤ C f˜H 0,0 (S (ω)) + G + u H H (SR (ω)) R R R β,1

β,1

β,1

2 Remark 5.4.3 The term GH 1,1 (SR (ω)) can be rewritten as follows: By mapβ,1 ping arguments, the Hardy inequality Lemma A.1.7 allows us to conclude, for arbitrary ﬁxed R0 ∈ (0, R), the existence of C depending only on R0 , R, β ∈ (0, 1), and ω such that ˆ0,β,1 ∇GL2 (S (ω)) + GL2 (S (ω)\S (ω)) . (5.4.31) GH 1,1 (SR (ω)) ≤ C Φ R R R0 β,1

5.4 Neumann and transmission problems

191

Proposition 5.4.4. Let 0 < R < R, ω ∈ (0, 2π), δ > 0. Assume that the coeﬃcients A, b, c satisfy (5.3.7). Deﬁne E by (5.3.8). Then there exists β ∈ [0, 1) depending only on ω and A(0), and there exists C > 0 independent of E such that 0,0 each solution u ∈ H 1 (SR (ω)) of (5.4.27) with right-hand side f ∈ Hβ,E (SR (ω)), 1,1 0,δ 1,1 ω,δ and boundary data G1,0 ∈ Hβ,E (SR (ω)), G1,ω ∈ Hβ,E (SR (ω)), G2,0 , G2,ω ∈ 1,∞ ((0, R)) satisﬁes W ˆ0,β,E ∇2 uL2 (S (ω)) ≤ CE −2 (E/ε)2 f 0,0 Φ Hβ,E (SR (ω)) + (E/ε)CG1 R + E∇uL2 (SR (ω)) + Cc (E/ε)2 uL2 (SR (ω)) + CG2 (E/ε)uH 1,1 (SR (ω)) , β,E

where CG1 := G1,0 H 1,1 (S 0,δ (ω)) + G1,ω H 1,1 (S ω,δ (ω)) , β,E

R

β,E

R

CG2 := G2,0 W 1,∞ ((0,R)) + G2,ω W 1,∞ ((0,R)) . Proof: For simplicity of notation, we introduce the functions G1 , G2 as in Lemma 5.4.1 such that Gi,0 = Gi |Γ0 , Gi,ω = Gi |Γω and G1 H 1,1 (SR (ω)) ≤ CCG1 , β,E

G2 L∞ (SR (ω)) + |x|∇G2 L∞ (SR (ω)) ≤ CCG2 ,

for a constant C > 0 that depends only on δ, β ∈ (0, 1), R, ω. We may therefore formulate the proof for function G1 , G2 that are deﬁned on SR (ω) and have these regularity properties. The proof follows very closely that of Proposition 5.3.2. Since we will use the same covering B by balls, the same values of h, H, and the same abbreviations, we refer the reader to the proof of Proposition 5.3.2 for these notions. Let χ be a cut-oﬀ function satisfying χ ≡ 1 on Bh (0), χ ≡ 0 on S \ SH , ∇j χL∞ (S) ≤ CE −j , j ∈ {0, 1, 2}, ∂nA χ = 0 on ∂SR (ω) \ ∂BR (0) as given by Lemma A.1.2. We then observe that χu satisﬁes −∇ · (A∇(uχ)) = f˜ := χε−2 [f − b · ∇u − cu] − 2∇χ · (A∇u) − u∇ · (A∇χ), ˜1 + G ˜ 2 := ε−1 (χG1 + G2 (χu)) on ∂SR . ∂n (uχ) = G A

Lemma 5.4.2 gives the existence of β ∈ [0, 1) and C > 0 such that ˜1 + G ˜ 2 1,1 rβ ∇2 (uχ)L2 (SR ) ≤ C f˜H 0,0 (SR ) + G H (SR ) . β,1

β,1

˜ is supported by BH (0) with H < R, we may apply In view of the fact that G (5.4.31) to arrive at ˜ 1 L2 (S ) + rβ ∇G ˜ 2 L2 (S ) . rβ ∇2 (uχ)L2 (SR ) ≤ C rβ f˜L2 (SR ) + rβ ∇G R R −1 ˜ 2 L2 (S ) ≤ Cε−1 CG (χu) 1,1 Since rβ ∇G CG2 rβ ∇(χu)L2 (SR ) 2 R Hβ,1 (SR ) ≤ Cε by Lemma A.1.7 and the support properties of χ, we arrive at

192

5. Regularity Theory in Countably Normed Spaces

rβ ∇2 (uχ)L2 (SR ) ≤ C rβ f˜L2 (SR ) + ε−1 rβ ∇(χG1 )L2 (SR ) + ε−1 CG2 rβ ∇(χu)L2 (SR ) . Dividing both sides by E β and exploiting the support properties of χ gives ˆ0,β,E f˜L2 (S ) ˆ0,β,E ∇2 uL2 (S ) ≤ C Φ Φ H h ˆ0,β,E ∇(χG1 )L2 (S ) + CG2 Φ ˆ0,β,E ∇(χu)L2 (S ) . + ε−1 Φ H H ε Expanding the term involving f˜ as in the proof of Proposition 5.3.2 and expanding the terms involving χG1 , χu gives ˆ0,β,E ∇2 uL2 (S ) ≤ CE −2 (5.4.32) Φ h (E/ε)2 f H 0,0 (SH ) + E∇uL2 (SH ) + Cc (E/ε)2 uL2 (SH ) + uL2 (SH ) β,E + (E/ε)G1 H 1,1 (SH ) + CG2 (E/ε)uH 1,1 (SH ) , β,E

β,E

which has, with the exception of the term uL2 (SH ) , the desired from. In order to remove this term, we take u as the average of u over SH and note that u − u satisﬁes −ε2 ∇ · (A∇(u − u)) = f − b · ∇(u − u) − c(u − u) − cu, ε ∂nA (u − u) = ε(G1 + G2 (u − u)) + εG2 u 2

on SR

on Γlat .

Hence, we may apply (5.4.32) to this problem with G1 replaced with G1 + G2 u and f replaced with f + cu; employing then the Poincar´e inequality u − uL2 (SH ) ≤ CE∇uL2 (SH ) , the bound uL2 (SH ) ≤ uL2 (SH ) , and Lemma 4.2.10 allows us to remove the term uL2 (SH ) in (5.4.32). Paralleling the proof of Proposition 5.3.2, we now turn to the bound on SR \ Sh . By (5.5.22) for balls Bi with Bi ⊂ SR and by (5.5.35) for balls with xi ∈ Γlat we have for all i ∈ N \ IE , ri2 ∇2 uL2 (Bi ∩SR ) ≤ C ε−2 ri2 f − b · ∇u − cuL2 (Bi ∩SR ) + ri ∇uL2 (Bi ∩SR ) + ri ε−1 G1 L2 (Bi ∩SR ) + ri2 ε−1 ∇G1 L2 (Bi ∩SR )

+ CG2 ri ε−1 uL2 (Bi ∩SR ) + CG2 ri2 ε−1 ∇uL2 (Bi ∩SR ) . Dividing this estimate by ri2 , summing over all i ∈ N \ IE and exploiting the overlap properties of the covering B together with ri ≥ CE and gives ∇2 uL2 (SR \Sh ) ≤ CE −2 (E/ε)2 f H 0,0 (SR ) + E∇uL2 (SR ) β,E 2 + Cc (E/ε) uL2 (SR ) + (E/ε)G1 H 1,1 (SR ) + CG2 (E/ε)∇uH 1,1 (SR ) . β,E

β,E

2

5.4 Neumann and transmission problems

193

Proposition 5.4.5. Let 0 < R < R, ω ∈ (0, 2π), δ > 0. Assume that the coeﬃcients A, b, c satisfy (5.3.7). Let E be given by (5.3.8). Then there exists β ∈ [0, 1) depending only on ω and A(0), and there exist C, K > 0 independent of E, Cf , CG1 , CG2 such that each solution u ∈ H 1 (SR (ω)) of (5.4.27) with right-hand 0,δ 0 1 (SR (ω), Cf , γf ), and boundary data G1,0 ∈ Bβ,E (SR (ω), CG1 , γG2 ), side f ∈ Bβ,E ω,δ 1 G1,ω ∈ Bβ,E (SR (ω), CG1 , γG1 ), and G2,0 , G1,ω satisfying n n! Dn G2,0 L∞ ((0,R)) + Dn G2,ω L∞ ((0,R)) ≤ CG2 γG 2

∀n ∈ N0 ,

satisﬁes for all p ∈ N0 ˆp,β,E ∇p+2 uL2 (S (ω)) ≤ CK p+2 max {p + 1, E −1 }p+2 × Φ R 2 (E/ε) Cf + E∇uL2 (SR (ω)) + Cc (E/ε)2 uL2 (SR (ω)) + (E/ε)CG1 + (E/ε)CG2 uH 1,1 (SR (ω)) . β,E

Proof: The proof is very similar to that of Proposition 5.3.4; we will therefore merely point out the appropriate modiﬁcations. First, we note that the assumptions on G2,0 , G2,ω imply that they can be extended analytically to functions (again denoted G2,0 , G2,ω ) with ∇n G2,0 L∞ (Uδ (Γ0 )) + ∇n G2,ω L∞ (Uδ (Γω )) ≤ CCG2 γ n n! ∀n ∈ N0 , for appropriately chosen C, γ and δ > 0 suﬃciently small, where the neighborhoods Uδ (Γ0 ), Uδ (Γω ) are given by Uδ (Γ0 ) = ∪x∈Γ0 Bδ (x), Uδ (Γω ) = ∪x∈Γω Bδ (x). We will assume that the covering B that is chosen in Proposition 5.3.4 is such 0,δ ω,δ that the half-balls Bi ∩ SR (ω) are contained in SR (ω) ∪ SR (ω). We split the index set N into N = Ibdy ∪ Iint ,

Iint := {i ∈ N | Bi ⊂ SR (ω)},

Ibdy := N \ Iint .

The treatment of the balls Bi , i ∈ Iint , is completely analogous to that in the proof of Proposition 5.3.4. We will therefore restrict our attention to the half balls Bi ∩ SR (ω), i ∈ Ibdy . Reasoning as in the proof of Proposition 5.3.4 (cf. (5.3.21)), the application of 0 Lemma 4.2.17 applied to the function f ∈ Bβ,E (SR (ω), Cf , γf ) with the covering B yields the existence of C, K > 0 independent of E and i such that with the numbers Cf (i) as given by Lemma 4.2.17 we have ∇p f L2 (S∩Bi ) ≤

1 CK p Cf (i) max {(p + 1)/|xi |, E −1 }p . ˆ Φ0,β,E (xi )

Cf2 (i) ≤ CCf2

(5.4.33) (5.4.34)

i∈Ibdy

Completely analogously, we can exploit with Corollary 4.2.18 the fact that G1,k ∈ k,δ 1 (SR (ω), CG1 , γG1 ), k ∈ {0, ω}, to get the existence of numbers CG (i) as Bβ,E given by Corollary 4.2.18 such that for k ∈ {0, ω}

194

5. Regularity Theory in Countably Normed Spaces

∇p G2,k L2 (S∩Bi ) ≤ C

K p CG (i) max {(p + 1)/|xi |, E −1 }p , ˆ−1,β,E (xi ) Φ

2 2 CG (i) ≤ CCG 1

(5.4.35) (5.4.36)

i∈Ibdy

As in the proof of Proposition 5.3.4 we now employ a local analytic regularity result, namely, Proposition 5.5.3, to get r p+2 i

2

∇p+2 uL2 (Bi ∩S) ≤ Cu (i)K p+2 max {(p + 3), ri /E}p+2

∀p ∈ N0 ,

where we abbreviate Cu (i) := min{1, ri /E}E∇uL2 (Bi ∩S) + Cc (E/ε)2 min{1, ri /E}2 uL2 (Bi ∩S) 1 + CCf (i) min {1, ri /E}2 (E/ε)2 ˆ0,β,E (xi ) Φ + CCG (i)

1 ˆ−1,β,E (xi ) Φ

min {1, ri /E}(E/ε)

+ CCG2 (E/ε) min {1, ri /E} min {1, ri /E}E∇uL2 (Bi ∩S) + uL2 (Bi ∩S) . Reasoning as in the proof of Proposition 5.3.4, we arrive at ˆp,β,E ∇p+2 u2 2 ≤ Φ L (S ) CK 2p max {p + 1, E −1 }2p

(5.4.37) ∞

ˆ20,β,E (xi ) max {1/ri , E −1 }4 Cu2 (i). Φ

i=1

ˆ1,0,E (xi ), we further bound Using min {1, ri /E} ≤ C Φ ˆ1,β,E (xi )E∇uL2 (B ∩S) + Cc (E/ε)2 Φ ˆ1,β,E (xi )uL2 (B ∩S) ˆ0,β,E (xi )Cu (i) ≤ Φ Φ i i 2 ˆ2 2 ˆ + Cf (i)(E/ε) Φ1,0,E (xi ) + CG (i)(E/ε)Φ1,0,E (xi ) ˆ2,β,E (xi )E∇uL2 (B ∩S) + Φ ˆ1,β,E uL2 (B ∩S) . +CG2 (E/ε) Φ i i Inserting this in (5.4.37), appealing to Lemma 5.3.6, and employing Proposition 5.4.4 yields ˆp,β,E ∇p+2 uL2 (S ) ≤ CK p max {p + 1, E −1 }p E −2 E∇uL2 (S) Φ + Cc (E/ε)2 uL2 (S) + Cf (E/ε)2 + CG1 (E/ε) + CG2 (E/ε)uH 1,1 (S) . β,E

2 Remark 5.4.6 Proposition 5.4.5 considers the case of straight sectors. However, l since by Theorem 4.2.20 the spaces Bβ,E are invariant under analytic changes of

5.4 Neumann and transmission problems

195

variables, the case of case of a curvilinear sector can be directly inferred from Proposition 5.4.5 by a mapping argument. As in Remark 5.3.5, it is not essential that the data f , b, c, and ε be real. In the case of complex ε, it suﬃces to replace ε with |ε| in all estimates. The proof shows that the result could be slightly sharpened: Some of the norms uL2 (SR (ω)) , ∇uL2 (SR (ω)) , can be replaced with weighted norms. 5.4.2 Mixed corners The case of corners with mixed boundary conditions is treated in exactly the same way as the cases of Dirichlet or Neumann boundary conditions. We will therefore merely state the results and use some of the notation already introduced at the outset of Section 5.4.1. We consider −ε2 ∇ · (A(x)∇u) + b(x) · ∇u + c(x)u = f (x) on SR (ω), (5.4.38a) 2 ε ∂nA u = ε(G1 + G2 u) on Γ0 , (5.4.38b) u = 0 on Γω . (5.4.38c) Then the following result holds: Proposition 5.4.7. Let 0 < R < R, ω ∈ (0, 2π), δ > 0. Assume that the coeﬃcients A, b, c satisfy (5.3.7). Let E be given by (5.3.8). Then there exist β ∈ [0, 1), which depends only on ω and A(0), and C > 0, which is independent of E, such that each solution u ∈ H 1 (SR (ω)) of (5.4.38) with right-hand side 0,0 1,1 0,δ (SR (ω)) and boundary data G1 ∈ Hβ,E (SR (ω)), G2 ∈ W 1,∞ ((0, R)) f ∈ Hβ,E satisﬁes ˆ0,β,E ∇2 uL2 (S (ω)) ≤ CE −2 E∇uL2 (S (ω)) + Cc (E/ε)2 uL2 (S (ω)) Φ R R R ˆ0,β,E f L2 (S (ω)) + (E/ε)G1 1,1 + (E/ε)2 Φ R H (SR (ω)) + CG2 uH 1,1 (SR (ω)) , β,E

β,ε

where we set CG2 := G2 W 1,∞ ((0,R)) . 0,δ 0 1 If furthermore f ∈ Bβ,E (SR (ω), Cf , γf ), G1 ∈ Bβ,E (SR (ω), CG1 , γG1 ), and G2 satisﬁes Dn G2 L∞ ((0,R)) ≤ CG2 γ n n! ∀n ∈ N0 , then there exist C, K > 0 independent of E, Cf , CG1 , CG2 such that ∀p ∈ N0 ˆp,β,E ∇p+2 uL2 (S (ω)) ≤ CK p max {p, E −1 }p+2 E∇uL2 (S (ω)) Φ R R + Cc (E/ε)2 ∇uL2 (SR (ω)) + Cf (E/ε)2 + CG1 (E/ε) + CG2 uH 1,1 (SR (ω)) . β,E

Proof: The proof is completely analogous to that of Propositions 5.4.4, 5.4.5. 2

196

5. Regularity Theory in Countably Normed Spaces

5.4.3 Transmission problems To ﬁx the notation, we assume that a sector SR (ω) = {(r cos ϕ, r sin ϕ) | 0 < r < + − R, 0 < ϕ < ω} is divided into SR , SR by a line Γω passing through the origin. Speciﬁcally, we set for an ω ∈ (0, ω) + SR := {(r cos ϕ, r sin ϕ) | 0 < r < R, ω < ϕ < ω}, − SR := {(r cos ϕ, r sin ϕ) | 0 < r < R, 0 < ϕ < ω }, Γ0 := {(r, 0) | 0 < r < R}, Γω := {(r cos ω, r sin ω) | 0 < r < R}, Γω := {(r cos ω , r sin ω ) | 0 < r < R},

ω < ϕ < ω + δ}.

ω ,δ SR (ω) := {(r cos ϕ | r sin ϕ) | 0 < r < R,

We consider H 1 -solutions to the following diﬀerential equation: + − −ε2 ∇ · (A(x)∇u) + b(x) · ∇u + c(x)u = f (x) on SR ∪ SR , (5.4.39a) 2 [ε ∂nA u] = ε (G1,ω + G2,ω u) on Γω ,(5.4.39b)

where the expression [ε2 ∂nA u] denotes the jump across Γ . Concerning boundary conditions on the lateral sides Γ0 , Γω , we assume any of the following three types: u=0 on Γ0 ∪ Γω or 2 ε ∂nA u = ε(G1,0 + G2,0 u) on Γ0 ε2 ∂nA u = ε(G1,ω + G2,ω u) on Γω ∂nA u = ε(G1,0 + G2,0 u) on Γ0

(5.4.40a) or

(5.4.40b)

u = 0 on Γω .

(5.4.40c)

and

+ − 2 + − We assume ε ∈ (0, 1] and that the data A ∈ A(SR ∪ SR , S> ), b ∈ A(SR ∪ SR ), + − c ∈ A(SR ∪ SR ) satisfy, for some CA , Cb , γA , γb , Cc , γc ≥ 0, p ∇p AL∞ (S + ∪S − ) ≤ CA γA p! R

R

0 < λmin ≤ A ∇ bL∞ (S + ∪S − ) ≤ p

R

R

∇ cL∞ (S + ∪S − ) ≤ p

R

R

Cb γbp p! Cc γcp p!

on

∀p ∈ N0 ,

(5.4.41a)

− SR ,

(5.4.41b)

∀p ∈ N0 ,

(5.4.41c)

∀p ∈ N0 .

(5.4.41d)

+ SR

∪

The data G2,k , k ∈ {0, ω , ω}, are assumed analytic on Γk , k ∈ {0, ω , ω}, respectively; i.e., there exist CG2 , γG2 > 0 such that G2,k , k ∈ {0, ω , ω}, as functions parametrized by arclength satisfy n Dn G2,k L∞ ((0,R)) ≤ CG2 γG n! ∀n ∈ N0 . (5.4.41e) 2 k∈{0,ω ,ω}

In order to treat the diﬀerent types of lateral boundary conditions in a uniﬁed way, the following proposition assumes that a solution u to (5.4.39) is piecewise 2,2 in Hβ,E .

5.5 Local regularity

197

Proposition 5.4.8. Assume (5.4.41) and let the relative diﬀusivity E be deﬁned by (5.3.8). Assume that for a β ∈ (0, 1) a function u satisﬁes 2,2 + − 1. u ∈ H 1 (SR (ω)) ∩ Hβ,E (SR ∪ SR ); 2. u solves (5.4.39) together with any of the three lateral boundary conditions (5.4.40); k,δ + − 0 1 (SR ∪SR , Cf , γf ) and G1,k ∈ Bβ,E (SR (ω), CG1 , γG1 ), k ∈ {0, ω , ω}. 3. f ∈ Bβ,E + − ∪ SR ) and for every R < R there exist C, K > 0 independent Then u ∈ A(SR of E, Cf , CG2 , CG1 such that for all p ∈ N0

ˆp,β,E ∇p+2 u 2 + − ≤ CK p+2 max {p + 1, E −1 }p+2 × Φ L (SR ∪SR ) 2 ˆ ˆ0,β,E ∇uL2 (S (ω)) + (E/ε)2 Cc uL2 (S (ω)) E Φ0,β,E ∇2 uL2 (S + ∪S − ) + EΦ R R R R + (E/ε)2 Cf + (E/ε)CG1 + (E/ε)CG2 uH 1,1 (SR (ω)) . β,E

Proof: The proof is very similar to that of Proposition 5.4.5 and therefore omitted. For the treatment of the balls near the interface Γ , we employ the local regularity result Proposition 5.5.4. 2 Remark 5.4.9 Mutatis mutandis, the Remarks 5.3.5, 5.4.6 apply: The coeﬃcients b, c may be complex and curvilinear sectors can be treated with mapping arguments provided by Theorem 4.2.20.

5.5 Local regularity The aim of the present section is the proof of four local analytic regularity assertions, Propositions 5.5.1–5.5.4. The ﬁrst one is concerned with regularity on balls (thus leading to interior regularity results); the second and third ones on half balls (thus leading to regularity results at the boundary); the fourth one with regularity results on balls where the data are assumed to be only piecewise analytic. We start by introducing the following notation: For n ≥ 2 and R > 0 we denote + by BR := BR (0) ⊂ Rn the ball of radius R with center in the origin; BR and − BR denote the “upper” and “lower” parts: BR := BR (0), + BR := {x ∈ BR | xn > 0},

ΓR := {x ∈ BR | xn = 0} − BR := {x ∈ BR | xn < 0}.

(5.5.1)

The intersection of BR and the hyper-plane xn = 0 is denoted ΓR : ΓR := BR ∩ + or on the two half balls {xn = 0}. For problems deﬁned on the half ball BR + − BR ∪BR , the direction normal to ΓR is special. Hence, we will denote the variable xn by y. In this context, we will denote by ∇x diﬀerentiation with respect to the tangential variables x1 , . . . , xn−1 . The local analytic regularity results are based on the following notation:

198

5. Regularity Theory in Countably Normed Spaces

[p] = max (1, p), 1 NR,p (v) := sup (R − r)2+p ∇p+2 vL2 (Br ) , [p]! R/2≤r
p ∈ N0 ∪ {−2, −1},

1 sup (R − r)p+q+2 ∂yq+2 ∇px vL2 (Br+ ) , [p + q]! R/2≤r
p ≥ 0, q ≥ −2,

1 sup (R − r)p+q+2 ∂yq+2 ∇px vL2 (Br+ ∪Br− ) , p ≥ 0, q ≥ −2, [p + q]! R/2≤r
With these notions, we can formulate the following four local analytic regularity results. Their proofs are technical and relegated to the ensuing subsections. Proposition 5.5.1 (Interior analytic regularity). Let R ∈ (0, 1] and BR ⊂ Rn be a ball of radius R. Let A ∈ A(BR , Sn> ), b ∈ A(BR , Rn ), c ∈ A(BR ) satisfy λmin ≤ A ∇ bL∞ (BR ) ≤ p

on BR ,

p ∇p AL∞ (BR ) ≤ CA γA p!,

Cb γbp p!,

∇ cL∞ (BR ) ≤ p

∀p ∈ N0 , (5.5.2a) ∀p ∈ N0 .

Cc γcp p!.

(5.5.2b)

Let ε ∈ (0, 1], and deﬁne the relative diﬀusivity E by √ Cc Cb + 1. E −1 := 2 + ε ε

(5.5.3)

Let f ∈ A(BR ) satisfy for all p ∈ N0 ∇p f L2 (BR ) ≤ Cf γfp max {p/R, E −1 }p

(5.5.4)

and let u be a solution of −ε2 ∇ · (A∇u) + b · ∇u + cu = f

on BR .

Then for some K > 0 depending only on the constants of (5.5.2) and γf max {(p + 3), R/E}p+2 ∀p ∈ N0 ∪ {−1}, (5.5.5) [p]! Cu := min {1, R/E}E∇uL2 (BR ) (5.5.6) + (E/ε)2 min {1, R/E}2 Cc uL2 (BR ) + Cf .

NR,p (u) ≤ Cu K p+2

+ ⊂ Rn be Proposition 5.5.2 (Dirichlet conditions). Let R ∈ (0, 1] and BR + + + n n a semi-ball of radius R. Let A ∈ A(BR , S> ), b ∈ A(BR , R ), c ∈ A(BR ) satisfy

λmin ≤ A ∇ bL∞ (B + ) ≤ p

R

+ on BR ,

p ∇p AL∞ (B + ) ≤ CA γA p!,

Cb γbp p!,

∇ cL∞ (B + ) ≤

R

p

R

Cc γcp p!

∀p ∈ N0 , (5.5.7a) ∀p ∈ N0 .

(5.5.7b)

+ Let ε ∈ (0, 1], deﬁne the relative diﬀusivity E by (5.5.3). Let f ∈ A(BR ) satisfy

∇p f L2 (B + ) ≤ Cf γfp max {p/R, E −1 }p R

∀p ∈ N0 ,

(5.5.8)

5.5 Local regularity

199

and let u be a solution of −ε2 ∇ · (A∇u) + b · ∇u + cu = f

+ on BR ,

+ on ∂BR .

u=0

Then for some K1 , K2 > 0 depending only on the constants of (5.5.7) and γf there holds for all p ∈ N0 , q ∈ N0 ∪ {−2, −1} such that p + q = −2 max {(p + q + 3), R/E}p+q+2 , (5.5.9) [p + q]! Cu := min {1, R/E}E∇uL2 (B + ) R 2 + (E/ε) min {1, R/E}2 Cc uL2 (B + ) + Cf . (5.5.10)

(u) ≤ Cu K1p+2 K2q+2 NR,p,q

R

Proposition 5.5.3 (Neumann conditions). Assume the same hypotheses + + on A, b, c, ε, E as in Proposition 5.5.2. Let f ∈ A(BR ), G1 , G2 ∈ A(BR ) satisfy ∇p f L2 (B + ) ≤ Cf γfp max {p/R, E −1 }p R

∀p ∈ N0 ,

p max {p/R, E −1 }p ∇p G1 L2 (B + ) ≤ CG1 γG 1

∀p ∈ N0 ,

R

∇ G2 L∞ (B + ) ≤ p

R

p CG2 γG p! 2

∀p ∈ N0 ,

(5.5.11) (5.5.12) (5.5.13)

and let u ∈ H 1 (BR+ ) solve −ε2 ∇·(A∇u)+b·∇u+cu = f

ε2 ∂nA u = ε (G1 + G2 u)

on BR+ ,

on ΓR .

Then for some K1 , K2 > 0 depending only on the constants of (5.5.7) and γf , γG1 , γG2 , there holds for all p ∈ N0 , q ∈ N0 ∪ {−2, −1} such that p + q = −2 max {(p + q + 3), R/E}p+q+2 , [p + q]! Cu := min {1, R/E}E∇uL2 (B + ) R + (E/ε)2 min {1, R/E}2 Cc uL2 (B + ) + Cf

(u) ≤ Cu K1p+2 K2q+2 NR,p,q

(5.5.14) (5.5.15)

R

+ CG1 min {1, R/E}(E/ε)

+ CG2 (E/ε) min {1, R/E} min {1, R/E}E∇uL2 (B + ) + uL2 (B + ) . R

R

+ Proposition 5.5.4 (Transmission problem). Let R ∈ (0, 1], A ∈ A(BR ∪ − n + − + − n BR , S> ), b ∈ A(BR ∪ BR , R ), c ∈ A(BR ∪ BR ) satisfy + − on BR ∪ BR ,

λmin ≤ A

∇p bL∞ (B + ∪B − ) ≤ Cb γbp p!, R

R

p ∇p AL∞ (B + ∪B − ) ≤ CA γA p! ∀p ∈ N0 , (5.5.16a) R

R

∇p cL∞ (B + ∪B − ) ≤ Cc γcp p! R

R

∀p ∈ N0 . (5.5.16b)

+ ∪ Let ε ∈ (0, 1], and deﬁne the relative diﬀusivity E by (5.5.3). Let f ∈ A(BR − + BR ), G1 , G2 ∈ A(BR ) satisfy

∇p f L2 (B + ∪B − ) ≤ Cf γfp max {p/R, E −1 }p R

R

p ∇p G1 L2 (B + ) ≤ CG1 γG max {p/R, E −1 }p 1 R

∇ G2 L∞ (B + ) ≤ p

R

p CG2 γG p! 2

∀p ∈ N0 .

∀p ∈ N0 , ∀p ∈ N0 ,

(5.5.17) (5.5.18) (5.5.19)

200

5. Regularity Theory in Countably Normed Spaces

Let u ∈ H 1 (BR ) be a solution of the transmission problem: −ε2 ∇·(A∇u)+b·∇u+cu = f

[ε2 ∂nA u] = ε (G1 + G2 u)

on BR ,

on ΓR .

Then for some K1 , K2 > 0 depending only on the constants of (5.5.16) and γf , γG1 , γG2 , there holds for all p ∈ N0 , q ∈ N0 ∪ {−2, −1} such that p + q = −2 max {(p + q + 3), R/E}p+q+2 , (5.5.20) [p + q]! Cu := min {1, R/E}E∇uL2 (BR ) (5.5.21) 2 2 + min {1, R/E} (E/ε) Cf + Cc uL2 (BR ) + CG1 min {1, R/E}(E/ε) + CG2 (E/ε) min {1, R/E} min {1, R/E}E∇uL2 (BR ) + uL2 (BR ) .

,± NR,p,q (u) ≤ Cu K1p+2 K2q+2

5.5.1 Preliminaries: local H 2 -regularity Lemma 5.5.5 (H 2 -regularity). Let BR := BR (0). Let f ∈ L2 (BR ), A ∈ C 1 (BR , Sn> ) with 0 < λmin ≤ A(x) on BR . Then there exists CI > 0 depending only on λmin , AL∞ (BR ) , and RA L∞ (BR ) such that the weak solution u of −∇ · (A∇u) = f

on BR ,

u=0

on ∂BR

satisﬁes ∇2 uL2 (BR ) ≤ CI f L2 (BR ) . Remark 5.5.6 The proof shows that the constant CI depends only on lower bounds on λmin and upper bounds on AL∞ (BR ) , RA L∞ (BR ) . Furthermore, it is worth stressing that it is independent of R ≤ 1. Proof: Introduce the aﬃne map F : B1 → BR given by F (x) = Rx. Upon setting u ˆ = u ◦ F , Aˆ = A ◦ F , fˆ = f ◦ F , we get from Lemma A.1.1 that u ˆ solves ˆ u) = R2 fˆ −∇ · (A∇ˆ

on B1 ,

u ˆ=0

on ∂B1 .

Standard regularity theory (see, e.g., [49, 56]) gives the existence of CI > 0 ˆ = λmin , A ˆ L∞ (B ) = AL∞ (B ) , Aˆ L∞ (B ) = depending only on λmin (A) 1 1 R 2 ˆL2 (B1 ) ≤ CI R2 fˆL2 (B1 ) . Transforming back to RA L∞ (BR ) such that ∇ u BR yields the desired result. 2 Analogously, we have a priori estimates on semi balls: + Lemma 5.5.7 (H 2 -regularity, Dirichlet conditions). Let f ∈ L2 (BR ), A ∈ + + 1 n C (BR , S> ) with 0 < λmin ≤ A on BR . Then there exists CB > 0 depending only on λmin , AL∞ (B + ) , and RA L∞ (B + ) such that the weak solution u of R

−∇ · (A∇u) = f

R

on

+ , BR

u=0

satisﬁes ∇2 uL2 (B + ) ≤ CB f L2 (B + ) . R

R

on ΓR

5.5 Local regularity

201

Finally, we need a local regularity result for transmission problems (see also Lemma 5.5.22 below for a version on balls BR with R < 1). Lemma 5.5.8 (H 2 -regularity, transmission conditions). Let f ∈ L2 (B1 ), g ∈ H 1/2 (Γ1 ) (with Γ1 = {x ∈ B1 | xn = 0}), A ∈ L∞ (B1 , Sn> ) with 0 < λmin ≤ A on B1 . Assume that A|B + ∈ C 1 (B1+ , Sn> ), A|B − ∈ C 1 (B1− , Sn> ). Then for 1 1 every R ∈ (0, 1) there exists CR > 0 depending on R, λmin , AL∞ (B1 ) , and A L∞ (B + ∪B − ) such that the weak solution u ∈ H01 (B1 ) of 1

1

−∇ · (A∇u) = f satisﬁes

on B1 ,

[∂nA u] = g

on Γ1

∇2 uL2 (B + ∪B − ) ≤ CR f L2 (B1 ) + gH 1/2 (Γ1 ) . R

R

Proof: First, we note that the assumptions give the a priori bound uH 1 (B1 ) ≤ C f H −1 (B1 ) + gH −1/2 (Γ1 ) ≤ C f L2 (B1 ) + gH 1/2 (Γ1 ) . 2 Standard local regularity results also imply that u ∈ Hloc (B1+ ∪ B1− ), and it merely remains to obtain the bound. This is proved with the method of tangential diﬀerential quotients of Nirenberg (see, e.g., [49, 62]) in a manner completely similar to that of local regularity for the Neumann problem (note that R < 1 so that our bounds are really just assertions at ﬂats parts of the boundary of B1+ , B1− ). In particular, one checks that in this procedure, only tangential derivatives of A are needed to obtain bounds for the tangential derivatives of ∇u. Then, the + − diﬀerential equation is employed to get a bound on ∂x2n u on BR ∪ BR . 2

Lemma 5.5.9 (H 2 -regularity, Neumann conditions). Let f ∈ L2 (B1+ ), g ∈ H 1/2 (Γ1 ) (with Γ1 = {x ∈ B1 | xn = 0}), A ∈ C 1 (B1+ ) with 0 < λmin ≤ A on B1+ . Then for every R ∈ (0, 1) there exists CR > 0 depending on R, λmin , AL∞ (B + ) , and A L∞ (B + ) such that the weak solution u ∈ H01 (B1+ ) of 1

1

−∇ · (A∇u) = f satisﬁes

on B1+ ,

∂nA u = g

on Γ1

∇2 uL2 (B + ) ≤ CR f L2 (B + ) + gH 1/2 (Γ1 ) . R

1

Proof: The proof is very similar to that of Lemma 5.5.8.

2

Lemma 5.5.10. Let G ⊂ Rn be open, A ∈ C p+1 (G, Mn ), u ∈ C p+2 (G). Then there holds on G  1/2 p+1  |α|!  p+1 Dα ∇ · (A∇u) − ∇ · ADα ∇u 2 ≤ |∇q A| |∇p+2−q u|.   α! q q=1 |α|=p

202

5. Regularity Theory in Countably Normed Spaces

Proof: We start by noting that ∇ · (A∇u) = (∇ · A)∇u + A : ∇2 u,

where ∇ · A = ( i ∂i aij )nj=1 , ∇2 u = (∂i ∂j u)ni,j=1 . Hence, Dα ∇ · (A∇u) − ∇ · ADα ∇u = α D (∇ · A)∇u − (∇ · A)Dα ∇u + Dα A : ∇2 u + A : (Dα ∇2 u) . Applying Lemma A.1.4 we arrive at  1/2  |α|!  2 Dα ∇ · (A∇u) − ∇ · ADα ∇u   α! |α|=p

1/2    |α|! Dα (∇ · A)∇u − (∇ · A)Dα ∇u2 ≤   α! |α|=p

1/2    |α|! 2 D α A : ∇2 u − A : D α ∇2 u +   α! |α|=p

p p p |∇q+1 A| |∇p+1−q u| + |∇q A| |∇p+2−q u| q q q=1 q=1 p p+1 p p+1 p+1 = |∇ A| |∇u| + |∇q A| |∇p+2−q u| |∇A| |∇ u| + q 1 q=2

≤

≤

p

p+1 p+1 q=1

q

|∇q A| |∇p+2−q u|. 2

5.5.2 Interior regularity: Proof of Proposition 5.5.1 We introduce the additional notation 1 sup (R − r)2+p ∇p vL2 (Br ) , MR,p (v) := p! R/2≤r
p ∈ N0 .

Lemma 5.5.11 (interior regularity). Let A ∈ C 1 (BR , Sn> ) with 0 < λmin ≤ A(x) on BR . Then there exists C > 0 depending only on λmin , AL∞ (BR ) , RA L∞ (BR ) such that any solution u of −∇ · (A∇u) = f ∈ L2 (BR )

on BR

satisﬁes for all r, δ > 0 with r + δ < R, ∇2 uL2 (Br ) ≤ C f L2 (Br+δ ) + δ −1 ∇uL2 (Br+δ ) + δ −2 uL2 (Br+δ ) .

5.5 Local regularity

203

Proof: The proof is essentially the same as that of [98, Lemma 5.7.1]. The key ingredient is to employ a smooth cut-oﬀ function χ being identically one on BR−δ , vanishing on BR \ BR−δ/2 , satisfying |∇j χ| ≤ Cδ −j , j ∈ {0, 1, 2} and then to consider the function uχ. 2 The following lemma generalizes this result to higher order derivatives. It should be noted that control of the L2 -norm of u is not required—control of derivatives of u suﬃces. Lemma 5.5.12 (interior regularity). Let p ∈ N0 , A ∈ C p+1 (BR , Sn> ) with 0 < λmin ≤ A on BR , f ∈ H p (BR ). Then there exists CI > 0 depending only on λmin , AL∞ (BR ) , RA L∞ (BR ) , and n such that any solution u of −∇ · (A∇u) = f

on BR

satisﬁes for all p ∈ N0 NR,p (u) ≤

CI

q p+1 p+1 R [p − q]! MR,p (f ) + ∇q AL∞ (BR ) NR,p−q (u) 2 p! q q=1 + NR,p−1 (u) + NR,p−2 (u) .

For p = 0 we have the sharper bound , 2 R R NR,0 (u) ≤ CI f L2 (BR ) + ∇uL2 (BR ) . 2 2

(5.5.22)

Proof: Let p ∈ N0 . For all α ∈ Nn0 with |α| = p the function Dα u satisﬁes on BR . −∇ · (A∇Dα u) = Dα f + Dα ∇ · (A∇u) − ∇ · ADα ∇u By Lemma 5.5.5 we therefore get for r, δ > 0 with r + δ < R ∇2 Dα u2L2 (Br ) ≤ C Dα f 2L2 (Br+δ ) + Dα ∇ · (A∇u) − ∇ · (Dα ∇u)2L2 (Br+δ )

+ δ −2 Dα ∇u2L2 (Br+δ ) + δ −4 Dα u2L2 (Br+δ ) .

|α|! α!

and summing on α with |α| = p we get ∇p+2 u2L2 (Br ) ≤ C ∇p f 2L2 (Br+δ )

Multiplying by

|α|! Dα ∇ · (A∇u) − ∇ · (Dα ∇u)2L2 (Br+δ ) α! |α|=p + δ −2 ∇p+1 u2L2 (Br+δ ) + δ −4 ∇p u2L2 (Br+δ ) . +

204

5. Regularity Theory in Countably Normed Spaces

Together with Lemma 5.5.10, we then arrive at ∇p+2 uL2 (Br+δ ) ≤ C ∇p f L2 (Br+δ ) p+1 p+1 ∇q AL∞ (BR ) ∇p+2−q uL2 (Br+δ ) + q q=1 + δ −1 ∇p+1 uL2 (Br+δ ) + δ −2 ∇p uL2 (Br+δ ) with δ > 0 still at our disposal. From the deﬁnition of the quantities MR,p (f ), NR,p (u) we infer ∇p f L2 (Br+δ ) ≤ (R − r − δ)−(p+2) p!MR,p (f ), ∇p+2−q uL2 (Br+δ ) ≤ (R − r − δ)−(p+2−q) [p − q]!NR,p−q (u), ∇p+1 uL2 (Br+δ ) ≤ (R − r − δ)−(p+1) [p − 1]!NR,p−1 (u), ∇p uL2 (Br+δ ) ≤ (R − r − δ)−p [p − 2]!NR,p−2 (u). Choosing now δ :=

R−r , p+2

R − r − δ = (R − r)

p+1 , p+2

(5.5.23)

we arrive at

* ! p + 2 p+2 1 p+2 p+2 MR,p (f ) (R − r) ∇ uL2 (Br ) ≤ C p! p+1 ! p+2−q * p+1 p+2 p+1 [p − q]! q q (R − r) NR,p−q (u) ∇ AL∞ (BR ) + p! p+1 q q=1 ! p+1 * (p + 2)[p − 1]! p + 2 NR,p−1 (u) + p! p+1 p (p + 2)2 [p − 2]! p + 2 NR,p−2 (u) . + p! p+1

It is easy to see that the terms in curly braces can be bounded uniformly in p. Taking the supremum over r ∈ (R/2, R) and bounding (R − r)q ≤ (R/2)q , we get the desired result. We now turn to the special case p = 0. Replacing the function u by u − u, where u is the average of u over BR , we see that the same reasoning as above allows us to get with a Poincar´e inequality the desired bound (5.5.22) if we appropriately adjust the constant CI . 2 Lemma 5.5.13. Let b ∈ A(BR , Rn ), c, u ∈ A(BR ) and assume that b, c satisfy the estimates (5.5.2b). Then

5.5 Local regularity

MR,p (cu) ≤ Cc

p q=0

γc

R 2

q

R 2

2

205

[p − q − 2]! NR,p−q−2 (u) (p − q)!

q 2 R R [p − q − 2]! = Cc γc NR,p−q−2 (u) 2 2 (p − q)! q=0 p 2 R R + Cc γc NR,−2 (u), 2 2 q p R R [p − q − 1]! MR,p (b · ∇u) ≤ Cb γb NR,p−q−1 (u). 2 2 (p − q)! q=0 p−1

(5.5.24) (5.5.25)

Proof: We will only prove the second estimate. By Lemma A.1.3 we have 1 sup (R − r)p+2 ∇p (b · ∇u)L2 (Br ) p! R/2≤r
MR,p (b · ∇u) ≤

The desired bound (5.5.25) is now obtained by observing sup R/2≤r
(R − r)p+2 ∇p−q+1 uL2 (Br )

≤ =

R 2 R 2

q+1 sup q+1

R/2≤r
(R − r)(p−q−1)+2 ∇(p−q−1)+2 uL2 (Br )

[p − q − 1]!NR,p−q−1 (u). 2

Proof of Proposition 5.5.1: We start by choosing K > max {1, γf /2, γb R/2, γc R/2, γA R/2} such that for CI of Lemma 5.5.12 CI

1 4

K −2 + K −1

1/2 1/4 + K −2 1 − γb R/(2K) 1 − γc R/(2K) CA γA R/2 ≤ 1. + K −1 + K −2 + K −1 1 − γA R/(2K)

We ﬁrst note that the claim holds true for p = −1 since K ≥ 1 and

(5.5.26)

206

5. Regularity Theory in Countably Normed Spaces

R 1R ∇uL2 (BR ) ≤ E∇uL2 (BR ) 2 2E 1 1 ≤ max {1, R/E} min {1, R/E} E∇uL2 (BR ) ≤ max {1, R/E} Cu . 2 2

NR,−1 (u) ≤

We now show that the claim is also true for p = 0. From (5.5.22) and the diﬀerential equation in the form −∇ · (A∇u) = ε−2 [f − b · ∇u − cu] we get NR,0 (u) ≤ , 2 R Cb R2 Cc R2 R ∇uL2 (BR ) + 2 uL2 (BR ) + ∇uL2 (BR ) . CI f L2 (BR ) + 2 2ε ε 4 ε 4 2 From (5.3.9) and R/E = min {1, R/E} max {1, R/E} we get R4 1 1 ≤ (R/E)2 (E/ε)2 ≤ min {1, R/E}2 (E/ε)2 max {1, R/E}2 , 4ε2 4 4 1 Cb R2 1 ≤ E −1 R2 ≤ E min {1, R/E}2 max {1, R/E}2 , 2 ε 4 4 4 R ≤ min {1, R/E} max {1, R/E}E. Inserting these bounds into the estimate for NR,0 (u) gives NR,0 (u) ≤ max {1, R/E}2 CI Cu ≤ K 2 max {1, R/E}2 Cu K −2 CI . By our choice of K, we have K −2 CI ≤ 1 so that the desired bound for p = 0 holds. Let us next proceed by induction on p and assume that (5.5.5) holds for all −1 ≤ p < p. With CI of Lemma 5.5.12 we have NR,p (u) ≤ CI ε−2 MR,p (f − b · ∇u − cu) + NR,p−1 (u) + NR,p−2 (u) q p+1 p+1 R [p − q]! + ∇q AL∞ (BR ) NR,p−q (u) . 2 p! q q=1 Next, MR,p (f − b · ∇u − cu) ≤ MR,p (f ) + MR,p (b · ∇u) + MR,p (cu). We ﬁrst turn our attention to MR,p (f ). We have p+2 R 1 max {p/R, E −1 }p ε−2 MR,p (f ) ≤ ε−2 Cf γfp p! 2 R2 E 2 1 Cf γf p ≤ 2 2 max {p + 3, R/E}p . E ε p! 4 2 As R2 /E 2 ≤ max {p + 3, R/E}2 min {1, R/E}2 we infer max {p + 3, R/E}p+2 Cf γf p min {1, R/E}2 (E/ε)2 . p! 4 2 max {p + 3, R/E}p+2 1 −2 γf p K ≤ Cu K p+2 . p! 4 2K

ε−2 MR,p (f ) ≤

5.5 Local regularity

207

We now turn to ε−2 MR,p (b · ∇u). From p! [p − q − 1]! 1 NR,p−q−1 (u) ≤ [p − q − 1]!NR,p−q−1 (u) (p − q)! p! (p − q)! 1 ≤ pq Cu K p−q+1 max {p + 3, R/E}p−q+1 p! 1 ≤ Cu K p−q+1 max {p + 3, R/E}p+1 , p! we infer from Lemma 5.5.13 and the observation ε−2 Cb ≤ E −1 : q p γb R max {p + 3, R/E}p+1 R ε−2 MR,p (b · ∇u) ≤ Cb ε−2 Cu K p−q+1 2 2 p! q=0 q p p+2 1 −1 γb R p+2 max {p + 3, R/E} . ≤ Cu K K p! 2 2K q=0 In order to treat the term MR,p (cu), we ﬁrst note that ε−2 Cc R2 NR,−2 (u) = (E/ε)2 Cc (R/E)2 uL2 (BR ) ≤ Cc (E/ε)2 min {1, R/E}2 max {1, R/E}2 uL2 (BR ) ≤ Cu

p! max {p + 3, R/E}p+2 max {1, R/E}2 ≤ Cu . p! p!

Hence, observing that ε−2 Cc ≤ E −2 we can bound MR,p (cu) using Lemma 5.5.13 as follows: q 2 p−1 γc R max {p + 3, R/E}p R −2 −2 ε MR,p (cu) ≤ Cc ε Cu K p−q 2 2 p! q=0 p p+2 γc R 1 p+2 max {p + 3, R/E} + C K u 2K 4K 2 p! q p p+2 1 −2 γc R p+2 max {p + 3, R/E} K . ≤ Cu K p! 4 2K q=0 Estimating 1 p! max {p + 3, R/E}p+2 max {p + 2, R/E}p+1 ≤ , p! [p − 1]! p! p! max {p + 3, R/E}p+2 1 max {p + 1, R/E}p ≤ , p! [p − 2]! p! we can bound NR,p−1 (u) + NR,p−2 (u) ≤ Cu K p+2

max {p + 3, R/E}p+2 −1 K + K −2 . p!

208

5. Regularity Theory in Countably Normed Spaces

Finally, bounding

p+1 q

≤

q p+1 p+1 R 2

q

q=1

Cu CA

2

p+2 max {p

we derive

∇q AL∞ (BR )

q p+1 γA R q=1

Cu K

(p+1)q q!

[p − q]! NR,p−q (u) ≤ p!

K p−q+2 (p + 1)q

max {p − q + 3, R/E}p−q+2 ≤ p!

q p+1 γA R + 3, R/E}p+2 CA . p! 2K q=1

This allows us to conclude max {p + 3, R/E}p+2 1 −2 γf p CI K p! 4 2K 1/2 1/4 CA γA R/(2K) + K −2 + K −1 + K −2 + . + K −1 1 − γb R/(2K) 1 − γc R/(2K) 1 − γA R/(2K)

NR,p (u) ≤ Cu K p+2

As the expression in brackets is bounded by 1 uniformly in p by our choice of K in (5.5.26), the induction argument is completed. 2

5.5.3 Regularity at the boundary: Proof of Proposition 5.5.2 We will consider half-balls at the boundary and show analytic regularity on such half-balls. This will be done in two steps: First, the growth of the tangential derivatives (x denotes the tangential variables; derivatives of u with respect to x are denoted in this section by ∇x u) is controlled. In a second step, the derivatives of u in the normal direction (denoted y) are controlled. Most of the arguments are quite similar to analogous ones of Subsection 5.5.2; proofs will therefore only be provided for selected steps. We start by introducing the following auxiliary notation suitable for controlling tangential derivatives: 1 sup (R − r)p+2 ∇px vL2 (Br+ ) , p! R/2≤r
MR,p (v) =

R/2≤r
˜ R,p (v) = 1 M sup (R − r)p+2 ∇p vL2 (Br+ ) . p! R/2≤r
if p ≥ 0 if p = −2, −1,

5.5 Local regularity

209

Control of tangential derivatives. Analogous to Lemma 5.5.10 is the following result. Lemma 5.5.14. Let G ⊂ Rn be open, A ∈ C p+1 (G, Mn ), u ∈ C p+2 (G). Denote by Dxα diﬀerentiation with respect to the variables x. Then on G  1/2 p+1  |α|!  p+1 2 2 Dxα ∇ · (A∇u) − ∇ · ADxα ∇u |∇q A| |∇p−q ≤ x ∇ u|.   α! q |α|=p

q=1

The analog of Lemma 5.5.12 is the following lemma. + + Lemma 5.5.15. Let p ∈ N0 , A ∈ C p+1 (BR , Sn> ) with 0 < λmin ≤ A on BR , + p f ∈ H (BR ). Then there exists CB > 0 depending only on λmin , AL∞ (B + ) , R RA L∞ (B + ) , and n such that for all solutions u of R

−∇ · (A∇u) = f

+ on BR ,

u=0

on ΓR

there holds q p+1 p+1 R [p − q]! NR,p−q (u) MR,p (u) ≤ CB (f ) + ∇q AL∞ (B + ) NR,p R 2 p! q q=1 + NR,p−1 (u) + NR,p−2 (u) . For p = 0, we have the sharper bound , 2 R R NR,0 (u) ≤ C f L2 (B + ) + ∇uL2 (B + ) . R R 2 2

(5.5.27)

Proposition 5.5.16 (control of tangential derivatives). Let R ∈ (0, 1] and + + + + Let A ∈ A(BR , Sn> ), b ∈ A(BR , Rn ), c ∈ A(BR ), f ∈ A(BR ) satisfy (5.5.7), (5.5.8). Let ε ∈ (0, 1] and let E be deﬁned by (5.5.3). Finally let u be a solution of + −ε2 ∇ · (A∇u) + b · ∇u + cu = f on BR , u = 0 on ΓR . Then for some K1 > 0 depending only on the constants of (5.5.7) and γf of (5.5.8) there holds max {(p + 3), R/E}p+2 , p ≥ −1, (5.5.28) [p]! Cu := min {1, R/E}E∇uL2 (B + ) R 2 + (E/ε) min {1, R/E}2 Cc uL2 (B + ) + Cf . (5.5.29)

(u) ≤ Cu K1p+2 NR,p

R

Proof: The proof proceeds by induction analogous to the way Proposition 5.5.1 is proved. The cases p = −1 and p = 0 are shown, as in the proof of Proposition 5.5.1, by inspection and Lemma 5.5.15. The induction argument then parallels that of Proposition 5.5.1 using Lemma 5.5.15. 2

210

5. Regularity Theory in Countably Normed Spaces

Control of normal derivatives. We now turn to bounding the normal derivatives. This is done using the fact that u satisﬁes a diﬀerential equation. We stress the fact that boundary conditions are immaterial for this step. We start with the following lemma. + + + Lemma 5.5.17. Let R > 0, A ∈ C ∞ (BR , Mn ), u ∈ C ∞ (BR ), Ann ≡ 0 on BR . Then there holds

1 sup (R − r)p+q+2 ∇px ∂xq n (A : ∇2 u)L2 (Br+ ) ≤ [p + q]! R/2≤r
q q p q

A : ∇ u| ≤ 2

(i,j)=(n,n) r=0 s=0

q q p q ≤ r s r=0 s=0

≤

q q p q r=0 s=0

r

s

r

s

q−s |∇rx ∂xsn Aij ||∇p−r x ∂xn ∂i ∂j u|

1/2 |∇rx ∂xsn Aij |2

(i,j)=(n,n)

1/2 q−s 2 |∇p−r x ∂xn ∂i ∂j u|

(i,j)=(n,n)

|∇rx ∂xsn A| |∇p−r+2 ∂xq−s u| + |∇p−r+1 ∂xq−s+1 u| . x x n n

(u) we infer From the deﬁnition of Np,q,R

sup R/2≤r

≤ ≤

R 2 R 2

(R − r)p+q+2 ∇p−r+2 ∂xq−s uL2 (Br+ ) x n

r+s sup r+s

R/2≤r
(R − r)(p−r+2)+(q−2−s)+2 ∇p−r+2 ∂x(q−2−s)+2 uL2 (Br+ ) x n

[p − r + q − s]!NR,p−r+2,q−s−2 (u)

and also sup R/2≤r
(R − r)p+q+2 ∇p−r+1 ∂xq−s+1 uL2 (Br+ ) x n ≤

R 2

r+s

[p − r + q − s]!NR,p−r+1,q−s−1 (u).

The result now follows easily. We obtain by similar reasoning a result of the following form:

2

5.5 Local regularity

211

+ + + Lemma 5.5.18. Let R > 0, b ∈ C ∞ (BR , Rn ), c ∈ C ∞ (BR ), u ∈ C ∞ (BR ). Then there holds

1 sup (R − r)p+q+2 ∇px ∂xq n (b · ∇u)L2 (Br+ ) ≤ [p + q]! R/2≤r
0≤r≤p 0≤s≤q (r,s)=(p,q)

1 + ∂ q ∇p c ∞ + [p + q]! xn x L (BR )

R 2

p+q+2 NR,0,−2 (u).

Lemma 5.5.19. Let G ⊂ Rn be open, a ∈ A(G), a ≥ λ > 0 on G and let a satisfy for some CA , γA > 0 p ∇p aL∞ (G) ≤ CA γA p!

∀p ∈ N0 .

Then there are CA , γA > 0 depending only on CA , γA , and λ (in particular, they are independent of G) such that the function $ a := 1/a satisﬁes p ∇p $ aL∞ (G) ≤ CA (γA ) p!

∀p ∈ N0 .

Proof: Follows from Cauchy’s integral representation of derivatives. Lemma 5.5.20. Let G ⊂ Rn be open and let a, b ∈ A(G) satisfy p ∇p aL∞ (G) ≤ CA γA p!,

p ∇p bL2 (G) ≤ CB γB p!

∀p ∈ N0 .

Then the product ab satisﬁes ∇p (ab)L2 (G) ≤ CA CB (γA + γB )p p! Proof: By Lemma A.1.3, we have on G

∀p ∈ N0 .

2

212

5. Regularity Theory in Countably Normed Spaces

|∇ (ab)(x)| ≤ p

p p q=0

q

|∇ a(x)| |∇ b(x)| ≤ q

q

p p q=0

q

∇q aL∞ (G) |∇p−q b(x)|.

Hence, ∇p (ab)L2 (G) ≤ CA

p q=0

≤ CA CB

p! γ q ∇p−q bL2 (G) (p − q)! A

p

q p−q p!γA γB ≤ CA CB p!(γA + γB )p .

q=0

2 Proof of Proposition 5.5.2: We will choose K1 , K2 > 1 suﬃciently large below (however, both constants will only depend on the constants of (5.5.7) and γf of (5.5.8)). As we choose K2 ≥ 1 we see that the claim (5.5.9) holds true ≤ NR,p for all for q = 0 by Proposition 5.5.16 and the observation that NR,p,0 p ∈ N0 . For q = −2 and q = −1, we calculate directly for p ∈ N0 : NR,p,−2 (u) ≤

1 sup (R − r)p ∇px uL2 (Br+ ) ≤ NR,p−2 (u), [p − 2]! R/2≤r
(u) ≤ NR,p,−1

1 (u). sup (R − r)p ∂xn ∇px uL2 (Br+ ) ≤ NR,p−1 [p − 1]! R/2≤r
We now proceed by induction on q. Let us assume that (5.5.9) holds for all −2 ≤ q < q and all p ∈ N0 with p + q ≥ −1. Upon denoting by Aˆ the n × n matrix given by Aˆij = Aij for (i, j) = (n, n) and Aˆnn = 0 we note that u satisﬁes −Ann ∂n2 u = ε−2 [f − b · ∇u − cu] + (∇ · A)∇u + Aˆ : ∇2 u. Thus, after divided both sides by Ann , the function u satisﬁes $ cu + A∇u + B : ∇2 u, −∂n2 u = ε−2 f$ − $b · ∇u − $

(5.5.30)

where, by Lemmata 5.5.19, 5.5.20, 4.3.1 there are C , γ > 0 depending only on λmin , CA , γA , γf , γb , γc such that for all p ∈ N0 $ ∞ + ≤ C γ p p!, ∇p A L (B ) R

∇p BL∞ (B + ) ≤ C γ p p!, R

Bnn = 0,

∇p f$L2 (B + ) ≤ Cf C γ p max {p/R, E −1 }p , R

p$

∇ bL∞ (B + ) ≤ Cb C γ p p!, R

∇p $ cL∞ (B + ) ≤ Cc C γ p p!. R

We now use (5.5.30) in order to bound NR,p,q (u) =

1 uL2 (Br+ ) . sup (R − r)p+q+2 ∇px ∂xq+2 n [p + q]! R/2≤r
5.5 Local regularity

213

We start with the contribution from f$: For r ∈ [R/2, R) we have (R − r)p+q+2 ε−2 ∇px ∂xq n f$L2 (Br+ ) p+q+2 R −2 ≤ε Cf C γ p+q max {(p + q)/R, E −1 }p+q 2 γ p+q R 2 E 2 C Cf ≤ max {(p + q), R/E}p+q . 4 2 E ε Bounding (R/E)2 ≤ min {1, R/E}2 max {p + q + 3, R/E}2 we arrive at 1 (R − r)p+q+2 ε−2 ∇px ∂xq n f$L2 (Br+ ) [p + q]! 2 γ p+q E max {(p + q), R/E}p+q+2 C ≤ min {1, R/E}2 Cf 4 2 ε [p + q]! . p q / p+q+2 γ γ max {(p + q), R/E} C ≤ Cu K1p+2 K2q+2 K1−2 K2−2 [p + q]! 4 2K1 2K2 with Cu of (5.5.10). We now turn to the term involving $b · ∇u, which we handle using Lemma 5.5.18. By the induction hypothesis [p − r + q − s − 1]! NR,p−r,q−1−s (u) + NR,p+1−r,q−s−2 (u) ≤ Cu K1p−r+2 K2q−s+2 max {p + q + 3, R/E}p+q−r−s+1 K2−1 + K1 K2−2 . Next, we observe that q q −2 p r s $ −2 p ε ∇x ∂xn bL∞ (B + ) ≤ C Cb ε (r + s)!γ r+s R r s r s ≤ C E −1 pr q s (2γ)r+s . Thus, by Lemma 5.5.18 1 ε−2 sup (R − r)p+q+2 ∇px ∂xq n ($b · ∇u)L2 (Br+ ) [p + q]! R/2≤r
≤ Cu K1p+2 K2q+2

We now turn to the terms involving $ cu. First, we note that N0,−2 (u) = uL2 (B + ) ; thus, R

214

5. Regularity Theory in Countably Normed Spaces

ε

−2

p+q+2 R 1 q p ∂ ∇ $ c ∞ + NR,0,−2 (u) [p + q]! xn x L (BR ) 2 p+q 1 γR [p + q]! ≤ C (R/E)2 Cc (E/ε)2 uL2 (B + ) R 4 2 [p + q]! p+q γR 1 ≤ C Cc (E/ε)2 min {1, R/E}2 max {1, R/E}2 uL2 (B + ) R 4 2 p+q p+q+2 γR max {p + q + 3, R/E} 1 ≤ C Cu . 4 2 [p + q]!

This estimate implies for the term involving $ cu: 1 ε−2 sup (R − r)p+q+2 ∇px ∂xq n ($ cu)L2 (Br+ ) ≤ [p + q]! R/2≤r
r s q p γR max {p + q + 3, R/E}p+q+2 C −2 γR . K2 [p + q]! 4 K1 K2 r=0 s=0

$ It remains to bound the terms involving A∇u and B : ∇2 u. Reasoning analogously as in the treatment of the term involving $b · ∇u, we obtain 1 p+2 q+2 $ K2 . sup (R − r)p+q+2 ∇px ∂xq n (A∇u) L2 (Br+ ) ≤ Cu K1 [p + q]! R/2≤r
≤ Cu K1p+2 K2q+2

Next, we assume that K1 > max {1, γ/2, γR}, K2 > max {1, γ/2, γR}, so that all geometric sums arising in the above estimates are convergent series. Hence, by combing all the above estimates and observing that max {p+q +3, R/E}p+q+1 ≤ max {p + q + 3, R/E}p+q+2 , we get

5.5 Local regularity

215

max {p + q + 3, R/E}p+q+2 × [p + q]! C 1 C K1−2 K2−2 + (K −1 + K1 K2−2 ) 4 (1 − γR/K1 )(1 − γR/K2 ) 2 2 1 R + K2−2 + [K2−1 + K1 K2−2 ] + [K1 K2−2 + K12 K2−2 ] . 4 2

NR,p,q (u) ≤ Cu K1p+2 K2q+2

Choose now K1 > max {1, γ/2, γR} such that it is greater than the constant K1 given by Proposition 5.5.16. Next, choose K2 > max {1, γ/2, γR} such that the expression in brackets is bounded by one. This allows us to complete the induction argument. 2

5.5.4 Regularity of transmission problems: Proof of Prop. 5.5.4 The proof of Proposition 5.5.4 is very similar to that of Proposition 5.5.2. One proceeds in two steps by ﬁrst estimating the growth of the derivatives in the tangential direction and then controlling the derivatives in the normal direction. In order to do that, we need the analog of Lemma 5.5.11. To infer such a result from Lemma 5.5.22, we need the following lemma. Lemma 5.5.21. Let A ∈ L∞ (B1 , Sn> ) satisfy 0 < λmin ≤ A on B1 . Assume that the restrictions A+ := A|B + , A− := A|B − satisfy A+ ∈ C 1 (B1+ , Sn> ), A− ∈ 1

1

C 1 (B1− , Sn> ). Then there exists C > 0 depending only on λmin , AC 1 (B + ) + 1 AC 1 (B − ) such that for every δ ∈ (0, 1) we can ﬁnd a piecewise smooth cut-oﬀ 1 function χδ ∈ W 1,∞ (Rn ), χδ ≡ 1 on B1−δ (0), χ ≡ 0 on R2 \ B1−δ/2 (0) with the following properties: 1. ∂nA+ χδ = 0 and ∂nA− χδ = 0 on xn = 0; in particular, therefore, [∂nA χδ ] = 0 on xn = 0; 2. ∇j χδ L∞ (B + ∪B − ) ≤ Cδ −j for j ∈ {0, 1, 2}. 1

1

Proof: For simplicity of exposition, we construct χδ in the two-dimensional case n = 2 only. We construct χδ in polar coordinates (r, ϕ). Furthermore, we will only construct χδ for the right half-plane (i.e., ϕ > 0); the general case follows easily. Let χ be a smooth cut-oﬀ function satisfying χ ≡ 1 for r ∈ (0, 1 − δ), χ ≡ 0 for r > 1 − δ/2 and χ(j) L∞ (R) ≤ Cδ −j , j ∈ {0, 1, 2} for some C > 0 independent of δ. We compute for the co-normal derivatives on the line ϕ = 0. To that end, we write ∇χδ as ∇χδ = (n · ∇χδ )n + (t · ∇χδ )t = −

∂ϕ χδ n + (∂r χδ ) t, r

where n is the outer normal vector on ϕ = 0 and t is the (normalized) tangential vector. Inserting this decomposition in the deﬁnition of ∂nχA+ δ gives

216

5. Regularity Theory in Countably Normed Spaces

∂nA+ χδ = −(n A+ n)

∂ϕ χδ ∂ϕ χδ + (nT A+ t) ∂r χδ =: a+ (r) + b+ (r)∂r χδ (r), r r

where the functions a+ , b+ are smooth. Furthermore, from nT A+ n ≥ λmin > 0 we get that |a+ | is bounded from below by λmin . We now set for ϕ ∈ (0, π/2): χδ (r, ϕ) := χ(r) −

rb+ (r) χ (r) ϕ ρ(ϕ/δ), a+ (r)

where ρ is a smooth cutoﬀ function satisfying ρ ≡ 1 in a (ﬁxed) neighborhood of ϕ = 0 and ρ ≡ 0 on R\(−π/4, π/4). We see that χδ satisﬁes χδ ≡ 1 for r ≤ 1−δ, χδ = 0 for r > 1 − δ/2 and that χδ is smooth (for ϕ ∈ [0, π/2]). Furthermore, the conditions on ∇j χδ are satisﬁed because sup

|∂ϕj (ϕρ(ϕ/δ))| ≤ Cδ −j+1 ,

j ∈ {0, 1, 2}

ϕ∈[0,π/2]

in view of the support properties of ρ. Inserting χδ in ∂n+ χδ for ϕ = 0, we A

see that ∂n+ χδ = 0. This completes the construction for ϕ ∈ (0, π/2). For A

ϕ ∈ (−π/2, 0), the construction is completely analogous (note that choosing the same “base” function χ guarantees the continuity across ϕ = 0). 2 We can now prove the analog of Lemma 5.5.11. Lemma 5.5.22. Let R ≤ 1, A ∈ L∞ (BR , Sn> ) with 0 < λmin ≤ A, whose + , Sn> ), A− ∈ restrictions A+ := A|B + , A− := A|B − satisfy A+ ∈ C 1 (BR R

R

− n + , S> ). Let f ∈ L2 (BR ), G+ ∈ H 1 (BR ). Let u ∈ H 1 (BR ) be a solution of C 1 (BR the transmission problem

−∇ · (A∇u) = f

on BR ,

[∂nA u] = G+

on xn = 0.

Then there exists C > 0 depending only on λmin , AL∞ (BR ) , RA L∞ (B + ) + R RA L∞ (B − ) , and n such that for all r, δ > 0 with r + δ < R there holds R

∇2 uL2 (Br+ ∪Br− ) ≤ C f L2 (Br+δ ) + ∇G+ L2 (B +

+ δ −1 G+ L2 (B + ) r+δ −1 −2 + δ ∇uL2 (Br+δ ) + δ uL2 (Br+δ ) . r+δ )

Proof: Let F : x → Rx be the aﬃne mapping transforming the unit ball B1 to the BR . From Lemma A.1.1, we see that the transformed function u ˆ := u ◦ F satisﬁes the following transmission problem ˆ u) = R2 fˆ −∇ · (A∇ˆ

on B1 ,

ˆ ∂nAˆ u ˆ = RG

on xn = 0,

ˆ + := G+ ◦ F , Aˆ := A ◦ F . We note in passing that where we set fˆ := f ◦ F , G ˆ L∞ (B ) = AL∞ (B ) , Aˆ ∞ + − = RA ∞ + − . Let δ ∈ (0, 1) A 1

R

L

(B1 ∪B1 )

L

(BR ∪BR )

5.5 Local regularity

217

and let χ be the cut-oﬀ function given by Lemma 5.5.21. A calculation then shows that the function χˆ u satisﬁes the following transmission problem on B1 : ˆ u) · ∇χ − ∇ · (A∇χ) ˆ ˆ ˆ u ˆ − (A∇χ) · ∇ˆ u on B1 , −∇ · (A∇(χˆ u)) = R2 χfˆ − (A∇ˆ ˆ+ [∂nAˆ χˆ u] = RχG

on xn = 0.

Furthermore, as χˆ u and the right-hand side vanish in a neighborhood of ∂B1 , this equation in fact holds on B2 : ˆ ˆ u) · ∇χ − ∇ · (A∇χ) ˆ ˆ −∇ · (A∇(χˆ u)) = f˜ := R2 χfˆ − (A∇ˆ u ˆ − (A∇χ) · ∇ˆ u ˜ := RχG ˆ [∂nAˆ χˆ u] = G

on xn = 0.

(We implicitly extended Aˆ to B2 appropriately). Thus, the interior regularity assertion of Lemma 5.5.8 is applicable (with B2 taking the place of B1 ) and yields with the trace theorem ˜ H 1/2 (Γ ) ˆL2 (B + ∪B − ) ≤ ∇2 (χˆ u)L2 (B + ∪B − ) ≤ C f˜L2 (B2 ) + G ∇2 u 2 1 1 1−δ 1−δ −1 ˆ+ 2 + ˆ+ 2 + G ≤ C R2 fˆL2 (B1−δ /2 ) + R∇G L (B1−δ /2 ) + Rδ L (B1−δ /2 ) −1 −2 + δ ∇ˆ uL2 (B1−δ /2 ) + δ ˆ uL2 (B1−δ /2 ) . Scaling back to the original domain BR gives the desired bounds for r := R(1 − δ ), δ := Rδ /2. 2 In order to prove Proposition 5.5.4, we introduce the following notation, analogous to the notation used previously for the regularity assertions in the interior and at the boundary. We deﬁne for p ∈ N0 ∪ {−2, −1} 1  sup (R − r)p+2 ∇2 ∇px vL2 (Br+ ∪Br− )  p! R/2≤r
if p ≥ 0 if p = −2, −1,

R/2≤r
and for all p ∈ N0 we introduce

. / 1 R−r p p+1 p sup (R − r) ∇x ∇vL2 (Br+ ) , ∇x vL2 (Br+ ) + HR,p (v) := [p − 1]! R/2≤r
,± (f ) := MR,p

1 sup (R − r)p+2 ∇px f L2 (Br+ ∪Br− ) . p! R/2
The analog of Lemma 5.5.15 is the following lemma.

218

5. Regularity Theory in Countably Normed Spaces

Lemma 5.5.23. Let p ∈ N0 , A ∈ L∞ (BR , Sn> ) with 0 < λmin ≤ A, whose + , Sn> ), A− ∈ restrictions A+ := A|B + , A− := A|B − satisfy A+ ∈ C p+1 (BR R

R

− n + − + , S> ). Assume f ∈ H p (BR ∪ BR ). Let G ∈ H p+1 (BR ). Then there C p+1 (BR exists CB > 0 depending only on λmin , AL∞ (BR ) , RA L∞ (B + ∪B − ) , and n R R such that any solution u ∈ H 1 (BR ) of the transmission problem

−∇ · (A∇u) = f

on BR ,

[∂nA u] = G

on ΓR

satisﬁes ,± ,± MR,p (u) ≤ CB (f ) + HR,p (G) + NR,p * ! q q−1 p+1 p+1 R R q q−1 ∇ AL∞ (B + ∪B + ) + q∇ AL∞ (B + ∪B + ) × R R R R 2 2 q q=1 [p − q]! ,± ,± ,± NR,p−q (u) + NR,p−1 (u) + NR,p−2 (u) . p! For p = 0, we have the sharper bound ,± NR,0 (u) ≤ (5.5.31) R2 f L2 (BR ) + RGL2 (B + ) + R2 ∇GL2 (B + ) + R∇uL2 (BR ) . CB R

R

Proof: For brevity of notation, we set Br := Br+ ∪ Br− . One proceeds as in the proof of Lemma 5.5.15 by diﬀerentiating the diﬀerential equation in the tangential direction. Let p ∈ N0 and α ∈ Nn−1 with |α| = p be 0 given. Then Dα u solves the following transmission problem: f˜α := Dα f + Dα (∇ · (A∇u)) − ∇ · (ADα ∇u) on BR , + − ˜ ˜ ˜ Gα + Gα + Gα

:= Dα G + Dα (nT A+ ∇u+ ) − nT A+ Dα ∇u+

− Dα (nT A− ∇u− ) − nT A− Dα ∇u− on ΓR ,

−∇ · (A∇Dα u) = [∂nA Dα u] =

+ − , BR , where we used the notation u+ , u− to denote the restriction of u to BR respectively. From Lemma 5.5.22, we therefore have the existence of C > 0 such that for all r, δ > 0 with r + δ < R: ˜ α 2 2 + + ∇G ˜ α 2 2 + ∇2 Dα u2L2 (Br ) ≤ C f˜α 2L2 (B ) + δ −2 G ) ) L (B L (B r+δ

˜ + 2 2 + + ∇G α L (B

r+δ

+δ

−2

D ∇u2L2 (B + α

r+δ

r+δ

˜ + 2 2 + + δ −2 G α L (B ) − r+δ ∪Br+δ

˜ − 2 2 − + δ −2 G ˜ − 2 2 − + ∇G α L (B α L (B r+δ ) r+δ ) r+δ ) + δ −4 Dα u2L2 (B ) . ) r+δ

|α|! α 2 2 Recalling that ∇px ∇2 u2L2 (B ) = |α|=p α! D ∇ uL2 (Br ) , we see that we r ˜ 2 have to bound |α|=p |α|! α! fα L2 (B ) etc. We have r+δ

5.5 Local regularity

219

|α|! Dα f 2L2 (B ) = ∇px f 2L2 (B ) , r+δ r+δ α!

|α|=p

|α|! δ −2 Dα ∇u2L2 (B ) + δ −4 Dα u2L2 (B ) r+δ r+δ α!

|α|=p

= δ −2 ∇px ∇u2L2 (B ) + δ −4 ∇px u2L2 (B ) , r+δ r+δ |α|! α 2 −2 α 2 D ∇GL2 (B + ) + δ D GL2 (B + ) r+δ r+δ α!

|α|=p

= ∇px ∇G2L2 (B +

r+δ )

+ δ −2 ∇px G2L2 (B +

r+δ )

.

From Lemma A.1.4, we obtain (as in the proofs of Lemmata 5.5.12, 5.5.10), with the abbreviation ∇q A∞ := ∇q AL∞ (BR ) : |α|! Dα (∇ · (A∇u) − ∇ · (ADα ∇u))2L2 (B ) ≤ r+δ α! |α|=p * ! p 2 p q+1 p−q q p−q 2 ∇ A∞ ∇x ∇uL2 (Br+δ , C ) + ∇ A∞ ∇x ∇ uL2 (Br+δ ) q q=1 ˜+: and analogously (see Lemma A.1.4) for the terms involving G α |α|! 2 ˜+ ∇G α L2 (B + ) ≤ r+δ α! |α|=p * ! p 2 p q p−q 2 + + ∇q+1 A∞ ∇p−q C x ∇uL2 (Br+δ ) + ∇ A∞ ∇x ∇ uL2 (Br+δ ) q q=1 and |α|! 2 ˜+ G α L2 (B + ) ≤ C r+δ α!

|α|=p

! p p q=1

q

*2 ∇q A∞ ∇p−q x ∇uL2 (B +

.

r+δ )

˜ − . Combining the bounds ob˜ − , ∇G Completely analogous bounds hold for G α α tained so far, we have the existence of C > 0 (independent of δ) such that p −1 ∇px GL2 (B + ) ∇px ∇2 uL2 (Br ) ≤ C ∇px f L2 (Br+δ ) + ∇x ∇GL2 (B + ) + δ r+δ

+δ

−1

+ δ −1

+

∇px ∇uL2 (Br+δ )

p

q=1

+δ

r+δ

∇px uL2 (Br+δ )

p + ∇q A∞ ∇p−q x ∇uL2 (Br+δ ) q

q=1 p

p q

−2

∇q+1 A∞ ∇p−q x ∇uL2 (B +

r+δ

q p−q 2 ) + ∇ A∞ ∇x ∇ uL2 (B +

r+δ

. )

220

5. Regularity Theory in Countably Normed Spaces

The remainder of the proof is now similar to that of Lemma 5.5.12. We use ,± ,± , HR,p , NR,p and choose δ as in (5.5.23) to the deﬁnition of the quantities MR,p obtain with similar arguments as given there: 1 (R − r)p+2 ∇px ∇2 uL2 (Br ) ≤ p! ,± ,± ,± C MR,p (f ) + HR,p (G) + NR,p−1 (u) + NR,p−2 (u) q p p R [p − 1 − q]! ,± NR,p−q−1 (u) + (p + 1) ∇q A∞ 2 p! q q=1 +

p+1 p+1 q=1

q

∇q A∞

R 2

q

[p − q]! ,± NR,p−q (u) . p!

The claim of the lemma now follows by some further manipulations of this last ,± expression and observing that taking the supremum over r yields NR,p (u) on the left-hand side. We ﬁnally turn to (5.5.31), i.e., the case p = 0. This is done with a Poincar´e argument. The assertion for general p yields for some C > 0 ,± NR,0 (u) ≤ (5.5.32) 2 2 C R f L2 (BR ) + RGL2 (BR ) +R ∇GL2 (BR ) + R∇uL2 (BR ) + uL2 (BR ) .

Let u be the average of u over BR . We note that u − u satisﬁes the same diﬀerential equation as u; thus, (5.5.32) also holds with u replaced with u − u. Using the Poincar´e estimate u − uL2 (BR ) ≤ CR∇uL2 (BR ) then gives the desired (5.5.31) after suitably adjusting the constant CB . 2 + − Lemma 5.5.24. Let G2 satisfy (5.5.19) and assume u ∈ A(BR ∪ BR ). Then

1 1E min {1, R/E}× [p]! 2 ε , q q+1 p p γG2 R γG2 R ,± 2 q![p − q − 2]!NR,p−q−2 + (u) max {p + 1, R/E} 2 2 q q=0 q p p γG2 R ,± + max {p + 1, R/E} [p − q − 1]!NR,p−q−1 (u) . 2 q q=0

ε−1 HR,p (G2 u) ≤ CG2

Proof: The deﬁnition of HR,p consists of two terms, which we will estimate separately. We start with estimating H := ε−1

1 sup (R − r)p+1 ∇px (G2 u)L2 (Br+ ) . [p − 1]! R/2
Using Leibniz’ formula and the bounds (5.5.19), we get

5.5 Local regularity

221

q p R p q R −1 [p] CG γ q! H ≤ε sup (R − r)p−q ∇p−q x uL2 (Br+ ) [p]! 2 2 q=0 q G2 2 R/2
In a similar way, we estimate H := ε−1

1 sup (R − r)p+2 ∇px ∇(G2 u)L2 (Br+ ) , [p]! R/2
namely, we bound |∇px ∇(G2 u)| ≤

p p

q

q=0

and estimate

p q=0

1 ε−1 [p]!

S1,p,q :=

p q

sup R/2
S2,p,q :=

sup R/2
q p−q |∇qx ∇G2 | |∇p−q x u| + |∇x G2 | |∇x ∇u|

(S1,p,q + S2,p,q ), where

(R − r)p+2 |∇qx ∇G2 | |∇p−q x u| L2 (Br+ ) , (R − r)p+2 |∇qx G2 | |∇p−q x ∇u| L2 (Br+ ) .

For S1,p,q , we obtain q+1 S1,p,q ≤ CG2 γG (q + 1)! 2

sup R/2
≤

q+1 CG2 γG (q 2

≤

CG2 R(p + 1)q! 2

+ 1)!

R 2

(R − r)p+2 ∇p−q x uL2 (Br+ )

q+2

RγG2 2

,± [p − q − 2]!Np−q−2 (u)

q+1

,± [p − q − 2]!Np−q−2 (u).

For S2,p,q we bound q S2,p,q ≤ CG2 γG q! 2

sup R/2
q ≤ CG2 γG q! 2

R 2

(R − r)p+2 ∇p−q x ∇uL2 (Br+ )

q+1

,± [p − q − 1]!NR,p−q−1 (u).

222

5. Regularity Theory in Countably Normed Spaces

Since we can again estimate R (p + 1) ≤ ε R ≤ ε

E min {1, R/E} max {p + 1, R/E}2 , ε E min {1, R/E} max {1, R/E}, ε

we can combine the above bounds to arrive at 1 CG2 E min{1, R/E}× [p]! 2 ε q+1 p p RγG2 ,± 2 q! [p − q − 2]!NR,p−q−2 (u) max {p + 1, R/E} 2 q q=0 q p p RγG2 ,± + max {p + 1, R/E} q! [p − q − 1]!NR,p−q−1 (u) . 2 q q=0

H ≤

2 Proposition 5.5.25 (control of tangential derivatives). Let R ∈ (0, 1]. Let the coeﬃcients A, b, c satisfy (5.5.16). For ε ∈ (0, 1] deﬁne E by (5.5.3) and assume that f satisﬁes (5.5.17) and that G1 , G2 satisfy (5.5.18), (5.5.19). Finally let u be a solution of the transmission problem −ε2 ∇·(A∇u)+b·∇u+cu = f

on BR ,

[ε2 ∂nA u] = ε (G1 + G2 u)

on ΓR .

Then for some K1 > 0 depending only on the constants of (5.5.16) and γf , γG1 , γG2 of (5.5.17)–(5.5.19), there holds max {(p + 3), R/E}p+2 , p ≥ −1, (5.5.33) [p]! (5.5.34) Cu := min {1, R/E}E∇uL2 (BR ) 2 2 + (E/ε) min {1, R/E} Cc uL2 (BR ) + Cf + CG1 min {1, R/E}(E/ε) + CG2 (E/ε) min {1, R/E} min {1, R/E}E∇uL2 (BR ) + uL2 (BR ) .

,± NR,p (u) ≤ Cu K1p+2

Proof: The proof is very similar to the induction argument given in the proof of Proposition 5.5.16. We therefore merely outline the important points. First, we calculate ,± NR,−1 (u) ≤

R 1 ∇uL2 (BR ) ≤ min {1, R/E} max{1, R/E}E∇uL2 (BR ) , 2 2

which shows that the claim is true for p = −1, if K1 ≥ 1. Next, for p = 0, we write the diﬀerential equation as −∇·(A∇u) = ε−2 [f − b · ∇u − cu] with transmission conditions [∂nA u] = G := ε−1 (G1 + G2 u) and employ Lemma 5.5.23 to get

5.5 Local regularity ,± NR,0 (u) ≤ C

R 2

223

Cb R2 Cc R2 2 (B ) + ∇u uL2 (BR ) L R ε ε2 ε2 R R2 R ∇G1 L2 (BR ) + G2 L∞ (BR ) uL2 (BR ) + G1 L2 (BR ) + ε ε ε R2 R2 uL2 (BR ) + G2 L∞ (BR ) ∇uL2 (BR ) . + ∇G2 L∞ (BR ) ε ε f L2 (BR ) +

Exploiting R ≤ 1, R/E = min {1, R/E} max {1, R/E}, Cb /ε−2 ≤ E −1 , and the assumptions on f , G1 , G2 in (5.5.17)–(5.5.19), we obtain for a constant C > 0 with Cu as deﬁned in (5.5.34) ,± NR,0 (u) ≤ C max {1, R/E}2 Cu .

Hence, for suitable K1 , we see that (5.5.33) is indeed valid. The induction argument then proceeds along similar lines as in the proof of Proposition 5.5.16 and is based on Lemma 5.5.23 with G = ε−1 (G1 + G2 u). In addition to the terms that arise in the proof of Proposition 5.5.16 (where the induction is based on Lemma 5.5.15), we have to control the extra term HR,p (ε−1 (G1 + G2 u)) ≤ ε−1 HR,p (G1 ) + ε−1 HR,p (G2 u). The contribution ε−1 HR,p (G2 u) is estimated in Lemma 5.5.24; the estimate of Lemma 5.5.24 has a form that is amenable to the induction argument of the proof of Proposition 5.5.16 with the exception of one point: The ﬁrst sum in the estimate of Lemma 5.5.24 contains the term (corresponding to q = p there) , p p+1 CG2 E γG2 R γG2 R ,± 2 min {1, R/E} max {1, R/E} T := p!NR,−2 + (u), [p]! ε 2 2 which has to be treated separately as the induction hypothesis (5.5.33) does not ,± cover the case p = −2. However, since NR,−2 (u) = uL2 (BR ) and we can bound 2 p+2 p! max {1, R/E} ≤ max {p + 3, R/E} , we get, for any K1 > 0, max {p + 3, R/E}p+2 CG2 (E/ε) min {1, R/E} × T ≤ K1p+2 [p]! , p p+1 γG2 R γG2 R −2 −1 K1 + K1 . 2K1 2K1 This is of a form suitable for the induction argument if we choose K1 suﬃciently large so that the bracketed term is small than 1/2, say. For the other contribution, ε−1 HR,p (G1 ), we estimate

224

5. Regularity Theory in Countably Normed Spaces

ε−1 HR,p (G1 ) . / 1 R−r p sup (R − r)p+1 ∇px G1 L2 (Br+ ) + ∇x ∇G1 L2 (Br+ ) = ε−1 [p − 1]! R/2≤r
. / max {p + 3, R/E}p+2 γG1 p γG1 p+1 CG1 min {1, R/E} + p! 2 2 2

and can then proceeds just as in the proof of Proposition 5.5.16.

Proof of Proposition 5.5.4: Proposition 5.5.4 is now proved just as Proposition 5.5.2 with Proposition 5.5.25 as the induction hypothesis rather than Proposition 5.5.16. 2

5.5.5 Regularity of Neumann problems: Proof of Prop. 5.5.3 The proof of Proposition 5.5.3 is very similar to that of the transmission problem, Proposition 5.5.4, and therefore omitted. One merely has to check that each + − + occurrence of BR ∪ BR or Br+ ∪ Br− can be replaced with BR , Br+ , respectively. The operator [∂nA u] is simply replaced with the co-normal derivative operator ∂nA . For easy of citation, however, we repeat here the analog of Lemma 5.5.23: + + , Sn> ) with 0 < λmin ≤ A. Assume f ∈ L2 (BR ). Lemma 5.5.26. Let A ∈ C 1 (BR + 1 Let G ∈ H (BR ). Then there exists CB > 0 depending only on λmin , AL∞ (B + ) , R RA L∞ (B + ) , and n such that any solution u ∈ H 1 (BR ) of the Neumann probR lem + −∇ · (A∇u) = f on BR , ∂nA u = G on ΓR

satisﬁes R2 ∇2 uL2 (B + ) ≤ (5.5.35) R/2 CB R2 f L2 (B + ) + RGL2 (B + ) + R2 ∇GL2 (B + ) + R∇uL2 (B + ) . R

R

R

R

6. Exponentially Weighted Countably Normed Spaces

6.1 Motivation and outline 6.1.1 Motivation The present Chapter 6 is closely connected with the ensuing Chapter 7 as the regularity theory in exponentially weighted countably normed spaces developed here will be employed in Chapter 7 for the deﬁnition of the corner layers in the asymptotic expansions. The corner layers for problem (1.2.11) are introduced in Chapter 7 to remove an incompatibility between two boundary layer expansions that meet at a vertex. Speciﬁcally, near each vertex Aj , the corner layer is deﬁned as the solution of a transmission problem in a sector near Aj where the jump across an interface and the jump of the co-normal derivative across this interface is chosen so as to match the jumps of the boundary layer expansions. One essential feature of these jumps is that they decay exponentially away from Aj . Thus, we expect the solution to decay exponentially as well. This is indeed the case and in fact, all derivatives decay exponentially away from Aj . In order to make this more precise, the present chapter is devoted to a study of the analytic regularity properties of solutions of transmission problems with exponentially decaying input data. To illustrate the key ideas, let us again consider a sector SR (ω). This sector is divided into two subsectors S + , S − by the curve Γ = {(r cos ω , r cos ω ) | 0 < r < R} for some ﬁxed ω ∈ (0, ω). We then consider the transmission problem −ε2 ∆uε + uε = f [uε ] = h1 ε2 [∂n uε ] = εh2 uε = 0

on S + ∪ S − , on Γ , on Γ , on ∂SR (ω).

(6.1.1a) (6.1.1b) (6.1.1c) (6.1.1d)

Here, the bracket [ · ] represents the jump across Γ ; h1 , h2 are assumed to be smooth on Γ and h1 vanishes at the endpoints of Γ . One essential assumption on the data f , h1 , and h2 is that they decay exponentially away from the origin. This is formalized using the weight functions Ψˆp,β,ε,α deﬁned in (6.2.1) as: ˆp,β,ε (x)eα|x|/ε . Ψˆp,β,ε,α (x) = Φ l are then deﬁned in complete analogy to the Countably normed spaces Bβ,ε,α l l spaces Bβ,ε of Chapter 4. In fact, for α = 0, we recover the spaces Bβ,ε of

J.M. Melenk: LNM 1796, pp. 227–254, 2002. c Springer-Verlag Berlin Heidelberg 2002

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6. Exponentially Weighted Countably Normed Spaces

Chapter 4. As the results of this chapter are mostly of auxiliary nature for the deﬁnition of the corner layers in Chapter 7, we restrict our regularity theory for (6.1.1) with data hi , f for which we have pointwise control of the growth of the derivatives: Ψˆ0,0,ε,α Dp hi L∞ (Γ ) ≤ Ch K p max {p + 1, ε−1 }p

∀p ∈ N0 ,

(6.1.2a)

Ψˆp,1,ε,α ∇p f L∞ (S + ∪S − ) ≤ Cf K p max {p + 1, ε−1 }p

∀p ∈ N0 .

(6.1.2b)

Here, C, K, α > 0 and the diﬀerentiation operator D represents diﬀerentiation in the tangential direction on Γ . The solution uε of the transmission problem (6.1.1) with these input data is then analytic on S + ∪ S − , it is exponentially decaying away from the origin, and it has a singularity at the origin. More precisely, Theorem 6.4.13 asserts that for ﬁxed R < R the solution uε is in the 2 space Bβ,ε,α (S , Cε, γ) for some C, K > 0 and α ∈ (0, α) independent of ε, where S = (S + ∪ S − ) ∩ BR (0). 2 The regularity assertion uε ∈ Bβ,ε,α (S , Cε, γ) can be written more explicitly as

εΨˆ0,0,ε,α ∇uε L2 (S ) + Ψˆ0,0,ε,α uε L2 (S ) ≤ Cε, Ψˆp,β,ε,α ∇p+2 uε L2 (S ) ≤ Cεγ p max {p + 2, ε−1 }p+2

(6.1.3) ∀p ∈ N0 .

(6.1.4)

These estimates, in particular (6.1.3), show that the solution uε is small: (6.1.3) implies that the energy norm of uε is O(ε). A similar observation will be made about the corner layers in Chapter 7, which in turn has ramiﬁcations for the design of FE-meshes in Chapter 3. The proof of this regularity result is similar to that of Proposition 5.3.4. The key observation is that the weight functions Ψˆp,β,ε,α satisfy (6.2.7), i.e., on balls Br (x) with r = c|x| for some c ∈ (0, 1) we have for some C, K > 0 independent of ε, x, and α ≥ 0 min z∈Bc|x| (x)

max z∈Bc|x| (x)

Ψˆp,β,ε,α (z) ≥ CK p e−cα|x|/ε Ψˆp,β,ε,α (x) = CK p Ψˆp,β,ε,(1−c)α (x), (6.1.5a) Ψˆp,β,ε,α (z) ≤ CK p ecα|x|/ε Ψˆp,β,ε,α (x) = CK p Ψˆp,β,ε,(1+c)α (x). (6.1.5b)

These properties of the weight functions Ψˆp,β,ε,α allow us to proceed as in the proof of Proposition 5.3.4 by combing local regularity results into a global one; we proceed in several steps: 1. As a ﬁrst step, a suitable lifting H1 to S + ∪ S − is constructed for the datum h1 so that one may consider (6.1.1) with h1 = 0. Owing to the fact that a straight sector is considered, this lifting is most conveniently constructed in 2 polar coordinates. To show that the lifting are indeed in the spaces Bβ,ε,α , the change of variables from polar coordinates to Cartesian has to be analyzed very carefully. This is done in Section 6.3.

6.1 Motivation and outline

229

2. After the ﬁrst step, we may assume that h1 = 0. The “energy” estimate (6.1.3) is obtained by showing that the bilinear form associated with (6.1.1) satisﬁes an inf-sup condition on pairs of exponentially weighted H 1 -spaces, 1 1 i.e., on pairs Hε,α × Hε,−α . These spaces are the usual Sobolev spaces H 1 equipped with the norms · ε,α , · ε,−α given by uε,α = Ψˆ0,0,ε,α uL2 (S ) + εΨˆ0,0,ε,α ∇uL2 (S ) . Proposition 6.4.6 shows that the bilinear form associated with (6.1.1) sat1 1 isﬁes an inf-sup conditions on the pairs of spaces Hε,α × Hε,−α for every 0 < α < α suﬃciently small. This allows us to obtain in Proposition 6.4.8 for α < α < α uε,α ≤ C Ψˆ0,1,ε,α f L2 (S ) +Ψˆ0,0,ε,α H2 L2 (S ) +εΨˆ1,0,ε,α ∇H2 L2 (S ) , where H2 is an extension of h2 into the sector SR (ω). We note that the right-hand side involves L2 -based bounds on f whereas we stipulate L∞ based bounds on f on (6.1.2). We bound further using α < α Ψˆ0,1,ε,α f L2 (S ) ≤ CΨˆ0,1,ε,α f L∞ (S ) Ψˆ0,0,ε,−(α−α ) L2 (S ) ≤ Cε Ψˆ0,1,ε,α f L∞ (S ) ,

(6.1.6)

where we used Ψˆ0,0,ε,−(α−α ) (x) = e−(α−α )|x|/ε . Similar reasoning allows us to show in Lemma 6.4.11 for the extension H2

Ψˆ0,0,ε,α H2 L2 (S ) + εΨˆ1,0,ε,α ∇H2 L2 (S ) ≤ Cε.

(6.1.7)

Combining (6.1.6), (6.1.7) therefore yields uε ε,α ≤ Cε, for a constant C > 0 independent of ε. The reasoning for the presence of the factor ε can be best seen in our derivation of (6.1.6): We assume L∞ based bounds on the data instead of L2 -based ones. A similar mechanism is responsible for the factor ε in (6.1.7). 3. The next step consists in obtaining bounds in weighted L2 spaces for ∇2 uε . For suitable β ∈ [0, 1) and α < α we have by Proposition 6.4.9 for R ∈ (R , R) upon setting S = (S + ∪ S − ) ∩ BR (0) Ψˆ0,β,ε,α ∇2 uε L2 (S ) ≤ Cε−1 Ψˆ0,1,ε,α f L∞ (SR (ω)) + Ψˆ0,0,ε,α H2 L∞ (SR (ω)) + εΨˆ0,1,ε,α H2 L∞ (SR (ω)) . Comparing this result with the corresponding one for α = 0 in Proposition 5.3.2, we see that we gained again one power of ε. Just as above in our “energy” estimates, this is due to our assumption that L∞ -based bounds on the data f , h2 are available whereas merely L2 -based ones are required.

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6. Exponentially Weighted Countably Normed Spaces

4. The bounds on higher derivatives are now obtained from local regularity akin to our procedure in Proposition 5.3.4. We cover S by balls Bri (xi ) with the key property that ri = c|xi | for some ﬁxed c ∈ (0, 1). Next, we use local regularity results, Proposition 5.5.1 for balls in the interior of S + ∪ S − , Proposition 5.5.2 for half-balls near the outer boundary, and Proposition 5.5.4 for balls with center on Γ , and then sum over all balls to get Ψˆp,β,ε,α ∇p+2 uε L2 (S ) ≤ CK p max {p + 2, ε−1 }p+2 × εΨˆ−1,0,ε,(1+δ)α ∇uε L2 (S ) + Ψˆ0,0,ε,(1+δ)α uε L2 (S ) + εCf + εCh , where S = (S + ∪ S − ) ∩ BR (0) for some suitable R ∈ (R , R) and δ > 0 is essentially determined by the factor c ∈ (0, 1) as in the relations (6.1.5). Just as in the proof of Proposition 5.3.4, the properties (6.1.5) of the weight function Ψˆp,β,ε,α are instrumental for obtaining the above global result by summing local ones. Note that the contributions εCf and εCh from the righthand side f and the data h2 are of size O(ε). The reason for the presence of the factor ε is the same as in the “energy” estimate and the bound on ∇2 uε : For example, the L2 -bounds Ψˆp,β,ε,α ∇p f L2 (S ) are required whereas we stipulate pointwise control of the derivatives of f in (6.1.2). 5. The terms Ψˆ−1,0,ε,(1+δ)α ∇uε L2 (S ) , Ψˆ0,0,ε,(1+δ)α uε L2 (S ) are treated as in Chapter 5 using an embedding theorem, Lemma 6.2.3, to give εΨˆ−1,0,ε,(1+δ)α ∇uε L2 (S ) +Ψˆ0,0,ε,(1+δ)α uε L2 (S ) ≤ Cuε H 2,2

β,ε,(1+δ)α

(S ) .

Finally, this last term can be controlled in terms of f by the bounds above if we choose α so small that (1 + δ)α is suﬃciently small. 6.1.2 Outline of Chapter 6 m,l and the countably Section 6.2.1 starts with the deﬁnition of the spaces Hβ,ε,α l normed spaces Bβ,ε,α . Their main properties are also collected in Section 6.2.1. m,l l and Bβ,ε for the special choice α = 0. These spaces reduce to the spaces Hβ,ε They are deﬁned completely analogously to those introduced in Chapter 4 and share many of their properties. As the proofs of these properties are very similar to those of Chapter 4, most of the arguments are merely outlined. As before, the key results are Hardy-type embedding results (Lemma 6.2.3) and the fact l that the countably normed spaces Bβ,ε,α are (up to a modiﬁcation of α) invariant under analytic changes of variables (Theorem 6.2.6). Results for curvilinear sectors will be inferred from straight sectors via a mapping argument (by appealing to Theorem 6.2.6). As in Chapter 4, pointwise results for the growth of l the derivatives of functions from Bβ,ε,α are made available in Theorem 6.2.7 and Corollary 6.2.8. For the analysis in straight sectors, it is often convenient to introduce polar coordinates. Section 6.3 therefore collects some results concerning the change of variables from polar to Cartesian coordinates.

m,l l 6.2 The exponentially weighted spaces Hβ,ε,α and Bβ,ε,α in sectors

231

In Section 6.4.1 we formulate the transmission problem that we analyze and prove unique solvability of the corresponding variational formulation in exponentially weighted spaces. Section 6.4.4 ﬁnally contains the main result of this chapter, Theorem 6.4.13. This result is again obtained by proving the corresponding result on straight sectors and using a mapping argument by appealing to Theorem 6.2.6. Inhomogeneous boundary and jump conditions are treated using special lifting results in Lemma 6.4.15. m,l l 6.2 The exponentially weighted spaces Hβ,ε,α and Bβ,ε,α in sectors

6.2.1 Properties of the exponentially weighted spaces On a sectors S we deﬁne for p ∈ Z, α ∈ R, β ∈ [0, 1], and ε > 0 the weight functions Ψˆp,β,ε,α by ˆp,β,ε (x), Ψˆp,β,ε,α (x) := eα|x|/ε Φ

(6.2.1)

ˆβ,ε are deﬁned in (4.2.2). We note that where the weight functions Φ ˆp,β,ε , Ψˆp,β,ε,0 = Φ

Ψˆ0,0,ε,α (x) = eα|x|/ε ,

1 Ψˆ0,0,ε,α (x)

= Ψˆ0,0,ε,−α (x).

ˆp,β,ε and an The deﬁnition of the weight function Ψˆp,β,ε,α as a product of Φ exponential function allows us to infer properties of the weights Ψˆp,β,ε,α from ˆp,β,ε . The analog of Lemma 4.2.2 reads as follows: those of the functions Φ Lemma 6.2.1. Let S ∈ R2 be a sector. Then there holds for all p ∈ N0 , ε ∈ (0, 1], β ∈ [0, 1], α ≥ 0, l ∈ N, and x ∈ S: ˆp,0,ε (x)Ψˆ0,β,ε,α (x) ∼ Ψˆp,0,ε,α (x)Φ ˆ0,β,ε (x), Ψˆp,β,ε,α (x) ∼ Φ ˆ−l,β,ε (x) ∼ Φ ˆp,0,ε (x)Ψˆ−l,β,ε,α (x), Ψˆp−l,β,ε,α (x) ∼ Ψˆp,0,ε,α (x)Φ Ψˆp−l,β,ε,α (x) ∼ Ψˆp,β−l,ε,α (x), , p+β 1 min {1, ε(p + 1)} −α|x|/ε 1+ , ∼e |x| Ψˆp,β,ε,α (x) max {p + 1, ε−1 }p ∼ e−α|x|/ε max {(p + 1)/|x|, ε−1 }p . Ψˆp,0,ε,α (x)

(6.2.2) (6.2.3) (6.2.4) (6.2.5) (6.2.6)

Here, the relationship a ∼ b means that there exist C, K > 0 independent of p ∈ N0 , ε ∈ (0, 1], and x ∈ S such that C −1 K −p a ≤ b ≤ CK p a. Furthermore, let c ∈ (0, 1) be given. Then there holds for all balls Bc|x| (x) with x∈S

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6. Exponentially Weighted Countably Normed Spaces

min z∈Bc|x| (x)

max z∈Bc|x| (x)

Ψˆp,β,ε,α (z) ≥ CK p e−c|x|α/ε Ψˆp,β,ε,α (x) = CK p Ψˆp,β,ε,(1−c)α (x),

(6.2.7)

Ψˆp,β,ε,α (z) ≤ CK p ec|x|α/ε Ψˆp,β,ε,α (x) = CK p Ψˆp,β,ε,(1+c)α (x),

(6.2.8)

and also ˆp,β,ε (x) ≤ C −1 K −p e(1−c)α|x|/ε Φ max z∈Bc|x| (x)

min z∈Bc|x| (x)

Ψˆp,β,ε,α (z),

(6.2.9)

ˆp,β,ε (x). Ψˆp,β,ε,α (z) ≤ CK p e(1+c)α|x|/ε Φ

(6.2.10)

l,m The spaces Hβ,ε,α (S) are deﬁned as in Section 4.2: For m, l ∈ N0 , m ≥ l, m,l β ∈ [0, 1), ε ∈ (0, 1], α ≥ 0, and a sector S, the spaces Hβ,ε,α (S) are deﬁned as the completion of the space C ∞ (S) under the norm

u2H m,l

β,ε,α (Ω)

l−1

:=

(6.2.11)

ε Ψˆ0,0,ε,α ∇k u2L2 (Ω)+ε2l 2k

k=0

m

Ψˆk−l,β,ε,α ∇k u2L2 (Ω) ,

k=l

where for l = 0 we used the implicit assumption that an empty sum takes the m,0 value 0; i.e., for l = 0, the spaces Hβ,ε,α are the completion under the norm u2H m,0

β,ε,α (Ω)

:=

m

Ψˆk,β,ε,α ∇k u2L2 (Ω) .

k=0

l (S, Cu , γu ) are For a given sector S and constants Cu , γu > 0, the spaces Bβ,ε,α deﬁned as l,l l (S, Cu , γu ) = {u ∈ Hβ,ε,α (S) | uH l,l Bβ,ε,α

β,ε,α (S)

≤ Cu and

(6.2.12)

Ψˆk,β,ε,α ∇k+l uL2 (S) ≤ Cu γuk max {k + 1, ε−1 }k+l ∀k ∈ N0 }. l,l l and Bβ,ε,α share some invariance properties under smooth The spaces Hβ,ε,α (analytic) changes of variables:

ˆ S be two sectors. Let g ∈ C 2 (S, ˆ S) be a C 2 -diﬀeomorphism Lemma 6.2.2. Let S, satisfying additionally g(0) = 0. Then for ε ∈ (0, 1] and α ∈ R there exist C, ˆ g, and the sign of α such that c > 0 depending only on S, C −1 uH l,l

β,ε,cα (S)

≤ u ◦ gH l,l

ˆ

β,ε,α (S)

≤ CuH l,l

β,ε,α/c

(S) ,

l ∈ {0, 1, 2}.

Proof: The proof is similar to that of Lemma 4.2.6. The key observation is that ˆ there are c1 , c2 > 0 such that c1 |x| ≤ |g(x)| ≤ c2 |x| for all x ∈ S. 2 In order to prove the analog of Theorem 4.2.20, we start by noting that ˆ0,β,ε replaced by Ψˆ0,β,ε,α : Lemma 4.2.10 holds with Φ

m,l l 6.2 The exponentially weighted spaces Hβ,ε,α and Bβ,ε,α in sectors

233

Lemma 6.2.3. Let S be a C 2 -curvilinear sector, β ∈ (0, 1), α ≥ 0, l ∈ {1, 2}. Then there exists C > 0 depending only on S, β, α, and l such that for every l,l ε ∈ (0, 1] and every u ∈ Hβ,ε,α (S) there exists u ∈ R such that: (i) if l = 2 the constant u may be taken as u = u(0) and there holds Ψˆ0,β−2,ε,α (u − u) L2 (S∩B2ε (0)) ≤ εΨˆ0,β−1,ε,α ∇uL2 (S∩B2ε (0)) , , εΨˆ0,β−1,ε,α ∇uL2 (S∩B (0)) ≤ Cu 2,2 2ε

Hβ,ε,α (S∩B2ε (0))

Ψˆ0,β−2,ε,α uL2 (S\Bε (0)) + εΨˆ0,β−1,ε,α ∇uL2 (S\Bε (0)) ≤ CuH 2,2

β,ε,α (S)

;

(ii) if l = 1: Ψˆ0,β−1,ε,α (u − u) L2 (S∩B2ε (0)) ≤ CεΨˆ0,β,ε,α ∇uL2 (S∩B2ε (0)) ≤ CuH 1,1

β,ε,α (S)

Ψˆ0,β−1,ε,α uL2 (S\Bε (0)) ≤ CuH 1,1

β,ε,α (S)

.

Proof: The proof consists in noting that on S ∩ B2ε (0), the weights Ψˆ0,β,ε,α and ˆ0,β,ε are equivalent and then appealing to Lemma 4.2.10. The assertions on the Φ sets S \ Bε (0) are obvious. 2 l Lemma 6.2.4 (local characterization of Bβ,ε,α ). Let S be a sector, l ∈ N0 , β ∈ (0, 1), ε > 0, α ≥ 0. Let B = {Bi | i ∈ N} be a collection of balls Bi = Bri (xi ) with the following properties:

1. there exists c ∈ (0, 1) with ri = c|xi | for all i ∈ N; 2. there exists N ∈ N such that card {i ∈ N | x ∈ Bi } ≤ N for all x ∈ S. l Let f ∈ Bβ,ε,α (S, Cf , γf ). Then there exist C, γ > 0 depending only on γf such that for all p ∈ N0 , i ∈ N there holds

Ψˆp,β,ε,(1−c)α (xi )∇p+l f L2 (S∩Bi ) ≤ CC(i)γ p max {p + 1, ε−1 }p+l , ∞ 4 C 2 (i) ≤ Cf2 N < ∞, 3 i=1

(6.2.13) (6.2.14)

where the constants C(i) are given by C 2 (i) :=

∞ Ψˆp,β,ε,α ∇p+l f 2L2 (S∩Bi ) p=0

max {p +

1, ε−1 }2(p+l)

1 4 ≤ Cf2 < ∞. 2p (2γf ) 3

Proof: The proof is very similar to that of Lemma 4.2.17.

(6.2.15) 2

Next, we formulate the analogue of Lemma 4.2.19. l Lemma 6.2.5. Let S be a sector, c0 > 0 be given. Let u ∈ Bβ,ε,α (S, Cu , γu ) for some l ∈ {0, 1, 2}, β ∈ (0, 1), α ≥ 0, ε > 0, and Cu , γu > 0. Then there exist constants C, γ > 0 independent of ε and a constant u ∈ R such that

,

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6. Exponentially Weighted Countably Normed Spaces

Ψˆp−l,β,ε,α ∇p (u − u)L2 (S∩Bc0 ε (0)) ≤ CCu max {p + 1, ε−1 }p

∀p ∈ N0 ,

−1 p

∀p ∈ N0 .

Ψˆp−l,β,ε,α ∇ uL2 (S\Bc0 ε (0)) ≤ CCu max {p + 1, ε p

}

Moreover, in the case l = 2, the constant u may be taken as u = u(0) and for l = 0 we may take u = 0. Proof: The proof is analogous to that of Lemma 4.2.17.

2

l For functions from the countably normed spaces Bβ,ε,α , we have a result analogous to Theorem 4.2.20: l Theorem 6.2.6 (invariance of Bβ,ε,α under changes of variables). Let S 2 be a C -curvilinear sector and g : S → g(S) ⊂ R2 be analytic on S, g(0) = 0, and assume that g −1 is analytic on g(S). Let Cu , γu > 0, β ∈ (0, 1), α ≥ 0. Then there exist constants C, γ > 0, c > 0 depending only on g, S, β, and γu (in particular, they are independent of ε) such that for l ∈ {0, 1, 2} l l u ∈ Bβ,ε,α (g(S), Cu , γu ) =⇒ u ◦ g ∈ Bβ,ε,cα (S, CCu , γ).

Proof: The proof is similar to that of Theorem 4.2.20.

2

Finally, we have the analogue of Theorem 4.2.23: l Theorem 6.2.7 (pointwise control of Bβ,ε,α -functions). Let S be an anl alytic sector, l ∈ {0, 1, 2}, u ∈ Bβ,ε,α (S, Cu , γu ) for some Cu , γu > 0 and β ∈ (0, 1), ε > 0, α ≥ 0. Then for every neighborhood U of Γ3 there exist C, γ > 0, c ∈ (0, 1) independent of ε and Cu such that on S := S \ U ) ) ) Ψˆ ) ) p−l+1,β,ε,cα p ) ≤ CCu γ p max {(p + 1), ε−1 }p+1 ∀p ∈ N. ∇ u) ) ) max {1, |x|/ε} ) ∞ L

(S )

This estimate is also valid for p = 0 if either l = 0 or l = 2 together with u(0) = 0. A more convenient form of Theorem 6.2.7 is the following corollary. l -functions). Assume the hyCorollary 6.2.8 (pointwise control of Bβ,ε,α potheses of Theorem 6.2.7 and additionally α ∈ (0, 1]. Then there exist C, γ > 0, c ∈ (0, 1) independent of ε such that for all p ∈ N and x ∈ S there holds p −(p+2)

|∇ u(x)| ≤ Cp!γ α p

−(p+1)

|x|

l−β |x| min 1, e−cα|x|/ε . (6.2.16) ε

For l = 0, this estimate is also valid for p = 0. For the special case l = 2 and u(0) = 0, we have |u(x)| ≤ Cα−2 ε−1 min {1, |x|/ε}1−β e−cα|x|/ε .

(6.2.17)

6.3 Change of variables: from polar to Cartesian coordinates

235

Proof: From Lemma 6.2.1, we have that max{1, |x|/ε}

max {p + 1, ε−1 }p+1 Ψˆp−l+1,β,ε,α (x) ∼

max {1, |x|/ε} −α|x|/ε max {(p + 1)/|x|, ε−1 }p+1 . e ˆ−l,β,ε (x) Φ

Next, we bound max {1, |x|/ε} max {(p + 1)/|x|, ε−1 }p+1 ≤ |x|−(p+1) max {1, |x|/ε} max {p + 1, |x|/ε}p+1 ≤ |x|−(p+1) max {p + 1, |x|/ε}p+2 and

e−α|x|/(2ε) max {p + 1, |x|/ε}p+2 ≤ Cγ p α−(p+2) p!

for some appropriate C, γ > 0 independent of x, ε, α. Combining the above considerations, we obtain |∇p u(x)| ≤ Cγ p α−(p+2) e−α|x|/(2ε) |x|−p−1

1 . ˆ Φ−l,β,ε (x)

The bound (6.2.16) now follows for p ≥ 1. It also holds for p = 0 together with l = 0 and p = 0 together with u(0) = 0 for the case l = 2 by Theorem 6.2.7. In the latter case, we simplify the estimate as follows: |u(x)| ≤ Cγα−2 |x|−1 min {1, |x|/ε}2−β e−cα|x|/ε ≤ Cγα−2 ε−1 (ε/|x|) min {1, |x|/ε}2−β e−cα|x|/ε ≤ Cγα−2 ε−1 min {1, |x|/ε}1−β e−cα|x|/ε . 2 Finally, the following lemma is useful. Lemma 6.2.9. Let S be a sector, β ∈ (0, 1), δ > 0. Then there exists C > 0 depending only on β, δ, and S such that Ψˆ0,−β,ε,−δ L2 (S) ≤ Cε. Proof: The proof follows by a direct calculation.

2

6.3 Change of variables: from polar to Cartesian coordinates Lemma 6.3.1. Let S be a sector and let g : S → PS be the change of variables from Cartesian to polar coordinates, (r, ϕ) = g(x, y). Let f1 , f2 be two functions, analytic on PS , that satisfy for all (p, s) ∈ N0 and (r, ϕ) ∈ PS the estimates:

236

6. Exponentially Weighted Countably Normed Spaces

p s ∂r ∂ϕ fi (r, ϕ) ≤ CK p+s p!s!r−p max {1, r/(ε(p + 1))}p rti e−αi r/ε ,

i = 1, 2,

where C, K > 0, and αi , ti ∈ R are independent of ε. Then there exist C , γ > 0 depending only on C, K, and S such that the product (f1 · f2 ) ◦ g satisﬁes for all p ∈ N0 and all (x, y) ∈ S: |∇p(x,y) ((f1 · f2 ) ◦ g)(x, y)| ≤ C r−p γ p p! max {1, r/(ε(p + 1))}p rt1 +t2 e−(α1 +α2 )r/ε where, as usual, r = r(x, y). Proof: The proof is based the Cauchy integral theorem for derivatives. We introduce the polydisc D(κr, κ) := {(zr , zϕ ) ∈ C2 | |zr | < κr, |zϕ | < κ},

where κ :=

1 , 2K

and claim that the fi are in fact holomorphic on the polydisc (r, ϕ) + D(κr, κ) for (r, ϕ) ∈ PS . In order to see this, we ﬁrst note that we have for |zr | ≤ κr, |zϕ | ≤ κ the bounds 1 K p p!r−p max {1, r/(ε(p + 1))}p |zr |p ≤ 3eK|zr |/(rε) , p!

p∈N0

1 1 K s s!|zϕ |s ≤ ≤ 2. s! 1 − Kκ

s∈N0

(The ﬁrst bound is obtained by splitting the sum into a part from 0 to r/ε and one from r/ε to ∞—the ﬁrst sum can be majorized by an exponential series, the second one by a geometric series). From these two estimates it follows immediately that the Taylor series of the functions fi at the point (r, ϕ) converge on the polydisc (r, ϕ) + D(κr, κ); furthermore, we get the bounds |fi (r + zr , ϕ + zϕ )| ≤ Crti e−αi r/ε 6eK|zr |/ε

∀(zr , zϕ ) ∈ D(κr, κ),

i = 1, 2.

Next, the map g is an analytic map and there exists a Cg > 0 suﬃciently large and a C˜g > 0 suﬃciently small such that for (x, y) ∈ S and all zx , zy ∈ C with |zx | + |zy | ≤ C˜g r(x, y): |r(x + zx , y + zy ) − r(x, y)| ≤ Cg (|zx | + |zy |) , |zx | + |zy | |ϕ(x + zx , y + zy ) − ϕ(x, y)| ≤ Cg . r(x, y) We are now in position to apply Cauchy’s integral theorem for derivatives of (f1 · f2 ) ◦ g: Let (x, y) ∈ S and let δ > 0 (to be chosen below) be such that 0 < 2δ < κr/Cg ,

where r = r(x, y).

(6.3.18)

This choice of δ guarantees that the function (f1 · f2 ) ◦ g is holomorphic on the polydisc {(zx , zy ) ∈ C2 | |zx − x| < δ, |zy − y| < δ}. Cauchy’s integral formula for derivatives gives for (s, t) ∈ N20

6.3 Change of variables: from polar to Cartesian coordinates

237

∂xs ∂yt ((f1 · f2 ) ◦ g)(x, y) = s!t! ((f1 · f2 ) ◦ g)(x + z1 , y + z2 ) − 2 dz1 dz2 4π |z1 |=δ |z2 |=δ (−z1 )s+1 (−z2 )t+1 and therefore s t ∂x ∂y ((f1 · f2 ) ◦ g)(x, y) ≤ C s!t! e−(α1 +α2 )r/ε rt1 +t2 e2Kδ/ε . δ s+t Setting p := s + t and choosing δ such that 2δ =

1 min {(p + 1)ε, κr/Cg }, 2

we obtain for some C, γ > 0 independent of r, ε, and p: . / s t ∂x ∂y ((f1 · f2 ) ◦ g)(x, y) ≤ Crt1 +t2 e−(α1 +α2 )r/ε γ p p! + max (p, ε−1 )p . rp This last estimate can readily be brought to the desired form.

2

Lemma 6.3.2. Let U ⊂ R be a bounded neighborhood of 0 and G ⊂ R be a bounded open set. Let gi , i ∈ {1, 2}, be analytic on U × G and satisfy for some ε ∈ (0, 1], C, K > 0, αi ∈ R, i ∈ {1, 2}: p s (∂r ∂ϕ gi )(r, ϕ) ≤ CK p+s s! max {p, ε−1 }p e−αi r/ε ∀(p, s) ∈ N20 ∀(r, ϕ) ∈ U ×G. Then the following holds: (i) The product g1 · g2 is analytic on U × G and there exist C , γ > 0 depending only on C, K such that for all (p, s) ∈ N20 and all (r, ϕ) ∈ U × G p s (∂r ∂ϕ (g1 · g2 ))(r, ϕ) ≤ C γ p+s s! max {p, ε−1 }p e−(α1 +α2 )r/ε . (ii) If g1 (0, ϕ) = 0 for all ϕ ∈ G, then the function h(r, ϕ) := 1r g1 (r, ϕ)g2 (r, ϕ) is analytic on U × G and there exist C , γ > 0 depending only on C, K, and α1 , such that for all (p, s) ∈ N20 and all (r, ϕ) ∈ U × G p s ∂r ∂ϕ h(r, ϕ) ≤ ε−1 C γ p+s s! max {p, ε−1 }p e−(α1 +α2 )r/ε . (iii) If α1 + α2 > 0 then for every 0 < α < α1 + α2 there exist constants C, K > 0 such that the function h(r, ϕ) := ε−1 rg1 (r, ϕ)g2 (r, ϕ) satisﬁes for all (p, s) ∈ N20 p s ∂r ∂ϕ h(r, ϕ) ≤ C γ p+s s! max {p, ε−1 }p e−α r/ε ∀(r, ϕ) ∈ U × G. Proof: We will only show assertion (ii); the other two are proved similarly. Deﬁne the polydisc D := {(zr , zϕ ) ∈ C2 | |zr | < 1/(2K), |zϕ | < 1/(2K)}. As

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6. Exponentially Weighted Countably Normed Spaces

in the proof of the Lemma 6.3.1, we conclude by Taylor expansions that the functions gi satisfy for (r, ϕ) ∈ U × G and (zr , zϕ ) ∈ D |gi (r + zr , ϕ + zϕ )| ≤ |gi (r, ϕ)| + 6C|zr |e−αi r/ε eK|zr |/ε ,

i ∈ {1, 2}. (6.3.19)

Without loss of generality, assume that Bε (0) ⊂ U . Then, as g1 (0, ϕ) = 0 for all ϕ ∈ G, we can bound 1 | g1 (r, ϕ)| ≤ Cε−1 ≤ Cε−1 e−α1 r/ε r 1 | g1 (r, ϕ)| ≤ Cε−1 e−α1 r/ε r

for (r, ϕ) ∈ Bε (0) × G, for (r, ϕ) ∈ (U \ Bε (0)) × G.

Combining this with (6.3.19), we obtain for all (r, ϕ) ∈ U × G and (zr , zϕ ) ∈ D: |zr | |r|/ε + . |h(r + zr , ϕ + zϕ )| ≤ Ce−(α1 +α2 )r/ε e2K|zr |/ε |r + zr | |r + zr | Hence, from Cauchy’s integral formula for derivatives we obtain for δ1 , δ2 > 0 to be chosen suﬃciently small below: h(r + zr , ϕ + zϕ ) p!s! p s ∂r ∂ϕ h(r, ϕ) = − 2 dzr dzϕ , 4π |zr |=δ1 |zϕ |=δ2 (−z1 )s+1 (−z2 )t+1 p s |r|/ε |zr | −(α1 +α2 )r/ε 2Kδ1 /ε ∂r ∂ϕ h(r, ϕ) ≤ C p!s! . e e sup + δ1p δ2s |r + zr | |r + zr | |zr |=δ1 We now choose δ1 and δ2 . For ﬁxed r ∈ U , we set 1 , 4K δ := min {ε(p + 1), 1/(12K)}, ! δ if 0 < δ < r/2 δ1 := if δ ≥ r/2. 3δ δ2 :=

It easy to see that then for |zr | = δ1 , we have the bounds |zr | |r|/ε + ≤ Cε−1 |r + zr | |r + zr | for some C > 0 independent of r, ε, and p. The remainder of the proof is the same as in Lemma 6.3.1. 2

6.4 Analytic regularity in exponentially weighted spaces 6.4.1 Transmission problem: problem formulation Let S be a sector (cf. Deﬁnition 4.2.1) and Γ be a (smooth) curve passing through the origin such that Γ divides S into two Lipschitz domains S + , S − . In the

6.4 Analytic regularity in exponentially weighted spaces

239

present section, we are interested in (analytic) regularity results for transmission problems of the following type: on S + ∪ S − , on ∂S,

−ε2 ∇ · (A(x)∇u) + c(x)u = f u=g [u] = h1

on Γ , on Γ ,

ε[∂nA u] = h2

(6.4.1a) (6.4.1b) (6.4.1c) (6.4.1d)

where f ∈ H −1 (S), g ∈ H 1/2 (∂S), h1 ∈ H00 (Γ ), h2 ∈ H −1/2 (Γ ). We will make additional regularity assumptions on these data shortly. The bracket operator [ · ] stands for the jump across the curve Γ , and the co-normal operator ∂nA u is shorthand for nT A∇u where n stands for the outer normal vector of S + . In view of the analytic regularity results that we seek later, we assume that the coeﬃcients A ∈ A(S + ∪ S − , S2> ), c ∈ A(S + ∪ S − ) satisfy 1/2

p ∇p AL∞ (S + ∪S − ) ≤ CA γA p! ∀p ∈ N0 , + 0 < λmin ≤ A on S ∪ S − , p p ∇ cL∞ (S + ∪S − ) ≤ Cc γc p! ∀p ∈ N0 ,

0<λ≤c

−

on S ∪ S . +

(6.4.2a) (6.4.2b) (6.4.2c) (6.4.2d)

The weak formulation of (6.4.1) reads: Find u ∈ H 1 (S + ∪ S − ) such that bε (u, v) = ε h2 v ds + f v dx ∀v ∈ H01 (S) (6.4.3a) Γ

u=g [u] = h1

S

in H 1/2 (∂S + \ Γ ),

u=g

in H 1/2 (∂S − \ Γ ),

in H 1/2 (Γ ),

(6.4.3b) (6.4.3c)

where the bilinear form bε is given by ε2 ∇u · (A(x)∇v) + c(x)uv dx. bε (u, v) :=

(6.4.4)

S + ∪S −

In order to see that solutions of (6.4.1) exist, we note that the assumptions on g and h1 imply the existence of u0 ∈ H 1 (S) with u0 = g on ∂S and u1 ∈ H 1 (S + ) with u1 = h1 on Γ , u1 = 0 on ∂S + \ Γ . Hence, the solution u of (6.4.1) can be sought in the form ! u1 on S + u=u ˜ + u0 + 0 on S − , where u ˜ ∈ H01 (S) satisﬁes for all v ∈ H01 (S) bε (˜ u, v) = f v dx + ε h2 v ds − bε (u0 , v) − S

Γ

S+

ε2 ∇u1 · (A(x)∇v) + c(x)u1 v dx.

It is now easy to see that the Lax-Milgram Lemma can be applied to infer the existence of a unique solution u ˜ ∈ H01 (S) to this last variational problem. We

240

6. Exponentially Weighted Countably Normed Spaces

2,2 now show that a Hβ,1 regularity result holds as well. To do so, we consider 0,0 1,1 (6.4.1) with g = 0, h1 = 0, f ∈ Hβ,1 (S), and h2 = H|Γ , where H ∈ Hβ,1 (S + ). f 1 generates a bounded linear functional on H (S) by Lemma 5.3.7. From [15], we have that also h2 ∈ H −1/2 (Γ ):

Lemma 6.4.1. Let S be a straight sector, β ∈ (0, 1), and let be Γ1 one of the sides of S passing through the origin. Then there exists C > 0 such that for all 1/2 1,1 (S) there holds for the trace h := H|Γ1 ∈ Hloc (Γ1 ): H ∈ Hβ,1 ˆ0,β/2,1 hL2 (Γ ) ≤ CH 1,1 , Φ 1 Hβ,1 (S) hv ds ≤ CHH 1,1 (S) vH 1 (S)

∀v ∈ H 1 (S).

β,1

Γ1

2

Proof: The lemma is taken from [15, Lemmata 2.9, 2.11]. In this situation, we have the following regularity assertion.

Lemma 6.4.2. Let S := SR (ω) be a straight sector, Γ = {(r cos ω , r sin ω ) | 0 < r < R} for some ω ∈ (0, ω). Γ splits SR (ω) into two sectors S + , S − . Let A+ , A− ∈ S2> and deﬁne A by A|S + = A+ , A|S − = A− . Let R < R. Then there exist 0,0 1,1 β ∈ [0, 1) and C > 0 such that for every f ∈ Hβ,1 (S) and every H ∈ Hβ,1 (S + ), the problem (6.4.1) with c = 0, g = 0, h1 = 0, and h2 = H|Γ has a unique solution u ∈ H01 (S), which satisﬁes uH 1 (S) + uH 2,2 ((S + ∪S − )∩B (0)) R β,1 ≤ C f H 0,0 (S) + HH 1,1 (S + ) . β,1

(6.4.5)

β,1

Proof: We ascertained above that the weak solution u ∈ H01 (S) exists. From the Lax-Milgram theorem and Lemma 6.4.1 we get for every β ∈ [0, 1) the existence of a constant Cβ > 0 such that uH 1 (S) ≤ Cβ f H 0,0 (S) + HH 1,1 (S + ) . (6.4.6) β,1

β,1

By our local regularity results, (cf. Propositions 5.5.1, 5.5.2, 5.5.4), it suﬃces to show the weighted H 2 -estimate in a neighborhood of the origin that may be chosen suﬃciently small. To that end, let χ be a smooth cut-oﬀ function supported by B2δ (0) with χ ≡ 1 on Bδ (0) for δ > 0 suﬃciently small. A calculation then shows that u ˜ := uχ satisﬁes the transmission problem −∇ · (A∇˜ u) = f˜ := χf − 2∇χ · A∇u − u∇ · (A∇χ) u ˜ = 0 on ∂S, ˜ 2 := χh2 + u[∂ χ] on Γ . [∂nA u ˜] = h nA

on S,

Clearly, f˜H 0,0 (S) ≤ Cf H 0,0 (S) + CHH 1,1 (S + ) . In view of a) [∂nA χ] = 0 β,1

β,1

β,1

near the vertex, b) u is in H 2 ((S + ∪ S − ) \ Bδ (0)), and c) the bound (6.4.6), we ˜ 2 can be written in the form h ˜ 2 = H| ˜ Γ , where see that h

6.4 Analytic regularity in exponentially weighted spaces

241

˜ 1,1 + ≤ C f 0,0 H H (S ) H (S) + HH 1,1 (S + ) . β,1

β,1

β,1

It suﬃces to establish the bound (6.4.5) for u ˜ instead of u. We achieve this by transforming the problem to the particular form considered in Proposition A.2.1. We refer to Lemma A.1.1 for how diﬀerential equations change under bi-Lipschitz changes of variables. As the matrices A+ , A− are both elements of S2> , they can be simultaneously diagonalized by similarity transformations; i.e., there is an aﬃne transformation F : x → F (x) := F x such that F T A+ F = Id and F T A− F = D for a diagonal matrix D = diag (d1 , d2 ) (cf., e.g., [57]). The images of S + , S − , Γ under this transformation may again be denoted by S + , S − , Γ ; also the transforms of the ˜ 2 are denoted again f˜, h ˜ 2 . The original problem problem has been data f˜, h transformed to one of the form −p1 ∆˜ u+ = f˜ ˜2 p1 ∂n u+ − nT D∇˜ u− = h

−∇ · (D∇˜ u− ) = f˜

on S + , on Γ .

on S − ,

In order to obtain the form of Proposition A.2.1, we use the aﬃne change of variables T + , T − on the sectors S + and S − with the additional property that T + |Γ = T − |Γ . They are chosen as follows: T − is deﬁned by (x, y) → (x, (d1 /d2 )1/2 y) on S − . This transforms the diﬀerential operator on S − to one of the form p2 ∆u, which is the desired form. For S + , it is easy to ﬁnd d > 0 and an orthogonal matrix O such that the map T + (x) = dOx satisﬁes T + |Γ = T − |Γ . Thus, the diﬀerential equation on S + preserves its form, and we get the desired form for an application of Proposition A.2.1. The existence of β ∈ [0, 1) and the bound (6.4.5) then follows for the transformed function u ˜. Since all changes of variables were piecewise aﬃne, the desired weighted H 2 -bound for the original u ˜ follows. 2 Before extending the regularity assertion of Lemma 6.4.2 to variable coeﬃcients, we make the following observation: Remark 6.4.3 Assume the hypotheses of Lemma 6.4.2. Let for some δ > 0 the two sectors S +,δ := S + ∩ {(r cos ϕ, r sin ϕ) | 0 < r < R, ω − δ < ϕ < ω + δ}, S −,δ := S − ∩ {(r cos ϕ, r sin ϕ) | 0 < r < R, ω − δ < ϕ < ω + δ} be given. Let β be given by Lemma 6.4.2. Then for h2 of the form h2 = H + |Γ + 1,1 1,1 (S +,δ ), H − ∈ Hβ,1 (S −,δ ), we have the bound (for a constant H − |Γ for H + ∈ Hβ,1 C depending additionally on δ) uH 1 (S) + uH 2,2 ((S + ∪S − )∩B (0)) R β,1 + ≤ C f H 0,0 (S) + H H 1,1 (S +,δ ) + H − H 1,1 (S −,δ ) . β,1

β,1

β,1

To see this, we observe that “reﬂecting” the data H − at the line Γ yields a ˜ − with H ˜ − |Γ = H − |Γ and H ˜ − 1,1 +,δ = H − 1,1 −,δ . If function H H (S ) H (S ) β,1

β,1

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6. Exponentially Weighted Countably Normed Spaces

˜ − to an H 1,1 (S + )-function with δ < ω − ω , we can extend the function H + + H β,1 the aid of Lemma 5.4.1. The stated bound now follows from Lemma 6.4.2. We can now extend this result to the case of variable coeﬃcients A, c: Proposition 6.4.4 (weighted H 2 -regularity of transmission problems). Assume that (6.4.2) holds and let 0 < R < R , ω ∈ (0, 2π), ω ∈ (0, ω). Set S := SR (ω) and Γ = {(r cos ω , r sin ω ) | 0 < r < R}. Then there exist β ∈ [0, 1) 0,0 1,1 and C > 0 such that for every f ∈ Hβ,1 (S), H ∈ Hβ,1 (S + ) the problem (6.4.1) with g = 0, h1 = 0, h2 = H|Γ has a unique solution u ∈ H01 (S), which satisﬁes uH 1 (S) + uH 2,2 ((S + ∪S − )∩B (0)) ≤ C f H 0,0 (S) + HH 1,1 (S + ) . R

β,1

β,1

β,1

Proof: The proof consists in a perturbation argument akin to the one in the proofs of Proposition 5.2.2 and Lemma 5.4.2. First, we note that the solution u ∈ H01 (S) exists by Lax-Milgram. Next, as in the proof of Proposition 5.2.2, we have for an H01 -solution u of the transmission problem −∇ · (A∇u) = f

[∂nA u] = h2 = H + |Γ + H − |Γ

on S,

on Γ

1,1 (S + ), the following weighted H 2 -bound, if R ∈ (R, R ) is ﬁxed and H + ∈ H1,1 1,1 H − ∈ H1,1 (S − ) (see also Proposition 5.5.4 for suitable local estimates):

|x|∇2 uL2 (S + ∪S − ) ≤ C |x|f L2 (SR ) + ∇uL2 (SR )

(6.4.7) + H + L2 (S + ) + |x|∇H + L2 (S + ) + H − L2 (S − ) + |x|∇H − L2 (S − ) . R

R

R

R

R

We introduce the piecewise constant matrix  + A˜ := lim A(x)    x→0 x∈S + A˜ := ˜−  lim A(x)  A := x→0 x∈S

R

for x ∈ S + , for x ∈ S − .

−

Since by assumption A+ := A|S + ∈ C 1 (S + , S2> ), A− := A|S + ∈ C 1 (S + , S2> ), there exists C > 0 such that ˜ |A(x) − A(x)| ≤ C|x| ∀x ∈ S + ∪ S − , |n+ · (A+ (x) − A˜+ )| ≤ C|x|, |n− · (A− (x) − A˜− )| ≤ C|x|

∀x ∈ Γ.

where n+ , n− denote the outer normal vectors for S + , S − , respectively. Next, we observe that the solution u ∈ H01 (S) satisﬁes the equation ˜ ˜ ∇u + (A − A) ˜ : ∇2 u −∇ · (A∇u) = f˜ := f − cu + ∇ · (A − A) on S + ∪ S − , ˜ 2 := H ˜ + |Γ + H ˜ − |Γ [∂nA˜ u] = h

on Γ ,

6.4 Analytic regularity in exponentially weighted spaces

243

˜ + , deﬁned on S + , and H ˜ − , deﬁned on S − , are given by where H ˜ + := H − n+ · (A+ − A˜+ )∇u+ , H

˜ − := −n− · (A− − A˜− )∇u− ; H

as in Lemma 5.4.2, the functions n+ , n− are suitably deﬁned on the sectors S + , S − . The argument can be completed using Remark 6.4.3 and by bounding ˜ + 0,0 + , H ˜ − 0,0 − with the aid of (6.4.7). 2 f˜H 0,0 (S ) , H H (S ) H (S ) β,1

R

β,1

R

β,1

R

6.4.2 Transmission problem in exponentially weighted spaces We are interested in solutions u of (6.4.1) that decay exponentially away from the origin. In order to treat such solutions in a variational framework, it is convenient 1 to introduce the exponentially weighted spaces Hε,α . For ε > 0, α ∈ R, and a 2 1 sector S ⊂ R we deﬁne the spaces Hε,α (S) as the usual space H 1 (S) equipped with the norm · ε,α given by u2ε,α := ε2 Ψˆ0,0,ε,α ∇u2L2 (S) + Ψˆ0,0,ε,α u2L2 (S) . Similarly, the space H01 (S) equipped with the norm · ε,α is denoted by 1 H0,ε,α (S). The following lemma gives a convenient characterization of the spaces 1 1 Hε,α (S), H0,ε,α (S): Lemma 6.4.5. Let S be a sector, α ∈ R, ε ∈ (0, 1]. Then there exists C > 0 independent of ε such that for all u ∈ H 1 (S) there holds C −1 uε,α ≤ εΨˆ0,0,ε,α uH 1 (S) + Ψˆ0,0,ε,α uL2 (S) ≤ Cuε,α . Proof: Denoting r = |x|, we have Ψˆ0,0,ε,α u = eαr/ε u. Hence, |∇(eαr/ε u)| ≤ |α|/ε|eαr/ε u|+|eαr/ε ∇u|. This gives the upper bound. The lower bound is proved similarly. 2 The key tool for our regularity theory in exponentially weighted spaces is the assertion that the bilinear form bε of (6.4.4) satisﬁes an inf-sup condition on 1 1 appropriate spaces H0,ε,α × H0,ε,−α : 1 1 Proposition 6.4.6 (inf-sup condition on H0,ε,α ×H0,ε,−α ). Let S be a sector, bε be the bilinear form deﬁned in (6.4.4) with coeﬃcients A, c satisfying (6.4.2). Then for α ≥ 0 with 2 0 ≤ α2 CA < λmin λ (6.4.8) 1 1 there holds for u ∈ H0,ε,α (S) and v ∈ H0,ε,−α (S):

inf sup

0=u 0=v

2 2 bε (u, v) λmin λ − CA α 2 # ≥ √ , 2 2 2 uε,α vε,−α 3 + 4α λmin + λ + 4CA α + (λmin − λ)2 |bε (u, v)| ≤ max {CA , Cc }uε,α vε,−α .

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6. Exponentially Weighted Countably Normed Spaces

0,0 1,1 Furthermore, for every f ∈ H1,ε,α (S) and every h ∈ Hε,α+δ (S) with δ > 0, we have 1 f v dx ≤ CΨˆ0,1,ε,α f L2 (S) vε,−α ∀v ∈ H0,ε,−α (S), S 1 ε ≤ Cδ h 1,1 hv ds ∀v ∈ H0,ε,−α (S), H (S) vε,−α Γ

1,ε,α+δ

where C > 0 depends only on S and Cδ depends only on S, α, and δ > 0. 1 Proof: We start with the proof of the inf-sup condition. Given u ∈ Hε,α (S), we set v := Ψˆ0,0,ε,2α u = e2αr/ε u, where r = r(x) = |x|. We note that ∇v = e2αr/ε ∇u + 2α/ε∇re2αr/ε u. From this, it is easy to get # (6.4.9) vε,−α ≤ 3 + 4α2 uε,α .

Next, we compute 2 (A(x)∇u)∇v + c(x)uv dx = bε (u, v) = ε S ε2 e2αr/ε (A(x)∇u)∇u dx + 2αε e2αr/ε (A(x)∇u)(∇r)u dx+ c e2αr/ε u2 dx S

S

S

≥ λmin ε Ψˆ0,0,ε,α ∇u2L2 (S) 2

− 2αCA εΨˆ0,0,ε,α ∇uL2 (S) Ψˆ0,0,ε,α uL2 (S) + λΨˆ0,0,ε,α u2L2 (S) ≥ (λmin − αCA δ) ε2 Ψˆ0,0,ε,α ∇u2L2 (S) + (λ − αCA /δ) ε2 Ψˆ0,0,ε,α u2L2 (S) ≥ min {(λmin − αCA δ) , (λ − αCA /δ)}u2ε,α for all δ > 0. It remains to choose δ appropriately. It is not hard to see that α needs to satisfy the constraint (6.4.8) in order for the minimum to be nonnegative. Next, given α satisfying this constraint, we choose δ such that the minimum is as large as possible, i.e., we choose δ such that λmin − αCA δ = λ − αCA /δ. Elementary calculations then show that this choice of δ leads to min {(λmin − αCA δ) , (λ − αCA /δ)} =

2 2 α ) 2(λmin λ − CA # , 2 2 λmin + λ + 4CA α + (λmin − λ)2

which, combined with (6.4.9), gives the desired inf-sup condition. The continuity 1 1 of the bilinear form bε on the spaces Hε,α × Hε,−α follows readily from the Cauchy-Schwarz inequality. For the last two estimates, we start by noting that Lemma 4.2.2 implies 1 ε 1 ˆ Ψˆ0,0,ε,−α (x), (x) ≤ C 1 + = Ψ ˆ0,1,ε (x) 0,0,ε,−α d(x) Ψˆ0,1,ε,α (x) Φ 1 where d(x) = dist(x, ∂S). Hence, we get with Lemma 4.2.12 for v ∈ H0,ε,−α (S):

6.4 Analytic regularity in exponentially weighted spaces −1 Ψˆ0,1,ε,α vL2 (S)

245

/ . 1 ≤ C Ψˆ0,0,ε,−α vL2 (S) + ε (Ψˆ0,0,ε,−α v)L2 (S) d ≤ C Ψˆ0,0,ε,−α vL2 (S) + εΨˆ0,0,ε,−α vH 1 (S) ≤ Cvε,−α ,

where we employed Lemma 6.4.5 in the last step. We can now conclude that f v dx ≤ Ψˆ0,1,ε,α f L2 (S) Ψˆ −1 vL2 (S) ≤ Ψˆ0,1,ε,α f L2 (S) vε,−α . 0,1,ε,α S

For the last estimate, we proceed similarly: In view of Lemma A.1.8, we have −1 vH 1 (S) . ε hv ds ≤ εΨˆ0,0,ε,α hH 1,1 (S) Ψˆ0,0,ε,α 1,1

Γ

−1 vH 1 (S) ≤ Cvε,−α . Using As above, Lemma 6.4.5 gives εΨˆ0,0,εα

sup max {1, |x|/ε}e−δ|x|/ε ≤ max {1, 1/(eδ)} < ∞, x∈S

we compute for the other factor Ψˆ0,0,ε,α hL2 (S) ≤ hH 1,1 (S) ≤ hH 1,1 , 1,ε,α 1,ε,α+δ (S) r∇(Ψˆ0,0,ε,α h)L2 (S) ≤ C Ψˆ0,0,ε,α+δ hL2 (S) + εΨˆ0,1,ε,α+δ ∇hL2 (S) ≤ ChH 1,1

1,ε,α+δ (S)

,

where the constant C > 0 is independent of ε but depends on δ > 0.

2

Remark 6.4.7 We note that the proof of Proposition 6.4.6 actually shows that 1 1 the inf-sup condition also holds for the pair Hε,α (S)×Hε,−α (S)–the homogeneous boundary conditions are not essential. Proposition 6.4.6 is the basis for the solution theory in exponentially weighted spaces: 1 Proposition 6.4.8 (existence and uniqueness in H0,ε,α ). Let S be a sector and Γ be a curve as given at the outset of Section 6.4.1. Let A, c satisfy (6.4.2) 0,0 1,1 and α satisfy (6.4.8). Let δ > 0. Then, for every f ∈ H1,ε,α (S), h ∈ H1,ε,α+δ (S) the problem: Find u ∈ H01 (S) s.t. bε (u, v) = ε hv ds + f v dx ∀v ∈ H01 (S) Γ

S

has a unique solution u, and for a C > 0 independent of ε, f , h, we have uε,α ≤ C f H 0,0 (S) + hH 1,1 (S) . 1,ε,α

1,ε,α+δ

Proof: Proposition 6.4.6 shows that the bilinear form bε satisﬁes an inf-sup 1 1 (S) × H0,ε,−α (S), and that the functions condition on the pair of spaces H0,ε,α 1 f , h generate bounded linear functionals on H0,ε,−α (S); the proof now follows from a well-known result, see, e.g., [11, Thm. 5.2.1], [31, Sec. II.1.1]. 2

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6. Exponentially Weighted Countably Normed Spaces

6.4.3 Analytic regularity in exponentially weighted spaces 2,2 We start with an Hβ,ε,α estimate for solutions to transmission problems.

Proposition 6.4.9. Let S = SR (ω) and let Γ be a straight curve passing through the origin that splits S into S + , S − . Let A, c satisfy (6.4.2), and assume that α > 0 satisﬁes (6.4.8). Let furthermore gi = 0, i ∈ {1, 2}, h1 = 0. Let β ∈ (0, 1) be given by the statement of Proposition 6.4.4. Then there exist 0,0 C > 0, α ∈ (0, α) independent of ε such that for every f ∈ Hβ,ε,α (SR (ω)) and 1,1 every h2 ∈ Hβ,ε,α+δ (SR (ω)), δ > 0, the solution u of the transmission problem 2 (S + ∪ S − ) and satisﬁes (6.4.3) is in Hloc Ψˆ0,β,ε,α ∇2 uL2 ((S + ∪S − )∩BR (0)) ≤ Cε−2 f H 0,0 (S) + HH 1,1 (S) . β,ε,α

β,ε,α

Proof: The bound is obtained with the aid of Proposition 6.4.4; the actual proof 2,2 regularity result of Propositions 5.3.2, 5.4.4. is very similar to the analogous Hβ,ε We will therefore merely outline the major diﬀerences. We start by introducing the shorthand S ± := (S + ∪S − )∩BR (0). Furthermore, we use α < α repeatedly in the proof but will specify it toward the end of the proof. In order to account for the fact that a transmission problem is considered, the collection of balls B = {Bi | i ∈ N} of the proof of Proposition 5.3.2 is now taken such that additionally the balls Bi satisfy the dichotomy that either Bi ⊂ S + ∪ S − (implying in fact that either Bi ⊂ S + or Bi ⊂ S − ) or that the center of the ball Bi is located on Γ , Γ1 , or Γ2 (implying that Bi ∩ S + or Bi ∩ S − is a half disc). In this way, the local regularity results Lemmata 5.5.5, 5.5.7, 5.5.23 are applicable. Following the proof of Propositions 5.3.2, 5.4.4 we then obtain bounds on Sh ∩ S ± and S ± \ Sh separately (see the proof of Proposition 5.3.2 for the precise choice of h—the essential requirement is that h ∼ ε). We note that on Sh the weight functions ˆ0,β,ε and Ψˆ0,β,ε,α are equivalent, i.e., there is c > 0 independent of ε such that Φ ˆ0,β,ε (x) ˆ0,β,ε (x) ≤ Ψˆ0,β,ε,α (x) ≤ cΦ c−1 Φ

∀x ∈ Sh .

Thus, we can conclude in the same fashion as in the proof of Proposition 5.4.4 (merely replacing the appeal to Lemma 5.4.2 with that to Proposition 6.4.4) Ψˆ0,β,ε,α ∇2 uL2 (S ± ∩Sh ) ≤ C ε−2 f H 0,0 (S ± ∩SH ) + ε−2 h2 H 1,1 (S ± ∩SH ) β,ε,α β,ε,α −1 −1 1 + ε ∇uL2 (S ± ∩SH ) + ε uL2 (S ± ∩SH ) . |x| From Lemma 4.2.12, we see that, using the homogeneous boundary conditions, 1 we may bound |x| uL2 (S∩SH ) ≤ C∇uL2 (S∩SH ) . Since Ψˆ0,0,ε,α (x) ∼ 1 for all x ∈ SH , we obtain Ψˆ0,β,ε,α ∇2 uL2 (S ± ∩Sh ) ≤ C ε−2 f H 0,0 (S ± ∩SH ) (6.4.10) β,ε,α + ε−2 h2 H 1,1 (S ± ∩SH ) + ε−1 Ψˆ0,0,ε,α ∇uL2 (S ± ∩SH ) . β,ε,α

6.4 Analytic regularity in exponentially weighted spaces

247

Next, we consider S ± \ Sh . We observe that for all balls Bi = Bc|xi | (xi ) with i ∈ N \ Iε we have for some C, κ ≥ 1 independent of ε and δ > 0 |xi | ≥ C −1 ε, C −1 Ψˆ0,β,ε,κ−1 δ (xi ) ≤ min Ψˆ0,β,ε,δ (z)

(6.4.11) (6.4.12)

z∈Bi

≤ max Ψˆ0,β,ε,δ (z) ≤ C Ψˆ0,β,ε,κδ (xi ). z∈Bi

We now employ the local regularity results Lemmata 5.5.5, 5.5.7, 5.5.23 to conclude as in the proofs of Propositions 5.3.2, 5.4.4 that for i ∈ N \ Iε ∇2 uL2 (Bˆi ∩S ± ) ≤ C ε−2 f L2 (Bi ∩S ) + ε−2 h2 L2 (Bi ∩S ) + ε−1 ∇h2 L2 (Bi ∩S ) + ε−1 ∇uL2 (Bi ∩S ) + ε−2 uL2 (Bi ∩S ) , where we abbreviated S := SR (ω) ∩ (S + ∪ S − ). Multiplying this last estimate with Ψˆ0,β,ε,κα (xi ) we obtain with (6.4.12) (taking δ = α and δ = κ2 α ) Ψˆ0,β,ε,α ∇2 uL2 (Bˆi ∩S ± ) ≤ C ε−2 Ψˆ0,β,ε,κ2 α f L2 (Bi ∩S ) + ε−2 Ψˆ0,β,ε,κ2 α h2 L2 (Bi ∩S ) + ε−1 Ψˆ0,β,ε,κ2 α ∇h2 L2 (Bi ∩S ) + ε−1 Ψˆ0,0,ε,κ2 α ∇uL2 (Bi ∩S ) + ε−2 Ψˆ0,0,ε,κ2 α uL2 (Bi ∩S ) . We choose now α < α/κ2 and set α := α κ2 < α. We then get Ψˆ0,β,ε,α ∇2 uL2 (Bˆi ∩S ± ) ≤ Cε−2 Ψˆ0,β,ε,α f L2 (Bi ∩S ) + Ψˆ0,β,ε,α h2 L2 (Bi ∩S ) + εΨˆ0,β,ε,α ∇h2 L2 (Bi ∩S ) + εΨˆ0,0,ε,α ∇uL2 (Bi ∩S ) + Ψˆ0,0,ε,α uL2 (Bi ∩S ) . Squaring and summing on i we get, after including the contribution from Sh and observing that α ≤ α Ψˆ0,β,ε,α ∇2 uL2 (S ± ) ≤ Cε−2 f H 0,0 (S ) + h2 H 1,1 (S ) (6.4.13) β,ε,α β,ε,α + εΨˆ0,0,ε,α ∇uL2 (S ) + Ψˆ0,0,ε,α uL2 (S ) . Since α < α, there exists δ > 0 such that α + δ ≤ α, so that we can use the a priori estimate of Proposition 6.4.8 to bound εΨˆ0,β,ε,α ∇uL2 (S ) + Ψˆ0,β,ε,α uL2 (S ) ≤ C f H 0,0 (S) + HH 1,1 (S) ≤ C f H 0,0 1,ε,α

1,ε,α +δ

β,ε,α

(S) + HH 1,1

1,ε,α (S)

. 2

A bootstrapping argument then allows us to control in a similar fashion all derivatives of the solution:

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6. Exponentially Weighted Countably Normed Spaces

Theorem 6.4.10. Let S = SR (ω) and let Γ be a straight curve passing through the origin that splits S into S + , S − . Let R ∈ (0, R), let A, c satisfy (6.4.2), and assume that α > 0 satisﬁes (6.4.8). Let furthermore gi = 0, i ∈ {1, 2}, h1 = 0. Let β ∈ (0, 1) be given by the statement of Proposition 6.4.4. Then for every 0 1 f ∈ Bβ,ε,α (S + ∪ S − , Cf , γf ) and every h2 ∈ Bβ,ε,α (S + , Ch , γh ) the solution u of + the transmission problem (6.4.3) is analytic on S ∪ S − and satisﬁes for some C, K, α ∈ (0, α) independent of ε, Cf , Ch Ψˆp,β,ε,α ∇p+2 uL2 ((S + ∪S − )∩BR (0)) ≤ CK p+2 max {p + 1, ε−1 }p+2 [Cf + Ch ] . If additionally, the data A, c, are analytic on SR (ω), h2 = 0, and f ∈ B 0 (S, Cf , γf ), then u is analytic on SR (ω) and Ψˆp,β,ε,α ∇p+2 uL2 (SR (0)) ≤ CCf K p+2 max {p + 1, ε−1 }p+2 for all p ∈ N0 . Proof: The proof follows the arguments given in the proof of Proposition 5.4.5. The key observation for handling the exponential decay is (6.4.12). 2

6.4.4 Analytic regularity for a special transmission problem We now consider the regularity of (6.4.1) with analytic data where—contrasting Theorem 6.4.10 in which L2 -based bounds on the data h2 , f were required— pointwise estimates on the data gi , hi , f are available. This will lead to an improvement in the estimate by a factor ε. Straight sectors. For simplicity, we will ﬁrst consider the case of a straight sector S = SR (ω). The case of a general analytic sector will then be obtained by a mapping argument. Let Γ be a straight line of the form Γ = {(r cos ω , r sin ω ) | r ∈ (0, R)} for some ω with 0 < ω < ω. Next, we denote by Γ1 = {(r cos ω, 0) | r ∈ (0, R)}, Γ2 = {(r cos ω, r sin ω) | r ∈ (0, R)} the two straight sides of ∂SR (ω) and by Γ3 the curved side Γ3 = {(R cos ϕ, R sin ϕ) | 0 < ϕ < ω}. The curve Γ divides SR (ω) into two components S + , S − . We introduce four functions g1 , g2 , h1 , h2 : [0, R] → R and f : S + ∪ S − → R satisfying for some C, K, α > 0, R ∈ (0, R): g1 (R) = g2 (R) = 0, g1 (0) = g2 (0), h1 (0) = h1 (R) = 0 D gi L∞ ((0,R)) ≤ Cε

−p

e D hi L∞ ((0,R)) ≤ Cε Ψˆ0,1,ε,α f L∞ (S + ∪S − ) ≤ C.

−p

e

αr/ε

αr/ε

p

p

(6.4.14a) (6.4.14b) (6.4.14c)

,

p ∈ {0, 1, 2},

i ∈ {1, 2}, (6.4.14d)

,

p ∈ {0, 1, 2},

i ∈ {1, 2}, (6.4.14e) (6.4.14f)

Additionally, we stipulate for all p ∈ N0 : eαr/ε Dp gi L∞ ((0,R )) ≤ CK p max {p, ε−1 }p , D hi L∞ ((0,R )) ≤ CK max {1, ε

−1 p

Ψˆp,1,ε,α ∇ f L∞ ((S + ∪S − )∩BR (0)) ≤ CK max {1, ε

−1 p

e

αr/ε

p

p

p p

} , } .

i ∈ {1, 2}, (6.4.14g) i ∈ {1, 2}, (6.4.14h) (6.4.14i)

6.4 Analytic regularity in exponentially weighted spaces

249

We are now in position to formulate the transmission problem that we consider: Find u ∈ H 1 (S + ∪ S − ) such that bε (u, v) = ε h2 v ds + f v dx ∀v ∈ H01 (SR (ω)), (6.4.15a) S + ∪S −

Γ

u = gi on Γi , i ∈ {1, 2},

u = 0 on Γ3 ,

[u] = h1 on Γ ,

(6.4.15b)

where, by a slight abuse of notation, the functions gi , hi depending on one variable only, the “radial variable r”, are deﬁned on the straight lines Γi , Γ as, e.g., gi |Γi (x) = gi (|x|) for x ∈ Γi . We note that the assumption (6.4.14) guarantees that the boundary data g are indeed an element of H 1/2 (∂S). Similarly, the con1/2 ditions on the hi imply that h1 ∈ H00 (Γ ) and h2 ∈ H −1/2 (Γ ). Thus, existence and uniqueness of the solution of (6.4.15) is ensured. We note the following lemma: Lemma 6.4.11. Let α > 0, β ∈ (0, 1), α ∈ (0, α), and f , h2 satisfy for some Cf , Ch , Kf , Kh > 0 Ψˆp,1,ε,α ∇p f L∞ (S + ∪S − ) ≤ Cf Kfp max {1, ε−1 }p

∀p ∈ N0 ,

−1 p

∀p ∈ N0 .

Ψˆp,0,ε,α D h2 L∞ ((0,R)) ≤ p

Ch Khp

max {1, ε

}

Then there exists a function H2 such that H2 |Γ = h2 and there exist constants 0 + C, γ > 0 independent of ε, Cf , Ch such that f ∈ Bβ,ε,α ∪ S − , CCf ε, γ), (S 1 H2 ∈ Bβ,ε,α (S, CCh ε, γ). Proof: The assertion for f follows by a direct computation; the factor ε in the expression CCf ε stems from the fact that α < α. The function H2 is constructed in polar coordinates as H2 (r, ϕ) := h2 (r); clearly, H2 |Γ = h2 . Writing r = |x|, we have by Lemma 6.2.1 for some C, K > 0 e−αr/ε Ψˆp,0,ε,α (x) ≤ CK p max {(p+1)/r, ε−1 }p . max{p + 1, ε−1 }p (6.4.16) Lemma 6.3.1 and the assumptions on h2 imply the existence of K, C > 0 such that |∇p H2 (x)| ≤ CCh K p e−αr/ε max {(p + 1)/r, ε−1 }p ∀p ∈ N0 .

C −1 K −p max {(p+1)/r, ε−1 }p ≤

The bound (6.4.16) therefore implies again with Lemma 6.2.1 Ψˆp,0,ε,α ∇p H2 L∞ (S) ≤ CCh K p max {(p + 1), ε−1 }p

∀p ∈ N0 .

Since 0 < α < α, this last estimate can be used to show the existence of C, 1 γ > 0 such that H2 ∈ Bβ,ε,α 2 (S, CCh ε, γ). Our main result of this section are the following Theorems 6.4.12, 6.4.13.

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6. Exponentially Weighted Countably Normed Spaces

Theorem 6.4.12 (analytic regularity, transmission problem). Let S = SR (ω) be a straight sector, Γ = {(r cos ω , r sin ω ) | r ∈ (0, R)}, ω ∈ (0, ω), be a straight line dividing S into S + and S − . Assume that A, c satisfy (6.4.2) and that α > 0 satisﬁes (6.4.8). Let the data gi , hi (i ∈ {1, 2}), f satisfy (6.4.14). Finally, let R ∈ (0, R ). Then the solution u ∈ H 1 (S + ∪ S − ) of (6.4.15) exists and is analytic on (S + ∪ S − ) ∩ BR (0). Furthermore, there exist C, γ > 0, α ∈ (0, α), and β ∈ [0, 1) independent of ε such that for all p ∈ N0 Ψˆ0,0,ε,α uL2 (S + ∪S − ) + εΨˆ0,0,ε,α ∇uL2 (S + ∪S − ) ≤ Cε, Ψˆp,β,ε,α ∇

p+2

uL2 ((S + ∪S − )∩BR (0)) ≤ Cεγ max {p + 1, ε p

−1 p+2

}

.

(6.4.17) (6.4.18)

If additionally h1 = h2 = 0 and the coeﬃcients A, c, and the right-hand side f are analytic on SR (ω), then the solution u of (6.4.15) is analytic on SR (ω) and the set (S + ∪ S − ) ∩ BR (0) may be replaced with SR (ω) in (6.4.17), (6.4.18). Proof: The proof of Theorem 6.4.12 is based on Theorem 6.4.10, Lemma 6.4.11, and Lemmata 6.4.15 below: The lifting result Lemma 6.4.15 is used to reduce the problem to one with g1 = g2 = h1 = 0; the right-hand side f and the jump h2 still satisfy the bounds (6.4.14f), (6.4.14i), (6.4.14e), (6.4.14h). We conclude with 0 + 1 + Lemma 6.4.11 that f ∈ Bβ,ε,α ∪ S − ) ∩ BR (0), Cε, γ), h2 ∈ Bβ,ε,α ∩ ((S (S BR (0), Cε, γ) for α ∈ (0, α). An appeal to Theorem 6.4.10 ﬁnally allows us to conclude the argument. 2 Curved sectors. With a mapping argument, Theorem 6.4.12 can be extended to curvilinear sectors. To ﬁx ideas, let S be a curvilinear sector (with sides Γi , i ∈ {1, 2, 3}) and let Γ be an analytic arc passing through the origin and dividing S into two subsectors S + , S − . The arcs Γi , Γ may be parametrized by analytic maps Λi : (0, R) → Γi and Λ : (0, R) → Γ for some R > 0. We assume that the maps Λi , Λ satisfy C −1 r ≤ |Λi (r)| ≤ Cr,

C −1 r ≤ |Λ(r)| ≤ Cr

∀r ∈ (0, R).

(6.4.19)

1/2

Concerning the data g ∈ H 1/2 (∂S) ∩ C(∂S), h1 ∈ H00 (Γ ), h2 ∈ H 1/2 (Γ ), f , we make the following assumptions: There are constants C, K, α > 0 such that upon setting gi := g|Γi we have for p ∈ {0, 1, 2} g ∈ C(∂S), g3 = 0, eα|Λi (r)|/ε Dp (gi ◦ Λi )(r)L∞ ((0,R)) ≤ Cε−p , eα|Λ(r)|/ε Dp (hi ◦ Λ)(r)L∞ ((0,R)) ≤ Cε−p , Ψˆ0,1,ε,α f L∞ (S + ∪S − ) ≤ C,

i ∈ {1, 2},

(6.4.20a) (6.4.20b)

i ∈ {1, 2},

(6.4.20c) (6.4.20d)

and additionally the analyticity properties for all p ∈ N0 : eα|Λi (r)|/ε Dp (gi ◦ Λi )(r)L∞ ((0,R )) ≤ Cε−p , e

α|Λ(r)|/ε

D (hi ◦ Λ)(r)L∞ ((0,R )) ≤ Cε p

−p

,

i ∈ {1, 2},

(6.4.20e)

i ∈ {1, 2},

(6.4.20f)

Ψˆp,1,ε,α ∇ f L∞ ((S + ∪S − )∩BR (0)) ≤ Cγ max {p + 1, ε p

p

−1 p

} .

(6.4.20g)

6.4 Analytic regularity in exponentially weighted spaces

251

We then consider the transmission problem: Find u ∈ H 1 (S + ∪ S − ) such that h2 v ds + f v dx ∀v ∈ H01 (S), (6.4.21a) bε (u, v) = ε S + ∪S −

Γ

u=g

on ∂S,

[u] = h1

on Γ .

(6.4.21b)

Theorem 6.4.13 (analytic regularity, transmission problem). Let S be a curvilinear sector and let Γ be an analytic curve passing through the origin that divides S into two curvilinear sectors S + , S − . Assume that A, c satisfy (6.4.2). Let the data gi , hi , (i ∈ {1, 2}), f satisfy (6.4.20). Then the transmission problem (6.4.21) has a unique solution u ∈ H 1 (S + ∪ S − ), and there exist constants C, γ > 0, α ∈ (0, α), β ∈ [0, 1), R > 0 independent of ε such that for all p ∈ N0 Ψˆ0,0,ε,α uL2 (S + ∪S − ) + εΨˆ0,0,ε,α ∇uL2 (S + ∪S − ) ≤ Cε, Ψˆp,β,ε,α ∇p+2 uL2 ((S + ∪S − )∩B (0)) ≤ Cεγ p max {p + 1, ε−1 }p+2 . R

(6.4.22) (6.4.23)

If additionally h1 = h2 = 0 and the coeﬃcients A, c, and the right-hand side f are analytic on S ∩ BR (0), then the solution u is analytic on S ∩ BR (0); the set (S + ∪ S − ) ∩ BR (0) may then be replaced with S ∩ BR (0) in (6.4.22), (6.4.23). Proof: The proof follows from Theorem 6.4.12 by a mapping argument. Let ω1 , ω2 be the angles between the lines Γ1 , Γ2 and Γ at the origin. It is then easy to construct an invertible Lipschitz mapping F : SR (ω1 + ω2 ) (for some appropriate R) such that F is analytic on SR (ω1 ) and SR (ω1 + ω2 ) \ SR (ω1 ) (e.g., by the “blending method,” [58–60]). This mapping allows us to apply Theorem 6.4.12 in a neighborhood of the origin. Mapping back to the original variables yields the desired result with Theorem 6.2.6. 2 A diﬀerent way of putting the result of Theorem 6.4.12 is to state that the restric2 tions to the two subsectors S + , S − are in exponentially weighted spaces Bβ,ε,α , 2 + 2 − that is, u|S + ∈ Bβ,ε,α (S ∩ BR (0), Cε, γ), u|S − ∈ Bβ,ε,α (S ∩ BR (0), Cε, γ). Hence, applying Corollary 6.2.8 allows us to state the following corollary. Corollary 6.4.14. Under the assumptions of Theorem 6.4.13, the solution u of the transmission problem (6.4.21) is analytic on the two sectors S + ∩ BR (0), S − ∩ BR (0). Furthermore, for β ∈ [0, 1) of Theorem 6.4.12 and some C, γ, α > 0 independent of ε there holds for x ∈ (S + ∪ S − ) ∩ BR (0) |u(x) − g(0)| ≤ C

|x| ε

1−β

|∇p u(x)| ≤ Cp!|x|−p

e−α |x|/ε ,

|x| ε

1−β

e−α |x|/ε

∀p ∈ N.

Proof: We will only show the statements on S := S + ∩ BR (0). The claim 2 for the other sector is proved analogously. As u|S + ∈ Bβ,ε,α (S , Cε, γ), we can apply Corollary 6.2.8 with l = 2, to get that u satisﬁes on S for some C, γ > 0 independent of ε:

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6. Exponentially Weighted Countably Normed Spaces

|∇p u(x)| ≤ Cεγ p p! |x|−1−p min {1, |x|/ε}2−β e−α |x|/ε

∀p ∈ N.

After properly adjusting the constants C, γ, α , this estimate can be brought to 2 the desired form. For the case p = 0, we ﬁrst note that u|S ∈ Bβ,ε,α implies u|S ∈ C(S ) by Lemma 4.2.9. Next, as the boundary data g1 is continuous up to the origin, we conclude that u(0) = g1 (0) = g2 (0). Thus, appealing again to Corollary 6.2.8 gives the desired result. 2 Lifting results for inhomogeneous boundary conditions. We prove a lifting result to deal with inhomogeneous boundary conditions: Lemma 6.4.15 (lifting). Let SR (ω) be a straight sector that is divided into two sectors S + , S − by the straight line Γ . Let R ∈ (0, R) and set ± SR := (S + ∪ S − ) ∩ BR (0),

± + SR ∪ S − ) ∩ BR (0). := (S

Let gi (i ∈ {1, 2}), h1 satisfy (6.4.14a)–(6.4.14e) and the analyticity require± ments (6.4.14g), (6.4.14h). Let β ∈ (0, 1). Then there exist u ∈ H 1 (SR ), u ± analytic on SR , and constants C, γ > 0, α ∈ (0, α) independent of ε such that i ∈ {1, 2},

u|Γi = gi , [u] = h1

u|Γ3 = 0,

on Γ ,

(6.4.24a) (6.4.24b)

εΨˆ0,0,ε,α ∇uL2 (S ± ) ≤ Cε,

(6.4.24c)

Ψˆ0,0,ε,α uL2 (S ± ) ≤ Cε,

(6.4.24d)

R

R

Ψˆp,β,ε,α ∇p+2 uL2 (S ± ) ≤ Cεγ p max {p + 1, ε−1 }p+2

(6.4.24e)

R

for all p ∈ N0 . Furthermore, for A, c satisfying (6.4.2) the function f := −ε2 ∇ · (A∇u) + cu and the jump h2 := ε[∂nA u] across Γ satisfy Ψˆ0,1,ε,α f L∞ (S ± ) ≤ C,

(6.4.25a)

R

Ψˆp,1,ε,α ∇p f L∞ (S ± ) ≤ Cγ p max {p + 1, ε−1 }p R

Ψˆ0,0,ε,α Dp h2 L∞ ((0,R)) ≤ CK p max {p + 1, ε−1 }p Ψˆ0,0,ε,α Dp h2 L∞ ((0,R )) ≤ CK p max {p + 1, ε−1 }p

∀p ∈ N0 ,

(6.4.25b)

∀p ∈ {0, 1},

(6.4.25c)

∀p ∈ N0 .

(6.4.25d)

Proof: The proof is lengthy and therefore broken up into three pieces: First, we construct u satisfying (6.4.24). Next, we ascertain the bounds (6.4.25a), (6.4.25b) concerning f . In the ﬁnal step, we check the bounds on ∂nA u on Γ . 1. step: For the ﬁrst assertions concerning u, we construct u satisfying (6.4.24a), (6.4.24b) and the following conditions: Ψˆ0,0,ε,α uL∞ (S ± ) + εΨˆ0,0,ε,α ∇uL∞ (S ± ) ≤ C, R

R

Ψˆ1,0,ε,α ∇2 uL∞ (S ± ) ≤ Cε−2 , R

(6.4.26a) (6.4.26b)

Ψˆp,0,ε,α ∇p+1 uL∞ (S ± ) ≤ Cγ p max {p + 1, ε−1 }p+1 . (6.4.26c) R

6.4 Analytic regularity in exponentially weighted spaces

253

The bounds (6.4.24c)–(6.4.24e) then follow easily from Lemma 6.2.9: For example, (6.4.24e) follows from Ψˆp,β,ε,α ∇p+2 uL2 (S ± ) R

≤ CK p Ψˆp+1,0,ε,α ∇p+2 uL∞ (S ± ) Ψˆ0,−(1−β),−(α−α ) L2 (S ± ) R

R

≤ CK p εΨˆp+1,0,ε,α ∇p+2 uL∞ (S ± ) . R

Recalling the notation Γ1 = {(r, 0) | r ∈ (0, R)}, Γ2 = {(r cos ω, r sin ω) | r ∈ (0, R)}, Γ = {(r cos ω , r sin ω ) | r ∈ (0, R)}, we claim that the following function (deﬁned in polar coordinates) has the desired properties (6.4.26): u := ug + uh ,

(6.4.27)

ϕ ug (r, ϕ) := g1 (r) + (g2 (r) − g1 (r)) , ω ! ϕ if ϕ ∈ (0, ω ], h1 (r) ω uh (r, ϕ) := 0 else.

(6.4.28) (6.4.29)

By construction, u satisﬁes the desired boundary conditions (6.4.24a) and the jump condition (6.4.24b). The assumptions on the functions gi , i ∈ {1, 2}, and h1 readily imply that u satisﬁes Ψˆ0,0,α,ε uL∞ (SR (ω)) ≤ C for some C > 0 independent of ε, giving the ﬁrst part of (6.4.26a). It suﬃces therefore to consider higher derivatives of u to prove (6.4.26). We restrict our attention to the sector S + = {(r cos ϕ, r sin ϕ) | r ∈ (0, R), ϕ ∈ (0, ω )}, the other one being handled analogously. We write u in the form u(r, ϕ) = u1 (r) + l(ϕ)u2 (r) where the analytic functions u1 , u2 depend on r only and l is a linear function. We note that u2 (0) = 0 in view of g1 (0) = g2 (0) and h1 (0) = 0. Using the formulae ux = ur cos ϕ − uϕ

sin ϕ , r

uy = ur sin ϕ + uϕ

cos ϕ , r

(6.4.30)

we infer from Lemma 6.3.2 that ∇(x,y) u1 satisﬁes for ϕ ∈ (0, ω ), r ∈ (0, R ) |∂ϕs ∂rp ∇(x,y) u1 (r, ϕ)| ≤ Cε−1 γ p+s s! max {p + 1, ε−1 }p e−αr/ε ∀(p, s) ∈ N20 . (6.4.31) Hence, applying Lemma 6.3.1, we get that u1 satisﬁes in Cartesian coordinates ± on SR for some C, γ > 0 independent of ε: |∇p ∇u1 (x)| ≤ Cε−1 γ p max {(p + 1)/r, ε−1 }p e−αr/ε ,

r = |x|.

Using Lemma 6.2.1, we see Ψˆp,0,ε,α ∇p ∇uL∞ (S + ) ≤ Cε−1 γ p max {p + 1, ε−1 }p

∀p ∈ N0 ,

from which (6.4.26c) follows. For u2 , we note that u2 (0, ϕ) = 0 and that thus (6.4.30) and (ii) of Lemma 6.3.2 imply that ∇(x,y) u2 satisfy for ϕ ∈ (0, ω), r ∈ (0, R )

254

6. Exponentially Weighted Countably Normed Spaces

|∂ϕs ∂rp ∇(x,y) u2 (r, ϕ)| ≤ Cε−1 γ p+s s! max {p + 1, ε−1 }p e−αr/ε

∀(p, s) ∈ N20 ,

which is a bound that has the same structure as (6.4.31). Thus, we may reason as in our treatment of u1 to see that u2 also satisﬁes (6.4.26c). The bound (6.4.26a), (6.4.26b) follow by inspection. 2. step: The bound (6.4.26c) implies with Lemma 6.2.1 for ∇u: |∇p ∇u(x)| ≤ Cε−1 γ p max {(p + 1)/r, ε−1 }p e−αr/ε

∀p ∈ N0 .

Applying Lemma 4.3.3 (with the change of variables taken as the identity) to the product A∇u, we get the existence of C, γ > 0 such that |∇p (A∇u)(x)| ≤ Cε−1 γ p max {(p + 1)/r, ε}p e−αr/ε

∀p ∈ N0 .

Thus, we conclude for ∇p (∇ · (A∇u)) |∇p (∇ · (A∇u)) (x)| ≤ Cε−1 γ p max {(p + 1)/r, ε}p+1 e−αr/ε

∀p ∈ N0 .

Hence, with Lemma 6.2.1 Ψˆp,1,ε,α ∇p (∇ · (A∇u)) L∞ (S ± ) ≤ Cε−1 γ p max {p + 1, ε−1 }p+1 . R

After adjusting the constants C, γ, we arrive at Ψˆp,1,ε,α ∇p ε2 ∇ · (A∇u) L∞ (S ± ) ≤ Cγ p max {p + 1, ε−1 }p R

∀p ∈ N0 ,

which is part of the desired bound for f . Controlling ∇p (cu) is done analogously. Thus (6.4.25b) holds. In order to prove (6.4.25a), one has to verify Ψˆ0,1,ε,α f L∞ (S ± \S ± ) ≤ C, which follows from (6.4.26b). R

R

3. step: We ﬁnally turn to the bound for the jump of the co-normal derivatives. We start by noting that the co-normal derivatives operator ∂nA on the curve Γ for the subsectors S + , S − can be written as ∂nA =

a± (r) ∂ϕ + b± (r)∂r r

(6.4.32)

for some analytic functions a+ , a− , b+ , b− . Here, the + and − sign indicate whether the co-normal derivative corresponds to S + or S − . We ﬁx one of the sectors, S + , say, and consider the co-normal derivative ∂nA u|Γ . With the functions u1 , u2 , l(ϕ) of the ﬁrst step, we then get ∂nA u|Γ =

a+ u2 (r, ω )l (ω ) + b+ (r)∂r u2 (r, ω )l(ω ) + b+ (r)u1 (r). r

Lemma 6.3.2 then allows us to conclude Ψˆ0,0,ε,α Dp (∂nA u|Γ )L∞ ((0,R )) ≤ Cε−1 max {p + 1, ε−1 }p

∀p ∈ N0 ,

from which we get (6.4.25d) in view of h2 = ε[∂nA u]. The bound (6.4.25c) is obtained by inspection. 2

7. Regularity through Asymptotic Expansions

7.1 Motivation and outline 7.1.1 Motivation Preliminaries. In Chapter 5, we expressed regularity of the solution uε of l and obtained a shift theorem, (1.2.11) through the countably normed spaces Bβ,ε 0 Corollary 5.3.12. This shift theorem states that for right-hand sides f ∈ Bβ,ε , 2 0 the solution uε of (1.2.11) is in Bβ,ε . The condition for f to be in Bβ,ε is not restrictive if ε is small as the derivatives of f may be very large everywhere in Ω. For example, a function f such as sin(x/ε) or, more generally, a function satisfying ∇p f L∞ (Ω) ≤ Cγ p max {p + 1, ε−1 }p ∀p ∈ N0 0 is an element of Bβ,ε . For highly oscillatory right-hand sides f such as f (x) = sin(x/ε), one has to expect that the solution uε is highly oscillatory on Ω as well; i.e., one has to expect that the derivatives of the solution uε are large 0 2 the statement uε ∈ Bβ,ε is everywhere in Ω. For right-hand sides f ∈ Bβ,ε therefore the best one can expect. In practice, however, the right-hand side f may be more “regular” in the sense bounds on its derivatives are available that are independent of ε:

∇p f L∞ (Ω) ≤ Cf γfp p!

∀p ∈ N0 .

(7.1.1)

0 and thus Corollary 5.3.12 is applicable. Clearly, such a function is still in Bβ,ε However, Corollary 5.3.12 is no longer sharp. In this situation, the typical behavior of the solution uε of (1.2.11) is that it is smooth (with bounds on the derivatives independent of ε) in the interior of Ω and that uε has boundary layer character with sharp gradients near the boundary only. A precise characterization of this behavior is achieved through asymptotic expansions. This is the approach taken in the present chapter. In order to illustrate our claim that for smooth right-hand sides f the typical solution behavior is indeed to be smooth in the interior and to have boundary layer character near the boundary, we consider a one-dimensional model problem in the following lemma. As in the introductions to the preceding chapters, it is mostly the construction employed in the proof that is of interest here. We mention that closely related analysis can be found in [92].

J.M. Melenk: LNM 1796, pp. 255–295, 2002. c Springer-Verlag Berlin Heidelberg 2002

256

7. Regularity through Asymptotic Expansions

Lemma 7.1.1 (regularity of asymptotic expansions). Let Ω = (−1, 1) and assume that f is analytic on Ω with f (p) L∞ (Ω) ≤ Cf γfp p!

∀p ∈ N0 .

(7.1.2)

Let uε be the solution to Lε uε := −ε2 uε + uε = f

on Ω,

uε (±1) = 0.

(7.1.3)

Then there exist C, γ, α > 0 independent of ε such that for each ε ∈ (0, 1] the solution uε can be decomposed as uε = wε + uBL + rε ε with the following properties (we set ρ(x) = dist(x, ∂Ω)): wε(p) L∞ (Ω) ≤ Cγ p p! ∀p ∈ N0 , BL (p) (x)| ≤ Cγ p max {p!, ε−p }e−ρ(x)/ε | uε rε L2 (Ω) +

εrε L2 (Ω)

−α/ε

≤ Ce

∀p ∈ N0

∀x ∈ Ω,

,

rε (±1) = 0. Proof: The functions wε , uBL ε , and rε are constructed with the aid of the classical asymptotic expansions for (7.1.3). In a ﬁrst step, we construct for each M ∈ N0 functions wM , uBL M , and rM such that uε = wM + uBL M + rM . In a second step, we choose the expansion order M appropriately in dependence on ε. Let M ∈ N0 be given. We deﬁne the outer expansion wM by wM (x) :=

M

ε2i f (2i) (x).

i=0

We note that the functions wM are good approximations to particular solutions of (7.1.3) as the defect f − Lε wM is small for small ε: f (x) − Lε wM (x) = ε2M +2 f (2M +2) (x). The outer expansion wM , however, does not conform to the boundary conditions. In order to correct this, we introduce the boundary layer function uBL as the solution of Lε uBL on Ω, uBL M =0 M (±1) = −wM (±1). uBL M has the form − −(1+x)/ε −(1−x)/ε uBL + A+ , M = AM e Me

7.1 Motivation and outline

257

+ where the constants A− M , AM are bounded by − + A + A ≤ C (|wM (−1)| + |wM (1)|) ≤ CwM L∞ (Ω) M M

for some constant C independent of ε and M . Using the deﬁnition ρ(x) = dist(x, ∂Ω), we see that BL (p) (x) ≤ Cε−p e−ρ(x)/ε wM L∞ (Ω) , uM where again C > 0 is independent of ε and M . The remainder rM is simply deﬁned by rM := uε − wM − uBL M . By construction we have rM (±1) = 0. For bounds on rM , we observe that it solves the following equation Lε rM = Lε uε − Lε wM = f − Lε wM = ε(2M +2) f (2M +2)

in Ω,

rε (±1) = 0.

Appealing to the Lax-Milgram theorem, we obtain in the energy norm · ε deﬁned as in (1.2.7) 2M +2

rM ε ≤ ε(2M +2) f (2M +2) L2 (Ω) ≤ Cf (γf ε(2M + 2))

. (p)

Using the elementary fact (2i + p)2i+p ≤ (2i)2i pp e2i+p , we can bound wM by (p)

wM L∞ (Ω) ≤ Cf pp ep S(M ) S(M ) :=

M

∀p ∈ N0 , 2i

(eγf ε(2i)) .

i=0

We have thus deﬁned for each M a decomposition uε = wM + uBL M + rM with the following properties (p)

wM L∞ (Ω) ≤ Cf ep pp S(M ) ∀p ∈ N0 , BL (p) −p −ρ(x)/ε (x)| ≤ Cε e S(M ) ∀p ∈ N0 | uM rM L2 (Ω) + εrM L2 (Ω) ≤ Cf (γf ε(2M + 2))

2M +2

(7.1.4c) 2M +2

2α3 ε

with

∀x ∈ Ω, (7.1.4b)

.

We now choose M so as to minimize (γf ε(2M + 2)) choosing M such that 2M + 2 =

(7.1.4a)

α :=

. Speciﬁcally, upon

1 , e2 γf

we get eγf ε(2M + 2) ≤ e−1 and (2M + 2) ≥ α/ε − 1. Thus, 2M +2

(eγf ε(2M + 2))

≤ e−(2M +2) ≤ ee−α/ε ,

S(M ) ≤ Cf

M i=0

2i

(γf ε(2M ))

≤ Cf

∞ i=0

e−2(2i) ≤ Cf

1 . 1 − e−4

258

7. Regularity through Asymptotic Expansions Γ5

A5

A4

Ω5

Γ6 = Γ0 Ω = Ω 6 0 Ω1 A6 = A0 Γ 1

Ω4 A1 Γ2

Γ4

Ω2

Ω3 A2

Γ3

A3

Fig. 7.1.1. Scheme of the supports of χuBL and χu ˆ CL ε ε .

Inserting these bounds in (7.1.4) shows that this speciﬁc choice of the expansion order M implies the statement of the lemma. 2 Lemma 7.1.1 illustrates several things. First of all, neglecting the exponentially small contribution rε , the exact solution uε has indeed two components: a smooth (analytic) component wε and a boundary layer component uBL that decays exε ponentially away from ∂Ω. We will meet these two solution components in the two-dimensional context again. Next, when using asymptotic expansions one often does not have bounds for the remainder that are explicit in both the perturbation parameter ε and the expansion order. However, this is necessary if one wants to choose the expansion order for a given ε so as to minimize the remainder. In our case, the analyticity of the datum f permits such explicit control and thus allows us to get an “optimal” expansion order M ∼ ε−1 . This relationship will pervade much of our two-dimensional analysis. A further observation to be made about Lemma 7.1.1 is that, while it provides precise analytic regularity assertions for the terms wε and uBL ε , the remainder rε is merely asserted to be small in some norm. This is typical for techniques from asymptotic expansions. It implies in particular that in the context of numerical analysis, decompositions of the form given in Lemma 7.1.1 can only be used up to a certain “error level”, namely, the bound on rε . Beyond that, additional information about the remainder rε is required. Outline of the construction of the decomposition. Lemma 7.1.1 provides a decomposition of the solution uε of (7.1.3) that captures the main features of uε , namely, the smooth behavior in the interior and the boundary layer behavior near the boundary. The goal of the present chapter is to provide an analogous decomposition for (1.2.11) in the two-dimensional case. Speciﬁcally, solutions uε

7.1 Motivation and outline B

259

Γ

Γ Bj Ωj

Ωj Γj

Γj

Ωj+1

Aj

Aj

Ωj+1

Γj+1 Γj+1 Fig. 7.1.2. Scheme near vertex Aj for the deﬁnition of uCL M .

of (1.2.11) on curvilinear polygons Ω are decomposed as + χu ˆ CL + rε . uε = wε + χuBL ε ε

(7.1.5)

The terms wε , uBL ε , and rε are deﬁned in a similar way as in Lemma 7.1.1 and are introduced to capture the smooth part and the boundary layer behavior near the boundary. However, neither the smooth part wε nor the boundary layer part uBL capture the corner singularities that the solution uε must have. These ε eﬀects are captured by the corner layer part uCL ε . As the boundary layer part CL uBL and the corner layer part u can be deﬁned in a meaningful way in a ε ε neighborhood of ∂Ω and the vertices only, we employ cut-oﬀ functions χ, χ ˆ for suitable localizations. The actual construction of the terms in (7.1.5) is rather lengthy and technical; the whole of Chapter 7 is devoted to this undertaking. The main results are collected in Theorem 7.4.5 and Corollary 7.4.6. In order to give the reader a guideline for the proceedings of this chapter, we outline the main steps (which parallel those of Lemma 7.1.1) for the following model problem: Lε uε := −ε2 ∆uε + uε = f

on Ω,

uε = g

on ∂Ω.

(7.1.6)

Here, Ω is assumed to be the polygon shown in Fig. 7.1.1. The right-hand side f is assumed analytic on Ω and satisﬁes (7.1.1). The Dirichlet data g are analytic on each portion Γi , i ∈ {1, . . . , 6}, of ∂Ω. 1. As in the proof of Lemma 7.1.1, we ﬁrst seek classical asymptotic-expansionsbased decompositions for the solution uε of (7.1.6) ˆ CL uε = wM + χuBL M + χu M + rM

(7.1.7)

for each expansion order M . At the end, we will choose M ∼ ε−1 to obtain the desired decomposition (7.1.5). 2. Deﬁnition of the outer expansion wM in Section 7.2. In the present constant-coeﬃcient case, we deﬁne wM as

260

7. Regularity through Asymptotic Expansions

wM =

M

ε2i ∆i f.

i=0

wM can be viewed as a particular solution to the equation that satisﬁes the diﬀerential equation (up to a small defect) but does not conform to the boundary conditions. A simple calculation shows that the defect f − Lε wM is given by f − Lε wM = ε2M +2 ∆M +1 f . Hence, assuming that f satisﬁes (7.1.1), one can show that the defect satisﬁes 2M +2

f − Lε wM L∞ (Ω) ≤ C (γε(2M + 2))

(7.1.8)

for some C, γ > 0 independent of ε and M . It can also be shown that there are C, γ, K > 0 independent of ε and M such that under the assumption ε(2M + 2)K ≤ 1,

(7.1.9)

the outer expansion wM is analytic on Ω and satisﬁes ∇p wM L∞ (Ω) ≤ Cγ p p!

∀p ∈ N0 .

3. Deﬁnition of the boundary layer expansion uBL M in Section 7.3. We assume that ε and M satisfy (7.1.9). Then the outer expansion wM yields (up to the small defect) a particular solution to (7.1.6). However, it fails to conform to the boundary conditions. This is corrected in a second step by means of boundary layer expansions uBL M , which are deﬁned in a neighborhood of the boundary ∂Ω. They are most conveniently deﬁned in boundary-ﬁtted coordinates. However, as ∂Ω is only piecewise smooth, the use of boundary-ﬁtted coordinates implies that we can deﬁne the boundary layer expansion uBL M only in a piecewise fashion. Speciﬁcally (cf. Fig. 7.1.1), on each subdomain Ωj , j ∈ {1, . . . , 6}, we introduce boundary-ﬁtted coordinates (ρj , θj ) where ρj measures the distance to the curve Γj and θj is the “tangential” coordinate in this boundary-ﬁtted system. The boundary layer expansion uBL M is then deﬁned on each subdomain Ωj separately in terms of boundary-ﬁtted coordinates (ρj , θj ). It is an (approximate) solution to Lε uBL = 0

on Ωj ,

uBL = g − wM |Γj .

(7.1.10)

The boundary layer expansions uBL M then have the following key properties: There are C, γ, α, K > 0 independent of ε, M such that under the assumption ε(2M + 2)K ≤ 1 (7.1.11) (which is the same as (7.1.9) after properly adjusting the constant K) there holds uBL M |Γj = g − wM , sup ∂ρsj ∂θtj uBL (ρ , θ ) ≤ Cγ s+t t!ε−s e−αρj /ε , i j M

(7.1.12) (7.1.13)

θj

2M +2 −αρj /ε

|Lε uBL M (ρi , θj )| ≤ (γε(2M + 2))

e

.

(7.1.14)

7.1 Motivation and outline

261

For those combinations of ε and M for which γε(2M + 2) is small, (7.1.12), (7.1.14) expresses the fact that the function uBL M is indeed an approximate solution of (7.1.10). (7.1.12) shows that the functions uBL M have indeed the typical boundary layer behavior: They decay exponentially away from the boundary curves Γj , and they behave smoothly (in fact, analytically) in the tangential variable θj . As the boundary layer functions uBL M are deﬁned in a meaningful way only near the boundary ∂Ω, their eﬀect is conﬁned to ∪6j=1 Ωj with the cut-oﬀ function χ. 4. Deﬁnition of the corner layer uCL M in Section 7.4.3. We assume that ε and M satisfy (7.1.9) and additionally (7.1.11). The boundary layer expansions uBL M are deﬁned on each subdomain Ωj separately. In general, they do not match where two subdomains Ωj , Ωj+1 meet. Their common boundary is denoted Γ in Fig. 7.1.2. This incompatibility is removed by corner layers uCL M that are deﬁned in the shaded regions of Fig. 7.1.1 near the vertices Aj (see also Fig. 7.1.2 for a detailed view near Aj ). Speciﬁcally, each such shaded region is interpreted as a sector Sj that is divided into two subsectors Sj+ , Sj− by Γ . The corner layer is then deﬁned as the solution of a transmission problem of the following type: on Sj+ ∪ Sj− ,

Lε uCL M = 0

BL on Γ , [uCL M ] = −[χuM ] CL BL [∂n uM ] = −[∂n (χuM )] on Γ , CL uM = 0 on ∂Sj .

As the boundary layer expansions uBL M decay exponentially away from Aj on the common interface Γ , this transmission problem is of the form analyzed in Chapter 6. One therefore obtains regularity assertions of the following type: + − For a suitable ball B = Bκ (Aj ) (see Fig. 7.1.2), uCL M satisﬁes on (Sj ∪Sj )∩B 1−βj −p −αrj /ε

p βj −1 rj |∇p uCL M (x)| ≤ Cγ ε

e

∀p ∈ N0 ,

where rj = dist(x, Aj ), and the constants C, γ > 0, βj ∈ [0, 1) are independent of ε and M . We note that the solution uCL M of this transmission problem decays exponentially away from Aj and has the typical corner singularity behavior near Aj . It should be stressed, however, that it is only piecewise CL smooth and that its jump across Γ matches that of χuBL M . As uM is only deﬁned as the solution of a local problem near the vertices Aj , we conﬁne it to a neighborhood of the vertices Aj with the aid of a cut-oﬀ function χ. ˆ 5. Deﬁnition of the remainder rM in Section 7.4.3. We still assume that ε CL and M satisfy (7.1.9), (7.1.11). Once wM , uBL M , and uM are deﬁned, the remainder rM is determined by (7.1.5). As in Lemma 7.1.1, one obtains bounds on rM by observing that it solves a diﬀerential equation. From the bounds on the residuals f − Lε wM and Lε uBL M (and some technicalities involving the cut-oﬀ functions χ, χ), ˆ we obtain from the shift theorem Theorem 5.3.8 2M +2 −α/ε ≤ C (γε(2M + 2)) + e rM H 2,2 (Ω) β ,ε

262

7. Regularity through Asymptotic Expansions

for some α, C, γ > 0 and β ∈ [0, 1)J independent of ε, M . Here, the term e−α/ε stems from our treatment of the cut-oﬀ functions and the term 2M +2 (γε(2M + 2)) reﬂects the residuals f − Lε wM and Lε uBL M . 6. In the ﬁnal step, we choose M = λε−1 for λ suﬃciently small depending only on the input data. For this choice CL of M we write wε = wM , uBL = uBL = uCL ε M , uε M , rε = rM and get a decomposition of the form (7.1.5) with the following properties: ∇p wε L∞ (Ω) ≤ Cγ p p!, s+t (ρ , θ ) t!ε−s e−αρj /ε , sup ∂ρsj ∂θtj uBL j j ≤ Cγ ε θj

1−βj −p −αrj /ε

p βj −1 |∇p uCL rj ε (x)| ≤ Cγ ε

rε H 2,2

β ,ε

(Ω)

e

,

x ∈ (Sj+ ∪ Sj− ) ∩ Bκ (Aj ),

≤ e−α/ε ,

for all p ∈ N0 ; all constants are independent of ε. A few general comments about asymptotic expansions and our decomposition (7.1.5) are in order. First, asymptotic expansions do not converge in general, i.e., including more terms (in our notation: increasing M ) does not necessarily increase the accuracy for a ﬁxed ε. Realizing this, it is natural to seek the expansion order M that minimizes the remainder rM for a given ε. This is achieved in our construction by choosing M ∼ ε−1 . Instrumental for this procedure, however, is the piecewise analyticity of the input data. Next, while we deﬁne outer expansion wM and the boundary layer expansion uBL M in essentially the classical way, we introduce the corner layer uCL with our application in mind, namely, the M convergence analysis of the hp-FEM in Chapter 3. One of the aims of classical asymptotic expansions is to obtain smooth (diﬀerentiable) expansions. For the convergence analysis of the FEM, however, it is suﬃcient that the function to be approximated be smooth on each element. Hence, we may use piecewise smooth functions in our decomposition of the solution u provided that the curves of discontinuity coincide with mesh-lines. We exploit this observation in our deﬁnition of the corner layer uCL M as solutions of transmission problems of the form considered in Chapter 6. The restriction that the corner layers are only piecewise smooth is more of an aesthetic restriction than of practical importance as the line of discontinuity, the curve Γ in Fig. 7.1.2, can be chosen somewhat arbitrarily. 7.1.2 Outline of Chapter 7 The outline of this chapter is as follows. We start in Section 7.2 by deﬁning the outer expansion wM . These functions are (up to a small defect) particular solutions of the equation but ignore the boundary conditions. Theorem 7.2.2 in Section 7.2 makes precise analytic regularity statements for the outer expansion wM . As we consider (1.2.11) with variable coeﬃcients, the outer expansion wM

7.2 Regularity of the outer expansion

263

cannot be deﬁned in as explicit a way as was done above. Rather, it is deﬁned in a recursive way and regularity results as stated above have to be proved by induction. In a second step, the boundary conditions are corrected. This is done in Section 7.3 by means of the boundary layer expansion uBL M . Technically, this is achieved by introducing boundary-ﬁtted coordinates (ρj , θj ) for each boundary curve Γj and then seeking a solution of the homogeneous diﬀerential equation with inhomogeneous boundary data on the subdomain Ωj . More precisely, the solution uBL is expanded formally as uBL (ρj , θj ) ∼

∞

εi Ui (ρj /ε, θj ),

i=0

and then inserted into the diﬀerential equation. Equating like powers of ε ultimately yields a recurrence relation for the functions Ui , each being the solution of an ordinary diﬀerential equation. The proper treatment of these recursively deﬁned functions Ui is again achieved by induction arguments. The boundary BL layer expansion uBL . M is then obtained by truncating the formal sum for u BL The main results concerning the boundary layer expansions uM are collected in Theorem 7.3.3. As Ω is a curvilinear polygon, this boundary layer expansions lead to incompatibilities near the vertices Aj . These incompatibilities are removed with the aid of corner layers in Section 7.4.3. Analysis of the remainder rM is also done in Section 7.4.3. We leave some freedom in the choice of the curve Γ : For concave corners Aj , this smooth curve can be chosen arbitrarily as long as the angle between Γ and each of the two boundary curves Γj , Γj+1 is less than π. At convex corners Aj , an alternate corner layer may even be deﬁned without taking recourse to an auxiliary curve Γ . A corner layer deﬁned in this way in convex corners is then analytic instead of being merely piecewise analytic. Section 7.4.3 includes the construction in this case as well. The construction of this alternate corner layer in convex corners is closely related to the choice of the subdomains Ωj . In the example of this introductory section, the subdomains Ωj in Fig. 7.1.1 do not overlap, and the corner layers uCL M are only piecewise smooth. These corner layers are deﬁned near each vertex Aj and depend on the particular choice of the auxiliary line Γ passing through Aj ; this line is ∂Ωj ∩ ∂Ωj+1 . If a vertex Aj is a convex corner, then an alternative deﬁnition of a corner layer is possible that is analytic and not merely piecewise analytic. For such a construction, it is more convenient to allow sudomains Ωj , Ωj+1 to overlap at convex corners. Our technical assumptions on the subdomains Ωj and the cut-oﬀ functions χ, χ ˆ at the outset of Section 7.4.1 reﬂect this.

7.2 Regularity of the outer expansion The outer expansion for a problem of the form (1.2.11a) is obtained by seeking a (formal) particular solution of the diﬀerential equation in the form

264

7. Regularity through Asymptotic Expansions

∞

u ∼ j=0 εj uj . This (formal) power series in ε is inserted into (1.2.11a) and like powers of ε are equated, yielding the following recurrence relation for the unknown functions uj : u0 (x) :=

1 f (x), c(x)

u2j+2 (x) :=

1 ∇ · (A(x)∇uj (x)), c(x)

u2j+1 = 0,

j ∈ N0 . (7.2.1)

For each M ∈ N0 we can then deﬁne the outer expansion as wM :=

2M +1

j

ε uj (x) =

j=0

M

ε2j u2j (x).

(7.2.2)

j=0

From this deﬁnition of the outer expansion, we compute Lε wM = f − ε2M +2 ∇ · (A∇u2M ).

(7.2.3)

We remark the particularly simple structure of wM for the special case A = Id: wM =

M

ε2j ∆j f.

(7.2.4)

j=0

The following lemma (cf. also [92, 96]) is useful for controlling the functions uj deﬁned by (7.2.1): Lemma 7.2.1. Let Ω ⊂ R2 be a domain, A, c be analytic on Ω and assume that c ≥ c0 on Ω for some c0 > 0. Assume that u0 is analytic on Ω and satisﬁes there ∇p u0 L∞ (Ω) ≤ Cu γup p! ∀p ∈ N0 . Deﬁne the functions u2j , j ∈ N0 , recursively by uj+2 (x) :=

1 ∇ · (A(x)∇uj (x)). c(x)

Then there exist C, K, γ > 0 depending only on A, c such that there holds ∇p uj L∞ (Ω) ≤ CCu K j j!p!(γ + 2γu )j+p

∀j ∈ N0 ,

p ∈ N0 .

Proof: By our assumptions, the functions A, 1c , u0 have holomorphic exten˜ := {(x + z1 , y + z2 ) ∈ C2 | (x, y) ∈ sions (again denoted A, 1/c, u0 ) to a set G Ω, zi , ∈ C, |zi | < δ0 } with δ0 < min {1, 1/γu } appropriately chosen. Moreover, by replacing δ0 with δ0 /2, we may assume that u0 L∞ (G) ˜ ≤ 2Cu , ) ) ) ) ) ) )1 ) ) ) )1 )1 ) Aij ) ) ) ) ) )c ) ∞ ˜ + ) c ∂x Aij ) ∞ ˜ + ) c ∂y Aij ) ∞ ˜ ≤ CA , L (G) L (G) L (G)

i, j ∈ {1, 2}

for some CA depending only on the data A, c. For δ ∈ (0, δ0 ), we deﬁne the sets

7.2 Regularity of the outer expansion

265

˜ | (x, y) ∈ Ω, |zi | < δ0 − δ}. ˜ δ := {(x + z1 , y + z2 ) ∈ G G We now prove a stronger statement than required by the statement of the lemma, namely, that for K 2 > 10eCA uj L∞ (G˜ δ ) ≤ δ −j K j j!u0 L∞ (G) ˜

∀j ∈ 2N0

∀δ ∈ (0, δ0 ).

(7.2.5)

The claim of the lemma then follows easily from (7.2.5) with Cauchy’s integral formula for derivatives (with the path of integration being a circle with radius ∼ δ0 /2) and the choice δ ∼ δ0 /2. For the proof of (7.2.5), we note that it holds for j = 0 and proceed by induction on the even j. To that end, assume that for some j ∈ 2N0 (7.2.5) is already proven for all even 0 ≤ j ≤ j. From Cauchy’s integral theorem for derivatives, we have for all δ ∈ (0, δ0 ) and all κ ∈ (0, 1): Dα uj L∞ (G˜ δ ) ≤

|α|! uj L∞ (G˜ (1−κ)δ ) , (κδ)|α|

∀α ∈ N20 .

Thus, we get from the deﬁnition of the recurrence relation, the fact that δ < 1, κ < 1, and the induction hypothesis: . / 2 4·2 uj+2 L∞ (G˜ δ ) ≤ CA uj L∞ (G˜ (1−κ)δ ) (7.2.6) + κδ (κδ)2 1 uj L∞ (G˜ (1−κ)δ ) ≤ 10CA (κδ)2 ≤ 10CA δ −(j+2) j!K j (1 − κ)−j κ−2 u0 L∞ (G) ˜ . 10CA K −2 ≤ δ −(j+2) K j+2 (j + 2)!u0 L∞ (G) ˜

/ 1 −j . (1 − κ) (j + 1)(j + 2)κ2

Next, choosing κ = 1/(j + 2) and observing that this choice implies j+1 (j + 2)j+1 1 (1 − κ)j = = 1 + ≤ e, κ2 (j + 1)(j + 2) (j + 1)j+1 j+1 we can bound the expression in brackets [ · ] in (7.2.6) by 1 and thus conclude the induction argument. 2 Lemma 7.2.1 enables us to formulate the following for the outer expansion wM : Theorem 7.2.2 (regularity of the outer expansion). Let G ⊂ R2 be an open set and let f ∈ A(G) satisfy ∇p f L∞ (G) ≤ Cf γfp p!

∀p ∈ N0 .

Assume that A, c are analytic on G and that c > 0 on G. Then, for each M ∈ N0 the function wM given by (7.2.4) is analytic on G and there exist C, K, γ > 0 depending only on the coeﬃcients A, c such that under the constraint 0 < 2M ε(γ + 2γf )K ≤ 1 there holds for all p ∈ N0 ∇p wM L∞ (G) ≤ Cf C(γ + 2γf )p p!, 2M +2

∇p (Lε wM − f ) L∞ (G) ≤ Cf C {(2M + 2)ε(γ + 2γf )K}

(γ + 2γf )p p!.

266

7. Regularity through Asymptotic Expansions

Proof: We employ Lemma 7.2.1. From the deﬁnition u0 = f /c, we readily ascertain that there exists C, γ > 0 depending only on c such that ∇u0 L∞ (G) = ∇p (f /c)L∞ (G) ≤ CCf (γ + 2γf )p p!

∀p ∈ N0 .

˜ > 0 depending only on A, c Thus, Lemma 7.2.1 yields the existence of C, γ, K such that the functions u2j deﬁned in (7.2.1) satisfy ˜ 2j ∇p u2j L∞ (G) ≤ CCf (γ + 2γf )p+2j (2j)!p!K

∀p ∈ N0 , j ∈ N0 .

(7.2.7)

Hence, the deﬁnition of wM implies ∇p wM L∞ (G) ≤ CCf (γ + 2γf )p p!

M

˜ 2j ε2j (γ + 2γf )2j (2j)!K

j=0

≤ CCf (γ + 2γf )p p!

M

2j ˜ ε2j 2M (γ + 2γf )K

j=0

≤ CCf (γ + 2γf ) p!; p

˜ for the constant K of the statement of this in the last step, we took K := 2K theorem in order to bound the sum by a geometric series converging to 2. This proves the ﬁrst claim of the theorem. For the second one, we observe that (7.2.3) implies by (7.2.7) ∇p (Lε wM − f )L∞ (G) = ∇p ε2M +2 ∇ · (A∇u2M )L∞ (G) 2M +2

≤ CCf ((2M + 2)ε(γ + 2γf )K)

(γ + 2γf )p p!. 2

Remark 7.2.3 Theorem 7.2.2 gives bounds on the derivatives of the outer expansion wM that are explicit in their dependence on the domain of analyticity of the datum f (measured in terms of γf ). This could be of interest if singular right-hand sides f are considered. In what follows, we will use the outer expansion only as a means to obtain one particular solution to (1.2.11a) that is analytic and whose residue is small. It is therefore convenient to choose the expansion order M in dependence on ε such that the residue is small. This is achieved in the ensuing corollary. Corollary 7.2.4. Assume the hypotheses of Theorem 7.2.2. Then there exist C, K, γ > 0 independent of ε such that for every ε ∈ (0, 1] there exists wε ∈ A(G) satisfying ∇p wε L∞ (G) ≤ CK p p!

∀p ∈ N0 , −γ/ε

∇ (Lε wε − f )L∞ (G) ≤ CK p!e p

p

∀p ∈ N0 .

7.3 Regularity of the boundary layer expansion

267

Proof: Let γ, K be given by Theorem 7.2.2. We then choose 5 4 1 M := 2K(γ + 2γf ) and set wε := wM . The desired result then follows from Theorem 7.2.2. We remark that this proof also gives explicit bounds on the constants C, K, γ appearing in the statement of Corollary 7.2.4. 2

7.3 Regularity of the boundary layer expansion 7.3.1 Deﬁnition and properties of the boundary layer expansion The purpose of the present subsection is the deﬁnition of boundary layer expansions. For our purposes, let Ix , Iy ⊂ R be bounded intervals with the additional assumption that the left endpoint of Iy is the origin. Deﬁne R = Ix × Iy and let A ∈ A(R, S2> ), c ∈ A(R) be given satisfying 0 < λmin ≤ A

on R,

c>0

on R

(7.3.1)

for some λmin > 0. On the rectangle R, we consider the diﬀerential operator Lε u = −ε2 ∇ · (A(x, y)∇u) + c(x, y)u and construct (approximate) solutions to Lε u = 0

on R,

u|y=0 = g,

where g is assumed to be analytic on Ix and satisﬁes ∇p gL∞ (Ix ) ≤ Cg γgp p!

∀p ∈ N0 .

(7.3.2)

The solution u that we construct furthermore satisﬁes a decay condition in y. Let us denote the components of the matrix A by aij (i, j ∈ {1, 2}) with a12 = a21 . The expression ∇ · (A∇u) then takes the form ∇·(A∇u) = a11 ∂x2 + 2a12 ∂xy + a22 ∂y2 + (a11,x + a12,y )∂x + (a12,x + a22,y )∂y u where we abbreviated akj,x = ∂x akl , akj,y = ∂y akl . Next, we introduce the stretched variable yˆ := y/ε. In the new coordinates (x, yˆ), the diﬀerential operator Lε reads Lε u = − ε2 a11 ∂x2 + 2εa12 ∂xˆy + a22 ∂y2ˆ (7.3.3) + ε2 (a11,x + a12,y )∂x + ε(a12,x + a22,y )∂yˆ u + cu. The coeﬃcients akl , akl,x and c are expanded as power series in y as follows:

268

7. Regularity through Asymptotic Expansions

akl (x, y) =

∞

aikl (x)y i ,

akl,x (x, y) =

i=0

∞

aikl,x (x)y i ,

c(x, y) =

i=0

∞

ci (x)y i .

i=0

Our assumptions on the data imply convergence of these power series on the ball {y ∈ C | |y| < y0 } for some y0 > 0. Inserting these Taylor expansions into (7.3.3) and equating like powers of ε, we obtain Lε =

∞

εi Li ,

(7.3.4)

i=0

where the diﬀerential operators Li are given by i−1 i−1 i−2 i−2 2 yˆ ∂xˆy − b11 yˆ ∂x − b1i−2 yˆi−2 ∂x − b2i−1 yˆi−1 ∂yˆ + ci yˆi , Li = −bi22 yˆi ∂y2ˆ − b12 (7.3.5) with coeﬃcients bikl , bik deﬁned by

bi22 (x) = ai22 (x), bi12 (x) = 2ai12 (x), bi11 (x) = ai11 (x), bi1 (x) = ai11,x (x) + ai+1 12 (x)(i + 1) , bi2 (x) = ai12,x (x) + ai+1 22 (x)(i + 1) ,

(7.3.6a) (7.3.6b)

bi11 (x)

(7.3.6d)

=

bi12 (x)

=

bi22 (x)

=

bi1 (x)

=

bi2 (x)

=0

(7.3.6c)

for i < 0.

In particular, the operator L0 has the form L0 = −a022 (x)∂y2ˆ + c0 (x).

(7.3.7)

Remark 7.3.1 The formal series in (7.3.4) in fact converges for |εˆ y | < y0 because the power series deﬁning akl , akl,x , c converge. We will make use of this observation below. The constant y0 > 0 depends only on the data A, c. In order todeﬁne the boundary layer expansion, we make the formal ansatz ∞ j (x, yˆ). Upon inserting this ansatz into (7.3.4), we obtain u(x, y) ∼ εj U ∞ j i j=0 j=0 ε j=0 Lj Ui−j = 0. Next, setting the coeﬃcients of this formal power series in ε to zero, yields a recurrence relation of ordinary diﬀerential equations j : (in yˆ) for the unknown functions U

i 6

i = L0 U i = − i−j := Fi = Lj U Fik , −a022 (x)∂y2ˆ + c0 (x) U j=1

Fi1 =

i−1

i−1−j , bj+1 ˆj+1 ∂y2ˆ U 22 y

Fi2 =

j=0

Fi3 =

i−2

i−1 j=0

i−1−j , bj12 yˆj ∂xˆy U

(7.3.8b)

j=0

i−2−j , bj11 yˆj ∂x2 U

j=0

Fi5 =

i−1

(7.3.8a)

k=1

Fi4 =

i−2

i−2−j , bj1 yˆj ∂x U

(7.3.8c)

j=0

i−1−j , bj2 yˆj ∂yˆ U

Fi6 = −

i−1 j=0

i−1−j , cj+1 yˆj+1 U

(7.3.8d)

7.3 Regularity of the boundary layer expansion

269

where we used the tacit convention that empty sums take the value zero. In order to complete this system of ordinary diﬀerential equation, we have to prescribe i . As we want u to decay for yˆ → ∞ and as two boundary conditions for each U we want to satisfy the boundary condition u(x, 0) = g(x), we prescribe i → 0 U for yˆ → ∞, ! i (x) = g(x) for i = 0 U 0 for i > 0.

(7.3.9a) (7.3.9b)

i , i = 0, 1, . . . of functions. The bound(7.3.8), (7.3.9) deﬁne a unique sequence U BL ary layer expansion uM is taken as uBL M (x, y) :=

2M +1 i=0

i (x, yˆ) = εi U

2M +1

i (x, y/ε). εi U

(7.3.10)

i=0

∞ i Remark 7.3.2 The series i=0 ε Ui is a formal series, which cannot be ex2M +1 pected to converge. We truncated this formal series after the term ε2M +1 U 2M +2 so that the ﬁrst neglected term is of order ε . This is the same (formal) error as the one introduced by truncating the outer expansion. We are now in position to state our main result concerning the regularity of the boundary layer expansion (7.3.10). Theorem 7.3.3 (regularity of the inner expansion). Let A ∈ A(R, S2> ), c ∈ A(R) satisfy (7.3.1), and let g satisfy (7.3.2). Then the function uBL M of (7.3.10) is analytic on R and satisﬁes uBL M (·, 0) = g on Ix . Moreover, there exist C, γ, K, K , y0 , λ > 0 depending only on the data A, c such that for all ε ∈ (0, 1], M ∈ N0 with 0 ≤ ε(2M + 2)(γ + 2γg )K ≤ 1 the following holds: for all (p, q) ∈ N20 p+q −p |∂xq ∂yp uBL ε (γ + 2γg )q e−λy/ε M (x, y)| ≤ CCg q!K

∀(x, y) ∈ Ix × R+ 0

and for the residual, we have for all (x, y) ∈ Ix × (0, y0 ) |∂xq ∂yp Lε uBL M (x, y)| ≤ CCg K p+q p!q!ε−p (γ + 2γg )q (ε(2M + 2)(γ + 2γg )K )

2M +2 −λy/ε

e

.

The proof is lengthy and therefore relegated to the next subsection. As in the case of the outer expansion, we can extract from Theorem 7.3.3 a corollary by choosing the expansion order M proportional to ε−1 . Corollary 7.3.4. Under the hypothesis of Theorem 7.3.3 there exist C, K, γ, λ > 0, and y0 > 0 independent of ε such that for every ε ∈ (0, 1] there exists 2 uBL ∈ A(R) with uBL ε ε (·, 0) = g on Ix and there holds for all (p, q) ∈ N0 p+q |∂xq ∂yp uBL q!ε−p e−λy/ε , ε (x, y)| ≤ CK p+q |∂xq ∂yp Lε uBL p!q!e−γ/ε e−λy/ε ε (x, y)| ≤ CK

∀(x, y) ∈ Ix × R+ , ∀(x, y) ∈ Ix × (0, y0 ).

Proof: The proof follows by the same reasoning as in the proof of Corollary 7.2.4 2 by choosing the expansion order M proportional to ε−1 .

270

7. Regularity through Asymptotic Expansions

7.3.2 Proof of Theorem 7.3.3 Let Ix be the interval deﬁned at the outset of Section 7.3.1. For X > 0, we deﬁne complex neighborhoods of Ix as SX := {z ∈ C | dist(z, Ix ) < X}, SX (δ) := {z ∈ SX | dist(z, ∂SX ) > δ},

δ > 0.

(7.3.11)

By geometric considerations, it is easy to see that SX (δ) = SX−δ . In view of the form of the operator L0 , it is convenient to introduce the function 0 c0 (x) λ(x) := , (7.3.12) a022 (x) which is positive on Ix and which can be extended holomorphically to a (complex) neighborhood of Ix . Lemma 7.3.5. Let the coeﬃcients A, c be analytic on R and let g ∈ A(Ix ) satisfy (7.3.2). Then there exist constants CA , γA , λ0 > 0 depending only on A, c such that for X := (γA + 2γg )−1 > 0 the functions g, λ, and bi11 , bi12 , bi22 , bi1 , bi2 , ci of (7.3.5) have holomorphic extensions to SX and satisfy for all z ∈ SX |g(z)| ≤ 2Cg , 1 a0 (z) ≤ CA , 22

|λ(z)| ≤ CA ,

Re λ(z) ≥ λ0 ,

Re λ2 (z) ≥ λ20 ,

i |bi11 (z)| + |bi12 (z)| + |bi22 (z)| + |bi1 (z)| + |bi2 (z)| + |ci (z)| ≤ CA γA

∀i ∈ N0 .

Proof: For z ∈ S1/(2γg ) , the bound on g follows from power series expansions around points of Ix . The second bounds follow easily from the assumptions on A, c. For the last bound, we note that there exists a complex neighborhood ˜ ⊂ C × C of R on which A and c are holomorphic. The result follows from R Cauchy’s integral formula for derivatives; for example, c(x, y) 1 i dy c (x) = 2π i C (−y)i+1 for some closed loop C around the origin (in the complex plane).

2

Lemma 7.3.6. Let λ ∈ C with Re λ > 0, Re λ2 > 0. Let f be an entire function satisfying for some Cf > 0, j ∈ N0 , q ≥ (j + 1/2)/|λ| > 0 |f (z)| ≤ Cf e− Re(λz) (q + |z|)j

∀z ∈ C.

Let g ∈ C and let u : (0, ∞) → C be the solution to −u + λ2 u = f

on (0, ∞),

u(0) = g,

lim u(x) = 0.

x→∞

Then u can be extended to an entire function (again denoted u), which satisﬁes . / 1 j+1 −1 |u(z)| ≤ Cf ∀z ∈ C. (q + |z|) (j + 1) + |g| e− Re(λz) |λ|

7.3 Regularity of the boundary layer expansion

271

Proof: For z ∈ (0, ∞), the use of a Green’s function gives the following representation of the solution u(z): ∞ 1 −λz λz y λz u(z) = e e f (y/λ) dy + e e−y f (y/λ) dy 2λ2 0 λz ∞ −λz −e e−y f (y/λ) dy + ge−λz . 0

Analytic continuation then removes the restriction to (0, ∞). In order to get the desired bound, we estimate each of these four terms separately. For the ﬁrst integral, we use as the path of integration the straight line connecting 0 and λz to get λz 1 −λz y − Re λz e f (y/λ) dy ≤ e Cf (q + t|z|)j |λz| e− Re tλz eRe tλz dt e 0 0 ≤ Cf e− Re λz

|λ| (q + |z|)j+1 − q j+1 . j+1

For the second term, we calculate ∞ λz ∞ −y −y e = e f (y/λ) dy e f (z + y/λ) dy λz 0 ∞ ≤ e− Re λz Cf |λ|−j e−2y (|λ|q + |λz| + y)j dy 0

= Cf e− Re λz |λ|−j 2−(j+1) e2|λ|(q+|z|) Γ (j + 1, 2|λ|(q + |z|)), where Γ (·, ·) denotes the incomplete Gamma function, and we used [61, eq. 8.353.5] in the last step. We observe that 2|λ|q ≥ 2j + 1 ≥ j. Thus, we may employ the estimate −ξ α e ξ , α0 = max {α − 1, 0}, Re ξ ≥ 0, |ξ| > α0 |Γ (α, ξ)| ≤ |ξ| − α0 (see, e.g., [103, Chap. 4, Sec. 10]) to ﬁnally arrive at λz ∞ −y (q + |z|)j+1 e e f (y/λ) dy ≤ Cf e− Re λz |λ| 2|λ|(q + |z|) − j λz (q + |z|)j+1 ≤ Cf e− Re λz |λ| . j+1 +∞ For the third term, we observe that the integral 0 f (y)e−y dy is precisely the second term with z = 0. We conclude therefore that for the third term j+1 −λz ∞ −y e ≤ Cf e− Re λz |λ| q f (y/λ)e dy . j+1 0 Hence, we arrive at

272

7. Regularity through Asymptotic Expansions

λz ∞ ∞ −λz y λz −y −λz −y e f (y/λ) dy + e e f (y/λ) dy − e e f (y/λ) dy ≤ e 0 λz 0 2Cf e− Re λz (q + |z|)j+1

|λ| . j+1

Combining this estimate with the obvious one for the fourth term, we arrive at the desired bound. 2 Lemma 7.3.7. Let X > 0 and let λ be a function holomorphic on SX satisfying |λ(x)| ≤ CA for all x ∈ SX . Let U be holomorphic on SX × C and assume that there exist CU > 0, λ0 , i ∈ N0 such that for all δ ∈ (0, X) and all (x, z) ∈ SX (δ) × C |U (x, z)| ≤ CU {(2i + 1)/λ0 + |z|}2i e− Re(λ(x)z) δ −i . Then for all δ ∈ (0, X) and for all (x, z) ∈ S(X − δ) × C there holds 2i 2(i + 1) + 1 λ0 2CA /λ0 e |∂z U (x, z)| ≤ CU + |z| e− Re(λ(x)z) δ −i , 2 λ0 2i 2 2 ∂z U (x, z) ≤ λ0 e2CA /λ0 CU 2(i + 1) + 1 + |z| e− Re(λ(x)z) δ −i , 2 λ0 2i+1 2i + 2 2CA /λ0 |∂x U (x, z)| ≤ 4λ0 e CU + |z| e− Re(λ(x)z) δ −(i+1) , λ0 2i+2 2 ∂x U (x, z) ≤ 4λ20 e2CA /λ0 CU 2i + 2 + |z| e− Re(λ(x)z) δ −(i+2) , λ0 2i+1 2(i + 1) + 1 2 4CA /λ0 |∂xz U (x, z)| ≤ 2λ0 e CU + |z| e− Re(λ(x)z) δ −(i+1) . λ0 Proof: For the ﬁrst estimate, we use Cauchy’s integral theorem for derivatives: 1 U (x, z + t) dt |∂z U (x, z)| = 2π i |t|=2/λ0 t2 ≤ λ0 /2CU {(2i + 1)/λ0 + |z| + 2/λ0 }2i e− Re λz+|λ|2/λ0 δ −i . The second estimate is proved similarly. For the derivatives with respect to the x-variable, we ﬁrst note that for all δ ∈ (0, X), there holds |λ (z)| ≤ CA δ −1

∀z ∈ SX (δ),

(7.3.13)

which can easily be ascertained with Cauchy’s integral theorem for derivatives, taking as the path of integration the (complex) circle of radius δ < δ around x and then letting δ → δ. In order to get the third estimate, we use Cauchy’s integral theorem for derivatives but choose the path of integration as ∂Bκδ (x)

7.3 Regularity of the boundary layer expansion

273

with κ ∈ (0, 1) to be chosen below. Noting that this path is completely contained in SX ((1−κ)δ), we have for t ∈ C with |t| = κδ that Re(λ(x+t)z) = Re(λ(x)z)+ Re ((λ(x + t) − λ(x))z) and arrive 1 U (x + t, z) |∂x U (x, z)| = dt 2π i |t|=κδ t2 2i −1 2i + 1 1 ≤ CU + |z| e− Re(λ(x)z) eκδCA ((1−κ)δ) i (κδ)((1 − κ)δ) λ0 2i 2i + 1 1 −(i+1) ≤ CU eCA κ/(1−κ) δ + |z| e− Re(λ(x)z) . κ(1 − κ)i λ0 Choosing κ = 1/(2i + 2 + λ0 |z|) and observing that this choice implies 1 ≤ 1 − κ, 2

1 ≤ (1 − κ)2i+2+λ0 |z| ≤ (1 − κ)i ≤ 1 4

∀i ∈ N0 ,

we obtain

2i 2i + 1 + |z| e− Re(λ(x)z) λ0 2i+1 2i + 2 2CA /λ0 −(i+1) ≤ CU e 4δ λ0 + |z| e− Re(λ(x)z) . λ0

|∂x U (x, z)| ≤ CU e2CA /λ0 δ −(i+1)

4 κ

Finally, the fourth and ﬁfth estimate are proved completely analogously.

2

i deﬁned by (7.3.8), (7.3.9). We note that We now turn to bounding the terms U i satisfy the equation with λ deﬁned in (7.3.12), the functions U i = i + λ2 (x)U −∂y2ˆ U

1 Fi a022 (x)

with the boundary conditions (7.3.9). From F0 = 0 and (7.3.9), we obtain 0 (x, yˆ) = g(x)e−λ(x)ˆy . U

(7.3.14)

We compute F1 (x, yˆ) = {a(x) + b(x)ˆ y } e−λ(x)ˆy , 0 a(x) := b2 (x)λ(x)g(x) − b012 (x)(λ (x)g(x) + λ(x)g (x) − λ(x)λ (x)g(x)) , b(x) := λ2 (x)b122 (x) − c1 (x) g(x), 1 is then given by and ﬁnd that the function U . / a(x) b(x) 2 −λ(x)ˆy 1 b(x) 1 (x, yˆ) = y ˆ − e . U − y ˆ a022 (x) 2λ(x) 4λ2 (x) 4λ(x)

(7.3.15)

274

7. Regularity through Asymptotic Expansions

It can be shown inductively (in fact, this is shown in Corollary 7.3.9 below) that i (x, yˆ) is of the form U i (x, yˆ) = P2i (x, yˆ)e−λ(x)ˆy where for all i ∈ N0 the function U the functions P2i are polynomials of degree 2i in the variable yˆ with coeﬃcients analytic in x. This observation also motivates the induction hypothesis for the following proposition. Proposition 7.3.8. Let the coeﬃcients A ∈ A(R, S2> ), c ∈ A(R) satisfy (7.3.1) i be deﬁned by (7.3.8), and let g ∈ A(Ix ) satisfy (7.3.2). Let the functions U (7.3.9). Then there are CU , K, γA > 0 depending only on the coeﬃcients A, c such that with X = (γA + 2γg )−1 there holds for all i ∈ N0 and all (x, yˆ) ∈ SX (δ) × C i (x, yˆ)| ≤ CU Cg K i δ −i 1 |U i!

2i 2i + 1 + |ˆ y| e− Re(λ(x)ˆy) . λ0

(7.3.16)

Proof: Let X and the constants CA , γA be given by Lemma 7.3.5. The choice of X in Lemma 7.3.5 and Cauchy’s integral theorem for derivatives imply for the holomorphic extension of g to SX : |g(z)| ≤ 2Cg

on SX ,

|g (z)| ≤ 2δ −1 Cg

on SX (δ),

δ > 0.

These bounds allow us to ﬁnd a constant C > 0 depending only on the coeﬃcients 0 , U 1 given in (7.3.14), A and c such that for all (x, yˆ) ∈ SX (δ)×C the functions U (7.3.15) satisfy 0 (x, yˆ)| ≤ CCg e− Re(λ(x)z) , |U

1 (x, yˆ)| ≤ CCg δ −1 |ˆ y | + |ˆ y |2 e− Re(λ(x)z) . |U

We note that for some C > 0 depending only on the coeﬃcients A, c we can bound: |ˆ y | + |ˆ y |2 ≤ C(2/λ0 + |ˆ y |)2 for all (x, yˆ) ∈ SX × C. We conclude that (7.3.16) holds true for i = 0 and i = 1. In order to proceed by induction on i, we now deﬁne the constants CU , K, whose existence is asserted in the statement of the proposition. CU is chosen such that (7.3.16) holds true for i = 0 and i = 1. For the deﬁnition of K, we introduce C1 :=

λ20 2CA /λ0 e , 2

C4 := 4λ0 e2CA /λ0 ,

C2 := 2λ20 e4CA /λ0 , C5 :=

λ0 2CA /λ0 e 2

C3 := 4λ20 e2CA /λ0 ,

(7.3.17a) (7.3.17b)

and then choose K > 1 such that / . C1 λ0 2K C4 λ30 C5 λ20 λ0 C2 λ0 C3 λ20 2 ≤ 1. CA + + + + + 2K − λ0 2K 2K 2K 4K 2 8γA K 2 4γA K (7.3.18) Next, we have for i ≥ 2

7.3 Regularity of the boundary layer expansion

275

j+1

j+1 ij+1 1 λ0 2i 1 ≤ ≤ + |ˆ y| , (i − j − 1)! i! i! 2 λ0 j+2 j+2 ij+2 1 λ0 2i 1 ≤ ≤ + |ˆ y| , (i − j − 2)! i! i! 2 λ0 −1 λ0 2i λ0 ≤ . + |ˆ y| ≤ λ0 4 2

j ≤ i − 1, j ≤ i − 2,

This allows us to bound for b ∈ {0, 1} 2(i−j−1) j+1 2i 2i 1 1 2i λ0 |ˆ y| + |ˆ y| ≤ + |ˆ y| , (7.3.19) (i − 1 − j)! λ0 2 i! λ0 2(i−j−1)+1−b j+1+b 2i 2i 1 1 2i λ0 |ˆ y |j + |ˆ y| ≤ + |ˆ y| ,(7.3.20) (i − 1 − j)! λ0 2 i! λ0 2(i−j−2)+2−b j+2+b 2i 2i 1 1 2i λ0 j |ˆ y| + |ˆ y| ≤ + |ˆ y| .(7.3.21) (i − 2 − j)! λ0 2 i! λ0 j+1

We can now prove the statement of the proposition by induction on i. To that end, we assume that (7.3.16) holds true for all 0 ≤ i < i for i ≥ 2 and show that it holds for i as well. This is achieved with the aid of Lemma 7.3.6. By i by considering 6 the linearity of the operator L0 , we can obtain a bound on U (slightly) diﬀerent types of subproblems. First, let us consider solutions u1j of −∂y2ˆ u1j + λ2 u1j =

bj+1 22 i−1−j , yˆj+1 ∂y2ˆ U a022

u1j (x, 0) = 0,

lim u1j (x, yˆ) = 0.

yˆ→∞

λ2

Noting that 20 e2|λ(x)|/λ0 ≤ C1 on SX , we obtain from Lemma 7.3.7, the induction hypothesis, and (7.3.19) that there holds on SX (δ) × C j+1 2 y ∂yˆ Ui−1−j ˆ 2(i−j−1) 2i 1 ≤ |ˆ y |j+1 C1 CU Cg δ −(i−1−j) K i−1−j + |ˆ y| e− Re(λˆy) (i − 1 − j)! λ0 2i 2i −(i−1−j) i−1−j j+1 1 ≤ C1 CU Cg δ K (λ0 /2) + |ˆ y| e− Re(λˆy) . i! λ0 0 Applying now Lemma 7.3.6 and noting the bounds on bj+1 22 , a22 from Lemma 7.3.5, we get on SX (δ) × C ! j+1 * 2i λ0 γA δ 1 2i j i −i − Re(λˆ y) 2 |u1 (x, yˆ)| ≤ K δ CU Cg CA C1 . + |ˆ y| e i! λ0 2K

We immediately note that a similar reasoning applies to the functions uj6 deﬁned i−1−j with the corresponding as the solutions to −∂y2ˆ uj6 +λ2 uj6 = −cj+1 /a022 yˆj+1 U boundary conditions:

276

7. Regularity through Asymptotic Expansions

|uj6 (x, yˆ)|

1 ≤ K δ CU Cg i! i −i

! j+1 * 2i 2i λ0 γA δ − Re(λˆ y) 2 CA . + |ˆ y| e λ0 2K

The remaining 4 cases are treated similarly. We bound with C2 of (7.3.17) and i−1−j as follows: (7.3.20) for the solution uj2 of −∂y2ˆ uj2 + λ2 uj2 = bj12 /a022 yˆj ∂xˆy U j i−1−j y ∂xˆy U ˆ 2(i−j−1)+1 2i 1 ≤ |ˆ y |j C2 CU Cg K i−1−j δ −(i−j) + |ˆ y| e− Re(λˆy) (i − 1 − j)! λ0 2i 2i i−1−j −(i−j) j+1 1 ≤ C2 CU Cg K δ (λ0 /2) + |ˆ y| e− Re(λˆy) . i! λ0 Hence, reasoning as before, we conclude that uj2 satisﬁes ! j * 2i 2 2i CA C2 λ0 γA δλ0 j i −i 1 − Re(λˆ y) |u2 (x, yˆ)| ≤ CU Cg K δ . + |ˆ y| e i! λ0 2K 2K i−2−j , we bound with C3 For the solution uj3 of −∂y2ˆ uj3 + λ2 uj3 = bj11 /a022 yˆj ∂x2 U of (7.3.17) and (7.3.21) j 2 y ∂x Ui−2−j ˆ 2(i−j−2)+2 2i 1 j i−2−j −(i−j) ≤ |ˆ y | C3 CU Cg K δ + |ˆ y| e− Re(λˆy) (i − 2 − j)! λ0 2i 2i i−2−j −(i−j) j+2 1 ≤ C3 CU Cg K δ (λ0 /2) + |ˆ y| e− Re(λˆy) . i! λ0 Thus, we conclude that uj3 satisﬁes |uj3 (x, yˆ)|

i −i

≤ CU Cg K δ

1 i!

! j * 2i 2 2i CA C3 λ20 γA δλ0 − Re(λˆ y) . + |ˆ y| e λ0 4K 2 2K

i−2−j , we bound with C4 of For the solution uj4 of −∂y2ˆ uj4 + λ2 uj4 = bj1 /a022 yˆj ∂x U (7.3.17) and (7.3.21) j y ∂x Ui−2−j ˆ 2(i−j−2)+1 2i 1 j i−2−j −(i−1−j) ≤ |ˆ y | C4 CU Cg K δ + |ˆ y| e− Re(λˆy) (i − 2 − j)! λ0 2i 1 2i ≤ C4 CU Cg K i−2−j δ −(i−1−j) (λ0 /2)j+3 + |ˆ y| e− Re(λˆy) . i! λ0 This leads to a bound for u4j of the form

7.3 Regularity of the boundary layer expansion

|uj4 (x, yˆ)| ≤ CU Cg K i δ −i

1 i!

277

! j * 2i 2 2i CA C4 λ30 δ γA δλ0 − Re(λˆ y) . + |ˆ y| e λ0 8K 2 2K

i−1−j , we bound Finally, for the solution uj5 of −∂y2ˆ uj5 + λ2 uj5 = bj2 /a022 yˆj ∂yˆ U with C5 of (7.3.17) and (7.3.20) j y ∂yˆ Ui−1−j ˆ 2(i−j−1) 2i 1 j i−1−j −(i−1−j) ≤ |ˆ y | C5 CU Cg K δ + |ˆ y| e− Re(λˆy) (i − 1 − j)! λ0 2i 1 2i ≤ CU Cg K i−1−j δ −(i−1−j) (λ0 /2)j+2 + |ˆ y| e− Re(λˆy) . i! λ0 Thus, reasoning as before, we conclude that uj5 satisﬁes ! j * 2i 2 2i CA C5 λ20 δ γA δ j i −i 1 − Re(λˆ y) |u5 (x, yˆ)| ≤ CU Cg K δ . + |ˆ y| e i! λ0 4K K −1 We notice that δ ≤ X ≤ γA . Thus, for 2K > λ0 we can bound

j ∞ γA λ0 δ j=0

∞ j=0

≤

j+1 ∞ λ0 2K ≤ , 2K 2K − λ0 j=0

2K j+1 2K γA λ0 δ λ0 ≤ , 2K 2K 2K − λ0

i on SX (δ) × C by combining the above six and we obtain for the function U estimates: 2i 2i i −i 1 |Ui (x, yˆ)| ≤ CU Cg K δ + |ˆ y| × i! λ0 / . λ0 C2 λ0 C3 λ20 C1 λ0 2K C4 λ30 δ C5 λ20 δ 2 + + + . CA + + 2K − λ0 2K 2K 2K 4K 2 8K 2 4K −1 Since δ ≤ γA , Our choice of K in (7.3.18) implies that the expression in brackets [ · ] is bounded by 1; this concludes the induction argument. 2

i Corollary 7.3.9. Under the hypotheses of Proposition 7.3.8, the functions U deﬁned by the recursion (7.3.8), (7.3.9) are of the form   2i   i (x, yˆ) = αi,j (x)ˆ y j e−λ(x)ˆy U   j=0

for some functions αi,j holomorphic on SX .

278

7. Regularity through Asymptotic Expansions

Proof: The proof follows from a variation of Liouville’s theorem. To that end, i eλˆy . From Proposition 7.3.8, v we consider for ﬁxed i, x the function v := U 2i is an entire function that is O(|ˆ y | ) at ∞. With the aid of Liouville’s theorem that entire bounded functions are constant, it is now easy to see that v is a polynomial of degree 2i. The holomorphy of the coeﬃcients αi,j now follows i in the ﬁrst variable x. We remark that for i ≥ 1, the from the holomorphy of U i (x, 0) = 0 implies that αi,0 (x) ≡ 0 for i ≥ 1. condition U 2 Lemma 7.3.10. Let a, b ≥ 0 with a + b ≥ 1. Then the function j →

1 (2j + a + b)2j b−j Γ (j + 1)

is monotonically increasing on R+ 0. Proof: We deﬁne the function 1 (2j + a + b)2j b−j = 2j ln(2j + a + b) − ln Γ (j + 1) − j ln b, f (j) := ln j! 2j f (j) = 2 ln(2j + a + b) + − ln b − ψ(j + 1), 2j + a + b d where ψ(x) = dx Γ (x). Using the bound ψ(x) ≤ ln x (cf., e.g., [61, eq. 8.361.3]), we can estimate 2j + a + b 2j + a + b 2j f (j) ≥ ln + ln + ≥0 b j+1 2j + a + b

by our assumptions b > 0 and a + b ≥ 1.

2

Lemma 7.3.11. Assume the hypotheses of Proposition 7.3.8 and let X > 0 be as given there. Then there exist constants C, K, γ > 0 depending only on the data A, c such that for all i ∈ N0 and (x, yˆ) ∈ SX (δ) × C there holds for k, l ∈ N0 with k + l ≤ 2 2i+k 2i + 3 k l i −(i+k) 1 + |ˆ y| e− Re(λ(x)ˆy) . ∂x ∂yˆ Ui (x, yˆ) ≤ CCg K δ i! λ0 Proof: The proof follows immediately from Proposition 7.3.8, Lemma 7.3.7, slightly enlarging the constant K of Proposition 7.3.8, and appropriately choosing C. 2 Lemma 7.3.12. Let uBL M be given by (7.3.10). Then there exist constants C, K, λ0 > 0, y0 > 0 depending only the coeﬃcients A, c such that on SX (δ)×By0 (0) ⊂ C × C there holds 2M +2 2M +2 −(2M +2) Lε uBL ε δ × M (x, y) ≤ CCg K 4M +3

(4M + 3 + λ0 |y/ε|) (2M + 1)!

e− Re(λ(x)y/ε) 1 + εδ −1 (4M + 3 + λ0 |y/ε|) .

7.3 Regularity of the boundary layer expansion

279

Proof: In the proof we will write yˆ for yˆ = y/ε whenever notationally convenient. ∞ i From Remark 7.3.1 there is y0 > 0 such that Lε = i=0 ε Li uniformly on compact subsets of SX × By0 (0) ⊂ C × C. Next, combining this with the deﬁning i , namely, i Lj U i−j = i Li−j U j = 0 for all property of the functions U j=0 j=0 i ∈ N0 , we get on compact subsets of SX × By0 (0) Lε uBL M (x, y) =

∞

εi Li

2M +1

i=0

=

j (x, y/ε) = εj U

j=0

∞ i=2M +2

ε

i

2M +1

∞

min {i,2M +1}

εi

i=0

j (x, y/ε) Li−j U

j=0

j (x, y/ε). Li−j U

j=0

j . In view of the deﬁnition of the operator Li in We now have to bound Li−j U j = T1 + T2 + · · · + T6 . In order to bound (7.3.5), (7.3.6), we can write Li−j U these six terms, we introduce the abbreviation 2j 2j + 3 1 Fj := + |ˆ y| e− Re(λ(x)ˆy) . Γ (j + 1) λ0 For the constant γA of Lemma 7.3.5 and with the aid of Lemma 7.3.11, we get the existence of constants C, K > 0 (K suitably larger than K given in Lemma 7.3.11) depending only on the data A, c such that i−j |T1 | ≤ CCg γA |ˆ y |i−j K j δ −j Fj , i−1−j |T2 | ≤ CCg γA |ˆ y |i−1−j K j δ −j−1 Fj+1/2 , i−2−j |T3 | ≤ CCg γA |ˆ y |i−2−j K j δ −j−2 Fj+1 , i−2−j |T4 | ≤ CCg γA |ˆ y |i−2−j K j δ −j−1 Fj+1/2 , i−1−j |T5 | ≤ CCg γA |ˆ y |i−1−j K j δ −j Fj , i−j |T6 | ≤ CCg γA |ˆ y |i−j K j δ −j Fj ;

furthermore, T3 = T4 = 0 for i − j − 2 < 0. Let us ﬁrst consider the contribution of the terms T1 to Lε uBL M . We have   2M +1 2M +1 ∞ ∞ i −j i −j j −j   ε |T1 | ≤ CCg γA |ˆ y | K δ Fj (εγA |ˆ y |) . j=0 i=2M +2

j=0

i=2M +2

Assuming that y0 in the statement of the lemma is so small that γA y0 < 1/2, ∞ i y |) ≤ 2(εγA |ˆ y |)2M +2 and therefore arrive at we can bound i=2M +2 (εγA |ˆ 2M +1

∞

j=0 i=2M +2

2M +2

εi T1 ≤ CCg (εγA |ˆ y |)

2M +1

−j γA |ˆ y |−j K j δ −j Fj .

j=0

Next, from Lemma 7.3.10, we obtain for 0 ≤ j ≤ 2M + 1

280

7. Regularity through Asymptotic Expansions

1 2j y |) e− Re(λˆy) (2j + 3 + λ0 |ˆ Γ (j + 1) 1 −(2M +1)−j 2(2M +1) − Re(λˆ y) |ˆ y |−(2M +1) y |) e . ≤ λ0 (4M + 5 + λ0 |ˆ (2M + 1)!

|ˆ y |−j Fj ≤ λ−j y |)−j 0 (λ0 |ˆ

Thus, by properly adjusting the constant K, and assuming that δ ≤ 1 (as we may since δ ≤ X = (γ + 2γg )−1 and γ is at our disposal), we get 2M +1

∞

εi |T1 |

j=0 i=2M +2

|ˆ y| 2(2M +1) − Re(λˆ y) y |) e (4M + 5 + λ0 |ˆ (2M + 1)! 1 4M +3 − Re(λˆ y) ≤ CCg K 2M +2 δ −(2M +1) ε2M +2 y |) e . (4M + 5 + λ0 |ˆ (2M + 1)! ≤ CCg K 2M +2 δ −(2M +1) ε2M +2

The contributions due to T2 , T5 , T6 are treated similarly. We have 2M +1

∞

εi |T2 |

j=0 i=2M +2

≤ CCg K 2M +2 δ −(2M +2) ε2M +2 2M +1

∞

1 4M +3 − Re(λˆ y) (4M + 5 + λ0 |ˆ y |) e , (2M + 1)!

εi |T5 |

j=0 i=2M +2

≤ CCg K 2M +2 δ −(2M +1) ε2M +2 2M +1

∞

1 4M +2 − Re(λˆ y) (4M + 5 + λ0 |ˆ y |) e , (2M + 1)!

εi |T6 |

j=0 i=2M +2

≤ CCg K 2M +2 δ −(2M +1) ε2M +2

1 4M +3 − Re(λˆ y) y |) e . (4M + 5 + λ0 |ˆ (2M + 1)!

For the last two contributions, due to T3 , T4 , we observe that they vanish for i − 2 − j < 0, i.e., for (j, i) = (2M + 1, 2M + 2). Hence, we bound 2M +1

∞

j=0 i=2M +2

We compute

ε |T3 | ≤ i

2M

∞

j=0 i=2M +2

ε |T3 | + i

2M +1

∞

j=2M +1 i=2M +3

εi |T3 |.

7.3 Regularity of the boundary layer expansion 2M

∞

281

εi |T3 |

j=0 i=2M +2

≤ CCg K 2M +2 δ −(2M +2) ε2M +2 2M +1

∞

1 4M +2 − Re(λˆ y) (4M + 5 + λ0 |ˆ y |) e , (2M + 1)!

εi |T3 |

j=2M +1 i=2M +3

≤ CCg K 2M +2 δ −(2M +3) ε2M +3

1 4M +4 − Re(λˆ y) y |) e . (4M + 5 + λ0 |ˆ (2M + 1)!

The contribution from T4 is bounded analogously. We conclude that (again, after appropriately adjusting the constants C, K) there exist C, K > 0 depending only on A, c such that for (x, y) ∈ SX (δ) × By0 (0) ⊂ C × C there holds 2M +2 −(2M +2) 2M +2 Lε uBL δ ε × M (x, y) ≤ CCg K

1 4M +3 − Re(λˆ y) y |) e y |) . 1 + εδ −1 (2M + 5 + λ0 |ˆ (4M + 3 + λ0 |ˆ (2M + 1)! 2 Lemma 7.3.13. For every M , a ≥ 0, α ∈ (0, 1) there holds sup (M + a + r)M e−αr ≤ r>0

M α

M e(1−α)M eαa .

Proof: We note that sup(M + a + r)M e−αr = sup(M + x)M e−αx eαa ≤ sup(M + x)M e−αx eαa . r>0

x>a

x>0

The claim now follows from elementary considerations and α < 1.

2

Proof of Theorem 7.3.3: We start by considering the derivatives of the funci . Let λ0 , CA , K, X = (γA + 2γg )−1 be given by Proposition 7.3.8. From tions U Lemma 7.3.5, we have that λ0 < Re λ and |λ| ≤ CA on SX . In order to calculate i (x, yˆ), we choose κ ∈ (0, 1) such that ∂ypˆ ∂xq U κ λ0 CA ≤ . 1−κ 2 This choice of κ guarantees that eκ/(1−κ)CA e−λ(x)ˆy ≤ e−(λ0 /2)ˆy

∀(x, yˆ) ∈ Ix × R+ 0.

(7.3.22)

Let (x, y) ∈ Ix × R+ ˆ = y/ε. Cauchy’s integral theorem for derivatives 0 and set y and Proposition 7.3.8 give the existence of C, K > 0 depending only on the coeﬃcients A, c such that for 0 < δ < X

282

7. Regularity through Asymptotic Expansions

p!q! i (x + t, yˆ + s) U p q ds dt ∂yˆ ∂x Ui (x, yˆ) = 2 q+1 p+1 4π (−s) |t|=κδ |s|=R (−t) 2i 1 2i + 1 −i ≤ CCg p!q!(κδ)−q R−p ((1 − κ)δ) K i + yˆ + R eκ/(1−κ)CA yˆe−λ(x)ˆy . i! λ0 Choosing now R = p + 1 and adjusting the constants C, K, we get 1 p q 2i γ p+q δ −(i+q) K i (2i + 1 + λ0 yˆ + λ0 (p + 1)) e−(λ0 /2)ˆy ∂yˆ ∂x Ui (x, yˆ) ≤ CCg q!˜ i! for some γ˜ > 0 depending only on the coeﬃcients A, c. Splitting e−λ0 /2ˆy = e−λ0 /4ˆy e−λ0 /4ˆy we get from Lemma 7.3.13 that (2i + 1 + λ0 yˆ + λ0 (p + 1))2i e−(λ0 /4)ˆy ≤ C(8i)2i e(λ0 /4)p and thus 1 p q γ p+q δ −(i+q) K i i2i e−(λ0 /4)ˆy . ∂yˆ ∂x Ui (x, yˆ) ≤ CCg q!˜ i! Readjusting again the various constants and letting δ → X = (γA +2γg )−1 (with γA , γg as in the statement of Proposition 7.3.8) gives p q γ p+q (γA + 2γg )q (γA + 2γg )i K i ii e−λ0 /4ˆy . ∂yˆ ∂x Ui (x, yˆ) ≤ CCg q!˜ where the constants C, K, γ˜ , γA depend only on the coeﬃcients A, c. With these constants γA , K, we now impose the assumption ε(2M + 2)(γA + 2γg )K ≤

1 2

(7.3.23)

to conclude that under this assumption on ε, M there holds +1 2M i p q ε ∂yˆ ∂x Ui (x, yˆ) ≤ CCg q!˜ γ p+q (γA + 2γg )q e−(λ0 /4)ˆy ∀(x, yˆ) ∈ Ix × R+ 0. i=0

The ﬁrst bound in Theorem 7.3.3 now follows. For the second bound, we proceed similarly, but base the proof on Lemma 7.3.12 rather than Proposition 7.3.8. For δ < X and R < y0 /2 (y0 as given by Lemma 7.3.12) we want to apply Cauchy’s integral theorem for derivatives. To that end, we observe that Lemma 7.3.12 gives the existence of C, K > 0 depending only on A, c such that for all (t, s) ∈ Bκδ (0) × BRε (0) ⊂ C × C there holds (cf. also (7.3.13)) −(2M +2) 2M +2 2M +2 Lε uBL ε ((1 − κ)δ) × M (x + t, y + s) ≤ CCg K 1 4M +3 −λ(x)ˆ y κ/(1−κ)CA yˆ y + R)) e e × (4M + 3 + λ0 (ˆ (2M + 1)! −1 1 + ε ((1 − κ)δ) (4M + 3 + λ0 (ˆ y + R)) .

7.4 Regularity through asymptotic expansions

283

We reason now as above in order to simplify this expression. First, we note that the assumption x ∈ Ix and the choice of κ guarantee (7.3.22). Next, Lemma 7.3.13 allows us to conclude 1 1 4M +4 −(λ0 /4)ˆ y (4M + 3 + λ0 (ˆ (16M )4 M y + R)) e ≤ CK M (2M + 1)! (2M + 1)! ≤ CK M M 2M for some appropriate constants C, K. Thus, we arrive at

2M +2 2M +2 −(2M +2) Lε uBL ε δ M 2M e−(λ0 /4)ˆy 1 + εδ −1 M (x + t, y + s) ≤ CCg K

2M +2 −(λ0 /4)ˆy ≤ CCg ε(2M + 2)δ −1 K 1 + εδ −1 . e Cauchy’s integral theorem for derivatives therefore yields BL p!q! p q L u (x + t, y ˆ + s) ε BL M ∂y ∂x Lε uM (x, y) = ds dt 4π 2 |t|=κδ |s|=Rε (−t)q+1 (−s)p+1

2M +2 ≤ CCg p!q!(κδ)−q (εR)−p ε(2M + 2)δ −1 K e−(λ0 /4)ˆy 1 + εδ −1 . Next, observing that we may let δ → (γA + 2γg )−1 , we obtain for some suitable constants p q −p p+q ∂y ∂x Lε uBL γ˜ (γA + 2γg )q × M (x, y) ≤ CCg p!q!ε 2M +2 −(λ0 /4)ˆ y

(ε(2M + 2)(γA + 2γg )K)

e

{1 + ε(γA + 2γg )} .

As by assumption (7.3.23), ε(γA +2γg ) is bounded, the desired result now follows. 2

7.4 Regularity through asymptotic expansions 7.4.1 Notation and main result Let Ω be a curvilinear polygon satisfying the assumptions set out in Section 1.2. The boundary ∂Ω consists of J analytic arcs Γj , j = 1, . . . , J, whose endpoints are the vertices Aj−1 , Aj j = 1, . . . , J (we set A0 := AJ ). As described in Section 1.2, each analytic arc Γj is parametrized as Γj = {(xj (θ), yj (θ)) | θ ∈ (0, 1)} for some analytic functions xj , yj . Our assumptions imply furthermore the existence of Θ > 0 such that the functions xj , yj are analytic on (−2Θ, 1+2Θ). We write Γ$j := {(xj (θ), yj (θ) | θ ∈ (−Θ, 1+Θ)}. Without loss of generality, we may furthermore assume that the parametrization of Γj is done such that the normal vector (−yj (θ), x (θ)) = 0 points into Ω. Next, we deﬁne boundary-ﬁtted coordinates (ρj , θj ) through the mapping

284

7. Regularity through Asymptotic Expansions x3j (0) = x4j (1)

Γj4

Γj3

x2j (1) = x3j (1)

Ωj

Γj2

Γj1 = Γj Aj−1 = x1j (0) = x4j (0)

Aj = x1j (1) = x2j (0)

Fig. 7.4.1. Orientation of the curves Γjk comprising the boundary of Ωj .

ψj : [0, ρ0 ] × [−Θ, 1 + Θ] → Ω (ρ, θ) → (xj (θ), yj (θ))+

ρ 2 (−yj (θ), xj (θ)). x (θ) + y (θ)2 j

j

The maps ψj are real analytic and in fact invertible in a neighborhood of {0} × (0, 1) (with inverse ψj−1 being again real analytic). Without loss of generality, we may assume that ρ0 and Θ are chosen such that ψj is real analytic and invertible on [−ρ0 , ρ0 ] × [−Θ, 1 + Θ]. The inverse ψj−1 thus deﬁnes boundaryﬁtted coordinates (ρj , θj ) in a neighborhood of Γ$j via (ρj , θj ) = (ρj (x), θj (x)) = ψj−1 (x). Remark 7.4.1 ρj in the boundary-ﬁtted coordinates (ρj , θj ) has a geometric interpretation: for x ∈ Ω in a neighborhood of Γ$j , we have ρj (x) = dist(x, Γ$j ). Boundary layers and corner layers are phenomena that are restricted to a (small) neighborhood of the boundary ∂Ω. We therefore cover the half-tubular neighborhood U := {(x, y) ∈ Ω | dist(x, ∂Ω) < ρ0 } by sets Ωj with the following properties: 1. The domains Ωj ⊂ Ω are curvilinear rectangles, i.e., ∂Ωj consists of four analytic arcs Γjk = {xkj (t) | t ∈ (0, 1)}, k ∈ {1, . . . , 4}, where the functions xkj are analytic on [0, 1]. 2. The four angles of Ωj are strictly between 0 and π and there holds Aj−1 = x1j (0) = x4j (0), Aj = x1j (1) = x2j (0), x2j (1) = x3j (1), x4j (1) = x3j (0). In particular, therefore, we have Γj1 = Γj (cf. Fig. 7.4.1). 3. The arcs Γj4 , Γj2 meet Γj only at the points Aj−1 , Aj , respectively: There exists C > 0 such that ρj (x4j (t)) ≥ Ct, ρj (x2j (t)) ≥ Ct. 4. ∪Jj=1 Ωj covers the half-tubular neighborhood U of ∂Ω. 5. Ωj ⊂ ψj ((0, ρ0 ) × (−Θ, 1 + Θ)) for all j. 6. ρ0 is so small that Ωj ∩ Ωk = ∅ if and only k ∈ {j − 1, j, j + 1}. 7. If the angle ωj at the vertex Aj satisﬁes ωj ≥ π, then the intersection 4 Γj := ∂Ωj ∩ ∂Ωj+1 satisﬁes Γj = Γj2 = Γj+1 (i.e., at re-entrant corners Aj the sets Ωj , Ωj+1 do not overlap).

7.4 Regularity through asymptotic expansions

285

8. If the angle ωj at the vertex Aj satisﬁes ωj < π, then either the intersection 4 or Γj := ∂Ωj ∩ ∂Ωj+1 is an analytic arc and satisﬁes Γj = Γj2 = Γj+1 4 2 Γj ⊂ Γj−1 , Γj ⊂ Γj . Remark 7.4.2 These deﬁnitions are rather technical. They formalize the following idea (cf. Fig. 7.4.2): 1. At a re-entrant corner Aj (cf. Fig. 7.4.2), the two subdomains Ωj , Ωj+1 abutting on Aj share only a common analytic arc Γj ; the crucial features is that Γj divides the angle at Aj into two angles, each being smaller than π. 2. At the convex corners Aj−1 , the two subdomains Ωj−1 , Ωj meet in one of two ways. Either, the two subdomains meet in the same fashion as in the case of re-entrant corners (left panel of Fig. 7.4.2) or they overlap (right panel of Fig. 7.4.2). In the latter case there is the additional condition that two boundary curves of the subdomains Ωj−1 , Ωj lie on the boundary ∂Ω: For 1 2 ∂Ωj−1 we have Γj−1 = Γj−1 and Γj−1 ⊂ Γj and for ∂Ωj we have Γj1 = Γj 4 together with Γj ⊂ Γj−1 . It is noteworthy that we leave choices for the selection of the domains Ωj at convex corners. This freedom will later on correspond to diﬀerent choices in the decomposition of uε .

Γj−1

Ωj−1

Γj−1

Γj

Γj−1

Ωj−1

Γj

Ωj Aj−1

Γj

Ωj Aj

Ωj+1

Aj−1

Γj

Aj

Ωj+1

Γj+1

Γj+1

Fig. 7.4.2. Diﬀerent choices of subdomains Ωj at convex corners. CL Next, we choose cut-oﬀ functions χBL j , χj . We start with the cut-oﬀ functions CL χj , supported by neighborhoods of the vertices Aj . To do so, let R > 0 be so small that

B2R (Aj ) ⊂ Ωj ∪ Ωj+1 , B2R (Aj ) ∩ B2R (Ak ) = ∅

j = 1, . . . , J,

if j = k

and that the sets Sj := Ω ∩ B2R (Aj )

(7.4.2)

form curvilinear sectors (with apex Aj ) in the sense of Deﬁnition 4.2.1 whose two sides emanating from Aj lie on the arcs Γj , Γj+1 . Furthermore, we set

286

7. Regularity through Asymptotic Expansions

Sj− := Sj ∩ Ωj .

Sj+ := Sj ∩ Ωj+1 ,

(7.4.3)

Remark 7.4.3 In the case that Ωj , Ωj+1 do overlap, we have Sj = Sj+ = Sj− . In the case that Ωj , Ωj+1 do not overlap, the sector Sj is divided by the analytic arc Γj into two subsectors that are precisely Sj+ , Sj− . We choose cut-oﬀ functions χCL : R2 → R+ j 0 with supp χCL ⊂ BR (Aj ), j χCL j

≡1

j = 1, . . . , J,

on BR/2 (Aj ).

∩ Ω ⊂ Sj . Note that supp χCL j We now turn to the deﬁnition of the cut-oﬀ functions χBL j , which is most conveniently done in boundary-ﬁtted coordinates. To that end, we introduce a smooth cut-oﬀ functions χ = χ(ρ, θ) satisfying χ(ρ, θ) ≡ 1 for |ρ| ≤ R /2 and χ(ρ, θ) ≡ 0 for ρ ≥ R for some 0 < R < ρ0 to be determined shortly. We then set (the function EΩj denotes the characteristic function for the set Ωj ) χBL := (χ ◦ ψj−1 ) · EΩj . j

(7.4.4)

R > 0 is now chosen so small that χCL ≡1 j

on the set (supp χBL ∩ supp χBL j j+1 ).

(7.4.5)

Remark 7.4.4 The precise choice of the cut-oﬀ functions χBL is not very imj BL portant. The essential feature is that a) χj ≡ 1 in a neighborhood of Γj and b) χBL is supported by Ωj . Condition a) is enforced by deﬁning χBL in j j boundary-ﬁtted coordinates and condition b) is enforced by multiplying by the characteristic function EΩj . By means of the method of matched asymptotic expansions, the solutions u of (1.2.11a) with boundary conditions (1.2.11b) can be written as follows: For every M ∈ N0 , there are functions wM (the smooth part), uBL j,M , the boundary layer , the corner layer parts, and r , the remainder, such that the exact parts, uCL M j,M solution u admits the following decomposition: uε = wM +

J j=1

J BL CL uj,M ◦ ψj−1 + χBL χCL j j uj,M + rM .

(7.4.6)

j=1

In order to formulate the main result of this section, Theorem 7.4.5, which asserts regularity properties of the components of (7.4.6), we introduce, analogous to the deﬁnition of Φp,β,ε at the outset of Section 5.3.2, the weight function J Ψp,β,ε,α (x) := Πj=1 Ψˆp,βj ,ε,α (x − Aj ).

(7.4.7)

7.4 Regularity through asymptotic expansions

287

Theorem 7.4.5 (asymptotic expansion, regularity). Let Ω be a curvilinear polygon, f be analytic on Ω satisfying ∇p f L∞ (Ω) ≤ Cf γfp p!

∀p ∈ N0 ,

and let the piecewise analytic boundary data g satisfy (1.2.4). Then there exist C, γ, K, α > 0, R > 0, β ∈ (0, 1)J , β ∈ (0, 1)J independent of ε and γf , and there exist constants C , γ > 0 independent of ε such that the terms appearing in (7.4.6) satisfy the following: If 0 < ε(2M + 2)(1 + γf )K ≤ 1 then there holds for all p ∈ N0 and all q ∈ N0 p

∇p wM L∞ (Ω) ≤ C (γ(1 + γf )) p!, p+q

sup θj ∈[−Θ,1+Θ]

|∂ρpj ∂θqj uBL j,M (ρj , θj )| ≤ C (γ(1 + γf )) rM H 2,2 β

(uBL j,M

◦

ψj−1

(Ω) ,ε

q!ε−p e−αρj /ε ,

ρj ≥ 0,

2M +2 , ≤ C e−α/ε + (ε(2M + 2)γ(1 + γf ))

+ wM )|Γj = g|Γj , rM = 0 on ∂Ω.

+ − The corner layers uCL j,M are analytic on (Sj ∪ Sj ) ∩ BR (Aj ) and satisfy for all p ∈ N0 CL Ψ0,0,ε,α uCL j,M L2 (S + ∪S − ) + εΨ0,0,ε,α ∇uj,M L2 (S + ∪S − ) ≤ C ε, j

j

j

Ψp,β,ε,α ∇p+2 uCL j,M L2 ((Sj+ ∪Sj− )∩BR (Aj ))

p

(7.4.8a)

j

≤ C ε(γ ) max {p, ε

−1 p+2

}

. (7.4.8b)

In particular, if Sj+ = Sj− = Sj , then the corner layer function uCL j,M is anasatisfy the following lytic on Sj ∩ BR (Aj ). Furthermore, the corner layers uCL j pointwise estimates for p ≥ 0 and x ∈ (Sj+ ∪ Sj− ) ∩ BR (Aj ) 1−βj p CL −p p rj ∇ uj,M (x) − uCL (A ) ≤ C p!r (γ ) e−αrj /ε , j j,M j ε

(7.4.9)

where rj = dist(x, Aj ). In particular, if Sj+ = Sj− = Sj , then uCL j,M (Aj ) = −g(Aj ) + − CL whereas in the case Sj ∩ Sj = ∅ we have uj,M (Aj ) = 0. The proof of Theorem 7.4.5 is lengthy and therefore broken up into several steps handled in the subsequent subsections. Using the same arguments as in Corollaries 7.2.4, 7.3.4, we can extract from Theorem 7.4.5 the following result. Corollary 7.4.6 (asymptotic expansion, regularity). Under the assumptions of Theorem 7.4.5, there exist C, γ, R , α > 0 and β ∈ (0, 1)J , β ∈ (0, 1)J independent of ε such that for every ε the solution uε of (1.2.11) can be decomposed as J J −1 BL CL χBL (u ◦ ψ ) + χCL uε = wε + j j,ε j uj,ε + rε , j j=1

where the terms wε ,

uBL j,ε ,

uCL j,ε ,

j=1

rε satisfy the boundary conditions

288

7. Regularity through Asymptotic Expansions −1 (uBL j,ε ◦ ψj + wε )|Γj = g|Γj , on ∂Ω, rε = 0

and have the following regularity properties: For all p, q ∈ N0 ∇p wε L∞ (Ω) ≤ Cγ p p!, sup θj ∈[−Θ,1+Θ]

rε H 2,2

β ,ε

p+q |∂ρpj ∂θqj uBL q!ε−p e−αρj /ε , j,ε (ρj , θj )| ≤ Cγ

(Ω)

≤ Ce−γ

/ε

ρj ≥ 0,

,

CL Ψ0,0,ε,α uCL j,ε L2 (S + ∪S − ) + εΨ0,0,ε,α ∇uj,ε L2 (S + ∪S − ) ≤ Cε, j

j

j

j

p −1 p+2 Ψp,β,ε,α ∇p+2 uCL } , j,ε L2 ((Sj+ ∪Sj− )∩BR (Aj )) ≤ C εγ max {p, ε 1−βj p CL ≤ Cp!r−p γ p rj ∇ uj,M (x) − uCL e−αrj /ε , j,M (Aj ) j ε

for all x ∈ (Sj+ ∪ Sj− ) ∩ BR (Aj ); here rj = dist(x, Aj ). In particular, if Sj+ = + − Sj− = Sj , then uCL j,M (Aj ) = −g(Aj ) whereas in the case Sj ∩ Sj = ∅ we have uCL j,M (Aj ) = 0. Proof: As in the proof of Corollaries 7.2.4, 7.3.4, we merely choose M = λ/ε in Theorem 7.4.5 for some suitable λ > 0. 2

7.4.2 Proof of Theorem 7.4.5: smooth and boundary layer parts This section is devoted to the deﬁnition and the proof of the bounds on the smooth part, wM and the boundary layer components, uBL j,M . Theorem 7.2.2 gives the existence of C, K, γ > 0 depending only on the coeﬃcients A, c such that under the assumption 0 < ε(2M + 2)(1 + γf )γK ≤ 1

(7.4.10)

there holds for wM as deﬁned in (7.2.2) p

∇p wM L∞ (Ω) ≤ CCf (γ(1 + γf )) p!

∀p ∈ N0 , 2M +2

Lε wM − f L∞ (Ω) ≤ CCf (ε(2M + 2)(1 + γf )γK)

(7.4.11) .

(7.4.12)

(7.4.11) gives the desired bound on wM . Next, we proceed with deﬁning and analyzing the terms uBL j,M . This is done as in the classical asymptotic analysis of the problem under consideration; the technical tools for its execution have been provided in Section 7.3.1. We note that the outer expansion wM represents, up to a small defect, a particular solution to (1.2.11a). However, wM does not satisfy the correct boundary conditions on ∂Ω. In order to remedy that, one would like to introduce a boundary layer function uBL satisfying

7.4 Regularity through asymptotic expansions

uBL = g − wM

Lε uBL = 0 on Ω,

289

on ∂Ω,

for wM + uBL then satisﬁes the correct boundary conditions and, up to a small defect, also the diﬀerential equation. The boundary layer functions uBL j,M are now BL in neighborhoods of the arcs Γj . They are constructed approximations to u as follows. The function uj := uBL ◦ ψj satisﬁes on the rectangle R = (0, ρ0 ) × (−Θ, 1 + Θ) ε uj := −∇(ρ ,θ ) · A∇ ˆ (ρ ,θ ) uj + cˆuj = 0 on R, L (7.4.13a) j j j j uj |ρj =0 = gj := (g ◦ ψj − wM ◦ ψ)|ρj =0 ,

(7.4.13b)

where the subscript in the notation ∇(ρj ,θj ) emphasizes that diﬀerentiation takes ˆ place with respect to the boundary-ﬁtted coordinates (ρj , θj ). The functions A, cˆ are given by (cf. Lemma A.1.1) Aˆ = (det ψj )(ψj )−T (A ◦ ψj )(ψj )−1 ,

cˆ = (c ◦ ψj )det ψj .

By the analyticity of ψj , there holds Aˆ ∈ A(R, S2> ), cˆ ∈ A(R) with cˆ > 0 on R. Furthermore, from Lemma 4.3.4, there are C, γ > 0 independent of ε and M such that for gj of (7.4.13) p

Dp gj L∞ ((−Θ,1+Θ)) ≤ C (γ(1 + γf )) p!

∀p ∈ N0 .

(7.4.14)

The situation of (7.4.13) is therefore the one considered in Section 7.3.1. Deﬁning uBL j,M as in (7.3.10) we get from Theorem 7.3.3 that under assumption (7.4.10) (with appropriately modiﬁed constants γ, K, which are still independent of ε, M , γf ) there holds for some α > 0 independent of ε, M , γf and all (p, q) ∈ N20 and ρj ≥ 0: uBL j,M = (g ◦ ψj − wM ◦ ψj ) sup θj ∈(−Θ,1+Θ)

|∂ρpj

∂θqj

on ρj = 0,

uBL j,M (ρj , θj )|

(7.4.15) p+q

≤ C (γ(1 + γf ))

ε uBL (ρj , θj )| ≤ C (ε(2M + 2)γ(1 + γf )K) |L j,M

q!ε

−p −αρj /ε

e

2M +2 −αρj /ε

e

,

.

(7.4.16) (7.4.17)

Lemma A.1.1 implies that ε (v ◦ ψj ) = ((Lε v) ◦ ψj ) · (det ψ )−1 L j

for all smooth functions v. (7.4.18)

Thus, we obtain on the “physical” domain ψj (R) with ρj = dist(x, Γ$j ) −1 |Lε (uBL j,M ◦ ψj )| ≤ C (ε(2M + 2)γ(1 + γf )K)

2M +2 −αρj /ε

on ψj (R) ⊃ Ωj . (7.4.19) These estimates imply the desired bounds on the functions uBL . In fact, the j,M transition from (7.4.17) to (7.4.19) requires only adjusting the constant C—the constants γ and K are the same in both instances. The properties of the functions uBL j,M obtained so far lead to the following lemma. e

290

7. Regularity through Asymptotic Expansions

BL Lemma 7.4.7. Let uBL there j,M satisfy (7.4.16), (7.4.19). Then on supp χj holds for some C, γ, α > 0 independent of 2M +2 −1 BL −α /ε BL Lε χBL . u ◦ ψ ≤ C e + (ε(2M + 2)γ(1 + γ )) ∞ f L (supp χj ) j j,M j

= χ ◦ ψj−1 · EΩj , it suﬃces Proof: In view of (7.4.18) and the deﬁnition of χBL j to show 2M +2 −α /ε ε χuBL . + (ε(2M + 2)γ(1 + γf )) L j,M L∞ (supp χ) ≤ C e We observe that for smooth functions χ, u there holds ε (χu) = −ε2 ∇ · A∇(χu) ˆ L + cˆχu ε u − ε2 (∇χ) · (A∇u) ˆ ˆ = χL − ε2 (∇u) · (A∇χ). For u = uBL j,M , the bound (7.4.22) gives ε uBL L∞ (supp χ) ≤ C (ε(2M + 2)γ(1 + γf )) χL j,M

2M +2

.

Next, in order to treat the remaining terms, we observe that in a neighborhood of ρ = 0, the cut-oﬀ function χ ≡ 1, i.e., ∇χ ≡ 0. Furthermore, for ρ > R /2, the function uBL j,M and all its derivatives are exponentially small by (7.4.16), implying −α /ε ˆ BL ) − ε2 (∇uBL ) · A∇χ ˆ − ε2 ∇χ · (A∇u L∞ (supp χ) ≤ Cεe j,M j,M

for some α > 0 depending on α of (7.4.16) and χBL j .

2

7.4.3 Proof of Theorem 7.4.5: corner layer and remainder So far, we have deﬁned wM and the functions uBL j,M . Consider now the function uIO := wM +

J

BL uj,M ◦ ψj−1 . χBL j

(7.4.20)

j=1

are only piecewise The deﬁnition (7.4.4) implies that the cut-oﬀ functions χBL j smooth with possible jumps across ∂Ωj . Likewise, the function uIO may jump across ∂Ωj . We note that these regions are contained in the set V := ∪Jj=1 supp χCL j . First, we show that on (∂Ω) \ V , 2M +2 , ≤ C e−α /ε + (ε(2M + 2)γ(1 + γf ))

uIO = g Lε uIO − f L∞ (Ω\V )

(7.4.21) (7.4.22)

7.4 Regularity through asymptotic expansions

291

where the constants C, α , γ > 0 are independent of ε, M , and γf . The esti≡ 1 in a neighborhood of Γj , the mate (7.4.21) follows from the fact that χBL j construction of the functions uBL , and the assumptions on the supports of the j,M BL cut-oﬀ functions χj . (7.4.22) follows directly from Lemma 7.4.7. Bounds (7.4.21), (7.4.22) imply that the function uIO satisﬁes the diﬀerential equation (up to a small defect) and the boundary conditions except for a neighborhood of the vertices. There, the function uIO is not necessarily continuous and does not satisfy the boundary conditions. This last inconsistency is now removed with the aid of the corner layers uCL j,M . An additional side eﬀect of the is that the function introduction of the corner layers uCL j,M uIOC := wM +

J j=1

−1 BL χBL j (uj,M ◦ ψj ) +

J

CL χCL j uj,M

(7.4.23)

j=1

is an element of C 1 (Ω). When we deﬁned the subdomains Ωj in Section 7.4.1, we pointed out that two subdomain Ωj , Ωj+1 meet in one of two ways near a vertex Aj . In the ﬁrst situation, Ωj , Ωj+1 overlap, in which case we call the vertex Aj a “convex” corner. In the second situation, Ωj , Ωj+1 do not overlap, and the vertex Aj is called a “general” corner. We emphasize that a convex corner may also be treated as a general corner. Each of these two scenarios is dealt with in turn in the following two subsections. Corner layers at convex corners. We start with the setting at a convex corner because the construction of the corner layer is more intuitive and straight forward. We point out, however, that a convex corner may also be treated as a general corner, elaborated in the ensuing subsection. Let Aj be the convex corner under consideration and recall the deﬁnition of the sector Sj in (7.4.2). Its two sides emanating from Aj are denoted Γ 1 , Γ 2 with Γ 1 ⊂ Γj , Γ 2 ⊂ Γj+1 (see Fig. 7.4.3). By our assumption that Aj is a convex corner (in the sense that the domains Ωj , Ωj+1 overlap) we see that on Sj both boundary-ﬁtted coordinate systems (ρj , θj ), (ρj+1 , θj+1 ) are available; uIO of (7.4.20) is smooth on Sj and can be written as −1 −1 BL BL BL uIO = wM + χBL j (uj,M ◦ ψj ) + χj+1 (uj+1,M ◦ ψj+1 )

on Sj .

Lemma 7.4.7 therefore implies that 2M +2 Lε uIO − f L∞ (Sj ) ≤ C e−α /ε + (ε(2M + 2)γ(1 + γf )) . BL Furthermore, by the construction of the functions uBL j,M , uj+1,M , it is easy to see that BL −1 uIO = χBL on Γ 1 ⊂ Γj , (7.4.24a) j+1 uj+1 ◦ ψj+1 −1 BL BL 2 uj ◦ ψj on Γ ⊂ Γj+1 . (7.4.24b) uIO = χj

292

7. Regularity through Asymptotic Expansions Γj Sj supp χBL j

supp χBL j+1

Γ1

Γ2

Γj+1

Aj Fig. 7.4.3. Situation at a convex corner.

We are now in position to apply Theorem 6.4.13 to the sector Sj with homogeneous transmission conditions h1 = h2 = 0 (in fact, we have not speciﬁed the arc Γ that subdivides Sj into two subsectors!), homogeneous right-hand side f but inhomogeneous boundary data g. On ∂Sj , the boundary data g are given by −1 BL g1 = g|Γ1 := −χBL j+1 uj+1,M ◦ ψj+1 ,

−1 BL g2 = g|Γ2 := −χBL j uj,M ◦ ψj .

It remains to check that the boundary data g satisfy the assumptions of Theorem 6.4.13, i.e., (6.4.20). These follow by combining the following facts: BL 1. χBL j , χj+1 are identically one in a neighborhood of Aj ; 2 2. the assumptions on the arcs Γj4 , Γj+1 guarantee that (6.4.19) is satisﬁed; BL BL 3. uj,M , uj+1,M satisfy the desired estimates in the boundary-ﬁtted coordinate systems (ρj , θj ), (ρj+1 , θj+1 ).

These three facts together with Lemma 4.3.3 (dealing with changes of variables) give (6.4.20). Thus, Theorem 6.4.13 yields the existence of a function uCL j,M , analytic in a neighborhood of Aj (more precisely: on Sj ∩ BR (Aj ) for suitable R > 0), with the following properties: Lε uCL j,M ≡ 0 uCL j,M uCL j,M

= =

on Sj ,

−1 BL −χBL j uj,M ◦ ψj −1 BL −χBL j+1 uj+1,M ◦ ψj+1

(7.4.25) on Γj+1 ∩ ∂Sj ,

(7.4.26)

on Γj ∩ ∂Sj ,

(7.4.27)

together with a priori bounds for all p ∈ N0 CL Ψ0,0,ε,α uCL j,M L2 (Sj ) + εΨ0,0,ε,α ∇uj,M L2 (Sj ) ≤ Cε,

Ψp,β,ε,α ∇p+2 uCL j,M L2 (Sj ∩BR (Aj ))

≤ Cε max {p, ε

(7.4.28) −1 p+2

}

. (7.4.29)

for some constants C, γ, R , α > 0 independent of ε. We note that (7.4.28), (7.4.29) are the desired estimates. Finally, by the assumptions on the supports BL of the cut-oﬀ functions χCL we have j , χj

7.4 Regularity through asymptotic expansions

293

Sj Γ˜j+1 supp χBL j

Γj

supp χBL j

Γ ⊂ Γj

Γ˜j Γj

Aj

Γj

Γ1

supp χBL j+1

Aj supp χBL j+1

Γ2

Γj+1

Γj+1

Fig. 7.4.4. Situation at a general corner. BL χCL = χBL j χj j

on Γj+1 ,

BL BL χCL j χj+1 = χj+1

on Γj .

CL Thus, the function χBL j uj,M also satisﬁes −1 CL BL BL χCL j uj,M = −χj uj,M ◦ ψj CL χCL j uj,M

=

BL −χBL j+1 uj+1,M

◦

on Γj+1 ,

−1 ψj+1

on Γj .

We conclude that on Sj the function uIOC of (7.4.23) satisﬁes on ∂Ω ∩ ∂Sj , 2M +2 . ≤ C e−α /ε + (ε(2M + 2)γ(1 + γf ))

uIOC = g Lε uIOC − f L2 (Sj )

(7.4.30)

Remark 7.4.8 Why might it be advantageous to choose to treat even a convex corner as a general one? The above arguments show that, in order for Theo−1 −1 BL rem 6.4.13 to be applicable, we need the functions uBL j,M ◦ ψj , uj+1,M ◦ ψj+1 to decay exponentially on the the arcs Γj+1 , Γj , respectively. As long as the angle between Γj , Γj+1 is strictly less than π, this is guaranteed (in a neighborhood of Aj at least). However, the decay rate deteriorates as the angle tends to π. Treating a convex corner as a general one avoids this problem. Corner layers at general corners. We now turn to the construction of the corner layer in a general corner. Let Aj be a general corner in the sense that the subdomains Ωj , Ωj+1 abutting on Aj do not overlap. Recall the deﬁnition Γj = ∂Ωj ∩ ∂Ωj+1 and the deﬁnition of the sector Sj in (7.4.2). Denote by Γ 1 , Γ 2 the two sides of Sj emanating from Aj with Γ 1 ⊂ Γj , Γ 2 ⊂ Γj+1 . Next, we set Γ := Sj ∩ Γj (see Fig. 7.4.4). We recognize that by our assumptions on the supports of the cut-oﬀ functions χBL j , χBL , the function u of (7.4.20) restricted to S has the form IO j j+1 uIO = wM

! −1 BL χBL j uj,M ◦ ψj + −1 BL χBL j+1 uj+1,M ◦ ψj+1

on Sj− on Sj+ .

(7.4.31)

294

7. Regularity through Asymptotic Expansions

BL By construction of the functions uBL j,M , uj+1,M , it is then clear that uIO satisﬁes the correct boundary conditions on ∂Ω ∩ ∂Sj , i.e., uIO = g on ∂Ω ∩ ∂Sj . From Lemma 7.4.7, we also see that 2M +2 . Lε uIO L∞ (S + ∪S − ) ≤ C e−α /ε + (ε(2M + 2)γ(1 + γf )) j

j

The representation (7.4.31) shows that uIO is discontinuous across Γ . The corner layer function uCL j,M is now chosen such that this discontinuity (and the jump of the co-normal derivative) is corrected. The jumps are −1 −1 BL BL BL h1 := χBL j+1 uj+1,M ◦ ψj+1 − χj uj,M ◦ ψj −1 BL ) A∇(χBL j+1 uj+1,M ◦ ψj+1 ) −1 BL + ε(n− )T A∇(χBL j uj,M ◦ ψj )

on Γ ,

(7.4.32a)

+ T

h2 := ε(n

on Γ ,

(7.4.32b)

where n+ , n− represent the outer normal vector of Sj+ , Sj− on the curve Γ . We note that the continuity of g and wM at the vertex Aj implies that h1 (Aj ) = BL 0. Furthermore, because for some κ > 0 the cut-oﬀ functions χBL j , χj+1 are identically one on Bκ (Aj ) ∩ Γ , the data h1 , h2 are analytic on Bκ (Aj ) ∩ Γ . BL In fact, one can check using the properties of the functions uBL j,M , uj+1,M and Lemma 4.3.3 that (6.4.20) hold. Thus Theorem 6.4.13 is applicable (with g1 = g2 = 0 and h1 , h2 given by (7.4.32)) and yields the existence of uCL j,M such that Lε uCL j,M ≡ 0 uCL j,M CL [uj,M ] ε[∂nA uCL j,M ]

=0

on Sj+ ∪ Sj− ,

(7.4.33a)

on ∂Sj ∩ ∂Ω,

(7.4.33b)

= h1 = h2

on Γ ,

(7.4.33c)

on Γ ,

(7.4.33d)

together with the following a priori estimates for all p ∈ N0 : CL Ψ0,0,ε,α uCL j,M L2 (S + ∪S − ) + εΨ0,0,ε,α ∇uj,M L2 (S + ∪S − ) ≤ Cε, j

j

Ψp,β,ε,α ∇p+2 uCL j,M L2 ((Sj+ ∪Sj− )∩BR (Aj ))

j

j

≤ Cε max {p, ε

−1 p+2

}

(7.4.34) (7.4.35)

for some constants C, γ, R , α > 0 independent of ε. Thus, the function uCL j,M satisﬁes the desired bounds. We observe that on Sj , the function uIOC of (7.4.23) takes the form −1 −1 CL BL BL BL BL CL CL uIOC = uIO +χCL j uj,M = wM +χj uj,M ◦ψj +χj+1,M uj+1,M ◦ψj+1 +χj uj,M . BL As uCL j,M ≡ 0 on ∂Sj ∩ ∂Ω, we have by construction of the functions wM , uj,M , uBL j+1,M that uIOC = g on ∂Sj ∩ ∂Ω. Thus, the correct Dirichlet boundary conBL ditions are satisﬁed. By the assumptions on the cut-oﬀ functions χBL j , χj+1 we have BL BL BL χCL = χBL on Γ , χCL on Γ . j χj j j χj+1 = χj+1

7.4 Regularity through asymptotic expansions

295

We therefore obtain that uIOC is smooth on Sj+ ∪ Sj− and satisﬁes [uIOC ] = 0,

[∂nA uIOC ] = 0

on Γ .

By the smoothness of the coeﬃcient matrix A, we conclude that uIOC ∈ C 1 (Sj ). Because additionally uIOC ∈ H 1 (Sj+ ∪ Sj− ), we get 2 uIOC ∈ H 1 (Sj ) ∩ C 1 (Sj ) ∩ C ∞ (Sj+ ∪ Sj− ) ⊂ H 1 (Sj ) ∩ Hloc (Sj ) ∩ C ∞ (Sj+ ∪ Sj− ).

This allows us to verify that uIOC is a weak solution to Lε uIOC = f˜ ∈ L∞ (Sj ) ! Lε uIOC ˜ f := Lε uIOC

on Sj , on Sj+ on Sj− .

Using (7.4.35) and the properties of χCL j , we infer furthermore Lε uIOC − f L∞ (Sj ) = f˜ − f L∞ (Sj ) 2M +2 . ≤ C e−α /ε + (ε(2M + 2)γ(1 + γf ))

(7.4.36)

Remainder rM and pointwise estimates. As uIOC has now been deﬁned, the remainder rM may be deﬁned through (7.4.6). In order to obtain an estimate for rM , we start by observing that combining (7.4.22) with (7.4.36) implies 2M +2 . (7.4.37) Lε uIOC − f L∞ (Ω) ≤ C e−α /ε + (ε(2M + 2)γ(1 + γf )) We observe that the function uIOC solves the following (weakly posed) problem: Lε rM = Lε (uε − uIOC ) = frM = f − Lε uIOC on ∂Ω, rM = 0

on Ω,

where the right hand-side frM ∈ L∞ (Ω) satisﬁes (7.4.37). Thus, appealing to Theorem 5.3.8 gives the existence of β ∈ (0, 1)J and C > 0 such that 2M +2 −α /ε rM H 2,2 , ≤ C e + (ε(2M + 2)γ(1 + γ )) f (Ω) β ,ε

allowing us to conclude the proof of Theorem 7.4.5 with the exception of the pointwise estimates for the corner layer functions uCL j,M . These are obtained from 2 the L -based estimates using Corollary 6.2.8. The proof of Theorem 7.4.5 is now complete.

Appendix

A.1 Some technical lemmata A.1.1 Transformations of elliptic equations Lemma A.1.1. Let G ⊂ Rn be open, A = (aij (·))ni,j=1 ∈ L∞ (G, Sn ), and f ∈ 1 L2 (G). Let u ∈ Hloc (G) be a weak solution of −∇ · (A∇u) = f i.e.,

on G,

(A∇u) · ∇v dx =

G

∀v ∈ C0∞ (G).

fv G

(A.1.1)

(A.1.2)

ˆ → G be a bi-Lipschitzian mapping from the open set G ˆ to G. Then Let F : G 1 ˆ is a weak solution of the function u ˆ := u ◦ F ∈ Hloc (G) ˆ −∇ · (det F )(F )−T Aˆ (F )−1 ∇ˆ u = fˆ det F on G, where Aˆ = A ◦ F , fˆ = f ◦ F . Proof: We have to show that ∇ˆ u · (det F )(F )−T Aˆ (F )−1 ∇ϕ dx = fˆ det F ϕ dx ˆ G

ˆ G

ˆ ∀ϕ ∈ C0∞ (G).

Equivalently, by transforming to the domain G (using the fact that F is biLipschitzian and [136, Theorem 2.2.2]), we have to show that $ (A∇u) · ∇ϕ dx = f ϕ dx ∀ϕ ∈ C(G), G

G

$ ˆ $ where C(G) = {ϕ(x) = ϕˆ ◦ F −1 (x) for some ϕˆ ∈ C0∞ (G)}. Clearly, C(G) ⊂ 1 1 1 Ccomp (G), where Ccomp (G) denotes the set of all C -functions on G that have 1 compact support. As C0∞ (G) is dense (in the H 1 -topology) in Ccomp (G), assumption (A.1.2) readily implies that 1 ∇u · A∇ϕ dx = f ϕ dx ∀ϕ ∈ Ccomp (G). G

This completes the argument.

J.M. Melenk: LNM 1796, pp. 297–310, 2002. c Springer-Verlag Berlin Heidelberg 2002

G

2

298

Appendix

Lemma A.1.2. Let R > 0, ω ∈ (0, 2π), A ∈ C 2 (SR (ω), S2> ) and 0 < λmin ≤ A on SR (ω). Then there exists C > 0 such that for every δ ∈ (0, 1) we can ﬁnd a function χδ ∈ C 2 (SR (ω)) with the following properties: 1. 2. 3. 4.

χδ ≡ 1 on Bδ/2 (0) ∩ SR (ω); χδ ≡ 0 on ∩SR (ω) \ Bδ (0); |∇j χδ | ≤ Cδ −j , j ∈ {0, 1, 2}; ∂nA χδ = 0 on ∂SR (ω) \ ∂BR (0).

Proof: In polar coordinates (r, ϕ), the co-normal derivatives operator ∂nA takes the following form on each of the two lateral parts Γ1 ⊂ {(r, 0) | r > 0}, Γ2 ⊂ {(r cos ω, r sin ω) | r > 0} of the sector SR (ω): 1 ∂nA v = ai (r) ∂ϕ v + bi ∂r v, r where the functions ai , bi , i ∈ {1, 2}, are C 2 . Additionally, we may reason as in the proof of Lemma 5.5.21 to see |ai (r)| ≥ a0 > 0, i ∈ {1, 2} for all r ∈ (0, R). Choose a ﬁxed function χ ∈ C ∞ (R) with the following properties χ ≡ 1 on B1/2 (0),

χ ≡ 0 on R \ B1 (0).

We then deﬁne the cut-oﬀ function χδ in polar coordinates by χδ (r, ϕ) = χ(r/δ) −

r b2 (r)χ (r/δ) r b1 (r)χ (r/δ) (ϕ/ω)2 (ϕ − ω) − ϕ(1 − ϕ/ω)2 . δ a2 (r) δ a1 (r) 2

A calculation shows that χδ has all the desired properties.

A.1.2 Leibniz formulas Lemma A.1.3. Let U , V be C ∞ tensors. Then there holds p p p |∇ (U · V )(x)| ≤ |∇q U (x)||∇p−q V (x)|. q q=0 Proof: The proof is an extended version of the proof of [98, Lemma 5.7.4]. By arranging the set of common indices and the set of the remaining indices into single sequences, we may assume that V = {Vij }, U V = {ωij }, ωij = Uik Vjk . U = {Uij }, k

The claim for p = 0 is obvious. For p = 1, we calculate 0 Uij Vjk )| = | ∂l Uij Vjk + Uij ∂l Vjk |2 |∇( j

i,k,l

j

i,k,l

j

j

0 0 2 ≤ | ∂l Uij Vjk | + | Uij ∂l Vjk |2 . i,k,l

j

A.1 Some technical lemmata

299

Now, | Uij ∂l Vjk |2 ≤ |Uij |2 |∂l Vjk |2 l

i,k

j

i,k,l

=

j

|Uij |

2

i,j

j

|∂l Vjk |2 = |U |2 |∇V |2 .

l,k,j

2 2 2 Completely analogously, we obtain i,k,l | j ∂l Uij Vjk | ≤ |∇U | |V | . This proves the case p = 1. In order to prove the claim for p > 1, we ﬁrst note that |∇p+1 U | = |∇p (∇U )|. We are now in position to prove the claim by induction on p. Suppose, the claim of the lemma holds for p. We then estimate |∇p+1 U V | = |∇p ∇(U V ) | ≤ |∇p (∇U )V | + |∇p U ∇V | p p p p q+1 p−q ≤ |∇ U | |∇ V | + |∇q U | |∇p+1−q V | q q q=0 q=0 = |∇

p+1

U | |V | +

p−1 p q=0

+

p p q=1

q

q

|∇q+1 U | |∇p−q V | + |U | |∇p+1 V |

|∇q U | |∇p+1−q V |

= |∇p+1 U | |V | + |U | |∇p+1 V | + +

=

p+1 q=0

p p q=1

q

p p |∇q U | |∇p+1−q V | q − 1 q=1

|∇q U | |∇p+1−q V |

p+1 |∇q U | |∇p+1−q V |. q 2

Lemma A.1.4. Let U , V be C ∞ tensor. Then for all p ∈ N0 there holds p 2 p |α|! α α 2 q p−q |∇ U (x)| |∇ V (x)| . |D (U V )(x) − (U D V )(x)| ≤ α! q q=1 |α|=p

Proof: Without loss of generality, the point x may be taken as x = 0. We will also drop the explicit dependence on x for the remainder of the proof. Furthermore, for simplicity of notation, we will also assume that U and V are scalar functions. We start by noting that by Lemma A.1.3 there holds

300

Appendix

p p

|α|! |Dα (U V )|2 = |∇p (U V )|2 ≤ α!

|α|=p

q=1

q

2 |∇ U | |∇ q

p−q

V|

(A.1.3)

provided that ∇p V = 0 at the ﬁxed point x = 0. Next, introduce the auxiliary function Dα V (0) zα. H(z) := α! |α|=p

! 0 if |β| < p D H(0) = Dβ V (0) if |β| = p.

It satisﬁes

β

From (A.1.3) we get |∇p

U (V − H) |2 ≤

p p q=1

=

q

p p q=1

q

2 |∇q U | |∇p−q (V − H)| 2 |∇ U | |∇ q

p−q

V|

.

(A.1.4)

Next, for |α| = p, we calculate α α α β α−β D U (V − H) = D UD Dβ U Dα−β V (V − H) = β β 0≤β≤α

0=β≤α

= Dα (U V ) − U Dα V.

(A.1.5)

Combining (A.1.4), (A.1.5), we arrive at |α|! |α|! |Dα (U V ) − U Dα V |2 = |Dα U (V − H) |2 α! α! |α|=p |α|=p p 2 p p 2 q p−q |∇ U | |∇ V | , = |∇ U (V − H) | ≤ q q=1 which proves the lemma.

2

Lemma A.1.5. Let U , V be C ∞ tensor ﬁelds deﬁned on G ⊂ Rn . Denote by ∇x diﬀerentiation with respect to the variables x1 , . . . , xn−1 . Then there holds for all (x, xn ) ∈ G, p, q ∈ N0 : p q p q |∂xq n ∇px ∇p (U · V )| ≤ |∂xsn ∇rx U ||∂xq−s ∇p−r x V |. n r s r=0 s=0 Proof: The proof is completely analogous to that of Lemma A.1.3 by induction on q. The case q = 0 is handled by Lemma A.1.3. 2

A.1 Some technical lemmata

301

A.1.3 Hardy inequalities We have the following well-known Hardy inequality in one dimension: Lemma A.1.6. Let β < 1/2 and x0 ∈ (0, 1). Then there exists C > 0 depending only on β and x0 such that . 1 / 1 1 x−2β u2 (x) dx ≤ C x2(1−β) |u (x)|2 dx + u2 (x) dx 0

0

x0

for all functions u such that the right-hand side is ﬁnite. Proof: The proof follows from the Hardy inequality [68, Thm. 330].

2

This one-dimensional result can be extended to sectors: Lemma A.1.7. Let β ∈ [0, 1), T = {(x, y) | 0 < x < 1, 0 < y < 1 − x}, and T ⊂ T . Then there exists C > 0 depending only on β and T such that r(x) := dist(x, (0, 0)), rβ−1 uL2 (T ) ≤ C rβ ∇uL2 (T ) + uL2 (T ) , for all functions u such that the right-hand side is ﬁnite. Proof: The result follows easily from Lemma A.1.6 if all integrals are expressed in polar coordinates. 2 Lemma A.1.8. Let SR (ω) = {(r cos ϕ, r sin ϕ) | r ∈ (0, R), ϕ ∈ (0, ω)} , ω > 0 be a sector. Let ω ∈ (0, ω) and let Γ := {(r cos ω , r sin ω ) | r ∈ (0, R)} divide SR (ω) into two sectors S + , S − . Then there exists C > 0 such that Hv ds ≤ CH 1,1 + vH 1 (S (ω)) ∀v ∈ H01 (SR (ω)), H ∈ H 1,1 (S + ), 1,1 R H (S ) 1,1

Γ

where the norm · H 1,1 (S + ) is given by HL2 (S + ) + |x|∇HL2 (S + ) . 1,1

Proof: An analogous result for v ∈ H 1 (SR (ω)) is given in Lemma 6.4.1. It is the Dirichlet boundary conditions imposed on v that allow us to take β = 1. We may restrict our attention to a neighborhood of the origin. Then, using smooth changes of variables, it is easy to see that it suﬃces to show the following analogous bound on a square: 1 1 H(x, 0)v(x, 0) dx ≤ CHH 1,1 (S) vH 1 (S) ∀v ∈ H(0) (S), (A.1.6) 0

1,1

1 where S = (0, 1)2 , H(0) (S) = {v ∈ H 1 (S) | v(x, ·) = 0, x = 0 or x = 1}, and HH 1,1 (S) = HL2 (S) + |x|∇HL2 (S) . We deﬁne on the interval (0, 1) the 1,1

1 weight function d(x) := x(1 − x). The assumption v ∈ H(0) (S) implies

302

Appendix

0

1

1 2 v (x, 0) dx ≤ Cv(·, 0)2H 1/2 ((0,1)) ≤ Cv2H 1 (S) . d(x) 00

(A.1.7)

˜ : (x, y) → d(x)H(x, y) is also in Next, a calculation shows that the function H 1 H(0) (S) and satisﬁes

˜ 2 (x, 0) H ˜ 0)2 1/2 ˜ 21 1,1 dx ≤ CH(·, ≤ CH H (S) ≤ CHH1,1 (S) . H00 ((0,1)) d(x) 0 # # ˜ Hence, since d(x)H(x, 0) = 1/ d(x)H(x, 0), we get 1 1 1/2 −1/2 H(·, 0)v(·, 0) dx = d H(·, 0)d v(·, 0) dx 1

0

(A.1.8)

0

≤ d1/2 H(·, 0)L2 ((0,1)) d−1/2 v(·, 0)L2 ((0,1)) ≤ CHH 1,1 (S) vH 1 (S) . 1,1

2

A.2 Kondrat’ev’s theory for a special transmission problem A.2.1 Problem formulation and notation The present section is devoted to the regularity analysis of a transmission problem in inﬁnite sectors. Let −2π < ω1 < 0 < ω2 < 2π be given and deﬁne the two inﬁnite sectors S1 := {(r cos ϕ, r sin ϕ) | 0 < r < ∞, ω1 < ϕ < 0}, S2 := {(r cos ϕ, r sin ϕ) | 0 < r < ∞, 0 < ϕ < ω2 }. We denote by Γ1 , Γ2 , Γ the three rays Γi = {(r cos ωi , r sin ωi ) | 0 < r < ∞}, Γ = {(r, 0) | 0 < r < ∞}. For ﬁxed p1 , p2 > 0 we consider the following transmission problem: −pi ∆ui = fi ui = 0 u1 = u2 p1 ∂n u1 − p2 ∂n u2 = G|Γ

on Si ,

i ∈ {1, 2},

on Γi , i ∈ {1, 2}, on Γ , on Γ ,

(A.2.1a) (A.2.1b) (A.2.1c) (A.2.1d)

0,0 1,1 where the data fi and G are assumed to satisfy fi ∈ Hβ,1 (Si ), G ∈ Hβ,1 (S1 ) for some β ∈ [0, 1). We point out that |ω1 | + |ω2 | may be larger than 2π. In order to clarify the notion of solutions, we introduce the space V01 of pairs of functions deﬁned on S1 × S2 as follows:

V01 := {(u1 , u2 ) | ui ∈ H 1 (Si ∩ BR (0)) ∀R > 0, ui = 0 on Γi and u1 = u2 on Γ }.

A.2 Kondrat’ev’s theory for a special transmission problem

303

Solutions of (A.2.1) are understood in a weak sense, i.e., a pair (u1 , u2 ) ∈ V01 is a solution of (A.2.1) if p1 ∇u1 ∇v1 dx + p2 ∇u2 ∇v2 dx = f1 v1 dx + f2 v2 dx + Gv1 S1

S2

for all (v1 , v2 ) ∈

1 V0,comp ,

S1

S2

Γ

where

1 = {(v1 , v2 ) ∈ V01 | vi = 0 on Si \ BR (0) for some R > 0}. V0,comp

We can now formulate our main result: Proposition A.2.1. There exist β ∈ [0, 1) and C > 0 depending only on the parameters ωi and pi such that the following regularity assertion holds: Let fi ∈ 0,0 1,1 Hβ,1 (Si ) and G ∈ Hβ,1 (S1 ) with fi = 0 on Si \ B1 (0) and G = 0 on S1 \ B1 (0). Assume that a pair (u1 , u2 ) ∈ V01 is a solution of (A.2.1) and satisﬁes ui = 0 on Si \ B1 (0). Then , 2 2 ui H 2,2 (Si ) ≤ C fi H 0,0 (Si ) + GH 1,1 (S1 ) . β,1

i=1

β,1

β,1

i=1

Similar results have been proved and announced in [64, 101, 102]. In the ensuing subsection, we present an outline of a proof that follows closely the argument given in [15], where such a result was proved for a single sector with Dirichlet, Neumann, or mixed boundary conditions. The actual proof is lengthy and therefore relegated to the ensuing subsection. Essentially, the proof consists in two 2 2 steps: In a ﬁrst step, a solution (w1 , w2 ) ∈ Wβ,0 (Wβ,0 will be deﬁned shortly) of (A.2.1) is constructed by the classical techniques of Kondrat’ev. In the second step, it is ascertained that (w1 , w2 ) in fact coincides with the given solution (u1 , u2 ). A.2.2 Proof of Proposition A.2.1 We will follow closely [15] and use their notation as much as possible. The Kondrat’ev spaces (cf., e.g., [39, 79]) are deﬁned for k ∈ N0 and β ≥ 0 by Wβk (Si ) := {u ∈ L2loc (Si ) | rp−k+β ∇p uL2 (Si ) < ∞}. 0≤p≤k

For β ∈ [0, 1) and k = 2 we deﬁne furthermore 2 Wβ,0 := {(w1 , w2 ) ∈ Wβ2 (S1 ) × Wβ2 (S2 ) | wi = 0 on Γi and w1 = w2 on Γ }.

Next, we set D1 = {(τ, θ) | τ ∈ R, D2 = {(τ, θ) | τ ∈ R, D = {(τ, θ) | τ ∈ R,

θ ∈ (ω1 , 0)}, θ ∈ (0, ω2 )}, θ ∈ (ω1 , ω2 )},

304

Appendix

and for h ≥ 0 and k ∈ N0 , we set Hhk (D)

= {u |

e2hτ D

k

|∇p u|2 dτ dθ =: u2Hk (D) < ∞} h

p=0

with analogous deﬁnitions for Hhk (Di ), i = 1, 2. We use the notation Lh (D) := Hh0 (D) for the special case k = 0. We write the transmission problem (A.2.1) in polar coordinates (r, ϕ) and then introduce the new variable τ = ln(1/r) to arrive at the following transmission problem on the strip D ˜i + ∂θ2 u ˜i = f˜i (τ, θ), i ∈ {1, 2}, −pi ∂τ2 u u ˜i |θ=ωi = 0, i ∈ {1, 2}, u ˜1 |θ=0 = u ˜2 |θ=0 , ˜ θ=0 , p1 ∂θ u ˜1 |θ=0 − p2 ∂θ u ˜2 |θ=0 = G|

(A.2.2a) (A.2.2b) (A.2.2c) (A.2.2d)

where we abbreviated (thinking of the functions ui , fi , G as given in polar coordinates) u ˜i (τ, θ) = ui (e−τ , θ), f˜i (τ, θ) = e−2τ fi (e−τ , θ), ˜ θ) = e−τ G(e−τ , θ). G(τ,

(A.2.3a) (A.2.3b) (A.2.3c)

We now turn to the question of solvability of (A.2.2) and that of uniqueness and regularity of the solutions. It is convenient to combine the two functions u ˜1 , u ˜2 into a single function u ˜ deﬁned on D: ! u ˜1 on D1 , u ˜ := u ˜2 on D2 In complete analogy to [15, Lemma 2.3], we have Lemma A.2.2. There exists h0 > 0 depending only on pi , ωi such that for each 0 < h < h0 there exists C > 0 with the following properties: If f˜i ∈ Lh (Di ) ˜ ∈ H1 (D1 ), then there exists a unique solution u ˜ ∈ {v ∈ Hh1 (D) | v|Di ∈ and G h 2 Hh (Di )} of (A.2.2), which satisﬁes , 2 2 ˜ 21 ˜ u2 1 . (A.2.4) + ˜ u|D 2 2 ≤C f˜i 2 + G Hh (D)

i

i=1

Lh (Di )

Hh (Di )

Hh (D1 )

i=1

Proof: We follow [15, Lemma 2.3] closely and may therefore be brief. We deﬁne on the strip D the piecewise constant function p by p|Di = pi and the function f˜ by f˜|Di = fi . By means of partial Fourier transform (in the τ variable), i.e.,

A.2 Kondrat’ev’s theory for a special transmission problem

√ fˆ(λ, θ) = 1/ 2π

∞

305

e−iλτ f˜(τ, θ) dτ,

−∞

ˆ equation (A.2.2) is transformed into a system and an analogous formula for G, of parameter dependent transmission problems, and we get for λ = ξ +ih, ξ ∈ R, −p∂θ2 u ˜ + λ2 p˜ u = fˆ(λ, ·) on (ω1 , 0) ∪ (0, ω2 ), u ˜(λ, ωi ) = 0, i ∈ {1, 2}, u ˜1 (λ, 0) = u ˜2 (λ, 0), ˆ 0) [p∂ u ˜(λ, ·)] = G(λ, θ

on θ = 0.

The next step is to obtain bounds for the solution u ˜ that are explicit in the parameter λ. This transmission problem has the form considered in Lemma A.2.5 below, which provides the required bounds and in particular the existence of h0 . Proceeding analogously to the proof of [15, Lemma 2.3], we obtain the bound (A.2.4). The ﬁnal step of the proof of Lemma 2.3 consist in the uniqueness assertion of the solution u ˜. In the present situation of a transmission problem, we therefore have to show that a function u ˜ ∈ {v ∈ Hh1 (D) | v|Di ∈ Hh2 (Di )} satisfying u=0 −pi ∆˜ u ˜i |θ=ωi = 0 [p∂n u ˜] = 0

on Di , i ∈ {1, 2}, on θ = 0

vanishes identically. We note that u ˜ has to be piecewise smooth by the local regularity assertions Propositions 5.5.1, 5.5.2, 5.5.4. Let now (ϕj , µj ) ∈ H01 (I) × C, j ∈ N, be the eigenfunctions and eigenvalues µj of (A.2.12) in the proof of Lemma A.2.5. The eigenfunctions ϕj are taken orthonormal with respect to the +ω ˜ may be expanded as inner product (u, v)p = ω12 puv dx. The solution u u ˜(τ, θ) =

∞

aj (τ )ϕj (θ),

(A.2.5)

j=1

aj (τ ) =

p˜ u(τ, θ)ϕj (θ) dθ.

(A.2.6)

I

Hence, aj (τ ) =

p∂τ2 u ˜(τ, θ)ϕj (θ) dθ = −µ2j aj (τ ). I

Thus, we conclude that each aj satisﬁes the diﬀerential equation aj (τ ) + µ2j aj (τ ) = 0. ˜j with µ ˜j > 0 Recalling that the eigenvalues µj are purely imaginary, i.e., µj = i µ and that µ ˜j → ∞ as j → ∞, we see that the coeﬃcients aj (τ ) have the form aj (τ ) = cj e−˜µj τ + dj eµ˜j τ

(A.2.7)

306

Appendix

for some constants cj , dj . As in [15], we now get for A > 0 and DA := {(τ, θ) | − A < τ < A, θ ∈ (ω1 , ω2 )}

e2hτ |˜ u|2 dτ dθ ∼

DA

=

A

ω2

p −A

ω1

∞

A

j=1

−A

 ∞  

j=1

2  aj (τ )ϕj (θ) e2hτ dτ dθ 

e2hτ |aj (τ )|2 dτ.

(A.2.8)

(A.2.9)

As u ˜ ∈ Hh1 (D) the left-hand side of (A.2.8) is ﬁnite, and it can be seen from this as in [15] that the ﬁniteness of each term in the sum in (A.2.9) implies cj = dj = 0 for all j. This proves uniqueness and completes the proof of the lemma. 2 [15, Lemma 2.4] also carries over almost verbatim: Lemma A.2.3. Let the assumptions of Lemma A.2.2 hold. Let in addition ˜ θ) = 0 for τ < 0. Then for ε > 0 and 0 ≤ γ = h + h0 − ε, f˜i (τ, θ) = 0, G(τ, ˜ ˜ 1 := {(τ, θ) | τ < 0, ω1 < θ < 0}, D := {(τ, θ) | τ < 0, ω1 < θ < ω2 }, D ˜ D2 := {(τ, θ) | τ < 0, 0 < θ < ω2 } we have for |α| ≤ 2 with α = (0, 2) α1 α2 2 2(h−γ)τ |∂τ ∂θ u ˜| e dτ dθ ≤ C(ε) |∂τα1 ∂θα2 u ˜|2 e2hτ dτ dθ, ˜ ˜ D D |∂θ2 u ˜|2 e2(h−γ)τ dτ dθ ≤ C(ε) |∂θ2 u ˜|2 e2hτ dτ dθ. ˜2 ˜ 1 ∪D D

˜2 ˜ 1 ∪D D

Proof: The proof is almost identical to the proof of [15, Lemma 2.4]. One uses ˜ = 0 for the expansion (A.2.5), (A.2.6) and shows that the assumptions f˜i = 0, G τ < 0 imply that the coeﬃcients cj = 0 for all j ∈ N. The result then follows by elementary considerations. A slight diﬀerence to the procedure in [15, Lemma 2.4] is that the eigenfunctions ϕj are only piecewise smooth. This forces us to split ˜ 2 when considering ∂ 2 u ˜ into D ˜ 1 and D 2 D θ ˜. We now proceed with the analog of [15, Lemma 2.6]. Lemma A.2.4. Let h0 be as in Lemma A.2.2 and let β ∈ (0, 1) satisfy β > 1 − h0 . Then there exists C > 0 depending only on ωi , pi , and β such that for 0,0 1,1 all fi ∈ Hβ,1 (Si ), G ∈ Hβ,1 (S1 ) with fi = 0 on Si \ B1 (0), G = 0 on S1 \ B1 (0), there exists a solution w of (A.2.1) with 2,2 2 (i) w ∈ Wβ,0 , wi ∈ Hβ,1 (Si ∩ B1 (0)), 2 2 ∇w < ∞, (ii) i L (Si ) i=1 , 2 2 wi H 2,2 (Si ∩B1 (0)) ≤ C fi H 0,0 (Si ) + GH 1,1 (S1 ) . (iii) β,1

i=1

β,1

i=1

β,1

A.2 Kondrat’ev’s theory for a special transmission problem

307

Proof: Follows by imitating all the steps of [15, Lemma 2.6]. In essence, it ˜ as deﬁned in (A.2.3) do satisfy the asis checked that the transforms f˜i , G sumptions of Lemma A.2.2. The pair (w1 , w2 ) given by Lemma A.2.2 then satisﬁes (i) and (iii). In order to see that (ii) also holds, we have to check that ∇wi L2 (Si \B1 (0)) < ∞. This is done with the aid of Lemma A.2.3. 2 Proof of Proposition A.2.1: This is essentially [15, Lemma 2.8]. Let (u1 , u2 ) ∈ V01 be a solution of (A.2.1) and assume that ui = 0 on Si \ B1 (0). Let (w1 , w2 ) ∈ 2 Wβ,0 be the solution of (A.2.1) given by Lemma A.2.4. It suﬃces to show ui = wi . We start by noting that the diﬀerence (u1 − w1 , u2 − w2 ) ∈ V01 and 2 i=1

pi ∇(ui − wi ) · ∇vi dx = 0

1 ∀vi ∈ V0,comp .

Si

(A.2.10)

$ 1 (Si ) = {v | ∇vL2 (S ) < ∞, v = 0 on Γi and Γ }. By our Introduce H 0 i assumptions on ui we have ∇ui ∈ L2 (Si ). Lemma A.2.4 also implies that ∇wi ∈ L2 (Si ). Hence, as u1 = u2 on Γ and w1 = w2 on Γ , we obtain that $ 1 (Si ). Finally, as ui − wi ∈ H 0 {(v1 , v2 ) ∈ V01 | v1 = 0 = v2 on Γ } $ 1 (S1 ) × H $ 1 (S2 ), we may choose (v1 , v2 ) = (u1 − w1 , u2 − w2 ) in is dense in H 0 0 (A.2.10) to conclude that ∇(ui − wi ) = 0 on Si . The boundary conditions now imply that ui = wi . 2 Lemma A.2.5. Let −∞ < ω1 < 0 < ω2 < ∞ (in particular, |ω1 |, |ω2 | may be bigger than 2π) and set I1 := (ω1 , 0), I2 := (0, ω2 ), I := (ω1 , ω2 ). Deﬁne the piecewise constant function p by p|Ii := pi > 0, i ∈ {1, 2}. Then there exist h0 , C > 0 such that for all λ ∈ Λ := {λ ∈ C | |Im λ| < h0 } the transmission problem: Find u ∈ H01 (I) such that −(pu ) + pλ2 u = f ∈ L2 (I)

on I,

[pu ] = g ∈ C

at x = 0,

has a unique solution satisfying u2H 2 (I1 ∪I2 ) +(1+|λ|2 )u 2L2 (I) +(1+|λ|4 )u2L2 (I) ≤ C f 2L2 (I) + (1 + |λ|)|g|2 . + Proof: Introduce the weighted L2 -inner product (u, v)p := I puv dx. The weak formulation of our problem then reads: Find u ∈ H01 (I) such that (u , v )p + λ2 (u, v)p = (f /p, v)p + gv(0)

∀v ∈ H01 (I).

(A.2.11)

Consider the eigenvalue problem −(pu ) + pµ2 u = 0

on I,

[pu ] = 0

at x = 0.

(A.2.12)

308

Appendix

By the spectral theorem, there is a countable sequence (ϕj , µj )j∈N ⊂ H01 (I) × C such that the functions ϕj form an orthonormal basis of L2 (I) (with respect to (·, ·)p ) and an orthogonal basis of H01 (I) (equipped with the inner product (u , v )p ). Furthermore, the eigenvalues µj are purely imaginary, i.e., they are of ˜j with µ ˜j → ∞. Without loss of generality, we may therefore the form µj = i µ assume that the µ ˜j are sorted in ascending order, that is, µ ˜j+1 ≥ µ ˜j for all j ∈ N. Finally, µ ˜1 > 0 implies (u , u )p ≥ µ ˜21 (u, u)p

∀u ∈ H01 (I).

We now choose h0 ∈ (0, µ ˜1 ) and set Λ := {λ ∈ C | |Im λ| < h0 }. Hence, for all λ ∈ Λ there holds Re λ2 > −h20 > −˜ µ21 and thus the bilinear form generated by the left-hand side of (A.2.11) is coercive on H01 (I) × H01 (I) for λ ∈ Λ; in particular the coercivity constant is independent of λ ∈ Λ. It is now easy to see that the desired bound holds for λ ∈ Λ bounded. We will therefore restrict our attention to the case λ ∈ Λ, |λ| → ∞. We consider the case g = 0 ﬁrst. We write f /p = j∈N fj ϕj with fj = (f /p, ϕj )p and note that there exists C > 0 such that |fj |2 ≤ Cf 2L2 (I) . j∈N

The solution of (A.2.11) can be represented in the form u = uj = −

j∈N

uj ϕj where

fj . µ2j − λ2

Thus, we get |λ|4 u2L2 (I) ≤ C|λ|4 (u, u)p = C|λ|4

|uj |2 = C

j∈N

|λ|2 u 2L2 (I) ≤ C|λ|2 (u , u )p = C|λ|2

j∈N

|uj |2 |µj |2 = C

j∈N

≤C

j∈N

|fj |2

|fj |2

|λ|4 , |µ2j − λ2 |2

|fj |2

j∈N

|λ|2 |µj |2 |µ2j − λ2 |2

|λ| + |µj | . |µ2j − λ2 |2 4

4

Elementary considerations show sup sup

2 j∈N λ∈Λ |µj

|λ|4 < ∞, − λ2 |2

sup sup

|µj |4 < ∞. − λ2 |2

2 j∈N λ∈Λ |µj

The desired bounds for |λ|4 u2L2 (I) and |λ|2 u 2L2 (I) now follow. To see that the bound for uH 2 (I1 ∪I2 ) also holds, we note that the diﬀerential equa 2 tion is satisﬁed 2pointwise a.e. on I1 ∪ I2 . Hence, we get u L (I1 ∪I2 ) ≤ 2 2 C f L (I) + |λ| uL (I) .

A.3 Stability properties of the Gauss-Lobatto interpolant

309

It remains to consider the case g = 0 and f = 0. This is done by constructing a special function ug satisfying the homogeneous boundary conditions, the diﬀerential equation with homogeneous right-hand side, and the correct jump condition at x = 0. We make the ansatz ! α sinh(λ(x − ω1 )) for x ∈ I1 , ug := β sinh(λ(x − ω2 )) for x ∈ I2 . The requirement that ug be continuous at x = 0 and that the jump condition be satisﬁed yields two conditions for α, β. This solution is given by sinh(λω1 ) sinh(λω2 ) , β=g , λW λW W = − [p1 sinh(λω2 ) cosh(λω1 ) − p2 cosh(λω2 ) sinh(λω1 )] . α=g

Elementary consideration now show that for |λ| large (implying that | Re λ| is large), ug satisﬁes the desired bound. This concludes the proof. 2

A.3 Stability properties of the Gauss-Lobatto interpolant Theorem A.3.1. Let I = [−1, 1]. Then there exists C > 0 such that for every p ∈ N the Gauss-Lobatto interpolation operator ip : C(I) → Pp satisﬁes ip uH 1 (I) ≤ CuH 1 (I) ∀u ∈ H 1 (I), ip uq L2 (I) ≤ C(1 + q/p)uq L2 (I) ∀u ∈ Pq , −1 ∀u ∈ H 1 (I), ip uL2 (I) ≤ C uL2 (I) + p uH 1 (I) √ ∀u ∈ C(I), ip uH 1/2 (I) ≤ C puL∞ (I) √ ip uH 1/2 (I) ≤ C puL∞ (I) ∀u ∈ C(I) with u(±1) = 0. 00

Proof: For the ﬁrst three estimates, see, e.g., [28, Sec. 13]. The last two follow from Theorem A.3.2. For example, for the fourth inequality, we have ip uH 1/2 (I) ≤ C

! p

*1/2 2

(ip u(xi ))

√ ≤ C puL∞ (I) .

i=0

2 Theorem A.3.2. Let I = [−1, 1]. Then there exists C > 0 with the following property. For p ∈ N let (xi )pi=0 be the Gauss-Lobatto points on I, i.e., the zeros of the polynomial x → (1 − x2 )Lp (x). Then for all u ∈ Pp there holds with

p 2 1/2 u2 = : i=0 |u(xi )| 1. uL∞ (I) ≤ u2 ,

310

Appendix

uL2 (I) ≤ Cp−1/2 u2 , uH 1/2 (I) ≤ Cu2 , uH 1 (I) ≤ Cpu2 , uH 1/2 (I) ≤ Cu2 if additionally u(±1) = 0, 00 +1 1 6. u22 ≤ Cp −1 √1−x u2 (x) dx ≤ Cpu2H 1/2 (0,1) . 2

2. 3. 4. 5.

Furthermore, the results are sharp with respect to the spectral order p. Proof: See [93, Thm. 4.1]. Closely related results can be found in [34].

2

A.4 L∞ projectors The following theorem concerning the existence of projection operators onto ﬁnite dimensional subspaces is due to Kadec-Snobar, [72]. We present it in the form given in [130] and [41, Chap. 9, Sec. 7]. Theorem A.4.1. Let Y be any Banach space and Xn ⊂ Y be a subspace of dimension n ∈ N. Then there exists a (bounded linear) projector Π : Y → Xn with Πv = v ∀v ∈ Xn , √ ΠvY ≤ nvY ∀v ∈ Y. If Y is an Lp -space with p ∈ (1, ∞), then the norm estimate of Π can be improved 1 1 to Π ≤ n| 2 − p | . An easy application is therefore the following corollary. Corollary A.4.2. Let K ⊂ R2 be a bounded open set. Then for every p ≥ 1 and every subspace V of the space of polynomials of degree p there exists a bounded linear operator Π : L∞ (K) → V with the following properties: 1. Πv = v for all v ∈ V ; 2. ΠvL∞ (K) ≤ (p + 1)vL∞ (K) . Proof: The space of polynomials of degree p is clearly a subset of Qp , which has dimension (p + 1)2 . The result now follows from Theorem A.4.1. 2 Corollary A.4.3. Let K ⊂ R2 be a bounded open set. Let | · | be any norm on W 1,∞ (K) (equivalent to the standard W 1,∞ (K) norm). Then for every p ≥ 1 and every subspace V of the space of polynomials of degree p there exists a bounded linear operator Π : W 1,∞ (K) → V with the following properties: 1. Πv = v for all v ∈ V ; 2. |Πv| ≤ (p + 1)|v| for all v ∈ W 1,∞ (K).

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Index

hp – space, 39 – – one-dimensional, 26 admissible boundary layer mesh, 117 admissible boundary mesh, 40 analytic – piecewise, 3, 189 analytic arc, 1 analytic regularity, 145 – asymptotic expansion, 35 – boundary conditions – – Dirichlet, 178, 200 – – mixed, 197 – – Neumann, 190, 201 – – Robin, 190, 201 – – transmission conditions, 198, 201 – curvilinear polygon, 186, 188 – in countably normed spaces, 33 – one-dimensional, 24, 146 – pointwise estimate, 178 – polygon (Babuˇska & Guo), 9, 178 – transmission conditions, 252 analyticity – on closed set, 24 approximation – on reference interval, 92 – on reference square, 93 – on reference triangle, 95 asymptotic expansion, 10, 257, 285, 288 – expansion order, 14, 264 – inner, 11, 269 – inner expansion, 290 – one-dimensional, 25 – outer, 10, 265 – outer expansion, 290 – regularity, 35, 288 – remainder, 13, 292 boundary layer – approximation, 79, 132 boundary layer mesh – one-dimensional, 26

boundary-ﬁtted coordinates, 10, 35 change of variables, 152, 166, 237 coercive, 2 compatibility conditions, 14, 18 convection-diﬀusion – regularity, 25 corner layer, 35, 292 – approximation, 82, 132 countably normed space, 8, 143, 257 – deﬁnition, 151 – exponentially weighted, 234 – invariance under transformation, 161 – local characterization, 158 – pointwise estimate, 165 – properties, 156 covering theorem, 179 – Besicovitch, 157 cusp, 2 degree of freedom (DOF), 30 element map, 39 – one-dimensional, 26 element size, 57, 113 ellipse, 92, 94 embedding – compact, 153 – in exponentially weighted space, 235 – in weighted spaces, 153, 303 energy estimate, 3, 146 energy norm, 3, 26 error estimate – H 1 -projection, 111 – Πp∞ , 106 – Πp1,∞ , 110 – anisotropic, 110 – exponential, 110 – – on minimal mesh, 132 – – on reference interval, 92 – – on reference square, 93 – – on reference triangle, 95 – linear interpolant, 92

318

Index

existence and uniqueness, 3 – in exponentially weighted spaces, 247 extension, 91, 107 FEM, 5, 26, 43 – low order, 5, 15, 16, 55, 65 ﬁnite element – space, 26, 39, 115 ﬁnite element method, 5 Gauss-Lobatto – approximation property, 112 – points, 89 – stability, 90, 112, 311 geometric mesh – one-dimensional, 29 – two-dimensional, 53 Gevrey regularity, 9, 178 Hardy inequality, 146, 156, 303 interpolant – linear, 60, 92 inverse estimate, 91 Legendre series, 93, 94 – truncated, 93 length scale – characteristic, 171 local elliptic regularity, 199 – Dirichlet conditions, 200 – interior, 200 – Neumann conditions, 201 – transmission conditions, 201 locking, 16 macro triangulation, 118 mesh, 39 – (CM , γM )-regular, 113 – admissible boundary layer, 40, 117 – normalizable, 114 – of Bakhvalov type, 16 – of Gartland type, 16 – of Shishkin type, 16, 55, 62 – one-dimensional, 26 – radical, 15, 58 – regular admissible, 117 mesh design principles, 5, 39 normalizable, 114 normalizable triangulation, 57, 69 patch, 118 – boundary layer, 120 – corner layer, 120 polygon, 1

– curvilinear, 1 polynomial – extension, 91, 107 – interpolation operator Πp∞ , 105 – interpolation operator Πp1,∞ , 110 – inverse estimate, 91 – spaces, 89 projector – H 1 -projection, 111 – H 1 -projector, 116 – Πp∞ (L∞ -based), 105 – Πp1,∞ (W 1,∞ -based), 110 ∞ – Πp,T , 115 – anisotropic, 107 – existence in Banach spaces, 312 – linear interpolant, 60, 92 reference – interval, 26, 89 – square, 38, 89 – triangle, 38, 89 regular – (CM , γM ), 113 regular admissible, 117 regular normalizable, 69 Reissner-Mindlin plate, 53 relative diﬀusivity E – deﬁnition, 179 – one-dimensional, 171 robust exponential convergence, 4, 5, 44, 45, 132 sector, 148 shift theorem, 7, 145, 180, 187 Shishkin mesh, 16, 54 singular functions, 8 solver – direct, 6 – multigrid, 5 stability – Stokes equation, 53 stretched variables, 11, 269 stretching map, 40 transition point, 17, 55 transmission problem, 229, 240, 304 triangulation, 39 – (CM , γM )-regular, 113 – normalizable, 114 weak formulation, 3, 23 weighted Sobolev space, 143 – deﬁnition, 151 – exponentially weighted, 234 – properties, 152