Numerical_Methods for Nonlinear Variable Problems Episode 1 docx

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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To my wife Angela and to Mrs Madeleine Botineau

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

viii Preface

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

Preface ix

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

x Preface

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

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

xii Contents

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

Some Preliminary Comments

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-" The Buddha, the Godhead, resides quite as comfortably in the circuits of a digital

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."

CHAPTER I

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 I Generalities on Elliptic Variational Inequalities and on Their Approximation

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

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

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

Let i>!, v 2 e V Then since P K is a contraction we have

— 2pa(v 2 — v lt v 2 — Vi).

Hence we have

HWp(oi) " W p (v 2 )\\ 2 < ( 1 - 2pa + p 2 \\A\\ 2 )\\v 2 - v t \ \ 2 (3.7)

Thus W^, is a strict contraction mapping if 0 < p < 2a/\\A\\ 2 By taking p i n this range, we

have a unique solution for the fixed-point problem which implies the existence of a solution

4 I Generalities on Elliptic Variational Inequalities and on Their Approximation

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)

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

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.

6 I Generalities on Elliptic Variational Inequalities and on Their Approximation

(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)

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)