Solution of Variational Inequalities in Mechanics (Applied Mathematical Sciences)

1. HIavacek J. Haslinger J. Necas, J. Lovisek Applied Mathematical Sciences 66 Solution of Variational Inequalities in...

Author: Ivan Hlavacek | Jaroslav Haslinger | Jindrich Necas | Jan Lovisek

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1. HIavacek

J. Haslinger J. Necas, J. Lovisek Applied Mathematical Sciences 66

Solution of Variational Inequalities in Mechanics

Springer-Verlag

Applied Mathematical Sciences I Volume 66

Applied Mathematical Sciences 1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. 4. Percus: Combinatorial methods. 5. von Mises/Friedrichs: Flid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 11. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions. 15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II. 25. Davies: Integral Transforms and Their Applications, 2nd ed. 26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems 27. de Boor: A Practical Guide to Splines.

28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics.

31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz? Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III. 34. Kevorkian/Cole: Perturbation methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/Ghil/Kallt n: Dynamic Meterology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Stochastic Motion. 39. Piccini/Stampacchia/Vidossich: Ordinary Differential Equations in R". 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. 43. Ockendon/Tayler: Inviscid Fluid Flows.

(continued on inside back cover)

I. Hlavacek J. Haslinger J. Necas J. Lovisek

Solution of Variational Inequalities in Mechanics

Springer-Verlag

New York Berlin Heidelberg London Paris Tokyo

I. Hlavacek

J. Haslinger, J. Necas

Mathematical Institute of the Czechoslovak Academy of Sciences Zitna 25

Faculty of Mathematics and Physics of the Charles University Prague

115 67 Praha 1

Czechoslovakia

Czechoslovakia

J. Lovfsek Faculty of Civil Engineering Slovak Technical University Bratislava Czechoslovakia

Mathematics Subject Classification (1980): 73K25 Library of Congress Cataloging-in-Publication Data Solution of variational inequalities in mechanics. (Applied mathematical sciences; v. 66) Translated from the Slovak. Bibliography: p. Includes index.

1. Continuum mechanics. 2. Variational inequalities (Mathematics) I. Hlavai`ek, Ivan. II. Series: Applied mathematical sciences (SpringerVerlag New York Inc.); v.66. QA1.A647 vol. 66 [QA808.21

510 s

87-20767

[531]

This book is a translation of the original edition: Riesenie variacnych nerovnosti v mechanike.

© 1982 by Alfa Publishers. © 1988 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.

987654321 ISBN 0-387-96597-1 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-96597-1 Springer-Verlag Berlin Heidelberg New York

Preface The idea for this book was developed in the seminar on problems of continuum mechanics, which has been active for more than twelve years at the Faculty of Mathematics and Physics, Charles University, Prague. This seminar has been pursuing recent directions in the development of mathematical applications in physics; especially in continuum mechanics, and in technology. It has regularly been attended by upper division and graduate students, faculty, and scientists and researchers from various institutions from Prague and elsewhere. These seminar participants decided to publish in a self-contained monograph the results of their individual and collective efforts in developing applications for the theory of variational inequalities, which is currently a rapidly growing branch of modern analysis. The theory of variational inequalities is a relatively young mathematical discipline. Apparently, one of the main bases for its development was the paper by G. Fichera (1964) on the solution of the Signorini problem in the theory of elasticity. Later, J. L. Lions and G. Stampacchia (1967) laid the foundations of the theory itself. Time-dependent inequalities have primarily been treated in works of J. L. Lions and H. Brezis. The diverse applications of the variational inequalities theory are the topics of the well-known monograph by G. Duvaut and J. L. Lions, Les inequations en mecanique et en physique (1972). Analyse numerique des inequations variationnelles (1976), by R. Glowinski, J. L. Lions, and R. Tremolieres, deals with the numerical analysis of various problems formulated in terms of variational inequalities. 1980 heralded the appearance of the excellent book An Introduction to Variational Inequalities and Their Applications, by D. Kinderlehrer and G. Stampacchia. Our intention was to loosely follow the ideas of these and other works, especially those of some chapters of the book Mathematical Theory of Elas-

vi

Preface

and I. Hlav£Lek, 1981). Thus, the tic and Elasto-Plastic Bodies (J. second and the third chapter develop, both theoretically and from the point of view of numerical analysis, the solution of the Signorini generalized problem and two models of the theory of plastic flow, respectively. In both the engineering and physics literature we can find many problems solved by intuitive, ad hoc methods, regardless of the fact that there exists a possibility of formulating and solving these problems in the framework of the theory of variational inequalities, and thus penetrating more profoundly to the very core of the problems. The general purpose of our book is to acquaint the wider reading public with the progress of modern mathematics in this field, and to help this readership find an adequate formulation of the problem as well as an economic method of computation. Our specific desire when writing this book was to share experience gained in the course of solving individual problems for some industrial organizations. Our thanks are due to our colleagues Dr. O. John and L. TravnMek for their careful reading of the original version of the book, and to Dr. J. Jarnik for the translation into English. I. Hlava&k J. Haslinger J.

J. Loviek

Contents Preface

v

Unilateral Problems for Scalar Functions Unilateral Boundary Value Problems for Second Order Equations Primal and Dual Variational Problems 1.1.1 1.1.11 Dual Variational Formulation 1.1.12 Relation Between the Primal and Dual Variational Formulations Mixed Variational Formulations 1.1.2 Solution of Primal Problems by the 1.1.3 Finite Element Method and Error Bounds 1.1.31 Approximation of Problem P1 by the Finite Element Method 1.1.32 The General Theory of Approximations for Elliptic Inequalities 1.1.33 A Priori Bound for Problem P1 Solution of Dual Problems by the 1.1.4 Finite Element Method and Error Bounds 1.1.41 Problems with Absolute Terms 1.1.411 A Priori Error Bounds 1.1.412 A Posteriori Error Bounds and the Two-Sided Energy Bound 1.1.42 Problems Without Absolute Terms 1.1.421 A Priori Error Bound 1.1.422 A Posteriori Error Bounds and the

1

1.1

2

4 7 10 13 18

18

21 24 27 27 29 34 36 41

Contents

viii

1.1.5 1.1.51

1.1.52 1.1.53 1.1.6 1.1.61

1.1.62 1.1.63 1.1.7 1.1.71 1.1.72

1.1.73

1.2

1.2.1 1.2.2 1.2.3

1.2.4

1.2.41 1.2.42

1.2.43 1.2.5

Two-Sided Energy Estimate Solution of Mixed Problems by the Finite Element Method and Error Bounds Mixed Variational Formulations of Elliptic Inequalities Approximation of the Mixed Variational Formulation and Error Bounds Numerical Realization of Mixed Variational Formulations Semicoercive Problems Solution of the Primal Problem by the Finite Element Method and Error Bounds Solution of the Dual Problem by the Finite Element Method and Error Bounds Convergence of the Dual Finite Element Method Problems with Nonhomogeneous Boundary Obstacle Approximation of the Primal Problem Solution of the Dual Problem by the Finite Element Method A Posteriori Error Bounds and Two-Sided Energy Estimate

Problems with Inner Obstacles for Second-Order Operators Primal and Dual Variational Formulations Mixed Variational Formulation Solution of the Primal Problem by the Finite Element Method Solution of the Dual Problem by the Finite Element Method Approximation of the Dual Formulation of the Problem with an Inner Obstacle Construction of the Sets Qrh and Their Approximate Properties A Priori Error Bound of the Approximation of the Dual Formulation Solution of the Mixed Formulation by the Finite Element Method

48 49 51

54 59 62

66 69 75 82 85 87 88

89 89 93 94 98 98

99 99 104

Contents

ix

One-Sided Contact of Elastic Bodies

2 2.1 2.1.1 2.1.2 2.1.3

Formulations of Contact Problems Problems with Bounded Zone of Contact Problems with Increasing Zone of Contact Variational Formulations

109 110 112 114 116

2.2 2.2.1 2.2.2

Existence and Uniqueness of Solution Problem with Bounded Zone of Contact Problem with Increasing Zone of Contact

121 121 130

Solution of Primal Problems by the Finite Element Method 2.3.1 Approximation of the Problem with a Bounded Zone of Contact 2.3.2 Approximation Problems with Increasing Zone of Contact 2.3.3 A Priori Error Estimates and the Convergence 2.3.31 Bounded Zone of Contact 2.3.3.11 Polygonal Domains 2.3.312 Curved Contact Zone 2.3.32 Increasing Zone of Contact 2.3

2.4

2.4.1 2.4.11 2.4.12 2.4.13

Dual Variational Formulation of the Problem with Bounded Zone of Contact Approximation of the Dual Problem Equilibrium Model of Finite Elements Applications of the Equilibrium Model Algorithm for Approximations of the Dual Problem

Contact Problems with Friction The Problem with Prescribed Normal Force Some Auxiliary Spaces Existence of Solution of the Problem with Friction Algorithms for the Contact Problem with Friction for Elastic Bodies 2.5.41 Direct Iterations 2.5.42 Alternating Iterations 2.5.421 Unilateral Contact with a Given Shear Force 2.5.422 Realizability of the Algorithm of Alternating Iterations 2.5 2.5.1 2.5.2 2.5.3 2.5.4

134 134 136 138 138 139 148 154

164 170 173 175

176 182 187 192 194

196 196 207 208 212

Contents

x

Problems of the Theory of Plasticity

3 3.1 3.1.1 3.1.2 3.1.21

Prandtl-Reuss Model of Plastic Flow Existence and Uniqueness of Solution Solution by Finite Elements A Priori Error Estimates

3.2 3.2.1

Plastic Flow with Isotropic or Kinematic Hardening Existence and Uniqueness of Solution of the Plastic Flow Problem with Hardening Solution of Isotropic Hardening by Finite Elements A Priori Error Estimates A Priori Error Estimates for the Plane Problem Convergence in the Case of Nonregular Solution

3.2.2 3.2.21 3.2.22 3.2.23

References Index

221 226 229 233 235

238 241 247 250 258 262

267 273

Chapter 1

Unilateral Problems for Scalar Functions In this chapter we will consider boundary-value problems with one-sided conditions in the form of inequalities, where the unknown is a single real scalar function in a given domain of an n-dimensional space R', n = 2, 3. First we pay attention to problems in which the inequalities are prescribed only on the boundary of the domain considered. Then we deal with problems which involve inequalities inside the domain, i.e., with an obstacle inside the domain. For the sake of simplicity we restrict our exposition to the second order elliptic operators. We will show how to formulate these problems in terms of variational inequalities, which in the case of a symmetric elliptic operator are equivalent to the variational problem of finding the minimum of a quadratic functional on a certain closed convex set. Thus, we will in fact obtain a generalization of two dual variational principles of the minimum of the potential energy or the minimum of the complementary energy, well known in mechanics. Moreover, we will also present the mixed variational formulations-a generalization of the Hellinger-Reissner-type canonical variational principles. Each of the above-mentioned three variational formulations may serve as the basis for deriving approximate variational methods of solution. We will show here applications of one of the simplest variants of the finite element methods, namely, the elements with linear polynomials on a triangular net. We will establish some a priori error bounds provided the exact solution is sufficiently regular. We will also prove that the finite element method converges even if the solution fails to be regular. The use of dual analysis-that is, the simultaneous solution by means

1. Unilateral Problems for Scalar Functions

2

of both dual variational formulations-makes it possible to find even a posteriori error bounds, as well as two-sided bounds for the energy of the exact solution.

1.1

Unilateral Boundary Value Problems for Second Order Equations

Let us first consider elliptic equations of the second order with boundary conditions in the form of inequalities. Such problems describe steady-state phenomena in some branches of mathematical physics, as for example, thermics, fluid mechanics, and electrostatics (see Duvaut and Lions (1972)). In order to grasp the crucial points of the solution of this class of problems as easily as possible, let us first choose several representative model problems of a relatively simple character. Thus, let n c R" be a bounded domain with a Lipschitzian boundary (for the definition, see (1967) or Hlava ek and (1981)), and let us consider "the equation with an absolute term"

- Au + u = f

in 11,

(1.1)

with boundary conditions of the so-called Signorini type

U>0'

au ->0' (9M

uau =0 av

on an = r'

(1.2)

where au/av stands for the derivative with respect to the outer normal v to the boundary r. Problem (1.1) and (1.2) will briefly be called the "problem P1" Secondly, let us consider "the equation without an absolute term"

- Au = f

in n,

with boundary conditions of two types:

u=0 on rucr, U>0' au/av > 0 and u au/av = 0

on ra = r-ru.

(1.5)

Problem (1.3), (1.4), and (1.5) will be briefly called "problem P2". The results which will be established in what follows can be easily extended to equations of the form i i,;= 1

(aij (x) 2

+ ao(,)u = t>

1.1. Unilateral Boundary Value Problems

3

where aid E L°° (f2), ao E LOO (Q), aid is a symmetric matrix that is positive definite in 12, and ao(x) > c > 0

or ao(x) = 0,

hold almost everywhere in SZ for the class of problems which generalize P1 or P2, respectively.

In what follows we shall make use of the Sobolev spaces Wk,p(c2) of functions which possess generalized derivatives integrable with the p-th power up to and including the k-th order. For p = 2, we shall write Wk,2(ft) = Hk(f2), III (fl) = L2 (Q). Further, we introduce the scalar product in L2 (1Z) by

(f, 9)0 =

o f (x)g(x)dx

(x = x1 .. , xn, dx = dxl... dxn). The norm in Hk(fl) will be denoted by IIk,n, with the subscript fl omitted if no misunderstanding can occur. The symbol (ulk,n will stand for the seminorm, II

-

7k

1/2

lulk,n =

Ilu'Ilk,n - E [uIj,ri

IIDaull°,o

° =k

1/2

9=o

In H1(0) we introduce the scalar product (u, v)1 = (u, v)o + (Ou, Ov)o, where U

n v

i.l

au, a (axi axi ) 0

The norm fulfills

Ilulil,n = (u,u)1/2 If U E [Hk(f2)]n, then IIuIlk,n means the usual norm 1/2

n

llullk,n = Ilu'llk =

lluillk,n)

,

k = 1,2,...

;

,_1 (>

similarly for the seminorm.

Let us always assume that the right-hand side of equations (1.1) and (1.3) fulfills f E L2(f2).

1. Unilateral Problems for Scalar Functions

1.1.1 Primal and Dual Variational Problems Problem P1 may be formulated as a variational problem. To this end, let us define the set K1 = {v I v E H'(12), ryv > 01, where ryv denotes the trace of the function v on the boundary r (e.g., Nel as (1967)), and the functional of potential energy ,CI(v) = 1 IIvII1 - (f, v)o.

Then the problem: find u E K1 such that L1(u) < L1(v)

Vv E K1i

(1.6)

is a variational formulation of problem P1. A function u E K1, which is a solution of (1.6), is called a generalized solution of problem P. The relation between problem Pi and problem (1.6) is fundamentally seen from:

Theorem I.I. Let V be a Banach space, let a functional jr : V ---' R possess the Gateaux differential in V and let K be a convex subset of V. Then u minimizes 3 on the set K only if

DI(u,v-u)>0 bvEK.

(1.7)

If, moreover, 3 is convex, then (1.7) is a necessary and sufficient condition for u to minimize .7 on K.

Proof. See Cea (1971). As K1 is a convex set (indeed, ry(Au + (1 - A)v) = A-yu + (1 - .\)ryv > 0, VA E (0, 1), du, v E K1), and L1 is a convex functional in V = Hi(ll), the problem (1.6) is equivalent to the variational inequality

DL1(u,v-u)>0 VvEK1i which may be written in the form

(u,v-u)1>(f,v-u)o VvEK1. Choosing here

v=u±tp

rpECO (R),

we obtain (u, co)1 = (f, V) o

t/

(1.8)

1.1. Unilateral Boundary Value Problems

5

which implies that the equation (1.1) is satisfied in the sense of distributions and

Du = u - f E L2(fl)In order to find out in what sense the boundary conditions are satisfied, we first introduce the space of traces of functions from H' (f2) on the boundary F.

Definition 1.1. If fl is a domain with a Lipschitzian boundary F, then the image of the space H1 (Q) is denoted by

7(H1(0)) = H1/2(r), and the norm in H1/2(F) is defined as

-

I1WIl1/2,r

vEHI( n)

Ijvlll>n.

(1.9)

1v =w

Remark 1.1. In Nei=as (1967), the reader will find another definition of H1/2(F) together with a norm which is equivalent to the norm (1.9). H1/2(F) is a linear subspace of L2(r).

Theorem 1.2. Let us denote by

H-1/2(r) = [Hl/2(r)I' the space of linear continuous functionals over the space H'/2(F). Further, let u E H1(cl) satisfy Du E L2(fl) and let w c H1/2(F), v e H' (0), ryv = w. Then the formula

(au, w) _

(vu, 0v)o + (v, /u)o

(1.10)

determines an element from H-1/2(F), which will be denoted by au/av. Remark 1.2. The reader will easily verify that the left-hand side of formula (1.10) for functions u E C00((1) reduces to the integral au

8U

(a v' w) = ,Ir av

wdF.

Consequently, theorem 1.2 extends the notion of the derivative with respect to the normal to a more general class of functions.

Proof of Theorem 1.2. 10 The right-hand side of (1.10) is independent of the choice of v E H1(fl). Indeed, the identity (Du, 0'P)o + (V, Du)o = 0 V(p E Ca (1)

1. Unilateral Problems for Scalar Functions

6

directly follows from the definition of Du in the sense of distributions.

2° As the mapping ry is additive and homogeneous, the functional 8u/8v is linear. 3°

au

12 (aU,w) < (11- + IlAul10)I'Zllvlll Vu E H1(fi), 'yv = w.

Passing to the infimum we find that 8u/8v is bounded in H1/2(1'). Now let us substitute v = 0 and v = 2u into inequality (1.8). We obtain (u, u)1 = (f, u)o

(1.11)

and, since Du = u - f, we have au,'YU)

=

(vu, pu)o + (u, L\u)o

_ (Du,Vu)o+(u,u)o-(f,u)o=0.

(1.12)

Substituting (1.11) into (1.8), we find 0

(u, v)1 - (f, v)o = (ou, pv)o + (u - f, v)o

(vu, pv)o + (zu, v)o = (au, -IV) Vv E K1.

(1.13)

Inequality (1.13) states that the functional 8u/av is nonnegative, as will be seen later on. Thus, we conclude that a solution u of problem P1 fulfills the second and last condition from the triplet of boundary conditions (1.2) in the functional sense. Conversely, each sufficiently smooth solution of problem Pi solves problem (1.6). Indeed, it suffices to multiply (1.1) by a function v E K1 and to integrate by parts, thus deriving (1.12) and (1.13). Hence, the variational inequality (1.8) follows. The same method may also be applied to problem P2. Let us define the set K2 = {v I v E H1(12), ryv

0 on ru,, 7v > 0 on ha},

and the functional of potential energy .C2(v) = 2lvl1 - (f,v)o,

where Iv112

= (Vv, Vv)o.

1.1. Unilateral Boundary Value Problems

7

Then the problem: find u c K2 such that C2(u) < C2(u) Vv E K2

(1.14)

is a variational formulation of problem P2.

The relation between the solution of problem P2 and the variational problem (1.14) is quite analogous to the case of problem P1. Problems (1.6) and (1.14) are called the primal variational formulations of the one-sided problems P1 and P2, respectively. Although the methods of solution for both of these primal problems differ only insignificantly, we

shall see that the difference between the corresponding dual variational formulations is essential.

1.1.11. Dual Variational Formulation. We can formulate problems P1 and P2 in such a way that the unknown is not the function u but rather its gradient, which is often an important and interesting quantity from the physical point of view. However, instead of the direct transformation of problems P1 and P2, we will derive variational formulations which are dual with respect to the primal variational formulations (1.6) and (1.14). Let us introduce the set

H(div, fl) = {q I q E [L2(fl)]", div q E L2(fl)),

where the divergence operator is understood in the sense of distributions:

Jo

q vcp dx = -

Jn

p div q dx `dip E Co (fl).

Theorem 1.3. Let q E H(div,12), w E H1/2(r). Then the formula (q - v,

w).=J(q .v v+vdivq)dx,

with v E H'(ll) and ryv = w, defines a functional q v E H-1/2(r). Proof. Almost identical with that of theorem 1.2. O We shall write s > 0 on r (or I',) for a functional s E H-1/2(r) if (s, iv) > 0 Vv E K, (K2, respectively). Let us introduce the class of admissible functions U1 = {q I q E

[L2(tl)]"+1,

q = [q, qn+1), q E H(div, fl),

q"+1 = f + div q, q v > 0 on I'},

1. Unilateral Problems for Scalar Functions

8

and the functional of the complementary energy 1 n+1 S, (q)

_

Ilgi IIo

i=1

The problem: find q° E tl1 such that

S1(4)

S, (q)

(1.15)

Vq E u1

will be called the dual variational formulation with respect to the problem (1.6).

Let us consider problem P2. Introduce the set tl2 = {g I q E H(div, f2), div q + f = 0,

q v > 0 on I'a}

and the functional of complementary energy S2(q) _

II4iII00

i=1

The problem: find q° E tl2 such that

S2(9) < S2(q)

(1.16)

Vq E U2

will be called the dual variational formulation with respect to the problem (1.14).

Theorem 1.4. Both of the primal problems (1.6) and (1.14), as well as both of the dual problems (1.15) and (1.16), has a unique solution.

Proof. Based on the following general result.

0

Theorem 1.5. Let 3 be a strictly convex, continuous functional defined on a reflexive Banach space X. Let K c X be a closed convex set and let 3 be coercive on K, i.e.,

vEK, IIvII-'oo=* 1(v)-++oo.

(1.17)

Then there is one and only one solution of the problem:

3(u) = min on K. (Proof is found in the book by FOik and Kufner (1978). See theorem 26.8.)

1.1. Unilateral Boundary Value Problems

9

The functionals Li, i = 1, 2 are continuous, strictly convex, and coercive in the spaces V1 = H1(1Z) and V2 = {v E H1(11) I7v = 0 on I'T1 respectively.

Indeed, take for example L2. It has a second differential which satisfies D2L2(u, v, v) = IvI

>

- CIIvjI,

Vv E V21 (mes ru > 0),

where C = const > 0. Hence, L2 is strictly convex and satisfies (1.17). It is not difficult to verify that the sets Ki are closed and convex in Vi. In order to prove existence and uniqueness for problem (1.15), we first transform it to an equivalent problem, formulated only for "reduced" vector functions q E H(div,11). Namely, after the substitution qn+1 = f + div q we have a new equivalent problem: find q E llo such that I (4) < I (q)

(1.18)

bq E Uo,

where

on F), n

I(q) =

2

1: Ilgllo+Ildivgllo

+(f,div')o.

i-t

If the norm in the space H(div, fl) is introduced by

/n

\ 1/2 IIgIIH(div,n) = t Ilgillo + IIdivgllo) i=1

then H(div, ll) is a Hilbert space. Uo is closed and convex in H(div, ll). To prove for example the closedness of 110, it is sufficient to realize that the mapping q - q v of the space H(div,11) into H-1/2(r) is linear and continuous. The functional I is continuous on H(div,1Z), strictly convex, and coercive. Hence, according to theorem 1.5 we obtain the existence and uniqueness of the solution to problem (1.15). We proceed analogously with problem (1.16). 112 is closed and convex in the space H(div,11), and the functional $2 is continuous and strictly convex

in H(div,1l). Since q E 112, IIgIIH(a1n)

o0

= i-1

Ilgillo + Ilfllo

S2 (q) - +oo, $2 is also coercive in 112.

-' oo

1. Unilateral Problems for Scalar Functions

10

1.1.12. Relation Between the Primal and Dual Variational Formulations. We will now show how the solutions of the dual problems are related to those of the primal problems. To this end we will use the saddlepoint method (the min-max method), the sequel of which will also enable us to derive the mixed variational formulation of the original problem P1 or P2, corresponding to the canonical variational principles of Hellinger-Reissner type (see Hlav£tek and (1981)). Let us consider problem P1 together with its variational formulations.

Theorem 1.6. If u is a solution of (1.6) and q° a solution of (1.15), then qi°

8u = a zi ,

2.

(

= 1, 2, ... n,

1.19

)

q0

n+1 - u,

S1(q°) +.L1(u) = 0.

(1.20)

Proof. For v E H1(Q) let us denote [L2(fl)]n+1,

C'(v) = [Vv, v] E

M = [L2(jj)]n+1'

=K1xM and let us introduce the Lagrangian functional )((v, N; q) =

2

II NII2 - (f, v)o + (q, ((v) - N),

where v E K1, N E M, q E M, and II

denote the norm and the

' II,

scalar product in [L2(c2)]n+1, respectively. We easily find that sup (q, e(-) - M) qEM

0

forty=C(v)

+oo

for N

e (v),

and thus

inf .£1(v) =

vEK1

inf

sup )-((v, N; q).

(v,NJE'W qEM

(P)

In the theory of duality (e.g., Ekeland and Temam (1974)), the problem P'

sup

inf

qEM (v,N)EW

)((v, N; q)

1.1. Unilateral Boundary Value Problems

11

is called the dual problem to P. The question arises when both values coincide.

Let us denote So (q) =

inf

[v,NJEW

X (v, N; q)

and let us try to explicitly calculate this functional. First of all, we have So(q) <

.(v, N; q) = vEK1 inf .C1(v) = C1(u)

inf [v, .A/ J E W

u=c (.)

for all q E M. This implies sup So(q)

,(,'1(u).

(1.21)

M

Furthermore, So(q)

Nnf , EM X1(X,q)+vinf X2(v,q)

where

X,(X,q) = 211X112-(q,X), )(2 (v, q) = (q, E(v)) - (f, v)o.

It is easy to calculate

i f1

XI(.u,q) = -2118112.

Lemma 1.1. We have inf X2 (v, q) _

vEK1

0

for q E 111i

-00 for q 0 U1.

Proof. 10 X2 q) is a continuous linear functional in H1(0). Let there exist vo E K1 such that 12(vo,q) < 0. Then if t - +oo, tvo E K1 and lim X2 (tvo, q) = -oo; hence the infimum is -oo. Consequently, if the infimum is to be greater than -oo, then necessarily X2(v, q) > 0 or

n

8v t

8x; J 0

Vv E K1i

+ (9n+1, v)o - (f, v)o > 0 Vv E Kl.

1. Unilateral Problems for Scalar Functions

12

Substituting in the inequality v = +(p E Co (f1), we obtain

qn+l - f = div q c L2 (f1). Hence, q E H(div, f1). Further, according to theorem 1.3 we have for all v E K1

0 < M2 (v, q) = J (q . Vv + v div q)dx = (q v, ryv). n

Hence, q v > 0 on r, which means q E u1. 2° Let q E U1, v E K1. Then (q, e(-,))

v+vdiv+vf)dx

= (q

v, 7v) + (f, v)o >_ (f, v)0.

Hence, for q E U1 we have 12(v, q) > 0 Vv E K1. Substituting v = 0 E K1 we derive that the infimum is equal to zero, which completes the proof of lemma 1.1. Altogether, we can write So (q)

So(q)

2

jjgjj2 = -Sl(q)

for g E u1,

=-oo for g0U1.

Thus, by virtue of (1.21) we have

C1 (u) > sup So(q) = sup[-S,(q)) = inf Si(q) _ -S1 (q°). M

(1.22)

U

Setting q = cj = t(u) we obtain q E U1. Indeed, at the beginning of section 1.1.1 we found that Au = u - f E L2(fl). Hence:

div q=div pu=LuEL2(fl) =qEH (div,f ), div q = qn+1 - f Finally, it follows from (1.13) that

q

- S1(q)

q into $1 and using equation (1.11), we obtain 2 114112 = -2[IuI1i = 21Ju11i - (f,u)o = L1 (u).

(1.23)

1.1. Unilateral Boundary Value Problems

13

Hence, -S1 assumes its maximum at the point q, as is seen in (1.22). However, the uniqueness of the solution of problem (1.15)-see theorem 1.4-implies that q = q°. This, together with (1.23), yields relation (1.20). Remark 1.3. The point {[u, S(u)], q°} E w x M is called a saddle point of the functional X([v, N]; q) in the Cartesian product 1U x M. The following identity holds for this point as a consequence of (1.23) and (1.22): X([u, t(u)]; q°)

= mina, supM X([v, A(]; q) maxM inf3y X ([v, X]; q).

(1.24)

An analogous theorem holds for problem P2 and its variational formulations:

Theorem 1.7. If u is a solution of (1.14) and q° a solution of (1.16), then

q° = Vu, S2(q°) + £2(u) = 0.

Proof. Can be carried out analogously to that of theorem 1.6.

1.1.2

O

Mixed Variational Formulations

In this section we will derive new, equivalent variational formulations of problems P1 and P2. The difference between the primal formulations and the new ones consists of the fact that our new formulations do not require the one-sided boundary conditions to be explicitly fulfilled. Let us restrict our considerations to problem P1. The symbol H+ 112(I") (I' = 8fl) denotes the set of all nonnegative linear functionals over H1i2(I') (in the sense of section 1.1.11). It is evident that v c K1 G-a - (,p, -iv) < 0

Vcp E H+ 112 (r).

Moreover,

sup(r){-
0

+oo

+ 1/3

provided v E K1, provided v 0 K1.

Indeed, let v E K1. Then,

-(`p, 7v) < 0 Va E H+1/2(I') and

(B, -IV) = 0,

(2.1)

1. Unilateral Problems for Scalar Functions

14

where B is the zero element of H-1/2(r). On the other hand, if v 0 K1, then there exists p E H+1/2 (r) such that -(µ, ryv) = c > 0.

It is easily verified that the elements of the form pp belong to H+1/2(r) for every p > 0. Consequently, PC

H

-' +00,

Thus, we can formally write

i fC1(v)=Hnf

sup H+

(r

)

{Li(v)-(p,ryv)}.

Let us write X (v, µ) = C1(v) - (lp,'Yv). We shall establish the relation between the solution of P1 and the saddle point of X in H1 (n) x

H+1/2(I').

Theorem 2.1. A pair (w, A) E H1(1) X H+-1/2 (r) is a saddle point of X in H1(n) x H+1/2(I') if and only if w = u,

a = 8u/8v,

where u E K1 is a solution of Pi.

Proof. (i) Let (w, A) be a saddle point of X in H1(1) X H+ 1/2(I,): (w, A) E H1(n) X H+1/2(I') :

X(w,µ) :5X(w,A) s X(v;A)

d(v,µ) E H1(1) x H+112(r).

(2.2)

Substituting into this inequality first u = B, then p = 2A, we obtain (A, "w) = 0,

(2.3)1

and consequently -(ip, ryw) < 0

V11

E

H+-112

(r).

(2.3)2

Hence, of necessity w E K1. Taking into account (2.1), the second inequality in (2.2) and (2.3)1,2, as well as the definition of X, we conclude C1(w)

C1(v) - (µ,'Yv) < C1(v)

Vv E K1.

1.1. Unilateral Boundary Value Problems

15

This means that w E K1 is an element which minimizes Cl in K1. Such an element, as we know, is unique; hence, w = u. The functional assumes its minimum in H1(fl) at the point v = w. Therefore, (w, v)1 - (A, 7v) = (f, v)o

Vv E H'(1).

Using Green's formula (1.10) together with (1.1), we finally obtain

(aw/av, "v) - (A, ryv) = 0 Vv E H1(0),

which implies aw/av = A. (ii) Let u be a solution of P1. Then, (1.13) implies (u,au/av) E H1(0) x H+1/2(r). Further,

X (u, au/av)

='C1 (u) - (au/av, 7u) >- £1(u) - (,U, 7u)

= X(u,µ) Vp E H; 1/2(r). Similarly,

X(u,au/av) - X(v, au/ail) = C1(u) -£1(v)+ (aujav,'yv-'yu)

-

2 II u

- v 11

1

+ (u, u - v)1 + (f, v - u)o + (au/av, -1v - 7u)

_ -2Iju - v11i < 0 VV E H'(fl) by virtue of Green's formula (1.10) and also (1.1). Thus, we have verified that (u, (9 u/av) is a saddle point of X in Hl (0) X H+-1/2 (r). Remark 2.1. By using partial variations 8u X, 8µX with respect to the variables v, p, it is possible to equivalently characterize (2.2) by the following relations '(see Ekeland-Temam (1974)): (w, A) E H1(fl) x H+1/2(r), Vv E H1(0) 4=a (w, v) 1 - (A, v)

8 X (w, A; v) = 0

(f, v)o 8,4X(w, A;,u- A)

<

Vv E H1(f2),

(2.2')

0 Vp E H+1/2(r) (µ - A, yw) > 0 Vp E H+1/2(r).

The relations (2.2) or (2.2') will be called the mixed formulation of Pl. Similar formulation is available for P2.

1. Unilateral Problems for Scalar Functions

16

It is seen from the definition of the mixed variational formulation that the first component ranges over the whole space H'(f3), and not only over the set K1, as was the case with the primal variational formulation. The fact that the first component of the saddle point is eventually an element of K1 is a consequence of the mixed formulation itself; more precisely of the first inequality in (2.2) or (2.2'). We introduce one more example, which serves as a simpler model problem for the so-called one-sided boundary value friction problems. Such problems will be studied in more detail in Chapter 2. Let

f E L2(fl),

£3(V) = 2IIvII1 - (f, v)0 + J lvi ds,

where r = 812 is the boundary of a bounded domain 12 with Lipschitz boundary. Let us define the following problem:

find u E H1 (fl) such that

£3(u) < £3(v) Vv E H' (0).

(2.4)

Since L3 is strictly convex, weakly lower semicontinuous, and coercive in H1(fl), there exists a unique u satisfying (2.4). However, unlike ,C1, .C2, the functional C3 is not differentiable in H1(fl). In order to be able to interpret (2.4), we have to make use of the following generalization concerning theorem 1.1:

Theorem 2.2. Let V be a Banach space, let K C V be a nonempty convex closed subset, and let 3 = 31 + 32 : V i-- R where 31, 32 are convex, and, moreover 31 has its Gdteaux differential in V. Then, the assertions (i) u E K : 3(u) < 3(v) Vv E K;

(ii) D31(u,v-u)+32(v)- 32(u) >0 Vv EK are equivalent.

Proof. Can be found in C4a (1971). This theorem implies that u E H1(f1) is a solution of (2.4) if and only if

(u, v - u)1 + f(IvI - Iul)ds > (f, v - u)o Vv E H1(Il). Hence, using Green's formula and proceeding similarly to the case of problem Pi, i = 1, 2, we derive that u is a solution of the problem

-'L u + u = f a.e. in fI, Iau/,9vl < 1, (8u/(9v)u + Jul = 0,

a.e. on r.

1.1. Unilateral Boundary Value Problems

17

We omit the proof, since in Chapter 2 we will prove an analogous assertion for one-sided contact problems with friction. One of the main difficulties, particularly from the practical point of view,

is the fact that C3 is not differentiable in H1(1). A number of methods have been developed to avoid this difficulty. One of these methods is based on an idea similar to that exploited in the mixed formulation of problem p1.

We evidently have

flvlds = sup J 1u vds, A

where

A={,u EL2(I')1 11,1 <1 a.e. onl'} is a convex, closed, and bounded subset of L2(r). Therefore, we may formally write inf C3 (v) = inf sup X (v, µ),

H1(0)

where

H1(Q) x A I

HIM A

; R is given by the formula

(v) 14) = 2llvlli - (f,v)o+J jA vds.

r

Instead of problem (2.4), let us now consider the problem of finding a saddle point (w, A) of the functional X in H1(fl) x A: (w, A) E H1(O) x A,

V(v,.u) E H' (0) x A. (2.5) It is now possible to prove a result analogous to theorem 2.1. Indeed, the (w) µ) < X (w, A) < X (v, A)

following theorem holds:

Theorem 2.3. A pair (w, A) E H1(12) x A is a saddle point of X in Hl(fl) X A if and only if

w=u, A=-8u/8v, where u is a solution of problem (2.4).

Problem (2.5) will be called the mixed variational formulation of problem (2.4). We see that the problem of minimization of a nondifferentiable functional is reduced to the problem of finding a saddle point of the functional X, which already is differentiable in the variables v, /.c. This is one of the main advantages of the mixed formulation (2.5). An analogous approach will be used in Chapter 2 when solving the Signorini problem with friction.

1. Unilateral Problems for Scalar Functions

18

1.1.3

Solution of Primal Problems by the Finite Element Method and Error Bounds

Before we start to study the general theory of approximations of the primal formulations of variational inequalities of elliptic type, we shall present an example illustrating how to solve approximately problems of this type. In doing so, we shall restrict ourselves to problem P1.

1.1.31. Approximation of Problem P1 by the Finite Element Method. For the sake of simplicity, let us assume that fl C R 2 is a bounded domain with a polygonal boundary r. In what follows, a triangulation of the domain fl will mean any finite collection of closed triangles {Ti}iEJ such that

fl=UTi t

0

Ti nT3=

A

t

J

Vi,jET,i#j,

where A or 1 denote respectively, a common vertex or a whole common side of Ti, T2 .

Each triangulation will be characterized by two parameters: the longest

side h, and the least inner angle B among all the triangles of the given triangulation, which will be denoted by {Th,B}. Of course, we will consider not a single triangulation but rather a system of triangulations {Th,9} for h -+ 0+. For our purposes, we shall consider only regular systems of triangulations.

We say that a given system {Th,e}, h -+ 0+ is regular if there exists Bo > 0 independent of h such that 0 > Bo. In other words, when refining the partition of fl, the triangles of the given triangulation do not reduce to segments. Thus, any regular system of triangulations is actually characterized by the single parameter h > 0. In the case of a regular system of triangulations, therefore, we shall use the simpler symbol {Th} instead of {7h,01-

Remark 3.1. When solving problems with various types of boundary conditions prescribed on certain parts of the boundary r, we subject the system of triangulations to the additional assumption of being compatible

with the partition of r. That is, each point of r at which the boundary condition changes must be a vertex of a certain triangle Ti E Th.

Let {Th}, h -+ 0+ be a regular system of triangulations of (2. The nodes of a triangulation (i.e., vertices of Ti's) lying on r will be denoted by

1.1. Unilateral Boundary Value Problems

19

a1, ... a,.,,,(h). Each Th will be associated with a finite-dimensional space Vh of piece-wise linear functions:

Vh={vIvEC(n), VTEP1(T) VTETh}, where Pk(T) (k > 0 is an integer) denotes the set of all polynomials of degree at most k with the definition domain T. Let us further define

Kh={vhEVh Ivh(ai)>0 Vi=1,2,...m(h)}. It is easily seen that Kh is a convex closed subset of Vh, Kh C K Vh > 0. To obtain an approximation of P1, we will make use of a modification of the Ritz method, with which the reader is familiar as one of the possible methods of numerical solution of variational equations. Problem (1.6) will be replaced by the problem of minimization of L'1 in Kh. That is, we look for uh E Kh such that £1(uh) : C1(vh)

VVh E Kh.

A complex natural question arises, which is what is the relation between u and uh, and more precisely, whether I (u - uh 11 1 -+ 0, h -+ 0+, or possibly, what is the rate of this convergence if expressed in powers of h. We shall study these issues in the subsequent text.

Remark 3.2. Problem (3.1) is already suitable for computer realization. It is easily seen that dim Vh = M(h), where M(h) is the number of all nodes of the given triangulation Th. Let {(Pi}t_1 be such elements of Vh that (pi(A1) = 6i; (Kronecker's symbol). Then, evidently, {,Pt}Mlh) forms a basis of Vh. Moreover, {Ai}i_ll1h

M(h) vh(A1)cpi(x)

vh(x)

VVh E Vh.

i-1

Let T : Vh --+ RM(h) be the isomorphism given by

T vh = a = (a1, a2) ... , aM(h)) E RM(h)

VVh E Vh,

where the components of the vector a are the coordinates of Vh with respect to {cpi }; ih). By means of T we can identify the set Kh with a convex closed subset KM(h) C RM(h): KM(h) = T (Kh)

{a E RM(h) I at > 0 Vi E I},

1. Unilateral Problems for Scalar Functions

20

where I c {1,. .. , M(h) } is the set of indices, which in the given numeration, correspond to the vertices a; E I'. Problem (3.1) is then equivalent to the problem of finding a` = (ai, ... , am* (h)) E KM(h) such that

y(a*) < Y(a) Va E KM(h).

(3.2)

Here,

Y (a) = Li(T -la) =

2 (a, Act) RM(h) - (3, a)RM(h),

T-1 : R`u(h) -+ Vh is the mapping inverse to T, A = ((cPi,coj)1)M(hi is the stiffness matrix, 3 = ((f,'P1)o,

..., (f, PM(h))O) is the vector resulting

by integrating the right-hand side f, and

denotes the scalar product in RM(h). The desired solution uh E Kh is then determined from the formula

uh =

T_la*

M(h) j=1

Thus, (3.1) leads to the problem of quadratic programming seen in (3.2). At this point, let us briefly mention the algorithm which facilitates the approximate solution of minimization problems of the type (3.2). The definition of the set KM(h) implies that we can write

KM(h) = Kl x K2 x ... x KM(h), where K' = [0, +oo) for Z 'E I, K' = R1 provided %'V I. It follows from this formula for the convex set KM(h) that each of the variables a1, ... , aM(h) is subjected to at most one constraint (provided i E I), and, moreover, that this constraint involves no other variable. In this sense, the variables are

separated. In order to find the minimum of the quadratic function / on the convex closed set KM(h) of the above-mentioned type, it is advantageous to adopt the following generalization of the well-known SOR method (see Glowinski, Lions, and I'Wmolieres (1976)): choose ao E KM1 arbitrarily; if a(n) E KM is already known, we successively correct its individual components in accordance with the following scheme: a(n+1/2)

+>aija1 -

_ah y
a!,+ 1)

ail

Px: ((1 -w)a(n)

'Here we write M instead of M(h).

j>i

1.1. Unilateral Boundary Value Problems

21

and then set a("+1) = (,,i"+1),

Here, PK; stands for the

projection of R' to K' and w > 0 is the relaxation parameter, whose proper choice increases the rate convergence of a(') to a* (the minimum point of in KM). In the case just considered, we have l,.

PK: (a) =

if a E [0, +oo),

a

ifa<0

0

for i E I and PK; (a) = a bra E Rl

provided i 0 I. In the look mentioned above, the authors proved that for a quadratic function given by a symmetric, positive definite matrix A, the above algorithm for w E (0, 2) converges for an arbitrary choice a(0) E KM. The method just described will be called the superrelaxation method with an additional projection.

1.1.32. The General Theory of Approximations for Elliptic Inequalities. This section is devoted to the problems of approximation of variational inequalities from a general point of view. Let V be a real Hilbert space with a norm 11 11, V' the space of continuous

linear functionals over V and (f, v) the value of the functional f E V' at the point v E V. Let IJ : V H R1 be the general quadratic functional I, (v) = 2 a(v, v) - (f, v),

f E V',

(3.3)

where a : V x V H R1 is a continuous, V-elliptic, and symmetric bilinear form, i.e.,

3M = const > 0: Ja(u,v)l < MIJull

lull

Vu,v E V (continuity);

3a = const > 0 : a(v, v) > a11v[12 Vv E V (V-ellipticity);

a(u, v) = a(v, u)

du, v E V (symmetry).

(3.4) (3.5) (3.6)

Let K C V be a nonempty, convex, closed subset. Let us introduce the problem

find u E K such that Y(u) < y(v) Vv E K.

(P)

If (3.4)-(3.6) hold, then P has exactly one solution, since 1, satisfies all the assumptions of theorem 1.5. Moreover, theorem 1.1 implies that P is equivalent to the variational inequality

uEK: a(u, v - u) >(f, v - u) dvEK.

(P')

1. Unilateral Problems for Scalar Functions

22

Remark 3.3. If the bilinear form a is merely continuous and V-elliptic, then problem P' has exactly one solution (e.g., Glowinski, Lions, and Tremolieres (1976)).

If moreover, a is symmetric, then P' is equivalent to the problem of minimization of the quadratic functional 1Y on K.

Problem P or P' is hardly solvable in most cases; therefore, it must usually be replaced by another sequence of problems, which we are able to deal with algorithmically. To this end, we will use the same method we have sketched in the previous section for the approximation of problem Pl. Let {Kh}, h E (0, 1) be a collection of nonempty, convex, closed subsets of V. Also, in all cases to be considered, each Kh will be embedded into its own finite-dimensional space Vh, i.e., Kh is a convex and closed subset of Vh. Now we replace problem P by the sequence of problems

find uhEKh: Y(uh) !5 Y(Vh) Vvh E Kh.2

(Ph)

or equivalently,

find uhEKh: a(uh, Vh - Uh) > (f, vh - Uh)

VVh E Kh.

Remark 3.4. The method which replaces P by problems Ph is known in the literature as the Ritz method. If a fails to be symmetric, then Ph is an approximation of P' and we speak of the Galerkin method. In the subsequent text, if the fact that the form is symmetric or not is immaterial, we will use the term Ritz-Galerkin method, and the solution uh of problems Ph, Ph will be called the Ritz-Galerkin approximation of the solution u in Kh.

Definition 3.1. We say that the Ritz-Galerkin method (applied to P or P) converges, if (3.7) c(h) = Jlu - u'hll - 0, h , 0+. We will formulate sufficient conditions for {Kh}, h E (0, 1), to guarantee (3.7).

Theorem 3.1. Let a bilinear form a fulfill (3.4) and (3.5), and let the following assertions hold:

VvEK2vhEKh:Vh-sv, h-'0+ mV; vh E Kh, vh

v,

h --* 0+ (weakly) in V implies v E K.

(3.8) (3.9)

2Thus, Ph leads to the problem of generally nonlinear programming in a finite dimension.

1.1. Unilateral Boundary Value Problems

23

The Ritz-Galerkin method then converges.

Proof. For a symmetric bilinear form, the proof is found in Cea (1971). For a general bounded and V-elliptic form, the proof is found in Glowinski, Lions, and Tremolieres (1976).

Remark 3.5. Let K C K be another convex closed subset. If we know in advance that the solution u of problem P lies in k, then it suffices to consider the condition in V.

(3.8')

Remark 3.6. Generally, Kh need not be subsets of K. If Kh C K Vh E (0, 1), we say that Kh are internal approximations of K. In the opposite case we speak of external approximations of K.

Remark 3.7. If {Kh}, h E (0, 1) form an internal approximation of K, then (3.9) is automatically fulfilled by virtue of the weak closedness of K,

i.e., xn f x, n - oo, xn E K = x E K. To get a deeper insight into the matter, it is also useful to know the rate of convergence of uh to u. There are two approaches which provide this information, on the one hand, there exists the method of so-called one-sided approximations, which will be explained in detail in the next section, and

on the other hand there exists the method suggested for the first time by Falk. We will describe it now.

Lemma 3.1. The inequality allu - uh ll2 < a(u - uh, u - uh) _< Y, U - Vh) + (f, uh - v)

+a(uh - u, Vh - u) + a(u, v - uh) + a(u, vh - u)

(3.10)

holds for every v E K, Vh E Kh, h e (0, 1).

Proof. P and Ph imply that a(u,u) < (f, u - v) + a (u, v) Vv E K, a(uh, Uh)

(f, Uh - Vh) + a(uh, Vh)

VVh E Kh.

Hence,

- uh, u - uh)

= a(u, u) + a(uh, uh) - a(uh, u) - a(u, uh) < (f, u - V) + a(u, v) + (f, uh - Vh) + a(uh, uh) a(uh, u) - a(u, uh)

(f,U-Vh)+(f,uh-v)+a(u,v-uh) a(uh - u, Vh - u) + a(u, vh - u),

1. Unilateral Problems for Scalar Functions

24

which is the same as (3.10).

Remark 3.8. If Kh c K Vh E (0, 1), then we can write (3.10) in a simpler form, namely

allu-uh112
(3.10')

which follows from (3.10) by setting v = Uh E K.

Remark 3.9. Let

I

I be a seminorm in V, and let the bilinear form a

merely fulfill 3a = const > 0 : a(v,v) > aIvI2

Vv E V

(3.5')

instead of (3.5). In this case, a solution of problems P, Ph generally need not exist, and even if it does exist, it need not be unique (the reason why this is so is concretely illustrated in section 1.1.6. Nevertheless, if a solution of P, Ph exists, then it suffices to replace the symbol 11 11 in (3.10), (3.10') by the symbol I I in order to obtain a bound for the seminorm of the error uh - U. We can analogously generalize theorem 3.1 on the convergence of the approximate solutions Uh to u. Let us formulate the variant, which will be used in the subsequent text. Let V, H be two Hilbert spaces. Let V c H and let the embedding be totally continuous. Let -

IIv112=Iv12+IIvIIH

where II

- IIH

VV EV,

denotes the norm in H. If / is coercive on UhE(o,1)Kh, that

is,

hl1o Y(vh) = +00, IIvh!I -- +00,

vh E Kh,

and if (3.5'), (3.8), and (3.9) hold, then

provided u, uh exist and u is uniquely determined.

1.1.33. A Priori Bound for Problem P1. Let us now go back to the approximation of P1 in the form described in section 1.1.31. We will show how to apply the results of the previous section to this particular case.

1.1. Unilateral Boundary Value Problems

25

Theorem 3.2. Let a solution u E H2(f2) n K,u c W1,°° (r)3 satisfy au/av E L°°(r), and let the set of points from r at which u changes from u = 0 to u > 0 be finite. Then llu - uhlll < c(u)h,

h --+ 0+,

(3.11)

where c(u) is a positive constant that depends only on u.4

Proof. As Kh C K Vh E (0, 1), we can use the relation (3.10') to obtain the required bound. Let us set Vh = rhu, where rhu denotes the piecewise linear Lagrange interpolation of the function u. As

rhu(a,) = u(a,) > 0 bpi = 1,..., we have rhu E Kh. Using Green's formula and (1.1), we obtain

a(u, rhu - u)

= (u, rhu - u)1

u+u,rhu-u)o+fr (f, rhu - u)o +

C7v(rhu-u)ds

Jr av (rhu - u) ds.

Substituting into the right-hand side of (3.10') and using classical interpolation properties of the function rhu, we conclude that

llu-Uhll1

= (u-uh,u-uh)1

(uh - u, rhu - u)1 + Jr av (rhu - u)ds 2lluh - ulll + 21lrhu - ulll + fr av(rhu - u)ds 2 lluh - ull1 + ch2lu12 p +

fr

av(rhu - u)ds.

(3.12)

It remains to estimate the integral over r. We divide the boundary points into two groups:

ro={xEr:u(x)=0},

'Let r = U!"_1A,A,+1 be the boundary of a polygonal domain U. We define: u E i.e., the trace u on A; A,+1 and its first derivative in the direction of the side A,A,+1 is a bounded measurable function of one variable (the parameter of the side A,A,+1) 41n the sequel, c denotes a general positive constant, which may assume different W 1.O° (r) # UI A. A, E W 1,oo (A,A,+1) Vi =

values at different points of our exposition. If we want to explicitly express its dependence on parameters t 1, t2, .. , t we write c = c(t1, l2, .. , t,).

1. Unilateral Problems for Scalar Functions

26

r+ = (x E r : u(x) > o). Let alai+1 C ro. Then, rhu(ai) = u(ai) = rhu(ai+l) = u(ai+1) = 0, and consequently rhu - 0 on alai+1. Hence,

au

(rhu - u)ds = 0.

(3.13)

If alai+1 c r, then (1.2) implies that au/am =_ 0 on alai+1, and (3.13) again holds. Let T be the set of all alai+l whose interiors contain points from both r+, ro. Making use of the assumptions on the smoothness of u, au/av on the boundary, and of the interpolation properties of rhulr we obtain (rhu - u)ds 09V

< <

11

av II L-(a:a;+l)II rhu - UIIL-(aia.+l)h au

ch2Iul2,a.a.+l II

am

IIL 0(a:a:+l).

According to the assumptions of the theorem, the number of elements of the set T is bounded from above independently of h. Hence, am

KU - u)ds = a.a.+lET

J a,+l ,

am

(rhu - u)ds = 0(h2).

This together with (3.12) completes the proof of the theorem.

O

The rate of convergence 0(h) is obtained under comparatively strong assumptions on the smoothness of u. However, the actual situation is usually such that assumptions analogous to those formulated in the preceding theorem are unrealistic. Therefore we shall try as far as possible, to prove the convergence of Uh to u without additional assumptions on the smoothness of u. Naturally, we shall pay for it by obtaining no information about the rate of convergence. Let us again use problem P, to illustrate. Theorem S.S. For an arbitrary regular system of triangulations {Th}, h 0+, we have

IIu-uhIIl-'0,

h-+0+.

Proof. We now will verify the assumptions of theorem 3.1. Since Kh c K Vh E (0, 1), it is sufficient to verify (3.8). To this end we shall use the following auxiliary result, which is proven in Haslinger (1977):

C- (fl) n K = K,

(3.14)

1.1. Unilateral Boundary Value Problems

27

where C°° (11) is the set of all infinitely differentiable functions in 0, which together with their derivatives are continuously extensible to fl. The closure in (3.14) is taken in the norm of H1(Il). Let v E K be arbitrary. Then, (3.14) implies that there exists a sequence vn E K n CCO (0) such that v,, --+ v, n -* oo

in H1(fl).

(3.15)

To every function vn we can construct its piecewise linear Lagrange interpolation rhun E Kh. Then, llvn - rhvnlll < chlvn12,0.

(3.16)

The triangule inequality 11v - rhvn l l l 5 11V - vn l l i+ l l vn - rhvn it 1,

together with (3.15), (3.16) yields a sequence vh E Kh satisfying (3.8) (it suffices to set vh = rhvn, where n is sufficiently great while h > 0 is sufficiently small).

1.1.4

Solution of Dual Problems by the Finite Element Method and Error Bounds

In dealing with dual problems we have to distinguish problems Pl and P2, since they essentially differ in the construction of approximations of admissible functions. Indeed, problem P1, for equations with absolute terms, can be dually formulated in terms of the equivalent variational problem (1.18) in the set UO, in which we do not require-in contradistinction to U2-any differential equation to be fulfilled.

1.1.41. Problems with Absolute Terms. We shall start with the dual equivalent formulation (1.18). Let the domain fl c R2 have a polygonal boundary. Let us consider a triangulation Th and a space of piecewise linear finite elements Vh = {v i v E C(O),

viT E P1(T) VT c 7h}.

We introduce an approximation of the set 11o by

Uoh=uon[Vh12As [Vh]2 C [H'(0)12 c H(div, 0), we evidently have Uoh = {q e [Vh]2

1 q v> 0 on F}.

1. Unilateral Problems for Scalar Functions

28

A vector function qh E Uoh will be called an approximation of the dual problem (1.18), provided I(qh) <_ I(q)

Vq E Uoh.

(4.1)

Since Uoh is a convex and closed subset in H(div, Cl), problem (4.1) has a unique solution (see theorem 1.5).

Algorithm for Solution of the Problem (4.1). Let {w', w2, ... , wN} be a basis of the space [Vh]2. Then, the equivalence N {q=iwui

gEUoh

By>0

i=1

evidently holds, where B is a (p x N)-matrix with a rank p, p < N. The condition By > 0 results from the boundary condition q-v > 0 on r. Notice that, by virtue of the fact that q v is linear on each side of the polygonal boundary, it is sufficient to fulfill this condition solely at the nodes of the triangulation Th. Thus, problem (4.1) can be written in the form

3(y)= 1YTAy-yTb=min in the set Y = {y E RN, By > 0}.

(4.2)

where A is the Gramm matrix and b a fixed vector. Problem (4.2) is a problem of quadratic programming. It can be solved, for example, in the following way: 10 At the nodes of the triangulation on r, we apply a local transformation of the form Zi

= yi

zi+1

= yiV1 +Y3+1v2

Zi

= yi v1 + Yi+1V2

Zi+1

= yi+1

for v2 54 0,

for vi # 0,

setting Zi = yi v1(1) + yi+1v2(1)

Zi+1 = yivi(2) + yi+iY2(2) if the node coincides with a vertex of r with normals v(i), j = 1, 2. At the nodes inside fl we set Zk = yk

1.1. Unilateral Boundary Value Problems

29

Altogether, we have a substitution

Ry=z, with a regular matrix R, where

By>0BR-'z>0, and BR-1 has only one nonzero element in each row. Instead of problem (4.2) we now have the problem

G(z) = 3(R-'z) = min in the set

Z = {z E RN, BR-1 > 01,

which we are able to effectively solve, for example, by the superrelaxation method with an additional projection (see section 1.1.3).

1.1.411. A Priori Error Bounds. In order to make an a priori estimate of the error q° - qh, we will use the method of so called one-sided approximations (see Mosco and Strang (1974)).

Lemma 4.1. Let Y(v) be a real-valued functional defined on a convex closed subset M of a Banach reflexive space X. Let us assume that lY has both the first and the second differential (in a Gateaux sense) and that there exist positive constants ao, c such that

aollzll2 < D2Y(u;z,z) < cllzll2 Vu E M, dz E X.

(4.3)

Let Mh C M be a convex closed subset. Denote by u and Uh the elements minimizing / in M and Mh, respectively. Let us assume that there exists wh E Mh such that 2u - Wh E M. Then 1/2

Ilu-uhll <

(-

Ilu - whIl.

(4.4)

Proof. The Taylor theorem implies that there exists 8 E (0, 1) such that Y (uh) = Y (u) + D y (u, uh - u) +

2

D2.7 (u + O(uh - u); uh - u, uh - u)

> /(u') + 1aoIluh - ull2, as

DtY(u,uh-u)>0.

1. Unilateral Problems for Scalar Functions

30

For any v E Mh we may write Y (v)

= y (u) + Dy (u, v - u) + 2 D2y (u + 91(v - u); v - u, v - u) y(uh).

Substituting V = Wh and v = 2u - wh into the condition Dy (u, v - u) > 0, we obtain Dy (u, wh - u) = 0, hence

y(uh)

<_

Y(wh)

y(u) + 2 D2y(u + B2(wh - u); wh - u, wh - u) Y(u) + Zcllwh -

U112.

Combining (4.5) with this inequality we obtain (4.4). Let us apply lemma 1.2 with y I, M = 110, Mh = 11oh, X = H(div, ft), ceo = c = 1. If we find a vector th E 11oh such that 2q - th c 110, then in virtue of (4.4) we shall have IIq° - ghIIH(div,n) < IIq° - thIIH(div,n).

(4.6)

An answer to this question is given in the following theorem.

Theorem 4.1. Let us assume that q° E IH2(fl)I2 and q° - v E H2(rm), rn = 1, 2,... , m, where r n denotes any one of the sides of the polygonal boundary r. Then there is th E Uoh such that

a.e. on r

(4.7)

and, if the system of triangulations {Th},O < h < h0i is (o:,/3)-regular's then 2

m

IIq° - thIIH(div,n) < Ch E IIq°112 + E IIq - vll2,rj=1

m=1

We shall need two lemmas to prove the theorem. 5A system of triangulations {Th}, 0 < h < ho will be called (a, g)-regular, if positive

constants a,0 exist independent of h and such that (i) no inner angle among all the triangles is less than a (i.e., regularity), and (ii), the ratio of any two sides in Th is less than 0.

1.1. Unilateral Boundary Value Problems

31

Lemma 4.2 (One-Sided Approximation of the Flow on the Boundary). Let q° satisfy the assumptions of Theorem 4.1. Then there exist piecewise linear functions o c C(rm) with vertices determined by the triangulations Th, such that

on1'n Vm,

0<0h

(4.9)

m

2 Ilgi v-1hllr2 ,.