Tài liệu Scientific Computation Editorial Board pdf

The first version of Numerical Methods for Nonlinear Variational Problems was, in fact, part of a set of monographs on numerical mathe-matics published, in a short span of time, by the

Scientific Computation

Editorial Board

J.-J Chattot, Davis, CA, USA

P Colella, Berkeley, CA, USA

W E, Princeton, NJ, USA

R Glowinski, Houston, TX, USA

M Holt, Berkeley, CA, USA

Y Hussaini, Tallahassee, FL, USA

P Joly, Le Chesnay, France

H.B Keller, Pasadena, CA, USA

J.E Marsden, Pasadena, CA, USA

D.I Meiron, Pasadena, CA, USA

O Pironneau, Paris, France

A Quarteroni, Lausanne, Switzerland

and Politecnico of Milan, Italy

J Rappaz, Lausanne, Switzerland

R Rosner, Chicago, IL, USA

P Sagaut, Paris, France

J.H Seinfeld, Pasadena, CA, USA

A Szepessy, Stockholm, Sweden

M.F Wheeler, Austin, TX, USA

Roland Glowinski

Numerical Methods for Nonlinear

Variational Problems

With 82 Illustrations

123

Scientific Computation ISSN 1434-8322

Library of Congress Control Number: 2007942575

© 2008, 1984 Springer-Verlag Berlin Heidelberg

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When Herb Keller suggested, more than two years ago, that we update ourlectures held at the Tata Institute of Fundamental Research in 1977, and thenhave it published in the collection Springer Series in Computational Physics,

we thought, at first, that it would be an easy task Actually, we realized veryquickly that it would be more complicated than what it seemed at first glance,for several reasons:

1 The first version of Numerical Methods for Nonlinear Variational

Problems was, in fact, part of a set of monographs on numerical

mathe-matics published, in a short span of time, by the Tata Institute of mental Research in its well-known series Lectures on Mathematics andPhysics; as might be expected, the first version systematically used thematerial of the above monographs, this being particularly true for

Funda-Lectures on the Finite Element Method by P G Ciarlet and Funda-Lectures on Optimization—Theory and Algorithms by J Cea This second version

had to be more self-contained This necessity led to some minor additions

in Chapters I-IV of the original version, and to the introduction of achapter (namely, Chapter Y of this book) on relaxation methods, sincethese methods play an important role in various parts of this book Forthe same reasons we decided to add an appendix (Appendix I) introducinglinear variational problems and their approximation, since many of themethods discussed in this book try to reduce the solution of a nonlinearproblem to a succession of linear ones (this is true for Newton's method,but also for the augmented Lagrangian, preconditioned conjugategradient, alternating-direction methods, etc., discussed in several parts

of this book)

2 Significant progress has been achieved these last years in computationalfluid dynamics, using finite element methods It was clear to us that thissecond version had to include some of the methods and results whoseefficiency has been proved in the above important applied field This led

to Chapter VII, which completes and updates Chapter VI of the originalversion, and in which approximation and solution methods for someimportant problems in fluid dynamics are discussed, such as transonicflows for compressible inviscid fluids and the Navier-Stokes equations

for incompressible viscous fluids Like the original version, the main goal

of this book is to serve as an introduction to the study of nonlinearvariational problems, and also to provide tools which may be used fortheir numerical solution We sincerely believe that many of the methodsdiscussed in this book will be helpful to those physicists, engineers,and applied mathematicians who are concerned with the solution ofnonlinear problems involving differential operators Actually this belief

is supported by the fact that some of the methods discussed in this bookare currently used for the solution of nonlinear problems of industrialinterest in France and elsewhere (the last illustrations of the book repre-sent a typical example of such situations)

The numerical integration of nonlinear hyperbolic problems has not beenconsidered in this book; a good justification for this omission is that thissubject is in the midst of an important evolution at the moment, with manytalented people concentrating on it, and we think that several more years will

be needed in order to obtain a clear view of the situation and to see whichmethods take a definitive lead, particularly for the solution of multidimen-sional problems

Let us now briefly describe the content of the book

Chapters I and II are concerned with elliptic variational inequalities (EVI),more precisely with their approximation (mostly by finite element methods)and their iterative solution Several examples, originating from continuummechanics, illustrate the methods which are described in these two chapters.Chapter III is an introduction to the approximation of parabolic variationalinequalities (PVI); in addition, we discuss in some detail a particular PVIrelated to the unsteady flow of some viscous plastic media (Bingham fluids) in acylindrical pipe

In Chapter IV we show how variational inequality concepts and methodsmay be useful in studying some nonlinear boundary-value problems which can

be reduced to nonlinear variational equations

In Chapters V and VI we discuss the iterative solution of some variationalproblems whose very specific structure allows their solution by relaxationmethods (Chapter V) and by decomposition-coordination methods via aug-mented Lagrangians (Chapter VI); several iterative methods are describedand illustrated with examples taken mostly from mechanics

Chapter VII is mainly concerned with the numerical solution of the fullpotential equation governing transonic potential flows of compressible inviscidfluids, and of the Navier-Stokes equations for incompressible viscous fluids

We discuss the approximation of the above nonlinear fluid flow problems

by finite element methods, and also iterative methods of solution of theapproximate problems by nonlinear least-squares and preconditionedconjugate gradient algorithms In Chapter VII we also emphasize the solution

of the Stokes problem by either direct or iterative methods The results of

numerical experiments illustrate the possibilities of the solution methodsdiscussed in Chapter VII, which also contains an introduction to arc-length-continuation methods (H B Keller) for solving nonlinear boundary-value

problems with multiple solutions.

As already mentioned, Appendix I is an introduction to the theory andnumerical analysis of linear variational problems, and one may find in itdetails (some being practical) about the finite element solution of suchimportant boundary-value problems, like those of Dirichlet, Neumann,Fourier, and others

In Appendix II we describe a finite element method with upwinding which may be helpful for solving elliptic boundary-value problems with large

first-order terms.

Finally, Appendix III, which contains various information and resultsuseful for the practical solution of the Navier-Stokes equations, is a comple-ment to Chapter VII, Sec 5 (Actually the reader interested in computationalfluid mechanics will find much useful theoretical and practical informationabout the numerical solution of fluid flow problems—Navier-Stokes equa-

tions, in particular—in the following books: Implementation of Finite Element

Methods for Navier-Stokes Equations by F Thomasset, and Computational Methods for Fluid Flow by R Peyret and T D Taylor, both published in

the Springer Series in Computational Physics.)

Exercises (without answers) have been scattered throughout the text;they are of varying degrees of difficulty, and while some of them are directapplications of the material in this book, many of them give the interestedreader or student the opportunity to prove by him- or herself either some tech-nical results used elsewhere in the text, or results which complete those ex-plicitly proved in the book

Concerning references, we have tried to include all those available to usand which we consider relevant to the topics treated in this book It is clear,however, that many significant references have been omitted (due to lack ofknowledge and/or organization of the author) Also we apologize in advance

to those authors whose contributions have not been mentioned or have notreceived the attention they deserve

Large portions of this book were written while the author was visiting thefollowing institutions: the Tata Institute of Fundamental Research (Bombayand Bangalore), Stanford University, the University of Texas at Austin, theMathematical Research Center of the University of Wisconsin at Madison,and the California Institute of Technology We would like to express specialthanks to K G Ramanathan, G H Golub, J Oliger, J T Oden, J H Nohel,and H B Keller, for their kind hospitality and the facilities provided for usduring our visits

We would also like to thank C Baiocchi, P Belayche, J P Benque, M.Bercovier, H Beresticky, J M Boisserie, H Brezis, F Brezzi, J Cea, T F.Chan, P G Ciarlet, G Duvaut, M Fortin, D Gabay, A Jameson, G

Labadie, C Lemarechal, P Le Tallec, P L Lions, B Mercier, F Mignot,

C S Moravetz, F Murat, J C Nedelec, J T Oden, S Osher, R Peyret,

J P Puel, P A Raviart, G Strang, L Tartar, R Temam, R Tremolieres,

V Girault, and O Widlund, whose collaboration and/or comments andsuggestions were essential for many of the results presented here

We also thank F Angrand, D Begis, M Bernadou, J F Bourgat, M O.Bristeau, A Dervieux, M Goursat, F Hetch, A Marrocco, O Pironneau,

L Reinhart, and F Thomasset, whose permanent and friendly collaborationwith the author at INRIA produced a large number of the methods andresults discussed in this book

Thanks are due to P Bohn, B Dimoyat, Q V Dinh, B Mantel, J Periaux,

P Perrier, and G Poirier from Avions Marcel Dassault/Breguet Aviation,whose faith, enthusiasm, and friendship made (and still make) our collabor-ation so exciting, who showed us the essence of a real-life problem, and whoinspired us (and still do) to improve the existing solution methods or todiscover new ones

We are grateful to the Direction des Recherches et Etudes Techniques(D.R.E.T.), whose support was essential to our researches on computationalfluid dynamics

We thank Mrs Francoise Weber, from INRIA, for her beautiful typing ofthe manuscript, and for the preparation of some of the figures in this book, andMrs Frederika Parlett for proofreading portions of the manuscript

Finally, we would like to express our gratitude to Professors W Beiglbockand H B Keller, who accepted this book for publication in the Springer Series

in Computational Physics, and to Professor J L Lions who introduced us tovariational methods in applied mathematics and who constantly supportedour research in this field

September 1982

Some Preliminary Comments xiv

3 Existence and Uniqueness Results for EVI of the First Kind 3

4 Existence and Uniqueness Results for EVI of the Second Kind 5

5 Internal Approximation of EVI of the First Kind 8

6 Internal Approximation of EVI of the Second K i n d 12

7 Penalty Solution of Elliptic Variational Inequalities of the First Kind 15

8 References 26

CHAPTER II

Application of the Finite Element Method to the Approximation of

Some Second-Order EVI 27

1 Introduction 27

2 An Example of EVI of the First Kind: The Obstacle Problem 27

3 A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion

Problem 41

4 A Third Example of EVI of the First Kind: A Simplified Signorini Problem 56

5 An Example of EVI of the Second Kind: A Simplified Friction Problem 68

6 A Second Example of EVI of the Second Kind: The Flow of a Viscous

Plastic Fluid in a Pipe 78

7 On Some Useful Formulae 96

CHAPTER III

On the Approximation of Parabolic Variational Inequalities 98

1 Introduction: References 98

2 Formulation and Statement of the Main Results 98

3 Numerical Schemes for Parabolic Linear Equations 99

4 Approximation of PVI of the First Kind 101

5 Approximation of PVI of the Second Kind 103

6 Application to a Specific Example: Time-Dependent Flow of a Bingham

Fluid in a Cylindrical Pipe 104 CHAPTER IV

Applications of Elliptic Variational Inequality Methods to the Solution

of Some Nonlinear Elliptic Equations 110

2 Some Basic Results of Convex Analysis 140

3 Relaxation Methods for Convex Functionals: Finite-Dimensional Case 142

4 Block Relaxation Methods 151

5 Constrained Minimization of Quadratic Functionals in Hilbert Spaces by

Under and Over-Relaxation Methods: Application 152

6 Solution of Systems of Nonlinear Equations by Relaxation Methods 163

CHAPTER VI

Decomposition-Coordination Methods by Augmented Lagrangian:

Applications 166

1 Introduction 166

2 Properties of (P) and of the Saddle Points of i ? and i ? , 168

3 Description of the Algorithms 170

2 Least-Squares Solution of Finite-Dimensional Systems of Equations 1 9 5

3 Least-Squares Solution of a Nonlinear Dirichlet Model Problem 198

4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 211

5 Numerical Solution of the Navier-Stokes Equations for Incompressible

Viscous Fluids by Least-Squares and Finite Element Methods 244

APPENDIX I

A Brief Introduction to Linear Variational Problems 321

1 Introduction 321

2 A Family of Linear Variational Problems 321

3 Internal Approximation of Problem (P) 326

4 Application to the Solution of Elliptic Problems for Partial Differential

Operators 330

5 Further Comments: Conclusion 397

APPENDIX II

A Finite Element Method with Upwinding for Second-Order Problems

with Large First- Order Terms 399

1 Introduction 399

2 The Model Problem 399

3 A Centered Finite Element Approximation 400

4 A Finite Element Approximation with Upwinding 400

5 On the Solution of the Linear System Obtained by Upwinding 404

2 Finite Element Approximation of the Boundary Condition u = g o n F i f g # 0 415

3 Some Comments On the Numerical Treatment of the Nonlinear Term (u • V)u 416

4 Further Comments on the Boundary Conditions 417

5 Decomposition Properties of the Continuous and Discrete Stokes Problems

of Sec 4 Application to Their Numerical Solution 425

6 Further Comments 430

Some Illustrations from an Industrial Application 431 Bibliography 435 Glossary of Symbols 455 Author Index 463 Subject Index 467

To those who might think our approach is too mathematical for a bookpublished in a collection oriented towards computational physics, we wouldlike to say that many of the methods discussed here are used by engineers inindustry for solving practical problems, and that, in our opinion, mastery

of most of the tools of functional analysis used here is not too difficult foranyone with a reasonable background in applied mathematics In fact, most

of the time the choice of the functional spaces used for the formulation andthe solution of a given problem is not at all artificial, but is based on well-known physical principles, such as energy conservation, the virtual workprinciple, and others

From a computational point of view, a proper choice of the functional spaces

used to formulate a problem will suggest, for example, what would be the

"good" finite element spaces to approximate it and also the good ditioning techniques for the iterative solution of the corresponding approxi-mate problem

precon-computer or the gears of a cycle transmission as he does at the top of a mountain or

in the petals of a flower."

Robert M Pirsig

Zen and the Art of Motorcycle Maintenance,

William Morrow and Company Inc., New York, 1974

"En tennis comme en science, certains ecarts minimes a la source d'un phenomene peuvent parfois provoquer d'enormes differences dans les ejfets qu'ils provoquent." *

Phillipe Bouin,

UEquipe, Paris, 2-26-1981

* " I n tennis, as in science, certain tiny gaps at the very beginning of a phenomenon can occasionally produce enormous differences in the ensuing results."

Generalities on Elliptic Variational Inequalities

and on Their Approximation

1 Introduction

An important and very useful class of nonlinear problems arising from

mechanics, physics, etc consists of the so-called variational inequalities.

We consider mainly the following two types of variational inequalities,namely:

1 elliptic variational inequalities (EVI),

2 parabolic variational inequalities (PVI)

In this chapter (following Lions and Stampacchia [1]), we shall restrict our

attention to the study of the existence, uniqueness, and approximation of the

solution of EVI (PVI will be considered in Chapter III)

2 Functional Context

In this section we consider two classes of EVI, namely EVI of the first kind and EVI of the second kind.

2.1 Notation

• V: real Hilbert space with scalar product (•, •) and associated norm || • ||,

• V*: the dual space of V,

• a(-,-): V x V -» U is a bilinear, continuous and V-elliptic form on V x V.

A bilinear form a{-, •) is said to be V-elliptic if there exists a positive constant

a such that a(v, v) > oc\\v\\ 2 , V v e V.

In general we do not assume a{-, •) to be symmetric, since in some

applica-tions nonsymmetric bilinear forms may occur naturally (see, for instance,Comincioli [1])

• L: V -> U continuous, linear functional,

• K is a closed convex nonempty subset of V,

• j(-): V -* U = U \J {ao} is a convex lower semicontinuous (l.s.c.) and

proper functional (;'(•) is proper ifj(v) > — oo, V v e V and j ^ + oo).

2.2 EVI of the first kind

Find u e V such that u is a solution of the problem

2.3 EVI of the second kind

Find u e V such that u is a solution of the problem

2.4 Remarks

Remark 2.1 The cases considered above are the simplest and most important.

In Bensoussan and Lions [1] some generalization of problem ( P J called

quasivariational inequalities (QVI) are considered, which arises, for instance,

from decision sciences A typical problem of QVI is:

Find ue V such that

a(u, v - u) > L(v — u), V P G K(U), U e K(u),

where v -> K(v) is a family of closed convex nonempty subsets of V.

Remark 2.2 If K = V and j = 0, then problems ( P J and (P2) reduce to theclassical variational equation

a(u, v) = L(v), VveV, ueV.

Remark 2.3 The distinction between (Px) and (P2) is artificial, since (Px)can be considered to be a particular case of (P2) by replacing _/(•) in (P2) by

the indicator functional I K of K denned by

O tiveK,

+ 00 if v $ K.

Even though ( P J is a particular case of (P2), it is worthwhile to consider(Pj) directly because in most cases it arises naturally, and doing so we willobtain geometrical insight into the problem

EXERCISE 2.1 Prove that I K is a convex l.s.c and proper functional

EXERCISE 2.2 Show that ( P ^ is equivalent to the problem of finding ue V such that a(u, v — u) + I (v) — I (u) > L{v - U ) , V P G F

3 Existence and Uniqueness Results for EVI of the First Kind

3.1 A theorem of existence and uniqueness

Theorem 3.1 (Lions and Stampacchia [1]) The problem ( Pt) has a unique solution.

PROOF We first prove the uniqueness and then the existence.

(1) Uniqueness Let u x and u 2 be solutions of ( P ^ We then have

a(u u v — u t ) > L(v — «i), VveK, u x e K, (3.1) a(u 2 , v-u 2 )> L(v - u 2 ), VneiC, u 2 e K (3.2)

Taking v = u 2 in (3.1), v = u t in (3.2) and adding, we obtain, by using the F-ellipticity

a | | « 2 - " i l l 2 ^ a ( " 2 - « i , «2 - « i ) ^ 0,

which proves that u t = u 2 since a > 0.

(2) Existence We use a generalization of the proof used by Ciarlet [ l ] - [ 3 ] , for example, for proving the Lax-Milgram lemma, i.e., we will reduce the problem (Pj) to a fixed- point problem.

By the Riesz representation theorem for Hilbert spaces, there exist A e JS?(F, V) (A = A' if a{-, •) is symmetric) and / e V such that

(Au, v) = a(u, v), V u , o e F a n d L(v) = (/, v), V c e V ( 3 3 )

T h e n t h e p r o b l e m ( P i ) i s e q u i v a l e n t t o finding u e V s u c h t h a t

(u - p(Au - I) - u, v - u) < 0, V D E X , ueK, p > 0 (3.4)

This is equivalent to finding u such that

u = P K (u — p(Au — /)) for some p > 0, (3.5)

where P K denotes the projection operator from F to K in the || • |j norm Consider the mapping W p : V -* V defined by

3.2 Remarks

Remark 3.1 If K = V, Theorem 3.1 reduces to Lax-Milgram lemma (see

Ciarlet [ l ] - [ 3 ] )

Remark 3.2 If a(-, •) is symmetric, then Theorem 3.1 can be proved using

optimization methods (see Cea [1], [2]); such a proof is sketched below

Let J: V -> R be defined by

J(v) = Hv, v) - L(v) (3.8)

Then

(i) l i m |M H + 00 J(y) = +co

since J(v) = \a{v, v) - L(v) > (a/2)|M|2 - ||L|| \\v\\.

(ii) J is strictly convex.

Since L is linear, to prove the strict convexity of J it suffices to prove that

the functional

v -> a(v, v)

is strictly convex Let 0 < t < 1 and u,veV with u=£v; then 0 < a(v — u, v — u)

= a(u, u) + a(v, v) — 2a(u, v) Hence we have

2a(u, v) < a(u, u) + a(v, v) (3.9)

Therefore v ->• a(v, v) is strictly convex.

(iii) Since a{-, •) and L are continuous, J is continuous.

From these properties of J and standard results of optimization theory

(cf Cea [1], [2], Lions [4], Ekeland and Temam [1]), it follows that the

minimization problem of finding u such that

J(u) < J(v), \/veK, ueK (n)

has a unique solution Therefore (n) is equivalent to the problem of finding u

such that

(J'(u), v - u) > 0, VveK, ueK, (3.11)

where J'(u) is the Gateaux derivative of J at u Since (J'(u), v) = a(u, v) — L(v),

we see that ( P J and (n) are equivalent if a( •, •) is symmetric.

EXERCISE 3.1 Prove that (J'(u), v) = a(u, v) — L(v), V«, v e V and hence deduce that J'(u) = Au — /, V ue V.

Remark 3.3 The proof of Theorem 3.1 gives a natural algorithm for solving

(Pt) since v -»P K (v — p(Av — /)) is a contraction mapping for 0 < p < 2a/\\A\\ 2

Hence we can use the following algorithm to find u:

Let u° e V, arbitrarily given, (3.12) then for n > 0, assuming that u" is known, define u" +1 by

Then u" ->u strongly in V, where u is the solution of ( P ^ In practice it is not easy to calculate / and A unless V = V* To project over K may be as

difficult as solving ( P J In general this method cannot be used for computing

the solution of ( P ^ if K =£ V (at least not so directly).

We observe that if a( : , •) is symmetric then J'(u) = Au - I and hence (3.13)

becomes

This method is known as the gradient-projection method (with constant step p).

4 Existence and Uniqueness Results for EVI of the Second Kind

Theorem 4.1 (Lions and Stampacchia [1]) Problem ( P2) has a unique solution.

PROOF AS in Theorem 3.1, we shall first prove uniqueness and then existence.

(1) Uniqueness Let u x and u 2 be two solutions of (P 2 ); we then have

a(u u v - » , ) + j(v) - j(ui) > L(v - « , ) , V c e F , u t eV, (4.1)

a(u 2 , v - u 2 ) + j(v) - j(u z ) > L(v - u 2 ), V c e F , u 2 e V (4.2)

Since /(•) is a proper functional, there exists v o eV such that — oo < j(v 0 ) < oo.

Hence, for i = 1, 2,

- oo < X«i) < j(v 0 ) - L(v 0 ~ ">) + a{u h v 0 - u t ) (4.3) This shows that j(u t ) is finite for i — 1,2 Hence, by taking v = u 2 in (4.1), v = u l in (4.2), and adding, we obtain

a|[«i - u 2 \\ 2 < a{u x - u 2 , «i - u 2 ) < 0 (4.4) Hence u = u

(2) Existence For each ueV and p > 0 we associate a problem (n") of type (P2 ) defined as follows.

Find w e V such that

(w, v — w) + pj(v) — pj(w) >{u,v — w)

+ pL(v - w) - pa(u, v - w), V c e F , weV (4.5) (n p )

The advantage of considering this problem instead of problem (P 2 ) is that the bilinear

form associated with (n p ) is the inner product of V which is symmetric.

Let us first assume that (TI") has a unique solution for all u e V and p > 0 For each p define the mapping f p : V -* V by f p (u) = w, where w is the unique solution of (n").

We shall show that f p is a uniformly strict contraction mapping for suitably chosen p Let u u u 2 e V and wf = //«;), i = 1, 2 Since j(-) is proper we haveX«i) finite which

can be proved as in (4.3) Therefore we have

(W 1; W 2 - Wt) + pj(w 2 ) - pj(Wi) > («!, W2 - W,)

+ pL(w 2 - Wj) - pa(«!, w 2 - w j , (4.6)

(w 2 , w t - w 2 ) + p;(wi) - p;(w 2 ) > (« 2 wi - w 2 )

+ pL{w^ - w 2 ) - pa(u 2 , Wj - w2 ) (4.7) Adding these inequalities, we obtain

< ((/ - pA)(u 2 - «i), w2 - Wj)

< ! | / - p A | ! ! [ « 2 - w 1 | | | | w 2 - w 1 | | (4.8) Hence

It is easy to show that ||/ — pA\\ < 1 if 0 < p < 2a/\\A\\ 2 This proves that f p is

uni-formly a strict contracting mapping and hence has a unique fixed point u This u turns out

to be the solution of (P 2) since f p (u) = M implies (M, V - u) + pj(v) - pj(u) >{u,v - u) + pL(v — u) — pa(u, v — u), V v e V Therefore

a(u, v-u)+ j(v) - ; ( « ) > Up - «), V o e K (4.9)

The existence and uniqueness of the problem {n up ) follows from the following

lemma.

Lemma 4.1 Let b:Vx V ->• U be a symmetric continuous bilinear V-elliptic

form with V-ellipticity constant p Let LeV* and j : V -» U be a convex, l.s.c proper functional Let J(v) = jb(v, v) + j(v) — L{v) Then the minimization problem (n):

Find u such that

J(u) < J(v), V v e V, u e V (n)

has a unique solution which is characterized by

b(u, v - u) + j(v) - j(u) >L(v - u), V v e V, ueV (4.10) PROOF (1) Existence and uniqueness ofu: Since b(v, v) is strictly convex, j is convex, and

L is linear, we have J strictly convex; J is l.s.c because i>(-, •) and L are continuous and j

(2) Necessity of (4.10): Let 0 < t < 1 Let u be the solution of {%) Then for all v e V

we get

0 < (J' 0 (u), i > - « ) + j(v) - j(u), VveV (4.15)

Since b(-, •) is symmetric, we have

(J' 0 (v), w) = b(v, w) - L(w), V t , w e K (4.16)

From (4.15) and (4.16) we obtain

b(u, v - u) + j(v) - j(u) > L(v - u), V v e V.

This proves the necessity.

(3) Sufficiency of (4.10): Let u be a solution of (4.10); for v e V,

J{v) - J(u) = #b(v, v) - b(u, «)] + j(v) - j(u) - L(v - u) (4.17)

See also Ekeland and Temam [1].

By taking b(-, •) to be the inner product in V and replacingX^) and L(v) in Lemma 4.1

by pj(v) and (u, v) + pL(v) — pa(u,v), respectively, we get the solution for (n" p ) •

Remark 4.1 FromtheproofofTheorem4.1 we obtain an algorithm for solving

(P2) This algorithm is given by

u° e V, arbitrarily given, (4-20)

then for n > 0, u" known, we define M"+1 from u" as the solution of

at each iteration is then a problem of the same order of difficulty as that of the

original problem (actually, conditionning can be better provided that p has been conveniently chosen) If a(-, •) is not symmetrical the fact that (•, •) is

symmetric can also provide some simplification

5 Internal Approximation of EVI of the First Kind

5.1 Introduction

In this section we shall study the approximation of EVI of the first kind from

an abstract axiomatic point of view

5.2 The continuous problem

The assumptions on V, K, L, and a{-, •) are as in Sec 2 We are interested in

the approximation of

a(u, v-u)>L(v-u), VveK, ueK, (PJ

which has a unique solution by Theorem 3.1

5.3 The approximate problem

5.3.1 Approximation of V and K

We suppose that we are given a parameter h converging to 0 and a family

{V h } h of closed subspaces of V (In practice, the V h are finite dimensional and

the parameter h varies over a sequence) We are also given a family {K h } h of

closed convex nonempty subsets of V with K h a V h , V h (in general, we do not

assume K h a K) such that {K h } h satisfies the following two conditions:

(i) If {v h } h is such that v h e K h ,V h and {v H } h is bounded in V, then the weak cluster points of {v h } h belong to K.

(ii) There exists x c V, x = K and r h : % -*• K h such that limA^0 r h v = v

strongly in V, V v e

x-Remark 5.1 If K h a K, V h, then (i) is trivially satisfied because K is weakly

closed

Remark 5.2 f] h K h a K.

Remark 5.3 A useful variant of condition (ii) for r h is

(ii)' There exist a subset x<^V such that x = K and r h :x~* V h with the

property that for each vex, there exists h 0 = h o (v) with r h v e K h for all

h < h o (v) and limh^0 r h v = v strongly in V.

5.3.2 Approximation o/(P1)

The problem ( P:) is approximated by

a(u h , v h - u h ) > L(v h - u h ), V c j e K h , u h e K h (Plft )

Theorem5.1 (P lh ) has a unique solution.

PROOF In Theorem 3.1, taking V to be V h and K to be K h , we have the result.

Remark 5.4 In most cases it will be necessary to replace a(-, •) and L by

a h (-, •) and L h (usually defined, in practical cases, from a(-, •) and L by a

numerical integration procedure) Since there is nothing very new on that

matter compared to the classical linear case, we shall say nothing about thisproblem for which we refer to Ciarlet [1, Chapter 8], [2], [3].

5.4 Convergence results

Theorem 5.2 With the above assumptions on K and {K h } h , we have

lim,,^0 \\u h — u\\ v = 0 with u h the solution o / ( Pu) and u the solution of

PROOF For proving this kind of convergence result, we usually divide the proof into

three parts First we obtain a priori estimates for {u h } h , then the weak convergence of {u h } h , and finally with the help of the weak convergence, we will prove strong convergence.

(1) Estimates for u h We will now show that there exist two constants C t and C 2

independent of h such that

Since u h is the solution of (P l t ), we have

a(u h , v h - u h ) > L(v h - u h ), V v h e K h (5.2) i.e.,

Hence there exists a subsequence, say {u h }, such that u h converges to u* weakly in V.

By condition (i) on {K h } h , we have u* e K We will prove that u* is a solution of ( P J

Also we have

0 < a(u h - u*, u h - u*) < a(u hl , u h ) - a(u hi , u*) - a(u*, u h ) + a(u*, u*)

i.e.,

a(u ht , u*) + a(u*, u h ) - a(u*, u*) < a(u h , u h ).

By taking the limit, we obtain

/ii->0

From (5.6) and (5.7), we obtain

a(u*, u*) < lim inf a{u hi , u h ) < a(u*, v) — L(v — u*), V v e /.

u* = u is the unique solution Hence u is the only cluster point of {u h } h in the weak topology

of V Hence the whole {u h } h converges to u weakly.

(3) Strong convergence By F-ellipticity of a( •, •), we have

0 < cc\\u h - u\\ 2 < a(u h - u,u h - u) = a(u h , u h ) - a(u h , u) - a(u, uj + a(u, «), (5.10)

where u h is the solution of (P lh ) and u is the solution of (Pt) Since u h is the solution of (P lft) and r h v e K h for any vex, from (P1(1 ) we obtain

a(u h , u h ) < a(u h ,r h v) - L(r h v - u h ), V U G / (5.11) Since lim h ^ 0 u h = u weakly in V and limft ^ 0 r h v = v strongly in V [by condition (ii)],

we obtain (5.11) from (5.10), and after taking the limit, V v e x, we have

0 < a lim inf \\u h - u\\ 2 < a lim sup||uft - u\\ 2 < a(u, v - u) - L(v - u) (5.12)

By density and continuity, (5.12) also holds for VveK; then taking v = u in (5.12),

we obtain

lim | K - u\ 2 = 0, i.e., the strong convergence •

Remark 5.5 Error estimates for the EVI of the first kind can be found in

Falk [1], [2], [3], Mosco and Strang, [1], Strang [1], Glowinski, Lions, andTremolieres (G.L.T.) [1], [2], [3], Ciarlet [1], [2], [3], Falk and Mercier [1],Glowinski [1], and Brezzi, Hager, and Raviart [1], [2] But as in manynonlinear problems, the methods used to obtain these estimates are specific

to the particular problem under consideration (as we shall see in the following

sections) This remark also holds for the approximation of the EVI of thesecond kind which is the subject of Sec 6.

Remark 5.6 If for a given problem, several approximations are available, and

if numerical results are needed, the choice of the approximation to be used isnot obvious We have to take into account not only the convergence properties

of the method, but also the computations involved in that method Someiterative methods are well suited only to specific problems For example,some methods are easier to code than others

6 Internal Approximation of EVI of the Second Kind

6.1 The continuous problem

With the assumptions on V, a(-, •), L, andj(-) as in Sec 2.1, we shall consider

the approximation of

which has a unique solution by Theorem 4.1

6.2 Definition of the approximate problem

Preliminary remark: We assume in the sequel that j : V -»• U is continuous.

However, we can prove the same sort of results as in this section under lessrestrictive hypotheses (see Chapter 4, Sec 2)

6.2.1 Approximation of V

Given a real parameter h converging to 0 and a family {V h } h of closed subspaces

of V (in practice, we will take V h to be finite dimensional and h to vary over a sequence), we suppose that {V h } h satisfies:

(i) There exists U c V such that U = V, and for each h, a mapping rA: U-+V h such that l i m ^ o r h v = v strongly in V, V v e U.

6.2.2 Approximation of j(-)

We approximate the functional^-) by {j h } h where for each h,j h satisfies

jh- Vh -* $&• jh is convex, l.s.c, and uniformly proper in h (6.1)

The family {j h } h is said to be uniformly proper in h if there exist 1 e V* and

ji e IR such that

Furthermore we assume that {j H } h satisfies:

(ii) If v h -* v weakly in V, then

lim Mj h (v h ) > j(v),

Jl->0

(iii) lim h ^ 0 j h (r h v) = j(v), VveU.

Remark 6.1 In all the applications that we know, if ;(•) is a continuous

functional, then it is always possible to construct continuous j h satisfying (ii)and (iii)

Remark 6.2 In some cases we are fortunate enough to have j h (v h ) — j(v h ),

V I),,, V ft, and then (ii) and (iii) are trivially satisfied

6.2.3 Approximation o/(P2)

We approximate (P2) by

a(u h , v h - u h ) + j h (v h ) - j h (u h ) > L(v h - u h ), V v h eV h , u h eV h (P2 h )

Theorem 6.1 Problem (P2J,) has a unique solution.

PROOF In Theorem 4.1, taking V to be V h ,j{-) to bej h (-), we get the result.

Remark 6.3 Remark 5.4 of Sec 5 still holds for (P 2 ) and (P 2h ).

6.3 Convergence results

Theorem 6.2 Under the above assumptions on {V h } h and {j h }h> we have

lim \\u h — u\\ = 0,

(6.3)

\imj h (u h )=j(u).

PROOF AS in the proof of Theorem 5.2, we divide the proof into three parts

(1) Estimates for u h We will show that there exist positive constants C t and C 2

inde-pendent of h such that

Since u h is the solution of (P2j,), we have

a(u h , u h ) + j h (u h ) < a(u h , v h ) + j h (v h ) - L(v h - u h ), Vv h eV h (6.5)

By using relation (6.2), we obtain

«KII < I|A|| ||«J| + \n\ + \\A\\ K | | H^ll + \j (v )\ + \\L\\(\\v \\ + ||uj|) (6.6)

Let v o e U and v h = r h v 0 By using conditions (i) and (iii), there exists a constant m, independent of h, such that \\v h \\ < m and \j h (v h )\ < m Therefore (6.6) can be written as

KH2 < - ( M l + \\A\\m + IILIDRH + - ( 1 + ||L||) + ~

a a a

= C J u J + C2where

C, = - (||A|| + \\A\\m + \\L\\) and C, = - (1 + ||L||) + —

a a a

and (6.4) implies

Kll < c, v fc,

where C is a constant.

(2) Weak convergence of {u h } h Relation (6.4) implies that u h is uniformly bounded.

Therefore there exists a subsequence {u h } h such that u h -» «,, weakly in V.

Since « A is the solution of (P lfc) and r k v e V h , V h and V v e U, we have

a("/,,, «*,) + AX"*,) ^ a ( u h t > r h,v) + Jh,(rh,v) - L(r hl v - u h ) (6.7)

By condition (iii) and from the weak convergence of {u h }, we have

limm{[_a(u hi ,u h )+j hl (u h )-]<a(u*,v)+Kv)-L(v-u*), VveU (6.8)

»,—o

As in (5.7), and using condition (ii), we obtain

a(u*, u*) + j(u*) < lim inf [a(«fcf, u h ) + ;»,(«*,)]• (6 ' 9 )

/ij->0

From (6.8) and (6.9), and using the density of U, we have

a(u*, v-u*)+ j(v) - j(u*) > L(v - u*), V v e V, u* e V.

This implies that u* is a solution of (P2) Hence u* = u is the unique solution of (P2 ),

and this shows that {u h } h converges to u weakly.

(3) Strong convergence of {u h } h From the F-ellipticity of a(-, •) and from (P2/l ) we have all"* - "II 2 + /*(«*) ^ a("» - « , « » - « ) + A(M»)

= a(« h, « 0 - a(u, u h ) - a(u h , u) + a(u, u) + j h (u h )

< a(u h , r h v) + j h (r h v) - L(r h v - u h ) - a(u, u h )

The right-hand side of inequality (6.10) converges to a{u, v — u) + j(v) — L{v — u)

as h -> 0, V v e U Therefore we have

lim inf j h (u h ) < lim inf \a\\u h - u\\ 2 + j h {u h )~]

*->0 h-0

< lim sup [a||uh - u||

< a(u, v-u)+ j(v) - L(y - u), V v 6 U (6.11)

By the density of U, (6.11) holds, V c e K Replacing v b y u in (6.11) a n d using condition

(ii), we obtain

j(u) < lim infj h (u h ) < lim sup [a||t< - u h \\ 2 + j h (u k y] < j(u).

This implies that

limj h (u h )=j(u)

and

lim \\u h — u\\ = 0.

This proves the theorem •

7 Penalty Solution of Elliptic Variational Inequalities of the First Kind

7.1 Synopsis

In this section we would like to discuss the approximation of elliptic variational

inequalities of the first kind by penalty methods In fact these penalty techniques

can be applied to more complicated problems as shown in Lions [1], [4](see also Chapter VII, Sec 4, where a penalty method is applied to the solution

of transonic flow problems)

7.2 Formulation of the penalized problem

Consider the EVI problem

Find ue K ( c F ) such that

a(u, v -u)>L(v -u), \/ve K, (7.1)

where the properties of V, a(-, •), L(-), and K are those given in Sec 2.1 Now suppose that a functional;': V -> U has the following properties:

j is convex, proper, l.s.c, (7.2) j(v) = 0oveK, (7.3) j(v) > 0, V c e K (7.4)

Let e > 0; we define j E : V -> U by

The penalized problem associated toy'(") is defined b y :

Find u e e V such that

a(u e , v - u s ) + ; » - /.(M.) > L(v - u e ), V C E K (7.6)

(7.6) is definitely an EVI of the second kind, and from the properties of V,

a{-, •), L(-), and;(-X it has (see Sec 4) a unique solution according to Theorem

4.1

Remark 7.1 Suppose that j£ is differentiable; the solution u c of (7.6) is thencharacterized by the fact that it is the unique solution of the following non-linear variational equation:

where j' e (v) (e V*, the topological dual space of V) denotes the differential

of j e at v, and where < •, • > is the duality pairing between V* and V That

differentiability property (if it exists) can be helpful for solving (7.6), (7.7) byefficient iterative methods like Newton's method or the conjugate gradientmethod (see Chapter IV, Sec 2.6 and Chapter VII for references and also someapplications of these methods)

7.3 Convergence of {H E } E

Concerning the behavior of {u e } e as e -> 0, we have the following:

Theorem 7.1 If the hypotheses on V, K,a{-,-), L(•),./(•) are those of Sees 2.1

and 7.2, we have

8->0

where u (resp., u e ) is the solution o/(7.1) (resp., (7.6)).

PROOF This proof looks very much like the proof of Theorems 5.2 and 6.2.

(1) A priori estimates From (7.6) we have

a(K, tO + MuJ < a(u e , v) - L(v - u e ) + j E (v), V v e V (7.10) Since j c (v) = s~ l j(v) = 0, V v e K [property (7.3)], we have, from (7.10),

a(u., iO + Mu c ) < a(u E , v) - L(v - u 8 ), V » e K (7.11) Consider v o e K (since K # 0, such a v 0 always exists) Taking v = v 0 in (7.11), from the

properties of a{-, •) and from (7.4), (7.5) we obtain

«II«J 2 < U\\ ||u,|| ||»oll + IH-IKIM + boll), (7-12)

0 <j(u ) < E ( M | | | U J K | | + ||L||(|«.|| + ||»|D) (7.13)

(a is the ellipticity constant of a(-, •))• Then it clearly follows from (7.12) that we have

which combined with (7.13) implies

0 < j ( u £ ) < C 2 f i ) (7.15) where in (7.14), (7.15), C x and C 2 denote two constants independent of s.

(2) Weak convergence It follows from (7.14) that we can extract from {u s } t a quence—still denoted {uj e —such that

which implies, at the limit as e —> 0,

a(u*, u*) < lim inf a(u E , u E ) < a(u*, v) - L(v - u*), V v e K (7.19)

Combining (7.17) and (7.19), we finally obtain

a(u*, v-u*)> L(v - «*), V v e K, u* e K; (7.20)

we have thus proved that u* = u and that the whole {u E } t converges weakly to u.

(3) Strong convergence From (7.3), (7.4), and (7.6) we have

0 < a\\u s - u\\ 2 + j s (u s ) < a{u c -u,u E -u)+ j£u E )

< a(u e , u e ) + j t (u c ) — a(u, u s ) — a(u e , u) + a(u, u)

< a{u c , v) - L(v - u e ) - a(u, u e ) - a(u E , u) + a(u y u), V t i e K (7.21)

The weak convergence of {u e } E to u implies that at the limit in (7.21) we have

0 < lim inf [a||« £ - u|| 2 + j£uj] < lim sup [a||uE - u\\ 2 + j E (uJ\

which clearly implies the convergence properties (7.8) and (7.9) •

Remark 7.2 If a(-, •) is symmetric, then the penalized problem (7.6) is

equivalent to the minimization problem:

Find u E e V such that

In this section we discuss, in some detail, applications of the penalty method to

the solution of some simple model problems in U N In Sec 7.4.2 we consider

(resp., Sec 7.4.3) a situation in which K is denned from linear equality

con-straints (resp., convex inequality concon-straints).

In the following A is a N x JV real matrix, positive definite, possibly symmetric, and b e U N To A and b we associate

non-a:U N x U N -*U and L:M N ->U

The form L(-) is clearly linear and continuous on U N ; similarly a(-, •) is bilinear

and continuous on U N x U N Since A is positive definite, «(•>•) is R^-elliptic,

and we have

a(v,v)>A0||v||2, V v e R " , (7.26)where Ao is the smallest eigenvalue of the symmetric positive-definite matrix

A + A'/2 (with A' the transpose matrix of A).

7.4.2 A first example

Let B e i?(RN, RM); B can be identified to a M x JV matrix We define R(B) (the range of B) by

R(B) = {q|q e R , 3 v e R* such that q = Bv}

and then X c i " b y

where, in (7.27), we have

c e K(BX=> K * 0) (7.28)

From the above properties of a(-, •)> L(-), and K, the EVI problem:

Find ue K such that

(Au, v - u) > (b, v - u), V v e K (7.29) has a unique solution since we can apply (with V = U N ) Theorem 3.1 of Sec 3.1.

Remark 7.3 If A = A', then problem (7.29) is equivalent to the minimization

Before going on to the penalty solution of (7.29), we shall prove some properties

of the solution u of (7.29); more precisely we have the following proposition

Proposition 7.1 The solution u of (7.29) is characterized by the existence of

Taking v = u + w in (7.22), we obtain

(Au - b, w) > 0, V w e Ker(B), (7.33) and (7.33) clearly implies

(Au - b, w) = 0, V W E Ker(B) i.e.,

(2) (7.31) implies (7.20) The second relation (7.31) implies that u e K Letting v e X , w e

then have

v - u e Ker(B), (7.35) and from (7.35) and from the first relation (7.31) it follows that

(b, v - u) = (Au, v - u) + (B'p, v - u) = (Au, v - u) + (p, B(v - u))

= (Au, v - u) (7.36)

We have thus proved that (7.31) implies (7.29) •

Remark 7.4 Suppose that A = A'; then the vector p of Proposition 7.1 is a

Lagrange multiplier vector for the problem (7.30), associated with the linear

equality constraint Bv — c = 0 defining K.

The following proposition and its corolaries state results quite easy toprove, but of great interest in studying the behavior of the solution of thepenalized problem to be defined later on

Proposition 7.2 Problem (7.31) has a unique solution in U N x jR(B) if

{b, c j e l f x R(B).

Let us denote by {u, p} this solution; then all solutions of (7.31) can be written

{u, p + q}, where q is an arbitrary element o/Ker(Br)

Corollary 7.1 The above vector p has the minimal norm among all the p e U M

such that {u, p} solves (7.31).

Corollary 7.2 The Hnear operator

A B'

B 0

is an isomorphism from U N x R(B) onto U N x R(B).

EXERCISE 7.2 Prove Proposition 7.2 and Corollaries 7.1 and 7.2

EXERCISE 7.3 Prove that p = 0 if and only if c = BA"Jb

In order to apply the penalty method of Sec 7.2 to the solution of (7.29),

we define;: R*-> U by

;(v) = i | B v - c |2, (7.37)

where |-| denotes the usual Euclidean norm of U M We can easily see thatj(-)

obeys (7.2)-(7.4); moreover, j(-) is a C°° functional whose differential / is

given by

/(v) = B'(Bv - c) (7.38)

The penalized problem associated with (7.29) and (7.37) is defined by:

Find u£ e U N such that

(An,, v - u£) + jjy) - j£(uE) > (b, v - u£), V v e B " , (7.39)

where j e = (1/e)/ (with e > 0)

From Remark 7.1 and (7.38) the penalized problem (7.39) is equivalent tothe linear system

A + - B B | u£ = - B e + b (7.40)

e / ewhose matrix is positive definite (and symmetric if A is symmetric) It followsfrom Theorem 7.1 (see Sec 7.3) that

lim ||ue - u|| = 0, (7.41)

£->0

where u is the solution of (7.29) We have, in fact, ||u£ — u|| = O(s); several

methods can be used to prove this result; we have chosen one of them based on

the implicit function theorem.

Define p£ e U M by

p£ = i (Bu£ - c) (7.42)Problem (7.40) is then equivalent to the following system:

Au£ + B'p£ = b,

- B u£ + ep£ = - c , (7.43)whose matrix

isaJV + MbyJV + M positive-definite matrix

EXERCISE 7.4 Prove that s/ e is positive definite

Since c e R(B), we have (from (7.42)) p8 e R(B) About the behavior of

{u£, p£} as s -> 0, we have the following theorem

Theorem 7.2 Let u£ be the solution of (139) and let p£ be defined by (7.42);

We have {u(0), p(0)} J= {0, 0}, unless p = 0, which corresponds (see Exercise 7.3) to the

trivial situation u = A~ :b This proves that the estimates (7.45) and (7.46) are of optimal order in general •

EXERCISE 7.5 Prove that |p £| < \p\, V e > 0.

Remark 7.5 It follows from Theorem 7.2 that ue and p E will be good

approxi-mations of u and p, respectively, provided that we use a sufficiently small s But in this case the condition number v(A£ ) of the matrix

occurring in (7.40) will be large; we indeed have (we suppose A = A' for

simplicity 2 )

We also suppose that Ker(B) i= 0 (which is the usual case).

where, in (7.53), lim8^0 j8(e) = 0, p(B'B) is the spectral radius of B'B (i.e., the

largest eigenvalue of B'B) and where a is defined by

, (Av,v)

vsKer(B)-{0(

For small s it clearly follows from (7.53) that A e is ill conditioned Actually that

ill-conditioning property that we pointed out for the model problem (7.29)

is the main drawback of penalty methods An elegant way to overcome thisdifficulty has been introduced by Hestenes [1] and Powell [1]: the so-called

augmented Lagrangian methods in which the combined use of penalty and

Lagrange multiplier methods allow larger e and moreover produces the exactsolution u instead of an approximated one.3 In Chapter VI, we will discuss thesolution of a particular class of variational problems by these augmentedLagrangian methods; for more details and a substantial bibliography, seeFortin and Glowinski [1] and Gabay [1]

EXERCISE 7.6 Prove (7.53)

7.4.3 A second example

Let G: U N - > ( 1 )M; we then have G = {g t }^ =1 , where g t are functionals from

U N to U We suppose that the following properties hold:

V i = 1 , , M, g t is a convex, l.s.c, and proper functional; (7.54)

the convex set K = {\ e U N , g t (\) < 0, V i = 1 , , M} is nonempty (7.55)

Suppose that the properties of a(-, •) and L(-) are those of Sec 7.4.1 From

these properties and from (7.55), the EVI problem:

Find u e K such that

Remark 7.7 If G obeys some convenient conditions (usually called

qualifica-tion condiqualifica-tions), we can generalize Proposiqualifica-tion 7.1 and associate to (7.56) the

so-called F John-Kuhn-Tucker multipliers; we shall not discuss this matter

here4 (the interested reader may consult Rockafellar [1], Cea [1], [2], Ekelandand Temam [1], and Aubin [1])

In order to apply the penalty method of Sec 7.2 to the solution of (7.56),

we define;': U N —• IR by

fi>) = i X > J | 0 I + ( V ) I 2 > (7.58)

where, in (7.58), a; are strictly positive and gf = sup(0, g t ) Since ;'(•) satisfies

(7.2)-(7.4), the associate penalized problem (with;'£ = (1/e)/, s > 0) is defined by:

Find u£ e U N such that

(An,, v - u£) + ;£(v) - ;£( u£) > (b, v - u£), V v e R " (7.59)

Remark 7.8 Suppose that g t e C1, V i = 1 , , M We then have |gf(+ |2 e C \

V i = 1 , , M, implying that j e C1 We have

/(v) = £ a i gt(y)g' i (y), (7.60)

and from Remark 7.1, (7.59) is equivalent to the nonlinear system in R":

Au£ + i / ( i g = b (7.61)

It follows from Theorem 7.1 (see Sec 7.3) that

lim ||u£ - u|| = 0 (7.62)

£">0

where u is the solution of (7.56)

To illustrate the above penalty method, we consider its application to the

solution of & discrete obstacle problem (see Chapter II, Sec 2 for a mathematical and mechanical motivation) For example (with M = N), we have

See, however, Exercise 7.7

problem (7.64) can be solved by the methods described in Chapter IV, Sec 2.6and also in Chapter VI, Sec 6.4

EXERCISE 7.7 Prove that the solution u of (7.56), with G defined by (7.63),

is characterized by the existence of p = {p;}f= x e IR^ such that

Au — p = b,

Also prove that limc^0 (l/e)(c — u£)+ = p

Hint: Observe that:

Actually similar ideas can be applied to the solution of EVI of the secondkind; for example, we can replace the solution of :

Find u e V such that

a(u, v-u)+ j(v) - j(u) > L ( v - u ) , V v e V (7.66)

(where V, a(-, •), L(-),j(-) obey the hypotheses of Sec 2.1) by the solution of:

Find ME e V such that

a(u s , v-u s )+ j e (v) - j £ (u e ) > L(v - u E ), V » e K , (7.67)

where j e is an "approximation" of; which is more regular For example, if wesuppose that j is nondifferentiable, it may be interesting from a computational

point of view to replace it by j E differentiate Such a process is called—for

obvious reasons—a regularization method.

If 78 is differentiate, (7.67) is clearly equivalent to the variational equation:

Find u e e V such that

where j' e (v) denotes the differential of;£ at v An application of these

regulariza-tion methods is given in Chapter II, Sec 6.6; we refer to G.L.T [1], [2], [3]for further details and other applications of these regularization methods

8 References

For generalities on variational inequalities from a theoretical point of view,see Lions and Stampacchia [1], Lions [1], Ekeland and Temam [1], Baiocchiand Capelo [1], [2], and Kinderlherer and Stampacchia [1]

For generalities on the approximation of variational inequalities fromthe numerical point of view, see Falk [1], G.L.T [1], [2], [3], Strang [1], Brezzi,Hager, and Raviart [1], [2], Oden and Kikuchi [1], and Lions [5]

For generalities and applications of the penalty and regularization methodsdiscussed in Sec 7, see Lions [1], [4], Cea [1], [2], G.L.T [1], [2], [3], andOden and Kikuchi [1] (see also Chapter II, Sec 6.6 and Chapter VII, Sec 4 ofthis book) Some additional references will be given in the following chapters