Numerical Analysis of Variational Inequalities (Studies in mathematics and its applications)

Numerical Analysis of Variational Inequalities

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 8

Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T. ROCKAFELLAR, Seattle

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD

NUMERICAL ANALYSIS OF VARIATIONAL INEQUALITIES

ROLAND GLOWINSKI Universiti Pierre et Marie Curie, Paris and INRIA JACQUES-LOUIS LIONS Collkge de France, Paris and INRIA RAYMOND TREMOLIERES Universiti dAix-Marseille

English version edited, prepared and produced b y TRANS-INTER-SCIENTIA P.O. Box, 16, Tonbridge, TN11 8 D Y , England

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD

North-Holland Publishing Company, 1981 All rights reserved. No part of this publication may he reproduced, stored in a rerrievalsystem, or transmirred. in any form or by any means. electronic. mechanical, photocopying, recording or orherwise, wirhout theprior permission of (he copyright owtier.

ISBN0444861998

Translarion and revised edition of: Analyse Numerique des Inequations Variationnelles (Tome I et 2 ) QBordas (Dunod),Paris, I976

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM . NEW YORK . OXFORD Sole disrriburorsfor rhe U.S.A.and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

Glowinski, R. Numerical analysis of variational inequalities. (Studies in mathematics and its applications ; v. 8) Revised translation of: Analyse numgrique des in6quations variationelles Bibliography: p. 1. Differential inequalities. 2. Numeripal analysis. I. Lions Jacques Louis. II. Tremolihres Raymond. IfI. Title. IV. Series. QA374.65713 1981 515.3'6 81-38345 ISBN 0-444-86199-8 AACR2

.

PRINTED IN THE NETHERLANDS

INTRODUCTION

1.

INTRODUCTION TO CHAPTERS 1-3 (VOL. I OF THE FRENCH EDITION) : GENERAL THEORY AND FIRST APPLICATIONS

Numerous problems i n Mechanics, Physics and Control Theory l e a d t o t h e study of systems o f p a r t i a l d i f f e r e n t i a l i n e q u a l i t i e s , t h e s o l u t i o n of which l e a n s h e a v i l y on t h e techniques of so-called

variational inequalities

.

1.1 We s t a r t by g i v i n g a very simple example, t h e physical motivation f o r which i s discussed i n t h e book by Duvaut-Lions /l/. (l)

On a bounded open domain 0 of R ( n = 2 , 3 i n p r a c t i c a l applica t i o n s ) with boundary r , w e seek a real-valued function x + u(x) which i n S2 s a t i s f i e s t h e c l a s s i c a l equation:

- Au

(1) Au =

azu Cax:

+ u = f,

f given i n a ,

, with boundary conditions i n the form of inequalit-

i=l

ies : (2)

u 2 0 ,

au av

-30,

au av

u-=O

on

r,

where alav denotes d i f f e r e n t i a t i o n along t h e outward normal t o

r

A similar time-dependent problem c o n s i s t s of f i n d i n g u = u(x, I) defined f o r ~ € 6 2and t > 0, which i s a s o l u t i o n of (3)

aU

--h=f, f at

given i n a x { t > O } ,

with inequality conditions i d e n t i c a l t o ( 2 ) on with t h e i n i t i a l condition

(I)

r

( f o r t .> 0 ) and

A number enclosed between s l a s h e s corresponds t o t h e main l i s t of r e f e r e n c e s , which follows Chapter 6.

Introduction

vi

(4) i s an inequality problem of parabolic Other i n e q u a l i t i e s encountered i n p r a c t i c a l a p p l i c a t i o n s are of "hyperbolic type" o r o f "Petrowski type", t o g e t h e r with numerous v a r i a n t s o f t h e s e . The problem ( 3 ) y ( 2 ) ,

type.

I n f a c t problem (l), ( 2 ) i s a very simple problem i n t h e calculus of variations with constraints. Indeed, i f w e i n t r o d u c e the functional (')

i n which

and i f we i n t r o d u c e t h e convex ( c l o s e d ) set K d e f i n e d by (6)

K = { u l u d O on

r},

t h e n problem (l), ( 2 ) i s equivalent t o f i n d i n g U E K such t h a t

(7)

J(u) = inf J(Y),

Y

EK

.

Problem ( 7 ) admits a unique solution u , c h a r a c t e r i s e d by:

The problem i n t h e form (8) i s c a l l e d an e l l i p t i c variational

inequality problem. "he formulation ( 8 ) allows problem ( 3 ) , ( 2 1 , (4) t o be s t a t e d i n a very convenient manner; denoting by u ( t ) . t h e f u n c t i o n x u(x,t) w e need t o determine u(t) such t h a t

(9)

where now

The c l a s s e s (Sobolev spaces) i n which t h e s e f u n c t i o n a l s must be taken w i l l b e defined i n t h e t e x t .

Chapters 1-3

vii

This is a time-dependent variational inequality problem. It may be shown that problem ( 9 ) aohits a unique solution (see Lions-Stampacchia 111; we shall give in Chapter 6 an existence proof based on d i s c r e t i s a t i o n )

.

Without claiming to be exhaustive, we indicate below some 1.2 areas in which steady-state and time-dependent variational inequalities arise.

(i) E l a s t i c i t y with unilateral conditions (see Signorini 111, Fichera /1/ and Fr6mond 111) and e l a s t i c i t y with f r i c t i o n (see . Duvaut-Lions /1/ and Kalker 111, 121) lead to elliptic variational inequalities and in the time-dependent case to hyperbolic variational inequalities (see Duvaut-Lions 111); similarly the theory of thin plates leads to inequality problems of a new type (see Duvaut-Lions 141). (ii) Elasto-plastic materials lead to constrained minimisation problems (see Koiter 111, Mandel 111, Prager /l/,..., 181, and Prager and Hodge 111) which have been studied comprehensively, using inequality techniques, in BrCzis and Sibony 111, DuvautLions / l f ,Lanchon 111, Lewy and Stampacchia 111, 121, Moreau 121, 131 and Ting /1/, 121, 131. (iii) Problems of hydrodynamics i n porous media lead to steady state or parabolic inequalities (see Duvaut-Lions 111). (iv) Problems of the flow of certain non-Newtonian fluids of the Binghm f l u i d type (see Duvaut-Lions 111, Mosolov and Miasnikov 111) lead to inequalities which contain the NavierStokes equations as a special case. (v) Various problems arising in servo-cmtroi! and in etectromagnetism i n unstable media lead to time-dependent inequalities of various types.

(vi) Numerous situations in which several of the above phenomena are present simultaneously, such as for example thermal phenomena in plastic media or in Bingham fluids (see Prager 131, Duvaut-Lions / 2 / , 131), also lead to inequalities.

(vii) In the foregoing applications, inequalities (steady state or time-dependent) on the “physical variable“ (displacement , pressure, temperature, etc. ) are obtained. Classical f r e e boundary problems, however, are not expressed in the ‘formof variational inequalities in the physical variables; nonetheless, Biaocchi, in his study of free-boundary seepage problems, has observed (see Biaocchi 111) that a f t e r a suitable (and non-trivial!) change of unknam function , variational

viii

Introduction

inequalities or problems using the theory of inequalities are obtained. This has been developed in a series of investigations at the C.N.R. Laboratory in Pavia: see Baiocchi, Comincioli, Magenes and Pozzi /1/ and the bibliography of the latter. (viii) Using methods inspired by that of Biaocchi 111, classical "free-boundary" problems may be reduced to variational inequalities; these include steady flows (see Br6zis and Stampacchia 111), wakes (see Br6zis and Duvaut /l/) , problems of Stefan type (see Duvaut /l/) and cavitation problems (see Bayada 111). For numerical details of wake problems solved using the methods of this book, see Bourgat and Duvaut 111. (ix) Similarly, optimal st'opping-time problems may be reduced to time-dependent variational inequalities (see Bensoussan-Lions /1/ and Friedman 111). (x) Impulse control problems may be reduced to problems which call upon variational inequality techniques: these are the quasi-variationaZ inequalities introduced in Bensoussan-Lions /I/, / 2 / and Bensoussan, Goursat and Lions /l/, and will not be considered here; from a numerical point of view, however, their treatment is based, in particular, on techniques described in the present book: see Goursat /1/ and Bensoussan-Lions 111. (xi) Control Theory for distributed systems leads to numerous problems of similar type, notably in the consideration of "a priori feedback" (see Lions / 2 / ) .

1.3 The main objective of the present work is to describe the principal numerical methods for computing the solution of steadystate and time-dependent variational inequalities, and to present some of the numerical calculations which have been performed. Most steady-state problems reduce to one of the two following cases:

(i) Minimisation of a differentiable functional J over a convex closed set K (called the "convex set of the constraints"); this is the case, for example, with problem (7) stated earlier.

(ii) Minimisation over the entire space (unconstrained) of a non-differentiable functional J. The principle behind the approximation is clear: we "approximate" the functional J(u) (or the form (J'(u),v) in the differentiable case) by means of discretisation, using either a finite

Chapters 1-3

ix

difference method o r a f i n i t e element method ( i . e . using t h e classical methods of Numerical A n a l y s i s ) , The s p e d f i c d i f f i c u l t i e s (which fundament a l l y d i s t i n g u i s h t h e numerical a n a l y s i s of e l l i p t i c inequalities from t h a t of e l l i p t i c equations) a r e concerned with: (j) I n case ( i ) ,t h e approximation by d i s c r e t i s a t i o n of t h e convex s e t K ; (jj)

The solution of the discretised problem;

this is a

( finite-dimensional) nonlinear programming problem. I n t h e case i n which J(u) i s "approximated" by Jh(u,,) (where h denotes t h e d i s c r e t i s a t i o n parameter - f o r example t h e mesh s i z e ) and t h e convex s e t K i s approximated by t h e convex s e t Kh , t h e approximate problem ( i n c a s e ( i ) )is:

and i n case ( T i ) i s :

inf Jh(u,,) , unconstrained , J,, being n o n d i f f e r e n t i a b l e o r i n any case t h e " l i m i t " of Jh not being d i f f e r e n t i a b l e . The question of t h e convergence of t h e methods does not present The problem o f t h e error estimation any p a r t i c u l a r d i f f i c u l t y . i s l e s s c l e a r c u t ; s e e Falk /I/, Mosco and Strang 111. The fundamental i s s u e i s t h e n t h a t of e f f e c t i v e l y solving (lo), (11). I n our experience, d i r e c t a p p l i c a t i o n of t h e "standard" methods o f nonlinear programming g e n e r a l l y l e a d s , i n t h e cases t h i s observconsidered h e r e , t o p r o h i b i t i v e computation times; a t i o n has prompted us t o r e c o n s i d e r t h e question i n a systematic manner, t a k i n g i n t o account t h e particular structure o f t h e f u n c t i o n a l s t o be minimised and of t h e convex sets of t h e constraints.

1.4 The f i r s t t h r e e c h a p t e r s of t h i s book (which formed Volume I of t h e o r i g i n a l French e d i t i o n ) d e s c r i b e t h e general methods r e l a t i n g t o steady s t a t e problems, and s t u d i e s i n some depth a t y p i c a l example taken from t h e t h e o r y of e l a s t o - p l a s t i c ity

.

Chapter 1 d e s c r i b e s t h e v a r i o u s ways i n which t h e o r i g i n a l problem may be approximated by a finite-dimensional problem. F i r s t , however, a b r i e f r6sum6 of i n e q u a l i t i e s i s given, which

X

Introduction

should enable the reader t o tackle t h i s book without having t o possess any specialised knowledge of i n e q u a l i t i e s . Chapter 2 i s devoted t o t h e methods used i n t h e s o l u t i o n of finite-dimensional problems. This chapter may be considered as p a r t of a book on t h e methods of nonlinear programing. Chapter 3 t r e a t s , by means of examples, t h e d e t a i l e d implementa t i o n of t h e g e n e r a l methods, and g i v e s t h e results of a f a i r l y t h i s makes it p o s s i b l e l a r g e number of numerical a p p l i c a t i o n s ; f o r comparisons t o be drawn between t h e v a r i o u s methods. The problem of t h e e k s t o - p Z a s t i c torsion of cylindrical bars i s This w a s chosen as a s t u d i e d i n a r e l a t i v e l y exhaustive manner. "model" problem not only because it i s o f both p h y s i c a l and t h e o r e t i c a l i n t e r e s t , b u t a l s o because i t s numerical s o l u t i o n i l l u s t r a t e s many of t h e fundamental d i f f i c u l t i e s of t h e t h e o r y ; i n f a c t , t h e convex set K i s i n t h i s case defined by

and t h e approximation of

K raises a number of i n t e r e s t i n g p o i n t s .

2.1 The previous t h r e e c h a p t e r s presented t h e g e n e r a l methodology: Chapter 1 considered t h e infinite-dimensional case and Chapter 2 t h e finite-dimensional c a s e , w h i l s t Chapter 3 i n v e s t i g a t e d a steady-state e l a s t o - p l a s t i c i t y problem which i l l u s t r a t e s some of t h e d i f f i c u l t i e s which are t y p i c a l o f t h i s s u b j e c t . Chapters 4 t o 6 (which formed Vol. I1 o f t h e o r i g i n a l French e d i t i o n ) continue t h e study of t h e Numerical Analysis of Variati o n a l I n e q u a l i t i e s , by analysing s e v e r a l c l a s s e s of steady-state problems occurring i n Mechanics and Physics, and a number of timedependent problems. 2.2 The steady-state case i s taken up i n Chapters 4 and 5 . These c h a p t e r s i n v e s t i g a t e some problems which a r e of p r a c t i c a l i n t e r e s t and which contain a number of typical d i f f i c u l t i e s i n s t e a d y - s t a t e i n e q u a l i t y problems, concerning i n p a r t i c u l a r nond if f erentiab t e functionuts. It would appear t h a t t h e following p r a c t i c a l conclusions may be drawn: when t h e convex set K i s of s u i t a b l e form, t h e "opti m a l " methods (with regard t o ease of programming, s t o r a g e r e q u i r e ) are o f t h e overrelaxation ment, computational speed, e t c ,

. . ..

chapters 4-6

type ( s e e Chapter 2 ) ; problems o f o r d e r 2.

xi

t h i s is t r u e i n particular for e l l i p t i c

Duality methods (see Chapters 1 and 2 ) have a more general range of a p p l i c a t i o n ( i n p a r t i c u l a r , t h e y are w e l l s u i t e d t o t h e s o l u t i o n of 4th o r d e r e l l i p t i c problems and of c e r t a i n nondiffere n t i a b l e problems) and have a high performance i n terms of computa t i o n a l speed; however, all o t h e r t h i n g s being equal, t h e y a r e more d i f f i c u l t t o implement and more c o s t l y i n terms of s t o r a g e requirement (and o f t e n o f computation t i m e ) t h a n o v e r r e l a x a t i o n methods when t h e s e a r e a p p l i c a b l e . I n t h e problems considered, penalty methods (see Chapter 2 ) a r e , i n g e n e r a l , more complicated t o implement and more c o s t l y both i n computation time and i n s t o r a g e requirement than t h e above methods. However , t h e y should not be dismissed completely, since c e r t a i n problems, p a r t i c u l a r l y t h o s e a r i s i n g i n optimal c o n t r o l , l e a d t o t h e i d e a t h a t t h e s e methods could prove t o be u s e f u l i f t h e y were employed simuZtaneously with o v e r r e l a x a t i o n and d u a l i t y methods. Methods based on decomposition and s p l i t t i n g of t h e c o n s t r a i n t s ( s e e Lions-Tham /l/) w i l l not be presented i n t h e p r e s e n t book; a g e n e r a l d e s c r i p t i o n of t h e s e i s given i n Bensoussan-Lions-T6mam t h e i r numerical a p p l i c a t i o n and, i n p a r t i c u l a r , t h e search /l/; f o r "optinal" decompositions of t h e right-hand s i d e , are described i n Lavainne 111.

2.3

Time-dependent problems form t h e s u b j e c t of Chapter 6.

If , i n t h e i n e q u a l i t y ( 9 ) , au(t)/at is replaced by - u")/At , where u" denotes an approximation of u(nAt) , t h e n it i s p o s s i b l e t o "approximate" ( 9 ) by using semi-discretisation: (u"+l

This i s o f t h e same form as a s t e a d y - s t a t e v a r i a t i o n a l inequa l i t y , and i s t h e r e f o r e amenable t o t h e methods mentioned i n 2.2 above. Hence, by discretisation o f c r e t i s a t i o n of problem ( 9 ) .

(lo),

we achieve t h e complete d i s -

The choice l e a d i n g t o (10)i s obviously t h e s i m p l e s t , but it i s not t h e only one p o s s i b l e ! The following p o i n t s r e q u i r e consideration:

xii

Introduction

(i) (it) (iii)

choice of the methods of discretisation in t analysis of the stability and convergence numerical applications allowing comparisons to be made.

From this viewpoint, Chapter 6 investigates time-dependent inequalities connected with operators which may or may not be linear: a a

parabolic inequalities of type I and type I1 (see chapter 6); “hyperbolic“ inequalities, or those corresponding to operators of the Petrowski type; then in Section 11, the analogous problems for nonlinear operators of the Bingham type are studied briefly - we refer the reader to Fortin /1/ and BCgis /1/ for further technical details.

There already exist in the literature numerous works 2.4 devoted to the Numerical Analysis of problems in Mechanics, with constraints. This is especially true in the field of elastoplasticity; here we may cite the works of Prager-Hodge 111, HCrakovitch and Hodge 111, Capurso /I/, 121, 131, Capurso and Maier /I/, Maier 111, /2/ and works using finite elements in elasto-plasticity for which we refer to Argyris 111, 121, Argyris, Scharpf and Spooner 111, Yamada /I./, Zienkiewicz 111, Zienkiewicz, Valliapan and King /1/ and their associated bibliographies. Some general methods for elliptic problems are described in Haugazeau 111, /2/ and Aubin 111. Herein we have primarily used the works of CCa and Glowinski /1/ and CCa, Glowinski, NCdClec 111. In the case of time dependent problems, the first results were given in a paper by Lions 151; herein we have primarily used the works of Trholieres / b / , Viaud /1/ and Fortin 111. We have also used, throughout the book, the results obtained by BCgis 111, Bourgat /l/, Goursat /2/, Lemonnier 111, Leroy 111, Marrocco 111, Mercier /I/, Thomasset /I/ to whom we are indebted. For other methods relating to the steady-state case, we refer the reader to the articles by Auslender /1/ and Mosco 111. The following works amongst others may be consulted for further applications: Baiocchi, Comincioli, MagPnes and Pozzi 111, Comincioli, Guerri and Volpi /1/, Fremond /1/, F’uscardi , Mosco, Scarpini and Schiaffino /l/. We would like to express our thanks to Messrs. C6a, NCd6iec and Strang for several interesting discussions on the material contained within this book.

Supplement

xiii

Since the final drafting of Volumes I and I1 of the original French version of the present work (June 1974) certain progress has been made in the Numerical Analysis of variational inequalities, principally in connection with the problems of estimating the approximation errors related to the use of the f i n i t e element method. We may cite here, amongst others, the works of BrezziHager-Raviart, Falk, Falk-Mercier, Glowinski, Johnson, JohnsonMercier , Mosco-Scarpini , Nitsche , etc. Some of the references corresponding to these investigations may be found in Glowinski

/9/. We should also emphasise the potential, confirmed by numerical experiments, of the augmented Lagrangian methods mentioned in Chapter 2, Section 6, in Chapter 3, Section 10 and in Chapter 5, Section 9. The reader may refer to Gabay-Mercier J I JGlowinski, Marrocco /I/,/ 2 / and Mercier /l/ for some applications of these methods to nonlinear partial differential equations and variational inequalities.

3.

SUPPLEMENTARY INTRODUCTION TO THE ENGLISH-LANGUAGE EDITION

Progress which has taken place since the publication of Volumes I and I1 of the original French edition of this book has led to developments under the following headings:

1.

Introduction and investigation of new algorithms.

2.

New applications of Variational Inequalities to freeboundary problems.

3.

Variants of the Variational Inequalities: the QuasiVariational Inequalities.

4.

Optimal Control of systems governed by Variational Inequalities.

5.

Application of the methodology of Variational Inequalities to the solution of other problems.

We shall now give a few brief comments on these points. 3.1

Introduction and investigation of new algorithms

This topic is the one most directly related to the theme of the present book. The new work has prompted us to produce Appendices 1 to 6, in which we present those contributions which seem to us to be the most important (see also the bibliography corresponding to these appendices ( ) )

.

6. The corresponding references in the text are followed by a Letter A.

(I) The bibliography for the appendices follows Appendix

Introduction

xiv 3.2

New applications of Variational Inequalities to freeboundary problems

It is especially important here to point out the work of the team at the Numerical Analysis Laboratory of the University of Pavia, and in particular that of C. Baiocchi, V. Comincioli and their associates. Using the techniques of steady-state and timedependent Variational Inequalities, the authors have solved progressively more complicated problems related to steady and timedependent seepage. Bibliographic references are cited in Appendices 1 and

3.3

Variants of the Variational Inequalities: Variational Inequalities

6.

the Quasi-

The theory of Impulse Control brings in new problems, similar in some ways to the obstacle problem but with "an obstacle which We thus have "implicit" Variational depends upon the solution". Inequalities, which A. Bensoussan and J.L. Lions have termed QuasiVariational Inequalities (abbreviated to Q.V.I.). These problems give rise to some quite serious theoretical difficulties (in particular, it seems difficult to apply these techniques to operators which do not possess a maximum principle) ; fundamental difficulties also arise in connection with the dimension of the problems to be investigated. A brief introduction to the investigation of this type of problem is given in Appendix 1, Section 7.

3.4

Optimal Control of systems governed by Variational Inequalities

Some recent and important applications, notably in the control of steel rolling mills (I), have led to a renewed interest in the problem (which was posed in a purely "academic" manner in Lions / 2 / ) of Variational Inequality Control and also, more generally, in free-boundary control. A number of recent theoretical investigations have been directed at this question, in particular those of V. Barbu and F. Mignot. A f u l l account of these problems will be presented at a later date.

('ISee Ch. Saguec /lA/.

xv

Supp Zeme n t

3.5

Application of the methodoloPy of Variational Inequalities to the solution of other problems

The techniques introduced for the numerical solution of Variational Inequalities have also proved useful for, amongst other things, problems of the following type: a a

Nonlinear probZems i n FZuid Mechanics (see in particular Appendix 4, Section 4) Free-boundary problems, even when it is not known whether these problems actually fall within the scope of Variational Inequalities.

Some interesting issues, not addressed in the present book, emerge in connection with all the headings outlined above; some of these have already given rise to books or monographs, and it is reasonable to conjecture that eventually the same will apply for those which are at present unresolved or not solved in full. Finally, we would like to acknowledge our particular indebtedness to Mrs. F. Weber of INRIA who painstakingly typed the French version of the Appendices, to M r . Barry Hunt and his colleagues in Bans-Inter-Scient{a who translated, edited and typed this English edition, and to the North-Holland Publishing Company for agreeing to publish this translation in the series

Studies in Mathematics and i t s AppZications Paris , France January 1981.

Roland Glowinski Jacques-Louis Lions Raymond TrBmoliSres

This Page Intentionally Left Blank

T A B L E OF CONTENTS

Chapter I :

G e n e r a l methods of a p p r o x i m a t i o n f o r steady-state i n e q u a l i t y problems

....... Synopsis ................................................... 1 . Exavples .............................................. 1.1 Fluid mechanics problems in media with semipermeable boundaries ............................. 1.2 Model elasto-plastic problem ..................... 1.3 Model friction problem ........................... 1.4 A f l o w problem ................................... 1.5 Synopsis ......................................... 2. Genera2 formulation o f steady-state variationa2 inequazities .......................................... 2.1 The symmetric case ............................... 2.2 The nonsymmetric case ............................ 2.3 Synopsis ......................................... 3 . Infinite-dimensiona2 approximation methods ............ 3.1 Successive approximations ........................ 3.2 Penalisation ..................................... 3.3 Regularisation ................................... 3.4 Duality (I) ...................................... 3.5 Duality (11) ..................................... 3.6 Duality (111) .................................... 3.7 Various remarks .................................. 3.8 Synopsis ......................................... 4. Interior approximations ............................... 4.1 Interior approximations of V . Finite elements ... 4.2 Interior approximation schemes in the case of equations ..................................... 4.3 Approximations of K .............................. 4.4 Approximation schemes for the initial problem 4.5 Approximation schemes for the penalised or regularised problems

....

.............................

4.6 Approximation schemes for the dual problem ....... 4.7 Open problems Estimates of the approximation

.

............................................ 5. Exterior approximations ............................... 5.1 An example ....................................... error

1 1

1 1

6 7 9 10 10 10

13 15;

15 2.5 19 21 22

26 31 36

37

37

37

40 41 41 43

44 46 49 49

Contents

xviii

5.2 5.3

6.

........... ................... ...........................................

E x t e r i o r approximations f o r V. a. K. j E x t e r i o r approximation schemes

Conclusions

52 55 57

................... 59 Synopsis .................................................... 59 1. The relaxation method ................................. 59 1.1 D e s c r i p t i o n of t h e p o i n t r e l a x a t i o n method . Unconstrained c a s e ............................... 59 1 . 2 P o i n t r e l a x a t i o n . Constrained c a s e .............. 64 1 . 3 Block r e l a x a t i o n ................................. 66 1 . 4 O v e r - r e l a x a t i o n and u n d e r - r e l a x a t i o n ............. 67 1 . 5 A c l a s s of n o n d i f f e r e n t i a b l e f u n c t i o n a l s which can be minimised u s i n g r e l a x a t i o n ................ 7 1 2. Methods of the gradient and gradient projection type .. 75 75 2 . 1 General remarks .................................. 2 . 2 Methods of t h e g r a d i e n t type ( u n c o n s t r a i n e d 75 c a s e ) ............................................ 2 . 3 Methods o f t h e c o n j u g a t e - g r a d i e n t t y p e 76 ( u n c o n s t r a i n e d c a s e ) ............................. 2.4 Constrained c a s e ................................. 79 3. Penalty methods and variants .......................... 81 3 . 1 General remarks .................................. 81 3.2 I n t e r i o r methods ................................. 81 3 . 3 E x t e r i o r methods ................................. 83 3.4 Method o f c e n t r e s w i t h v a r i a b l e t r u n c a t i o n ....... 85 4. Duality methods ....................................... 89 4 . 1 General remarks .................................. 89 90 4.2 Examples ......................................... 4 . 3 A saddle-point s e a r c h a l g o r i t h m .................. 91 4 . 4 A second saddle-point s e a r c h a l g o r i t h m ........... 94 Application o f relaxation and duality methods t o the 5. numerical analysis of a model variational problem ..... 97 5.. 1 General remarks .................................. 97 5.2 The continuous problem ........................... 97 5.3 The approximate problem .......................... 99 5.4 Convergence o f t h e approximate s o l u t i o n as h + 0 . 7.01 Chapter 2 :

5.5

6

.

5.6 5.7

Optimisation algorithms

S o l u t i o n o f t h e approximate problem over-relaxation with c o n s t r a i n t s S o l u t i o n by a d u a l i t y method A n a l y s i s o f t h e numerical r e s u l t s

Discussion

by p o i n t

................. 104 ..................... 104 ................ 108 ............................................ 1 1 3

xix

Contents

Chapter 3

.-

Numerical analysis o f the problem o f the elasto-plastic torsion o f a cyl i n d r i c a l b a r ...........................

117

...............................................

117

Introduction 1. statement of the continuous problem Physical motivation . Synopsis 1.1 Statement of the continuous problem ............. 1.2 Physical motivation 1.3 Synopsis Some properties of the solution o f the probZem (Po) .. 2 2.1 Regularity results

.

. .

3

................................ ............................. ........................................

.............................. 2.2 An equivalent variational problem ............... 2.3 Some particular cases where the solution is known ........................................... Nwnerical analysis of t h e problem (PI) ............... 3.1 Synopsis ........................................ 3.2 Exterior approximation of problem (PI) .......... 3.3 Convergence of the approximate solution as h 0 ........................................ 3.4 Solution of the approximate problem by point over-relaxation with projection ................. 3.5 Applications . Example 1 ........................ 3.6 Applications . Example 2 ........................ Interior approximations of (pol ...................... Synopsis ............................................. 4.1 A finite-element method ......................... 4.2 An interior approximation method using the eigenfunctions of the operator - A in HA(S2) ..... Exterior approximations of (Po) ...................... 5.1 Approximations of J ............................. 5.2 Exterior approximations of KO ................... 5.3 Formulation of the approximate problem .......... 5.4 Solvability of the approximate problem .......... +.

4.

5

. .

6

7

.

Convergence of the i n t e r i o r and exterior approximations

....................................... ............................................. ...................... ...... ......

117 117 118 119

120 120 120 121

123 123 123 125 131 132 135 136

136 137 142 145

145 146 150 150

Synopsis 6.1 A lemma concerning density 6.2 Convergence of the interior approximations 6.3 Convergence of the exterior approximations

150 150 151 152 157

Numerical solution of the approximate problems relating t o ( P ), by relaxation and point overrelaxation w i t 8 sequential projection 7.1 Synopsis 7.2 Description of the method

175

................ ........................................ .......................

175

175

............................ ....................... 7.5 Applications . Example 2 ....................... 7.6 Applications . Example 3 ....................... 7.7 Various remarks ................................ Solution of the approximate problems r e l a t i n g t o (Po) by penalty methods .................................. 8.1 Synopsis ....................................... 8.2 A first penalty method ......................... 8.3 A second penalty method ........................ 8.4 Comparisons .................................... 7.3

Convergence results

7.4 Applications . Example 1

8.

.

Solution of the approrimate problems r e l a t i n g t o (P, ) by duality methods 9.1 Introduction and synopsis

9

9.2 9.3

9.4

.................................. ...................... Application of the duality method . Case of the finite-element approximation of Section 4.1 .... Application of the duality method . Case of the interior approximation by eigenfunctions of - A in H ~ ( Q )................................... Application of the duality method . Case of the exterior approximations of Section 5 ...........

179 188 192

193 194

196 196 197 206 218

219 219 224 231 237

9.5 Application of the duality method of Section 9.1.3 to the solution of the approximate problems

.

10

Discussion

Chapter 4 :

....................................... ..........................................

Thermal control problems. boundary unilateral problems. and elliptic variational inequalities o f order 4

.

2

.

.

................................................... Problems of thermal control and of the d i f f u s i o n of f l u i d s through semi-permeable walls ................. 1.1 Formulation of the problem ..................... 1.2 Existence and uniqueness results for problem (1.4),(1.5) ............................ 1.3 Approximation by a finite element method ....... 1.4 Convergence. of the approximate solutions. (I).. The case q = 1 ................................. 1.5 Convergence of the approximate solutions . (11). The case q = 2 ................................. 1.6 Numerical solution of the approximate problems . 1.7 Examples .......................................

Synopsis 1

241

246

Friction problems

...................................

2.1 Formulation of the problems

....................

249

249 249

249 251 253

255 268 273 278

288 288

xxi

Contents

2.2 Existence and uniqueness results for problem (2.1)> (2.2) 2.3 Relationship with the problems of Section 1 2.4 Dependence of the solution on g 2.5 Duality properties 2.6 Approximation by finite elements 2.7 Numerical solution o f the approximate problems 2.8 An example

3

.

.................................. ... ............... ............................ .............. . ....................................

boundary

...........................................

...................................... .................................. ............................ ............................... ....................................... . .................................... Numerical analysis of variational inequalities of order 4 ............................................ 4.1 Synopsis

......................................

4.2 An iterative method for solving certain variational problems of order 4 4.3 A new variational formulation of the Dirichlet problem for A 2 4.4 An iterative method for the solution of the Dirichlet problem for A’ based on the variational formulation given in Section 4.3 4.5 An approximation of problem (4.2), (4.3) by mixed finite elements 4.6 An iterative method f o r the solution of the approximate problem (4.72). Generalisation to inequalities of order 4 4.7 Example and numerical application

.. .................

................................

........ .........................

5

.

....................... .............

Discussion

Chapter 5 :

.........................................

Numerical a n a l y s i s o f t h e s t e a d y f l o w of a Bingham f l u i d i n a c y l in d r i c a l d u c t

.......................

1

.

288 290

294 297 298 301

A problem with unilateral constraints a t the

3.1 Synopsis 3.2 Existence and uniqueness results for problem (3.1)¶ (3.2) 3.3 Regularity results 3.4 Duality results 3.5 Approximation by finite elements of order one and two 3.6 Numerical solution of the approximate problems 3.7 An example

4.

288

. Physical .............................

Statement of the continuous problem Synopsis motivation

.

308 308

308 308

309 310

313

315

317 317 317 328 330 332 335 339 345

347

1.1 Statement of the continuous problem

347 347

1.2 1.3

348 348

........... Physical motivation ........................... Synopsis ......................................

Contents

xxii

2

.

Some properties of the soZution of the continuous problem

............................................ ............................ ............... ......................................... I n t e r i o r approximation of (P ) by a f i n i t e element method ............................................. Synopsis ........................................... 3.1 Triangulation of Q . Definition of V h ......... 3.2 Definition of the approximate problem ........ 3.3 Solvability of the approximate problem ........ 2.1 Regularity results 2.2 Dependence of the solution on g 2.3 Some particular cases for which the solution is known

3

.

0

3.4

Explicit formulation of the approximate problem

348 348

348 352 355 355 355 355 355 355

3.5 On the use of finite elements of order greater

........................................ Exterior approximations of ( P o ) .................... 4.1 Approximation of J o ........................... 4.2 Exterior approximations of j .................. 4.3 Formulation of the approximate problem ........ 4.4 Solvability of the approximate problem ........ than 1

4.

5.

Convergence of the i n t e r i o r and e x t e r i o r approximations

...........................................

........................................... .....................................

Synopsis 5.1 Convergence of the finite element method of Section 3 5.2 Convergence of the exterior approximations

.

6

.... Methods of solution by r e g u l m i s i n q j .............. Synopsis ...........................................

6.1 Regularisation of the continuous problem ( P o ) . 6.2 Regularisation of the approximate problems .... 6.3 Solution of the problems (I) ation 6.4 Solution of the problems . (11) operator

.

7

.................................... ........................................... .......................................

Duality methods

Synopsis 7.1 Application to the solution of the continuous problem 7.2 Application to the solution of the approximate problems (I) Case of the exterior approximations of Section 4 7.3 Application to the solution of the approximate problems (11) The case of finite-element approximations

.

.

.

357 357

357 359 359 359 359 360 361 363 363 364 371

regularised approximate

. . Method of point over-relax......................................... regularised approximate . Gradient method with auxiliary ......................................

.

356

374 378 382 382 382

.........................

383

................................

391

Contents

8.

xxiii

Application t o the solution of the elasto-plastic torsion problem o f Chapter 3

....................... .............................. ....... ........... ....... (8.8), (8.9) .......................... ....... Application to an example ............. ....... A variant of algorithm (8.13), (8.14), 8.15) .

8.1 Synopsis 8.2 Reformulation of algorithm (9.5), (9.6) (9.7) of Chapter 3. Section 9.1.1 8.3 Approximate implementation of algorithm (8.7)

9

.

8.4 8.5

Discussion

Chapter 6 :

.........................................

398 398 398 399 401 401

402

General methods for the approximation and solution o f time-dependent v a r i a t i o n a l i n e q u a l i t i e s ............. 405

............................................... .........................................

405 406

Introduction 1. Background

1.1 Spaces of vector-valued distributions and

2

.

3

.

..................................... ............................ Introduction t o parabolic time-dependent inequali t i e s of type 1 .................................... 2.1 Examples of parabolic inequalities of type I .. 2.2 Abstract formulations ......................... Approximations o f parabolic inequalities of type I . Synopsis ........................................... 3.1 Fundamental assumptions for the approximation . functions 1.2 Functional setting

3.2 Approximation schemes for parabolic inequalities of type I 3.3 Convergence of the approximate inequalities Nwnerical solution o f some parabolic inequalities of type I 4.1 A model problem 4.2 A model problem of time-dependent friction 4.3 A model problem of the deformation of a membrane

........................

4.

5

.

.

6

.......................................... ...............................

406 407 408 408 411

414 414 415

...

420 422

....

431 431 439

...................................... Introduction t o parabolic inequalities o f type I1 .. 5.1 Example I ..................................... 5.2 Abstract formulation and existence theorem ....

447 453 453

. .

454 456 456

...

456 458

Approximation o f parabolic inequalities o f t y p e - I 1 6 . 1 Fundamental assumptions for the approximation

6.2 Approximation scheme for parabolic inequalities of type 11

....................................

6.3 Convergence of the approximate inequalities

Contents

xxiv

7

8

.

. .

9

10

11

Numerical solution of parabolic i n e q u a l i t i e s of type I1 7.1 Solution of Example I 7.2 Solution of Example I1 7.3 Conclusions Introduction t o time-dependent inequazities of the second order i n t 8.1 Example I 8.2 Example I1 8.3 Abstract formulation Approximation of i n e q u a l i t i e s of the second order i n t 9.1 Assumptions 9.2 Approximation schemes 9.3 Convergence of the approximate inequalities ... Numerical solution of i n e q u a l i t i e s of the second order i n t 10.1 Solution of Example I 10.2 Solution of Example I1 Numerical computation of the flow of Bingham fluids

......................................... ......................... ........................ ...................................

467 469 473 474

.................................. ..................................... .................................... ..........................

476 476 477 477

......................................... ................................... .........................

.

.

.......................................... ......................... ........................

............................................. ......... ............................. ............................. Discussion .........................................

11.1 Notation and statement of the problem 11.2 Numerical schemes 11.3 Numerical results

.

12

References Appendix 1 : 1

.

2.

2.3

.

Further discussion of steady-state i n e q u a l i t i e s ............................

........................................... ..................... ........................ Existence and uniqueness results for (1.1A) and (1.2A) ............................. On the interior approximation of (1.1A) and (1.2A) ....................................

Synopsis Existence. uniqueness and approximation r e s u l t s f o r problems (1.1A) and (1.2A) 2.1 The functional setting 2.2

3

...............................................

.

. General remarks . Con............................. ......................................

The obstacle problem (I) forming approximations 3.1 Synopsis

479 479 481 483 493 493 495 499 499 501

509 517 521

541 541 542

542 542

545 553 553

Contents

xxv

. .................................. 3.3 ......................................... 3.4 ................................ 3.5 ...................... 3.6 . ... 3.7 . .................................. The obstacle problem . (11). Non-conforming approximations using mixed f i n i t e elements ........... 4.1 Synopsis ........................................ 4.2 A dual formulation of the obstacle problem (3.1A), (3.2A) .................................. 4.3 An approximation of the dual problem (4.6A) (4.7A) by mixed finite elements ................. 3.2

Formulation of the problem Physical interpretation Other phenomena related to the obstacle problem Interpretation of (3.1A), (3.2A) as a freeboundary problem Existence. uniqueness and regularity of the solution of (3.1A), (3.2A) Finite-element approximations of problem ( 3.1A) (3.2A). (I) Piecewise-linear approximations Finite-element approximations of problem (3.1A), (3.2A). (11) Piecewise-quadratic approximations

.

4

.

.

5.

.

6

On a stamp problem which leads t o a variational inequality f o r a pseudo-differential operator 5.1 Statement of the problem

........ ........................ 5.2 Functional formulation .......................... 5.3 Finite-element approximation .................... SoZution of nonlinear Xirichlet problems by reduction t o variational inequalities .......................... 6.1 Synopsis ........................................ 6.2 The continuous problem .......................... 6.3 Existence and uniqueness results for (6.2~1,(6.3A) .................................. 6.4 The finite-element approximation of (6.2~) (6.3A) and (6.8A) ............................... Introduction t o numerical algorithms for quasivariational inequalities ....................... ..... 7.1 Quasi-variational inequalities ............ ..... 7.2 Iterative scheme .......................... .....

.

7

.

Appendix 2

. 2.

1

.-

Synopsis

F u r t h e r d i s c u s s i o n o f o p t i m i s a t on algorithms

.............................. .............................................

The method of block overrelaxation with projection

........................ ....................

2.1 Statement of the problem 2.2 Description of the algorithm

...

553

554 555

556

556 561 562 562

562 563

565 565 566 568

569 569 570 571 572 581 581 583

587

587 587 588 589

Contents

xxvi

3

.

4.

5.

.

(2.11A) ....... ................................. Duality methods - a further discussion ............. Introduction t o complementarity methods .............. 4.1 General remarks . Synopsis ...................... 4.2 The obstacle problem from the point of view of complementarity methods ......................... 4.3 Discussion and references ....................... 2.3 Convergence of algorithm (2.10A) 2.4 Various remarks

590 595 595

597 602

Minimisation of quadratic functionals over the products of i n t e r v a l s . using conjugate gradient methods 5.1 Synopsis 5.2 Description of the method . Convergence results

603

5.3

609

..............................................

6.

589 590

........................................ . Discussion ......................................

.

603

604

Alternating direction methods Application t o the solution of nonlinear variational problems

...........

609 6.1 Formulation of the problem . Synopsis ........... 610 6.2 Convergence of algorithms (6.3A), (6.4A) and 611 ( 6.5A) ( 6.6A) .................................. 6.3 Application to the solution of the obstacle 616 problem .........................................

.

.

7

Re lationship between the alternuting direction methods and augmented Lagrangian methods

.....................

617 7.1 A model problem in Hilbert space ................ 617 7.2 Decomposition-coordination of (7.lA), (7.2A)

..................

618 using the augmented Lagrangian Solution of (7.1A), (7.2A) via duality algorithms for L p 619 7.4 An "alternating direction" interpretation of 621 algorithms (7.12A)-(7.15A) and (7.16A)-(7.20A) 7.3

..........................................

..

Appendix 3 :

1

.

2

.

Synopsis

F u r t h e r d i s c u s s i o n of t h e n u m e r i c a l a n a l y s i s of t h e e l a s t o - p l a s t i c t o r s i o n problem

.......................

623

.............................................

623

The finite.-element approximation of problem (1.1A) (I). Error estimates f o r pTecewise-linear approximations

............................................. ............................

2.1 One-dimensional case 2.2 Two-dimensional case

3.

............................

624 624 626

Finite-element approximation of problem (1.1A). (11). @timl-order estimates through the use of an equiv630 alent f o m Z a t i o n

....................................

Contents

xxvii

........ .................................. Further discussion of i t e r a t i v e methods f o r solving the elasto-plastic torsion problem ................... 4.1 General discussion. Synopsis ................... 4.2 Solution of problem (3.1A) by a duality method .. 4.3 On conjugate-gradient-type variants of algorithm (4.3A)-(4.5A) ................................... 3.1 Equivalent formulation of problem (1.1A) 3.2 Approximation of problem (3.1A) by a finiteelement method

4.

Appendix 4 :

Further discussion of boundary u n i l a t e r a l p r o b l e m s a n d el 1 i p t i c variational inequalities o f order 4. A p p l i c a t i o n t o f l u i d mechanics

.............................. .............................................

1.

Synopsis

2.

Conforming and non-conforming approximations of the boundary uniZateraZ problem (l.lA), (1.2A)

...........

630 635

640 640 641 646

653 653

654

2.1 Approximation of the unilateral problem (1.1A) (1.2A) by first-order conforming finite elements. 654 2.2 Approximation of the boundary unilateral problem (l.lA), (1.2A) using non-conforming finite elements of mixed type 658

..........................

3.

Further discussion on the approximation of fourthorder variational problems using mixed finite-element methods 3.1 Synopsis

.............................................. ........................................

3.2 Further discussion of the convergence of the mixed finite-element method of Chapter 4, Section 4.5 3.3 Discussion supplementing Chapter 4, Section 4.6 on the solution of the approximate biharmonic problem (4.72) 3.4 Application to the numerical solution of problem (1.3A), (1.h)

..................................... .................................. ..................................

4.

Nwnerical simulation of the transonic potential f h of i d e a l compressible f l u i d s using variational inequality methods

................................... ........................................ Mathematical formulation ........................ Least-squares formulation of the continuous problem .........................................

4.1 Synopsis 4.2

4.3

662 662 662 663

666 690 690 691 693

4.4 Finite-element approximation and least-squaresconjugate-gradient solution of the approximate problems

........................................

695

Contents

xxviii

4.5 Numerical implementation of the entropy

...................................... .......................... Supplementary bibliography . . . ,. .. , , , .. . . . . . . . .. . . . . . condition

4.6 Numerical experiments

5,

702 709

716

I

Appendix 5

1. 2.

:

Further discussion o f the numerical analysis o f the steadv flow o f a Bingham f l u i d i n a c y l i n d r i c a l duct

................................... Synopsis ............................................

3.

717

Finite-element approximation of problem (1.1A). (I). Error estimates f o r piecewise-linear approximations 2.1 One-dimensional case 2.2 Two-dimensional case

........................... ...........................

717

.

718 718 721

Finite-element approximations of problem (1.lA). (11). Optimal-order error estimates through the use of an equivalent formulation 721

..............................

. . . . .. .

3.1 Equivalent formulation of problem (1.1A) 3.2 Approximation of problem (3.1A) by a finiteelement method

.................................

4.

Supplementary information on iterative methods f o r solving the problem of the steady flow of a Bingham fluid in a cylindrical duct

......................... ..................

4.1 General discussion. Synopsis

. .. ... .. . . .. ... . .. . . . ...... . . .. . .

4.2 Solution of problem ( 3 . U ) by a duality method 4.3 On conjugate-gradient type variants of algorithm (4.3A)-(4.5A) , ,

Appendix 6 :

1.

2.

3.

Further discussion o f the numerical a n a l y s i s o f time-dependent v a r ia t io n a 1 in e q u a 1 it i e s

. ..... . .. . . Synopsis .............................................. Supplementary bibliography ...........................

722

725

727 727 728 730

733 733 733

On the steady flow of a Bingham fluid in a cylindrical duct. Asymptotic properties of the continuous and 736 discrete problems

................................... ............................. .....................................

3.1 Formulation of the problem. Existence and uniqueness results 3.2 On the asymptotic behaviour of the continuous problem

736 737

xxix

Cont e n t s

3.3

Approximation of (3.1A). Asymptotic properties of the discrete solution 3.4 Further remarks

4.

............ ................................

Numerical simulation of the flow of a Bingham f l u i d i n a two-dimensional c a v i t y . using a stream function method 4.1 Synopsis 4.2 Review of the velocity formulation of the Ringham f l o w problem 4.3 A stream-function formulation of the Bingham flow problem 4.4 Approximation of the steady-state problem 4.5 Approximation of the time-dependent problem

..............................................

4.6 4.7

....................................... ........................... ................................... ...... (4.15A) ........................................ Solution of (4.19A), (4.54A) by augmented Lagrangian methods ............................. Numerical experiments ..........................

Bi b l iography o f t h e Appendi c e s

......................

739 743

743 743 743

745 747

754 756 761 767

This Page Intentionally Left Blank

Chapter 1 G E N E R A L METHODS OF APPROXIMATION FOR STEADY-STATE

I N E Q U A L I T Y PROBLEMS

SYNOPSIS A f t e r f i r s t g i v i n g some examples o f steady-state inequality problems ( r e f e r r i n g t o Duvaut-Lions /1/ f o r a d e t a i l e d motiva t i o n ) t h i s c h a p t e r d e a l s w i t h t h e general methods f o r approximating t h e s e i n e q u a l i t i e s . I n t h e s u c c e e d i n g f o u r c h a p t e r s , t h e s e methods w i l l t h e n be a p p l i e d , and t h e main t e c h n i c a l d i f f i c u l t i e s which a r i s e disc u s s e d i n some d e t a i l , f o r t h r e e c l a s s e s o f examples. The r e a d e r who simply r e q u i r e s an o v e r a l l view of s t e a d y s t a t e and time-dependent problems may proceed s t r a i g h t t o Chapter 6 a f t e r r e a d i n g Chapter 1; t o be a b l e t o a p p l y t h e methods i n v e s t i g a t e d i n t h i s book e f f e c t i v e l y , however, it i s e s s e n t i a l t o r e a d C h a p t e r s 2 t o 5.

1.

EXAMPLES 1.1

F l u i d mechanics problems i n media w i t h semi-permeable boundaries

Consider a medium 61 ofqW" ( n = 1, 2 o r 3 i n normal a p p l i c a t i o n s ) w i t h boundary we assume t h a t 61 i s occupied by a f l u i d f o r which t h e p r e s s u r e , i n t h e s t e a d y s t a t e , i s u(x) ( l ) and we assume t h a t t h e boundary r a l l o w s f l u i d e n t e r i n g 6 1 t o p a s s f r e e l y , but prevents f l u i d leaving. Suppose t h a t we a p p l y , on t h e boundary s u r r o u n d i n g 61, a f l u i d p r e s s u r e h = h(x). Under s u i t a b l e c o n d i t i o n s , it can be assumed t h a t , i n a, U satisfies:

r;'

r

(1.1)

- AU = f ,

where

A

=

1ax: ' a2

i=l

(')

f

=

f(x)

g i v e n i n 61,

W e d e n o t e a p o i n t i n 61 by x = {xi, p u t dx = dx, ....dx"

.

.

..., x n }

.

We s h a l l always

(CHAP. 1)

Approximation of steady-state inequalities

2

the boundary conditions being as follows: essary to distinguish two situations: ( i1

for

x ~ rit

14x1> h(x) :

the fluid thus has a tendency to leave a; the wall %\his, however, so that the flux is zero at x, i.e: au -(x) an

prevents

.

The fluid tends to enter fore the flux satisfies

where - ( x )

r

=0

( ii 1 u(x) d h(x)

au an

is nec-

is finite.

a;

the wall

r

allows this and there-

Hence, u is continuous in the neighbour-

hood of x, which, if the wall f is infinitely thin (Duvaut-Lions /l/), implies that u(x) = h(x)

.

S m i n g up, the boundary conditions are:

Remark 1.1. There are thus a priori two regions TI and r2 on f, for which u = h and &/an = 0 , respectively. However, the location of these two regions is not known a priori, and finding these regions is in fact equivalent to solving the problem; we thus have a "free boundary" type problem ( * ) . Transformation of the problem. We .shall now transform the problem (l.l), (1.4) into the form of a (simple) problem in the caZcuZus of variations. This will make it possible: (i) to show that (under suitable assumptions) the problem is "we11 posed" ; (ii) to l a y a foundation for constructive.rnethods f o r approximating the solution.

(l)

(')

d/dn = the derivative normal to r , directed t o m r d s the outside of (though this choice is arbitrary).

This observation applies to all the problems investigated in this book.

3

Examples

(SEC. 1)

We shall use several functional analysis tools; the basic principles behind these ideas may be found in Duvaut-Lions 111, Chapter 1; here we simply recall the definitions (for a complete study of Sobolev spaces see for example Necas 111, Lions-Magenes 111). We denote byH’(62) the Sobolev space (of order 1) of functions u ( I ) such that (see Sobolev 111): (1.5)

ao axi

D E L ~ ( R ) , -~L’(62),

i = 1,

..., n

where L2(62)is the space of (classes of) square summable functions on 62 ( 2, , and where in (1.5) the derivatives au/axi are taken in the sense of distributions on 61. H’(62)is a Hilbert space equipped with the inner product:

It may be shown (see the references cited) that, if the boundary f of 62 is finite and sufficiently regular, we can uniquely define the trace yv of D E H’(62)on f , and yu e L 2 ( f ), the mapping being linear continuous from H’(61)--t L 2 ( r ) (3 ) . D + yv If th’e function h i s given on f, then we can define the subset K of H’(62) by: (1.7)

K={DIuEH’(~~), y u > h aecmf}.

K can be shown to be a closed convex subset of H’(Q). We now introduce (1.8)

a(u,u) =

1

i=

au a0 -h, axi axi

(l)

A l l the functions considered in this book are real-vahed.

(*)

Hence p ~ L ’ ( 6 2 ) o p measurable on 61 and

(3)

Since the image of H’(62)by y i s a smaller space than L * ( f ) , namelyH”’(f),this result can be extended (see e.g. LionsMagenes 111, Chapter 1).

J*

(P(x)~dx < co.

4

Approximation of steady-state inequalities

(CHAP.l)

We s h a l l now v e r i f y t h a t the problem (l.l), ( 1 . 4 ) is equivalent to the minimisation of J(u)over K. This i s based on t h e following g e n e r a l observations: 1) The f u n c t i o n a l u + J(u) i s convex and differentiable, t h e d e r i v a t i v e J’(u) ( ’ ) , given by:

being expressed by:

If u € K c o n s t i t u t e s a minimum of J(u) over

2)

(1.13)

J(u) d J(u)

K, i . e .

Vu E K ,

then (1.14)

(J’(u),u

- U) 2

VU E K ,

0

and conversely. Therefore, u s i n g (1.12), it may be seen t h a t t h e problem i n t h e c a l c u l u s of v a r i a t i o n s : infJ(u), U E K , is equivalent to the

variational inequality: (1.15) 0

- u) 2 (A D - U)

VU E K .

We s h a l l now show t h a t (l.l), ( 1 . 4 ) are equivalent to (1.15). If we denote by 9 ( Q ) .t h e space o f i n d e f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s with compact support i n Q, we see t h a t i f cp~O(s2) then (1.16)

u = u + cp

is i n K. gives

The choice (1.16) i n t h e i n e q u a l i t y (1.15) t h e r e f o r e

4 4 cp) = (f,cp)

vv E 9 ( Q )

and hence equation (1.i) (by d e f i n i t i o n of t h e d e r i v a t i v e s i n t h e sense o f d i s t r i b u t i o n s on Q ) . However, by t a k i n g t h e i n n e r product of (1.1)with u - u and i n t e g r a t i n g by p a r t s , t h i s

( l ) . I n (1.111, J‘(u) denotes an element of t h e d u a l space (H1(Q))’

of If’@) and (J’(u),D) denotes t h e i n n e r product between (H’(Q))’ and H1(P)

.

Examples

(SEC. 1)

leads t o

-

S, @

(u - u) dT

an

+ a(u, u - u) = ( A u - u) ,

so t h a t ( 1 . 1 5 ) is equivalent to (1.1)and (1.17)

By t a k i n g u = u then deduce t n a t

+ $, $ 2 0

t o be a r b i t r a r y ( r e g u l a r ) on

r,

we

and t h e r e f o r e t h a t

au

0 onr

Taking s u c c e s s i v e l y , u = h and u = 2 u - h on t o be s u f f i c i e n t l y r e g u l a r on r ) , we deduce t h a t

r (assuming

h

au

-(u - h) = 0 o n r , an and hence (1.4), and conversely ( 1 . 4 ) i m p l i e s (1.17). This t h e r e f o r e proves t h e s t a t e d equivalence. To swmnarise: A p h y s i c a l problem gave r i s e t o t h e inequali t y problem (l.l), ( 1 . 4 ) and we have j u s t shown t h a t t h i s problem i s equivalent t o a problem i n t h e c a l c u l u s of v a r i a t The l a t t e r i t s e l f has two equivalent formulations: ions. 1)

inf J(u) ; V€K

2)

Find u i n K which i s a s o l u t i o n of a(u, u

- U) >, (fi u - U)

VU E K .

Remark 1.2. The above equivalences, do n o t , a t t h e moment, show the existence of a solution. However, we s h a l l g i v e some i n d i c a t i o n s on t h i s p o i n t i n S e c t i o n 2 , u s i n g t h e formulation corresponding t o a problem i n t h e c a l c u l u s o f v a r i a t i o n s . Synopsis

.

We s h a l l now g i v e some examples, formulated d i r e c t l y i n vari a t i o n a l form.

6

Approximation of steady-state inequalities 1.2

(CHAP. 1)

Model elasto-plastic problem

We shall now give a formulation of a model problem encountered in elasto-plasticity (see Annin /l/, Lanchon /l/, Duvaut-Lions /l/, Chapter 5 and the bibliography cited therein, and Mandel /l/), directly in the form of a problem in the calculus of variations. We assume 62 to be an open domain, as in Section 1.1. We introduce (1.18)

Hi@)

=

{ u I u E H'(62), p

=0}

,

which is a closed vector subspace of H'(62). We then introduce (1.19)

K = { U ) U E H ; ( R ) I, g r a d v ( x ) ) < l ae.in62).

It may easily be shown that (1.20)

K is a closed convex s e t in HA@).

J(u) is defined by (1.9). Hence, the problem becomes (1.21)

inf J(u) ,

,u E K

.

As in Section 1.1, we see that (1.21) is equivalent to seeking u such that (1.22)

-':,;{!

u)

( L o - u)

VUEK.

When 62 is simply connected ( I ) the problem (1.21)- or (1.22)is an elasto-plasticity problem. In Section 2 it will be shown that problem (1.21) Remark 1.3 admits a unique solution.

Remark 1.4 The problem can be interpreted as follows ( 2 ) : the open domain R consists of two regions and 62, (the elastic region and the plastic region, respectively). In Qe (situated

(I)

In the case of a multiply-connected section, it is necessary to modify the formulations (1.21), (1.22) (see e.g. Lanchon / 2 / , Glowinski-Lanchon /l/).

(2)

This will be dealt with in more detail in Chapter 3.

Examples

(SEC. 1)

7

" i n t h e i n t e r i o r " of 0) we have t h e c l a s s i c a l e q u a t i o n :

- AU = f ,

(1.23) and i n

QP

we have:

I grad u I =

(1.24)

1

.

t o which we add: u = O on and some t r a n s m i s s i o n c o n d i t i o n s which i n c i d e n t a l l y a r e somewhat awkward t o e x p r e s s e x p l i c i t l y ( s e e Br6zis 111) - a t t h e i n t e r f a c e between Sl, and a,,. These o b s e r v a t i o n s l e a d t o two remarks:

-

( i ) We once more encounter t h e "free-boundary" n a t u r e of t h i s t y p e o f problem - as mentioned e a r l i e r i n Remark 1.1 f o r t h e problem i n S e c t i o n 1.1; ( i i ) The "global" f o r m u l a t i o n i n t h e form ( 1 . 2 1 ) or ( 1 . 2 2 ) i s much more amenable t o caZcuZation t h a n t h e "pointwise" formula t i o n u s i n g ( 1 . 2 3 ) , ( 1 . 2 4 ) and t r a n s m i s s i o n c o n d i t i o n s at t h e i n t e r f a c e (or " f r e e boundary"). 8

Remark 1 . 5 . Many o t h e r s t e a d y - s t a t e or time-dependent inequ a l i t y probl'ems arise i n p l a s t i c i t y t h e o r y ; i n p a r t i c u l a r , we 8 r e f e r t h e r e a d e r t o Duvaut-Lions 111, Chapt'er 5. 1.3

Model f r i c t i o n problem

We now p r e s e n t a h i g h l y i d e a l i s e d c a s e o f one o f t h e problems encountered i n t h e t h e o r y o f e l a s t i c i t y w i t h f r i c t i o n , or w i t h u n i l a t e r a l c o n s t r a i n t s ( s e e S i g n o r h i 111, F i c h e r a 111). F u r t h e r examples of t h i s t y p e w i l l be given i n Chapter 4. We a g a i n t a k e t h e space H ' ( 0 ) i n t r o d u c e d i n S e c t i o n 1.1 and define t h e functional (1.25)

Au) =

Ir

g Iu 1 d r

where g i s a c o n s t a n t > 0 We d e f i n e (1.26)

a(u, U ) =

In+ 1 au (UU

n

i=l

(1.27)

J(u) = f

(I).

U(U, U )

- av

axi axi

- (A U)

+j ( ~ ) .

The problem c o n s i d e r e d i s t h e n t o f i n d :

( l ) The c a s e where g i s a f u n c t i o n which, f o r example,

inuous on

r

i s cont-

and 2 0, c o u l d e q u a l l y w e l l be considered.

8

Approximation of steady-state inequalities

(1.28)

infJ(u),

(CHAP. 1)

UEH'(Q).

This problem differs from those discussed earlier in that we now seek the lower bound over a vector space, and not over a convex set Kwhich is not a vector space. Moreover, the problem (1.28) does not reduce to the "Euler equation" J'(u) = 0

since the functional j (and hence also J ) is not differentiable. It may easily be shown that problem (1.28) is equivalent to seeking u such that (1.29)

{

, - u) + j ( v ) - j(u) 2 ( J v

U E H'(Q)

4u, u

Remark 1.6. (1.30)

.

- u) Vv E W ( Q )

Generally, if

J(v) = H(u)

+j ( v ) ,

where H i s differentiable and convex, and where j is continuous and convex but not necessarily differentiable, the condition (1.13) is equivalent to (1.31)

(H'(u),u

- u) + j ( u ) - j(u) 2

0 VUEH'(Q).

Remark 1.7. The inequality (1.29) is a variational inequality. It is possible to interpret (1.29). By first taking u = u f rp, rp E ~ ( Q ) , we find (1.32)

- AU + u = f in Q .

F r o m (1.32) and Green's formula, we deduce that

so

g

(v

- u) d f

+ a(u, u - u ) = cf, u - u) ,

that (1.29) is equivalent to (1.32) and

Replacing u by f Iu, 1 > 0, we deduce that

ExqZes

(SEC. 1)

9 -

and hence

However (1.34)

au an

u-

s

+glul>

equivalent t o

1 aupn I s g on

r;

hence

0

which i n combination with (1.35) shows t h a t

To sunanarise, ( 1 . 2 9 ) i s equivalent t o ( 1 . 3 2 ) with the bounda r y conditions on r :

Remark 1.8.

Conditions ( 1 . 3 6 ) a r e equivalent t o t h e follow-

ing (1):

Remark 1.9. Much more complicated problems, corresponding t o r e a l i s t i c p h y s i c a l s i t u a t i o n s , may be found i n t h e a r t i c l e s and l i t e r a t u r e a l r e a d y c i t e d , and i n Chapter 4 of t h e present book. 8

1.4

A flow problem

The laminar flow of a Bingham f l u i d i n a c y l i n d r i c a l duct ( s e e Duvaut-Lions 111, Chapter 6; Mosolov and Miasnikov /l/) l e a d s t o t h e following problem. On t h e space HA@) w e d e f i n e glgradoldx,

~~~~

(l)

g>O,

~~

I n t h i s form t h e "free-boundary" considered i s again e v i d e n t .

n a t u r e of t h e problems

A p p r o x h a t i o n of s t e a d y - s t a t e inequalities

10

(CHAP. 1)

( s o t h a t as i n S e c t i o n 1 . 3 above, j i s a n o n - d i f f e r e n t i a b l e funct i o n a l ) . W e d e f i n e u(u,v) u s i n g (1.8) and, as i n (1.27), we put

J(u) = $ u(v, v ) - (I,u )

(1.39)

+ j(u) .

The problem considered i s t h e r e f o r e analogous t o (1.28) with H'(i2) replaced by H i ( i 2 ) :

infJ(u),

(1.40)

v~Hi(i2),

which i s equivalent t o f i n d i n g u such t h a t

The i n t e r p r e t a t i o n of t h i s problem i n "standard" terms i s complicated; an i n t e r p r e t a t i o n u s i n g m u z t i p z i e r s i s p o s s i b l e - s e e Theorem 3.3, l a t e r . Synopsis

1.5

I n t h e following s e c t i o n ( S e c t i o n 2 ) w e s h a l l p r e s e n t a generaZ framework f o r all t h e above problems which w i l l t h e n enable u s (Sections 3 onwards) t o develop systematic methods f o r approxima t i n g t h e s e problems. GENERAL FORMULATION OF STEADY-STATE VARIATIONAL INEQUALITIES

2.

The symmetric c a s e

2.1

We consider a H i l b e r t space v ( o n t h e f i e l d of real numbers W ) we denote by 11 1) (1 t h e norm o f v i n V . On V we t a k e a continuous b i z i n e a r form u, v - + u ( u , u ) , which t h u s s a t i s f i e s

We assume t h a t t h e form (2.1)

u(u. 0 ) = u(v, u) vu, IJ E

u(u,u)

is symmetric, and hence

v.

We t h e n t a k e K c V with (2.2)

K

= c l o s e d convex subset of

V.

If V' denotes t h e d u a l of Vand ( J v ) t h e i n n e r product between f E V' and D E V , we consider t h e f u n c t i o n a l

General formulation

(SEC. 2 )

11

The problem c o n s i d e r e d i s :

(2.4)

infJ(v),

V E

K.

Remark 2.1. The examples c o n s i d e r e d i n S e c t i o n s 1.1 and 1 . 2 can be shown t o come w i t h i n t h i s c a t e g o r y . As i n S e c t i o n 1, problem ( 2 . 4 ) i s equivalent t o t h e following: (2.5)

{

f i n d U E K such t h a t u - U) 3 ( J u - U) V U E K .

U(U,

Problem ( 2 . 5 ) i s a g e n e r a l variational inequality problem. We can a l s o c o n s i d e r a f u n c t i o n a l v + j ( u ) such t h a t

(2. 6 )

u +J(u)

i s continuous from

v + R , convex

( 1).

We t h e n c o n s i d e r

(2.8)

infJ(u),

UEV.

Remark 2.2. The examples c o n s i d e r e d i n S e c t i o n s 1 . 3 and can be shown t o come w i t h i n t h i s c a t e g o r y . 8 Problem ( 2 . 8 ) i s e q u i v a l e n t t o t h e variational inequality problem: find U E V such t h a t

(2.9)

{

- u) + j ( v )

a(u, 0

- j(u)

> (f,v - u )

vu E

1.4

v.

Concerning t h e existence o f s o l u t i o n s , w e have t h e following classical result: Theorem 2.1.

(2.10)

J(u)+

+ co

If the function when

II 0 II

+

00,

u+

J(u) satisfies

V E K

(2)

then there exists a solution u of ( 2 . 5 ) o r ( 2 . 9 ) . A sufficient c o n d i t i o n f o r ( 2 . 1 0 ) t o be s a t i s f i e d i s t h a t t h e r e e x i s t s u > 0 such t h a t (2.11) a(u, u ) 2 a 11 v 11' V U E V .

('1

I n p r a c t i c e it i s s u f f i c i e n t t o assume t h a t j i s convex, proper and lower semi-continuous; j i s s a i d t o be proper i f j f + c o , j ( u ) > - co V U E v. (2) T h i s c o n d i t i o n i s redundant i f K is bounded in V, which i s t h e c a s e , f o r example, i n t h e c o n t e x t o f t h e example i n Section 1.2.

Approximation of steady-state inequalities

12

(CHAP. 1)

Remark 2.3. Condition (2.11) i s s a t i s f i e d i n t h e examples considered i n S e c t i o n s 1.2 t o 1 . 4 , but not i n t h e example of It can be shown t h a t i n t h i s case (2.10) i s satSection 1.1. i s f i e d i f we assume t h a t (2.12)

l * f ( x ) d x c 0 . rn

For t h e p o s s i b l e uniqueness of t h e s o l u t i o n , t h e following result i s immediate:

If the function u -+ J(v) is strictzy convex ( l ) Theorem 2.2. problem ( 2 . 5 ) (or (2.9)) a h i t s at most one solution. This c o n d i t i o n i s m e t when (2.11) i s t r u e , and t h i s shows t h a t t h e probZems considered in Sections 1 . 2 to 1 . 4 admit a unique so Zution. The uniqueness of the soZution of the probtem of S e c t i o n 1.1 Suppose may be v e r i f i e d directZy, under t h e assumption (2.12). ul and u2 are two p o s s i b l e s o l u t i o n s of ( 1 . 1 5 ) ; by t a k i n g u = u2 ( r e s p . v = ul ), i n t h e i n e q u a l i t y a s s o c i a t e d with u1 ( r e s p . u2 ) and adding, w e g e t , p u t t i n g w

=

u1 - u2 :

- a(w, w ) 2 0 ,

hence a(w, w ) = 0 and t h e r e f o r e w = constant = c ( * ) . If f o r s i m p l i c i t y w e put u2 = u , we t h u s have t h e s o l u t i o n s u and (u c) Hence

+

.

(2.13)

- AU = f ,

(2.14)

u-h>O,

(2.15)

u+c-h20,

aU

~

2

0

au an

,(u-h)-=O

on

aU

au an

(u+c-h)-=O an

-20,

r, on

r.

From (2.13) and (2.12) we deduce t h a t

so t h a t t h e r e e x i s t s a set

au ->O an

on

rl c r , measure (r,)> O',

rl.

However, (2.14) and (2.15) t h e n g i v e

u-h=Oand

u+c-h=O

onr,,

with

Genera1 f o m l a t i o n

(SEC. 2 )

and hence

c = 0,

which proves the uniqueness.

13 rn

Remark 2.4. It is possible to render the hypotheses considerably less stringent by assuming that V i s a r e f l e x i v e Banach space and that the form u,u + a(u, u) is continuous over V, linear in u, but nonlinear in u ; then (2.16)

~ ( uU) , = (A(u), U)

,

V' .

A(u) E

If K i s a closed convex subset of V, the problem can be considered d i r e c t l y i n the form (2.5): find u E K such that (2.17) (A(u) - f,u - U) 2 0 VU E K . It may be shown (see Br6zis /l/, /2/ Browder that if A is monotone, i.e. (2.18)

(+(U)

- A(u), u - U) 2

1 -(A(u),u)-r+oo

II u II

/l/)

0 VU, u E V

and coercive, i.e. (with 1) u 1) (2.19)

111, Lions

again denoting the norm of u in

if I I u l I + o o ,

v):

UEK(')

then probZem (2.17) admits a solution. Clearly, we have uniqueness in the case where A is s t r i c t l y monotone i.e. when we have the s t r i c t inequality in (2.18) for u # u.

We note that (2.17) is not necessarily a problem in the calculus of variations, since the assumption (2.18) does not require that A should be a gradient (on this point, see Rockafellar 111). For extensions and examples, see Lions 111. rn 2.2

The nonsymmetric case

We again consider the situation where V i s a Hilbert space and we consider a bilinear form a(u,v) which is continuous on v, but not necessarily symmetric i e.

.

(2.20)

-

n(u, u) # a(u, u)

in general.

The problem is then expressed d i r e c t l y in the form (2.5); we are thus no longer dealing with a problem in the calculus of variations.

(l)

This hypothesis is redundant if K is bounded.

14

Approximation of steady-state inequalities

(CHAP. 1)

Remark 2.5. I n t h e nonsymmetric c a s e , problem ( 2 . 9 ) can a l s o be expressed d i r e c t l y . It m a y t h e n be shown ( a c o n s t r u c t i v e proof of t h i s w i l l be given i n S e c t i o n 3.1 below) t h a t if (2.11) is satisfied, (where the form a is not symmetric) problem ( 2 . 5 ) (or problem ( 2 . 9 ) ) admits a unique solution. (Variant in the formulation of the problems).

Remark 2.6.

Problem ( 2 . 5 ) i s equivalent t o t h e following:

{

find u c K

such t h a t U(V, u - U) 2 (f,v - U) VV E K . ( s i m i l a r l y (2.9) i s equivalent t o seeking uEVsuch t h a t

(2.21)

(2.22)

u(v, v

- u) + j ( u )

- j ( u ) 2 (f,v

-4

VVE

V).

I n f a c t , i f (2.5) i s s a t i s f i e d then 4 0 ,v

- u) - (f,v - u) = u(u, - u) - (f,v - u) + u(v - u, 1)

- u) 2

0

conversely, using (2.21) and d i v i d i n g by 8 :

and hence ( 2 . 2 1 ) ;

a((i - e l u

1)

+ 8w,w -

- ( f , w- u)

2

o

we deduce ( 2 . 5 ) .

and by making 8 - 0

Remark 2.7. The following remark i s due both t o L. Nirenberg and t o R. Tr6moli8res; v a r i o u s v a r i a n t s and a p p l i c a t i o n s are given i n Tr6moli5res /4/. We introduce, f o r v, w E V: (2.23)

L(u, W ) = U(U, u

- w ) - (f,v - w ) .

If u i s t h e s o l u t i o n of ( 2 . 5 ) o r (2.21) we have:

L(u, u) 2 0 Vv E K

L(u, v )

<0

Vv E K

(

from (2.21)),

( from

(2.5)),

and t h e r e f o r e

(2.24)

L(u, V ) Q L(u, U) = 0 Q L(v, U) VV E K .

Hence

(2.25)

{ u, u }

is a saddle point of L(v, w ) on K x K

Conversely, i f { u , G } i s a saddle p o i n t of a v . w ) t h e n u = 12 = s o l u t i o n of ( 2 . 5 ) .

. on K x

K,

(SEC. 3)

2.3

Infinite-dimensional approximation

15

Synopsis

We s h a l l now study some c o n s t r u c t i v e methods f o r t h e infinite dimensional approximation of t h e s o l u t i o n u of e l l i p t i c v a r i a t ional inequalities. This w i l l form t h e s u b j e c t of Section 3. We s h a l l t h e n go on t o s e e ( i n Sections 4 and 5 ) how t h e equations obtained i n t h e infinite-dimensional c a s e can be approxima t e d by means of finite dimensional problems.

3.

INFINITE-DI~NSION& APPROXIMATION METHODS

3.1

Successive approximations

We d i r e c t our a t t e n t i o n t o t h e case of ( 2 . 5 ) , i n which a(u,V) i s not n e c e s s a r i l y symmetric, and s a t i s f i e s (3.1)

VUEV.

a ( u , u ) & ~ l l u l ) ~ u, > O ,

If K i s a closed convex subset of V , l e t

u be t h e s o l u t i o n i n

K of

(3.2)

U(U,

u

- U) 2

(Lu - U)

V UE

K.

The i d e a i s a s follows: assume t h a t a symmetric, continuous b i l i n e a r form b ( ~ , 2 )i)s known on V , such t h a t

(3.3)

b(u, u ) 2

B II 0 / I 2 , B > 0 ,

VU€

v

and f o r which t h e problem of f i n d i n g

(3.4)

~ ( w , u-

W)

2

G,u- W )

WEK

, the

s o l u t i o n of

VUEK,

( g given i n V ’ ) admits one and only one solution, which is “easy

to caZculater‘.

(1).

Remark 3.1. Since b i s symmetric, problem ( 3 . 4 ) i s equiva l e n t t o t h e minimisation over K o f

so t h a t (Theorem 2.1) problem ( 3 . 4 ) does indeed admit a unique solution. Now, f o r u given i n K (or i n V ) and f o r p > 0 f i x e d but arbi t r a r y f o r t h e moment, t h e r e e x i s t s a unique W E K such t h a t ~~~

(I)

~

N a t u r a l l y t h i s should be defined more p r e c i s e l y . . a l s o Remark 3.2 below.

.. see

Approximations of steady-state inequalities

16 (3.6)

b(w, O - w ) > b(u, u - w ) - ~ [ u ( u , v - w ) - ( J

(CHAP. i )

VUEK.

U-w)]

We t h e r e f o r e i n t r o d u c e a mapping

(3.7)

U +

w

=

S(u) from

K+ K

and we immediately see t h a t ( 3 . 2 ) i s equivalent t o seeking a fixed point o f ( 3 . 7 ) , i . e . a u such t h a t

(3.8)

u = S(u).

Now it w i l l be shown below ( u s i n g t h e method given i n LionsStampacchia /l/) t h a t p can be chosen such t h a t f o r a l l u , , u , ~ K or V we have

(3.9)

I S(UJ

- S(u2) 11 d 6 II u1 - u2 II ,

0 c 6 .c 1 ;

i f f o r t h e moment we assume ( 3 . 9 ) t o be t r u e , t h i s g i v e s :

( i ) t h e e x i s t e n c e and uniqueness o f t h e s o l u t i o n 2 o f ( 3 . 2 ) ; ( i i ) t h e p o s s i b i l i t y o f u s i n g successive approximations f o r t h e a p p r o x i m a t i o n , o f u ; s t a r t i n g from an a r b i t r a r y uo i n K , we c o n s t r u c t u " + I E K , as the s o l u t i o n i n K of (3.10)

b(u"+', u

- u"")

- p [ ~ ( u " tr,

- u"") - (fi v - U"+')]

>/ b(u", u

- u"")

VV E K

and hence u " + u i n V . L e t us now v e r i f y ( 3 . 9 ) . S i n c e b i s symmetric and s a t i s f i e s (3.3) we can t a k e b(v, w ) t o be a new inner product i n V and t a k e

II v II

=

(Nu,

w2.

We put wi = S(ui), i = 1, 2 and t a k e v = w2 ( r e s p . w l ) i n t h e i n e q u a l i t y ( 3 . 6 ) a s s o c i a t e d w i t h w1 (resp. w2 ) . Adding, we deduce:

(3.11)

-11

(3.12)

a(u, u) = ( d u , v)"

~

1

-

11'~ 22 -b(u,-uZ, w1-w2)+pa(u1-u2,

,

,

d

E

O ( V ;V) .

Then (3.11) i s expressed as

(3.13)

II

+ 112

Q ((1- p d ) cp. +)"

.

w I - w ~ ) .

Infinite-dimensional approximation

(SEC. 3 )

SO

17

t h a t (3.13) gives

(3.14)

II JI II

< (1 + p2 II .S7Z 11'

- 2 ap)1/2II cp

II .

S i n c e a > 0 w e can choose p such t h a t (1 and hence, (3.9) f o l l o w s .

+ p2 1) .S7Z (1,

- 2 0 r p ) ' / ~< 1

Remark 3.2. We s h a l l now g i v e a variant o f t h e above method; see a l s o Br6zis-Sibony 111. L e t W be a H i l b e r t space with (3.15)

Wc V,

(3.16)

{

Wdense i n V w i t h continuous i n j e c t i o n ,

KO = c l o s e d convex set i n W, such t h a t K = c l o s u r e of KO i n V

and l e t b(u,7>)be a continuous b i l i n e a r form, symmetric o r o t h e r wise, on W such t h a t (3.17)

b(u,u)&jil]uj&, /3>0,

> 0 , t h e r e e x i s t s a unique u C ~ K o such t h a t

Thus, f o r a l l (3.18)

VUEW.

&(u,, u - u,)

+ c~(u,,u - u,)

(Au

- u,) VU E KO

It may be shown (Huet 111) t h a t as a + O , U i s t h e solution of

.

u,+u

in

V , where

(2.5).

Using ( 3 . 1 8 ) it i s t h e n p o s s i b l e t o c o n s t r u c t an i t e r a t i v e scheme : (3.19)

&,b(U",o- U") 2

( X U- U") - u(U"-',U - u")

VUEK~

which may o f f e r some advantage i n t h e c a s e where ( 3 . 1 9 ) i s " s i m p l e r " t h a n t h e i n i t i a l problem. I n t h e c a s e o f equations , t h e f o l l o w i n g i s a u s e f u l choice ( l ) (Godunov-Prokopov 111); w e r e s t r i c t our a t t e n t i o n t o an example i n two dimensions. We assume t h a t V = H:(f2) , and t h a t a(u,zj) i s We t h e n i n t r o d u c e : as i n (1.8).

u,

(l)

W E

w.

A t l e a s t when r i s t h e union o f segments p a r a l l e l t o t h e c o o r d i n a t e axes.

(CHAP. 1)

Approximation of steady-state inequalities

18

The above t h e o r y a p p l i e s . Here, t h e advantage l i e s i n t h e f a c t t h a t t h e d i f f e r e n t i a l o p e r a t o r B corresponding t o b(u,w) can be decomposed:

B=

( ;:) (I $). I--

-

I n t h e c a s e o f e q u a t i o n s we t h e r e f o r e i n t r o d u c e t h e scheme:

(3.20)

b ( u " , ~ ) = b ( ~ " - ' , ~ ) + p [ ( f , ~ ) - a ( u " - ' ,VuU) E] W , P E W .

I f we assume t h a t t h e s o l u t i o n u' of a(u,u) = (AU) V U E V is an element o f W(which i n t h e a p p l i c a t i o n s w i l l come from a regularity theorem) t h e n

b(u, u ) = b(u, u)

+ p[(f,

1))

- a(u9 41

vv E

W.

T h e r e f o r e , i f w e put

we deduce from (3.20) t h a t

(3.21)

b(v"- @-',0 ) = - Pa(@"",

0)

By t a k i n g u = cp" - cp"-' and t h e n deduce s u c c e s s i v e l y (by p u t t i n g :

b(u, u)"'

=

111 v 111,

+ cp"-'

i n (3.21), we

t h e norm on W e q u i v a l e n t t o

and u s i n g

that :

and hence

Thus, i f w e choose p such t h a t 2a 0

M' c: '

v = cp"

(1

u

(Iw ,

(SEC. 3 )

Infinite-dimensional approximation

19

we have

Consequently: if we choose p in accordance with ( 3 . 2 2 ) scheme (3.20) is convergent in V towards the solution of the problem.

3.2

Penalisation

We now c o n s i d e r t h e c a s e o f ( 3 . 2 ) . The i d e a o f p e n a l i s a t i o n c o n s i s t s o f a p p r o x i m a t i n g ( 3 . 2 ) by equations i n which t h e r e l a t i o n a p p e a r i n g i n ( 3 . 2 ) , which expres s e s t h e f a c t t h a t u b e l o n g s t o K , i s r e p l a c e d by a penalisation t e r m which becomes p r o g r e s s i v e l y l a r g e r as t h e s o l u t i o n "moves away" from K and t h u s f o r c e s t h e l i m i t o f t h e approximate s o l u t i o n s t o b e l o n g t o K. S p e c i f i c a l l y , w e i n t r o d u c e a penalisation operator B which h a s t h e following p r o p e r t i e s :

i

(3.23)

B maps V i n t o V ' , B i s L i p s c h i t z c o n t i n u o u s Ker(B) = k e r n e l o f B = K B i s monotone.

(l)

It can be shown t h a t t h e r e exists a 6 which h a s t h e p r o p e r t i e s ( 3 . 2 3 ) . I n d e e d , t h e r e e x i s t s an infinite number of p e n a l i s a t i o n o p e r a t o r s ( f o r a g i v e n convex s e t K ) .

Example 3.1. We t a k e Example (1.7) o f S e c t i o n 1.1. For D, w E V = H'(f2) we p u t (3.24)

(P(D), W ) =

(3.25)

4-

- (D - h ) - w d T Sr where i n a g e n e r a l manner we have d e f i n e d SUP

(-

$9

0)

-

It may e a s i l y be shown t h a t ( 3 . 2 3 ) i s s a t i s f i e d . We may a l t e r n a t i v e l y d e f i n e by

(P(D), W ) = -

(3.26)

S,

M ( D- h)- w dT,

where M i s a c o n t i n u o u s f u n c t i o n > 0 on

Penalised equation. For (l)

E

> 0 w e c o n s i d e r the equation:

G l o b a l l y or l o c a l l y .

r.

20

(3.27)

Approximation of steady-state inequalities u(u,. U)

1 +; (B(uJ. U) = (5 U)

( C H A P . 1)

VV E V .

Equation (3.27) i s nonlinear. The e x i s t e n c e t h e problem comes from t h e g e n e r a l t h e o r y o f ( s e e Minty 111, Browder 111, Leray and Lions Equation ( 3 . 2 7 ) i s t h e penalised equation problem ( 3 . 2 ) .

of a s o l u t i o n u, o f monotone o p e r a t o r s

f1f 1. associated with

Using t h e c h o i c e ( 3 . 2 4 ) a p e n a l i s e d problem Example 3.2. a s s o c i a t e d w i t h (1.15) i s :

W e now show ( s e e Lions 111) t h a t : Theorem 3.1. (3.2).

As

-,0,

u, + u in

V,

u

being the solution of

The basic principle behind t h e proof i s as f o l l o w s : F i r s t , it i s v e r i f i e d t h a % u, r e s i d e s i n a bounded s u h s e t of V ( t a k e u = u, - uo, uOcK i n ( 3 . 2 7 ) , i n t r o d u c e A v o , u s e t h e c o e r c i v i t y and t h e c o n t i n u i t y o f A and

( B W 4 - uo) = (B(U.1

- B(vo), u,

- uo) 2

0).

It may t h e n be deduced from ( 3 . 2 7 ) t h a t

(B(u3,v) = &[(A0)

- 44,41

9

and, hence

(3.29)

)I B(u3 1) Y ‘ = WE)

A subsequence, also denoted by u, t h e sequence u, , such t h a t :

(3.30)

u,+u

, can

t h e n be e x t r a c t e d from

weakly i n V .

From t h i s it may be deduced (by u s i n g t h e monotonicity o f B ) that B(u) = 0 , and, hence, ( u s i n g ( 3 . 2 3 ) ) t h a t U E K . By r e p l a c i n g v i n (3.27) by v - u, v E K, it may be deduced that : 1 4u,, v - u,) - (I;0 - u3 = & (B(4- B(u3, 0 - u,) b 0 ,

-

and, hence

(3.31)

u(u,, U) - (f,u - U > b u(u,, u J .

(SEC. 3 )

Infinite-dimensional approximation

21

Since lim inf a(ucru,) 2 a(u. u), we deduce from (3.31) that c+ 0

a(u, u) - ( A u - u ) 2 a(u, u) V u s K, i.e. ( 3 . 2 ) . Remark 3.3. Numerous variants of the above method are poss-

ible; we shall meet some examples of these in the following chapters.

Remark 3.4. If we express the penalty method in the context of the "calculus of variations" - by assuming that a is symmetric and that 6 i s the gradient of a functional y (which is permissible) - problem (3.27) amounts to the minimisation of the penalised functional: (3.32)

1

1

5 U ( U , U) - ( A U) + -E Y(u )= J ~ u.)

In its present form, this method was introduced by Courant /l/. This idea has prompted numerous investigations in the theory of mathematical programming. We shall come back to this point in the following chapters. 3.3

Regularisation

We now consider a problem of the type (2.9) where the functional j is nondifferentiable. It is natural to "approximate" j by a family of functionals, j,which are convex and differentiable.

Example 3.3. (3.33)

jc(4=

Within the setting of Section 1.3 we may take

I

gCp,(u)

dT ,

where the functions d+q,(d) are convex differentiable approximations of d + I d I . For example, we can take

or, as is preferable in numerical applications

We then "approximate" problem (2.9) by (3.36)

a@,, u - u,)

+ j,(u)

- j,(uJ 2 ( A u - u3

Vu E V .

However, since j , is differentiable (3.36) is equivalent to the equation

(3.37)

a(u,, u)

+ (ji(uc).u) = (f,u)

Vu E V

.

Approximation of steady-state inequalities

22

(CHAP. 1)

This nonlinear equation admits a unique solution, by virtue of the general theory of monotone operators (since the operator is monotone from v 4 V ’ ) . 0 + j:(u) It may be shown that (3.38)

u , + u in V when e - 0 ,

where u is the solution of ( 2 . 9 ) .

This is achieved by first showing that-u,resides in a bounded subset of V ; a subsequence, also denoted by u, , can then be extracted, such that u , + u weakly in v. We deduce from (3.36), (3.37) that 4u,,

- u3 + j,(4 - jc(u3 - ( A 0 - u,) = ( ) = j,(o) - j,(u3 - (jl(U,), u - u*) 2 0

therefore a(u,, 1)) - ( A u - u,)

(3.39)

+ j.W

2 a(u,, u3

+jAU3 .

However (and these are the axiomatic approximation assumptions which are satisfied, for example, in (3.33)):

so that , since hm inf a(u,, u 3 2 u(u, u), we deduce from (3.39) that a(u, u ) - ( A 0 - u)

+ j ( u ) 2 a(u, u) + jtu)

i.e. ( 2 . 9 ) .

Remark 3.5. It is easy to extend (3.33) to the other situations considered. Some examples of applications are given in the following chapters. 3.4

Duality (I)

The theory of duality i s open to a number of approaches, which lead to problems with greatly differing analytic expressions. In this section and the two sections which follow, we shall present the three ideas which strike us as being the most important in this direction. The methods described in this section and the next relate to one example but we shall endeavour to extract from this example points which are of a general nature. Other examples and, in particular, numerous variants and numerical applications will be rn given in the following chapters. (l)

Using the convexity of j .

.

(SEC. 3 )

23

Infinite-dimensional approximation

The b a s i c concept behind a l l d u a l i t y t h e o r i e s i s t h a t a lower semi-continuous, proper, convex function is the upper envelope of the affine functions which are below it. Assuming t h e c o n d i t i o n s o f S e c t i o n 1 . 4 , t h e a p p l i c a t i o n of t h i s i d e a amounts t o s a y i n g t h a t t h e f u n c t i o n a l j g i v e n by (1.38) satisfies

,.d

(3.41)

j ( u ) = sup

gp.grad u d x ,

where t h e sup o v e r p i s t a k e n f o r p e B w i t h

( P I < 1 a.e.

B = { p l p = { p,,..., p , } ,

(3.42)

i n f2)

The problem ( 1 . 4 0 ) i s t h e r e f o r e e q u i v a l e n t t o

(3.43) We p u t ( 9 i s o f t e n termed a "Lagrangian"): 1 U ( u , p ) = za(u, u) - ( J u)

(3.44)

+

We t h e n have:

inf sup U ( u , p ) = J(u) .

(3.45)

u.H@)p.b

where u i s the s o l u t i o n o f ( 1 . 4 0 ) . From ( 3 . 4 5 ) w e deduce ( s i n c e w e a l s o have sup i n f that :

(3.46)

sup

<

i n f sup)

inf U ( u , p ) < J(u).

P € @ tI€H@)

Using a direct method ( '1, Theorem 3.2.

(3.47)

sup p€Lp

w e s h a l l now prove

We have

inf U ( v , p ) = J(u). v€Hd(n)

The proof l i e s i n t h e f o l l o w i n g r e s u l t , which i s i t s e l f o f interest:

m

Theorem 3.3. If u is the solution of ( 1 . 4 0 ) there exists ..., m, } with

= { m,, ~~

(I)

It i s a l s o p o s s i b l e t o u se t h e g e n e r a l r e s u l t s on min-max theorems.

Approximation of steady-state inequalities

24

<1

a.e. .in

D

(3.48)

lml

(3.49).

m.grad u = I grad u I a.e.

(CHAP. 1)

(l)

6-z D,

Conversely, if u,m satisfy (3.48),... ,(3.51) then u is the solution of (1.40). Remark 3.6. Theorem 3.3 gives an interpretation of the problem in Section 1.4. By successively taking

Proof of theorem 3.3.

V =

0 and

V = 2u in (1.41) we obtain

(3.52)

U(U,

U) -

then, with

(3.53)

(f,U) + j(u)

=0

k v and taking account of (3.52) , we obtain

=

1 a(u, 0 ) - (f,v ) I Q j ( v ) ,

E M a ).

(it will be noted that (1.41) is equivalent to (3.52), (3.53)). However, putting (3.54)

1

r=-(Au+f), 9

we see that (3.53) is equivalent to

(3.55)

I (r, u) I


I grad v I dx ,

Vu E Hd(Q).

We introduce the space

and the mapping

Thus (3.55) is equivalent to

from

H;(D)-r @

(l)

Therefore m e 9

(2)

C2 is assumed to be

(2).

(defined in (3.42)).

bounded.

(SEC. 3)

(3.56)

Infinite-dimensional approximation

1 (r, v ) I 6 I1

25

Ilo

and therefore, from the Hahn-Banach theorem, there exists m c @ ’ = ( L m ( Q ) ) n , such that

(3.57)

II m

(3.58)

(r, u ) = (m,nu)

1,

llw Q

=

and (3.52) is equivalent to

(3.59)

(r, u ) =

I,

I grad u I dx

However (3.57) is equivalent to (3.48), (3.58) to (3.50) and (3.59) to (3.49) (sincelml 6 lae.). Since ucHA(62) we therefore have (3.48) to (3.51). The inverse property may be verified by means of a direct calculation.

Proof of theorem 3.2.

inf

VEHd(D,

We can explicitly calculate

U(u,p )

which is met by the solution V = U ( p ) of

(3.60)

{ - Au u = o

gdivp onf.

=

+f,

We then have, for v = v ( p ) ,

(3.61)

U ( U ,=~ )f fa(^, u),

and consequently SUP

(3.62)

in{

p s l p V€H0(D)

U ( 0 ,p ) =

sup

PElp

- 4 .(u@).

u(p) solution of ( 3 . 6 0 )

.

u@))

,

If we take V = u , p = m, where u,m satisfy tne conditions of Theorem 3.3, we have p E 4, and SUP

inf

PESUEHb(D)

U ( u ,p ) B

- 4 a@, U) = J(U) ,

which, in combination with (3.46), proves

(3.47).

Remark 3.7. From the point of view of numerical applications, we therefore have the option of solving, instead of the initial (or primal) problem, the (dual) problem: (3.63)

inf P E P

t a(v(P), 4 ~ ) )

where u is a solution of (3.60).

26

Approximation of steady-state inequalities

(CHAP. 1)

In the following chapters we shall present some approximation aZgorithms for (3.63).

Remark 3.8. The simultaneous solution of the primal problem and a dual problem gives, as in classical theory, bounds for J(u):

J(u) d J(u) Q 3 a(u(p), u@)) vu, VP E 9 . We can consider (3.63) as an optimal control Remark 3.9. problem f o r a system governed by partiai! differential equations (see Lions / 2 / for this theory). In fact, we consider p as the control (or the command). The state of the system V = V ( p ) is given by the solution of (3.60) and the cost function, or criterion, is given by

(3* 64)

J(P) = 3 a(v(P), u(P,)

.

The optimal control probZem then consists of seeking

(3.65) 3.5

infJ(p),

PEP.

Duality (11)

Two other ideas, which are also fundamental in duality, are the following: (i) It is possible to "suppress constraints" by using "Lagrange multipliers"; (ii) The problem can be decomposed by the addition of "artificial constraints" which are then suppressed using method (i).. We shall now consider this in more explicit terms, using the same example as in Section 3.4. rn

Introduction of artificial constraints. The problem (1.40) is clearly equivalent to seeking

(3.66)

inf

;In

I w 1'dx

- (f,u)

+ Jng I w I dx

subject to the constraint (3.67)

w=gradu,

'uEH&?).

rn

Suppression of constraints by using multipliers. We use the fact that: SUP (P. w - Brad 4, P E (L'(Q))" P is equal to + 03 unless (3.67) is satisfied; this is an extension of the idea (3.41).

(SEC. 3 )

Infinite-dimensiona2 approximation

Hence we get the natural introduction of the problem:

(3.68)

sup inf A ( u , w , p ) P

v. w

where

(3.69)

A ( u , w, p ) =

:In

I w I’ dx - ( A u)

+ ( p , w - grad 0). 2,

+

rn

Remark 3.10. In (3.68) the functions V,W are independent; covers H;(bl) and w covers (L’(bl))”. We note (arid we have done what is necessary for this) that:

(3.70)

M u , grad u, p ) = J(o)

.

Hence

inf A ( u , w, p )

< inf J(u) = J(u) ”

v. w

and consequently

(3.71)

sup inf A ( u , w , p ) < J(u) . P

v, w

We shall now prove Theorem 3.5.

(3.72)

We have

sup inf A ( u , w, p ) = J(u) P

Proof.

u. w

We have

However

(3.73)

infA,(u,p) = U

0

if divp=f,

- OT) otherwise.

Then

I pi-* B

I n fact, this reduces to calculating

Approximation of steady-state inequazities

28

If

(CHAP. 1)

We immediately s e e t h a t t h i s lower bound i s zero i f IpI 2 g , w e n o t e t h a t

i?f[+ItI’ + s I t I

Ip 1 Q g

.

+ p t I = -+(IPI - d ’ +

+~~~[+(I~I-IPI+~)’+P~+IPII~II= = -f(IPI-9)’

which i s met by

t = - -( PUP PI 1

- g).

Consequently we have ( 3 . 7 4 ) and

We now t a k e { u , m } which s a t i s f y t h e c o n d i t i o n s o f Theorem 3 . 3 and l e t :

(3.76)

p = -gradu-

gm

Then, u s i n g ( 3 . 5 0 1 , we have divp=f ; moreover, i f gradu=O, t h e n Ip 1=g I m 1s g ; hence w e need o n l y be concerned w i t h t h e r e g i o n where gradu # 0 Now

.

(3.77)

therefore IPI=g+lgradul>g

and (3.78)

I

(lp I

- g)’ dx

=

IPI,Q

( s i n c e gradu = 0 a t i o n s ) . Hence SUP P

n

I grad u 1’ dx

=

1

5

I grad u 12 dx

IPOQ

i n t h e r e g i o n e x c l u d e d from t h e i n i t i a l i n t e g r -

inf &(u, w , p) 2 U. w

5I

-

f o r t h e c h o i c e ( 3 . 7 6 ) of p and t h e r e f o r e from ( 3 . 7 8 ) :

divp = 1 i n $2 and t h e i n t e g r a l i s e x t e n d e d t o t h e r e g i o n where I p ( x ) 12 g

( I ) The f o l l o w i n g s h o u l d be u n d e r s t o o d :

.

(SEC. 3 )

Infinite-dimensiona2 approximation

sup inf A ( u , w , p) 2 - 3 a(u, u) P

=

29

J(u) ,

h w

which, in combination with (3.71),proves the Theorem. Remark 3.11. In view of the ent subject matter to subsequent example of the above method. We find a vector u = { u , , ...,q,} and a (3.79)

- P A U = f - gradp,

(3.80)

div u = 0 ,

(3.81)

u = 0 on

0

extreme importance of the presarguments we shall give afirther consider the Stokes problem: scalar p such that:

r.

We are thus dealing with a problem of equations in which (3.80) is to be considered as a constraint. We introduce (3.82)

V = {ulu~(Hi(G?))”, d i v u = O in a }

Thus the problem (3.79),(3.80), (3.81) is equivalent to seeking : (3.84)

J(u) = 3 a(u, u )

inf J(u) , VE

v

- (f,u)

o r , equally, to seeking U E V with

(3.85)

U(U,U) =

(Ju)V U E V .

Taking (3.80)as a constraint, we introduce, by analogy with the above : (3-86)

1

M ( u . P ) = ~ a ( vU),

- (f,u) -

p(div u) dx

J”

where (3.87)

uE

W

=

(Hi(a))”,p E L z ( s l ) .

It may now be verified directly that (3.88)

sup inf N(U, p) P

V

=

J(u) .

Approximation of steady-state i n e q u a l i t i e s

30

inf m u , p ) , U E W

We c a l c u l a t e

u

bound i s a t t a i n e d f o r

V

.

( Cm.1)

For a given p , t h e lower

= V ( p ) , t h e solution of

- p A u = f - gradp,

(3.89)

u=O

on f ,

and t h u s

inf -4% p ) = - 4 a(u, u ) . v

We t h e n have: - inf

(3.90)

4 a(u, u ) = J(u)

P

where v = ~ ( p i)s t h e s o l u t i o n o f (3.89) (l). Algorithms f o r t h e approximation o f (3.90) a r e given l a t e r i n t h i s volume. rn The above methods a r e c l e a r l y not confined t o Remark 3.12. 2nd order o p e r a t o r s . I n Chapter 4 we s h a l l meet some examples o f t h e i r a p p l i c a t i o n f o r c e r t a i n 4 t h o r d e r o p e r a t o r s which appear i n t h e theory of t h i n p l a t e s .

Remark 3.13. A s w e have j u s t s e e n , t h e i d e a s o f d u a l i t y can l e a d t o a f o r m u l a t i o n which - i n appearance d i f f e r s widely from t h e i n i t i a l f o r m u l a t i o n . Moreover, more g e n e r a l l y , s t a r t i n g w i t h a problem i n t h e c a l c u l u s o f v a r i a t i o n s

-

inf J(u) v

f o r which the lower bound i s not necessarily reached (e.g. t h e theory o f m i n i m surfaces) w e can i n t r o d u c e d u a l i t y problems SUP @(PI P

suc h t h a t

(.i)

sup @@) = inf J(u) ; P

v

( i i ) t h e r e e x i s t s p which g i v e s t h e sup o f @ . For minimum s u r f a c e s , s e e T6mam /l/. We t h u s have a relaxat-

ion process

(2).

( l)

We a g a i n have a " d i s t r i b u t e d - s y s t e m o p t i m a l c o n t r o l ' ' type problem; s e e Remark 3.9.

(2)

Here, t h e t e r m " r e l a x a t i o n " i s t o be t a k e n i n a completely d i f f e r e n t sense from t h a t of t h e c l a s s i c a l relaxation methods o f numerical a n a l y s i s which, i n c i d e n t a l l y , we s h a l l a l s o use l a t e r .

Infinite-dimensional approximation

(SEC. 3)

31

Also, see Remark 3.17 later.

Various relaxation processes, particularly for nun convex functionals, are encountered in optimal control theory: see Ekeland /1/ and Bidaut /l/.

3.6

Duality (111)

Another possibility is offered by the use of conjugate functions; this classical idea, due originally to Legendre, is developed in Fenchel /l/, Mandelbrojt /l/, Hijrmander /l/, Moreau /l/, and Rockafellar / 2 / . As we shall now see, this does not assume that the form a(u,O) is symmetric, as was the case in the two previous sections. rn Over a Hilbert space V we say that a convex function u + # ( u ) is proper if:

(i) (ii)

is lower semi-continuous, with values in

@ @

is not identical to

+

00.

1- a,+Q)] ;

rn

Example 3.4. If K is a closed convex subset of the function given by

V, K # 0 ,

is a proper convex function. rn Using this concept, we see that the inequality (3.2) is equiualent to seeking u in V such that

(3.92)

U(U,

u - U) - (f, u -

U)

+ GK(u)- @ & ) > O ,

VUEV.

We also note that the function @ = j introduced in Section 2.1 is a proper convex function, so that all the inequalities mentioned so far come within the following general framework:

(3.93)

{

find U E V w i t h a(u, u - u)

- (f,u - u)

+ @(u) - @(u) 2 0

Vu E V .

In order to simplify notation in the following, we introduce the operator A where: A E U ( V ;V ' ) , (3.94) u(u, u) = (Au, u) Thus (3.93) is equivalent to

(3.95)

( ~ u - f , ~ - u ) + @ ( u ) - @ ( u ) , OVUEV.

We shall now transform (3.95), using sub-differentials. If the function @ is differentiable, then, from the convexity:

Approximation of steady-state inequalities

32

(3.96)

@(u)

- @(u) - (@'(u),u - u) 2

0

v.

VU€

We now i n t r o d u c e , i n a g e n e r a l manner, the set o f that

(3.97)

@(u) - Y u )

- ( L o - u) 2

(CHAP. 1)

< E V ' such

0 VUEV,

a concept which i s v a l i d whether @ i s d i f f e r e n t i a b l e a t u o r not. The s e t o f 5 such t h a t ( 3 . 9 7 ) i s s a t i s f i e d i s denoted by a@(u) and c a l l e d t h e sub-differential of @ a t u ( s e e Moreau /I/). We o f t e n w r i t e

(3.98)

-

@u) - @(u) - (ayu),

- u) 2

0 vv E

t h e c o r r e c t e x p r e s s i o n being ( 3 . 9 7 ) w i t h Clearly, i f @ i s d i f f e r e nt i a bl e at u :

(3.99)

a@(,)

=

{€a@@) .

{ @ '(4 1.

Using t h i s c o n c e p t , we s e e t h a t ( 3 . 9 5 ) is equivalent to - (h- f)E

(3.100)

a@@).

Remark 3.14. We observe t h a t ( 3 .loo) may be viewed a s an equ a t i o n r e l a t e d t o a multivalued operator (sometimes c a l l e d a setvalued operator). We s h a l l now t r a n s f o r m (3.100) u s i n g t h e concept o f a conjug-

ate convex function. If @ i s a p r o p e r convex f u n c t i o n , we d e f i n e the conjugate fun-

ction @*on t h e space V' by (3.101)

@*(u*) = sup [(u*, u ) - * u ) ]

.

1)

It may be shown ( s e e Moreau /l/) t h a t p i s a proper convex function.

Example 3.5. (3.102)

We t a k e V = Hd(Q),

K given by

K = { u I u E V ,u 2 0 8.e. in Q } . @=@I given by (3.91)

.

We i d e n t i f y L'(i2) with i t s dual; t h u s V c L'(Q) c where H-'(i2) i s th,e s p a c e ' o f d i s t r i b u t i o n s o f t h e form

Now t h e c o n j u g a t e f u n c t i o n o f @*(u*) = SUP (u*,U) ,

and hence

uEK

i s given by

V' = f f - ' ( Q ) ,

(SEC. 3 )

(3.103)

Infinite-dimensional approximation

@*(u*) =

{ + co

ifu*
0

33

(l)

If we introduce

K*= {u*[u*EV', u * < O }

(3.104)

=

{ cone of measures


in H - ~ ( Q 1) ,

we then have, using the notation (3.91): (3.105)

@* =

w

@p.

The following equivalence can be verified immediately:

(3.106)

u* E a@@)0 u E a@*(u*) .

This equivalence forms the basis of the duality in the present sense; here we follow the method described by Mosco /1/ ( 2 ) . From (3.100) and (3.106) we deduce that

We put: (3.108)

A-'J= F,

(3.109)

U* =

- AU

+f.

Thus (3.107) is equivalent to

- A -yu*

- f) = - ( A - 1 u*

- F ) E a@*(u*)

or (3.110)

( A - I U*

- F, U*

- u*) + @*(u*) - @*(u*) 2 0 ,

VU* E

V'

Inequality (3.110) is one "dual" inequality of (3.95).

W

Example 3.6. (see Fusciardi, MOSCO, Scarpin; and Schiaffino /l/). We consider the context of Example 3.5, with (3.111)

@,u)

= lngradu.gradudx.

Suppose J/ to be given in H,'(Q) (3.112)

,

and

f = - A+

(1)

in the sense of distributions in R , which is equivu*
('1

In which nonzinear operators d are also considered.

Approximation of steady-state i n e q u a l i t i e s

34

( C H A P . 1)

The i n e q u a l i t y :

{ :;,:'-

U ) 2

(ftu

- U)

VUEK,

t h e r e f o r e amounts t o :

u(u-$,u-u)>O

(3.113)

VUEK,

and hence i s e q u i v a l e n t t o f i n d i n g the projection of $ , i n the sense of the inner product ( 3 . 1 1 1 ) , on the cone of Hd(62) of functions 2 0. We p u t : A-' = G = Green's o p e r a t o r i n problem. Thus, from (3.105) and by n o t i n g t h a t is equivalent t o

(3.114)

{ '*'(Gu*- +,

(3.115)

On

u*

" - u*) 2

0 Vu*

<0

for the Dirichlet

F=

$,

problem (3.110)

on

Examples of numerical a p p l i c a t i o n s of (3.115) a r e given i n F u s c i a r d i , MOSCO, S c a r p i n t and S c h i a f f i n o 111.

Remark 3.15. In actual applications, a d i f f i c u l t y i n t h e implementation o f t h e above method r e s t s i n t h e e x p l i c i t calculation of t h e c o n j u g a t e f u n c t i o n @ * o f @. Remark 3.16. A s l i g h t l y d i f f e r e n t approach i s given i n Cea 111, which i s a l s o v a l i d f o r nonsymmetric a ( u , v ) . See a l s o t h e a r t i c l e by Cea

121.

Remark 3.17. The f u n c t i o n @ i s given on V. We have i n t r o d uced t h e c o n j u g a t e f u n c t i o n on V ' . However, i d e n t i f y i n g V w i t h V ' , we' can also i n t r o d u c e the dual function on V. Moreover, t h i s concept o f a d u a l f u n c t i o n depends on the inner product

.

chosen We assume t h a t a ( u , v ) i s symmetric and t a k e a ( u , v ) as a new i n n e r product on V ; (3.116)

U(U,U)

we t h e n put

= [U.v1.

We i n t r o d u c e . t h e "Green's o p e r a t o r " G E y(V' ; V ) by means o f : (3.117)

(ftu) = [Gf,u] V U E V .

Thus ( 3 . 9 3 ) i s equivalent t o (3.118)

[U

- G ~ ,u U] + @(u)

- @(u)

2 0 , V U EV .

Infinite-dimensional approximation

(SEC. 3 )

The c o n j u g a t e f u n c t i o n o f is:

(3.119)

Q,

35

on V which must now be i n t r o d u c e d

d*(u)= SUP ([u, ~ p 1- '@(PI). QEV

Then ( 3 . 1 1 8 ) i s e q u i v a l e n t i n t u r n t o

- (U - Gf)E ~ Q , ( u ) e u E a d * ( - (u - G f ) ) . We p u t :

(3.120)

w =

- u + Gf.

Then ( 3 . 1 1 8 ) i s e q u i v a l e n t t o

- w + GfEad*(W) - Gf, u - W ] + d * ( ~-) d*(w) 2 0 0 a(w, u - w ) - ( A u - w ) + d*(u)- $*(w) 3 0 0

(3.121)

[W

t h i s problem b e i n g i n t u r n e q u i v a l e n t t o t h e m i n i m i s a t i o n o f t h e functional:

(3.122)

+ d*(~),

+u(u,u) - (Ju)

UEV.

We t h u s have t h e v e r y "symmetric" result:

the problem of the

minimisation of the functional (3.123)

4 U(U, U) - ( A U) + Yu), 0 E

is "in duality" with ( 3 . 1 2 2 ) ; if u ( r e s p . w ) denotes the solutions of ( 3 . 1 2 3 ) ( r e s p . ( 3 . 1 2 2 ) ) . , then we have ( 3 . 1 2 0 ) , i . e . u

+ w = Gf.

( s e e a l s o , f o r example, Lions / 2 / , Chapter 3 , S e c t i o n 1 2 ) . W e now p r e s e n t an application of this result, t a k e n from B r d z i s 121, (Theorem 1 . 5 , C h a p t e r 1). We t a k e

K = { u ~ u E V I,v ( x ) I d 1 ae} @ ( u ) = O i f U E K , + a if u $ K . We t a k e f in Lz(Q) ( a n d n o t i n H-'(a)= V' ) . It may t h e n be shown (Brbzis, loc. cit.) t h a t t h e s o l u t i o n u of ( 3 . 1 2 3 ) sati s f i e s ( 1 ) (regularity theorem) :

(l)

Assuming t h e boundary

r

of Q t o be s u f f i c i e n t l y r e g u l a r .

Approximation of steady-state inequalities

36

(3.124)

(CHAP.

1)

u E H Z ( Q )n H i @ ) .

However, i f f E L2(Q) we have: Gf E HZ(P)n Hi ( P) and consequently t h e s o l u t i o n w o f t h e d u a l problem ( i n t h e above s e n s e ) s a t i s f i e s

(3.125)

w = Gf - u E H z ( P ) n H ; ( Q ) .

Hence, the lower bound of the functional (3.122) is reached on , and t h e r e f o r e

H z ( 0 ) n Hi(Q) (3.126)

The advantage o f t h e second f o r m u l a t i o n i n (3.126) i s t h a t c$*(v) immediately becomes explicit on H z ( Q ) n H;(Q) , whereas t h e same i s not t r u e on H i @ ) In fact, if

.

u E H Z ( Q )n HACQ), t h e n (3.119) g i v e s

The d u a l problem i s t h u s to find the lower bound ovey H2(Q) n H;(Q)

of. (3.127)

-21 U(U, U) - ( A U) +

(The noncoercive n a t u r e of t h e f u n c t i o n a l (3.127) on Sz(0)n HA(i2) may be noted; i f we i n t r o d u c e t h e space X o f U E H ; ( Q ) , such t h a t &JEL'(Q) , t h e n j ( u ) i s w e l l d e f i n e d on L'(Q) , but t h e space X is not reflexive; a direct s o l u t i o n of (3.127) i s not easy).

3.7

Various remarks

Remark 3.18.

Let u s c o n s i d e r an i n e q u a l i t y o f t h e t y p e ( 3 . 2 ) . V and uo

KO i s assumed t o be a n o t h e r c l o s e d convex subset of t h e s o l u t i o n i n KO o f (3.128)

u(u~,u - uO) 2 (5 u - uO)

E

KO .

Thus, it i s c l e a r l y p o s s i b l e t h a t f o r certain f we may have

(3.129)

u = UO

.

I n Chapter 3 w e s h a l l p r e s e n t an example (due t o B r & z i s

(SEC. 4)

37

Interior approximations

and Sibony 121) related to the situation in Section 1.2, K being given by (1.19), and where

(3.130)

KO = { u I UEH;(R), I u(x) I d

distance from x to

r}.

It is now clear that the calculation of u will be much more simple using (3.128), (3.130) than by using the initial formulat ion.

Remark 3.19. In the spirit of Remark 3.18, we can check the calculations by using, in particular, certain "comparison properties" relating the solution u of (3.2) to solutions of more simple problems. On this subject see Haugazeau 111, 121. Remark 3.20. Another method consists of approximating the steady-state problems by time-dependent inequality problems. We shall return to this point in Chapter 6. Remark 3.21. For applications of duality theory to approximation theory see Laurent 111.

3.8 Synopsis In the previous sections we have given a number of equivalent formulations of variational inequalities, leading to infinitedimensional approximation algorithms, i.e. the approximate solutions are themselves supplied by the solution of a problem supposedly "more simple" than the initial problem, but with infinite dimensions. It is clear that the practical development of these algorithms presupposes that they may be reduced to finite-dimensional problems. Therefore, in the remainder of this chapter, we shall prov-

ide the nwnericaZ tools which make this reduction to finite dimensions possible. For this there are two general types of method: interior approximations (Section 4) and exterior approximations (Section 5).

4.

INTERIOR APPROXIMATIONS

4.1

Interior approximations of V.

Finite elements

A family V,, of finite-dimensional vector spaces where h is a parameter €UP (or in the neighbourhood of the origin in w ) i s said to be an interior approximation to V when the following conditions are satisfied:

3a

Approximation of steady-state inequaZities

(4.1)

Vh,

(4.2)

I

(CHAP. 1)

V , c V ' ( l )

the V, "approximate" V in the following sense: we can find a space *Y which is dense in V ( 2 ) such that , V # E * Y , we can construct V,,E v h with u,+u

in V as h + O

in W". Let wl,w2, ..., w,,

ExampZe 4.1. (Galerkin method). "basis" of V in the following sense:

(4.3)

VJ, the wlr ..., wj are linearly independent;

(4.4)

the combinations

... be a

I j w j , I,EER, are dense in V ( 3 ) . jinite

Then putting h = l / p , where p is an integer,

we define V, = [w,,

(4.5)

..., wJ

=

the space generated by

wl,

..., w,, .

It can be shown that (4.1), (4.2) are satisfied (with *Y = V ) .

ExampZe 4.2. RZ, V = H;(Q).

(Finite elements).

,

We consider a function h - + c p ( h ) such that (4.6)

cp(h)+O

Let

, which

be an open domain of is real and continuous,

if h - 0 .

We consider a triangulation 5, consisting of a finite family of triangles T which have the following properties: T EJ, ,,measure of T Q cp(h) and Vh , the angles of the triangles are bounded below by a strictly positive number, independent of h ; T E5,* T c ; (4.9)

T,T' €5, * T n T' = 0 or T and T'have a common side or T and T'have a common vertex;

~

(l)

It is this condition which explains the terminology "interior"; this condition is not satisfied in the case of "exterior" approximations (see Section 5).

(*)

The introduction of Y is not absolutely essential. However, in order to satisfy the hypotheses in practice, it is clearly advantageous to have as weak as possible a definition of convergence.

(3)

Such bases always exist if V is separable.

(SEC.

4)

Interior approximatidns

39

i f a,= u T , T E Y , , R,+R i n t h e sense: f o r a l l compact E c R we have: R, 3 E f o r " s u f f i c i e n t l y s m a l l " h. It i s c l e a r t h a t such t r i a n g u l a t i o n s e x i s t , f o r any rp s a t i s f y -

(4.10)

{

ing ( 4 . 6 ) . We a s s o c i a t e t h e space v b w i t h Y k i n t h e f o l l o w i n g manner:. l e t M be an " i n t e r n a l " v e r t e x o f T E Y ~( s e e F i g u r e 1) with t h e adjacent v er t ic e s P I , , p k ; we d e f i n e

....

(4.11)

I

w, = an a f f i n e f u n c t i o n over each t r i a n g l e w i t h v e r t e x M, such t h a t w,(M) = 1 , w,(P,) = 0 , i = 1,..., k , and whe.re w, i s z e r o o u t s i d e t h e t r i a n g l e s w i t h v e r t e x M.

Fig. 1.

Now:

V, = t h e space g e n e r a t e d by t h e w,.

(4.12)

Clearly ( 4 . 1 ) i s s a t i s f i e d . t a k e u~B(62)and d e f i n e

(4.13)

0,

u(M) w,

=

I n o r d e r t o s a t i s f y ( 4 . 2 ) we

.

Af

The above c o n s t r u c t i o n i s t h e - m o s t elementary o f t h e finite - a c l a s s o f methods i n which c o n s i d e r a b l e developments have been made ( u s e o f b a s i s f u n c t i o n s o f degree g r e a t e r t h a n 1, d i s c r e t i s a t i o n o f t h e s t r u c t u r e i n t o r e c t a n g l e s , t e t rahedra, e t c ) We s h a l l come back t o t h i s l a t e r . We r e f e r f o r t h e moment only t o C i a r l e t and Wagschal /l/, F r a e i j s de Veubeke /l/, Zienkiewicz /1/, Zlamal /l/, / 2 / , Ph. C i a r l e t and P.A. R a v i a r t /1/ and t h e b i b l i o g r a p h i e s t h e r e i n .

element methods

... .

Approximation o f steady-state i n e q u a l i t i e s

40

4.2

( C H A P . 1)

I n t e r i o r approximation schemes i n t h e c a s e o f e q u a t i o n s

The i n t e r i o r approximation schemes f o r t h e v a r i a t i o n a l equation

a(u, u ) = (f, U)

(4.14)

VU E V .

Incan be deduced immediately from t h e concepts o f S e c t i o n 4 . 1 . f i n d U ~ E V such ,, that deed, l e t u s now i n t r o d u c e t h e problem: (4* ''1

a(uh, oh) = (f,oh)

vuh

E

v h

.

By assuming, a s u s u a l , t h a t (4.16)

a ( ~ , u ) 2 a I I u 1 1 ~a, > O ,

VUEV,

we immediately g e t : Theorem (4.17)

uh

4.1.

Using the hypotheses (4.1), (4.2) ue have: i n V as h

+ 1(

+ 0.

It may be deduced from ( 4 . 1 5 ) t h a t

Proof.

11 uh 11'

Q

= (f,'3

dUh9

d 11 f

IIY'

11 uh 11

and hence (4.18)

1 IIuhII d;Ilf 1 1 ~ ' .

We can e x t r a c t from uh a sequence, a l s o denoted by u,, such that (4.19)

u,

weakly i n V as h + 0.

+w

For U E Y , w e t a k e oh s a t i s f y i n g ( 4 . 2 ) ; i n V , for t h i s c h o i c e o f l]h we have:

a(uh, v&

+

)'

and hence, i n t h e l i m i t , a(w, u) =

s i n c e uh + u Strongly

(f,u )

( 4 . 1 5 ) gives

vu E Y ;

s i n c e Y i s dense i n V we deduce t h a t w = u and hence t h a t uh + u weakly i n V without having t o e x t r a c t a subsequence. It now remains t o show t h a t u h + u strongly i n V. For t h i s we consider xh =

a(Uh - u, uh

- u) .

From (4.15) , x h = (f,U& - a(% U, - u) - a(uh, u) + (f,u) - a(& u) = 0, and t h e r e f o r e . 11 uh - u (1 + 0 . rn

(SEC. 4)

41

Interior approximations

The problem is now to extend these considerations to variation-

a I inequa Zities.

4.3

Approximations of K

We consider, for each h, a set Kh with (4.20)

(4.21)

K,= closed convex subset of V,, Kh "approximates" K in the following sense: (i) VUE K, we can find uhek;, with u h + u in V ; (Ti) If u h e K h , u,+uweakly in v , then U E K . 8

I

ExampZe 4.3.

We consider the situation as in Example 4.2 and

let (4:22)

K = { u ~ u E Vu ,2 0

a.e. in P }

Thus if we take

where V, is defined by (4.12), then (4.21) is satisfied.

8

Remark 4.1. We can weaken (4.21) by introducing K n Y and assuming K n Y to be dense in K.

4

In practical applications, the construction of Remark 4.2. is one of the difficulties presented by inequality problems.

We shall give numerous examples i'n the following chapters. (We shall also use exterior approximations, see Section 5 below). 8

Remark 4.3. A difficulty in terminology should be pointed out. The approximation V, of V is interior since V h c V ; the approximation Kh of K has an interior character since KhtV but it is not assumed that K,cK.

8

For a general study of "the approximation of Remark 4.4. convex sets", we refer the reader to the works of Mosco /2/, Joly /l/. See also Aubin /l/, Tr6moliDres 141.

4.4.

Approximation schemes for the initial problem

We now consider the inequaZity probZems (4.24)

(utK9

a(U,u - U) 2

(J - U) VUEK,

Approximation of steady-state inequazities

42

(CHAP. 1)

The approximation schemes u s i n g t h e above concepts are:

respectively. We have Theorem 4.2. We assume ( 4 . 1 ) , ( 4 . 2 ) to be satisfied (and for problem ( 4 . 2 6 ) , the conditions ( 4 . 2 0 ) , ( 4 . 2 1 ) ) . Then if u,, is the soZution of (4.26) ( r e s p . ( 4 . 2 7 ) ) we have (4.28)

in V,

uh+u

where u is the soZution of ( 4 . 2 4 ) ( r e s p . ( 4 . 2 5 ) ) . Proof. For f i x e d u i n K , we t a k e uh s a t i s f y i n g (4.21) ( i ) ; with t h i s choice of vh i n (4.26) we g e t : (4.29)

a(uh,

(4.30)

11 uh 11

d a(uh,

- (f,v h - uh)


From uh we can t h e n e x t r a c t a subsequence such t h a t uh + w

weakly i n V.

From (4.21) ( i i ) ,W E K and from ( 4 . 2 9 ) we deduce: a(w, w ) d

l h infO(uh, u& d a(w9 0)

- (f,

- w,

9

t h e r e f o r e w = u , t h e s o l u t i o n of ( 4 . 2 4 ) . To demonstrate t h e strong convergence of u,, towards u we cons i d e r , as i n t h e proof of Theorem 4 . 1 , x h

= a(uh

- u, uh - u) .

From ( 4 . 2 6 ) w e have:

(SEC.

4)

I n t e r i o r approximations

43

and by choosing u, satisfying (4.21) (i) we deduce

lim supX, d a(u, u) - (f,u -

(4.31)

u)

- a(u, u) ;

in (4.31) u is fixed a r b i t r a r i l y in K . Taking v = u we thus have lkn sup X,=O and hence strong convergence. 8 The reasoning is the same in the case of (4.25), (4.27).

Remark 4.5. The problem has thus now been reduced to the e f f ective solution of (4.26) or (4.27). The algorithms for this solution are studied in the following chapters. 8 Remark 4.6. The schemes (4.26), (4.27) do not use the concepts introduced in Section 3. h ? are now going t o see how these concepts can be used f o r the approximation: in a general manner, which will later be defined more precisely, we shall no longer approximate the "initial" problem (4.24) or (4.25), but one of 8 the "transformed" problems obtained in Section 3.

4.5.

Approximation schemes for the penalised or regularised problems.

Pena Zised equation. We shall now approximate (3.27) using the "interior" method. Thus : 1) We choose E = dh), &(/I) --* 0 if h + 0 ; ( l ) 2) We define U,E v, as the solution of 1

(4'32)

u(uhy

'h)

+ E(h) (B(uh), uk)

= (f,vh)

vuh

E vh

.

We then have : Theorem 4.3. A s h (4.33)

where

u, + u u

-+ 0 ,

then:

i n V,

is the solution of (4.24).

The proof, which is based on principles presented earlier, will not be given here. 8

(l)

The practical choice of ~ ( h )poses certain difficulties.

44

Approximation of steady-state inequalities

( C H A P . 1)

Remark 4.7. We can consider the scheme (4.32) from a different viewpoint: first we consider scheme (4.26) and then we penalise probZem (4.26). The two procedures can lead to certain technical differences. rn Replarised equation. We consider a family jdh, of regularised functionals (cf. (3.33), for example). We then discretise (3.37), which gives the scheme: (4.34)

u&

+ (jib) (uh)t

uh)

=

(f,oh)

v'h

Vh

.

We again obtain the result (4.33).. Remark 4.8. This remark is analogous to 4.7; first, we can discretise as in (4.27) and then regularise scheme (4.27). rn or

Remark 4.9. Algorithms for the effective solution of (4.32) (4.34) will be given in the following chapters. rn

4.6

Approximation schemes for the dual problem

Form Zation (I)

.

Here we use Section 3.4. The initial problem has been transformed (see (3.63)) to: (4.35)

inf 3 a(m,4P))

(l)

P E P

where u = u@) is the solution of (4.36)

- A u = g d i v p +f.

u= 0

r.

on

Discretisation of problem (4.35), (4.36). A s a first stage, we consider V, = "interior approximation" of V = H,@) and we define, for P E B , the approximate solution l)h

Vk

(4 * 37)

by Ph)

= (g div P

+ f, (Ph)

VPh E

V h

.

We then consider (4.38)

id 5 a@,, u&

A second stage consists of "discretising" 9 . (l)

9 is defined in (3.42).

For this we

(SEC.

4)

45

Interior approximations

i n t r o d u c e a f a m i l y qC of s u b s e t s o f 9 , "approximating" 9 ( l ) and we c o n s i d e r ohmC t o be t h e s o l u t i o n i n V, of

(4*39)

a(oh.C,

(Ph)

= b div P< + f, (Ph)

tkPh

'h

which g i v e s t h e problem ( 2 )

(4.40)

inf 4 a(u,,<).

We t h e n have

(4.41)

inf f ~ ( u , , + ~ ) inf f a(u) PI

h,

as

< +O.

P

rn

AZgorithms f o r t h e e f f e c t i v e s o l u t i o n o f probRemark 4.10. lems o f t h e t y p e ( 4 . 4 0 ) a r e giver, i n t h e f o l l o w i n g c h a p t e r s . FormuZation (11). We now c o n s i d e r t h e s e t t i n g of S e c t i o n 3.5. The i n i t i a l problem h a s been, t r a n s f o r m e d ( s e e ( 3 . 7 5 ) ) t o :

(4.42)

infij

(Ip

1

- g)'dx,

R

where

(4.43)

divp

= f.

The method t h e n c o n s i s t s o f d i s c r e t i s i n g ( 4 . 4 3 ) , t h e exterior methods ( s e e t h e n e x t s e c t i o n ) b e i n g s i m p l e r h e r e . We t h e n subs t i t u t e t h e " d i s c r e t i s e d p" i n t o ( 4 . 4 2 1 , i . e . p,, A s a l r e a d y p o i n t e d o u t i n Remark 4.10, it now o n l y remains t o provide an algorithm f o r t h e c a l c u l a t i o n of

.

For a n u m e r i c a l a p p l i c a t i o n o f t h e c o n c e p t s of Remark 4.11. S e c t i o n 3 . 6 , s e e F u s c i a r d i , MOSCO, S c a r p i n i and S c h i a f f i n o 111.

(l)

I n t h e examples w e g e n e r a l l y t a k e

(')

We u s e t h e n o t a t i o n a(u) = a(u, u).

( = h.

46

Approximation o f steady-state inequalities

4.7

(CHAP. 1)

Open problems. Estimates of the approximation error

In the above, a significant problem, and one which is still largely open, is that of obtaining estimates of the error. It would be interesting to see to what extent the error estimates obtained for equations (see Aubin / 2 / , Bramble and Schatz /l/, Strang and Fix /l/, Ciarlet and Raviart /l/, /2/, etc.. . ) apply to inequalities. It appears that there are very few general results in this area, although we should point out the work of Falk /l/, /2/ and Mosco-Strang /1/ relating to certain particular problems; error estimates for finite element approximations will a l s o be found in the present book, in particular in Chapters 3 , 4 and 5; these estimates are probably not optimal, however. In order to conclude, for the time being, the discussion of interior approximations, we shall consider an example for which we can easily provide an error estimate which appears to be of

optimal order. We again consider Example 3.5 of Section 3.6, assuming C2 to be a bounded open domain of R2 , with a sufficiently regylar boundary r ; we then have (4.45)

v = Ht(l2)

(4.46)

K = { u l o ~ V u, > O a.e. in Q }

and in order to simplify things slightly we shall restrict our attention to the case where the bilinear form a is defined by (4.47)

a(u, u) =

I

grad u.grad u dx .

Let f E L Z ( R ) ; the variational inequality

(4.4)

[

~ ( uu,

- u) 2

In

f ( 0 - u) dx Vu E K

then admits one and only one solution and we recall (see BrCzisStampacchia /2/ ) that

with (4.50)

11 I I ~ ~G ( c~ iifnL2,m )

where, in (4.50), C is independent of f and u. rn We shall now approximate the solution of (4.48) by means of a

(SEC.

4)

47

I n t e r i o r approximations

t h e n o t a t i o n i s as i n S e c t i o n 4.1, Example I n o r d e r t o s i m p l i f y t h e argument we s h a l l assume t h a t C2 4.2. i s a convex polygon, s o t h a t r e l a t i o n s (4.49), (4.50) s t i l l h o l d , and we s a y t h a t Y h must s a t i s f y ,

f i n i t e element method;

U T=Z.

(4.51)

T E Y h

as w e l l as

(4.7)-(4.10).

We i n t r o d u c e

C, = { M I M ~ d l ,

(4.52)

M i s a vertex of T € . T h }

and we d e f i n e t h e "approximation" Vh o f Hi@) by (4.53)

v h

{ l)h 1 V h E H k ( 0 )

=

co(dl)z,, U h l E~

V T Er

h

}

where uhIT d e n o t e s t h e r e s t r i c t i o n o f vh t o T and Pi d e n o t e s t h e We t h e n space o f p o l y n o m i a l s i n two v a r i a b l e s o f d e g r e e < 1 . d e f i n e Kh, a n a p p r o x i m a t i o n o f K , by

& = v , ~ K ={ u , I ~ , E v ~ , v , 2 0 o n 51;

(4.54)

K,, i s a closed convex subset of

vh and

it i s immediate, s i n c e

belongs t o P I , t h a t

ohlT

(4.55)

Kh

={

vh

I uh E

vh,

vh(M) 3 0 V M E

Ch

}

*

We t h e n d e f i n e t h e approximate problem by

uhEKh

(4.56) admits one and only one solution. I n connection with t h e approximation e r r o r s h a l l now p r o v e

and

11 u h - u IIH;(0)

we a s s m e t h a t the, angles of P r o p o s i t i o n 4.1. below, uniformly i n h , by e0 > 0; we then have (4.57)

, we

F,, are bounded

11 uh - I( 11 d 11 f IIL'(f2) &

where C i s independent o f h,u,f and where we r e c a l l that /I = max(Area of T)

.

TEYh

Proof.

I n t h e following, C w i l l denote v a r i o u s co n s t an t s ;

we have U(U, uh

- U) 2 Jn

a(uh, uh

- uh) 2

f(Uh

f(vh

- U) dx

- uh) dx v o E K h

9

48

Approximations of steady-state inequalities

(CHAP. 1)

and hence by a d d i t i o n

We have u E HA(62) n H 2 ( @

- Au - f E L2(Q)

and hence

and i f we

Put (4.60)

I =

- AU - f ,

we g e t , u s i n g ( 4 . 5 0 ) , (4.60’)

c II f { IICI independent llL2(D)

Q

II”(f2)

of f .

From (4.60) w e deduce t h a t (4.61)

a(u, u) =

.

and from (4.591, 1

3 11 ‘h vuh E K b

I, +

(f I ) u dx

(4.61)

lliA(f2)

Vu E H h W ) ,

1

3 11 uh -

IlfrA(f?)

+

s.

I(Uh

- u) dx

9

and hence from (4.60’) (4*62)

f 11

uh

-

< 11 uh -

lli1(f2)

llil(f2)

+ 11 f llL’(f2) 11 uh -

llL2(f2)

Vu, E Kh . To e s t i m a t e uh - u w e s h a l l u s e ( 4 . 6 2 ) , choosing a s u i t a b l e u, : f i r s t we d e f i n e t h e interpolation o p e r a t o r nh: Hh(62) n Co(62)3 Vh by

(SEC. 5 )

Exterior approximations

{ nh

n,oE

(4.63)

v,

H;(o) n co(CSa, VME z, , vu

U ( M )=

49

V(M)

E

and, from ( 4 . 5 3 1 , ( 4 . 5 5 ) we have

nhu E K ,

(4.64)

vv

EK n

co(5).

I n ( 4 . 6 2 ) we s h a l l t a k e vh = n,u ; t h i s i s meaningful owing t o t h e f a c t t h a t , a s R i s a @ o - d i m e n s i o n a Z domain w i t h a L i p s c h i t z boundary, we have H'(62) c Co(0) a n d , from (4.64 ) nhu E K , Vh We r e c a l l t h a t h = m a x ( A r e a o f T ) ; t h u s from, f o r example,

.

T E F ~

Fix-Strang /l/, C i a r l e t - R a v i a r t /l/, we have t h e r e s u l t t h a t unde r t h e s p e c i f i e d c o n d i t i o n on t h e a n g l e s o f 9,we have

(4'65)

11 nh

- u llLZ(f2) < c 11 u IIH'(f2)h

(4.66)

11 nh

-

11

IIHA(f2)

IIHqI?)

9

fi

w i t h C independent o f h , u . The e s t i m a t e ( 4 . 5 7 ) t h e n f o l l o w s t r i v i a l l y from ( 4 . 5 0 ) , ( 4 . 6 2 ) w i t h v,=n,,u , ( 4 . 6 5 ) and ( 4 . 6 6 ) .

Remark 4.12. The r e s u l t s and t h e p r o o f o f P r o p o s i t i o n 4 . 1 a r e e a s i l y e x t e n d e d t o t h e c a s e where C2 i s a convEx polyhedron of R3 s i n c e i n t h i s c a s e t h e i n c l u s i o n H'(62) c Co(62) s t i l l h o l d s . Remark 4.13. I n Chapter 2 , S e c t i o n 5 we s h a l l s t u d y a problem similar t o problem ( 4 . 4 8 ) , u s i n g a f i n i t e d i f f e r e n c e method..

5.

EXTERIOR APPROXIMATIONS

5.1

An example

I n a p p l i c a t i o n s n o t o n l y a r e i n t e r i o r methods u s e d , b u t a l s o

exterior methods. B e f o r e g o i n g on t o t h e " g e n e r a l axioms", which a r e somewhat i n v o l v e d , ,we s h a l l p o i n t o u t t h e b a s i c i d e a s u s i n g a n example which h a s been chosen t o be as s i m p l e as p o s s i b l e . We c o n s i d e r t h e Dirichlet p r o b l e m :

(5.1) (5.2)

+ u =f, u=0 on r . -

Au

We i n t r o d u c e

(5.3) and

v = HA(62)

f

g i v e n i n L2(sZ),

Approximation of steady-state inequalities

50

(5.4)

a(u, u) =

I,

(uu

(CHAP. 1)

+ grad u.grad u ) dx .

The problem (5.1), ( 5 . 2 ) i s then equivalent t o (5.5)

a(u, 11) =

( A 1)) ,

vu E

v.

Our a i m i s t o approximate u using characteristic functions, i . e . f u n c t i o n s which do not belong to HI@); t h e use of t h e s e c h a r a c t e r i s t i c f u n c t i o n s corresponds t o finite difference methods. More s p e c i f i c a l l y : l e t h E W " ; we a s s o c i a t e with it t h e : grid

R , = { M I M E R " , M = { m, h l,..., m m h , } , m i E E } .

(5.6)

With each node M o f

R,, we a s s o c i a t e the panel with c e n t r e M :

and the cross ( w i t h c e n t r e M ) :

where ei denotes t h e i t h u n i t v e c t o r i n We t h e n d e f i n e (5.8)

a h

(5.9)

= =

{MIm,'(M) c a

}

R"

.

9

c h a r a c t e r i s t i c f u n c t i o n of m:(M)

,

and (5.10)

v,=

space generated by

e, M e a h .

The f u n c t i o n s #, are not i n H'(Q), so t h a t V, is not a subspace of H'(Q). Hence, we cannot d e f i n e a ( u , O ) on V h and we must t h e r e f o r e modify a(u,v). For t h i s we r e p l a c e t h e d e r i v a t i v e s by d i f f e r e n t i a l q u o t i e n t s . I n g e n e r a l , we put

Then f o r u, u, E V, we define

(l)

The i n d i c e s "0" and "1" which appear i n t h i s n o t a t i o n correspond t o t h e d i f f e r e n t orders o f t h e approximation; w e can introduce "crosses" o f a r b i t r a r i l y high o r d e r . See Aubin / 2 / .

Exterior approximations

(uh

'h

+ i=1

6,'h)

dx

it can be s e e n immediately t h a t ( 5 . 1 3 ) admits a unique s o l u t i o n . The fundamental r e s u l t i s t h e n Theorem 5 . 1 .

As

h

+ 0

we have:

We can t h u s e x t r a c t a subsequence, a l s o denoted by u h , such that (5.18)

uh + w

,

6,uh + wi

weakly i n L'(62)

.

We s h a l l now v e r i f y t h e two f o l l o w i n g p r o p e r t i e s : aw axi

-

(5.19)

wi =

(5.20)

w E Hi@).

I n f a c t , f o r q ~ B ( 0 and ) for sufficiently s m a l l

( l)

1 h ( we

We can i n f a c t d e m o n s t r a t e s t r o n g convergence. (5.15) i s a d e q u a t e f o r our p u r p o s e s .

have:

However,

Approximation of steady-state inequalities

52

and p r o c e e d i n g t o t h e l i m i t :

(wi, cp) =

( C H A P . 1)

-

strongly i n L z ( a ) ) , and hence (5.19). I n - o r d e r t o v e r i f y ( 5 . 2 0 ) , we c o n s i d e r Eh = e x t e n s i o n o f u, t o W" by 0 o u t s i d e a ; s i n c e ( 5 . 8 ) i s t r u e , t h e n :

(a%& 6j, a,(&)

and Ch -+

= -+

6i(ch)

weakly i n L z ( a )

6ji

(5.19), it

However, as for hence

=)ELZ(W") axi

.

may be v e r i f i e d t h a t w"

i

a --(q - ax,

and

vi,

t h e r e f o r e w " = O on r , which g i v e s ( 5 . 2 0 ) . It now r e m a i n s t o show t h a t w = u For t h i s we t a k e U E V and define

.

It may be v e r i f i e d t h a t :

r, u + u ,

(5.22)

Then, t a k i n g gives a(w, u) =

dirk u uh

=

( Au)

rh

-+

u

ao axi

, we

vu E

strongly i n have uh(uh,uh)

-+

LZ(a),

a(w, u)

vi .

and hence ( 5 . 1 3 )

v,

and t h e r e f o r e w = u , which p r o v e s t h e theorem. We s h a l l now " a x i o m a t i s e " t h i s p r o o f ( l ) , by a l s o i n t r o d u c i n g "approximations" o f K and j . E x t e r i o r approximations f o r V , a, K, j .

5.2

Approximation of V. We c o n s i d e r a H i l b e r t s p a c e F and a n o p e r a t o r a w i t h

(5.23)

,

u E U(V ;F ) ,

a i n j e c t i v e from V into F.

For example, i n t h e c o n t e x t o f S e c t i o n 5 . 1 , V = and a i s d e f i n e d by (5.24)

(l)

ow =

{ ,: u,-,

H,'(a), F = (Lz(a)y+1,

"> .

..., ax,

I n order t o avoid r e p e t i t i o n i n t h e a p p l i c a t i o n s .

(SEC. 5 )

Exterior approximations

53

We next consider a family of finite-dimensional spaces V , , equipped with (Hilbert) norms (1 vh I ) y h . We assume prolongation operators p , E Y ( V , ;F ) to be given, with

For example, in the context of Section 5.1, V, is defined by ph is defined by

(5.10)and ( I v h ( I Y h is defined by (5.16); (5.26)

P h trh

{

=

vh,

d,vh,

..., 6mvh 1 ;

we have:

We say that V, constitutes an exterior approximation ( I ) of V if (5.27),

V,E

V,,

p h v h converges weakly towards

i n F=. ( E ~ V

and if (5.27) 2

{ 11

E

v , 30, E < c.

vh

with p , vh + a V

strongly in F and

IIYh

In the context of Section 5.1, we shall take (5.21); (5.22) then corresponds to (5.27)2.

uh=rhv

defined by

Approximation of a. We now consider a bilinear form a,(',, uh) on V, ( 2 , and we assume that

We say that the ah constitute an approximation of a if:

(') (2)

(3,

Terminology justified by the fact that V , Q v. In fact, the whole of this may be extended to cer'tain nonlinear problems. Where, in general, a*(rp, $) = a($, rp), a:(rp, $) = ah($,rp) This assumption becomes redundant in the case where a is symmetric and where we then choose a, to be symmetric (which is possible, if not essential).

.

54

Approximation of steady-state inequalities

(CHAP. 1)

and a l s o i f if

(5.31)

weakly i n F , t h e n

uv

+

h infah(v,,

2

4 0 , 0)

I n t h e c o n t e x t o f S e c t i o n 5.1, 8 have ( 5 . 3 0 ) and ( 5 . 3 1 ) .

.

8

u, i s d e f i n e d by ( 5 . 1 2 ) and we

Approximation of K . We t a k e a f a m i l y o f sets K, where

K, i s a c l o s e d convex s u b s e t o f V,.

(5.32)

& constitute an approximation of K

We s a y t h a t t h e

(5.33)

E K,,

p, v, converges weakly towards ( i n

F*

if: (E uK,

and i f f o r a l l U E K , w e can f i n d V , E & such t h a t p, v,, + m s t r o n g l y i n F and II v h (IYh d C.

(5.34)

Remark 5.1.

If

K = V, K h =

Vhy

we recover (5.27)1, (5.27)2.

Remark 5.2. The construction o f lu, and the satisfaction of ( 5 . 3 3 ) , ( 5 . 3 4 ) form one o f t h e main d i f f i c u l t i e s i n a p p l i c a t i o n s . We s h a l l g i v e some examples l a t e r ; w e s h a l l t h e n see t h a t t h e u s e of exterior approximations i s o f t e n more convenient t h a n t h a t of interior approximations. 8 Approximation of j. Let j be a p r o p e r convex f u n c t i o n ( s e e S e c t i o n 3.6) on v. a p r o p e r convex f u n c t i o n on V,. We c o n s i d e r j , We say t h a t t h e j , constitute an approximation of j i f : (5.35)

oh E V, ,

P h vh +

(5.36)

0, E v h ,

0E

uu weakly i n F

=s

lim infj,(v,) 3 j ( v ) ,

V , ph v,, + ov s t r o n g l y i n F

Some examples w i l l be g i v e n i n Chapters

j

4 and 5.

jk(vb)+ j ( v )

.

8

Approximation of the right-hand side. L e t f E V' . We c o n s i d e r t h e c o n t i n u o u s l i n e a r form X on V, and we s a y t h a t constitutes an approximation of f i f (5.37) I fh(vb I 11 vh IIVh . (5.38) Ph v, --* m weakly i n F =s fh(vh) + (f,0). 9

(SEC.

5)

55

Exterior approximations

5.3 Exterior approximation schemes

Approximation of probZem (4.24). We seek u h e K h with

Approximation of probZem (4.25).

We now have Theorem 5.2. Assume that the hypotheses (5.23) to (5.34) hold. Let uh (resp. u ) be the solutions of (5.39) o r (5.40), (resp. of (4.24) o r (4.25)). Then (5.41)

strongly in F .

phur + ou

Remark 5.3. Result (5.41) again gives (5.15).

B

Proof of Theorem 5.2. We shall perform the proof for the case of (5.39). The same reasoning applies to the case (5.40). We take uEKand uheKh which satisfy (5.34). Then, with this choice of in (5.39) we have:

ub d

(5.42)

ah(uh9 v h )

and hence using (5.281, a 11 uh Ilf Q and therefore

11 uh

- f k ( u h - uh) (5.291, (5.34) and (5.37) ( l )

c(1 + 11 uh 113

d c.

From (5.25) we then have: '(5*44)

11 ph uh Ilp

< c.

Hence, we can extract a subsequence, also denoted by u h , such that ph uh + (

weakly in F

56

Approximation of steady-state inequalities

which w i t h ( 5 . 3 3 ) shows t h a t (5.45)

= aw,

w e K and hence:

w EK.

weakly i n F,

phuh+ a w

(CHAP. 1)

However, f r o m (5.31), ( 5 . 4 5 ) gives (5.46)

lim inf ah@,,uh) 3 a(w, w ) .

Mcreover, s i n c e p h u h + a , s t r o n g l y i n F and s i n c e we have ( 5 . 4 5 ) , it r e s u l t s from ( 5 . 3 0 ) t h a t (5.47)

ah(uh, v&

+

a(w7 v)

3

and f r o m ( 5 . 3 8 ) t h a t (5.48)

- u& -* (f,

fh(uh

- w, '

Using ( 5 . 4 6 ) , ( 5 . 4 7 ) , ( 5 . 4 8 ) i n (5.42) w e deduce t h a t a(w, w) Q a(w, D)

- (A u - W)

VV E K ,

hence w = u which shows t h a t ph u, + ou. I n o r d e r t o prove ( 5 . 4 1 ) we i n t r o d u c e pkiih+ au

(

4

ah€&

with

strongly i n F

e x i s t s , f r o m ( 5 . 3 4 ) ) and w e put = ah(uh

xh

- 5,uh - s) .

W e have : = a,+(%

x h

- ah(&, u& - ah(% 4) + uh(ah* iik) .

From ( 5 . 3 9 ) we g e t xh

d

ah(ghr

uh) - fh(l)h

-

Uh)

- ah(ah$ uh) - ah(%, a&

We t h e n u s e ( 5 . 3 0 ) ( l ) and assume t h a t ( 5 . 3 4 ) . We t h e n deduce t h a t (5.49)

I),,

+

ah)

'

i s chosen as i n

lim sup xhd a(u, v ) - (J D - u) - a(u, u) Vv E K .

Taking v

=u

i n ( 5 . 4 9 ) w e deduce t h a t

Xh + 0

and hence, f r o m ( 5 . 2 9 ) , t h a t

(l)

The 2nd assumption i n (5.30) i s only involved i n t h e c a s e of strong convergence.

(SEC. 6 )

57

Exterior approximations

and hence, from (5.25), phuh- ph Ch + 0

strongly in F which gives (5.41).

Remark 5.4. In applications it is sufficient that (5.34) be satisfied on a dense subset of K , i.e. that the following property be satisfied: (5.50)

{

there exists X c K , X dense in K , such that Q V E X we can find vheKhwith ph + ov strongly in F and 11 vh (Ih < C .

In fact (5.34) results from (5.50); that is, V E K a n d

(Pzc.f

with Ilcpe-vll

,<E,

E + O .

From (5.50) there exists h, and p h , ~ Kwith:

11 Ph. (Ph. -

IIF

<

;

thus :

11 Pha 'Phc - aV

[IF

11 Ph.

(Ph.

- a(Pc

[IF

+ 11

-

O V [IF

< CE

giving the required result.

Remark 5.5. The systematic theory of exterior approximation was introduced by Cea /3/, the special case of approximations An investusing "crosses" being introduced by J.L. Lions /3/. igation in the case of inequalities is made in Tr6moli3res /4/. Remark 5.6. Naturally, as for "interior" methods, it still remains to give aZgorithms for the solution of (5.39), (5.40). These algorithms, developed in the following chapters, will clearly be the same for the interior and exterior schemes. Remark 5.7. We can now go back, in the "exterior" case, to the considerations of Sections 4.6, 4.7. It is not necessary t o expand on this. Some examples will be given in the following chapters.

6.

CONCLUSIONS

The numerical approximation of steady-state variational inequalities may be described as follows:

58

Approximation of steady-state inequalities

(CHAP. 1)

Stage 1. Choice of the formulation: have mentioned are:

the main possibilities which we

formulation in the form of a problem in the calculus of variations; variational inequality formulations; penalised variational inequalities; regularised variational inequalities; dualised variational inequalities though there are also various possibilities within each of these types of method.

Stage 2. Choice of the discretisation. We Clearly a great exterior methods. sible within each of these methods: the choice of prolongation operators imation, etc...

have mentioned interior and number of variations is posthe choice of finite elements, of various orders of approx-

Stage 3. Having completed stages 1 and 2 (and Choice of the algorithm. so far we have been developing the tools which enable us to arrive at this point), we are faced with a nonlinear programming problem or with a problem in finite-dimensional nonlinear equations. It is now necessary to develop solution algorithms and then to choose between the various possibilities. This will form the main objective of the next four chapters, first, for general cases and then for some concrete examples and numerical applications.

Chapter 2 OPTIMISATION

ALGORITHMS

SYNOPSIS In this chapter, we shall develop various optimisation algorithms - concentrating especially on those suited to problems involving the inequalities (or functionals) encountered in Chapter 1. We shall illustrate the theory by using a number of very simple examples (or counter-examples). The fundamental examples will be developed in Chapters 3 to 5 (for the steady-state case).

1.

THE RELAXATION METHOD Description of the point relaxation method. Unconstrained case.

1.1

We consider a continuous convex functional v + J(v) from W N + R and we shall first examine the so-called “unconstrained” case in which we seek (1.1)

infJ(u),

V E WN.

We shall assume that

J

(1.3)

is strictZy conuex,

so that there exists a unique u such that (1.4)

J(u) = inf J(u) . Y

Starting from uo = { uy, ..., ug } and assuming ff to be known, we calculate$+’lsuccessively for i = 1, N. We define$+’ as the s o l u t i o n of (1.5)

J(u;+’, Q

...I

+:;,++‘,$+I,

J(u;+ I ,

...,

*:;,

...,

..., u 3 G ui, 4+1,..., 4) ,

vu,

,

for i = l , 2 ,..., N. Before going on to more general algorithms (constrained

@timisation

60

algorithms

(CHAP. 2 )

...

problems, b l o c k r e l a x a t i o n , e t c ), we s h a l l f i r s t s t u d y t h e problem o f t h e ( p o s s i b l e ) convergence o f a l g o r i t h m (1.5). Let u s begin w i t h a counter-example:

Counter-Exwnple 1.1. The point relaxation method (1.5) may not be convergent t o t h e s o l u t i o n of t h e problem. Suppose f o r example w e t a k e , with N = 2 , t h e f u n c t i o n a l : (1.6)

J ( u ~02) , = U:

+ U:

- 2 ( ~ 1+ 14 + 2 I ui - 02 I

which s a t i s f i e s (1.2), (1.3) and which i s nondifferentiable (due t o t h e term 2 I u 1 - u 2 1). We have u= {l,l}.

Let u s now a p p l y a l g o r i t h m (1.5), s t a r t i n g from uo= { O , O } .

W e t h u s d e f i n e ui by t h e minimisation w . r . t .

u1 of

+

J(u,, 0) = v: - 2 u1 2 I v1 I g i v i n g u: = 0 Next we d e f i n e ul by t h e minimisation w . r . t . of J(u:, u2) = J(0, u2) g i v i n g u: = 0 and so on:

.

d'= { O , O }

u,

vn

t h u s d' does not converge t o u.

Remark 1.1. I n S e c t i o n s 4.3 and 4.4 we s h a l l p r e s e n t v a r i o u s methods ( i n p a r t i c u l a r , t h e d u a l i t y method) which a l l o w u s t o minimise f u n c t i o n a l s o f t h e t y p e (1.6). Furthermore, i n S e c t i o n 1.5 below, it w i l l be seen t h a t a l g o r i t h m (1.5) i s convergent f o r certain nondifferentiabze f u n c t i o n a l s From t h e p o i n t o f view o f Counter-Example 1.1, it i s r e a s o n a b l e t o assume i n i t i a l l y t h a t t h e f u n c t i o n a l J i s differentiable; we have : Theorem 1.1. We assume that the function J satisfies (1.2), (1.3) and is of class C1. The point relaxation method (1.5) .then converges to the solution u of problem (1.4). The proof l i e s i n a number o f lemmas: Lemma 1.1. If J is of class C1 and strictZy convex, then, f o r any M > 0, there.exists a strictly increasing continuous function 1 + a&) from [0,2 M ]-B W+ with 6&l) = 0, such that

(1.7)

(J'(u) - J'(u), 1) - u) 2 SM(ll u - u II) vu, 0 w i t h IIuII G M , IIuII b M .

Proof. We denote by B,, t h e b a l l '5 E [o, 2 M ]w e Put

BM

=

{ u 111 u I( b M } and f o r

Relaxation methods

(SEC. 1)

(1.8)

6&(r)=

inf

(J’(u) - J‘(u), u

IIu-41 = r .

61

- u) .

I~VEBH

we have

6&(0)

0

=

and s i n c e J i s strictly convex, J ’ i s s t r i c t l y monotone, and hence

S&(r) >

o

if

>0.

T

S i n c e B~ x B~ i s compact in R~ x F IP and s i n c e J ’ i s of c l a s s f o r a g i v e n T t h e r e e x i s t s a p a i r uo, u o € B M x BM such t h a t

II uo - uo II

=

‘5

,

- J‘(UO),

(J’(U0)

Uo

cO,

- uo) = 6&(‘5).

We s h a l l now show t h a t t h e f u n c t i o n 6; i s s t r i c t l y i n c r e a s i n g . Consider T I , ~2 w i t h O<

T~

<

t2

d 2M

.

There e x i s t s u 2 , v 2 such t h a t

I1 02

-

u2

II

=

‘52

However, s i n c e J ’

(J’(u2) - J ’ ( U ,

6&(72)

1

=

(J’(u2)

- J’(Uz),

U2

(J’(u2)

+ r(u2 - u2)),

02

- J’(u2), 0 2 - u2) > ( J ’ ( U 2

- J’(U2).

02

- u2) > 0

+ r(u2 - U Z ) ) -

- uq), 0 d I < 1

.

However i f we i n t r o d u c e w by w

- u2

=

- (u2 - u 2 ) , ‘51

72

we have

II w

- uz

II

=

71

and i f , i n (1.91,we t a k e I = rl/t2 we g e t T2

6&(‘5,) > - (J’(w) - J ’ ( U , ) , w

- u2) >

71

> (J‘(w) - J ’ ( U 2 ) , w - u 2 ) 2 2 inf (J’(u) - J’(u), u - u) = 6&(r,) , IIu-uIl=r~

I~UEBH

which p r o v e s t h a t 6;

is s t r i c t l y increasing.

From t h e d e f i n i t i o n o f (1.10)

(J’(0)

u2).

i s s t r i c t l y monotone,

t h e r ef o r e (1.9)

-

S& i n (1.8) we have

- J‘(u), u - u) 2 s;(l

u -u

11).

if

0<1<1

@timisation algorithms

62

Applying (1.10) to the pair { u

+ t(u - u), u }, t E p, 11,

(J'(u + t(u - u)) - J'(u), 0 - u) 2

(1 .11)

and hence, with

7

(CHAP. 2 )

we get

1

7 6%t II 0 - u 11)

I

11 1) - u 11,

=t

and hence 1

(1.12)

-6L(7) 7

+0

7 +0

if

.

Integrating (1.11)we deduce

J(u) - J(u) 2 (J'(u), u - U)

(1.13)

By interchanging'u and

so that the function

, 6

V

+

II u

dt

- u 11) 7.

and adding, we get

defined by

(1. IS)

meets the required properties. Lemma 1.2.

Using the asswnptions and notation of Lemma 1.1, Qe

have : J(v) - J(u) 2 (J'(u),0

(1 -16)

- u) + 4 644 (1 u - u 11) ,

II u II, II II Q M . Proof. hence

In (1.11) and (1.13) we can replace :6

by 611

, and

which in particular, gives (1.16).

Proof of Theorem 1.1. Taking u, = u; in (1.5) and successively setting i = N, N - l,...,l, we have: J(u"+l ) < J(*+ *-Q J(u")

1,

..., .",'-4) :, Q J(4+ l , ...,4 t:, 4-1,G)Q

*-.

ReZaxation methods

(SEC. 1)

63

S i n c e ( 1 . 2 ) i s s a t i s f i e d , t h e r e t h e n e x i s t s a c o n s t a n t M such that

From Lemma 1 . 2 we g e t (1.19)

J(u~+',...,4?;,4,

..., u ; ) - J ( ~ + ' , ...,#?;,c1,4+1, ..., u;) 3 aJ 2 (4 - u;+')--(u;+l, ..., 4+', ..., 4)

+

h i

and s i n c e , by definition of

(1.20)

dJ -(#+I, 8%

#+I,

we have

..., # + ' , u ; + ' , ..., 4) = 0

we deduce from (1.19) t h a t

(1.21)

J ( 4 + 1 , ..., 4 : ; ,

2

+

4,..., u;;)-J(4+'. ..., q?;,#+', ad+I 4' - 4 I)

..., u;) 3

9

and hence

However, from ( 1 . 1 7 ) J ( f f ) - J ( f f + ' ) + O s t r i c t l y increasing, (1.22) gives

(1.23)

s o t h a t , s i n c e bnr i s

f f + ' - ff + O .

We s h a l l now show t h a t u"+u , where u i s t h e s o l u t i o n o f From (1.18) problem ( 1 . 4 ) which t h e r e f o r e s a t i s f i e s J'(u) = 0 we a l s o have 11 u (1 Q M a n d , i n (1.7), we can t a k e u = un+' and u , t h e s o l u t i o n o f problem ( 1 . 4 ) ; t h i s g i v e s

.

SO

t h a t ( 1 . 2 0 ) c a n a l s o be e x p r e s s e d by

(1.20')

aJ

--ff aui

'

+ I)

--0 .

64

Qptimisation algorithms

(CHAP. 2)

Thus (1.24) is equivalent to (1.25)

Since 11 uR+l - iU"+l 11 6 11 U"+' - u" 11 and J ' is uniformly continuous on bounded domains of RN , we see from (1.23) that the left-hand side of (1.25) tends to zero as n + a ; therefore

6,(l

U"+l

- 11)

+

0

which proves the theorem. 1.2

Point relaxation.

Constrained case

We now consider the case where J is defined on K with

(1.26) K = closed convex subset of RN

.

We assume that

n K,, N

(1.27)

K =

i= 1

K, = [a,, b,] c R (u,, b, finite or o t h e r w i s e ) .

We shall frequently state that under these circumstances K is defined by l o c a l c o n s t r a i n t s , since the membership of K is, at all points, expressed for each coordinate separately. The formal extension of algorithm (1.5) is obvious: we define # + l as t h e solution i n 4 of (1.28)

J(g+',...,#2i,4+1,#+l,..., u;) Q Q J(g+',..., U;?;, v,, U;+ ..., 4)

Vv, E Ki ,

i = 1, ..., N .

If we assume that (1.29)

J(v) +

+

00

if

II v II

+

+a,

VEK

(l)

and that (1.3) is satisfied, then t h e r e e x i s t s a unique u i n K such t h a t (1.30)

J(u) = i n f J ( v ) , V E K

We therefore have the following extension of Theorem 1.1:

(I)

This assumption becomes redundant if K is bounded.

Relaxation methods

(SEC. 1)

65

We assume that J is of class C1 and satisfies Then the point relaxation algorithm ( 1 . 2 8 ) converges to the solution u of problem (1.30). Theorem 1 . 2 .

(l.~?),(1.3).

The p r o c e d u r e i s t h e same as i n t h e proof o f Theorem Proof. 1.1. F i r s t , w e have ( 1 . 1 7 ) which g i v e s (1.181, ( 1 . 1 9 ) . On t h i s o c c a s i o n , i n s t e a d o f ( 1 . 2 0 ) ( o r (1.20')) w e have

(1.31)

aJ

aui (iU"+')

- uY+') 2

( ~ i

0 VO, E Ki ,

c 3 t h a t ( 1 . 2 1 ) and ( 1 . 2 2 ) s t i l l h o l d , and ( 1 . 2 3 ) is s a t i s f i e d . We deduce from ( 1 . 7 ) t h a t

and s i n c e

(1.32)

(J'(u), u -

U) 2

0 VO E K ,

we deduce ( 1 . 2 4 ) , which, u s i n g (1.31), g i v e s t h e i n e q u a l i t y ( 1 . 2 5 ) , and hence t h e s a m e r e s u l t as i n Theorem 1.1. A n a t u r a l q u e s t i o n a t t h i s p o i n t is t o e s t a b l i s h how f a r t h e method can be extended t o convex s e t s K which are not of t h e form (1.27). F i r s t of a l l , it i s n e c e s s a r y t o d e t e r m i n e What meaning i s t o be g i v e n t o t h e method; it would a p p e a r r e a s o n a b l e t o rep l a c e ( 1 . 2 8 ) by t h e same i n e q u a l i t y , f o r : { u ; + ' , ..., { u;+',

+I

ui, q+',..., u: } E K ,

..., u;?,', q", Ui.,', ..., u: } E K .

However, the method then becomes unsatisfactory. To i l l u s t r a t e t h i s , we may g i v e a counter-example:

Counter-Example 1 . 2 .

Let J be s t r i c t l y convex on K , r e p r e s -

e n t e d i n F i g u r e 1.

Fig. 1.

66

Optimisation algorithms We assume J t o be i n c r e a s i n g w . r . t . We start from

u1,u2

.

(CHAP. 2 )

each of t h e v a r i a b l e s

(dK = boundary of K ) .

uo = { uy, u$ } E dK

Then u:, u$ E Kl

for

must s a t i s f y J(u:,

~3Q J(vl, ~3

{ vlr u$ } E K ,

which g i v e s

The points on the boundary of K are thus "blockage" points of the algorithm, i . e . although convergence of t h e algorithm i s implied by t h i s l a s t r e s u l t , t h e l i m i t i s i n g e n e r a l not t h e solution.

H

Remark 1.2. I n Chapter 3, Section 7 , i n connection with t h e e l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l b a r ( s e e Chapter 1, Section 1.2)¶ w e s h a l l see t h a t a minimisation problem has t o be solved f o r a convex set K which does not s a t i s f y (1.27) ( l ) . Nevertheless, it w i l l be seen t h a t t h e r e l a x a t i o n method converges t o t h e s o l u t i o n , provided t h a t , amongst o t h e r t h i n g s , a suitable uo is chosen.

H

We can c a l c u l a t e i n two stages,, as follows: F i r s t we d e f i n e J;+lI2 as the s o l u t i o n of

Remark 1.3. dJ

(1.33)

-(u;+',

..., G?:,4+1/z,#+l, ..., 4)= 0 , and t h e n d e f i n e 4"

h i

by (1,34)*

u:+' = PK,(1(;+"')

9

where P, i s t h e o p e r a t o r which p r o j e c t s onto K i . , i . e .

P d f ) = max h,min (hf)).

1.3

Block r e l a x a t i o n

The decomposition (1.27) o f K i s assumed t o be performed coordinate by coordinate; it i s c l e a r t h a t t h i s i s not necessary and may e a s i l y be extended t o t h e case where we perform t h e decomposi t i o n u s i n g blocks of coordinates. and assume t h a t V i s decomposed as follows: We put V = R" (1)

We say:

f o r a nonlocal convex set K.

Relaxation methods

(SEC. 1)

v=

n vi, N

i= 1

Vi of dimension mi,

rn,

+ ... + rnN = M .

Let K be a closed convex subset of V of the form

n Ki , Ki a closed convex subset of N

(1.35)

K =

Vi.

i=l*

Method (1.28) i s readiZy extended t o t h i s s i t u a t i o n , a s i s Theorem 1 . 2 . Remark 1.4. A l l this also extends to the case where the spaces Vi are uniformly convex Hilbert or Banach spaces, provided the properties of Lemmas 1.1 and 1.2 are then e i t h e r included i n t h e hypotheses, or else deduced from hypotheses stronger than (1.2), (1.3) and the fact that J is of class C1. For considerations of this kind, see Levitin and Polyak /l/, 121, Polyak /l/, Glowinski / b / , and Cea and Glowinski /2/. 1.4

Over-relaxation and under-relaxation

Suppose A is a symmetric, positive-definite, (N,N) matrix A = (aij), aij = uji v i , j , and

fER".

The equation (1.36)

Au = f

is equivalent to the unconstrained minimisation of (1.37)

JW =

+

a@,u)

- (5 u) ,

where

c aij Al

(1.38)

a(u, u) =

uj ui

.

i.j= 1

Algorithm (1.5) then becomes N

(1.39)

4"

=

I a i( j q + l +' c? aijq - 1;: aii j= 1

j=i+l

which is the classical Gauss-Seidel algorithm (see Varga /l/) which, as is well known, can often be generalised to advantage using the following algorithm: i- 1

(1.40)

j= 1

q + l

=

(1

- w)q

j=i+l

+ 0 t4+1/2.

68

@timisation

algorithms

(CHAP. 2)

Algorithm (1.40) i s termed over-reZaxation, ( r e s p . under-relaxa t i o n ) if w > l ( r e s p . w < l), t h e i d e a n a t u r a l l y b e i n g to accezerate the convergence by choosing a s u i t a b l e w. We s h a l l now extend t h e above a l g o r i t h m t o t h e constrained c a s e , where N

fl K i .

K=

i= 1

We n o t e t h a t t h e f i r s t e q u a t i o n i n minimisation over 08 o f

ui 4

(1.40)i s e q u i v a l e n t t o t h e

J(G+',...,,u;-+;, ui, u ; + l , *..,4

9

i.e.

{

(1.41)

J(u;+',

..., 42;,

< J(G+1, ...,

< *-+;,u;+...,...,4)4) @+I,

I+, 1,

vu, €

R

We n e x t d e f i n e

g+? = PK,((l - w) U; + wu;+'").

(1.42)

W e t h e n have Theorem 1.3. J is asswned to be defined by (1.37), ( 1 . 3 8 ) , If it is asswned that where A is symetric positive-definite.

0cwc 2

(1.43)

then u" , defined by (1.41), (1.421, converges to u , the element of K minimising J over K. Proof.

F i r s t , we show t h a t t h e sequence J(lr) i s d e c r e a s i n g .

For t h i s , it w i l l be s u f f i c i e n t t o show t h a t

(1.44)

(--)

12-w

xi = J('-'u"+' 1 - J ( ' t f + ' ) > 7

uii I uy+1 - u; 1 2 ,

i = 1, ..., N , ( u s i n g t h e convention

Ou"+l

= u")

from which, by summation, w e g e t :

and s i n c e uIi > 0, i = 1,

We have, w i t h role) :

..., N, t h e n

w i t h S = inf { uIi 1 i = 1,

..., N

}

, we

have

i = 1 ( t h e o t h e r v a l u e s o f i having a s i m i l a r

(SEC. 1)

Relaxation methods

69

so that, using the first equation (1.40) (which results from (1.41)), we obtain XI = ta,1((U;)2- (u;+1)2)- a l l f(u;"

= all[-

- u;)Z

+ (u;

(1.44) provided

f r o m which we get

u;+"2(4

-

4+1)

- U Y + l ) ( u ; - ,+1'2)],

we ensure that

1' (u;-u;+l)(4 -u;+1'2)2-_Iu;+l-u;12, w

which is equivalent to

(4 - u;")

((1 - w ) 4

+ wu;+1/2- u;")

d 0,

the inequality which results from (1.42) (with i = 1). Thus the sequence J(u") is indeed decreasing, and since it is bounded below by J(u) it converges and we have

We have (J'(U"+') -.J'(u),

U"+1

- u) 2 A, 11 U " + l - u 112,

where 1, > 0 is the smallest eigenvalue of A. If u is the solution minimising J over K, we have (1.47)

( J ' ( u " + ~ )U," + I

- U) 2 A, 11 U"+' - u 112 .

We shall now prove that the left-hand side of this i.nequa1it.y tends to zero, which will give

11 U"+' - u 11

-+

0

which will prove the theorem. Let, be the minimum over Ki of the functional ui

and

-+

+I J(u;+', ..., 41 , ui, U l + l , ..., 4) 1

This Page Intentionally Left Blank

Relaxation methods

(SEC. 1) (1.49)

=

71

P , ( ( l - a,,)#+ w , # + " ~ )

I f we assume t h a t

(1.50) 0 < g l Q w, Q 2 - g 2 , ei f i x e d . we a g a i n have t h e r e s u l t o f Theorem 1.3 ( w i t h t h e same p r o o f ) .

Remark 1.6. C l e a r l y , e v e r y t h i n g we have s a i d can be extended t o t h e c a s e of block relaxation (as i n S e c t i o n 1.3); s e e Glowinski 141, Cea and Glowinski 121. Remark 1.7. The p r o o f s g i v e n , which a r e based on estimates of the energy, can be extended t o the case o f Hilbert spaces, see

141,

Glowinski

Cea and Glowinski

121.

Remark 1.8. I n t h e u n c o n s t r a i n e d c a s e , Theorem 1.3 i s proved by means o f d i f f e r e n t methods i n Varga 111. Remark 1.9. I n t h e c o n s t r a i n e d c a s e , Auslender proposed t h e scheme:

-4"

(1.51)

where

8

= (1

0
111, 121,

has

1

i s t h e minimum o v e r Ki o f t h e f u n c t i o n a l

u,

+ J(UY+',

+1 ..., 4-1, Dj, u:+l, ..., 4) .

T h i s method, i s i d e n t i c a l t o t h e one we p r o p o s e , as l o n g as t h e it a r e a l l o u t s i d e t h e K, ( a n d o f c o u r s e w i t h 0 c w Q 1 ) ; i s however, v e r y d i f f e r e n t when t h i s i s no l o n g e r t h e c a s e . I n i n t h e applications f a c t (1.51) does n o t a l l o w o v e r - r e l a x a t i o n ; which we have c o n s i d e r e d it i s , however, i m p o r t a n t t o be a b l e t o u s e w > 1 ( t h i s can be s e e n g e o m e t r i c a l l y ) .

1(;+'"

Remark 1.10. C e r t a i n v a r i a n t s o f t h e above a l g o r i t h m s a r e u s e f u l i n t h e s o l u t i o n o f systems o f e q u a t i o n s . See S c h e c h t e r 111, 121, / 3 / and O r t e g a and R h e i n b o l d t /l/, 121.

1.5

A c l a s s o f n o n d i f f e r e n t i a b l e f u n c t i o n a l s which can be minimised u s i n g r e l a x a t i o n

We have s e e n (Counter-Example 1.1) t h a t , i n general, t h e r e l a x a t i o n a l g o r i t h m i s n o t s u i t a b l e f o r t h e m i n i m i s a t i o n of nondifferentiable functions. We s h a l l however prove a r e s u l t which i s i n f a c t v a l i d f o r t h e c l a s s o f f u n c t i o n a l s o f t h e form: N

(1.52)

J(U) = J,(u)

+ C ai I ui I ,

cr, >/ 0 ,

1=1

where J ,

is

cl,

s t r i c t l y convex, J,(v)

+

+ a, if

11 D 11 + a,.

@timisation

72

Theorem

algorithms

(CHAP. 2 )

1.4. If J is given by (1.52), the point relaxation (1.5) converges to the solution u of problem ( 1 . h ) .

algorithm

W e introduce supplementary variables ( ) yf, i = 1, we put:

c

..., N,y

= { y, } ;

N

-

Jib, u) = Jo(4

(1 53)

+ i=1 a, Yi

9

which d e f i n e s J 1 , (not s t r i c t l y ) convex and of c l a s s C1 on R z N ; we introduce t h e closed convex subset of R2N: ' N

(1 .54)

inf J,(u, y ) = Jl(u, I u I) = J(u) .

(1.55)

L

We introduce :

( i ) The sequence d' (n >, 0)corresponding t o t h e r e l a x a t i o n algorithm f o r J ;

( i i )The sequence z " , corresponding t o t h e ( b l o c k ) r e l a x a t i o n algorithm, with r e s p e c t t o

It can be shown t h a t i f w e s t a r t from

then

and hence

(1.59)

J l ( f ) = J(U")

.

Since

J(P)>, J(u"+') (this does not a s m e that J is d i f f e r e n t i a b l e ) ,

('1 ( 2,

We Put Jl(u,Y) = J1(ulrYlruZ,YZr...ruN,~N) = J 1 ( Z 1 , Z 2 r - . . , Z N ) . SO as t o transform an "unconstrained nondifferentiable" problem i n t o a "constrained differentiable" problem.

Relaxation methods

(SEC. 1)

aJ"(u;+1,

au,

i=l

3 aui

( s i n c e we have

(u;+l,

73

..., u;+l,u;+l,..., G)(u;-u;+I)t

..., q+',U;+,,..., 4) (4- 4") +

- fl") 3 0

from t h e d e f i n i t i o n of ). From ( 1 . 6 5 ) and ( 1 . 6 4 ) we deduce t h a t :

(1.66)

u"+1-

d + O .

.

It now remains t o show t h a t u " + u The r e a s o n i n g i s t h e same as i n t h e d i f f e r e n t i a b l e c a s e ( s e e ( 1 . 2 4 ) ) ; we p u t I* =

{u,IuIJ

and w e s t a r t from

(J;(Z"+') - J;(z*), 9'' - Z*)R>N = (Ji(un+') -

JA(u), u"+' - u ) R N 2 6lu(ll fl+' - u 11).

S i n ee (J;(z*), f"

=

- z*)R>N 3 0 ,

74

@timisation algorithms

(CIIAP. 2 )

we deduce

which, by w r i t i n g out t h e left-hand s i d e e x p l i c i t l y , g i v e s

From t h e d e f i n i t i o n o f { l(;+',y;+' } and s i n c e { u,, I u, 1 } E KI , t h e second term on t h e left-hand s i d e of (1.69) i s G 0 , and hence

Since

*+'- # - 0 ,

we deduce t h a t

If+u.

Remark 1.11. Theorem 1.4 supplements a result due t o Auslender 111, / 2 / where t h e convergence of t h e algorithm t o wards a " c r i t i c a l point" ( l ) , which can be d i s t i n c t from t h e s o l u t i o n ?A, i s shown.

Remark 1 . 1 2 . The algorithms given i n t h i s s e c t i o n a r e o f t e n considered as coming w i t h i n t h e c l a s s of so-called " d i r e c t " methods, i . e . t h o s e not u s i n g t h e expression f o r t h e d e r i v a t i v e of t h e f u n c t i o n a l - a t l e a s t i n the writing o f t h e algorithm. There a r e many o t h e r " d i r e c t " methods: t h e c o o r d i n a t e r o t a t i o n method, Rosenbrock's method, t h e method of l o c a l v a r i a t i o n s or t h e Hooke and Jeeves method; see, f o r example, Cea 121. These algorithms can be modified i n such a way Remark 1.13. t h a t ' t h e y can be used on p a r a l l e l processors ( s e e Morice 111).

(l) .

"Blockage point" i n t h e sense o f t h e Counter-Examples 1.1 and 1 . 2 .

Gradient and gradient p r o j e c t i o n methods

(SEC. 2 )

75

SYNOPSIS

W e s h a l l now g i v e a b r i e f review of t h e methods which use t h e d e r i v a t i v e o f t h e f u n c t i o n a l t o be minimised.

2.

METHODS OF THE GRADIENT AND GRADIENT PROJECTION TYPE

Here we s h a l l only p r e s e n t t h e algorithms. For t h e proofs w e refer t h e r e a d e r t o Cea / 2 / ( s e e , i n p a r t i c u l a r , pp. 70-109 and pp. 118-156) and t h e bibliography t h e r e i n . 2.1

General remarks

Assuming zi" t o hsve been c a l c u l a t e d , we seek tP+'using:

(2.1)

= If

- P,,w",

where pn 3 0 i s t o be chosen, as i s t h e d i r e c t i o n w " . Assuming J t o be d i f f e r e n t i a b l e and u s i n g formal reasoning we have :

(2.2)

- pn w?

J(tP

- p,(J'(@,

= J(u?

w")

+ ...

so t h a t it would be o f advantage t o t a k e

(2.3)

(J'Ww") 3 0 .

The most n a t u r a l choice i s t o t a k e

(2.4)

W"

= J'(@

and t h i s l e a d s t o methods of t h e g r a d i e n t type.

2.2

Methods of t h e g r a d i e n t t y p e (unconstrained c a s e )

- Optimal s t e p method. We choose p,, so t h a t

(2.9

J(d

- p,, J'(u")) = infJ(tP - pJ'(u3). P

Assuming

(2.6)

J s t r i c t l y convex, o f c l a s s t h e method i s convergent.

- Fixed s t e p method. W e choose

c',

J(u)+ 8

+

a)

if

11 u 11 + a,

76

Uptimisation algorithms

(2.7)

(CHAP. 2)

p,, = p f i x e d .

The method i s c l e a r l y e a s i e r t o apply, but t h e c o n d i t i o n s f o r convergence a r e f a i r l y s t r i c t ; f o r example, i f w e assume J t o be of c l a s s ~2 and if we put

t h e method converges i f : 0 < 6 G p G 2 ~ -1 6 ,

(2.9)

PI

G PI.

- Variable step method. We seek p a p r i o r < i n t h e form

i

(2.10)

P =dp0. k E Z , po > 0 , a > O and k t o be a d j u s t e d i n t h e course of t h e i t e r at ions.

We choose p of the form ( 2 . 1 0 ) such t h a t J(u" - P J ' W

(2.11)

< JW)

J'(u9 # o ). The method i s again convergent

( w e assume

-

if po is sufficiently small.

Divergent series method.

We consider a sequence p,, such t h a t p,, > 0, p. + 0 and

p,, = n= 0

+ a).

We t h e n d e f i n e

The choice of a s u i t a b l e sequence pn i s very t r i c k y , and t h i s method g e n e r a l l y converges very slowly.

2.3

Methods of t h e conjugate g r a d i e n t t y p e (unconstrained case)

Consider t h e quadratic case

(2.13)

J(u) = ~ ( A uU), - (f,U) ,

where A i s a symmetric p o s i t i v e - d e f i n i t e (N,N) matrix. The d i r e c t i o n s { w " , . . . , # - I } of dy are s a i d t o be conjugate with

Gradient and gradient projection methods

(SEC. 2 )

77

respect t o A i f (2.14)

( A d , d )= 0 if

i#J,

# 0

otherwise.

S t a r t i n g from y o , we assume d ' t o have been c a l c u l a t e d and search f o r d'+' by dispzacement i n t h e d i r e c t i o n w " , t h u s (2.15)

uR+' =

U" + h " ,

1 being c a l c u l a t e d i n such a way a s t o minimise J(d'+l)

= ad')

+ 1 2 (Aw",w 3 + Y(AU",w") - ( A w")) , 7 J

and hence

A f t e r N i t e r a t i o n s , we thus a r r i v e ' a t

However a d i r e c t c a l c u l a t i o n shows t h a t , Vg'BE RN

, we

have

so t h a t (2.17) g i v e s (2.19)

d" = A - ' f

= u = s o l u t i o n o f t h e problem.

Hence t h e algorithm converges i n a f i n i t e number o f i t e r a t i o n s . The problem now becomes, f i r s t o f a l l , t o discover how t o generate t h e conjugate d i r e c t i o n s , wo, ..., d-l. S t a r t i n g from u o , w e t a k e as t h e f i r s t d i r e c t i o n t h e d i r e c t i o n of s t e e p e s t descent a t u o , bence (2.20)

wo =

- J'(U0)

i.e. i n t h e q u a d r a t i c c a s e (2.21)

wO'=

- AuO + f .

We t h e n seek u1

= uo

+ 1wo ,

c a l c u l a t e d s o a s t o minimise J(uo + form (2.22)

w1 =

1wy

; we t h e n seek w' i n t h e

- J'(U') + p w 0 ,

where p i s chosen i n such a way t h a t w'and

w1 are conjugate, i.e.

Optimisation algorithms

(CHAP. 2 )

(w', AwO) = 0 ,

and sc on. We hence arrive at the foZZowing algorithm: ro= -J'(uo), w o = r o ) We assume u", In, w" t o be known ( uo a r b i t r a r y , and w e d e f i n e :

then

(2.24)

In"

=

- J'(u"+'),

I n t h e non-quadratic c a s e , t h e algorithm i s adapted as follows. We d e f i n e :

(2.26)

u"+' = u " + K w " ,

where

i s chosen so as t o minimise J(u" p+' =

+ pw"), F E W

; th'en

- J'(u"+').

I n choosing w"" We put

it i s necessary t o t a k e c e r t a i n p r e c a u t i o n s .

We check whether t h e v e c t o r s In+' and P+'form an angle with s u f f i c i e n t l y s m a l l 6 > 0 . If so, we put

(2.28)

-II - 6, 2

w"+' = 3"

otherwise, t h e d i r e c t i o n 9""

(2.29)

Q

must be discarded and we t h e n t a k e

w"+' = I n + ' .

The algorithm i s convergent, by v i r t u e of t h e c l a s s i c a l assumptions (1.2), (1.3) and with J of c l a s s C1. From a numerical p o i n t of view, it i s advantageous t o t a k e t h e following a d d i t i o n a l precaution: i f , f o r N successive i t e r a t i o n s , j = i , i + l , ..., i + N - 1 , we have

d#r' we put w'+N= # + N .

Remark 2.1. The method described above i s t h e so-called conj u g a t e g r a d i e n t method. There a r e v a r i o u s methods which allow

79

Gradient and gradient projection methods

(SEC. 2 )

t h e c o n s t r u c t i o n of N conjugate d i r e c t i o n s . I n p a r t i c u l a r , w e mention t h e method o f F l e t c h e r and Powell which a l s o allows t h e i n v e r s e m a t r i x o f J r r ( u )t o be obtained. An a p p l i c a t i o n of t h e so-called conjugate d i r e c t i o n method t o t h e problem of t h e minimisation of

J(u) = M

u , v ) - (f,0)

s u b j e c t t o t h e c o n s t r a i n t s Bv = b , w i l l be given l a t e r ( s e e Remark 4.10). Constrained case

2.4

We now consider t h e system

(2.30)

infJ(u),

UEK,

K = closed convex subset of RN.

Here, w e s h a l l adapt t h e above methods by using t h e operator

Pr p r o j e c t i n g onto K. - Point projection method

S t a r t i n g from uo E K u"+' = PJu"

(2.31)

, we

d e f i n e u"+' from u" using

- P.J'(u"))-

Theorem 2.1. Consider the case (2.13) (quadratic functional). We can now choose po and p1 so that with

(2.32)

0 < Po Q

Pn

Q

PI

the method (2.31) converges t o the solution u of problem ( 2 . 3 0 ) . Proof. (2.33)

If u i s t h e s o l u t i o n of t h e problem we have:

u = PA.

- p,J'(u))

vp,

s i n c e (2.33) i s equivalent t o (u

- p, J'(u) - u, u .-

By p u t t i n g w" = u"

u) Q 0 Vu E K , i.e. (J'(u), u

- u and u s i n g t h e f a c t t h a t Pr is

c t i o n we deduce from ( 2 . 3 1 ) , (2.33) t h a t (2.34)

- u) 2 0

11 w'+' (1 G 1 w" - p,(J'(u") - J'(u))

1

Vu E K .

a contra-

80

@timisation However, J'(u")

(2.36)

aZgorithms

- J'(u) = A(u" - u) = Aw"

II w"+' 11' d

(1

- 2 ap,

(CHAP. 2 )

so t h a t (2.35) g i v es

+ CpX) 11 w" [ I z .

We c a n choose po and p1 i n s u c h a way t h a t ( 2 . 3 2 ) g i v e s

1 - 2ap,

+ CpX Q B < 1

and t h e n ( 2 . 3 6 ) shows t h a t

11 w" \I

+ 0.

Remark 2 . 2 . The above p r o o f i s r e a d i l y e x t e n d e d t o t h e nonquadratic case with Lipschitz J ' .

- Gradient

projection method.

I n t h e u n c o n s t r a i n e d i t e r a t i o n method o f ( 2 . 3 1 ) , t h e p o i n t s We a r e now g o i n g t o project the gradients. were p r o j e c t e d o n t o K. We assume:

(2.37)

t h e convex s e t K i s bounded and d e f i n e d by a f i n i t e number o f a f f i n e l i n e a r c o n s t r a i n t s

K = { u I Gj(u) = 0, 1 and t h a t f o r a l l p o i n t s CJu)

Qj Q

I, C,(u) Q 0, 1

+ 1 Q k Q rn }

U E K ,t h e v e c t o r s

, j E I(u) , I(u) = ( j I 1

Q j Q rn, Cj(u) = 0 } ,

are l i n e a r l y i n d e p e n d e n t . Then t h e p r o j e c t i o n o n t o K o f t h e s t r a i g h t l i n e p r o c e e d i n g from , i . e . t h e s e t o f p o i n t s w such t h a t

v i n t h e d i r e c t i o n - J'(u) (2.38)

w = PJu

-~J'(u)),

p 30

i s a segmented l i n e , t h e f i r s t segment b e i n g d e n o t e d by [u, u + ] The a l g o r i t h m i s t h u s as f o l l o w s : having o b t a i n e d ~ E E K w, e define by: (2.39)

J(u"+ ') Q J(u) , Vu E [u", (u")']

.

.

The d i r e c t i o n [ u , u + ] i s o b t a i n e d u s i n g a n o p e r a t o r which p r o j e c t s onto an a f f i n e l i n e a r manifold. If t h e m a n i f o l d i s d e f i n e d by

s=

{uIAu=b}

t h e p r o j e c t i o n o f a v e c t o r a o n t o S i s g i v e n by

P&)= [ I - A * ( A A * ) - ' A ] u + A * ( A A * ) - ' b . where A* i s t h e t r a n s p o s e o f t h e m a t r i x A . The method i s c o n v e r g e n t ( e . g . u s i n g t h e assumption ( 2 . 3 7 ) and w i t h J i s s t r i c t l y convex and of c l a s s Cl) ( s e e Rosen /l/, /2/ and Canon, C u l l u m and Polak /l/).

(SEC. 3 )

Penalisation methods

81

Remark 2.3. Apart from t h e gradient projection method, it i s a p p r o p r i a t e t o mention t h e reduced gradient method introduced by Wolfe /1/ i n t h e context o f t h e minimisation o f convex functions s u b j e c t t o linear constraints, by e x t e n s i o n of t h e simplex method. This method has been g e n e r a l i s e d t o t h e c a s e of nonlinear constraints by Abadie-Carpentier /1/ ( s e e a l s o F l e t c h e r 111). A method somewhat similar t o t h e above i s t h a t of FrankWolfe 111. 3.

PENALTY METHODS AND VARIANTS

3.1

General remarks

The i d e a of p e n a l i s a t i o n ( a l r e a d y encountered i n Chapter 1, Sect i o n 3.2; s e e i n p a r t i c u l a r Remark 3.4) c o n s i s t s of “approxima t i n g ” t h e problem,

inf J(u) , ueK, K = a c l o s e d convex subset of RN, by unconstrained problems ( a s w i l l be t h e c a s e i n t h e exterior method d e s c r i b e d i n S e c t i o n 3.3 below), or by problems with passive constraints, i . e . where t h e i n f i s a t t a i n e d in the interior o f t h e set o f c o n s t r a i n t s (as w i l l be t h e c a s e i n t h e interior method ( S e c t i o n 3.2 below) and i n t h e method of c e n t r e s with v a r i a b l e t r u n c a t i o n described i n W Section 3.4 below) (I). I n t h e followi’ng w e s h a l l assume t h a t K i s defined by

(3.2)

{ G,=

K={ulG,(u)>O, I G j G m } , a concave continuous f u n c t i o n on

RN.

We s h a l l assume t h a t J s ’ a t i s f i e s :

(3.3) 3.2

J i s d e f i n e d and continuous on

w,

s t r i c t l y convex.

I n t e r i o r methods

We i n t r o d u c e t h e f u n c t i o n a l

We make t h e assumptions:

(l)

C l e a r l y it then becomes necessary t o choose an algorithm f o r the solution of the penalised problem.

8

@timisation algorithms

82

{ KK

(3.5)

0

, w i t h a nonempty = { u I GLu) > 0, j = 1, ..., m 1. i s bounded ( I )

(CHAP. 2 )

interior

i , given

by

This l a s t c o n d i t i o n i s a s s u r e d when t h e r e e x i s t s uo such t h a t G,(uo) > 0 ,

(3.6)

-

j = 1, ..., m .

The $unction u + I(u, E) i s t h e r e f o r e n o t i d e n t i c a l t o +a or t o on K. I n f a c t ,

a)

Lema 3.1. Using the hypotheses ( 3 . 2 ) , (3.31, ( 3 . 5 ) , there e x i s t s a unique u, such t h a t : (3.7).

U,EK,

(3.8)

Z(U,, E ) Proof.

< I(u, 8)

Vu E

2.

With t h e element u,chosen as i n ( 3 . 6 ) , we i n t r o d u c e

the set

(3.9)

S, = { u I u E K, Z(u,

E)

< Z(u0, E) } ,

which i s nonempty s i n c e u,ES,

.

It can be shown t h a t S, i s

closed (and hence compact s i n c e S, c K , where K i s bounded); s i n c e I(u,E) i s bounded on S, t h e Cju) remain > 0 on S, and I(u,&) i s continuous on S,., which g i v e s t h e r e q u i r e d r e s u l t . Furthermore, 0

(3.10)

S, c

K, rn

and hence t h e lema.

W e can now s t a t e and prove:

Under the assumptions (3.21, ( 3 . 3 ) , ( 3 . 5 ) i f u, Theorem 3.1 i s the element defined by (3.71, ( 3 . 8 ) we have (3.11)

u,+u

i n RN

where u i s the soZution of (3.12)

uEK,

J(u) = i d J(u) . veK

Remark 3.1. T h e o p e n a l t y method t h u s d e f i n e d i s s a i d t o be interior since u , E K . (I)

The assumption " K bounded" i s i n t r o d u c e d o n l y w i t h a view t o s i m p l i f y i n g t h e p r o o f s . I n p r a c t i c e , it i s s u f f i c i e n t t o assume t h a t t h e m i n i m u m o f J ( V ) o v e r K i s reached f o r a p o i n t u a t a f i n i t e d i s t a n c e , which i s t h e c a s e if, f o r example, J s a t i s f i e s ( 1 . 2 ) .

(SEC. 3 )

Penalisation methods 0

83

Since u,EKand K i s bounded, we can e x t r a c t a subsequence, a l s o denoted by u , , such t h a t

Proof.

(3.13)

wsK

u,4winW,

We have

which, i n t h e l i m i t , g i v e s

J(w)

< J(u)

vu E

i,

and hence

J(w) d J(u) Vu E K and s i n c e J i s s t r i c t l y convex, w = u , g i v i n g t h e r e q u i r e d result.

Remark 3.2.

We have t h e following a d d i t i o n a l property: The f u n c t i o n E -D J(u3 i s decreasing towards J(u) as E + 0. To show t h i s , l e t 0 < q < E . We p u t :

From t h e d e f i n i t i o n s of u, and u,, we have:

J(u,)

+

1 E-

and

G(u3

Q J(u$

+E

1

(3.15)

so t h a t , by a d d i t i o n ,

and hence

1 ---GO.

G(u3

1

G(u,)

Thus (3.15) t h e n g i v e s

3.3

E x t e r i o r methods

We introduce t h e f u n c t i o n a l

(3.16)

E(u,E )

= J(u)

+ -1 G(u)E

G

(4

84

Gptimisation algorithms

(CHAP. 2 )

where by d e f i n i t i o n

(3.17)

G(D)- =

2 Gj(u)- 2 sup(- Gl(u),0). =

]= 1

j= 1

We n o t e ( c f . Chapter 1, S e c t i o n 3 . 2 ) t h a t

G(D)- = 0 0 D E K .

(3.18)

We do not make any assumption analogous t o ( 3 . 5 ) . Since, i n p a r t i c u l a r , K i s n o t assumed t o be bounded, it i s n e c e s s a r y t o assume t h a t (3.19)

if 1 1 u I I + c o ,

J(D)-,+co

u€RN.

Then, s i n c e

(3.20)

E(D, E ) 2

J(D),

w e h a v e , a f o r t i o r i , E(D,E ) +

+ co

if

11 D 11 + co and hence, w i t h f l u , & )

s t r i c t l y convex i n V ,

t h e r e e x i s t s a u n i q u e u, i n @ s u c h t h a t = inf E(o, E ) .

(3.21)

V€RN

We t h e n have: Theorem 3.2.

fied.

Then, if

(3.22)

u,+ u

We asswne t h a t ( 3 . 2 ) , ( 3 . 3 ) , ( 3 . 1 9 ) are s a t i s is d e f i n e d by (3.21) and u by (3.12) we have:

u,

i n RN.

Remark 3.3. The element a p p r o x i m a t i n g u, is not n e c e s s a r i l y hence t h e t e r m i n o l o g y : e x t e r i o r method. rn in K; Proof. (3.23)

We have:

J(uJ

< flu,,

E)

< uinf E(D,E ) sK

= inf J(u) = J(u) V € K

from which it r e s u l t s t h a t u, l i e s i n a bounded s u b s e t o f RN. We can , t h e r e f o r e e x t r a c t a subsequence, a l s o d e n o t e d by u, , such t h a t u, + w i n RNand w e deduce from ( 3 . 2 3 ) t h a t

(3.24)

J(w) < J(u) .

Moreover from ( 3 . 2 3 )

1 ;G(u,)-

d J(u) - J(u,)

therefore G(w)- = limG(uJ- = 0 t-0

Penalisation methods

(SEC. 3 )

85

and hence ( s e e ( 3 . 1 8 ) : W E K which, along with (3.24) shows t h a t w = U. w

Remark 3.4. I t o s h o u l d be emphasised t h a t od s t i l l holds i f K = 0 . This allows t h i s f o r t h e s o l u t i o n of problems involving linear ints: t h e p e n a l i s a t i o n term a s s o c i a t e d with Gj(u) = 0 t h e n t a k e s t h e form

Remark 3.5. G(v)- =

c

[SUP(-

t h e e x t e r i o r methmethod t o be used

equality constraa constraint

I n (3.17) we can a l s o d e f i n e G(u)- by G,(u), 0)F with

Q

3 1.

j= 1

Remark 3.6. I n t h e context of e x t e r i o r methods, t h e following v a r i a n t can a l s o be used: with each c o n s t r a i n t Gku) 2 0 ( j = 1,m) we a s s o c i a t e a v a r i a b l e ti >/ 0 ( c a l l e d a s l a c k v a r i a b l e ) from which we g e t t h e equivalent c o n s t r a i n t p a i r { G,(u) - ti = 0, ti 2 0 ) and t h e p e n a l i s e d f u n c t i o n a l

F(u,t , E )

= J(u)

+ -1 1(Gj(U) "I

tj)'

E j=l

where t = (II, ..., 1,) E R ; t ; t h e p e n a l i s e d problem t h e n c o n s i s t s of minimising f l u , t , E ) over R" x R;t . The advantage of t h i s approach over t h a t described above i s t h a t t h e o r d e r of d i f f e r e n t i a t i o n of J and of t h e Cj i s conserved a t t h e c o s t of i n t r o d u c i n g t h e v e c t o r t ( t h e c o n d i t i o n t h a t t h e t j must be p o s i t i v e does not i n t r o d u c e any a d d i t i o n a l p r a c t ical difficulties). The r e s u l t s of S e c t i o n 3.3, concerning f l u , & ) as defined by (3.1-6), a r e e a s i l y adapted ( s e e , f o r example, Cea / 2 / ) . An a p p l i c a t i o n r e l a t i n g t o t h e model e l a s t o - p l a s t i c problem defined i n Chapter 1, Section 1 . 2 i s given i n Chapter 3, Section 8.2.2. 3.4

Method of c e n t r e s with v a r i a b l e t r u n c a t i o n

General remarks. We i n t r o d u c e , a g a i n f o r problem ( 3 . 1 ) : (3.25)

K(I)=(ulu~K,J(~)brl}

which d e f i n e s a nonempty c l o s e d convex s e t i f :

I B J(u) ( and K(J(u)) = { u

1).

The formal i d e a i s t h e n t o c o n s t r u c t an algorithm which determines : .

86

@I timisation a Zgorithms

(CHAP. 2 )

( i ) a d e c r e a s i n g sequence of & (- J(u)) ( i i ) a " c e n t r e " of K ( & ) a s s o c i a t e d w i t h e a c h & ( i n f a c t An+] i s chosen u s i n g t h e " c e n t r e " of K(A,J). rn

"Centre of a conuex set". The c o n c e p t o f t h e " c e n t r e " o f a convex s e t is not intrinsic, and numerous b a s i c a l l y e q u i v a l e n t c o n c e p t s c a n be d e f i n e d . Here w e s h a l l g i v e t h e two c o n c e p t s which seem t o be t h e most n a t u r a l and t h e most u s e f u l . I n g e n e r a l , l e t 9 b e a c l o s e d convex s u b s e t o f R N d e f i n e d by

(3.26)

9 = { v I LAu) 3 0, 0 Q j Q m }

where, V j :

(3.27) 9

Lj i s a c o n t i n u o u s concave f u n c t i o n .

We t h e n c o n s i d e r t h e " c e n t r e " o f 9 t o be either t h e p o i n t i n , i n f a c t i n g o , where

(3.28)

fi LAv)

is maximum

1-0

or t h e p o i n t i n 9 , i n f a c t i n Y o , where 1

i s minimum.

8

AppZication of the concepts (3.28), (3.29) to K ( h ) . We u s e t h e above c o n c e p t s f o r

K(X), w i t h

Lo(v) = A - J(v) , t , ( v ) = Gkv) , 1 Q j Q rn

.

I n o r d e r t o s i m p l i f y t h e n o t a t i o n , we p u t :

n G,(v) , $1

(3.30)

D(u, A) = (1 - J(v))

I= 1

rn

(3.31)

AZgorithm.

We c o n s i d e r

VoEK

.

We t a k e :

A, = J ( v 0 ) . Then we d e f i n e vl t o be t h e " c e n t r e " of (I), i.e.: (l)

K(A,) u s i n g method (3.28)

If w e u s e (3.29), u1 ( d i f f e r e n t from t h e above) i s d e f i n e d by

Penalisation methods

(SEC. 3 ) (3.32)

D(u,, 1,) =

SUP

87

D(u, 11). ( ')

v e 4111

We next d e f i n e

(3.33)

J(w,) d

w, E K ( A , )

( s e e Remark 3.7 below) such t h a t

J(U1)

and then d e f i n e A2 by (3.34)

1,

=

1, - pl(l, - A w l ) )

0 < p1 < 1

Assuming un-, and 1, t o be known w e d e f i n e v, by (3.35)

D(U., 1,) = sup D(u, 1"). u E 41.1

We d e f i n e

(3.36)

K(AJ such t h a t

W,E

d

J(WJ

J(Vn)

and w e choose (3.37)

A,,

1

=

1, - P.[A. - J(wJ1

where (3.38)

0<6
It may be noted t h a t

(3.39)

-

u, E K(A,)

We have ( 2 )

Theorem 3.3. We assume t h a t ( 3 . 2 ) , ( 3 . 3 ) , ( 3 . 5 ) are s a t i s f i e d . Then t h e algorithm of t h e method o f centres with variable t m n c a t i o n defined by ( 3 . 3 5 ) , ( 3 . 3 6 ) , (3.37) i s convergent i n the f o l lowiag sense :

J(4

(3.40)

A,

(3.41)

u, + u .

-+

9

where u i s t h e solution of problem ( 3 . 1 ) .

(l)

A similar algorithm r e s u l t s u s i n g (3.29) i n s t e a d of (3.28).

(2)

There i s a similar r e s u l t ( a n d p r o o f ) i f we use (3.29).

88

o p t h i s a t i o n aZgorithms

(CHAP. 2 )

Proof. From t h e c o n s t r u c t i o n o f t h e a l g o r i t h m , & and J(u3are d e c r e a s i n g sequences and from (3.37) and (3.36) we have: 1 ,-

=

An+,

P J- ~ J(wn)] 3 Pn[An - J(un)J

and hence

Therefore

(3.42)

lim 1, = lim J(uJ = p 2 J(u) .

We s h a l l now show t h a t

(3.43)

p = J(u)

(which p r o v e s t h e theorem s i n c e u, i s t h e n a m i n i m i s i n g sequence and, s i n c e t h e f u n c t i o n J i s s t r i c t l y convex, we have (3.41)). Let u s assume t h a t e x i s t s u, such t h a t

m,,

.

Then

Ko1)

#

0 , so t h a t t h e r e

> 0.

However, s i n c e 1, of u, we h a v e : Wu.9

p'> J(u)

> p we

have

Nu,, A,)

> D(u,, p)

a n d , by d e f i n i t i o n

1,) 2 D(U,,13

hence

However

m

D(v,,

A,,)

=

(A, - 403)

n Gj(v3

j= 1

which c o n t r a d i c t s ( 3 . 4 4 ) .

Remark 3.7.

2)

c(1, - 403)

+

0

m

(On t h e c h o i c e o f t h e w , , ) .

1) F i r s t we c a n t a k e w,

-

Q

= u, :

i f p, = 1, t h i s i s t h e method of centres (Huard /l/), i f p , .c 1, we c a n c o n s i d e r t h i s method t o be a n u n d e r - r e l a x a t i o n method. The convergence i s a c c e l e r a t e d s i g n i f i c a n t l y i f w e t a k e w, such t h a t

J(v3 .

J(W3

I n p r a c t i c e , we c a n s e e k w , , ~ K ( 1 3e i t h e r on t h e s t r a i g h t l i n e J'(u,) , or on t h e segp r o c e e d i n g from u,, w i t h d i r e c t i o n v e c t o r ment tun- u,]

.

-

4)

(SEC.

89

Duality methods

The method i s i n t e r p r e t e d h e r e as a n o v e r - r e l a x a t i o n method.

4.

DUALITY METHODS

4.1

G e n e r a l remarks

We now r e t u r n t o t h e c o n c e p t s i n t r o d u c e d i n Chapter 1, S e c t i o n s 3.4 and 3.5 and from them we s h a l l d e r i v e a l g o r i t h m s which p l a y a n i m p o r t a n t p a r t i n t h e remainder o f t h e book. We c o n s i d e r a Hilbert space V -which i n t h e c a s e o f n u m e r i c a l a p p l i c a t i o n s w i l l always be f i n i t e dimensional: V = I#and the functional

(4.1)

4 u(u, u ) - (5 4 n(u, u) = a ( u , u ) V u , u ~V , i s JJU)

=

where isfying :

(4.2)

a(u,u)>aIIu1l2,

a>O,

a c o n t i n u o u s b i l i n e a r form on V sat-

VUEV

(where IIuII = t h e norm o f 2, i n V ) , and where i n ( 4 . 1 ) a c o n t i n u o u s l i n e a r form on V. We a l s o t a k e :

(4.3) (4.4) (4.5)

M

{

u+(Ju)

is

= c l o s e d convex s u b s e t o f V.

L = H i l b e r t s p a c e , a n d Q a f u n c t i o n from V M -+ L , l i n e a r or otherwise.

-+

L or from

A = c l o s e d convex s u b s e t o f L .

If ( , )L d e n o t e s t h e i n n e r p r o d u c t i n L , w e assume f o r a l l functions q E L t h a t : (4.6)

{

u + (q, @(u))~ i s lower semi-continuous and convex on V .

We t h e n c o n s i d e r t h e problem r

Remark 4.1. Thus, by d e f i n i t i o n , w e c o n s i d e r t h e problem i n a form a d a p t e d t o d u a l i t y ( s e e Chapter 1, S e c t i o n 3.4).- W e s h a l l s e e i n S e c t i o n 4 . 2 below how a number of i m p o r t a n t problems come w i t h i n t h e framework o f ( 4 . 7 ) .

Remark 4.2.

The f u n c t i o n

Qptimisation algorithms

90

(CHAP. 2 )

i s convex, l o w e r semi-continuous f o r t h e weak t o p o l o g y o f V ; t h i s i s t h e r e f o r e a l s o t h e c a s e for J d e f i n e d by (4.8)

4.2

40) = Jo(u) + SUP (q, “ ( 1 ) ) ) ~ . ~ E A Examples

Example 4 . 1 .

Using t h e n o t a t i o n

V = H;(Q), a(u,u)=

J‘,

A4

=

(1.18), Chapter 1, we t a k e :

V,

gradu.gradudx,

(f,u)=~,jiudx, feL2(Q),

L = (L2(Q))”, #u = grad u hence # E U ( V ;L ) , A = { q I q E L , Iq(x)l Q g a . e . i n Q } . We have ( s e e C h a p t e r 1, S e c t i o n 3.4 a b o v e ) : r

Then ( 4 . 7 ) becomes the flow problem c o n s i d e r e d i n Chapter 1, S e c t i o n 1 . 4 , which h a s a l r e a d y been c o n s i d e r e d from t h e p r e s e n t s t a n d p o i n t i n Chapter 1, S e c t i o n 3.4.

Example 4 . 2 .

We t a k e V,M,a, (f,v) a s i n Example

L = L2(s2), A , and we d e f i n e

G1 : V + L

Gl(u) = I grad u I ’

= [q

I q E L,

4.1:

q2 0

by:

- 1.

The f u n c t i o n #, i s L i p s c h i t z on V. T h i s comes w i t h i n t h e c o n t e x t o f S e c t i o n

4.1 and we have:

I n t h i s example, ( 4 . 7 ) i s t h u s t h e e l a s t o - p l a s t i c i t y problem c o n s i d e r e d i n Chapter 1 , ’ S e c t i o n 1.2. i s nondifferentiabze, and s i n c e t h i s i s a d i s W e note t h a t a d v a n t a g e ( f r o m t h e n u m e r i c a l p o i n t o f view ( l ) ) w e i n t r o d u c e G2 : H{(s2) -+ L’(s2) and A2 d e f i n e d by

(I)

E a s i l y surmountable:

s e e Chapter

5 , S e c t i o n 8.

(SEC. 4 )

Duality methods

91

indeed we t h e n have: sup

q@'(u)dx =

4eAz

0 if I g r a d u l d 1 a.e.

+

oc)

otherwise.

but, amongst o t h e r d i f f i c u l t i e s , t h e Hilbert space s e t t i n g of Section 4 . 1 i s not a p p r o p r i a t e f o r a2, i n t h e infinite dimensional case ( ) , s i n c e aZ(u) E~'(0). We n o t e t h a t @' is i n d e f i n i t e l y F r e c h e t - d i f f e r e n t i a b l e (and hence l o c a l l y L i p s c h i t z ) on Hh(62). 4.3

A saddle-point s e a r c h algorithm

Let u s now i n t r o d u c e t h e Lagrangian (4.9)

U(V9

r

4) = J O ( d

+ (4. @(U))L .

We make t h e following assumptions: (4.10)

t h e r e e x i s t s a saddle p p i n t of P(u, q) on M x A, i . e . a point { u , p } ~ M x A such t h a t U(u,q)
i

t h e function @ : M + A i s Lipschitz, i . e . there exists a constant C, such t h a t 1 @(u) - @(v) (1' d C1 11 u - u 11 VU, u E M . A v a r i a n t of ( 4 . 1 1 ) which i s u s e f u l i n a p p l i c a t i o n s i s as follows: (4.11)

(4.11')

I

VqEA

, the

s o l u t i o n of

inf 9 ( u , q)

lies in a

UEM

bounded s e t independent of q , and @ - i sZocaZZy L i p s c h i t z on V ( i n t h e sense of being Lipschitz on a l l bounded s u b s e t s of

v).

Under t h e s e c o n d i t i o n s we define the following algorithm ( 2 ) : S t a r t i n g from po, we c a l c u l a t e u o , and t h e n p l , e t c . The general r u l e i s (4.12)

{

with p" known, (ELI), we d e f i n e u" as t h e element of M which minimises J0(u) + (p", @ ( u ) ) ~

(I)

We s h a l l r e t u r n t o t h i s i n Chapter 3.

(2)

This i s e s s e n t i a l l y an algorithm of t h e Uzawa t y p e /l/.

Optimisation algorithms

92

(CHAP. 2 )

s e e Remark 4 . 3

( t h i s c e r t a i n l y d e f i n e s u" i n a unique manner; below); we n e x t d e f i n e (4.13)

= P,(p"

p""

+ pn Nu"))

where

,

(4.14)

P A = p r o j e c t i o n o p e r a t o r from L + A

(4.15)

pn > 0 i s t o be chosen a p p r o p r i a t e l y ( s e e below).

rn

Remark 4.3. The f u n c t i o n u + Jo(u) + (p", @ ( u ) ) ~i s s t r i c t l y convex and it is " i n f i n i t e at i n f i n i t y " ; i n f a c t , from ( 4 . 1 1 ) where we t a k e u t o be f i x e d a r b i t r a r i l y i n M , we have

I1 @@IIIL Q C(ll v II + 1) and hence

(where t h e

c

denote v a r i o u s c o n s t a n t s ) .

Remark 4.4. (Motivation of t h e a l g o ri t h m ) . saddle point then

If { u , p }

is a

which i s e q u i v a l e n t t o (4.18)

.

p = pA(p +

P @( U) )

VP >

and (4.18) e x p l a i n s ( 4 . 1 3 ) .

Remark 4 . 5 . I n f a c t t h e a l g o r i t h m i s n o t completely d e f i n e d s i n c e i n ( 4 . 1 5 ) we have the choice of the algorithm f o r calculating u". I n S e c t i o n 4 . 4 we s h a l l meet a v a r i a n t i n which t h i s choice is sp ec if ie d. rn Convergence df t h e algorithm. We s h a l l now prove:

...,

We assume t h a t ( 4 . 3 ) , (4.6) are t r u e , along Theorem 4.1. with (4.10) and ( 4 . 1 1 ) ( o r ( 4 . 1 1 ' ) ) . Thus t h e algorithm defined by (4.12), ( 4 . 1 3 ) is convergent i n the sense t h a t

(SEC. 4 )

DuaZity methods

(4.19)

U"

93

strongly i n V ,

+u

where u i s the solution of (4.71, when (4.20)

0 < a. d p. d aI ,

with suitabZe a. and

1 a1 ( )

P r o p e r t i e s (4.12) , (4.16) a r e e q u i v a l e n t t o

Proof.

- U") + (p". @(u) - @(U"))L 3

(4.21)

(Jh(u"), u

(4.22)

(Jd.(u), u - u )

+ (p. @(u) - @(u));

0 Vv E M

3 0 vu E

9

;

t a k i n g u = u ( r e s p . u = u" ) i n (4.21) ( r e s p . ( 4 . 2 2 ) ) we t h e n deduce (4.23)

O(ff

- U, U" - U) + (p" - p, @(u") - @(t())L d

0.

W e put (4.24)

p"

-p

= r".

From (4.13) , (4.18) and t h e f a c t t h a t P A i s a contraction, w e deduce t h a t (4.25)

I1 r"" 1; d

11 r" + p,,(@b") - @(u))1;

+ 2 PAP, @(fi- @(&

+ PI I

11 r" 1; + Yu") - @(4Ilt ' =

However, u s i n g (4.23) and ( 4 . 1 1 ) ( o r ( 4 . 1 1 ' ) ) , we deduce from ( 4.2 5 ) t h a t

We t h e n choose a. and a1 i n such a way t h a t (4.20) implies t h a t (4.27)

2 ap,

- C: Pf 2 b > 0 .

Then ( 4 . 2 6 ) g i' v e s (4.28)

11 r"" 1;

+ fl 1I U" -

11' d 11 r" 1.;

From (4.28) we g e t t h a t t h e sequence n + 11 r" ;1 and hence t e n d s t o a l i m i t , s o t h a t BllU"-ull2+O

is decreasing

and hence (4.19)

Remark 4.6. For some g e n e r a l r e s u l t s on t h e e x i s t e n c e of saddle p o i n t s , s e e Ekeland-Temam /l/. (l)

E s t i m a t e s o f aoand a1 are s u p p l i e d i n t h e proof which follows.

Optimisation aZgorithms

94

4.4

(CHAP. 2 )

A second saddle-point search algorithm

We shall now supplement Remark 4.5, by assuming that

Then in (4.12)u" is defined by

JiY)+ @*p" = 0 , i.e. ( A E ~ ( V V? , (4.30)

Au"

is defined by a(u,u) = ( A u , ~ ) ) :

+ @*p" - f = 0

and if we introduce an iterative algorithm for the solution of (4.30) we naturally arrive at the following algorithm: with u" and p" assumed to have been calculated, we define #+'by ( l ) : (4.31)

I.4""

= u" - p1 S-'(Au"

+ @*p" - f)

where: S = identity if V ' = V = finite-dimensional space ( o r alternatively S = arbitrary symmetric positive-definite matrix) , S = the duality operator from V + V' if V is infinite-dimensional ( 2 ) , and we then definep"+'by: (4.32)

p"" = P,(p"

+ p2 @ d + ' ) ,

where in (4.31) and (4.32) p 1 and p 2 are two parameters > 0 to be suitably chosen. We have:

.

Theorem 4.2. We assume that ( 4 . 3 ) ,. . ,(4.6), (4.29) are true. we can then choose p 1 , p 2 > 0 ( 3 ) i n such a way that algorithm (4.31), (4.32) i s convergent i n the sense t h a t (4.33)

u" + u

strongly i n V

where u i s the solution o f (4.7).

Proof. (4.34)

If (u,p} denotes a saddle point of 9 ( v , q) we have:

u =u

- p1 S-'(Au

+ @*p

- f),

(l)

This is essentially an algorithm of the Arrow-Hurwicz type /l/.

(2)

If we know that the solution u of the problem is in a reflexive space W c V , we can take an isomorphism from W + W ' for S.

(3)

Estimates of p 1 and p 2 result from the proof below.

(SEC. 4 ) (4.35)

IiuaZity methods

+

p = P,(p

p2

95

@u) .

P u t t i n g w " = u " - u and using t h e n o t a t i o n (4.24) we g e t :

- p 1 S-'(Aw" + @ *

(4.36)

w"+'

(4.37)

I1 r"+' Ilf c 1I P 1;

= W"

r"),

+ 2 p2(r", @w"+')' + p;

11 @W"+' 1.;

We deduce from (4.36) t h a t SW"+' =

sw" - p,(Aw" + @* P )

and hence, t a k i n g t h e i n n e r product o f both s i d e s with w"+' i n t h e d u a l i t y between V ' and V :

11 w"+' 112 = ((S - p' A ) w", W " + ' ) - p l ( @ * P, w " + ' ) . Since

A * = A , (Au, u ) 2

OT

)I v 112,

w e have:

Thus :

so t h a t (4.39)

, if

0
< min (p0,p:):

II r" 1;

+ CP II w" 112 - (I1 P+' 112 + CP 1I fl+'11')

(1

+ CP 11 W" [I2 decreases

2 y II w"+' ( I 2 .

Hence f"

1:

with

which i n t u r n means t h a t t h e left-hand hence 1) w"+' 11 -P 0.

y2,

and t h e r e f o r e converges,

s i d e of (4.39) -+ 0 and

96

@timisation algorithms

(CHAP. 2 )

Remark 4.7. The behaviour o f p " depends on t h e c h a r a c t e r i s t i c s of t h e problem i n hand. We can show, f o r a l l t h e examples s o l v e d by means o f t h e above methods, t h a t t h e s e q u e n c e d remains bounded and t h a t a l l o f t h e c l o s u r e v a l u e s o f t h i s sequence a r e such t h a t { u , p } i s a s a d d l e p o i n t of le(u,q) on M x A . Remark 4.8.

The f i r s t method p r o v i d e s an e s t i m a t e from below

of

[

inf J&) u6M

1

+ SUP (4. @(v)), . aeA

I n f a c t , by d e f i n i t i o n o f ff

JW)+ (P".OW)), Q J(u)

Q

,

JW + (P",@(U))L

+ asup (4, Hu)),= inf 6A usM

(

Q

J,(L~)+SUP (4, a*A

Remark 4.9. The above methods apply t o t h e minimisation o f n o n d i f f e r e n t i a b l e f u n c t i o n a l s , and i n p a r t i c u l a r t o a l l t h e problems o f t h i s c a t e g o r y s t u d i e d i n t h i s book, such as t h a t o f Example 4 . 1 which can be w r i t t e n e x p l i c i t l y as f o l l o w s :

and which w i l l be s t u d i e d i n Chapter 5 . I n t h e following we s h a l l show t h a t t h e d i s c r e t i s a t i o n of (4.40) l e a d s t o f u n c t i o n a l s of t h e t y p e found i n Counter-Example 1.1 ( a l t h o u g h c l e a r l y more complex).

Remark 4.10.

The s o l u t i o n of t h e problem

minJ(u) = ~

A uu), - (f,u)

where .A i s a symmetric p o s i t i v e - d e f i n i t e N x N m a t r i x , s u b j e c t t o the constraints

BU = b where B i s a M x N m a t r i x o f r a n k M . 6 N, r e d u c e s , a f t e r i n t r o d u c t i o n o f t h e Lagrange v a r i a b l e s P E R M , t o t h e solution of the system

which i s a system o f t h e form d z = d , where I i s a symmetric, but not g e n e r a l l y p o s i t i v e d e f i n i t e m a t r i x . I n p a r t i c u l a r , t h e conj u g a t e g r a d i e n t method cannot be a p p l i e d i n t h i s c a s e . It w i l l be noted however t h a t t h e problem

d*dz=d*d

(sEc. 5)

Application of relaxation and duality methods

97

is equivalent to the above and that the matrix &*&is symmetric and positive definite. The application of the conjugate gradient method is therefore possible and convergence is achieved, in a finite-dimensional space, in a finite number of iterations. From the numerical point of view, the use of & * d introduces rounding errors, due to a deterioration in the conditioning of the matrix, and this can adversely affect the convergence of the algorithm. This has led a number of authors (see Beckmann /l/, Luenberger 121) to adapt the conjugate gradient method directly to the case in which it is required to solve a system d z = d , with I symmetric but not necessarily positive definite.

5.

APPLICATION OF RELAXATION AND DUALITY METHODS TO THE NUMERICAL ANALYSIS OF A MODEL VARIATIONAL PROBLEM.

5.1

General remarks

We shall cow apply the results of Sections 1 and 4, and those of Chapter 1, Section 5, to the solution of a variational problem which arises in lubrication theory ( ’ ) ; this is a variant of the problem considered in Chapter 1, Example 3.6. . This problem is solved by a variety of methods in Fusciardi, MOSCO, Scarpini and Schiaffino /1/ and Marzulli /1/ (2). 5.2 5.2.1 Let

The continuous problem

Statement of the continuous problem f2 = 10, 1[ x 10, b[, f l = 10, 1[ x { 0 },

v = [v I u E fZ1(f2),u I,-,

= 0, u

Ira

r2 = )o, 1[

x {b}

= 0, u(O, x2) = ~ ( 1 x2) ,

and ae. on lO,bO

Fig. 5.1. (l)

For the physical motivation, see Marzulli /1/ and the bibliography therein.

(2)

Cryer /2/ gives an application of the method of over-relaxation with projection to a lubrication problem of the same type as that considered in this section, the domain f2 considered being one-dimensional.

98

@timisation

algorithms

(CHAP. 2)

be a Hilbert space for the inner product and the norm: (u, u)“ =

(5.1)

Jb

grad u.grad u dx ,

Given . f € L 2 ( Q ) (which is sufficient for the applications which we have in mind), we define on Vthe functional J , strictly convex and continuous, by: J(u) = -

(5.3)

I,

1 grad u I’ dx -

Infudx

and we consider the problem (P): min J(u) , K = [ o l u ~ V , u > O a.e.on Q],

since K i s convex and closed in V and

lim

IIvIlv+

a single optimal solution;

J(u)

=

+ m, (P) admits

+m

we denote this solution by u .

8

Regularity results

5.2.2

According to BrCzis and Stampacchia 121, if fe L P ( Q )(2 we have u E V n W2*p(Q)

5.2.3


<

+ m),

( 1)

Characterisation of the soZution

It follows from Chapter 1, Section 2 that the optimal solution u is characterised by: (5.4)

Ingradu.grad(o - u ) d x 2

In

f(u

- u)dx V U E K , U E K .

In view of the regularity (see 5.2.2 above) we can integrate by parts in (5.4), which leads to the characterisation:

(’)

By definition and for p a 1

1

E LP(Q), i , j = 1, n

.

Application of relaxation a d duality methods

(SEC. 5 )

99

K being a cone with apex 0. The c h a r a c t e r i s a t i o n ( 5 . 5 ) i s fundamental s i n c e it introduces ( A€L2(Id) from 5 . 2 . 2 ) as a Lagrange muZtipZier f o r we s h a l l r e t u r n t o t h e above i n Section 5.6. t h e problem ( P ) ;

A= -Au- f

5.3

The approximate problem

Remark 5.1. It i s convenient, bearing i n mind t h e p e r i o d i c i t y i n x 1 of t h e elements of V , t o i d e n t i f y Id w i t h fi P y x ]o,M where y i s t h e c i r c l e of r a d i u s 1 / 2 n ( t o r u s of dimension 1) o r i e n t a t e d i n t h e u s u a l sense ( t h e p h y s i c a l problem i s i t s e l f expressed i n t h e boundary f of 0 i s t h u s such a domain); '7 x

{OlUY

x {bl

and hence .

v=

u(?=o}.

{UIUEH~(O),

( P ) i s t h e r e f o r e a ( c o n s t r a i n e d ) D i r i c h l e t problem on a cylind8 r i c a l manifold. Exterior approximation of ( P )

5.3.1

The c o n s i d e r a t i o n s of Chapter 1, S e c t i o n 5.1, adapt r e a d i l y t o ( P ) ; t a k i n g account of Remark 5 . 1 and t a k i n g N1, N2 t o be p o s i t i v e i n t e g e r s which w i l l tend towards i n f i n i t y , where hi = l / N i , h, = b/Nz, h = (hi, hz), we d e f i n e : (5.6)

Rk = { MI M E y x R, M = (m,h,, m, h,), 0 d m, 6 N , - 1, m , E E } .

With each node of

4 we

Y,, x ](m2 - 4) hz, (mz + $1 h2C

(5.7)

w;(M)

(5.8)

w i ( M ) = w;(M)u

with ,y (5.9)

=

(0 Q m, 6 N ,

{

yo = 11

a s s o c i a t e t h e "panel" with c e n t r e M :

W:

(M f 3.). -

- 1 ) defined by:

- h,/2,11 u [O, h, /2[ - 4) h,, (mi + t ) h , [ if 1 G m, G N , - 1 .

ym, = ](mi

Y

Qptimisation algorithms

100

(CHAP. 2 )

By analogy w i t h Chapter 1, S e c t i o n 5.1, we next d e f i n e :

(5.10)

$={'MIME&,

where:

( s e e ( 5 . 9 ) , Chapter 1):

(5.13)

#

W:(M)cdi},

= c h a r a c t e r i s t i c f u n c t i o n of

m;(M).

Once a g a i n p u t t i n g ,

(5.14)

-q(x-;e3]

6 i q ( x ) = hi I[q(x+;ei)

(2)

we "approximate" J by Jh: Vh + R d e f i n e d by:

which g i v e s t h e approximate problem:

5.3.2

SoZvabiZity of

(P,)

It i s immediate t h a t

(1) W e c o u l d , as i n Chapter

1, S e c t i o n 5.2, defineph:V,-*(L'(y x ~ 1 1 3 , u s i n g qh and t h e 6, d e f i n e d i n ( 5 . 1 4 ) ; however, i n view o f t h e elementary n a t u r e o f t h e convex set K c o n s i d e r e d i n t h i s s e c t i o n , t h i s would not be p a r t i c u l a r l y u s e f u l . hl i s d e f i n e d modulo 1 such t h a t ( 2 ) I n ( 5 . 1 4 ) , f o r i = 1, x f it belongs t o [0, l [ .

Application of relaxation and duality methods

(SEC. 5 )

101

defines a norm on Vh , which will be denoted by 11 ; the functional Jh is therefore continuous and strictly convex on Vhwith lim Ill*llh-+m

Jh(U,,) =

+ co ,

and since& is closed we have the existence and uniqueness of an optimal solution for (ph) , denoted by uh. rn

Explicit formulation of ( P 3

5.3.3

, it is necessary With a view to the numerical solution of (P,,) to make (5.15), (5.16) explicit; if M e a h , it is convenient to = (kh,, lhJ identify M by a double subscript (k, I ) which gives We can then write (5.15) in with 0 < k < N, - 1, 1 < I < N, - 1 explicit form, as follows:

.

where@

=

{

(5.18)

f i h u ( f n R & ; the explicit form of (5.16) is:

m h Jk(u,,), "kE&

Kh

=

[Vh

I uh E

vh, uu

20

VMk,E

dJ.

In (.5.17)it is necessary to take ukI = 0 if Remark 5.2. M u # $ , and in order to take account of the periodicity in xi, v-

11

5.4

= ' u N , - 11,

V N , ~= ~ 0 vl. 1

Convergence of the approximate solution as h

+

0

Preliminary remark. The convergence of uh (in fact of q r u r ) to the solution of problem ( P ) is a direct consequence of the However, since the problem results of Chapter 1, Section 5.2. considered in this section is the first example of the numerical solution of a variational inequality encountered in this book we rn have chosen to detail the various points of the proof. We have : Theorem 5..1. As h

('1

-t

0 we have

If f is continuous, we take f u = f(M,).

Uptimisation algorithms

102

(5.20)

(CHAP. 2 )

J(u) .

Jh(u&

Proof. The norm 11 oh [Ih Poincarg i n e q u a l i t y :

(a

> O)

3

vh, vuh E

defined i n 5.3.2

vh

satisfies the discrete

.

For t h i s r e s u l t we r e f e r t h e r e a d e r t o C k a / 3 / - Lions / 3 / . Since 0 E 4 we have Jh(uh)Q J,,(O) = 0 so t h a t , t a k i n g account of

(5.21):

and hence, by proceeding as i n t h e proof of Theorem 5.1, Chapter 1, S e c t i o n 5.1, it can be shown t h a t f o r a subsequence, a l s o denoted by uh , w e have:

(5.23)

{

weakly i n L2(@

qh u,, -+ w & h U h * G

aw

weakly i n L2(fi)

(i = 1,2),

with: (5.24)

WE

V.

Moreover q,,u,>O a . e . and s i n c e t h e set [ u l u ~ L ~ ( f i ) , u0&a.e.]is convex and closed i n L 2 ( a : and t h e r e f o r e weakly closed, w e deduce: (5.25)

a.e.

w>O

~ w E K . f o r t h i s , we d e f i n e

It now remains t o show t h a t w = u ;

r, : V + V, by:

and if u & Oa. e. w e c l e a r l y have (rh gives : (5.27)

uEK

=t.

3 0 VM E

r,, u E K,,.

It can be shown t h a t V U E V we have:

a,,=s r, u E 4

which

(SEC.

5)

Application of relaxation and. duality methods

qhrhu + u

103

strongly in ~ ~ ( 0 )

(5.28)

from which we deduce (5.29)

Jh(rh

v ) + J(u) '

In view of (5.27), since uh is optimal onKhwe then have: (5.30)

Jh(uh)

,< Jh(rh u )

vv E K

and in the limit, by weak lower semi-continuity: (5.31)

J(W)

Q l h infJh(u& Q J(U)

VUE

K,

and from (5.241, (5.25), ( 5 . 2 7 ) , ( 5 . 3 1 ) , we deduce that w = u . It may be noted that the uniqueness of u implies the convergence of the entire sequence qhuh To prove (5.20) it is sufficient to set v ' = u in (5.30), (5.31); from this the strong convergence is deduced; in fact:

.

and in the limit, from ( 5 . 2 0 ) , (5.23), ( 5 . 2 8 ) , ( 5 . 2 9 ) , (5.31):

Optimisation algorithms

104

5.5

(CHAP. 2 )

Solution of the approximate problem by point over-relaxation with constraints

We use the algorithm (1.41), (1.42) of Section 1.4 which, taking account of (5.17), (5.18), takes the form:

if we iterate with k, 1 increasing. Remark 5.2 is valid for (5.38). Since Theorem 1.3, Section 1.4 applies to (P& we have: Proposition 5.1.

The sequence ui-+ u,.if

0 < w < 2. 8

Remark 5.3. If uf = 0 and if 0 < o < 1 , we immediately get from relations (5.38) that the sequence (4). is increasing V M , E $ ; the convergence is thus monotone and the coordinates uu of u, are approached from below.

5.6

Solution by a duality method

Problem ( P ) comes within the setting of Section J,(u) = J ( u ) , M

J

V , @(u)

=

u, L

A = L!(Q) = { q 1 q E Lz(Q), q Q 0

=

4 if we take:

L2(Q),

a.e. }

.

In fact, we then have

It is somewhat simpler if we use only L:(Q) problem ( P ) then takes the form: (5.39)

[

inf J(u)

+

1

sup (- p, u) rsu+(n)

.

We therefore introduce the Lagrangian: (5.40)

U(U,

=

J(4- 01,u)

, replacing

q by - p

(SEC. 5 )

Application of relaxation and duality methods

Let u be t h e s o l u t i o n of ( P ) , or e q u i v a l e n t l y , of We put (5.41)

105

(5.53.

1 = - AU - f.

I n o r d e r t o show that the conditions of Theorem 4 . 1 are satisfied, it i s s u f f i c i e n t t o prove t h e analogue of ( 4 . 1 0 ) . Now, more p r e c i s e l y , we have: Theorem 5.2. The pair {u,1} (1 defined by ( 5 . 4 1 ) ) is the unique saddle-point of U ( u , u) on V x L ~ ( Q ) .

Proof. 1) From 5.2.2 and ( 5 . 5 ) furthermore :

we have L E L : ( Q ) and b - 0 a e . ;

and hence :

(5.42) 2)

(5.43)

Y ( u , p ) d Y ( u , A ) ( = J(u)) V ~ L:(L!) E ;

With 1 f i x e d , we consider t h e o p t i m i s a t i o n problem:

min g ( u , 1) . VEV

The unique optimal s o l u t i o n u, of ( 5 . 4 3 ) i s c h a r a c t e r i s e d by:

from which u, = u

(5.45)

9 ( u , 1)

, and

< U(v,1)

hence: vu E

v

which, with ( 5 . 4 2 ) , shows t h a t {u,1} i s a saddle p o i n t .

We observe t h a t : (5.46)

Jn.

- 1)Ud.x 3 0 v p E L : ( Q ) .

I n o r d e r t o prove t h e uniqueness, we n o t e t h a t any saddle3) p o i n t (u*,A*) of Y on V x Lt(C2) satisfies a r e l a t i o n of t h e tme (5.41); from which w e e a s i l y deduce t h e c h a r a c t e r i s a t i o n :

@timisation algorithms

106

I-

Au*=f + I *

(5.47)

ma,

I, F

v-ddT=O

( C H A P . 2)

VVEV,

U*EV,

I

r

01 - 1*)u* dx 2 0

v p E L:(a),

A* J" E L:(a).

We then set p = 1* in (5.46), and p = 1 in the third relation in (5.47), which by addition gives: (5.48)

j"(1*

- 1)(u*

- u ) dx

Q 0;

now (5.49)

A* - 1 = - Nu* - u) ;

we therefore deduce from (5.47), (5.48) that: (5.50)

- ["A(.*

- u)(u*

- u)dx =

l"1

grad(u*

- u) I z d x ' < 0 ,

which gives I( u* - u 1; Q 0 and hence u* = u and 1* = 1.

Remark 5.4. It-is easy to prove an analogous result for the approximate problem ( P J , using the approximate Lagrangian ghdefined by:

with

5.6.2

Solution of the problem ( P ) by a dual method

The algorithm of Section 4.3 applied to the problem (P) takes the form: (5.52)

loE L:(Q)

chosen arbitrarily (e.g. zero),

(SEC. 5)

Application of relaxation and duaZity methods

P+l

(5.54)

=

107

(1"- p,, u"+ I ) + = = positive part of an - p n u " + ' , pn > O

Since the canonical injection from V - r L2(P) is continuous, there exists C > 0 such that:

II u I'(D) d c II 0

(5.55)

IIY

y

(this gives the Lipschitz constant o'f@; Theorem 4.1 gives (5.56)

strongly in V

u"+u

see Section 4.3).

(l)

under the condition

o < a,

(5.57)

Q p,, Q al c 2 / C 2 ,

where the constant C is that of (5.55). In the case of problem (P) it may easily be shown that the optimal value of C (i.e. the smallest possible), say Co, is the reciprocal of the square root of the smallest eigenvalue, 1, , associated with the sequence of eigenfunctions of the operator - A in V , this sequence being given by:

{eos2xmx1 sinnx--,sin2nmxl x2 s i n Xm ~ ; m 2 0, n 3 1 b b

(5.58)

I

and the sequence of corresponding eigenvalues by:

which gives 1,

=

x2/b2,C, = b/x and, for p,, the condition:

0 c a, Q P. Q a, < 2 a2/b2

(5.60)

which defines (5.57) more precisely.

5.6.3

Solution of the problem (P& b y a dual method

The analogue of the above algorithm for the problem (P& is given by : (5.61)

(I)

1; 2 0 V M , E ~ ~chosen ~, arbitrarily (e.g. zero) In fact we have strong convergence in H2-'(P)n V , for arbitrarily small E > 0 : indeed, the proof of Theorem 4.1 shows that An is bounded in L 2 ( P ) * m is bounded in L2(P) j u " is bounded in H2(P) n V , which gives the required r e s u l t . For the definition of t h e spaces Hs(R) , for noninteger s , see Lions-Magenes /l/.

Qptimisation algorithms

108

a;,+

(5.63)

=

max (0, ail - p. &+l ) ,

w i t h Remark 5.2 h o l d i n g f o r ( 5 . 6 2 ) . Under a c o n d i t i o n on pm of t h e t y p e ( 5 . 5 7 ) ,

4 = { u"u I M r i s k converges t o u,

5.7

, the

s o l u t i o n o f t h e problem ( P d ( l ) .

A n a l y s i s of t h e n u m e r i c a l r e s u l t s

All t h e c a l c u l a t i o n s were c a r r i e d o u t f o r :

(5.64)

f(Xl,Xz)

=4XzSh12Xxl,

u s i n g t h e C I I 10070 computer a t I R I A .

Nmerical values of the parameters

5.7.1

For b we u s e d t h e t h r e e v a l u e s :

b = 0.2,

0.34,

0.69:

h i = 1/10,

1/30,

1/50;

f o r hl:

With t h e t e r m i n a t i o n c r i t e r i o n cho-sen as:

1,

(5.65)

I 4 + ' - & I < E

Mki EQh

we t o o k : E

=

f o r b = 0.69, E

E

= lo-' f o r b = 0 . 2 .

= 5.10-'

rn

f o r b = 0.34,

(CHAP. 2 )

(SEC. 5)

Application of relaxation and duality methods

5.7.2

Lo9

Solution using overrelaxation

The results in Table 5 . 1 correspond to the solution of the problem (Pk)using algorithm (5.38) with: w = 1.6 for h, = 1/10,

1.7for h, = 1/30,

1.8 for h , = 1/50,

(5.66)

u:=o,

the parameters h,, h,, b, E having the values given in 5.7.1, and the termination criterion being given by (5.65). b

03

0.34

0.69

h,

Number of iterations

CII 10070 computation time in minutes

1/10

17

0.025

1/30

25

0.08

1/50

30

0.19

1/10

17

0.025

1/30

30

0.10

1/50

35

0.22

1/10

17

0.025

1/30

35

0.12

1/50

50

0.31

Table 5.1

Fig. 5.2 shows, for b = 0.69 and 321 = 1/50, the number of iterations as ti function of w , the termination criterion in all cases being given by (5.65); also shown on this figure is the curve giving the number of iterations as a function of w (again usin same termination criterion) for the unconstrained problem ( 1,the i.e.

?

(5.67)

(l)

I

-&=f

in a ,

In fact for the discretised form of (5.67).

@ t i m i s a t i o n algorithms

110

t

( C H A P . 2)

Number of iterations

\

Fig. 5.2.

0 = 10, I[ x ]0,0.69[ 1 0.69 h, = h2 - 10 A -

Influence of the parameter w on the rate of convergence.

We can therefore make the following remarks:

For a given w, the-over-relaxationmethod is Remark 5.5. more rapid, in terms of the number of iterations, for the constrained problem than for the unconstrained problem. H It may be noted that the maximal rate of convergRemark 5.6. ence is achieved for values of w much greater than 1; this justH ifies Remark 1.9 of Section 1.4. In the constrained case, the optimal value of w Remark 5.7. is a function of f (in contrast to the unconstrained case). More precisely, if we write 0 + = [XI XED, ~ ( x > ) 01, the value of w is essentially that which corresponds to the approximate solution of the Dirichlet problem: - Au = f on a + , the discretisation parameters

(SEC. 5 )

Application of r e h a t i o n and d u a l i t y methods

111-

h l , h2 r e m a i n i n g t h e same.

C l e a r l y f2+ i s n o t known a p r i o r i , so t h a t t h e above remark i s of t h e o r e t i c a l i n t e r e s t o n l y . rn 5.7.3

Solution using Uzawa’s algorithm

The p r o b l e m ( P & w a s s o l v e d u s i n g t h e a l g o r i t h m ( 5 . 6 1 ) ~ ( 5 . 6 2 ) , ( 5 . 6 3 ) w i t h a constant p a r a m e t e r p and 2; = 0 , t h e t e r m i n a t i o n c r i t e r i o n a g a i n b e i n g g i v e n by ( 5 . 6 5 ) . The approximate D i r i c h l e t sub-problems ( 5 . 6 2 ) were s o l v e d by p o i n t o v e r - r e l a x a t i o n w i t h w = 1 . 6 f o r h l = 1/10, 1.7 f o r h1 = 1/30 and 1.8 f o r hl = 1 / 5 0 . T a b l e 5.2 shows t h e number o f i t e r a t i o n s and t h e computation t i m e s f o r a l g o r i t h m (5.611, ( 5 . 6 2 ) , ( 5 . 6 3 ) , t h e p a r a m e t e r p having i t s optimal value (determined experimentally). F i g u r e 5 . 3 shows, f o r hl = 1 / 5 0 and b = 0.69, t h e v a r i a t i o n o f t h e number o f i t e r a t i o n s as a f u n c t i o n o f p , u s i n g t h e t e r m i n a t i o n c r i t e r i o n ( 5 . 6 5 ) ; t h e f o l l o w i n g remarks may b e made: The t o t a l number o f i t e r a t i o n s r e q u i r e d t o s o l v e Remark 5.8. (P& , w i t h a g i v e n p , i s more o r l e s s i n d e p e n d e n t o f t h e a c c u r a c y w i t h which t h e approximate D i r i c h l e t sub-problems

( 5 . 6 2 ) are solved.

rn

Remark 5.9.

For b = 0 . 6 9 , we have 2 ~ ~ l b ~ 1 . 4 2 ;. 5 t h e e s t i m a t e (5.60) ( I ) i s t h u s very r e a l i s t i c f o r t h e problem (Pd s i n c e f o r p = 42.5 convergence i s a t t a i n e d i n 522 i t e r a t i o n s , f o r p = 44 i n 698 i t e r a t i o n s , and f o r p = 45 t h e s o l u t i o n a c t u a l l y d i v e r g e s .

Remark 5.10. I n view o f t h e above r e s u l t s , it a p p e a r s t h a t i n t h e c a s e o f problem ( P ) t h e o v e r - r e l a x a t i o n method i s q u i c k e r t h a n t h e d u a l i t y method, a t l e a s t when t h e D i r i c h l e t sub-problems ( 5 . 6 2 ) are s o l v e d by o v e r - r e l a x a t i o n . It h a s a l s o been e s t a b l i s h e d t h a t t h e a d j u s t m e n t o f p t o o b t a i n s a t i s f a c t o r y convergence i s more c r i t i c a l t h a n t h a t o f w. It may be n o t e d , moreover, t h a t t h e d u a l i t y method i s more demanding i n t e r m s o f s t o r a g e r e q u i r e m e n t s t h a n t h e o v e r - r e l a x a t i o n method, s i n c e it i s n e c e s s a r y t o s t o r e t h e components o f 2, The rate o f convergence o f t h e d u a l i t y method c o u l d c l e a r l y be improved by s o l v i n g t h e D i r i c h l e t problems ( 5 . 6 2 ) by a method which i s q u i c k e r t h a n p o i n t o v e r - r e l a x a t i o n f o r t h i s t y p e o f problem ( a l t e r n a t i n g d i r e c t i o n , b l o c k o v e r - r e l a x a t i o n ) o r by a d i r e c t method ( e . g . Gauss o r C h o l e s k i , e t c ) ; however , t h i s would a l s o g e n e r a l l y i n c r e a s e t h e computer s t o r a g e r e q u i r e m e n t . The above i s a l s o v a l i d i f we r e p l a c e t h e c o n s t r a i n t

.

....

(I)

R e l a t i n g t o (P) r a t h e r t h a n (Ph)

.

(CHAP. 2 )

@timisation algorithms

112

Optimal

b

0.2

0.34

0.69

h,

P

Number of iterations

C I I 10070 computation time i n minutes

1/10

45

30

0.045

1/30

41

35

0.105

1/50

30

50

0.3

1/10

16

30

0.04

1/30

13

40

0.12

1/SO

10

50

0.3

1/10

13

40

0.09

1/30

8

70

0.27

1/50

5

100

0.6

Table 5.2

of t Number iterations

36 Fg 5.3.-

I n f l u e n c e of t h e parameter p on t h e r a t e of convergence.

(SEC.

6)

113

Discussion Shape of the f r e e surface

5.7.4

We have shown i n F i g u r e 5 . 4 , f o r b = 0.69 and u s i n g t h e s o l u t i o n o f problem (P,,)f o r hl = 1 / 5 0 , h, = 0.69/10, t h e r e g i o n where u = 0, a p a r t o f t h e r e g i o n where u > O ( i . e . a + ) and t h e boundary between t h e s e two r e g i o n s which w a s one o f t h e unknown o f ( P ) cons i d e r e d as a free boundary problem. We r e c a l l t h a t i n a + , we have - Au = f. W

0.69

0.345

0 0.5

0.75

1

Pip. 5.4. The h a t c h e d r e g i o n c o r r e s p o n d s t o u = 0 .

6.

DISCUSSION

We s h o u l d emphasise t h e f a c t t h a t t h i s Chapter d o e s not a i m t o For give an exhaustive i n v e s t i g a t i o n of minimisation algorithms. more comprehensive s t u d i e s of t h i s s u b j e c t , t h e f o l l o w i n g may be consulted :

- f o r unconstrained optimisation: Osborne

111, Spang

D a n i e l /2/, Kowalik and

Ill, Vainberg /l/;

- for constrained optimisation: Abadie /l/, /2/, B a l a k r i s h n a n and Neustadt /l/, Box 111, B r a m and S a a t y /l/, Cannon, Cullum and Polak / l / , . C C a /2/, F l e t c h e r /l/, Graves and Wolfe /l/, Hadley /l/, Kuhn and Tucker 111, Kunzi, K r e l l e and O e t t l i Ill, Kunzi, Tzschach and Zehnder /l/, Lasdon /I/, L e v i t i n and Polyak /l/, Luenberger /l/, Polak /l/, Tr6moliSres / 3 / , V a r a y i a /l/, Zangwill /l/, Z o u t e n d i j k

Ill. I d e a s similar t o t h o s e i n t r o d u c e d i n S e c t i o n 1 have been i n v e s t i g a t e d by a number of a u t h o r s : Crouzeix /l/, E l k i n Ill, C6a and Glowinski / 2 / , Glowinski /4/, /5/, Mezlyakov 111, M i e l l o u 111, Motzkin and Schoenberg /l/, O r t e g a and R h e i n b o l d t /l/, 121, 131, O r t e g a and Rockoff 111, P e t r i s c h y n 111, 121, S c h e c h t e r /l/, 121, 131, S o u t h w e l l /l/, S t i e f e l /l/, Comincioli Ill, Varga 111, Cryer

I l l , /2/*

@timisation

114

algorithms

(CHAP.

2)

- f o r various properties of convex functions, s e e L e v i t i n and Polyak 111. The c o n c e p t s i n t r o d u c e d i n S e c t i o n 2 a r e o f a c l a s s i c a l n a t u r e . For t h e u n c o n s t r a i n e d c a s e we r e f e r t h e r e a d e r t o Blum 111, Cauchy 111, C r o c k e t t and Chernoff 111, Curry Ill, G o l d s t e i n Ill, 121, Greenstadt 111, Polyak I l l . The convergence o f t h e method of d i v e r g e n t series, due t o Ermo l y e v , i s g i v e n i n Polyak /2/ and f o r t h e g e n e r a l u n c o n s t r a i n e d c a s e i n Tr6moliSres 121. A t r a n s p o s i t i o n o f t h i s method t o v a r i a t i o n a l i n e q u a l i t i e s i s made i n Auslender-Gourand-Guillet 111. The s o - c a l l e d c o n j u g a t e g r a d i e n t method, due t o Hestenes 111, has been s t u d i e d by Antosiewicz and Rheinboldt 111, Beckman 111, Durand 111, F l e t c h e r and Reeves 111, Hestenes and S t i e f e l 111. This method has been g e n e r a l i s e d t o i n f i n i t e dimensions by Daniel 111, 121. For methods o f t h e same t y p e , c o n s u l t Davidon 111, F l e t c h e r and Powell 111, Pearson 111. The e x t e n s i o n o f t h e method of c o n j u g a t e d i r e c t i o n s t o t h e c a s e of c o n s t r a i n e d m i n i m i s a t i o n has been c a r r i e d o u t by Goldfarb 111, Goldfarb and Lapidus Ill, Luenberger /2/ and TremoliPres 141. For methods o f t h e p o i n t p r o j e c t i o n o r g r a d i e n t p r o j e c t i o n t y p e s e e Altman Ill, Demyanov 111, G o l d s t e i n 111, L e v i t i n and Polyak I l l , 121, Polyak 131, 141, Rosen 111, 121, Sibony 111. The p e n a l i s a t i o n of c o n s t r a i n t s of t h e t y p e G ( V ) = 0 u s i n g t h e fu n c t i o n 1 J(u) ;G(u)’ , E + 0 ,

+

w a s i n t r o d u c e d by Courant /1/.

I n t h e case of i n e q u a l i t y const1 r a i n t s , t h e e x t e r i o r p e n a l t y f u n c t i o n J(u) ; ma[- G,(u),O]

+ f

l =1

i s due t o Ablow and Brigham /l/. The i n t e r i o r p e n a l t y f u n c t i o n

The method o f c e n t r e s u s i n g t h e d i s t a n c e - f u n c t i o n

w a s i n t r o d u c e d by Huard /l/ and extended t o t o p o l o g i c a l s p a c e s by Bui-Trong-Lieu

and Huard

111.

The second d i s t a n c e f u n c t i o n

1

+

,zl qq “

1

i s t h a t o f Fiacco

and McCormick Ill. The i d e a o f v a r y i n g t h e t r u n c a t i o n e r r o r s and t h e p o s s i b i l i t y o f a c c e l e r a t i n g t h i s method by o v e r - r e l a x a t i o n comes from Tr6moliSres 111. For a g e n e r a l p r e s e n t a t i o n o f p e n a l t y methods, s e e Fiacco and McCormick /1/ and Lootsma 111. The c o n c e p t s of S e c t i o n 4 a r e less c l a s s i c a l i n n a t u r e . For t h e g e n e r a l theorems on t h e e x i s t e n c e o f s a d d l e - p o i n t s s e e Berge 111, Browder 111, Ky-Fan 111, Sion Ill, / 2 / , Ekeland-T&nam 111. For d u a l i t y theorems i n mathematical programming s e e Arrow, Hurwi c z , and Uzawa 111, Bensoussan, Lions and Temam Ill, Mangasarian /1/, R o c k a f e l l a r / 3 / and Varayia 121.

(SEC. 6 )

Discussion

115

Some i n v e s t i g a t i o n s of t h e two p r i n c i p a l d u a l i t y algorithms may be found i n Arrow, Hurwicz and Uzawa /l/, Cga, Glowinski and NCdelec / 2 / ¶ C C a and Glowinski /1/ and i n Trgrnoli8res / 3 / . I n o r d e r t o b r i n g t h i s c h a p t e r t o a c l o s e , w e should mention t h e augmented Lagrangian methods which a r e obtained by combining penalisation and d u a l i t y ; we s h a l l meet some a p p l i c a t i o n s of t h i s i n Chapter 3 , Section 10 and i n Chapter 5, Section 9. This approach appears p a r t i c u l a r l y i n t e r e s t i n g for two fundame n t a l reasons :

1) t h e p o s s i b i l i t y of a c c e l e r a t i n g t h e convergence of t h e 2)

d u a l i t y methods, t h e p o s s i b i l i t y of improving t h e conditioning of t h e functi o n a l t o be minimised.

For t h e t h e o r e t i c a l a s p e c t s , we r e f e r t o Hestenes /2/, Powell / l j , Rockafellar /4/, / 5 / , e t c . . . Also, some a p p l i c a t i o n s of t h i s methodology t o t h e s o l u t i o n of nonlinear boundary-value problems a r e given i n Glowinski-Marrocco /l/, 121, Mercier /l/.

This Page Intentionally Left Blank

Chapter 3 NUMERICAL ANALYSIS O F T H E PROBLEM O F THE ELASTO-PLASTIC TORSION O F A CYLINDRICAL BAR INTRODUCTION I n t h i s c h a p t e r we s h a l l m a i n l y s t u d y t h e a p p l i c a t i o n of t h e r e s u l t s and methods o f C h a p t e r s 1 and 2 t o t h e n u m e r i c a l analysis o f t h e model e l a s t o - p l a s t i c problem c o n s i d e r e d i n Chapter 1, Section 1.2.

1.

STATEMENT OF THE CONTINUOUS PROBLEM. SYNOPSIS 1.1

PHYSICAL MOTIVATION.

S t a t e m e n t of t h e c o n t i n u o u s problem

We a g a i n adopt t h e formalism o f Chapter 1, S e c t i o n 1 . 2 ; w i t h 52 a bounded open domain i n w", w i t h boundary r , w e i n t r o auc e

(1.1)

H ; ( L 2 ) = { u ( u E H ~ ( n )y, u = O } ,

which i s a c l o s e d v e c t o r s u b s p a c e o f H'(L2) and a H i l b e r t space f o r t h e i n n e r p r o d u c t and t h e norm d e f i n e d by

(1.3)

II u II =

(1',

I grad u 1' dx)',',

t h e norm ( 1 . 3 ) b e i n g e q u i v a l e n t t o t h e norm induced by H ' ( f 2 ) .

We t h e n i n t r o d u c e

a cZosed convex s e t i n Hi@), and t h e f u n c t i o n a l

Elasto-plastic torsion o f a cylindrical bar.

118

(CHAP. 3 )

where L i s a c o n t i n u o u s l i n e a r form on H b ( Q ) . We t h e n c o n s i d e r t h e v a r i a t i o n a l problem ( P O )d e f i n e d by: (1.6)

(Po) Min J(u) . US&

S i n c e t h e f u n c t i o n a l J i s c o n t i n u o u s and s t r i c t l y convex on lim J(v) = + a , Theorem 2 . 1 o f Chapter 1,

Hb(Q), w i t h

-

IIvII + m

S e c t i o n 2 . 1 a p p l i e s , and t h i s i m p l i e s t h e e x i s t e n c e and uniquen e s s o f a s o l u t i o n for ( P O ) . We d e n o t e t h i s s o l u t i o n by u which i s a l s o t h e u n i q u e s o l u t i o n of t h e v a r i a t i o n a l i n e q u a l i t y

which i s e q u i v a l e n t t o ( P O ) . C e r t a i n p r o p e r t i e s of t h e s o l u t i o n u , f o r particular forms L , are d e t a i l e d i n S e c t i o n 2. 1.2

Physical motivation

With t h e set fi a simply connected, bounded open domain i n R 2 , w e c o n s i d e r a c y l i n d r i c a l b a r o f c r o s s - s e c t i o n fi, made o f an e l a s t i c - p e r f e c t l y p l a s t i c m a t e r i a l f o r which t h e t h r e s h o l d o f p l a s t i c i t y (b. t h e y i e l d s t r e s s ) i s d e f i n e d by t h e von Mises c r i t e r i o n ( s e e eg. Mandel 111). T h i s bar i s t h e n s u b j e c t e d t o a n i n c r e a s i n g t o r q u e , s t a r t i n g from a n u n s t r e s s e d i n i t i a l s t a t e , t h e t o r s i o n b e i n g c h a r a c t e r For e a c h v a l u e o f i s e d by t h e t w i s t a n g l e p e r u n i t l e n g t h C. C, and w i t h a s u i t a b l e system o f u n i t s , we c a n , u s i n g t h e HaarKarman p r i n c i p l e , r e d u c e t h e d e t e r m i n a t i o n o f t h e stress f i e l d t o t h e s o l u t i o n o f t h e v a r i a t i o n a l problem (where V d e n o t e s a "stress p o t e n t i a l " ) :

t h i s i s a s p e c i a l c a s e o f ( P O ) i n which t h e form L i s t h u s g i v e n by :

(1.9)

uu)= c

In

U(X)

bc

.

Remark 1.1. I n t h e c a s e o f a multiply-connected s e c t i o n R it i s c o n v e n i e n t t o employ a m o d i f i e d v e r s i o n o f f o r m u l a t i o n (1.8)j s e e Glowinski-Lanchon /l/, H. Lanchori 121. I n t h e c a s e where t h e i n i t i a l s t a t e i s n o t s t r e s s - f r e e , and/ or t h e t o r q u e i s n o t i n c r e a s i n g , f o r m u l a t i o n (1.8) i s i n c o r r e c t f o r t h e p h y s i c a l problem c o n s i d e r e d ; a s u i t a b l e approach i s

Statement of continuous problem

(SEC. 1)

119

then t o use time-dependent, or more p r e c i s e l y q u a s i - s t a t i c , mathematical models ( s e e Duvaut-Lions 111, Chapter 5 , f o r a 8 d e t a i l e d study of q u a s i - s t a t i c models i n p l a s t i c i t y ) .

Remark 1.2. We can supplement Remark 1 . 4 of Chapter 1, Section 1 . 2 by p o i n t i n g o u t t h a t s e v e r a l a u t h o r s , i n p a r t i c u l a r Shaw ill, have c a r r i e d out t h e numerical s o l u t i o n of t h e problem of t h e e l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l b a r , considered as a f r e e bouridmy problem, by using t h e p o i n t formu l a t i o n given i n t h e above-mentioned remark, ie. : (1.10)

- Au

(1.11)

Igradul = 1

=

C

Sa,

in in

a,,

with

(1.12)

R e = [xIxEP,Igradu(x)l <

(1.13)

a,,=9 - 9,;

11

it i s a p p r o p r i a t e t o add t o t h e above r e l a t i o n s

{

('*14)

uIr = 0

+ the continuity

of grad u at t h e i n t e r f a c e between

9, and 9,.

...,

Numerical methods based on t h e use of (l.lO), (1.14) give s a t i s f a c t o r y r e s u l t s ( s e e , f o r example, Hodge-StoutHerakovich 111) but appear t o be much more complicated t o apply than t h e m a j o r i t y of t h o s e given i n t h i s c h a p t e r , based on t h e 8 v a r i a t i o n a l formulation ( 1.8 ) . 1.3

Synopsis

I n S e c t i o n 2 w e s h a l l g i v e a number of p r o p e r t i e s of t h e s o l u t i o n of problem ( 1 . 6 ) ( i e . (Po)),and of t h e p a r t i c u l a r case (1.8),t h e s e p r o p e r t i e s having important consequences i n In particular (see t h e Numerical Analysis of t h e problem. Remark 3.18, Chapter 1, Section 3.7) t h e optimal s o l u t i o n of (1.8) i s a l s o t h e s o l u t i o n of

where 6 ( x , r ) = d i s t a n c e from x t o r . Problem ( P I ) ,which i s t h e s u b j e c t of Section 3 , i s very similar t o t h e l u b r i c a t i o n problem s t u d i e d i n Chapter 2 , Section 5 . From t h e p o i n t of view of a p p l i c a t i o n s t o mechanics, i t

,

Elasto-plastic torsion o f a cylindrical bar

120

(CHAP. 3 )

would appear t h a t i n view of t h e e q u i v a l e n c e of (1.81, (1.151, we could r e s t r i c t our a t t e n t i o n t o ( 1 . 1 5 ) ; however, i n view o f p o s s i b l e a p p l i c a t i o n s t o o t h e r s i t u a t i o n s , we have c o n s i d e r e d it worthwhile t o i n v e s t i g a t e t h e numerical a n a l y s i s o f ( P O )w i t h Q t W2 and f E L2(Q), by working d i r e c t l y with K O . S e c t i o n s h y 5 and 6 w i l l be devoted t o t h e approximation o f ( P O ) ,and Sections 7 , 8 and 9 t o t h e s o l u t i o n of t h e approxim a t e problems by r e l a x a t i o n , p e n a l i s a t i o n and d u a l i t y , resp8 ectively. 2.

SOME PROPERTIES OF THE SOLUTION OF PROBLEM (Pn) 2.1

Regularity r e s u l t s

We assume, i n ( 1 . 6 ) , t h a t t h e l i n e a r form L can be w r i t t e n : (2.1)

L(u) =

s.

f ( x ) u(x) dx .

Denoting t h e s o l u t i o n of (PO)by u , it i s shown i n B r B z i s Stampacchia /2/, S e c t i o n 3 t h a t : 1) If n i s a convex, bounded, open domain i n R " , w i t h Lipschitz boundary r , and i f f 6LP(R) n H - l ( Q ) ( l ) y p> 1, w e then

have :

{

(2.2)

+ 00) < + co).

AueLP(Q)

(1 < p <

u E W2*P(Q)

(1 < p

2 ) If i s a bounded open domain i n R" w i t h a s u f f i c i e n t l y r e g u l a r boundary I'( ) and if f E L p ( Q )n H - ' ( Q ) , p > 1 , we again have ( 2 . 2 ) .

Remark 2.1. I t w i l l be seen i n S e c t i o n 2 . 3 t h a t t h e above p r o p e r t i e s a r e o p t i m a l , at least i n Sobolev s p a c e s o f i n t e g e r order. 8 2.2

An e q u i v a l e n t v a r i a t i o n a l problem

I n Brezis-Sibony /l/, it i s shown t h a t i f n i s a bounded open domain i n R" w i t h a s u f f i c i e n t l y r e g u l a r boundary r , and i f L i s of t h e form

(')

This i s a u t o m a t i c a l l y s a t i s f i e d i f f ~ L ~ ( Q ) w i tph >, 2 .

( 2 ) We refer t o BrCzis-Stampacchia,

loc c i t . , f o r t h e

d e f i n i t i o n of a r e g u l a r i t y condition f o r s u f f i c i e n t t o ensure (2.2)

r

which i s

Properties of solution of ( P o )

(SEC. 2)

L(u) = C

(2.3)

121

I,

u(x)dx,

then t h e s o l u t i o n of ( P O )i s a l s o a s o l u t i o n o f ( P I ) , d e f i n e d i n (1.15), and o f t h e e q u i v a l e n t v a r i a t i o n a l i n e q u a l i t y :

( u E K I =[ulu~Ho'(R), Iu(x)I<6(x,~) a . e . 1 , t h e b i l i n e a r form b e i n g d e f i n e d as i n (1.2).

Remark 2 . 2 . I n f a c t , it i s easy t o show t h a t t h e s o l u t i o n of ( P I )has t h e same s i g n as C; i n t h e c a s e C > 0, K 1 can t h e r e fore be r e p l a c e d , i n ( 1 . 1 5 ) and ( 2 . 4 ) , by t h e even s i m p l e r convex s e t

K ; = [u 1 u E Hi(R),0 Q u(x) Q 6(x, f)

a.e.1.

Remark 2.3. From t h e numerical p o i n t of view, t h e s o l u t i o n of ( P I )i s e a s i e r t h a n t h a t o f ( P O ) ; t h e numerical a n a l y s i s of ( P I ) ,which i s a v a r i a n t of t h a t of t h e v a r i a t i o n a l problem of Chapter 2 , S e c t i o n 5, w i l l form t h e s u b j e c t o f S e c t i o n 3. Some p a r t i c u l a r c a s e s where t h e s o l u t i o n i s known(')

2.3 2.3.1.

A f i r s t example

We t a k e R = p, 1[ and L(u) = C i s d e f i n e d by:

with u ' = du/dx

v(x)dx w i t h C > 0 , so that ( P O )

.

I t can be v e r i f i e d t h a t t h e s o l u t i o n u o f ( P ~ ) ( a n o df ( P I ) ) i s given by: (2.6)

(')

C

u(x) = ?x(l

- x)

if C < 2

We r e s t r i c t o u r a t t e n t i o n h e r e t o very simple examples, designed t o a c t a s t e s t c a s e s for t h e v e r i f i c a t i o n o f t h e For v a r i o u s e x p l i c i t methods developed i n t h i s c h a p t e r . s o l u t i o n s s e e Mandel /I/.

E la s t o - p l as t i c t o r s i o n o f a cyZindrica2 bar

122

( C H A P . 3)

and i f C 2 2 by:

if 0 Q x Q

21 -

1

F i g u r e 2 . 1 shows t h e s o l u t i o n of ( P o ) (and ( P I ) )corresponding t o C = 4 (we t h u s have 112 - 1 / C = 114).

Fig. 2.1. 2.3.2

S o l u t i o n of ( P O )f o r C =

4.

A second example

We t a k e n = 2 ,

0=

[X

1x

L(u)=C

= (x1, xZ),

I

u(x)dx,

X:

+ 4 < R Z ],

c>o;

t h e s o l u t i o n of ( P O )(and ( P I ) ) is t h e n given by:

(2.8)

u(x) = (C/4)( R 2 -. r’)

where

r = (2.9)

u(x) =

C 6 2/R,

Jm, and i f C 2 2/R by

{ --

if R ’ G r G R Crz/4 + ( R - 1/C) i f 0 G r Q R’ ,

with

(2.10)

if

R‘ = 2 / C . rn

Remark 2.4. We can now supplement Remark 2.1; i n t h e two. above examples, f o r which t h e d a t a are very r e g u l a r , we have, f o r s u f f i c i e n t l y l a r g e C, u E c1(Sl> n~ ' ( 0 n )H ~ ( B ) ('1

but

.$CZ(E),

u#H3(62).

NUM3RICAL ANALYSIS OF PROBLEM ( P i 2

3.

3.1

Synopsis

I n t h i s s e c t i o n we s h a l l apply t h e r e s u l t s of Chapters 1 and 2 t o problem ( P I ) , with t h e open domain fi simply-connected i n R2 ; we s h a l l consider only t h e exterwr approximation of (Pi) (i.e. of t h e " f i n i t e d i f f e r e n c e ' ' t y p e ) and no a d d i t i o n a l d i f f i c u l t i e s a r i s e , r e l a t i v e t o t h e v a r i a t i o n a l problem of Chapter 2, Section 5, o t h e r t h a n t h e convergence of t h e approximate s o l u t i o n as h -+ 0, which i s r a t h e r more awkward t o prove. With regard t o algorithms, we s h a l l r e s t r i c t our a t t e n t i o n t o t h e over-relaxation method, which appears t o be t h e b e s t s u i t e d t o t h e t y p e of c o n s t r a i n t s m e t i n t h i s problem. 3.2 3.2.1.

E x t e r i o r approximation of problem ( P i )

Formulation of the approximate problem

We r e c a l l t h a t ( P I ) i s defined bv

Suppose h > 0, intended t o approach zero; proceeding as i n Chapter 1, Section 5 . 1 and Chapter 2 , S e c t i o n 5 , w e d e f i n e the g r i d k b y :

(3.2)

R,= { M i j ( M i j E R Z M , il= {ih,jh}, i,jEZ).

With each node Mil of R,we a s s o c i a t e t h e panel with c e n t r e Mi,

(3.3)

('1

m,O(Mtj) = ](i

- 4) h, (i + 4) h[

x

1(i - 4) h, (i+ 4) h [ ,

I n f a c t i n t h e case of t h e s e two examples w e have r a t h e r more, s i n c e av a2v ' ax,' ax,ax,

Elasto-plastic torsion of a cyZindrical bar

124

(3.8)

&j

= c h a r a c t e r i s t i c function of

(CHAP. 3)

mf(Mij).

Having p u t (as i n C h a p t e r 1, S e c t i o n 5 . 1 and i n C h a p t e r 2 , S e c t i o n 51, f o r k = 1, 2

we "approximate" J by

Jh

: Vh+ R

d e f i n e d by

and w e t a k e a s t h e a p p r o x i m a t e problem

M b Jh(vd

(P13[

(3.11)

9

Uh 6 K l h

Klh

=

{ uh I uh

vh,

I ufj I 6 q M f j ,

r),V M f j E Oh } .

It may be n o t e d t h a t Klh i s bounded and c l o s e d i n V,. rn 3.2.2

SoZvabiZity af ( P l h )

We proceed as i n C h a p t e r 2 , S e c t i o n 5 . 3 . 2 :

(l)

t h e mapping

We s h a l l u s e t h e same n o t a t i o n f o r t h e mapping V h + L z ( n ) d e f i n e d by u h - r r e s t r i c t i o n t o O . o f qhuh.

(SEC. 3 )

Numerical analysis o f ( P I)

125

d e f i n e s a norm over V, denoted by 11 0, Ilk; t h e f u n c t i o n a l Jh i s t h e r e f o r e continuous and s t r i c t l y convex on Vh and s i n c e K,, i s bounded and c l o s e d i n V, we have t h e e x i s t e n c e and uniqueness of an optimal s o l u t i o n f o r (Plh), say uI.

.

E x p l i c i t formulation of

3.2.3.

(Plh)

We can w r i t e ( 3 . 1 0 ) i n e x p l i c i t form as follows:

Remark 3.1. Mpq#%

3.3

'

I n ( 3 . 1 2 ) it i s necessary t o t a k e u r n = 0 i f

Convergence of t h e approximate s o l u t i o n as h

Preliminary remark.

-+

0

I n t h i s s e c t i o n we s h a l l prove t h a t

where u i s t h e optimal s o l u t i o n o f ( P I ) . This convergence r e s u l t can be proved d i r e c t l y by proceeding as i n Chapter 2 , Section 5 , though t h e r e a r e some a d d i t i o n a l t e c h n i c a l d i f f i c u l t i e s ; however, w e have p r e f e r r e d t o u s e t h e more g e n e r a l approach o f Chapter 1, S e c t i o n 5 even though t h i s makes t h e working somewhat more complicated. 3.3.1.

Reduction o f (P,d t o an equivalent variational inequality i n K , ,

I n o r d e r t o g e t back t o t h e s e t t i n g of Chapter 1, S e c t i o n 5 , it i s convenient t o formulate (P,) and (PI,,) i n terms of v a r i a tional inequalities;

t h i s gives

gradu.grad(u - u)dx 2 C

(u

- u)dx V U E K ,

Elasto-plastic torsion of a cylindrical bar

126

(CHAP. 3 )

I n ( 1 . 2 ) w e have p u t

In

grad u.grad u dx = a(u, v )

vh+ R

and we d e f i n e n, : Vh x (3 * 5 )

ah(uh, Oh)

3.3.2.

=

kil In

'k

qh

by

uh ' k

q h vh dx .

Exterior approximation f o r HA(Q), u, K,, L

We s h a l l now v e r i f y p o i n t by p o i n t t h a t t h e p r o p e r t i e s r e q u i r e d i n Chapter 1, S e c t i o n 5.2 a r e s a t i s f i e d :

Approximation of

HA(Q).

I n view o f C h a p t e r 1, S e c t i o n

(3.16)

V = HA(Q)

(3.17)

F = (L2(Q))'

and

a : V +F

5.1, we t a k e

i n j e c t i v e , d e f i n e d by:

I n S e c t i o n 3.2.1 we d e f i n e d a f a m i l y o f f i n i t e - d i m e n s i o n a l s p a c e s vh which are H i l b e r t s p a c e s f o r t h e norms 11 I(h d e f i n e d by:

.

we r e c a l l (see Cea / 3 I y L i o n s 131) t h a t t h e norm t h e d i s c r e t e Poincare inequality:

11. llh s a t i s f i e s

F i n a l l y , we d e f i n e a f a m i l y o f prolongation operators p h E 9 ( V h ;F) d e r i v e d from t h e o p e r a t o r qh : Vh+ L2(f2), d e f i n e d i n S e c t i o n 3.2.1, r e l a t i o n (3.7), by:

(3.21)

Ph uh = { q h %r

'1

qh h

r '2

q h uh

1

3

(SEC. 3 )

Numerical analysis of (PI)

127

and i n view of ( 3 . 2 0 ) , we indeed have:

This being t h e c a s e , w e s h a l l now prove:

The f a m i l y (V,), c o n s t i t u t e s an exterior

P r o p o s i t i o n 3.1. approximation of V.

Proof. It i s necessary f o r us t o v e r i f y t h a t ( 5 . 2 7 ) of ChapWe d e f i n e r, : V - r V, by: t e r 1, S e c t i o n 5 is s a t i s f i e d . (3*23)

(rh

v)Ml,

=

‘I

4x) dx,

VMij

E

4.

mO(Mij)

Thus ( s e e Chapter 1, Section 5 . 1 )

,

VvE

v=

and hence, t a k i n g account of ( 3 . 1 9 ) , (3.25)

(I r, u 1, 6 c

independent of h .

Approximation of a The form a, : V, x V, -r W d e f i n e d i n ( 3 . 1 5 ) s a t i s f i e s , f i r s t l y

(3-26)

I ah(% v h ) I

(3 * 27)

d V h ,

vk) =

d 11 uh 1, 11 uh

11 uh 1:

Ilk

9

and secondly ( t h i s may be proved without d i f f i c u l t y ) , (5.30) and ( 5 . 3 1 ) of Chapter 1, S e c t i o n 5.2, so t h a t we have: P r o p o s i t i o n 3.2.

The f a m i l y (a,&, c o n s t i t u t e s an approx-

imation of a. Approximation of K, With t h e family K,, d e f i n e d as i n (3.11) , w e have: P r o p o s i t i o n 3.3.

The family K , , c o n s t i t u t e s m appro*

mation of K , . Proof. It i s necessary t o prove (see ( 5 . 3 3 ) and ( 5 . 3 4 ) Chapter 1, S e c t i o n 5.2): (3.28)

if

vh

EK,,with limp, h-0

U, =

r

weakly i n F, t h e n { E O K ,

128

Elasto-plastic torsion of a c y l i n d r i c a l bar

(CHAP. 3 )

and

Verification of ( 3 . 2 8 ) .

[,,t2).

We p u t [ = (to, obtain that

from Chapter 1, S e c t i o n 5 . 1 w e t h e n

Moreover, s i n c e t h e f u n c t i o n x + &x, I') i s c o n t i n u o u s on it i s uniformly c o n t i n u o u s and t h u s , by d e f i n i t i o n O f Klh, gives

(3.32)

a,

lim q(h) = 0 ,

q(h) independent of x,

h-0

+

is, V E > 0 , S i n c e t h e set [u I UEL'(Q), 1 v(x) I Q 6(x, r ) E a.e] convex and c l o s e d i n L * ( Q ) , and t h e r e f o r e weakly c l o s e d , we have from ( 3 . 3 1 ) and ( 3 . 3 2 )

and we indeed have

<€OK1. 8

V e r i f i c a t i o n of ( 3 . 2 9 ) . I t ' i s s u f f i c i e n t ( s e e Chapter 1, S e c t i o n 5.3, Remark 5 . 4 ) t o v e r i f y t h a t ( 3 . 2 9 ) i s s a t i s f i e d for a subset X1of K,, which i s dense i n K, : Let U E K,; g i v e n E > 0 , w e d e f i n e v, = by:

(3.34)

T,(U)

= U, = (V

S i n c e rp + rp',

- E)'

- (U -k

E)-

.

rp- map H'(Q) + H1(Q), T,(u) E H'(Q) and

=0 Q 6(x,

if 6(x, I') Q E r )- E otherwise,

as may e a s i l y be v e r i f i e d .

,%unerical analysis of (PI1

(SEC. 3 )

129

Hence

(3.36)

VE > 0 .

~ ~ (E0K) ,

We then d e f i n e a subset X i of K , by

W e have

(3.38)

= K,

+

I n f a c t (u - 8)' ( r e s p . (u 8) - ) t e n d s s t r o n g l y towards u+ ( r e s p . u - ) i n H'(f2) as E -+ 0, t h e r e f o r e T ~ ( u )+ u s t r o n g l y i n H'(f2).

Variant. The above procedure shows t h a t t h e s e t of f u n c t i o n s V of K,, which a r e zero i n a neighbourhood of r a n d bounded above i n modulus by d(x,r) - ,&, i s dense in K l . This f a c t may a l s o be demonstrated by proceeding a2 follows : l e t U E K , and l e t 0, be a sequence of f u n c t i o n s of C'(n), (O,(x) Q 1, 0, = 0 i n t h e neighbourhood of r and 0,(x) + 1 uniformly over all compact s u b s e t s of f2,

I

<

constant ; thusemu+ u i n H'(f2) ( s i n c e u ~ H i ( f 2 ) )

and I 0, u(x) 1 Q I "<): 1 Q 6(x, r). Thus w e a r e already a b l e t o assume t h a t V i s zero i n t h e neighbourhood of r. However, i f 0 c I < 1 , 1 + 1 , w e have : Iu + u i n H'(f2) and h ( x ) < I q x , r), which combined with t h e f a c t t h a t u i s zero i n t h e neighbourhood of r shows t h a t Av(x) Q 6(x, r ) - E, , and hence t h e r e q u i r e d result. This being t h e c a s e , l e t U E X , ;with t h e o p e r a t o r r, : V + V, defined as i n ( 3 . 2 3 ) , i f we put tTh = rhu, we have, t a k i n g account of (3.35) and t h e c o n t i n u i t y of x + 6(x, r ) on R :

I

I

(3.39)

I

1

I

II,,,,,, I i2 I

I ufJ I = 7

I

v(x)dx Q -

Vh > 0 s u f f i c i e n t l y s m a l l

0

dx) I & Q 6(Mf,,r ) = a,,

v, E K,,

Vh > 0, s u f f i c i e n t l y s m a l l .

From (3.24) and ( 3 . 2 5 ) , p , r , u (=Phu&+m s t r o n g l y i n F and t h u s c o n d i t i o n (3.29) i s indeed s a t i s f i e d i n X1.

Elasto-plastic torsion of a cylindrical bar

130

(CHAP. 3 )

Remark 3.2. Following Raviart /l/, Chapter 0, we can v e r i f y (3.28) by u s i n g t h e f a c t t h a t i f p h v , + 5 weakly i n F, t h e n we have f o r a subsequence a l s o denoted by vh

(3.40)

q,

D,

-+

to s t r o n g l y i n L2(Q) and a . e . i n 62.

Remark 3.3. Basing our argument on ( 3 . 3 5 ) , w e s h a l l now prove t h a t 9(Q) n K, = K,;s i n c e t h e s e t xt c K, i s dense i n K , it w e s h a l l proceed by P e g u h . F i s s u f f i c i e n t t o show t h i s f o r xl;

isation : Consider then a r e g u l a r i s i n g sequence (p,,),

lR2

p,, 3 6(0),

p,(x)

dx

= 1, support (p,,) + 0); i f

DE

(i.e. pn E 9(R2), pa 2 0,

x,,

we extend v t o 8

ifcH'(W2), and w e d e f i n e i f , , ~ g ( R ' ) by

by 0 o u t s i d e f2;

r

We put

(3.42)

v, = r e s t r i c t i o n of

5, t o

a.

We then have

(3.43)

if,,+

if

strongly i n H'(R2)

and hence, by r e s t r i c t i o n t o 0 ,

(3.44)

v,+

v

s t r o n g l y i n H'(bl).

From (3.35) and f o r a s u f f i c i e n t l y l a r g e n , w e have

(3.45)

support (u;) c

n * 0, E 9(n),

It now remains t o show t h a t u , E K , f o r a s u f f i c i e n t l y l a r g e n . Suppose x E Q; we t h e n have:

we p u t :

(3.47) then x

(3.48)

o,(x) = [ y I y

E

R2, x - y E support @,,)I,

E w,,(x) and

diameter (a&)) = diameter support (p,,) -+ 0 as n

+

+

00.

Numerical analysis of (PI)

(SEC. 3 )

W e t h e r e f o r e have,

131

Vx'xELa,

I

(3.49)

It can r e a d i l y be deduced from (3.35) t o g e t h e r with t h e defi n i t i o n of 5 and (3.48) and ( 3 . 4 9 ) , t h a t

1 v,(x) I Q 6(x, r )

(3.50)

VX E Q provided

n i s sufficiently large ,

and hence vmsK1provided n i s s u f f i c i e n t l y l a r g e .

8

Approximation of L. I n t h e case considered, where

L(v) = cln4x)dX

, the

v e r i f i c a t i o n of (5.37), (5.38) of Chapter 1, Section 5.2 i s immediate f o r L , : Vh+W defined by: Lh(uk) =

In

4h

'h

dX .

A r e s u l t concerning strong convergence

3.3.3.

I n view of t h e p r o p e r t i e s demonstrated i n Section 3.3.2 f o r e x t e r i o r approximations of Hi(@, a, Kl,L we can apply t o problem (P,&, and more p r e c i s e l y t o t h e e q u i v a l e n t formulation ( 3 . 1 4 ) , Theorem 5.2 of Chapter 1, Section 5.3; t h i s g i v e s :

Let uh be the solution of problem ( P l h ) ; as h

Theorem 3.1.

-f

we have :

where u i s the solution o f problem ( P l ) . 3.4

8

Solution of t h e approximate problem by p o i n t overr e l a x a t i o n with p r o j e c t i o n .

We use algorithm ( 1 . 4 1 ) , (1.42) o f Chapter 2 , Section 1 . 4 ; t a k i n g account of (3.11), (3.12) and i t e r a t i n g with i n c r e a s i n g i, j t h i s t a k e s t h e form: I

(3.52)

u ; € K l h given uy"2=(l-w) ~ , + -0( ~ + l j + t r ; j + l + u ; - + : j + ~ ~ l l + h C) z 4

max (-

un i j+ l

=

Mi,

Oh .

a, min (#,+ ' I 2 ,

0,

Elasto-plastic torsion of a c y l i n d r i c a l bar

132

(CHAP. 3)

We again use t h e convention ukr = 0 i f Mk,#fZh. Since Theorem 1 . 3 of Chapter 2 , Section 1 . 4 a p p l i e s t o (P,,,) we have : Theorem 3.2. o < w < 2 .

The sequence

4 tends towards

uh

as n

+

+ oc,

if

Remark 3.4. If u: = 0, and i f 0 < w 6 1 , it r e s u l t s from (3.52) t h a t t h e sequence (Gj)" i s i n c r e a s i n g V M , , E f Z , ; t h e convergence i s t h e r e f o r e monotone and t h e coordinates uij of u, are approached from below. Remark 3.5. We could a l s o solve(P,,J by o t h e r methods and i n p a r t i c u l a r by a d u a l i t y method similar t o t h a t used i n Chapter 2 , Section 5.6; although t h e a p p l i c a t i o n o f such a method t o problem(P,& does not p r e s e n t any d i f f i c u l t i e s , w e have been content i n t h i s example, t o use t h e over-relaxation method, which seems t h e b e s t adapted t o t h e problem considered. 3.5.

Applications. Example 1.

We t a k e fZ = lo,1[ x ]0,1[ with C = 10.

3.5.1.

Numerical values o f the parameters

Mesh s i z e : h = 1/40. I n i t i a l i s a t i o n of algorithm (3.52)

: u: = 0.

Termination c r i t e r i o n f o r algorithm (3.52)

1

(3.53)

i.;,+l

:

- 4 1<

bfIIJ6ok

3.5.2.

Analysis o f the numerical r e s u l t s

Table 3.1 shows t h e r a t e of convergence as a function o f w :

w

1

1.4

1.5

1.6

1.7

1.8

1.9

2

Number of iterations

290

138

118

98

93

108

188

Divergence

Table 3.1 The execution time on an IBM 360191 i s 7.33 see. f o r w = 1 and 2.51 sec. f o r w = 1.7.

(SEC. 3)

FQ. 3.1.

Numerical anaZysis o f . ( P I )

133

Decrease o f R" as a f u n c t i o n o f t h e number of i t e r a t i o n s n. The c a s e where C = 10 (log-linear s c a l e ) .

XI

Fig. 3.2.

The p l a s t i c r e g i o n s a r e shown h a t c h e d ( C = 1 0 ) .

Elasto-plastic torsion of a cylindrical bar

134

(CHAP. 3)

Remark 3.6. We n o t e t h a t once again t h e r a t e of convergence i s g r e a t e s t w h e n o i s considerably l a r g e r t h a n 1. W Figure 3 . 1 shows t h e decrease of t h e q u a n t i t y

1

R"=

IU;,"

-

411,

M"SRk

as a f u n c t i o n of t h e number of i t e r a t i o n s , f o r w = 1.7 ; Figure 3.2 shows t h e p l a s t i c (I u(x) 1 = 6(x, r))( ) and e l a s t i c

(I u(x) I c 0)( 1 regions and r e l a t i n g t o t h e optimal v a l u e of o w e can s t a t e : Remark 3.7. For t h e unconstrained D i r i c h l e t problem on f2 = ]0,1[ x ]0,1[, d i s c r e t i s e d with a mesh s i z e h = 1 / 4 0 , t h e optimal value o f o i s very c l o s e t o 1.85, w h i l s t f o r problem (P,& w i t h C = 10 it i s c l o s e t o 1.7; t h i s v a l u e i s e s s e n t i a l l y t h a t which corresponds t o t h e s o l u t i o n of t h e D i r i c h l e t problem Au = C i n t h e e l a s t i c region f o r a d i s c r e t i s a t i o n with mesh s i z e h ; t h i s r e s u l t s from t h e experimentally observed f a c t t h a t t h e p l a s t i c region, i.e. t h e region where t h e c o n s t r a i n t s a r e s a t i s f i e d w i t h e q u a l i t y , i s located very quickly, and t h a t t h e computation time i s s p e n t mainly i n s o l v i n g - Au = C i n t h e e l a s t i c region. The above comments c o r r o b o r a t e Remark 5.7 of W Chapter 2 , S e c t i o n 5.7.2.

-

Figure 3.3 shows, u s i n g t h e numerical r e s u l t s , t h e f u n c t i o n s u ( z , 0.2) and x + u ( x , 0.5), where u i s t h e s o l u t i o n of problem ( P I )

x

-+

.

0.51

0.41

0.5

1

c

Fig. 3.3. f2 = 10, 1[ X 10, I[, C = 10. Representation of u ( x , 0.2) and u(x, 0.5).

i.e.

Igradul = 1.

(2) i.e.

Igradul < 1,

(l)

- AU = C .

(SEC. 3 )

3.6.

Numerical analysis of ( P I )

135

Applications. Example 2.

We t a k e f o r f2 t h e domain r e p r e s e n t e d i n Figure 3.4, w i t h C = 5 and 10.

I

, 1

I 1

0.4 0.6

Fig. 3.4.

3.6.1.

Numerical values of the parameters

Mesh s i z e : h = 1/20. I n i t i a l i s a t i o n of algorithm ( 3 . 5 2 )

:

U: =

Termination criterion for algorithm ( 3 . 5 2 )

0. : we again use

condition ( 3 . 5 3 ) . 3.6.2.

Analysis of the numerical r e s u l t s

The results in Table 3.2. show t h e r a t e of convergence a s a f u n c t i o n of w.

c w

5 1

5

10

10

1.5

1

1.5

81

64

38

Table 3.2 The computation time for C = 10, w = 1 i s 0.88 see. on an IBM 360191. These results confirm t h e advantage of u s i n g w > 1 i n algori t h m ( 3 . 5 2 ) ; moreover, s i n c e t h e p l a s t i c region i s l a r g e r Tor

Elasto-plastic torsion of a cylindrical bar

136

(CHAP. 3 )

C = 10 than f o r C = 5, more c o n s t r a i n t s a r e s a t i s f i e d with equality i n t h e former c a s e , which e x p l a i n s t h e faster rate of convergence of algorithm ( 3 . 5 2 ) . Figure 3.5 shows t h e e l a s t i c and p l a s t i c regions corresponding

to

c

= 10.

X,

Fig. 3.5.

4.

Representation of t h e e l a s t i c and p l a s t i c regions f o r C = 10 ( t h e p l a s t i c regions a r e shown h a t c h e d ) .

INTERIOR APPROXIMATIONS OF ( P a ) Synopsis.

I n t h e previous s e c t i o n w e s t u d i e d problem (Pi); we s h a l l now consider problem ( P O ! , f o r which t h e d i s c r e t i s a t i o n of t h e funct i o n a l J i s treated zn the same way whereas t h e a p p l i c a t i o n of t h e c o n s t r a i n t s (membership of K O ) poses new problems which vary In t h i s and widely with t h e mode o f approximation used f o r J . t h e following s e c t i o n s w e s h a l l t h e r e f o r e c a r r y out a systematic study of t h e approximation of J and the corresponding approxi-

mation of the convex s e t KO. I n t h i s s e c t i o n we s h a l l formulate two types of i n t e r i o r approximation i n t h e sense of Chapter 1, Section 4. I n Section 4 . 1 w e s h a l l consider a f i n i t e element method with approximation by a f f i n e l i n e a r f u n c t i o n s over t r i a n g l e s ( s e e Chapter 1, Section 4, Example 4.2), and i n Section 4 . 2 , a Galerkin approximation ( s e e Chapter 1, S e c t i o n 4 , Example 4 . 1 ) using t h e eigenfunctions of t h e o p e r a t o r - A i n H&2) as a b a s i s The convergence o f t h e s e approximations w i l l be s t u d i e d i n

(SEC.

4)

I n t e r i o r approximations o f (Po)

137

S e c t i o n 6.

4.1.

A f i n i t e element method

Triangulation of a.

4.1.1

Definition of

v h

We now go back t o Example 4.2 o f Chapter 1, S e c t i o n 4 ; suppose Y hi s a triangulation g e n e r a t e d by a f i n i t e number of t r i a n g l e s with t h e following p r o p e r t i e s : (4.1)

TEY~=-TC;

(4.2)

{ T,

T ' e Y h+ T n T' = 4

o r T and T' have a common s i d e

o r T and T' have a common v e r t e x ;

for rp ( s e e r e l a t i o n ( 4 . 6 ) o f Chapter 1, S e c t i o n we take

(4.3)

4, Example

4.2)

r p ( h ) = h = Max (Measure of T ) TcJ

h

which s a t i s f i e s t h e r e l a t i o n s S e c t i o n 4.

(4.6), ( 4 . 7 )

of Chapter 1,

We d e f i n e t h e s e t s (4.4)

-

a, =

u

T

f.Ch

(4.7)

z h = set

(4.8)

zh=[[M)MEzh,

of v e r t i c e s of yh

0

bf$r,,l.

If ME,$;, w e denote by M ( s e e Figure 4 . 1 ) ;

P,,...,Pk t h e n e i g h b o w i n g v e r t i c e s of

Fig. 4.1.

Elasto-plastic torsion of a cylindrical bar

138

(CHAP. 3 )

We d e f i n e :

(4.9)

[

w, = a f f i n e continuous f u n c t i o n over each t r i a n g l e with v e r t e x M, such t h a t w,(M) = 1, W A P , ) = 0, i = 1, ...,k and with w, zero on Q o u t s i d e t h e t r i a n g l e s with v e r t e x M,

then : (4.10)

vh

= t h e space spanned by t h e w,,

ME&,

and w e have: Proposition 4.1.

vh

i s a subspace of HA(Q) with

d im( V&=N=Card(&).

rn

Remark 4.1. Since t h e f u n c t i o n s u h e V h a r e a f f i n e over T , gradu, i s c o n s t a n t over T , V T E Y , . rn Definition of the approximate problem

4.1.2.

For ( P o ) , w e r e s t r i c t our a t t e n t i o n t o t h e case where (4.11)

L(u) = sD/udxdy, f ~ L ' ( n )

grad u.grad u dx dy ,

and hence, w i t h u(u, u ) = (4.12)

1

J(u) = 2 a(u, u ) - lnfu dx dy

and

I

Min J(u)

(Po)

OE=

KO =

[uI

U E H A ( ~lgradv ),

I < 1 aeJ.

We then d e f i n e t h e approximate problem (P,,,,) by: (4.13)

(Pd

Min

J(uJ.

rn

uhc&lnvh

Remark 4.2. The condition u , € K o n vh*Igradvh1' put ( s e e ( 4 . 1 0 ) ) : (4.14)

0,

=

,V

w,,

V,E

d 1 a.e. ; i f we

R

bf€&

then i n view of Remark 4 . 1 t h e approximate problem (4.13) reduces t o an o p t i m i s a t i o n problem i n RN with N ' ( = Card(YJ) q u a d r a t i c c o n s t r a i n t s with r e s p e c t t o t h e uy. rn

(SEC. 4 )

I n t e r i o r approximations of (PO )

139

Solvability of the approximate problem

4.1.3.

Since t h e f u n c t i o n a l J has t h e r e q u i r e d p r o p e r t i e s ( s e e Chapter 1, S e c t i o n 2 . 1 ) we have:

The approx$mate problem (4.13) admits a Proposition 4.2. unique solution u, characterised by : (4.15)

I

- ub) >/

a(uh,

u,

EK, n V,.

1.

f (0, - Uh) dX dy

VU, E KO A

vh,

rn

4.1.4. E x p l i c i t formulation of the approximate probtem L e t MI, M,,M3 be t h e v e r t i c e s o f a t r i a n g l e T e Y , with coordinates (xi,yi), i = 1, 2 , 3; t h e a f f i n e l i n e a r function defined by:

(4.16)

WJ

= vMl, i =

4

1,2,3

and 3

(4.19)

6=

z r i ,

I 6 I = 2 Measure (Triangle MI M 2M 3 ) . I n t h e following w e s h a l l use t h e n o t a t i o n m ( T ) = measure of f g r s i m p l i f i . c a t i o n , uMl = ui. If w e consideru,EVk,we have (see ( 4 . 1 4 ) ) :

T, and, (4.20)

q(x,

v) =

c

OM

wM(x9

Y).

M.PL

u s i n g t h e charIt i s a l s o p o s s i b l e (and u s e f u l ) t o d e f i n e a c t e r i s t i c f u n c t i o n s o f t h e t r i a n g l e s T E Y which, ~ t a k i n g account of (4.17) and (4.19) and u s i n g t h e obvious n o t a t i o n , gives:

(4.21)

Elasto-plastic torsion of a cylindrdcal bar

140

(CHAP.

3)

with (4.22)

Or = c h a r a c t e r i s t i c f u n c t i o n of T .

From t h i s w e deduce

which a l l o w s t h e approximate problem (Po& t o be expressed i n terms o f t h e u,.

Remark 4.3.

I,

In

(4.26) t h e i n t e g r a l

Ax, Y )W d X 9 Y )

dr

must a c t u a l l y be e v a l u a t e d on t h e s u p p o r t of w,.

Remark 4.4. I n t h e terminology o f f i n i t e element methods, t h e symmetric p o s i t i v e - d e f i n i t e m a t r i x r e l a t e d t o t h e q u a d r a t i c p a r t o f J(u& i s c a l l e d t h e s t i f f n e s s matrix a s s o c i a t e d w i t h t h e approximation considered; i n t h e approximate problem it w i l l p l a y t h e r o l e of -A i n t h e continuous problem.

4.1.5.

On the use of f i n i t e element methods of order greater than 1.

I n S e c t i o n s 4.1.1, 4.1.2, 4.1.3 an approximation w a s conside r e d which u s e s polynomials of maximum degree e q u a l t o u n i t y

(SEC. 4 )

I n t e r i o r approximations o f ( P o )

141

over each T EY h; t h i s g i v e s a constant g r a d i e n t over T, V T E Y , . The use of polynomials o f maximum degree k(> 2) over T, V T E . T , would, i n g e n e r a l , produce over T, V T E Y , a g r a d i e n t which i s a polynomial f u n c t i o n of x , y of degree k - 1 2 1 ( a n d t h u s not

constant).

The approximate f o r m u l a t i o n of t h e c o n d i t i o n I grad u(x, y ) I Q 1 a.e. could t h e n be c a r r i e d out i n various ways, i n p a r t i c u l a r ( t h i s l i s t i s not exhaustive): (4.28)

(4.31)

I { G,=

I

Brad Uh(Gr) Q 1 VT E Y h , c e n t r e of g r a v i t y of t h e t r i a n g l e T,

Igradu,(x,y))dxdy<

1 vTEYh

From a numerical p o i n t of view t h e f o r m u l a t i o n s ( 4 . 2 8 ) and (4.29) can, without g r e a t d i f f i c u l t y , be made e x p l i c i t i n terms of t h e c o e f f i c i e n t s of t h e polynomial approximation considered on T ; t h e s a t i s f a c t i o n o f t h e c o n d i t i o n ( 4 . 3 0 ) i s only made e x p l i c i t e a s i l y f o r k = 2; i n f a c t , i n t h i s p a r t i c u l a r c a s e , t h e mapping gr : T + R , d e f i n e d by: (4-32)

I

814x9 V) = grad uh(x,V) 1'

v(X,

y) E T ,

i s a polynomial of maximum degree 2 ( k - 1) = 2 and i t s Hessian m a t r i x ( i e . t h a t o f t h e second d e r i v a t i v e s ) i s c o n s t a n t and ( a t least) p o s i t i v e semi-definite; gr t h e r e f o r e a t t a i n s i t s maxi m u m at one of t h e v e r t i c e s of t h e t r i a n g l e T , so t h a t t h e r e a r e three constraints t o be s a t i s f i e d f o r each t r i a n g l e T E Y The ~ i n t e r m e d i a t e numerical c a l c u l a t i o n s n e c e s s a r y t o make e x p l i c i t (and a l s o t o s a t i s f y ) ( 4 . 3 0 ) makes t h i s f o r m u l a t i o n d i f f i c u l t t o use f o r k 2 3 ; t h i s i s a l s o t h e c a s e f o r ( 4 . 3 1 ) ( f o r k 2 2) s i n c e t h e s a t i s f a c t i o n of t h i s r e l a t i o n r e q u i r e s t h e c a l c u l a t i o n of t h e i n t e g r a l on t h e l e f t - h a n d s i d e by an approximate i n t e g r ation method, t h e e x a c t c a l c u l a t i o n b e i n g i n g e n e r a l impossible. L e t Vh be a subspace o f H&2) such t h a t U , E V, * uh i s a polynomial on T , of maximum degree k ; t h e approximation considered i s t h e r e f o r e i n t e r i o r i n t h e s e n s e of Chapter 1, S e c t i o n 4. We n o t e t h a t , except f o r k = 1(l ) , t h e f a c t t h a t u, s a t i s f i e s ( 4 . 2 8 ) , ( 4 . 2 9 ) , ( 4 . 3 1 ) does n o t i n g e n e r a l imply t h a t v h E K o = [ u ( u ~ H t ( R )(grad01 , Q 1 &el,

('1

I n t h i s c a s e t h e r e i s an e q u i v a l e n c e between ( 4 . 2 8 ) , ( 4 . 2 9 ) , (4.30), (4.31).

Elasto-plastic torsion of a cylindrical bar

142

t h e l a t t e r only being t r u e f o r (4.30). If w e use (4.30) , t h e approximate problem,

Min

( C H A P . 3)

J(uJ,

is

VrsKonVk

( f o r k = 2 ) a finite-dimensional o p t i m i s a t i o n problem with 3 C a d ( F & = 3 N ’ q u a d r a t i c c o n s t r a i n t s compared with N ’ i f k = 1 W (and with t h e same t r i a n g u l a t i o n ) ( ’). 4.2.

An i n t e r i o r approximation method u s i n g t h e eigen-

f u n c t i o n s of t h e o p e r a t o r - - A i n H i ( R ) . Synopsis.

We assume t h a t t h e e i g e n f u n c t i o n s i n H;(R) of t h e o p e r a t o r A are known though t h i s i s r a r e l y t h e case i n p r a c t i c a l applic a t i o n s ; t h e purpose o f t h i s approach i s t o demonstrate (and t h i s w i l l become apparent i n S e c t i o n 9 ) t h e d i f f i c u l t i e s involved i n using high-accuracy methods i n one-sided problems which have c o n s t r a i n t s of t h e point t y p e , as i s t h e c a s e with KO. 8

-

Approximation of HA(R)

4.2.1.

Since t h e domain R ’ i s bounded, t h e spectrum of we s h a l l use t h e n o t a t i o n :

-A

i n Ht(t2)

i s countable; (4.33)

(B,,JmEN= spectrum of

-A

i n H;(R)

with w, t h e corresponding eigenfunctions; (4.34)

{ w,- Aw,H i @ B,,,) w, =

E

(B, > 0, V m ) ,

hence

Vm E N , Vm E N .

We r e c a l l t h a t : (4.35)

I. In

w,(x) wm(x)cix = 0

if

m

+ n;

w e s h a l l assume t h a t : (4.36)

I w,,,(x) 1’

dx = 1 Vm E N .

I n p a r t i c u l a r s i n c e t h e family ( w , ) , . ~ c o n s t i t u t e s a Galerkin b a s i s f o r H;(R), w e approximate H ; ( R ) by t h e family ( V J M e N defined by:

(’)

The p r i n c i p a l motivation behind going from k = 1 t o k = 2 i s simply t o reduce t h e number o f t r i a n g l e s r a t h e r t h a n t o i n c r e a s e t h e accuracy.

(SEC. 4 )

VM = subspace of ' H i ( R ) generated by

(4.37)

143

I n t e r i o r approximations of (Po)

4.2.2.

(W,,,)~(,,,~Y.

Approximation of KO

Since i n p r a c t i c e it i s not p o s s i b l e t o use t h e convex s e t

KO n VM w e s h a l l be content i f U E VM t o s a t i s f y t h e condition Jgradul g 1 on a f i n i t e s e t o f p o i n t s of 0 ( o r of s), say ah.. (resp. by

ah);

f o r KO t h i s g i v e s t h e approximation K t . ( ' ) defined

K ~ = [ u I u E V M Iwdv(P)I< ; 1 VPER,,].

(4.38)

We g e t a similar d e f i n i t i o n i f we r e p l a c e

nh by Zh.

Remark 4.5. The method f o r approximating KO appears t o be of t h e colZocation t y p e ; a n a t u r a l method (which w i l l be employed i n Section g), i n s p i r e d by J . B . Rosen 131, c o n s i s t s of d e f i n i n g

R, = [x I x

(4.39)

=

(xl,xz) E W, xi = ih, xz

= jh,

i, j E Z] ,

where h > 0 i s given (and intended t o approach z e r o ) . S i m i l a r l y we can d e f i n e 3, = 0, u r h where r,. i s a f i n i t e subset of p o i n t s of r f o r which a t least one of t h e coordinates is an i n t e g e r m u l t i p l e of h.

4.2.3.

Definition of the approximate problem

With t h e f u n c t i o n a l J

:HA@)+ W once again o f t h e form:

with f EL'(^), t h e approximate problem i s defined by:

M.in J(u).

(4.41)

UE%Y

4.2.4.

Solvability of the approximate problem

As t h e f u n c t i o n a l J and K& have t h e r e q u i r e d p r o p e r t i e s (see Chapter 1, S e c t i o n 2 . 1 ) we have:

(l)

This approximation w i l l be j u s t i f i e d i n S e c t i o n

6.

144

Elasto-plastic torsion o f a cylindrical bar

(CHAP. 3)

P r o p o s i t i o n 4.3. The approximate problem ( 4 . 4 1 ) admits one and o n l y one solution, #, characterised by:

Explicit formulation o f the approximate problem

4.2.5. If

UE

V , w e use t h e n o t a t i o n :

.U. (4.43)

v=Cpiwi,

pi€R V i = 1 ,

..., M ,

i=l

and t a k i n g i n t o account t h e orthonormality of t h e ( w , ) , . ~ we have:

Moreover :

P u t t i n g p = (pl,p2,...,pM)and t a k i n g account of ( 4 . 4 4 ) , ( 4 . 4 5 ) , t h e approximate problem ( 4 . 4 1 ) can be w r i t t e n e x p l i c i t l y as

Remark 4.6. In view of ( 4 . 4 4 ) t h e m a t r i x a s s o c i a t e d with t h e q u a d r a t i c p a r t of J , considered as a f u n c t i o n of p , i s diagonal; t h e r e f o r e , from t h i s p o i n t of view t h e s i t u a t i o n i s optimal i n comparison with any o t h e r t y p e of approximation_; on t h e o t h e r hand t h e c o n s t r a i n t s lgrad.v(P) < 1, V P E ~( o r O h ) expZicitly involve a l l t h e c o o r d i n a t e s of p ( t h i s i s n o t t h e case f o r t h e i n t e r i o r approximation considered i n Section 4 . 1 o r i n t h e e x t e r i o r approximations s t u d i e d i n Section 5 ) and t h i s f a c t w i l l considerably l i m i t t h e a p p l i c a b i l i t y of t h i s t y p e of approximation.

I

(SEC. 5 )

5.

Exterior approximations of (Po

145

EXTERIOR APPROXIMATIONS OF ( P o )

5.1. Approximations o f J

5.1.1.

A f i r s t exterior approximation o f J

We r e t a i n t h e d e f i n i t i o n s and t h e formalism of S e c t i o n 3.2.1;

hence i f f EL'(P), t h e approximation o f J(v) = i s d e f i n e d by: J: : Vh+ W

t h e e x p l i c i t form o f ( 5 . 1 ) b e i n g given by:

with

t h e sets O h , z h are as d e f i n e d i n S e c t i o n 3.2.1 and Remark 3.1 a l s o holds f o r ( 5 . 2 ) . 5.1.2.

A

second exterior approximation of J

I n t h e f o l l o w i n g w e s h a l l a l s o u s e t h e approximation o f J d e f i n e d by

Before making ( 5 . 4 ) e x p l i c i t w e d e f i n e t h e grid Qh as f o l l o w s :

146

Elasto-plastic torsion of a cylindrica2 bar

(CHAP. 3 )

and w e s h a l l use t h e n o t a t i o n ( s e e Figure 5.1):

Fig. 5.1.

We d e f i n e & by: Zh = {

(5.7)

M 1 MEQ,, m&U)

a t l e a s t one of t h e

belonging t o

We can now express ( 5 . 4 )

J~(uJ=

(5.8)

hZ

c

4 v e r t i c e s of

ah1.

e x p l i c i t l y i n t h e form: x

Mi+iiai+iii6Ih

the

Lj 5.2.

s t i l l being defined by (5.3) and with u, = 0 i f

MPq$Dk

E x t e r i o r approximations of KO

It seems reasonable t o approximate KO by

+

t h e function I b,q, vh 1’ IbZq, v h 1’ being c o n s t a n t over square no. 1 ( r e s p . 2, 3, 4) of s i d e h / 2 , i n Figure 5.2, and having t h e value

(SEC. 5 )

Exterior upprodmations of (Po)

147

t h e e x p l i c i t form of ( 5 . 9 ) i s given by

with, i n (5.10), 0,=0 if M,#8,. rn It r e s u l t s from (5.10) t h a t t h e use of K& imately 4 Card (L?,,) q u a d r a t i c c o n s t r a i n t s . t h e number of c o n s t r a i n t s t o around Card(f2,J of t h e c o n s t r a i n t s (5.10) a t s u i t a b l y chosen which gives:

leads t o approxI n o r d e r t o reduce w e c8n take t h e mean p o i n t s of 4 or Qb

where t h e r e l a t i o n d e f i n i n g K& can be i n t e r p r e t e d as an approximation a t M i + l 1 2 j + l 1 2of t h e c o n s t r a i n t I grad u l2 < 1 ; K& can a l s o be defined as:

148

Elasto-plastic torsion of a cyZindrica2 bar

(CHAP. 3)

Before d e f i n i n g a t h i r d e x t e r i o r approximation o f K O we i n t r o duce t h e s e t Oh d e f i n e d by:

We now c o n s i d e r t h e e x t e r i o r approximation

The e x p l i c i t formulation of t h e membership of d i s t i n g u i s h between t h r e e c a s e s : I f M i j E Qh we have :

K & l e a d s us t o

with urn = 0 i f M,$Q,. If Mi, E rhw i t h a s i n g l e neighbour i n ah( i f t h i s neighbour i s for example M i + , , , s e e F i g u r e 5 . 3 ) we have

Fig. 5.3.

The e x t e r i o r o f O h i s shown hatched.

and, i n , f a c t , a s uij = 0, we have (5.17)

1

5; I “ + I , I d

1

. ah

If M,,Erk w i t h two neighbours i n ( i f t h e two neighbours a r e , f o r example M i + , , , Mi,-,, s e e F i g u r e 5 . 4 ) we have

Exterior approximations of (Po)

Fig. 5.4.

149

The e x t e r i o r of O h i s shown hatched.

and as uij = 0, we have (5.19)

2

2

+

# + + l j

d 1.

The r e l a t i o n s d e f i n i n g K& can be i n t e r p r e t e d as an approximation at M i j c z ho f t h e c o n s t r a i n t I grad u 1' Q 1. w We s h a l l a l s o use t h e e x t e r i o r approximation of KO defined by:

(5.20)

+

d1 a.e)

,

L

which i s expressed e x p l i c i t l y by:

Remark 5.1. Obviously t h e r e are o t h e r p o s s i b l e e x t e r i o r approximations f o r K O ; f o r example, i n o r d e r t o approximate I g d u 1 2 < 1 a t Mij we can use

Elasto-plastic torsion of a c y l i n d r i c a l bar

150

(CHAP. 3)

however, without going i n t o t h e d e t a i l s , we can s t a t e t h a t (5.22) p r e s e n t s s e r i o u s problems from t h e p o i n t of view o f numerical H stability.

5.3.

Formulation of t h e approximate problem.

With t h e f u n c t i o n a l s J l ( I = 1, 2) and t h e convex s e t s KG (m = 1, 2, 3,,4)defined as i n Sections 5 . 1 and 5.2, w e s h a l l t a k e as approxi m a t e problems : (5.23)

(P&

:

Mh

Ji(uh)

.

H

mlSG

5.4.

S o l v a b i l i t y of t h e approximate problem.

The reasoning of Chapter 2 , S e c t i o n 5.3.2 (or Chapter 3, Section 3.2.2) remains v a l i d i f we n o t e t h a t t h e mappings:

d e f i n e norms over V, and t h a t t h e convex sets KG (m = 1, 2, 3,4) are closed i n V,; we t h e r e f o r e have both t h e e x i s t e n c e and uniqueness of an optimal s o l u t i o n f o r (Pa),,,,. The convergence, as h -+ 0, of t h e s o l u t i o n of t h e approximate problem t o t h a t o f t h e continuous problem w i l l be i n v e s t i g a t e d i n Section 6.

6.

CONVERGENCE OF THE INTERIOR

AND EXTERIOR APPROXIMATIONS

Synopsis. I n t h i s s e c t i o n we s h a l l demonstrate t h e convergence of t h e approximations considered i n S e c t i o n s 4 and 5; w e s h a l l refer t o t h e methods of Chapter 1, S e c t i o n 4 f o r t h e i n t e r i o r approximations and o f Chapter 1, Section 5 f o r t h e e x t e r i o r approximations. The proof of t h e convergence of t h e v a r i o u s approximations H considered i s based e s s e n t i a l l y on a d e n s i t y r e s u l t .

(SEC. 6 )

151

Convergence of approximations

6.1.

A lemma concerning d e n s i t y

We s h a l l now prove:

6.1.

Lemma

We have

O(62)n KO = K O .

(6.1)

Proof.

We proceed (as i n Section 3.3.2, Remark 3.3) using

truncation mrd regularisation; w e t h u s have t h e following two stages :

1) L e t U E K , ; we d e f i n e u, = t,(u)(as T,(u) = U, = (U

(6.2)

i n ( 3 . 3 4 ) ) by:

- E)' - (U + 8)- ,

where E > 0 i s given; a . e . on n , and s i n c e :

w e t h u s have r,(u)EH1(f2) with Igradr,(u)

I v(x) I

KO c K , = { u I u E If&!),

Q 6(x,

r)

I< 1

a. e. }

we have (as i n (3.35)) : (6.3)

o if 6(~,r ) < & otherwise Q 6(x, r ) - E

u,(x) =

and hence T,(u) However (6.4)

T,U +

u

E

KO, V& > 0. i n H'(f2) as

E -+

0,

so t h a t :

{ it i s i s

s u f f i c i e n t t o approximate a function u c K o

(6.5)

which

i d e n t i c a l l y zero i n a neighbourhood of

r.

2 ) We extend v ( u ~ K a ) t oi7 i n Rz by 0 o u t s i d e n ; we have ~ E H ' ( Rand ~ ) (graddl < 1 a.e. on W2. I f @,Jn i s a regularising sequence, w e define C R ~ O ( R 2by: ) (6.6)

(li..)=

IR2 4~)-

4 = v"*

Pn

iTn+ i7

s t r o n g l y i n H'(Rz)

PAX

V ) dV) ;

we have : (6.7)

W e use t h e n o t a t i o n : (6.8)

u, = r e s t r i c t i o n of i7,, t o f2 ;

thus i f v satisfies ( 6 . 5 ) and provided n i s s u f f i c i e n t l y l a r g e ,

Elasto-plastic torsion of a cylindrical bar

152

(CHAP. 3 )

w e have : support (En) c

(6.9)

a=.

1)"

E w-4

s t r o n g l y i n Hi@) as n-+

v,+v

+ 00

We s h a l l now show t h a t v, E K O ;i n f a c t (6.10)

gradG,,=p,rgradE,

and hence

Remark 6.1. S i n c e t h e f u n c t i o n s of a (indeed equi-Lipschitz with constant

KO are equicontinuous on 1) and s i n c e KO i s bounded i n Lm(f2) , KO i s strongly compact i n Lm(Q (from A s c o l i ' s theorem), and hence i f v, E KO Vn w i t h v, -+ v i n 9'(62) then , v E KO and v, -+ v strongly i n Lm(61). ( i e . u n i f o d y ) . 8 6.2.

Convergence of t h e i n t e r i o r approximations.

6.2.1.

Convergence of the f i n i t e element method

Using t h e r e s u l t s o f Chapter 1, S e c t i o n s 4.3 and 4 . 4 , we s h a l l now i n v e s t i g a t e t h e convergence of t h e i n t e r i o r approximation o f S e c t i o n 4.1; w e r e c a l l t h a t h w a s d e f i n e d by:

cp(h) = h = Max (measure of 2').

(6.14)

TErh

Moreover l e t us s t a t e t h a t lim

(a -

= 0 if for all K c

a, K

h-0

compact, we have 0, 3 K provided h i s s u f f i c i e n t l y s m a l l . t h e n have :

We

Theorem 6.1. I f La - Q, -+ 0 with 0 2 go > 0 Vh, f o r any angle 0 of T , V T E Y,,, we have u, 4 u strongly i n Hi@) n Lm(Q)where uh i s the solution of the approximate problem (Po,,) defined i n Section 4.1.2, relation' ( 4 . 1 3 ) , and u i s the solution of (Po)

.

Proof. We s h a l l apply Theorem 4.2 o f S e c t i o n 4.4, Chapter 1. For t h i s it i s n e c e s s a r y t o show t h a t (i) V v E Hi@), t h e r e e x i s t s vh E V, such t h a t oh + v s t r o n g l y in

ma). (ii) Vv E K , t h e r e e x i s t s v4 E KO n v h such t h a t v, v s t r o n g l y i n of t h e s e p o i n t s it i s s u f f i c i e n t t o t a k e v i n a dense subset which i s i n H i ( 0 ) f o r ( i ) ,and i n KO f o r ( i i ) (see ( 4 . 2 ) and Remark 4 . 1 , Chapter 1). -+

H,@h f o r each

(Sec.

6)

Convergence of approximations

I n a g e n e r a l manner we t a k e

U E ~ ( Q )and

VT = { triangle M,M, M, } E g

153

introduce

h ,

t h e a f f i n e l i n e a r f u n c t i o n GT on T , such t h a t

and d e f i n e u, by (6.16)

in T .

uk = 8,

It can r e a d i l y be shown, using Taylor s e r i e s expansions of order 2 ( ' ) t h a t (6.17)

I

(6.18)

I grad @Ax) - grad u(x) I d c,(u) 4 Vx E T ,

@T(x)

-

Nx) I d

cI(u)

h VX E T ,

.

where t h e c o n s t a n t s c,(u), c2(u) depend on u From t h e s e estimates and from t h e d e f i n i t i o n (6.16) w e deduce s t r a i g h t away t h a t (6.19)

uh + u

i n H,'(Q),

which gives ( i ) (on 9(Q) dense i n H,'(Q) 1. I n o r d e r t o show ( i i ) ,w e use Lemma 6.1, which allows us t o take V i n K,,nB(Q) ; s i n c e (6.20)

u =

lim(Au) i n H,'(Q), A c 1 , A + 1 ,

we can confine our a t t e n t i o n t o u with (6.21)

u E 9(Q),

I grad u(x) 1 d A < 1 .

"hen, i n a l l T s Y h we have:

and hence (6.23)

uh E KO f o r s u f f i c i e n t l y s m a l l h (such t h a t A

which, i n conjunction with F i n a l l y , we have (6.24)

uh+ u

+ cZ(u) 4Q 1 )

(6.19)~ gives ( i i ).

s t r o n g l y i n Lm(Q)

(') For a systematic study see Ciarlet-Wagschal /l/.

154

Elasto-plastic torsion of a cylindrical bar

from Remark

(CHAP. 3 )

6.1.

Remark 6.2. Falk /1/ gives an estimate of t h e approximation e r r o r as a function of h ; we should p o i n t out t h a t t h e condit i o n s of v a l i d i t y ( r e l a t i n g t o GI and Y h ) of t h i s estimate are more r e s t r i c t i v e than t h e assumptions of Section 4.1.1 and of H, Theorem 6.1. Convergence of the i n t e r i o r approximation method using the eigenfunctions of -A i n H&?)

6.2.2.

We s h a l l now study t h e convergence of t h e i n t e r i o r a?proximation 4. defined i n S e c t i o n ( 4 . 2 ) and u s i n g t h e same n o t a t i o n . We assume t h a t t h e c o n s t r a i n t 1 grad u l Q 1 i s s a t i s f i e d on t h e set defined i n Section 4.2.2, Remark 4.5, by r e l a t i o n s (4.39), (4.40). We a l s o assume t h a t t h e spectrum (BT)meN of -A has been arranged i n o r d e r of i n c r e a s i n g eigenvalue, 1.e.:

Then, with u denoting t h e s o l u t i o n o f (Po) , w e have:

AsM++co'and h + O , with

Theorem 6.2. (6.26)

hS;,

+0

, real r > 1

we have : (6.27)

t#+u

strongly i n H&?).

Proof. We apply Theorem 4.2 of Chapter 1, Section 4.4, t a k i n g t h e only p o i n t s account of Remark 4 . 1 of Chapter 1, S e c t i o n 4.3; which are not immediate are t h a t we must v e r i f y t h a t under t h e s t a t e d conditions f o r M a n d h w e have: V V E Xc K O ,

X=

K O , there exists

(6.28)

#-+u

# E K$

such t h a t

s t r o n g l y i n H,'(bl)

and (6.29)

if

# EKZ, #

4

v

weakly i n H&2)

, then

V EKO.

Verification of (6.28) We proceed as i n t h e proof of Theorem 6.1, d e f i n i n g X (6.30)

x=

by:

U L{K0n9(Q)}. 0<1<1

L e t V E X ; we define # as t h e sum of t h e f i r s t M terms o f t h e F o u r i e r s e r i e s expansion of v with r e s p e c t t o t h e eigenfunctions (w,),.~ and hence, s i n c e t h e w, are orthonormal:

(SEC. 6 )

Convergence of approximations

155

We have o E X c 9(Q)and hence:

(6.32)

(- A ) q ~ ~ H i ( PVq) 3 0 .

from which it may be deduced (see Lions-Magenes

/l/) t h a t , as

M++m:

VM + u s t r o n g l y i n H'(O),

(6.33)

Vs 2 0

=-VM + u

strongly i n

Ck(5), V k ( 1 )

and i n p a r t i c u l a r :

(6.34)

grad I? + grad u s t r o n g l y i n C"(36) , Vu E x ;

t h e r e f o r e t h e r e exi s t s q ( M ) > 0 such t h a t f o r ~ € 3 6 ,as M + + Q) with :

(6.35)

q ( M ) + 0 , uniformly

I grad # ( x ) I 6 I grad u(x) I + q ( M ) d L + q ( M ) < 1

V x €5,

provided M i s s u f f i c i e n t l y l a r g e , and i n p a r t i c u l a r (grad f l ( x ) I < 1 on Qh, and hence U ~ E K :; it w i l l t h e r e f o r e be s u f f i c i e n t t o put .;" = 8 i n o r d e r t o v e r i f y ( 6 . 2 8 ) .

Verification of ( 6 . 2 9 ) . We s h a l l proceed i n two s t a g e s : 1) We d e f i n e a family o f open s e t s Oh c Q as follows: l e t

(6.36)

i h

=

{PIPE

z:(P)

t

n}

where we r e c a l l t h a t

if

P

w e t h e n d e f i n e Oh by:

= (xl,x2);

0

(6.37)

@h

u

=

z:(P),

P.&

and we have lim(P - O,,)= 0 i n t h e following sense: V K compact h-0

inside P w e have K c Oh provided h i s s u f f i c i e n t l y s m a l l . ( l ) This assumes

r

t o be s u f f i c i e n t l y regular.

156

Elasto-pZastic torsion of a cyZindricaZ bar

(CHAP. 3 )

Consider then uf E K z such t h a t u r -* u weakly i n Hi(61) ; i f P E8, t h e r e e x i s t s P* €61, with I PP* I < h/$ and t h e mean value theorem of D i f f e r e n t i a l Calculus, a p p l i e d t o grad # gives :

grad ur(P) = grad uf(P*)

+

(6.38)

with Q E [P. P*] ,

from which w e deduce:

I n t h e following C w i l l denote various c o n s t a n t s . Since t h e f u n c t i o n # E KY and s i n c e P* €61, w e have, by d e f i n i t i o n of K/’, Igad#(P*) 1 < 1 ; w e thereby deduce, t a k i n g account of (6.39) and t h e f a c t t h a t # belongs t o Cm(61): (6.40)

I grad #(P) I S 1 + ch (1 4 IIcqfi)

vp E 0,.

From Sobolev’s theorem ( ) we have Ha@) i n j e c t i o n being continuous, and hence: (6.41)

1 grad #(P) I < 1 + Ch (1 UF ((Ha(@

Moreover i f

#

.A4..

=

1 p,

(S

c

CZ@) i f

s

> 3 , the

> 3) VPE 8 h .

w, we have:

m= 1

-

We put r = (s - 1)/2, and hence s > 3 r > 1 ; moreover, s i n c e t h e sequence q i s weakly convergent i n Hi(61) it i s bounded i n and hence from ( 6 . 4 1 ) and (6.42) we f i n a l l y o b t a i n : (6.43)

(graduf(P)IGl+Ch&

VPEO,. w

(r>1),

( I ) More p r e c i s e l y i f 61 c W N and

r

i s s u f f i c i e n t l y r e g u l a r w e have

Ha@) c Ck@) with continuous i n j e c t i o n i f s >

2

+k

(see Lions-Magenes

/l/)

(SEC. 6 )

Convergence of approximations

157

e now show t h a t under t h e c o n d i t i o n h& + O as M + +a, 2) W L e t cp E ~ ( P )cp, 2 0; i n HA@) i s i n K O . t h e weak l i m i t u of the functional:

e

(6.44)

~ + ~ ~ d l g r a d-w1)dx I

From (6.43) and t h e i s convex and s t r o n g l y continuous i n H A @ ) . f a c t t h a t support (cp) c 0, provided h i s s u f f i c i e n t l y s m a l l , we f i r s t deduce t h a t (6.45)

Jncp(I g r a d e

I-

1 ) dx

< Ch&

I I

cp dx ,

and t h e n , under t h e assumption t h a t h& + 0 as M i n t h e l i m i t , by weak lower semi-continuity:

(6.46)

[

cp(l grad u I - 1 ) dx

< C(J'.cpdx)

< lim inf

rp(l grad

+

+ 00,

w e have,

e I - 1 ) dx <

limhp', = 0 Vcp~9(P), cp 2 0 ,

which implies : (6.47)

Igradul d 1

a.e.

*UEK~,

and proves ( 6 . 2 9 ) . 8 Taking account of (6.28) we t h u s have t h e s t r o n g convergence 8 of 44 u i n H k ( P ) .

Remark 6.3. Provided r i s s u f f i c i e n t l y r e g u l a r we can deduce from (6.43) (by using A s c o l i ' s theorem) t h a t we a l s o have # + I ( strongly i n L m ( P ) , t h e proof of t h i s p r o p e r t y being immediate when 0 i s convex. 8

6.3.

Convergence o f t h e e x t e r i o r approximations

I n t h i s s e c t i o n w e s h a l l i n v e s t i g a t e t h e convergence of t h e e x t e r i o r approximations of ( P O ) defined i n Section 5 , using t h e r e s u l t s of Chapter 1, Section 5 and a number of r e s u l t s from We s h a l l follow t h e same Section 3.3 of t h e p r e s e n t chapter. procedure a.s t h a t used i n Section 3.3 f o r t h e problem ( P I ) ;t h e notation i s as i n Section 5 .

6.3.1.

Reduction of (Po,),,,, t o a variational inequality

.We r e c a l l t h a t s o l v i n g ( P O ) i s e q u i v a l e n t t o f i n d i n g u which i s a s o l u t i o n of t h e v a r i a t i o n a l i n e q u a l i t y :

EZasto-pZastic torsion of a cyZindrica2 bar

158

grad u grad ( u - u) dx 3

(6.48)

(CHAP. 3)

f ( u - u) dx , V u E K O ,

f o r which t h e approximate analogue i s :

(6.49)

i

uh

-

3

fqh(Vh

- u& dx

3

vuh E '

KG

9

uh E Kzh where u, i s t h e s o l u t i o n of(PM),,,,, 2 = 1, 2, rn = 1, 2 , 3, 4 ( s e e ( 5 . 2 3 ) ) , and where t h e symmetric b i l i n e a r forms on Vh,a: ( b1, 2 ) , a r e defined by:

(6.so)

9

uh)

=

a&h

uh a k q k

0,

dx

k= 1

+61qhuh 2

( 4)). -xl,x

(6.51)

We r e c a l l t h a t here 6.3.2.

a(u, u ) =

In

grad u.grad v dx . 8

Exterior approximations f o r H&2),

a, KO,L

We s h a l l now v e r i f y f o r each p o i n t i n t u r n (as i n Section 3.3.2) t h a t t h e p r o p e r t i e s r e q u i r e d i n Chapter 1, Section 5.2, f o r Theorem 5.2 of Chapter 1, S e c t i o n 5.3, t o be a p p l i c a b l e , are indeed s a t i s f i e d :

Approximations of H { ( O ) . We s h a l l consider i n p a r t i c u l a r two cases which correspond t o t h e two approximations of a defined i n (6.50) and (6.51) , respectively.

1) A f i r s t exterior approximation of H;(O): Apart from a s l i g h t modification of t h e n o t a t i o n , t h i s reduces ve t h u s t o t h e approximation of HA(@ s t u d i e d i n S e c t i o n 3.3.2;

Convergence o f approximations

(SEC. 6 )

159

have :

(6.52)

V = HA(Q),

(6.53)

F1 = (LZ(n))’,

and t h e n u, : V

(6.55) (6.56)

Plh

+ F,,

:v h

+

F, and t h e norm 1 I

oh = { q h uh. 6lqh uh, 62qh uh 1

11 u l h

IIIh

=

[‘lh(uhr

uh)ll”

I1 1h on

vh,

defined by:

9

’

We can t h e n apply P r o p o s i t i o n 3.1 of Section 3.3.2: under (6.56) , t h e family Vh c o n s t i t u t e s conditions ( 6 . 5 2 ) , (6.53) , an e x t e r i o r approximation of H,’(Q).

...

2)

A second exterior approximation o f H A @ ) .

We once again t a k e V = HA(Q), and then:

(6.57)

F, = (L2(W2))’,

We have : Proposition 6.1.

Under conditions ( 6 . 5 7 ) , ( 6 .58),...,

(6.61)

Elasto-plastic torsion o f a cylindrical bar

160

(CHAP. 3 )

the f a m i l y (v& constitutes an exterior approximation of Hd(61). Proof. (6.62)

W e must v e r i f y ( s e e Chapter 1, Section 5 . 2 ) :

11 P 2 h

IIY(V,,.Fz)

d

Vh

3uh E

v h

and

(6.63)

Vu E H i @ ) ,

such t h a t p Z huh + a2 u strongly i n F2

with

W e s h a l l assume f o r t h e moment:

fi

which implies ( 6 . 6 2 ) (with c = ) I n o r d e r t o v e r i n (6.63), ( 6 . 6 4 ) , as i n Section 3.3.2 w e d e f i n e rh : H i ( n ) + Vh by (3.23) and f o r D E Hi(f2). w e take

(6.68)

uh

=

rho.

We have ( s e e Chapter 1, Section 5.1)

which gives

S t i l l denoting by 5 t h e element of H'(R2) obtained by extending u by 0 o u t s i d e R , we deduce from (6.69): qh uh + 5 s t r o n g l y i n L2(W2)

s t r o n g l y i n L ~ ( W Z()k = 1 , 2 ) .

(SEC. 63

Convergence of approximations

161

I n order t o prove (6.63) from (6.711, we s h a l l use t h e following r e s u l t . Given t h e two sequences gnE Lz(R2) and (ha,,, h, E R such t h a t f o r n + + co gn + g

(6.72)

s t r o n g l y ( r e s p . weakly) i n L z ( R z )

it is immediate t h a t (6.73)

{

s~t r o nQ g l y~ (resp. weakly) i n Lz(R2), TZhmgn + g s t r o n g l y ( r e s p . weakly) i n L ~ ( R ~ ) .

T

~

+g ~

T a k i n g account of ( 6 . 7 1 ) , ( 6 . 7 2 ) , (6.73) we then ha&

i

TZ(h/Z) '1qh ' h

(6.74)

Tl(h/Z) ' Z q h ' I

+ TZ(-h/Z)

'1qh ' h +

2

+ T1(-h/Z) 2

8x1 '2qh ' h

strongly i n L2(Rz), Strongly i n Lz(Rz)

~

8x2

which w i t h (6.59) and t h e f i r s t r e l a t i o n of (6.71) proves (6.63). A s regards ( 6 . 6 4 ) , t h i s r e s u l t s from (6.511, (6.61), (6.74) or more simply from (6.56), (6.65), (6.70). Hence w e have proved Proposition 6.1, subject t o t h e : Proof of Lema 6.2 1) F i r s t , w e s h a l l prove t h e i n e q u a l i t y on t h e r i g h t ; w e have

uh) c II (k = 1, 2) ( v h and qh being d e f i n e d i n Section 3.2.1 by 43.6) and ( 3 . 7 ) , r e s p e c t i v e l y ) ; from (6.75) we deduce

w i t h support

-

Elasto-plastic torsion of a cylindrical bar

162

(CHAP. 3 )

2 ) We s h a l l now prove t h e f i r s t i n e q u a l i t y o f (6.65) ( I ) n o t i n g t h a t , i f w e f i r s t extend Qh i n t o & ( s e e ( 4 . 3 9 ) ) and c a r r y out a t r a n s l a t i o n of t h e o r i g i n of coordinates, w e can always assume t h a t vh i s generated from ahdefined by:

Gh then being defined by:

t h e associated set

moreover

lim Nh = L c

(6.79)

h+O

+ co

t h e optimal choice of L being defined by

{L 2

(6.80)

= Minimum area of t h e squares with s i d e s p a r a l l e l

t o t h e coordinate axes and c o n t a i n i n g Q .

We c l e a r l y have:

(6.81)

dim

vh

=

.

(N -

We d e f i n e t h e symmetric, p o s i t i v e - d e f i n i t e l i n e a r o p e r a t o r

- A: (an "approximation" of - A ) by: (6.82) a;k2(vh,oh) = - h2(A:Uh, o h ) vvh E vh , where, i n (6.82) , (., .) denotes t h e s t a n d a r d Euclidean i n n e r product i n

(6.83)

Ba"-')*; i n a more e x p l i c i t form we have

- (A:

uJ,,=

- (ui+lj+l +

hJ

%-1J+1

+ ui+lj-l + ui-lJ-l)

2 h2

with, i n ( 6 . 8 3 ) , u , = O

if M,#Qh.

We s h a l l now v e r i f y , using (6.83), t h a t t h e eigenvectors of

- A: i n Vh are given by: (6,84)

1

(VJrY

=

{

x sin v( j ui, I vij = sin p(i - 1) -

i , j = 2 , 3 ,..., N

I

N

x. - 1) "

w i t h f i , v = 1 , 2,..., N - 1 ,

and t h e corresponding eigenvalues by:

(l)

This i s a " d i s c r e t e PoincarC" t y p e i n e q u a l i t y .

(SEC. 6 )

Convergence of approximations

163

The s m a l l e s t e i g e n v a l u e i s t h e r e f o r e o b t a i n e d f o r p = v = 1 ( a n d p = v = N - 1) and i s e q u a l t o :

(6.86)

2 1 r A:, = -sin2 -. h2 N

I n view of ( 6 . 8 2 ) and t h e o r t h o g o n a l i t y of t h e e i g e n v e c t o r s , we t h e r e f o r e have:

and a s h - 0 , N +

(6.88)

2 x lim-sinzhz N

which w i t h ‘o

+ co =

with

( 6 . 7 9 ) we have:

2rr2 > 0, L2

(6.87) p r o v e s Lemma 6 . 2

w

Approximation of a : We have; Proposition 6.2.

The families

(a:),

and (at), constitute exterior

approximations of .a. For t h e b i l i n e a r form a: as d e f i n e d i n ( 6 . 5 0 ) , and Proof. under t h e c o n d i t i o n s ( 6 . 5 2 ) , ( 6 . 5 3 ) , . , ( 6 . 5 6 ) , t h i s h a s alr e a d y been proved ( u s i n g d i f f e r e n t n o t a t i o n ) i n S e c t i o n 3.3.2, Proposition 3.2; it t h e r e f o r e o n l y r e m a i n s t o prove it f o r under t h e c o n d i t i o n s ( 6 . 5 7 ) , ( 6.58) , ( 6.59), ( 6.61) We c l e a r l y hav’e:

...

.

4

S6.91)

{

(6.92)

i f pzh vh + u2 v weakly i n Fz w e have lim infai(uh,u& 2 a(v, D)

as p z h uh + uz D weakly i n F2 ,p2hwh -P u2 w. strongly in Fz w e have at(vh, w&

+ U(D,w )

,

.

S i n c e t h e form

(6.93)

{ ( f i , f 2 ) 3 (el, ~

1

2 )

+

s.,

(fi 61 + fz

S J dx

i s b i l i n e a r c o n t i n u o u s o v e r ( L 2 ( R 2 ) r we deduce, from ( 6 . 5 1 ) ,

( 6 . 5 8 ) , ( 6 . 5 9 ) , t h a t under t h e a s s u m p t i o n s o f c o n d i t i o n (6.91) have :

we

164

Elasto-plastic torsion of a cylindrical bar

lima:(v,, w 3 =

(6.94)

k=l

h-0

I ~

%dx z

(CHAP.

3)

=

~

~

k

(6.91). Regarding (6.92),t h i s comes from (6.511, (6.58), (6.59)and t h e weak lower semi-continuity o f t h e convex

which p r o v e s

.

and strongly continuous f u n c t i o n a l on (L2(R2))’ d e f i n e d by

(6.95)

(fl,

f2)

I

-,

(f: + fz)dx.

R’

Exterior approximations of KO. We s h a l l d i s t i n g u i s h two c a s e s depending on whether we u s e t h e norm II 111, o r t h e norm 11 (12hover V,, , t h e a p p r o x i m a t i o n o f H;(Q> We s h a l l s t a r t w i t h t h e former c a s e which i s s l i g h t l y t h e s i m p l e r o f t h e two; i n t h i s c a s e we have

.

P r o p o s i t i o n 6.3. Under conditions (6.52), (6.53), ... , (6.56), the convex sets KG ( m = 1, 2, 3 , 4) defined in Section

5.2 constitute exterior approximations of K,. It i s n e c e s s a r y t o prove ( s e e Chapter 1, S e c t i o n

Proof. f o r m = 1,

(6.96)

5.2),

2, 3, 4:

i f v,

E KG

w i t h limp,, v, = h-0

t

weakly i n Fl

, then

ce a, K O ,

and

(6.97)

VPE K O , 3 v , ~KG such t h a t p,,u,,

11 vh

IIlh

a, v s t r o n g l y i n F, and

Q c.

Verification of (6.96). We p u t

5.1, t h a t :

t = (ro,tl,t2) ;

it t h e n r e s u l t s from Chapter 1, S e c t i o n

(SEC. 6 )

Convergence of approximations

and hence 5: E u1 Hi@) . It now remains t o prove t h a t m = 1; i n f a c t :

and from

{ E U ~KO

;

165

t h i s i s immediate f o r

(6.96), (6.98):

which, i n t h e l i m i t , s i n c e t h e convex s e t

[(fl,fz)lfl,~z~~z(Q f:) ,+f2 6 1 i s closed i n (LZ(Q))’, g i v e s :

a.4

and hence, w i t h (6.98), { E u l K o . We s h a l l now t r e a t simultaneously t h e c a s e s where m = 2, 3, noting t h a t from (5.14),S e c t i o n 5.2:

4,

(6.102)

If t h e f u n c t i o n to denotes t h e element o f H1(Rz) obtained by extending to by 0 o u t s i d e Q and qhub 6fihuh (k = 1,2) are considered t o be elements of L’(R’) ( l ) we have from (6.96), (6.98),

at o ax,

weakly i n L2(Rz) (k = 1,2).

We can t h u s apply

(6.72), (6.73), which g i v e s :

(6.103) 6&h

(6.104)

uh

-b

h-o lh ?k i ( f h / Z ) skzqh uh

=

Since t h e s e t s [(fl,fz) [(&1.2.3,4

IAELW),

at o ax,,

weakly i n L2(R’).

I fl, f2EL’(R2), f: i=1,2,3,4,

+ f:

6 1 a.e] resp..

1

$ f i 6 2 a.e.

I= 1

are convex and closed i n (Lz(Rz))2(resp. (L2(R’))q , we have i n t h e limit, t a k i n g account of (5.12), (6.1021, (5.201, (6.60), (6.104): (1)

we r e c a l l t h a t s i n c e t h e supports of qhu&, 6&4 (k = 1,2) are i n Q, t h e s e f u n c t i o n s a r e considered t o be elements of L2@) or L’(R’) , as t h e c a s e may be.

166

EZasto-plastic torsion of a cyZindricaZ bar

and hence, w i t h ( 6 . 9 8 ) ,

(CHAP. 3 )

(EulKo.

Verification of ( 6.97 ) . I t i s s u f f i c i e n t ( s e e Chapter 1, S e c t i o n 5.3, Remark v e r i f y t h i s for

5.4) t o

U A{ KO n %Q) }

(6.106)

OCI<1

dense i n KO, from Lemma 6.1 ( l ) . If v b e l o n g s t o t h e s u b s e t ( 6 . 1 0 6 ) , w e d e f i n e uh by: (6* lo7)

vh

=

{

I

vij vij

=

v(Mij),

Mi, E Qh }

9

and s i n c e v i s i n B(Q)we have ( 2 ) :

i

qhvh+ v

s t r o n g l y i n L2(Q)

ax,

s t r o n g l y i n L2(Q) (k = 1,2), av t h a t i s p l h vh -+ u1 v s t r o n g l y i n Fl ; we c l e a r l y have I( vh IIlh Q c from ( 6 . 5 0 ) ~( 6 . 5 6 ) and ( 6 . 1 0 8 ) . Moreover, a T a y l o r s e r i e s e x p a n s i o n o f o r d e r 2 would show t h a t t h e r e l a t i o n I g r a d v I < A c l i n n i m p l i e s , u s i n g (5.lO), ( 5 . l l ) , ( 5 . 1 5 ) , ( 5 . 2 l ) , t h a t v , E K Z f o r rn = 1, 2, 3 , 4 p r o v i d e d h i s s u f f iciently small. Concerning t h e second e x t e r i o r a p p r o x i m a t i o n c o n s i d e r e d w e have : (6.108)

6 f i h v h-+

...

P r o p o s i t i o n 6.4 Under conditions ( 6 . 5 7 ) , ( 6 . 5 8 ) , , (6.61), the convex sets KG (rn = 1, 2, 3, 4 ) defined in Section 5 . 2 eonstitute exterior approximations of KO.

I t i s n e c e s s a r y t o p r o v e ( s e e Chapter 1, S e c t i o n 5 . 2 ) , Proof. f o r rn = 1, 2, 3 , 4 , t h a t : (6.109)

If oh E Kth w i t h lim p Z hvh= h+O

< weakly

in F2

then

< E a2 KO,

and (6.110)

V V E K ~ 3, v , ~ K ; and

(')

11 vh

II2h

such t h a t

<

PZhVh

-+

a2v

s t r o n g l y i n F2

'

U E C1(@ and w i t h compact s u p p o r t i n

LI would b e s u f f i c i e n t .

( 2 ) I n f a c t t h e r e i s s t r o n g convergence i n L"(LI)

(SEC. 6 )

167

Convergence of approximations

Verification of ( 6 . 1 0 9 ) We p u t

(6.11 1)

4

=

r,, r,)

(to,

; w e t h e n have, from

(6.59);

qh uh + to weakly i n L2(R2) and f o r k,,k, = 1,2, k, # k, %'k

-k

1(h/2) 6k,qh

' k l ( - h / Z ) 6k,qh

-b

{k2

weakly i n L2(R2)

We s h a l l t r e a t t h e c a s e rn = 4 s e p a r a t e l y from t h e t h r e e o t h e r s : 0

Verification of ( 6 . 1 0 9 ) f o r rn = 1, 2 , 3 .

S i n c e t h e s u p p o r t o f 6dq,uh(k = 1,2) i s c o n t a i n e d i n R, it c a n e a s i l y b e shown t h a t t h e membership c o n d i t i o n s f o r K S ( s e e ( 5 . 9 ) f o r rn = 1, ( 5 . 1 2 ) f o r rn = 2, ( 5 . 1 4 ) and ( 6 . 1 0 2 ) f o r rn = 3 ) imply:

and hence, from Lemma 6 . 2 ( o r from S e c t i o n 3 . 3 . 2 ,

(6.113)

IIPlhUh

11 d c

relation (3.22))

i n Fl (= L2(a))3)v h .

T h e r e f o r e t h e r e e x i s t s a n e x t r a c t e d subsequence, a l s o d e n o t e d by uh and E F ] , such t h a t i f t* = 5:) , w e have:

.r*

{,"

(6.114)

(rt, c,

"+

"

6 k q h oh +

weakly i n L2(R) weakly i n L,(R).

From P r o p o s i t i o n 6 . 3 ,

( 6 . 1 1 4 ) and U ~ E K Simply:

Applying t h e t r a n s l a t i o n r e s u l t s ( 6 . 7 2 ) , ( 6 . 7 3 ) we deduce from (6.111), ( 6 . 1 1 4 ) , (6.115), t h a t :

where ?$ d e n o t e s t h e element o f H1(R2) o b t a i n e d by e x t e n d i n g 5: by z e r o o u t s i d e a ; from (6.116) and t h e f i r s t r e l a t i o n o f (6.11.5) we t h u s have ( E a, KO.

Verification of ( 6 . l o g ) f o r rn = 4 . The f i r s t r e l a t i o n o f

(6.117)

qhvh +

to

in

(6.111) i m p l i e s ( a f o r t i o r i ) :

B'(Wz)

168

E l a s t o p l a s t i c torsion of a cylindrical bar

(CHAP. 3 )

and hence as may b e v e r i f i e d immediately:

(kl, k2 = 1 , 2 ) , which, t o g e t h e r w i t h t h e second r e l a t i o n o f

(6.119)


=

at0 dx,

(6.111), i m p l i e s :

(k = 1 , 2 ) .

We t h e r e f o r e have :

(6.120)

toE H1(W2).

+

Moreover, t h e convex s e t [(fl, fz) 1 f,,f 2 E L2(R2),f : f: d 1 a .e .] b e i n g c l o s e d i n [ L 2 ( R 2 ) ] 2 , t h e second r e l a t i o n o f (6.111), and t h e r e l a t i o n e x p r e s s i n g t h e membership o f V, t o K& ( s e e ( 5 . 2 0 ) ) and (6.119) imply t h a t i n t h e l i m i t w e have:

~ is sufficient, i n I n o r d e r t o d e m o n s t r a t e t h a t < E U ~ K it view o f ( 6 . 1 2 0 ) and ( 6 . 1 2 1 ) , t o show t h a t to i s z e r o o u t s i d e f2; t h e proof o f t h i s i s immediate. I n d e e d , s i n c e q h u k h a s i t s s u p p o r t i n f2, it b e l o n g s t o t h e closed convex subset of L 2 ( R 2 ) d e f i n e d by:

(6.122)

[gIgEL2(R2), g = O

a.e.

in R2-s]

t h e f i r s t r e l a t i o n o f (6.111) t h e r e f o r e i m p l i e s t h a t , i n t h e l i m i t , to b e l o n g s t o t h e convex s e t ( 6 . 1 2 2 ) . 0

v e r i f i c a t i o n of ( 6 .uo).

As i n t h e proof o f t h e second p a r t o f P r o p o s i t i o n 6.3, we def i n e vh u s i n g t h e s e t ( 6 . 1 0 6 ) ; hence, p r o v i d e d h i s s u f f i c i e n t l y s m a l l , U ~ KG E Vm = 1 , 2 , 3 , 4 Furthermore (6.108) i m p l i e s , from (6.721, ( 6 . 7 3 ) t h a t :

.

[ qh uh +

(6.123)

1’

and hence

s t r o n g l y i n L2(W)

3 ( T k 1 ( y 2 ) ak,qh

uh

+ Tka(-h/2)

6k,qh

s t r o n g l y i n L2(W2)

u&

a5

+

ax,,

(kl, k, = 1,2),

(SEC. 6)

169

Convergence of approximations

Approximation of L. With t h e f u n c t i o n a l L : Hd(Q) + R defined by L(v) where f’EL’(R),and

4 : V, -+

R by Lh(u&=

=L

fvdx,

fq, u, dx, t h e v e r i f i c a t i o n of

( 5 . 3 7 ) , ( 5 . 3 8 ) of Chapter 1, S e c t i o n 5.2 i s immediate f o r t h e two types of approximation of HA considered. rn

Strong convergence results in F,,Fl.

6.3.3

I n view of t h e r e s u l t s proved i n S e c t i o n 6.3.2 f o r t h e exteri o r approximations of H;(Q), a, K,,,L , we can apply Theorem 5.2 of Chapter 1, Section 5.4 t o t h e problems (P&,(l= 1,2 ; m = 1, 2, 3, 4) and more p r e c i s e l y t o t h e equivalent formulations (6.49); t h i s gives :

Let Theorem 6.3 ution of (Po) ; as h

6 -+

be the solutions of we have:

o

1) Under conditions ( 6 . 5 2 ) , (6.531,

i

p l h u z + a1 u J,’(
(6..125)

(pOh)lm,

and u the sol-

.. .

, (6.56) :

...

, (6.61):

strongly in F1 9

Vm = 1, 2, 3 , 4 ;

2)

Under conditions (6.571, (6.581, Plh

e h

+

strongly ’in F,

al u

Vm = 1, 2 , 3 , 4 .

Remark 6.4. I n view of ( 6 . 7 2 ) , ( 6 . 7 3 ) , t h e convergence resu l t (6.125) implies:

(6- 127)

6.3.4

[

,

strongly in Fl

p,, l/;, + a, u J,’(

+

J(U)

9

Vm = 1,2, 3 , 4 .

Uniform convergence results

I n t h i s s e c t i o n w e s h a l l prove the uniform convergence of qhu”, to u , f o r 1 = 1, 2 and m = 1, 2, 3 , 4 ; t h e proofs a r e equa l l y v a l i d f o r I = 1, 2; i n c o n t r a s t , however, we s h a l l t r e a t t h e case’m = 4 s e p a r a t e l y from t h e o t h e r t h r e e c a s e s . A s i n t h e proof of t h e second p a r t of Lemma 6.2 we can assume t h a t V, i s generated from Qh defined by (6.77) and s a t i s f i e s (6.78), ( 6 . 7 9 ) , ( 6 . 8 0 ) ; t h u s , with Ohdefined by (5.13), we have in t h i s p a r t i c u l a r c a s e :

’

170 (6.128)

E l a s t o p l a s t i c torsion of a c y l i n d r i c a l bar

(CHAP. 3 )

0, = p, Nh[ x 10, Nh[ .

We c o n s i d e r o n

t h e t r i a n g u l a t i o n Zh shown i n F i g u r e 6.1:

and we d e f i n e

(We r e c a l l t h a t vij = O

if

M i j $ ah

).

We t h e n have:

Lemma 6.3. For m = 1, 2, 3 t h e s e t nhK&i s made up of functions which are bounded and Lipschitz on o h , uniformly i n h , and:

(6.130)

1) q h vh - nh vh 1-'

(Br)

Q Ch ,

C independent of h.

Proof. The g r a d i e n t o f n h v h i s c o n s t a n t a v e r each triangle o f Fhand t h e c o n d i t i o n s e x p r e s s i n g t h e membership o f Oh t o KG ( s e e S e c t i o n 5.2) imply:

which w i t h t h e c o n d i t i o n s

lim Nh = L c

+ co

and

nhvh = 0 on

88,

h-0

N-.+m

i m p l i e s t h a t t h e f u n c t i o n s nh 0, b e l o n g i n g t o nhK& (m = 1, 2, 3) a r e bounded and L i p s c h i t z , uniformly i n h. S i n c e t h e f u n c t i o n qhvh i s p i e c e w i s e c o n s t a n t and e q u a l t o vij on t h e p a n e l u$(M,~)it c a n b e shown, from ( 6 . 1 3 1 ) , t h a t we have:

(SEC. 6 )

which more p r e c i s e l y d e f i n e s , and proves, From Lemma 6.3, we deduce:

(6.130).

Theorem 6.4. Let 5 be the solution of ution of (Po) ; as h -* 0 we have: (6.133)

171

Convergence of approximations

,,+ u

{

8

and u the sol-

strongly in Lrn(Q)

V I = 1,2 and m = 1 . 2 , 3 .

a

c 0, ( a s i t u a t i o n which can always be Proof. We assume t h a t from Lemma 6.3, w e can applyAscoli's achieved by e x t e n s i o n of a,); Theorem t o t h e r e s t r i c t i o n of n,Um, t o 3 , and hence as h + 0 and i n view o f Theorem 6 . 3 and (6.132) w e g e t :

t h e r e s t r i c t i o n of ll, 4on q, 4 , +u V l = 1,2

a

converges strongly to u in Lrn(Q)

s t r o n g l y i n Lrn(Q) and m = 1 , 2 , 3 . 8

We s h a l l now prove a n analogous r e s u l t f o r ui. We i n t r o d u c e t h e two d i s j o i n t s u b s e t s o f R,, :

(6.135)

{ RZh

R , , = { Mii I Mii E R,, P a r i t y of i # p a r i t y of j } , = { Mii I Mii E R,, p a r i t y of i = p a r i t y of j } .

We t h u s have

R, = R,h

+

R2,.

We next i n t r o d u c e (6.136)

{

(6.137)

xii =

uh(Mij)= t h e open p a n e l w i t h ' c e n t r e M, and s i d e s of length

h f i

p a r a l l e l t o t h e b i s e c t o r s of

c h a r a c t e r i s t i c function of

x, Ox,,

uh(Mi,).

It may be noted t h a t

IJ a,(Mij) = R2

(6.138)

(r = 1, 2)

ER h

(6.139)

(l)

wt(Mij)c w:(MiJ

uh(Mi])

c wi(Mii)

( )

( s e e F i g u r e 6.2)

was d e f i n e d i n Chapter 1, S e c t i o n 5.1, r e l a t i o n ( 5 . 7 ' ) .

172

EZasto-plastic torsion of a cylindrical bar

Fig. 6.2.

R e p r e s e n t a t i o n of wf(Mf,), o#,(Mfj),w&+li,) ( wf(Mi,) i s shown h a t c h e d )

We t h e n have:

1) v e r i f i c a t i o n of ( 6 . 1 4 1 ) . We have ( V r = 1.2) :

from which, u s i n g t h e Schwarz i n e q u a l i t y , w e g e t :

(CHAP. 3 )

(SEC. 6)

2)

Convergence of approximations

Verification of ( 6 . 1 4 2 )

173

.

For g E O ( R z ) r e l a t i o n ( 6 . 1 4 2 ) i s immediate, u s i n g t h e argument of uniform c o n t i n u i t y on t h e s u p p o r t of g ; s i n c e 9(R2) is dense i n L z ( R 2 ) , and t a k i n g account o f (6.141), ( 6 . 1 4 2 ) i s t r u e i n LZ(R2). From t h i s we deduce:

.

C o r o l l a r y 6.1.

Lz(Rz) as

m+

+

00

Consider (SJm such that gm + g strongly in ; then as h - r 0,m + 00 we have Vq = 1,2 strongly in L 2 ( R 2 )

I,#,(S,)+g

vq = 1,2.

Before proving t h e uniform convergence of qhui t o u (1 = 1,2) w e s h a l l i n t r o d u c e ( b y analogy w i t h t h e c a s e s rn = 1, 2 , 3 ) :

(6.144)

q r h Oh

=

c

M,,6 nrh

VijdJ:'.

I t may be noted t h a t , from t h e d e f i n i t i o n o f Ph , we have w:(Mij) c P i f M U e P h; i n view o f ( 6 . 1 3 9 ) , w e t h e r e f o r e have: (6.145)

suppofi (qrh

oh)

c

n vt+, E vh ;

= 1, 2

f

We a l s o d e f i n e t h e t r i a n g u l a t i o n 9,+, (r = 1,2) a s shown i n F i g u r e 6.3 over&eaf, and

We t h e n have:

Lemma 6.5. The set nrhK$, is made up of functions which are bounded and Lipschitz on Wz , uniformly in h, and:

:

174

Elasto-plastic torsion of a cyZindrical bar

Fig. 6.3.

Definition of t h e triangulation

(CHAP. 3 )

irh.

Proof. We go back t o Lemma 6 . 3 ( b y r o t a t i n g t h r o u g h a14 and by r e p l a c i n g h by h $ ) , n o t i n g t h a t t h e membership c o n d i t i o n s f o r K& ( s e e S e c t i o n 5 . 2 ) imply (6.148)

( g r a d n r h u h< (

$

a.e.

VUhEKG;

r = 1,2.

From Lemma 6 . 5 , w e deduce: Theorem 6 . 5 . Let u: be t h e solution o f solution of (Po) ; as h + 0 we have:

(6.149)

{ $2. ; ;

(Poh)ld

, and

u the

strongly i n Lm(62)

Proof. 1) From Theorem 6 . 3 , q h u i + u s t r o n g l y i n L'(62) as h + 0; by a p p l y i n g C o r o l l a r y 6 . 1 o f Lemma 6 . 4 , and t a k i n g a c c o u n t o f ( 6 . 1 4 5 ) :

From Lemma 6 . 5 , w e c a n a p p l y A s c o l i ' s Theorem t o t h e res2) s t r i c t i o n t o 62 o f n,hui ; t h e r e f o r e , from ( 6 . 1 4 7 ) , ( 6 . 1 5 0 ) , we have :

7)

(SEC.

Solution of ( P o )by r e l a x a t i o n

from which, by restriction t o

175

IJ w&Ui3 ( r = 1,2) and reassembling, Mi, E %h

we o b t a i n t h e r e s u l t ( 6 . 1 4 9 ) .

7.

H

NUMERICAL SOLUTION OF THE APPROXIMATE PROBLEMS RELATING TO ( P o ) , BY RELAXATION AND POINT OVER-RELAXATION WITH SEQUENTIAL PROJECTION

7.1

Synopsis

I n t h i s s e c t i o n we s h a l l s t u d y t h e a p p l i c a t i o n o f t h e relaxation and over-relaxation methods o f Chapter 2, S e c t i o n 1, t o t h e numerical s o l u t i o n o f t h e f i n i t e - d i m e n s i o n a l problems o b t a i n e d by a p p l y i n g t o (P,,) t h e a p p r o x i m a t i o n s o f S e c t i o n s 4 . 1 and 5 ( l ) . We s h o u l d p o i n t o u t from t h e s t a r t t h a t o n l y i n t h e s p e c i a l c a s e of S e c t i o n 7 . 3 . 2 h a s convergence been proved f o r t h e a l g o r i t h m which w i l l b e d e f i n e d i n S e c t i o n 7 . 2 , s i n c e t h e convex sets a p p r o x i m a t i n g KO are n o t " l o c a l " i n t h e s e n s e o f t h e d e f i n i t i o n g i v e n i n Chapter 2, S e c t i o n 1 . 2 ( o n t h i s s u b j e c t , s e e CounterExample 1 . 2 o f Chapter 2 , S e c t i o n 1 . 2 ) . N o n e t h e l e s s , we t h i n k it p r o p e r t o d e s c r i b e t h i s method, inasmuch as i n t h e approximate s o l u t i o n o f (Po) , t h e r e s u l t s o b t a i n e d c o i n c i d e ( 2, w i t h t h o s e o b t a i n e d by o t h e r methods f o r which convergence h a s been e s t a b H lished ( 3 ) .

7.2

D e s c r i p t i o n o f t h e method

We u s e t h e p r o c e d u r e d e s c r i b e d i n Chapter 2 , S e c t i o n 1 . 2 , when t h e convex s e t K , i s n o t l o c a l .

7.2.1

Case of the interior approximation (by finite elements) of Section 4 .I.

We u s e t h e n o t a t i o n o f S e c t i o n

4.1.

We r e c a l l t h a t t h e

( l ) A s f a r as t h e a p p r o x i m a t i o n i n S e c t i o n 4 . 2 i s c o n c e r n e d , t h e f o l l o w i n g method i s n o t a p p l i c a b l e i n view o f t h e " g l o b a l " c h a r a c t e r o f t h e a p p r o x i m a t i o n o f KO ( s e e 4.2.5, Remark 4 . 6 ) . ( 2 ) A t least i n t h e examples which w e have c o n s i d e r e d . ( 3 ) For t h e r a t e o f convergence, s e e Remark 7.14.

176

Elastoplastic torsion of a cylindrical bar

(CHAP. 3 )

approximate problem d e f i n e d i n Sectloon 4 . 1 r e d u c e s t o a quadratic programming problem i n @ (N = Card (Z,))w i t h N' = Card (Y,J quadratic constraints; t h e v a r i a b l e s a r e t h e v a l u z s v,taken by t h e f u n c t i o n s v, (v,,E V,,c Hi@)) a t t h e nodes M E Zh I n t h e f o l l o w i n g , v, w i l l have i t s u s u a l meaning o r w i l l den-

.

o t e the vector ( l )

(uy)ME& ; by way o f s i m p l i f i c a t i o n we s h a l l

.. .

.

, N and p u t ui = vMr= v,,(Mi) index t h e M e & by i = I, The a l g o r i t h m o f r e l a x a t i o n w i t h p r o j e c t i o n i s d e f i n e d by u Z e K o n V,

(7.1)

and assuming f o r i = 1,

4

...

for

(7.3)

0,

3

chosen a r b i t r a r i l y (

t o b e known, N, by:

4''

uf

= 0 for example),

i s determined progressively,

such t h a t : {u:"

,..., #?;,ui,#+'

,..., I I ; ; } e K , , n V,

with

(7.4)

{ u;",

...,#+',#+', ..., 4 } E KO n V,,.

From a p r a c t i c a l p o i n t o f view we s h a l l make t h e f o l l o w i n g remarks w i t h r e g a r d t o a l g o r i t h m (7.11, , (7.4):

...

Remark 7.1. The number o f c o n s t r a i n t s t o b e t a k e n i n t o a c c o u n t i n t h e d e t e r m i n a t i o n o f #+' is e q u a l t o t h e number of triangles w i t h Mi a s v e r t e x ( s e e F i g u r e 7 .l)( * )

Fig. 7.1.

= r,v, where r,, is a n isomorphism

( )

R i g o r o u s l y , we have from oh + RN.

(2)

Moreover, t h e i r u n i o n i s t h e s u p p o r t o f wMl ( s e e S e c t i o n r e l a t i o n (4.9) f o r t h e d e f i n i t i o n o f w M ) .

(uM),,ih

4.1.1,

Solution o f ( P ) by r e l a x a t i o n

(SEC. 7 )

177

0

We s h a l l u s e t h e f o l l o w i n g n o t a t i o n :

(7.5)

J, = { T I T E J ~M, , i s a v e r t e x of T }

(7.6)

N,= Card (Fa) . The c o n d i t i o n s ( 7 . 3 ) and ( 7 . 4 ) imply t h a t u, belong t o a closed interval o f R which varies w i t h n , i n p r a c t i c e , & i s determined by n o t i n g t h a t

Remark 7 . 2 .

and 4" s a y K;;

(7.7)

K; = n I; r C rlh

where i n ( 7 . 7 ) I;! i s t h e cZosed i n t e r v a l of R d e f i n e d by:

where

I n ( 7 . 8 ) , t h e r e l a t i o n Igrad'$' Iz < 1 i s made e x p l i c i t , as a , by means of r e l a t i o n (4.27) f u n c t i o n of t h e c o o r d i n a t e s o f of S e c t i o n 4.1.4. It may b e noted t h a t t h e c o n d i t i o n u,"EK,,n V, l e a d s , progressively, t o K ; # 0 since 4 ~ K f .

'I$+'

Remark 7 . 3 . We c a l c u l a t e U;" as i n Chapter 2, S e c t i o n 1 . 2 ; t h u s we f i r s t d e f i n e G+'I2 as b e i n g t h e s o l u t i o n ( u n i q u e by s t r i c t monotonicity) o f : (7.10)

dJ (u;+', ..., G?;,

U;+'l2,

air,

U;+', ..., 4) = 0

(i = 1,

*

..., N ),

then:

(7.11)

4"

=

fir(#+'")

with:

(7.12) if

Kf

Sr( x ) = max (4,min (&, x)) Vx E R = [a;,bfl. rn

Remark 7.4. We s h a l l a l s o u s e t h e f o l l o w i n g v a r i a n t of t h e a l g o r i t h m (7.1), , ( 7 . 4 ) : t h e r e l a t i o n s (7.1), ( 7 . 4 ) rema i n i n g t h e same, we d e t e r m i n e U;+l/z u s i n g ( 7 . 1 0 ) and 4'' u s i n g :

...

(7.13)

u:+'

=

PEp((l- o)U;

+ oU;+'/~)

178

Elasto-plastic. torsion of a cylindrical bar

which i s of t h e t y p e o v e r - r e l a x a t i o n p r o j e c t i o n ( o n K,! )

.

7.2.2

(CHAP. 3)

( o r under-relaxation)

with

The case of the exterior approximations of Section 5

We u s e t h e n o t a t i o n of S e c t i o n

5;

t h e problems

(P&,,(l=1,2;

rn = 1, 2, 3 , 4) a r e quadratic programming problems i n WY (N = Card ( Q , ) ) ; t h e constraints a r e a l s o quadratic and t h e r e a r e approximThe a l g o r a t e l y 4N c o n s t r a i n t s f o r rn = 1 and N f o r rn = 2, 3, 4. ithm of r e l a x a t i o n w i t h p r o j e c t i o n ( a v a r i a n t of t h e a l g o r i t h m (7.l), , (7.4))a s s o c i a t e d w i t h (P&,,, i s t h e n d e f i n e d by:

...

(7.14)

u;k”E K

z,

chosen a r b i t r a r i l y ( f o r example uz = O), t h e n assuming 4 t o b e i s d e t e r m i n e d p r o g r e s s i v e l y , f o r M , , E R ~ by: known, #,+I

f o r a l l urj such t h a t

with

assuming t h a t t h e i t e r a t i o n s a r e performed w i t h i,j i n c r e a s i n g . The f o l l o w i n g remarks may b e made w i t h r e g a r d t o t h e a l g o r i t h m

(7.14), ... , (7.17):

Remark 7.5. From t h e d e f i n i t i o n of K Z (m = 1,2,3,4) ( s e e S e c t i o n 5.2) t h e number of c o n s t r a i n t s t o b e t a k e n i n t o a c c o u n t i n i s e q u a l t o N, (rn = 1, 2, 3, 4) w i t h : t h e d e t e r m i n a t i o n o f I#’

(7.18)

IN1 = N2= N3= IN4=

12 4 5 4.

Remark 7.6. C o n d i t i o n s (7.16), (7.17), imply t h a t ui, and #I’ belong t o a c l o s e d i n t e r v a l o f W which v a r i e s w i t h n, s a y K;; i n p r a c t i c e , K; i s d e t e r m i n e d by t a k i n g t h e i n t e r s e c t i o n o f t h e N, i n t e r v a l s o f W d e t e r m i n e d from t h e N, c o n s t r a i n t s which explicitly f o r t h e s e c a l c u l a t i o n s we s h a l l u s e t h e f o r m u l a s i n v o l v e uii ; ( 5 . 1 0 ) f o r rn = 1, (5.11) f o r rn = 2, (5.15), (5.16), (5.18) f o r rn = 3 and (5.21) f o r rn = 4. We n o t e t h a t t h e c o n d i t i o n u , ” E K G l e a d s p r o g r e s s i v e l y t o

K;; # 0

0

,

since

#,EK~;..

Remark 7.7. Proceeding a s i n S e c t i o n 7.2.1, , t h e s o l u t i o n of a t e d by f i r s t determining

#T1/2

aJ,:

(7.19)

179

Solution of ( P ) by r e l a x a t i o n

(SEC. 7 )

... # ~ : j , # ~ l ~ , # ’ l / z , @ + l j , # j + l ,

-(U;;l,

ao,,

q’l

i s calcul-

...) = 0

VMijE a,,,

giving:

it being assumed i n r e l a t i o n (7.2O)2 t h a t t h e t r a v e r s e i s f i r s t c a r r i e d o u t w i t h i n c r e a s i n g j , and t h e n w i t h j f i x e d and i n c r e a s ( 7 . 2 0 ) 2 t), = 0 i f M , 4 a, We t h e n determine: i n g i; i n ( 7 . 2 0 )

.

(7.21)

#T1 = PK,(#,’1/2).

I n t h e c a s e i n which a n o v e r - r e l a x a t i o n ( o r u n d e r - r e l a x a t i o n ) parameter w is used, ( 7 . 2 1 ) i s r e p l a c e d by:

(7.22)

= PKr,((l -

~ ) # j

+ w#j+’”).

Remark 7.8. I t f o l l o w s from (7.18) t h a t t h e u s e o f K& l e a d s t o a t l e a s t t w i c e as many c o n s t r a i n t s as t h e u s e o f KG ( m = 2, 3, 4 ) ; t h e u s e of t h e s e l a s t t h r e e convex s e t s i s t h u s p r e f e r r e d f o r s o l v i n g (P,,), i n an approximate manner, by r e l a x a t i o n with w projection.

7.3

Convergence r e s u l t s

A s was p o i n t e d o u t i n S e c t i o n 7.1, t h e problem o f t h e converg, ( 7 . 4 ) and (7.14), , (7.17) t o ence of a l g o r i t h m s (7.1), t h e s o l u t i o n s o f t h e corresponding approximate problems seems t o be open f o r t h e time b e i n g . A r e s u l t concerning t h e behaviour of t h e sequence 4 when f 3 0 and u:=O w i l l be g i v e n i n S e c t i o n 7.3.1; u s i n g t h e same assumptions f o r f and ut t h e convergence of t h e a l g o r i t h m , when $2 i s one-dimensional, w i l l b e proved i n Section 7.3.2.

...

7.3.1

...

A monotonic convergence result

Definition

7.1.

Given two v e c t o r s u and u e p w e say t h a t

Elasto-plastic torsion of a c y l i n d r i c a l bar

180

(CHAP. 3 )

P r o p o s i t i o n 7.1. I f f 3 0 and u: = 0, then when n + + 00 t h e sequence 4 defined by the algorithms ( 7 . 1 ) - ( 7 . 4 ) and (7.14)( 7 . 1 7 ) converges monotonically t o ut from below, i n the sense of Definition 7.1 ( 1 ) .

Proof.

We have:

I( 0, (1

(7.23)

Q C V v , E K , , n V, o r

K&

;

it i s t h u s s u f f i c i e n t t o prove t h a t

46 4+'

(7.24)

Vn 3 0 .

T h i s w i l l f i r s t b e proved f o r a l g o r i t h m r e l a t i n g t o t h e problems (P&,, : 1)

(7.14), . . . , (7.17)

Proof of ( 7 . 2 4 ) for algorithm ( 7 . 1 4 ) ,

,..

, (7.17).

By d e f i n i t i o n , w e have

thus

f 3 0 * A j 2 0 VMijE a,.

With t h e n o t a t i o n

we s h a l l prove t h a t we a l s o have:

We a r g u e by r e c u r r e n c e , showing t h a t ( 7 . 2 4 ) , ( 7 . 2 6 ) b e i n g t r u e up t o order n - 1 implies t h a t they a r e t r u e for order n . This p r o p e r t y i s i t s e l f proved by r e c u r r e n c e by assuming:

(7.27)

{

$il

f o r t h e coordinates of flM Q calculated befire

4"

4".

We have :

(l)

I n t h e c a s e of a l g o r i t h m ( 7 . 1 ) - ( 7 . 4 ) it is a l s o n e c e s s a r y t o assume t h a t t h e angles of t h e t r i a n g u l a t i o n Share Qx/2.

Solution of ,(Po)by relaxation

(SEC. 7 ) (7.28),

U;,?’/’ = i[U;?:j+ U;pl+ U;+l,

+ uYj+‘ + h2fu]

181

if

1 = 1

so t h a t , t a k i n g account of t h e above two r e c u r r e n c e hypotheses

Moreover :

so t h a t w i t h ( 7 . 2 9 ) :

(7.30)

4 ’;

2

4,.

I t may e a s i l y be shown t h a t ( 7 . 2 9 ) , ( 7 . 3 0 ) a r e t r u e f o r t h e G+112and #+’ and hence ( 7 . 2 4 ) , (7.26), (7.27) follow. I t t h u s remains t o be proved t h a t ( 7 . 2 4 ) , ( 7 . 2 6 ) a r e t r u e f o r n = 0: we have ~ : ( ~ = h ~i ff ~1 , =, 1, u : ( ’ = 2 h 2 f , , i f 1 = 2 , so t h a t w i t h 0 = uyl E K:, and f l l 2 0,

f i r s t c a l c u l a t e 3 coordinate of

(7.31)

u:(’ 2 uil 2 uy1 = 0 .

Using a step-by-step argument s t a r t i n g from ( 7 . 3 1 ) , and bearVM,,EQ, and t h a t 0 = u8.K; w e deduce ing i n mind t h a t f,>,O, from t h e formulas ( 7 . 2 8 ) , ( 1 = 1, 2 ) t h a t :

(7.32)

u:/’ 2

U:

2

U: = 0 ,

which completes t h e proof of ( 7 . 2 4 ) . 2)

Proof of ( 7 . 2 4 ) f o r a l g o r i t h m ( 7 . 1 ) ,

... , ( 7 . 4 ) .

We have

where i n (7.33) t h e v, r e l a t e t o t h e N, v e r t i c e s neighbouring M,, i . e . t h e v e r t i c e s of t h e t r i a n g l e s which have MI as comhon v e r t e x ( s e e Figure 7 . 1 ) . It may e a s i l y be shown ( i f t h e a n g l e s of r h are < rr/2 ) t h a t :

182

Elasto-plastic torsion of a c y l i n d r i c a l bar

(7.36)

f3 0* A 3 0.

(CHAP. 3 )

.

Taking a c c o u n t o f ( 7 . 3 3 ) , ( 7 . 3 4 ) , ( 7 . 3 6 ) , t h e proof o f ( 7 . 2 4 ) , ( 7 . 4 ) may b e a c h i e v e d by p r o c e e d i n g as above f o r a l g o r i t h m ( 7 . 1 4 ) , , (7.17).

f o r algorithm ( 7 . l ) ,

...

...

Remark 7 . 9 . The v e c t o r 4 a p p e a r i n g i n t h e p r e c e d i n g proposi t i o n i s a “blockage p o i n t ” i n t h e s e n s e o f t h e Counter-Examples 1.1 and 1 . 2 o f Chapter 2, S e c t i o n 1, f o r t h e a l g o r i t h m considered.. Remark 7 .lo. The r e s u l t s t a t e d i n P r o p o s i t i o n 7 . 1 s t i l l h o l d s i f we u s e t h e v a r i a n t ( 7 . 1 ) , ( 7 .lo), ( 7 . 1 3 ) ( r e s p . ( 7 . 1 4 ) , ( 7 . 1 9 ) , (7.22)) of algorithm ( 7 . 1 ) , , (7.4) (resp. (7.14) (7.1711, w i t h O < o < 1.

...

,...,

A particular case i n which the algorithm of relaxation

7.3.2

with projection converges t o the exact solution of t h e approximate problem Statement o f the problem and approximation. I n t h e p a r t i c u l a r c a s e f2 = ]0,1[

and we t a k e fcL’(0,l) ulat i o n :

i n (7.37);

we d e f i n e J : H i ( 0 , l ) + R by:

we t h u s have f o r (Po) t h e form-

Min J(u) ,

{ w e b

(7.38)

KO = [U 1 u E H ~ ( OI),, I u’ I < 1 a . e . on (0, l)] .

+

With N a p o s i t i v e i n t e g e r (which w i l l t e n d towards a),we p u t h = 1 / N and d e f i n e t h e i n t e r i o r approximations o f Hi(f2) and KO by:

(7.40)

K,

= [u,,

I uh E

W r i t i n g ; ui = ud(i

v h

, 1 u; I < 1 a.eJ .

- 1)h)

(i = 1,

..., N

+ 1)

we have:

183

We approximate J by Jk : v h + R d e f i n e d by:

w i t h ( f o r example):

(7.43)

I

f,=q (1 -4P

f(x)&,

- #M f , = f((i - 1) h) if (1

or

f c Co[O,11.

The e x p l i c i t form o f ( 7 . 4 2 ) i s g i v e n by

we d e f i n e t h e approximate problem (P,,)h by

(7.44)

Min Jh(uJ. V h G b

By a p p l y i n g t h e r e s u l t s of t h e preceding s e c t i o n s t o t h i s e l e mentary p a r t i c u l a r c a s e , it may b e shown t h a t (7.44) admits one and o n l y one s o l u t i o n , uh , such t h a t , when h + 0, u + u strongly i n H'(0,l) nLm(O,l) where u i s t h e ( u n i q u e ) s o l u t i o n of (7.38).

Solution of ( 7 . 4 4 ) by point relaxation with projection. I n t h e c a s e of problem ( 7 . 4 4 ) , and i f t h e i t e r a t i o n s a r e performed w i t h i n c r e a s i n g i , t h e a l g o r i t h m d e s c r i b e d i n S e c t i o n 7.2 t a k e s t h e form:

[ U: cKak chosen a r b i t r a r i l y (u:

The i n t e r v a l

K;

= 0 f o r example),

i s d e f i n e d e x p l i c i t l y by:

184

E l a s t o p l a s t i c torsion of a cylindrical bar

(CHAP. 3)

Convergence o f algorithm ( 7 . 4 5 ) i f f 2 0 and u: = 0. It w i l l be assumed i n t h e f o l l o w i n g t h a t f 2 0 and u: = 0 , so t h a t 5; 2 0 from ( 7 . 4 3 ) . The r e s u l t o f P r o p o s i t i o n 7 . 1 (which a l s o 'holds f o r a l g o r i t h m ( 7 . 4 5 ) ) w i l l b e supplemented by showing t h a t 4 = ub ; t h e proof o f t h i s p o i n t w i l l r e s u l t from t h e following p r o p o s i t i o n s . P r o p o s i t i o n 7.2.

We have:

the sequences

(7.47)

(7.48)

4+112

(7.49)

GEKw

Proof.

2

G+'

4

and

#+'I2

Vn 2 0 ,

Vn20.

See t h a t of P r o p o s i t i o n

Proposition 7.3.

have : (7.50)

uh 2 0

(7.51)

%ui+'

are increasing,

7 .l.

If rc, i s the (unique) solution of ( 7 . 4 4 ) we

+ ui-l) d u, d t(h2fi + ui,+' + u i - d ,

(2 d i d N).

Proof. 1)

Proof of ( 7 . 5 0 )

Let % be t h e s o l u t i o n o f ( 7 . 4 4 ) ; we d e f i n e

iih((i - 1 ) h ) = I u, I , i = 1,

(7.52) From

fi 2 0

i= 1

+ 1.

I I ui I - 1 ui-l I I 6 I ui - ui-l I

(IUi+I\-

(7.54)

so t h a t

and

..., N

IUiI)'

2N 15I

(

Ul+1

w e deduce

- ui) 2 ,

,'k

iih E

iih 2 0 by

So Zution of (Po) by r e Zaxation

(SEC. 7 ) 2)

Proof o f ( 7 . 5 1 ) .

L e t Ei = l(hzfi defined by: (7.57)

185

Ki

= [ui

+ u i + l + ui-l)

I ui E R, I

and K, be t h e nonempty i n t e r v a l of R

- U i + l I Q h, I Vi

- ui-1 I Q h] .

+

L e t wi = &ui+l ui-l) , t h e n f i 2 O - i i , 3 w, and s i n c e w, minimises t h e f u n c t i o n ui -t (ui - u,+~)' + (ui - ui-# over R , we have:

from which we deduce

-

which, with ui = PI,(&) and iii 3 wi l e a d s t o : (7.60)

(W(

- U i ) (iii - Ui) Q 0

wi Q ui

Q

G

hence ( 7 . 5 1 ) . P r o p o s i t i o n 7 . 4 . I f f o r an index io we have uio-uio-l=h (resp. Uio-Ub+l=h ) then ue have u, - u i - l = h V i , i = 2,...,i0 (resp. u, - u i + l = h Vi, i = io, ..., N )

.

Let io be such t h a t uio - uio-, = h ; t h e n t a k i n g accProof. ount of P r o p o s i t i o n 7.3 we have uia-l >l(uio+u,-J so t h a t ~ b - l - u b - 2 3 u , - u i o - l = h ; now U ~ E Kso~ t h a t u,O-l - uia-z = h , and so on u n t i l i = 2 . A similar argument i s used f o r t h e o t h e r family of r e l a t i o n s . P r o p o s i t i o n 7 . 5 . Let iM (2 < ,i < N) be such t h a t uiU = max u , ; 1S I G N + % then f o r h s u f f i c i e n t l y small we have: 0 Q uy, (7.61)

uiU

Proof. (7.62)

or : (7.63)

- uiy-l < h ,

- Ui,+l < h , = gh'fiU+ 4 , +

0 d uiy

zi,

1

ui,

- 1) *

With t h e n o t a t i o n of P r o p o s i t i o n 7.3 w e have: =I

Wfi,+ ui,-1 + uil+J

9

186

Elasto-plastic t o r s i o n o f a c y l i n d r i c a l bar

(CHAP. 3 )

putting e,

= ]
3 h, (iM - 4) h[ .

From P r o p o s i t i o n 7 . 3 we have

jiiM

2 uiy

,

so t h a t , with ( 7 . 3 6 ) ,

(7.64)

- UiX+l When h -+ 0 w e have

f(x)dx+O

E Kim

small,

(7.62), ( 7 . 6 3 )

with

proves

7.6.

Proposition

to

,

-

so t h a t f o r h s u f f i c i e n t l y u,, = iixwhich t o g e t h e r

(7.61).

The sequence

u,*EK,,, from below such t h a t

U; defined by ( 7 . 4 5 ) converges u,*

< u,.

Proof. We s h a l l f i r s t prove t h a t U; Q u, V n . W e s h a l l argue by r e c u r r e n c e and assume t h a t t h e p r o p e r t y i s t r u e up t o o r d e r n and t r u e f o r $+' up t o t h e i n d e x . i - 1: From P r o p o s i t i o n s 7 . 2 and 7 . 3 we have:

If u,

u, =

4

we have, from t h e r e c u r r e n c e assumptions, I f U, > u, we have from P r o p o s i t i o n 7 . 4 ,

2 #+'" 2 #+'

(7.66)

.

ui = min (u,,,, u i - l )

+h,

and l i k e w i s e we always have

(7.67)

#+I

Q

min (#-+;,#+1) + h .

Now ( r e c u r r e n c e assumptions)

6 ;:

Q ui-l

and

Q

so

that:

(7.68)

#+lQmin(#~~,#+l)+hQmin(u,~l,u,+,)+h=~u,.

Thus i f t h e p r o p e r t y i s assumed t o be t r u e t o t h e o r d e r n , = O it i s t r u e f o r s i n c e it i s t r u e t o t h e o r d e r n + 1 f o r U;+i ; now t h e p r o p e r t y i s t r u e f o r ut = 0 and i s t h u s t r u e f o r a l l n, t h a t i s :

<+'

(7.69)

U;<

u,,

Vn 2 0

(SEC. 7 )

187

Solution of ( P o ) by relaxation

S i n c e t h e sequence U; i s i n c r e a s i n g and bounded by v e r g e s from below t o U ~ E wKi t h~ ut Q uk.

u,

, it

con-

The function 4 satisfies the properties of , the subjects of Propositions 7 . 3 , 7 . 4 , 7.5.

P r o p o s i t i o n 7.7.

uk

From u t = 0

Proof. (7.70)

42 0

and P r o p o s i t i o n 7 . 2 we deduce:

Vn 2 0 ,

t h e n by a p p l y i n g t h e same method a s t h a t used a t p o i n t 2 o f t h e proof o f P r o p o s i t i o n 7.3:

+ q?:) Q 4" d i(h2f;-+ U;+l + u;-':),

(7.71)

so t h a t , proceeding t o t h e l i m i t i n ( 7 . 7 0 ) , ( 7 . 7 1 ) :

(7.72)

{

uk' 2 0 , t(u:+1 + u:-11

d u: Q i(h2f, + u:+1

+ Uf-1).

By s t a r t i n g from ( 7 . 7 2 ) and a r g u i n g as i n t h e proof of Propo s i t i o n 7.4 it may b e demonstrated t h a t t h e c o n c l u s i o n s of t h e l a t t e r a p p l y e q u a l l y t o uz ; t h e same a p p l i e s f o r t h e conclusi o n s o f P r o p o s i t i o n 7.5.

P r o p o s i t i o n 7.8.

we have

ut

= uh.

Proof. Assume t h a t t h e r e e x i s t s i,,, 2 < i , < N , such t h a t (resp. u$ u $ + ~= h) ; t h e n UZ = uro I n f a c t , from P r o p o s i t i o n 7.7, w e have: 1)

9 - U Z - ~= h

= (i,

.

-

- 1) h (rap. uf

= (N

+ 1 - i,) h) ;

now l u , + l - u f l < h Vi=l. ..., N so t h a t uio<(io-l)h, u,,d (N+l-i,)h hence ub Q UI', ; t h u s t a k i n g a c c o u n t of U$ < ub uio = u c and l i k e wise ui = @ V I , < i d i, 2)

(7.74)

Moreover, we have

(rap.Vi,i,

< i < N).

188

Elasto-plastic torsion o f a c y l i n d r i c a l bar

(CHAP. 3 )

(7.75) where

3) (7.76)

w, = u,

- e.

From P r o p o s i t i o n 7.7 we have:

- u:-, - * + I + 2u: - fi d 0 hZ

Vi = 2, ..., N

.

If t h e i n e q u a l i t y i s s t r i c t , we have I u:+~ - u:l = h a n d / o r I = h , so t h a t u, = @ from p o i n t 1. We t h u s have:

I @ - u:-l (7.77)

so t h a t

(7.78)

and s i n c e E K ~ we t h e n deduce We have t h u s proved:

u: = u,.

8

Theorem 7 .l. If f 3 0 and u: = 0 t h e sequence @ defined by algorithm ( 7 . 4 5 ) converges t o the s o l u t i o n u, o f problem ( 7 . 4 4 ) w i t h :

Remark 7.11.

The r e s u l t s t a t e d i n Theorem

7.1 s t i l l holds i f

w e u s e t h e v a r i a n t of a l g o r i t h m ( 7 . 4 5 ) d e f i n e d by:

7.4

Applications.

Example 1

,/m.<

We t a k e R = [ x I X E R2, r = i] and f = 10; t a k i n g account o f S e c t i o n 2.3.2 t h e s o l u t i o n of (PJ i s g i v e n by

<

u(x) = i - r if 0.2 r 0d r u(x) = 0.4 - 2.5 rz i f

f

< 0.2

( p l a s t i c zone) ( e l a s t i c zone)

(SEC. 7 )

Solution o f ( P o )b y reZaxation

189

Following Bourgat 111, w e approximate (Po) u s i n g t h e f i n i t e element method of S e c t i o n 4 . 1 , with t h e two t r i a n g u l a t i o n s F l y the J, shown ( f o r 1 / 1 2 of t h e d i s c ) on Figures 7.3 and 7.4; approximate problems a r e solved u s i n g t h e algorithm ( 7 . 1 ) , ( 7 .lo), (7.13) defined i n S e c t i o n 7.2.1, with u : = O , t h e t e r m i n a t i o n c r i t e r i o n being chosen as:

The main r e s u l t s r e l a t i n g t o t h e approximate s o l u t i o n of by t h e above techniques are shown i n Table

Triangulation

.

Number of v e r t i c e s

N

= Card

J l

121

(P,,)

7.1.

JZ

433-

(2,)

j = 1,2 w

Number of i t e r a t i o n s

1.6 7

1.6 20

Computation t i m e

(CII 10070)

10 s

50s

I n Table 7.1, E. r e p r e s e n t s t h e maximum a b s o l u t e v a l u e of t h e d i f f e r e n c e s ( e v a l u a t e d a t t h e v e r t i c e s of t h e t r i a n g u l a t i o n s cons i d e r e d ) between t h e exact s o l u t i o n and t h e approximate s o l u t i o n , c a l c u l a t e d from t h e approximation and u s i n g t h e above algorithm f o r t h e t e r m i n a t i o n c r i t e r i o n (7.82) . The d e t a i l s of t h e t r i a n g u l a t i o n s J , a n d Y, a r e shown i n Figures 7.2 and 7.3 f o r 1 / 1 2 of t h e d i s c ( t h e t o t a l t r i a n g u l a t i o n i s obtained by r o t a t i o n s of kx/6, k = 1, ..., 11 ) ; t h e s e f i g u r e s also show t h e approximations of t h e e l a s t i c zane and t h e p l a s t i c zone (hatched r e g i o n ) a s well as t h e i n t e r f a c e between t h e exact e l a s t i c and p l a s t i c zones ( i . e . t h e c i r c l e Of r a d i u s 0 . 2 depicted as a dashed l i n e ) . The r e a d e r may c o n s u l t Bourgat /1/ f o r

190

Elasto-pzastic torsion of a cyZindricaZ bar

(CHAP. 3 )

f u r t h e r d e t a i l s o f t h e approximate s o l u t i o n o f Example 1.

Fig. 7.2 (f = 10). D e t a i l o f t h e t r i a n g u l a t i o n y,and r e f r e s e n t a t i o n of t h e approximated e l a s t i c and p l a s t i c zones ( t h e p l a s t i c zone i s shown h a t c h e d ) .

Fig. 7.3. (f = 10). Detail of t h e t r i a n g u l a t i o n Y3and r e p r e s e n t a t i o n of t h e approximated e l a s t i c and p l a s t i c zones ( t h e p l a s t i c zone i s shown h a t c h e d ) .

Solution of ( P ) b y relaxation

191

0

Number of iterations

R e p r e s e n t a t i o n of t h e number of i t e r a t i o n s r e q u i r e d Fig. 7.4. f o r convergence, a s a f u n c t i o n o f w (Example No. 1 ) .

Remark 7.12. A d e t a i l e d a n a l y s i s ( s e e Bourgat 111) of t h e behaviour of t h e s u c c e s s i v e i t e r a t e s i n a l g o r i t h m ( 7 . 1 ) , ( 7 .lo), ( 7 . 1 3 ) shows t h a t t h e p l a s t i c zone and t h e p l a s t i c component of t h e s o l u t i o n are r a p i d l y a t t a i n e d , and t h a t t h e m a j o r i t y of t h e i t e r a t i o n s are s p e n t on t h e s o l u t i o n o f - Au = I i n t h e e l a s t i c zone, t h e convergence b e i n g a l l t h e more r a p i d ( f o r a g i v e n w ) as t h e s i z e of t h e p l a s t i c zone i n c r e a s e s ( i . e . a s t h e number o f t h i s b r i n g s t o g e t h e r Remarks 5.7, active constraints increases) ; Chapter 2, S e c t i o n 5.7.2 and 3 . 7 , Chapter 3, S e c t i o n 3.5.2. We n o t e ( s e e F i g u r e 7.4) t h a t once a g a i n t h e u s e of an w s i g n i f i c a n t l y g r e a t e r t h a n u n i t y a l l o w s t h e convergence t o be considrn erably accelerated. Remark 7.13.

The s o l u t i o n u o f t h e problem ( P ) of Example 1 0

satisfies

(7.83)

.

u E W,l*m(Q)n Wz*m(Q)

From t h i s it may b e deduced ( G . S t r a n g , p r i v a t e communication) t h a t under t h e assumptions o f Theorem 6.1, S e c t i o n 6.2.1, we have:

(7.84)

11 u - U, II,pCn, Q Ch1I4, C

with, w e r e c a l l , h = max. area Tcdr

independent of h ,

(T). rn

192

7.5

Elasto-plastic torsion of a cylindrical bar Applications.

(CHAP. 3 )

Example 2.

We c o n s i d e r a g a i n t h e example of S e c t i o n 3 . 5 (i.e. 0 = ]0,1[ x p,1[ and f = 1 0 ) s o l v e d p r e v i o u s l y by r e p l a c i n g t h e convex s e t KO by K, ( u s i n g t h e e q u i v a l e n c e r e s u l t of Brfkis-Sibony /1/ - s e e Section 2.2). Following Goursat /I./ we u s e , f o r t h e approximate s o l u t i o n of (Po) , a method o f exterior approx-irnation i n t h e s e n s e of S e c t i o n 5; more p r e c i s e l y , we approximate J by J: ( s e e S e c t i o n 5 . 1 . 1 , r e l a t i o n s ( 5 . 1 ) , ( 5 . 2 ) , ( 5 . 3 ) ) and KO by KA ( s e e S e c t i o n 5.2, relation (5.14)).

5

Fig. 7.5.

Decrease of R " a s a f u n c t i o n o f t h e number o f i t e r a t i o n s (Example No. 2 ) .

(sec. 7 )

Solution o f ( P o )by relaxation

193

Numerical values of the parameters

7.5.1

Mesh s i z e :

h = 1/20.

I n i t i a l i s a t i o n of algorithm (7.14), ( 7 . 1 9 ) , ( 7 . 2 2 ) :

ut = 0.

Termination c r i t e r i o n f o r algorithm (7.14), (7.19) , ( 7 . 2 2 ) :

c

- $1 <

1$+1

10-3.

Mil E nh

Analysis of the numerical r e s u l t s

7.5.2

For 0 < w 2, 4 converges t o a l i m i t which c o i n c i d e s t o withi n a r e l a t i v e v a l u e o f about lo-* w i t h t h e s o l u t i o n c a l c u l a t e d by s o l v i n g t h e corresponding problem (P,&( s e e S e c t i o n 3 . 5) by t h e method of S e c t i o n 3.4, for h = 1 / 2 0 . The o p t i m a l v a l u e o f w (determined e m p i r i c a l l y ) i s about 1 . 5 , which i s t h e o p t i m a l v a l u e o f w f o r t h e corresponding problem (Pl& ; Remark 3.7 ( s e e Section 3.5.2) therefore s t i l l holds. For w = 1 . 5 convergence of a l g o r i t h m (7..14), ( 7 . 1 9 ) , ( 7 . 2 2 ) i s reached i n 22 i t e r a t i o n s , corresponding t o 1.38. s e c on a n IBM 360191 or 22 s e c . on a C I I 10070; t h e d e c r e a s e o f

R"=

C Iq;'-@'I MIIJE

i s shown i n F i g u r e 7 . 5 as a f u n c t i o n o f t h e number o f i t e r a t i o n s , I t may b e s a i d , s c h e m a t i c a l l y , t h a t t h e c u r v i l i n e a r for w = 1.5. part o f t h e graph i s c h a r a c t e r i s t i c o f t h e nonlinearity o f t h e problem i n t h e c o u r s e o f t h e s o l u t i o n u n t i l t h e approximate p l a s t i c zones and p l a s t i c s o l u t i o n s have been determined; t h e remainder o f t h e graph, which i s r e c t i l i n e a r , corresponds t o t h e solu t i o n o f a linear problem w i t h - Au = f ( a p p r o x i m a t e l y ) i n t h e t h e above a p p l i e s e q u a l l y w e l l t o F i g u r e 3.1 of e l a s t i c zone; w S e c t i o n 3.5.2.

7.6

Applications.

We t a k e

ll = p,1[ x

Example 3.

p,1[

and f d e f i n e d by

With t h e corresponding problem (p,,) approximated by t h e exterior appro&mations o f t h e preceding example ( s e e S e c t i o n 7.5) w i t h h = 1 / 2 0 , convergence of t h e a l g o r i t h m (7.14), ( 7 . 1 9 ) , ( 7 . 2 2 )

for t h e i n i t i a l i s a t i o n u: = 0 and t h e t e r m i n a t i o n c r i t e r i o n

cml I%+,-

Ma1

$1

< lo-*,

194

Elasto-plastic t o r s i o n of a c y l i n d r i c a l bar

Fig. 7.6.

(CHAP. 3 )

R e p r e s e n t a t i o n of t h e zones i n which I grad u I < 1 and I grad u I = 1 ( t h e l a t t e r are shown h a t c h e d ) . (Example No. 3 ) .

i s r e a c h e d i n 57 i t e r a t i o n s ( c o r r e s p o n d i n g t o a c o m p u t a t i o n time of 1 min. 13 s e c . o n a C I I 10070), i f w = 1.7; t h e l i m i t t h u s o b t a i n e d c o i n c i d e s w i t h t h e s o l u t i o n of t h e c o r r e s p o n d i n g problem (Po)(,c d l c u l a t e d by means of o t h e r methods for which convergence From t h e r e s u l t of t h e above c a l c u l a t i o n , t h e c a n b e proved. blslnk r e g i o n i n F i g u r e 7 . 6 r e p r e s e n t s t h e zone i n which I grad u I < 1 ( a n d - Au = f ) and t h e h a t c h e d r e g i o n s r e p r e s e n t t h e zones i n which 1 gradu 1 = I .

7.7

V a r i o u s remarks

Reniurk 7.14. The a l g o r i t h m (7.14), (7.191, ( 7 . 2 2 ) a p p l i e d t o t h e s o l u t i o n of ( P o ) , u s i n g t h e a p p r o x i m a t i o n s of J and .KOemployed i n Examples 2 and 3, i s more r a p i d ( l ) i n terms of the nwn(l)

For a g i v e n w and w i t h t h e same i n i t i a l c o n d i t i o n s and terminetion c r i t e r i a .

so lution of ( P o by relaxation

(SEC. 7 )

195

ber of iterations t h a n t h e algorithm of over-relaxation with proj e c t i o n defined i n S e c t i o n 3.4, when t h e l a t t e r i s a p p l i e d t o t h e s o l u t i o n of t h e problem (P1)( l ) using t h e approximation of J defined i n S e c t i o n 3.2.1, r e l a t i o n s ( 3 . 1 0 ) , ( 3 . 1 2 ) , ( 2 ) and t h e approximation of Kl defined i n S e c t i o n 3.2.1, r e l a t i o n (3.11). This behaviour may be explained by noting t h a t t h e c o n s t r a i n t s defining KO (resp. K&) a r e more restrictive than t h o s e d e f i n i n g Kl (resp. K,,J s i n c e KO c Kl s t r i c t l y ( t h e analogous property f o r t h e approxmate convex s e t s i s somewhat more d i f f i c u l t t o e x p r e s s ) . A s f a r as t h e computation time i s concerned, t h e second algorithm has i n f a c t proved t o be somewhat more r a p i d than t h e f i r s t ( a l l o t h e r t h i n g s being equal, and f o r t h e examples which we have c o n s i d e r e d ) ; t h i s i s e a s i l y explained by t h e f a c t t h a t t h e proje c t i o n on t h e i n t e r v a l K/j , i n t h e c a s e o f t h e f i r s t algorithm, i n t h e c a s e of t h e seci s more complicated t h a n t h a t on [- a,, a,,] ond, t h e number o f r e l a t i o n s d e f i n i n g t h e s e two i n t e r v a l s being g r e a t e r i n t h e c a s e of t h e f i r s t algorithm. . I t should be added t h a t of a l l t h e methods t r i e d f o r solving (Po) i n a n approximate manner ( w i t h o u t making r e c o u r s e t o t h e equivalence result of S e c t i o n 2 . 2 ) , t h e method considered i n t h i s S e c t i o n 7 i s t h e one which has demonstrated t h e b e s t performance both i n terms of computation time and s t o r a g e requirement. W I n connection with t h e convergence of t h e algRemark 7.15. , ( 7 . 4 ) and (7.14), , (7.17) (and v a r i a n t s orithms (7.1), with under- o r over-relaxation parameters) it may be noted (though Q) ) t h i s does not c o n s t i t u t e a p r o o f ) t h a t when h - 0 ( o r N + t h e approximations of K,, defined i n S e c t i o n 4.1 ( f i n i t e elements) and i n S e c t i o n 5.2 ( f i n i t e d i f f e r e n c e s ) become progressively more local i n t h e following sense: t h e maximum number of c o n s t r a i n t s which e x p l i c i t l y involve a v a r i a b l e i s bounded above by No = 2rr/O0 i n t h e c a s e o f t h e f i n i t e element method and under t h e assumptions o f t h e convergence theorem 6.1, and by t h e i n t e g e r s N,,, (m = 1,2,3,4) given by ( 7 .is) i n t h e c a s e of t h e e x t e r i o r approximations of S e c t i o n 5 . 2 . We have, of course, l h ( N , , , / N ) = 0 (m = 0, 1, 2, 3, 4). where N i s t h e

...

...

+

k-0

number of v a r i a b l e s on which t h e approximate problem depends.

Remark 7.16.

w

I n order t o s o l v e (PJ i n a n approximateminner, it i s p o s s i b l e t o use t h e following method o f approximation and decomposition, t h e p r i n c i p l e of. which i s due t o 3 . Cea; t h i s method of a p p l i c a t i o n i s p a r t i c u l a r l y simple f o r exterior approximations (Po& (1 = 1,2; m = 2,4)

.

(’) (2)

When, of course, t h e equivalence r e s u l t of Section 2.2 holds. The approximation of J t h u s obtained t h e r e f o r e coincides with t h a t used f o r Example 2 i n S e c t i o n 7.5.

196

Elasto-plastic torsion of a cylindrical bar

(CHAP. 3 )

The p r i n c i p l e i s as f o l l o w s : With t h e f u n c t i o n a l J approximated by J: ( I = 1, 2) ( s e e 5.1 .l, 5.1.2), t h e r e l a t i o n s e x p r e s s i n g membership o f K& (m = 4) ( s e e S e c t i o n 5.2 ) a r e imposed a t t h e p o i n t s o f t h e mesh Qh ( s e e S e c t i o n 5.1.2 f o r t h e d e f i n i t i o n o f Q k ) which belong t o a mesh o f i n t e r v a l 2h c o n s t r u c t e d on Q, ( s e e F i g u r e 7 . 7 ) and hence i n p r a c t i c e a t approximately one p o i n t i n f o u r o f Q h .

z

T h i s method o f t r e a t i n g t h e c o n s t r a i n t s i m p l i e s t h a t t h e r e l a t i o n s e x p r e s s i n g membership o f t h e approximate convex s e t become uncoupled from each o t h e r , s i n c e t h e y do n o t e x p l i c i t l y i n v o l v e any common v a r i a b l e ; t h i s t h e r e f o r e l e a d s t o t h e vari a b l e s b e i n g grouped i n b l o c k s o f 4, and hence t o t h e approximate problem b e i n g solved by a metnod o f c o n s t r a i n e d under- o r overr e l a x a t i o n by b l o c k s ( o f 4 v a r i a b l e s ) , which i s convergent f o r 0 < w < 2, a s a consequence o f t h e aforementioned uncoupling o f t h e c o n s t r a i n t s ( s e e Chapter 2 , S e c t i o n 1 . 4 , Remark 1 . 7 and Glowinski 141, S e c t i o n 7 ) . 8

8.

SOLUTION OF THE APPROXIMATE PROBLEMS RELATING TO PENALTY METHODS

8.1

(P

Synopsis

I n t h i s s e c t i o n we s h a l l s t u d y t h e a p p l i c a t i o n o f t h e penalty methods o f Chapter 1, S e c t i o n 3.2 and Chapter 2 , S e c t i o n 3 , t o t h e s o l u t i o n o f (Po) and of t h e f i n i t e - d i m e n s i o n a l problems o b t a i n e d by a p p l y i n g t o (P,,) t h e finite-element approximation of S e c t i o n 4 . 1 and t h e exterior approximations o f S e c t i o n 5. We s h a l l c o n s i d e r two methods based on p e n a l i s a t i o n , founded r e s p e c t i v e l y on t h e f o l l o w i n g i d e a s :

SoZution of ( P o )by p e n a l t y methods

(SEC. 8 )

197

( i ) We suppress, by p e n a l i s a t i o n , t h e c o n s t r a i n t I grad v I < 1 a.e. o r , a t l e a s t , r e p l a c e it by c o n s t r a i n t s which a r e simpler from a numerical p o i n t o f view ( s e e S e c t i o n 8 . 2 b e l o w ) .

( i i ) We decompose t h e problem (Po) by adding supplementary f u n c t i o n s and " a r t i f i c i a l c o n s t r a i n t s " which a r e t h e n suppressed by p e n a l i s a t i o n ( s e e S e c t i o n 8 . 3 b e l o w ) .

We c o n s i d e r it a p p r o p r i a t e , b e f o r e a p p l y i n g t h e above methods t o t h e approximate problems, t o d e m o n s t r a t e them on t h e c o n t i n u o u s problem, which i s s i m p l e r i n formal t e r m s . We s h a l l t h e n p r e s e n t a number o f examples i l l u s t r a t i n g t h e r e s u l t s o b t a i n e d by t h e s e methods.

8.2

A f i r s t p e n a l t y method

P r i n c i p l e of t h e method

8.2.1

We a s s o c i a t e w i t h J t h e penalised f u n c t i o n a l d e f i n e d by ( I ) :

-J&(u)= J(u)

(8.1)

6,jn

+-

{ (1 - I grad 0 12)-

12 dx

J, : W;**(Q)+ W

.

We s h a l l assume t h e f o l l o w i n g results ( f o r t h e p r o o f s , s e e f o r example, Lions /l/, Chapter 3, S e c t i o n 5 ) : P r o p o s i t i o n 8.1.

(i) (ii) (iii)

(8.2)

We have t h e folZowing p r o p e r t i e s for J, :

J, i s continuous and s t r i c t l y convex on lim J,(o) = + "o,

II Ilw,ynt-

+

W;**(Q),

Q

i s Gateaux d i f f e r e n t i a b l e over J: being d e f i n e d b y : J,

(J:(v), W ) = U ( V, W ) -

(1

, i t s gradient

w;p4(f2)

- I grad u 12)-

x

x grad o.grad w dx , Vw E W;**(Q). rn

I n (8.2), t h e i n n e r p r o d u c t - o f J:(v) w i t h w i s t h a t o f t h e d u a l i t y between W-1*4'3(Q) ( i e . t h e d u a l o f WiB4(12)) , and W;**(Q).

.

(l)

We r e c a l l t h a t

198

Elasto-plastic torsion of a c y l i n d r i c a l bar

Proposition 8.2.

(CHAP. 3 )

The penalised problem (P3 associated with

(Pa):

admits one and only one s o l u t i o n ,

u,

, characterised b y :

u ( u , , ~ ) + f ~ ~-( Igradu,~')-gradu,.gradvdx l = (8.4)

[ u, E Wi*4(f2) .

We deduce from ( 8 . 4 ) t h a t u, i s t h e u n i q u e sol, of t h e monotone n o n l i n e a r e l l i p t i c problem:

Remark 8.1. Wi*'(ra>

ution, i n

Remark 8 . 2 . w;*4(ra)

rn

I t may b e shown t h a t t h e n o n l i n e a r o p e r a t o r -+

p:

w - '.4'3(ra)

d e f i n e d by:

(8.6)

(B(u), w ) =

J'. (1 - I grad

vu, w E

u

1')-

grad u.grad w dx

Wi*4(ra)

i s monotone ( b e i n g t h e g r a d i e n t of a convex f u n c t i o n ) , l o c a l l y L i p s c h i t z , and p(u) = 0 V V E KO ; p i s t h u s a penalisation operato r a s s o c i a t e d w i t h KO, i n t h e s e n s e o f Chapter 1, S e c t i o n 3 . 2 . rn I n c o n n e c t i o n w i t h t h e b e h a v i o u r o f u, a s E -+ 0 , we have: Theorem 8.1.

(8.7)

When

E

+ 0

strongZy i n

u,-u

wherae u i s the s o l u t i o n of

Proof. 3.2: For

(8.8)

we have: Hi(ra>

(Po).

T h i s i s a v a r i a n t o f Theorem 3.1, Chapter 1, S e c t i o n

u a s o l u t i o n o f (Po), w e have (1 - I grad u 1')J,(u> =S J,(u) Q J(u) V& > 0 .

We t h u s have:

=0

so t h a t :

S o l u t i o n of ( P o )by p e n a l t y ' methods

(SEC. 8 )

199

so t h a t :

(I u, llq(m Q C

(8.10)

VE > 0

I

There t h e r e f o r e e x i s t s u * ~ H t ( n )and a subsequence, a l s o deno t e d by u, , such t h a t :

(8.11)

lim u,

= u*

weakly i n

Hi(@

*+O

.

Relation (8.8) implies t h a t :

jn

(8.12)

{ (1

- I grad u, 12)-

}' dx Q 4 &Mu) - J(u3)

so t h a t , t a k i n g account of ( 8 . 1 0 ) ,

11 u, I I ~ ~ . Q. (c~ ,

(8.13)

- I grad u, 12)-

{ (1

(8.14)

}z dx = 0 .

For t h e above subsequence w e t h u s have:

limu, = u*

(8.15)

weakly i n Wt*'(Q).

,+O

From ( 8 . 9 ) , ( 8 . 1 1 ) , (8.14), semi-continuity :

(8.16)

J(u*) Q J(u) ,

(8.17)

1.((1 -

(8.15) we deduce by weak lower

(gradu*1*)-}'dx=O.

R e l a t i o n (8.17) i m p l i e s t h a t # * € K O ; we t h u s have, from (8.16) and t h e uniqueness of u, u* = u . We deduce from ( 8 . 9 ) , (8.11) .that for t h e above subsequence we have :

J(u) Q lim i d J ( u 3 Q lim sup J(u3 Q J(u) , so t h a t : (8.18)

lim J(u,) = J(u) , r+O

which implies :

Elasto-plastic torsion of a cylindrical bar

200

(CHAP. 3 )

There i s t h u s convergence o f t h e norm, and hence s t r o n g convergence s i n c e we a l r e a d y have weak convergence. I n view o f t h e uniqueness o f u , we have l h u , = u w i t h o u t r e s t r i c t i n g o u r s e l v e s t o a subsequence. '+'

A variant

8.2.2

The p e n a l t y method o f S e c t i o n 8 . 2 . 1 i n t r o d u c e s a truncation which has t h e e f f e c t o f l i m i t i n g t h e d i f f e r e n t i a b i l i t y o f t h e i n o r d e r t o g e t round t h i s d i f f f u n c t i o n a l J,, t o b e m i n h i s e d ; i c u l t y w e i n t r o d u c e a s l a c k f u n c t i o n 3 0 ( s e e Remark 3.6, Chapter 2, S e c t i o n 3 . 3 ) ; i n f a c t , t h e r e i s e q u i v a l e n c e between:

(8.20)

Igradul

< 1'

a.e.

and

(8.21)

(graduI2+q= 1

q,O

a.e.

and hence w e o b t a i n t h e problem a l i s i n g r e l a t i o n (8.21)

(n,), a

a.e.

(P8)., by pen-

v a r i a n t of

It may b e noted t h a t t h e p e n a l i s e d f u n c t i o n a l a p p e a r i n g i n I n view o f ( 8 . 2 2 ) is not convex w i t h r e s p e c t t o t h e p a i r ( u , q ) t h i s we have:

.

P r o p o s i t i o n 8.3. The problem (nJa h i t s a unique solution (u8,pJ where us i s the solution of (P3and where p8 is defined by:

(8.23)

p8 = SUP (0,l

Proof.

- I grad us 1')

(1

- I grad U, 12)' .

We w r i t e

and, g i v e n u , we d e f i n e qv by:

(8.24)

4. = SUP (0,l

- 1 grad u 1)' = (1 - I grad u 12)'

I

Solution o f ( P o )b y penalty methods

(SEC. 8 )

201

so t h a t with ( 8 . 2 3 )

i . e . t h e r e s u l t which w a s t o be proved. From Remark 8.1 and P r o p o s i t i o n 8.3, it i s easy t o deduce:

The p a i r ( u , , p ~ ,t h e s o l u t i o n of (II,), i s char-

P r o p o s i t i o n 8.4.

acterised by :

I

i z a

- Au, -; C -(I k=l

(8.27)

gradu, l2 + p a -

Application t o t h e s o l u t i o n o f t h e approximate problems (I). Formulation of the penalised approximate problem.

8.2.3

There i s ( i n p r i n c i p l e ) no d i f f i c u l t y i n FreZhinary remark. applying t h e p e n a l t y method of S e c t i o n s 8.2.1, 8.2.2 t o t h e varWe s h a l l i o u s a p p r d x h a t i o n s o f (Po) d e f i n e d i n S e c t i o n s 4 and 5 . c o n f i n e o q a t t e n t i o n t o t h e approximation (Pd13of (Po) ( s e e S e c t i o n s 5.1, 5.2, 5 . 3 ) , t h i s being t h e o n l y c a s e which has given r i s e t o numerical a p p l i c a t i o n s ; t h e method used i s t h e f i n i t e dimensional analogue of t h a t i n S e c t i o n 8.2.2.

Penalisation of t h e problem We d e f i n e t h e space

{ qh Iq h

4 and

(Pdl3. the. convex s e t Lh+by:

(8+28)'

Lh

(8.29)

Lh+={qhIqhELh, qijao

=

=

qij ) Y , j c a r

qijE

1

9

vMijEch},

and t h e f u n c t i o n a l I,, : Vh x Lh + R

a s s o c i a t e d with

with:

(l)

See S e c t i o n 5.1.1 f o r t h e d e f i n i t i o n o f

J,'

(Pod13 by

202

Elasto-plastic torsion of a c y l i n d r i c a l bar

where ( s e e S e c t i o n 5.2, F i g u r e s 5.2, 5.3,

(8.32)

I

Bij = 1 if MI, E a,, ;Pi, = 2 i f M , , E r , w i t h ,Bl, = 3. i f M l , ~ r , w , ith

(CHAP. 3 )

5.4):

a s i n g l e neighbour i n Q,,, two n e i g h b o u r s i n Q,,

( i t i s a p p r o p r i a t e t o t a k e urn = 0 i n ( 8 . 3 1 ) i f Mrn$Q&. W h i l s t it i s p o s s i b l e t o t a k e a l l t h e q, e q u a l t o u n i t y , a more n a t u r a l c h o i c e i s :

, t h e d i s c r e t e analogue of We t h e n d e f i n e t h e problem (Ha,) Y by:

(4)

and p r o c e e d i n g as i n S e c t i o n s 8 . 2 . 1 ,

8.2.2,

it i s e a s y t o p r o v e :

P r o p o s i t i o n 8.5. The problem ( H d a d m i t s one and only one solution,(uk,Pk). i n v h x &+ and when E + 0, uk + u, where 4 i s t h e soZution of (PM)13. P r o p o s i t i o n 8.6 . The pair ( u k , p d which i s the unique soZution of (Ha&i s characterised by:

Remark 8 . 3 . The r e s u l t s of P r o p o s i t i o n 8 . 5 a r e s t i l l v a l i d i f f o r t h e au w e t a k e s t r i c t l y p o s i t i v e s c a l a r s , which may b e chosen a r b i t r a r i l y

.

Remark 8.4. system ( 8 . 2 7 ) .

System ( 8 . 3 6 ) i s t h e d i s c r e t e a n a l o g u e o f t h e

(SEC. 8 )

Solution of ( P o )by penalty methods

20 3

Application t o the s o l u t i o n o f the approximate problem (11). Solution of the penalised approximate problem.

8.2.4

I n view of P r o p o s i t i o n 8 . 6 it i s s u f f i c i e n t , i n o r d e r t o s o l v e t o s o l v e t h e ( n o n l i n e a r ) system ( 8 . 3 6 ) ; f o r t h i s , we s h a l l u s e a v a r i a n t o f t h e method o f point over-relaxation with

(nab ,

projection. I n t h e f o l l o w i n g (by way o f s i m p l i f i c a t i o n ) we s h a l l w r i t e pk=(pi,)y,,eijk and s i n c e Gi,(ub expl i c i t l y i n v o l v e s , i n a d d i t i o n t o Mi, , o n l y t h e f o u r immediate neighbours o f Mi, , we s h a l l w r i t e : uk=(ui,)y,,sa*,

Gi,("b = G*,("i-l,9

"ij-19 "1,'

"i.11,

Vij+l).

I t i s t h e n p o s s i b l e t o p u t i n e x p l i c i t form t h e r e l a t i o n are,

-(ubrpb) 84,

=0;

assuming t h e QU t o be g i v e n by ( 8 . 3 3 ) , we have:

(8.37)

with upI = 0 i f

bY

M,,,#c1,

.

We d e n o t e t h e l e f t - h a n d s i d e o f (8.37)

Elasto-plastic torsion of a cylindrical bar

204

(CHAP. 3 )

and it may b e s e e n t h a t ( 8 . 3 7 ) c o r r e s p o n d s t o t h e d i s c r e t i s a t i o n , a t M f J c f & , o f t h e f i r s t r e l a t i o n i n ( 8 . 2 7 ) by means o f a 13p o i n t scheme, as shown i n F i g u r e 8.1:

Fig. 8.1.

Description o f t h e over-relaxation algorithm I n t h e f o l l o w i n g , it i s assumed t h a t t h e i t e r a t i o n s a r e p e r formed w i t h i n c r e a s i n g i, and t h e n w i t h i f i x e d and i n c r e a s i n g j. We t h u s u s e t h e a l g o r i t h m :

(8.38)

u,",p,"

given a r b i t r a r i l y ,

t h e n , 4.P;b e i n g assumed known, t h e c o o r d i n a t e s a r e determined u s i n g :

and having c a l c u l a t e d

(8.4)

fl;'

= MaX(O,1

#,+'

of

4"

#+' , &+' i s d e t e r m i n e d from:

- /$JG;(4+'))

vMfjEGh.

i s t h e s o l u t i o n o f a n equI t may b e noted t h a t i n ( 8 . 3 9 ) #' a t i o n o f t h e t h i r d degree ( w i t h a s i n g l e v a r i a b l e ) .

(SEC. 8 )

20 5

Solution of ( P o ) by penalty methods

Application to the solution of the approximate problems (111). Examples.

8.2.5

We s h a l l consider a g a i n t h e example of Section we thus have 51 = ]0,1[ x lo, 1[ and f = 10.

E q l e 1.

7.5;

Numerical values of the parameters Mesh size:

:

h = 1/20

Penalisation parameter:

E

= 0.625

K

Initialisation of algorithm ( 8 . 3 8 ) - (8.40) : ut = 0, p t Termination criterion f o r algorithm ( 8 . 3 8 ) - ( 8.40) :

= 0.

Solution of (8.39) : The 3rd-degree equation determining ' : 4 from (8.39) w a s solved t h i s is a by t h e Newton-Raphson method i n i t i a l i s e d with 4, j more r a p i d process than s o l u t i o n by Cardan's method.

Analysis of the numerical results: For w = 1 ( r e s p . w = 1.7) convergence i s reached i n 391 itera t i o n s ( r e s p . 186 i t e r a t i o n s ) corresponding t o a n execution time of 1 0 s e c . ( r e s p . 6.2 sec.) 011 a n IBM 360191.

E q l e 2. thus have

We a g a i n consider Example 3 o f S e c t i o n f i s given by:

a - p,l[ x P,1[ and 10

f(x)=

(8.41)

- 10 0

if

(XI,

xz) E lo,

tr x li, I[,

if ( x 1 , x Z ) ~ ] t1[, x

if

(XI, X J

7.6; w e

P, tr

x

lo,&, lo, tr u B, 1[

x

B, 1[.

Once again, we t a k e h = 1/20, E = 0.625 x u(: = 0 , = 0 and t h e same termination c r i t e r i o n f o r algorithm (8.38)

-

.40) as i n t h e preceding example. For w = '1, convergence i s then reached i n 560 i t e r a t i o n s , corresponding t o an execution time of 1 4 . 5 sec on a n IBM 360/91, : 4 c a l c u l a t e d , as above, by t h e Newton-Raphson method. m with '

.

Remark 8.5. The approximate s o l u t i o n s t h u s c a l c u l a t e d coinci d e i n both of t h e above examples ( t o w i t h i n b e t t e r than 1%) with t h e s o l u t i o n s of t h e same approximate problems, c a l c u l a t e d by w means of t h e o t h e r methods described i n t h i s c h a p t e r . Remark 8.6. ing

E;

The r a t e o f convergence may be improved by varya r e l a t i v e l y l a r g e v a l u e i s used t o s t a r t with, and E i s

Elasto-plastic torsion of a c y l i n d r i c a l bar

206

(CHAP. 3)

then reduced i n t h e course of t h e i t e r a t i o n s of algorithm (8.38) ( 8 . 4 0 ) ( s e e S e c t i o n 8 . 3 . 5 below)

.

Remark 8 . 7 .

Ji

V a r i a n t s of

(n,J

-

may b e d e f i n e d u s i n g t h e approx-

of J ( s e e S e c t i o n 5 . 1 . 2 ) a n d / o r t h e a p p r o x i m a t i o n s K& (m = 1,2,4) o f KO ( s e e S e c t i o n 5 . 2 ) ; t h e u s e o f K;;k (m = 1,2,4) would l e a d t o a v a r i a n t of ( 8 . 3 7 ) e x p l i c i t l y i n v o l v i n g t h e v a l u e s of urn c o r r e s p o n d i n g t o t h e 9 p o i n t s shown o n F i g u r e 8 . 2 and 1 2 ( r e s p . 4 ) s l a c k v a r i a b l e s for rn = 1 ( r e s p . rn = 2 , 4 ) . imation

Fig. 8.2.

We r e c a l l ( s e e S e c t i o n 5 . 2 ) t h a t f o r rn = 2, 3 , 4 t h e number of c o n s t r a i n t s i s a p p r o x i m a t e l y e q u a l t o t h e number o f v a r i a b l e s ui,, so t h a t t h e r e s h o u l d b e a p p r o x i m a t e l y t h e same number o f s l a c k v a r i a b l e s as of v a r i a b l e s u,, ; c o n v e r s e l y for m = 1 t h e r e would b e a p p r o x i m a t e l y f o u r t i m e s as many s l a c k v a r i a b l e s as v a r i a b l e s Uij.

8.3 8.3.1

A second p e n a l t y method

Principle of the method

We w r i t e :

There i s t h e n e q u i v a l e n c e between:

(Po) M h J(v) , US&

and

Solution of ( P o )by penalty methods

(SEC. 8 )

207

o r indeed:

We s h a l l p e n a l i s e o n l y t h e r e l a t i o n :

(8.45)

q = gradu,

so t h a t , f o r

(no)l,

and, f o r

We denote by fl,( r e s p . 9% ) t h e f u n c t i o n a l d e f i n e d by (8.46) ( r e s p . ( 8 . 4 7 ) ) and .Y = If;@) x (L2(0))' ; t h e f o l l o w i n g p r o p o s i t i o n s may t h e n e a s i l y b e proved: P r o p o s i t i o n 8.6.

we have the foZZowing properties f o r

ft,(k = 1, 2): (i) (ii) (iii)

fh i s continuous and strictZy convex on V , lim

II @,bII* - + O D

f,(u,q)

=

+ 00,

& i s & t e a m d i f f e r e n t i a b z e (and even Frechet sense) on V .

C" i n the

8

P r o p o s i t i o n 8 . 7 . The penalised problems (n,),(k = 1, 2) each a h i t one and only one soZution on If,'@) x A , say (.;",& , of which the respective characterisations arc:

Elasto-plastic torsion of a c y l i n d r i c a l bar

208

(CHAP. 3 )

for k = 1

and, f o r k = 2,

- Aui + divpi = ef

There i s e q u i v a l e n c e between t h e l a s t - t w o r e l a t Remark 8.8. i o n s o f ( 8 . 4 8 ) , (8.49) a n d , r e s p e c t i v e l y :

(8.50)

p,' = PA(gradUf),

where PAd e n o t e s t h e o r t h o g o n a l p r o j e c t i o n o p e r a t o r from L'(Q) x L'(Q) + A r e l a t e d t o t h e s t a n d a r d norm o f L'(Q) x L'(Q)

;

e x p l i c i t l y , w e have:

(8.52)

The u s e o f p e n a l i s a t i o n i s j u s t i f i e d by: Theorem 8 . 2 .

(8.53) (8.54)

When

E .+

4 - c ~ strongly -+

gradu

0 we have, Vk = 1 , 2 :

in H;(Q)

strongZy i n L2(C2) x L2(Q)

where u i s the soZution of (Po). T h i s i s a v a r i a n t o f t h e proof o f Theorem 3 . 1 , o f ChaProof p t e r 1, S e c t i o n 3 . 2 ; we s h a l l c o n s i d e r o n l y t h e c a s e k = 1, t h e c a s e k = 2 b e i n g t r e a t e d i n a similar manner.

Solution of ( P o ) by penalty methods

(SEC. 8 )

L e t u b e t h e s o l u t i o n o f (PJ and p = g r a d u ; (u,p) E H,'(n) x A and:

209

we have

J(u:) Q #1,,(uf,pf) < Yl,,(U,p)= J(u) V& > 0 .

(8.55)

We t h u s have

J(u:)

(8.56)

< J(u)

VE > 0

so t h a t

11 U:

(8.57)

(I,gdcn,


V& > 0 .

Moreover, pf ~ A * p f bounded i n L2(Q) x L'(Q) , so t h a t t h e r e e x i s t s a subsequence, a l s o d e n o t e d by (uf,pf), u* E H,'(n) and p* E L2(n) x L2((ra)such t h a t when E + 0:

(8.58)

u: * U, weakly i n H i @ )

(8.59)

pi + p *

weakly i n L 2 ( n )x L'(l2) ( I )

and s i n c e A i s convex and c l o s e d i n L2(12) x LZ(n) ( a n d t h u s weakly c l o s e d ) w e have:

~ * E A .

(8.60) Relation

(8.55) i m p l i e s t h a t :

so t h a t , t a k i n g a c c o u n t o f (8.57) :

From ( 8 . 5 6 ) , (8.58), ( 8 . 5 9 ) , ( 8 . 6 2 ) w e deduce by (weak) lower semi-continuity

(8.63)

J(u*) < J(u) ,

(8.64)

p* = gradu*,

.

and (8.60) i m p l i e s u*eKO We t h u s have u * = u . The s t r o n g convergence o f ut and t h e convergence o f t h e e n t i r e (u:). f a m i l y a r e proved by p r o c e e d i n g as i n t h e p r o o f o f

(l)

I n f a c t w e have p: + p *

weakly" i n L"(R) x L"(n)

E l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l bar

21 0

(CHAP. 3 )

Theorem 8.1, t h e s t r o n g convergence o f p: r e s u l t i n g from t h a t o f u,'. and from ( 8 . 6 2 ) . 8.3.2

Application t o t h e s o l u t i o n o f approximate problems (I). Formulation of t h e penalised approximate problem i n t h e case of t h e f i n i t e element method of S e c t i o n

4.1. We s h a l l c o n f i n e our a t t e n t i o n t o t h e f i n i t e - d i m e n s i o n a l a n a l ogue o f (Ift)l , t h e o t h e r c a s e b e i n g t r e a t e d i n a s i m i l a r manner ( b y way o f s i m p l i f i c a t i o n , t h e i n d e x 1 w i l l b e s u p p r e s s e d i n t h e t h e notation of Section 4 w i l l b e retained. following) ; Having d e f i n e d t h e s p a c e Vh i n S e c t i o n 4.1.1, we i n t r o d u c e t h e subspace Lh o f L 2 ( 9 ) x L ' ( l 2 ) :

(8.65)

h= =

{

q = ( q 1 . q2) I E Lz(12) x L2(Q),41 =

1

q1TeT,q2

T€dh

=

1

TeTh

q2TOT;

q17142TE

where

(8.66)

OT =

c h a r a c t e r i s t i c f u n c t i o n of

T,,

then

The f i n i t e - d i m e n s i o n a l a n a l o g u e o f (lZt)l i s t h e n d e f i n e d by:

where Y, i s d e f i n e d by prove:

(8.46) ; it i s a s t r a i g h t f o r w a r d matter t o

P r o p o s i t i o n 8.8. The problem (&) a h i t s one and only one s o l u t i o n , (uk,Pk) , and when E -+ 0 we have

Solution of ( P o ) by penalty methods

(SEC. 8 )

(8.73)

211

pka+ grad u,, ,

where uh is the (unique) solution of Min J(l)h). w uh E V k n PO

The i n t e r i o r a p p r o x i m a t i o n method o f S e c t i o n 4 . 2 Remark 8 . 9 . (using e i g e n f u n c t i o n s o f - A i n HA@)) i s n o t s u i t a b l e f o r t h e p e n a l i s a t i o n t e c h n i q u e o f S e c t i o n 8 . 3 , for t h e r e a s o n s i n d i c a t e d i n Remark 4.6 o f S e c t i o n 4 . 2 . 5 .

Application to the solution of the approximate problems (11). Formutation of the penalised approximate problem in the case of the exterior approximations of Sec-. tion 5 .

8.3.3

O f t h e e x t e r i o r a p p r o x i m a t i o n s o f KO c o n s i d e r e d i n S e c t i o n 5.2, i s t h e o n l y one i d e a l l y s u i t e d t o t h e u s e o f t h e p e n a l t y method d e s c r i b e d i n S e c t i o n 8 . 3 ( f o r a d i s c u s s i o n o f t h e d i f f i c u l t i e s c o n n e c t e d w i t h t h e u s e o f t h e o t h e r a p p r o x i m a t e convex s e t s o f S e c t i o n 5.2, s e e Marrocco /l/). Once a g a i n , we s h a l l c o n f i n e our a t t e n t i o n t o t h e f i n i t e - d i m e n s i o n a l a n a l o g u e o f (l7h1( s e e ( 8 . 4 6 ) ) and we s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n 5. We r e c a l l t h a t

K&

and w e i n t r o d u c e

We n e x t d e f i n e , f o r 1 = 1 , 2 t h e f o l l o w i n g a p p r o x i m a t i o n s of

m1 :

(8.76)

(nd

4

Elasto-plastic torsion of a c y l i n d r i c a l bar

212

(CHAP. 3 )

and it i s then easy to prove the following propositions: Proposition 8.9. The problem(llJadmits one and only one solu t i o n , ( u k , p 3 , and when E -+ o we have, for 1 = 1 or 2

dk+d

(8.77)

where

4 is the soZution

(see ( 5 . 2 3 ) ) of

u i + l j + l - uij+l + ui+lj - uij

2h

V l = 1,2.

.

Solution of ( P o )by penalty methods

(SEC. 8 )

213

Remark 8.9. R e l a t i o n s ( 8 . 7 8 ) , (8.791, (8.80) a r e t h e d i s c r e t e H analogues o f (8.48). Remark 8.10.

There i s e q u i v a l e n c e between (8.80) and:

vMi+ 1/2 j + I/2

'h

and where Pij i s t h e o r t h o g o n a l p r o j e c t i o n o p e r a t o r from R2 + 5 r e l a t e d t o t h e E u c l i d i a n norm of R2, or:

8.3.4

,

Application to the solution of the approximate problems (111). Solution of the penalised approximate problem by over-relaxation with projection.

We s h a l l c o n f i n e o w a t t e n t i o n t o t h e e x t e r i o r approximations of S e c t i o n 8 . 3 . 3 .

Preliminary remark:

There i s e q u i v a l e n c e between:

Description of the over-relaxation algorithm. I n t h e f o l l o w i n g , it w i l l b e assumed t h a t t h e i t e r a t i o n s a r e performed w i t h i n c r e a s i n g i, and t h e n w i t h i f i x e d and i n c r e a s i n g we t h u s u s e t h e a l g o r i t h m : j;

(8;86)

up,pf

given a r b i t r a r i l y ,

214

E l a s t o - p l a s t i c t o r s i o n o f a c y l i n d r i c a l bar

t h e n , 4.d b e i n g assumed known, t h e c o o r d i n a t e s a r e determined u s i n g :

- u;+ l j + 1 +U;:;j+ (8.87)

<

1 +u;+ 1j - 1

+Pli+ 1/2 j + 1 / 2 -PL-

+G:;j- 1 1/2J- 112 -Pli-

2h

-(1

U;+ 1j + 1 + U;:;, + I +@+ 1j +E)

Having c a l c u l a t e d

(8.90)

and

(8.91)

+

of

4"

1/2 j - l / Z

+

-

@ : I I$ -

2 h2

+pli+ 112 j + I p - ~ l i -112 j +

(8.88)

1

4 ';

+

2 hZ 11.7 j + 1/2+p;i+

(CHAP. 3 )

112 + P ~ ~ + jI+/ z 112 -Pli- I / Z j - 112

2h

i n t h i s way,

E

+

Ah i s determined u s i n g :

(SEC.

8)

Solution of ( P o ) by penalty methods

215

I n (8.89), (8.91)we t a k e O ~ w ~ , o ~ < w2 . The above a l g o r i t h m i s j u s t i f i e d through t h e c h a r a c t e r i s a t i o n of P r o p o s i t i o n 8.10, Remark 8.10 and t h e p r e l i m i n a r y remark made earlier

.

Convergence of the algorithm By u s i n g a block v a r i a n t of Theorem 1.3 o f Chapter 2, S e c t i o n ( s e e Glowinski /4/ Theorem 7.1, S e c t i o n 7 . 4 ) t h e following p r o p o s i t i o n may b e proved:

1.4

P r o p o s i t i o n 8.10 If 0 c wl c 2. 0 < w2 c 2 there i s convergence o f the algorithms (8.86), (8.87),(8.89),(8.go), (8.91)and (8.861, (8.88), (8.89),(8.90), (8.91)o r , when n - + + m , (8.92)

G + dk, V l = 1,2

(8.93)

p;+pi.

VI = 1,2.

W

8.3.5 Application t o the soZution o f the approximate problems (IV)

.

Examples.

Example 1. We c o n s i d e r Example 2 o f S e c t i o n 7 . 5 , which was t r e a t e d e a r l i e r ( u s i n g t h e o t h e r p e n a l t y method) i n S e c t i o n 8.2.5, Example 1; we t h u s have Q = ]0,1[ x ]0,1[ and f = 10. Approximation of (Po) : We use t h e approximation S e c t i o n 5.3, S e c t i o n 8.3.3).

(POJ14( s e e

Numerical values of the parameters: Mesh s i z e :

h = 1/20

I n i t i a l i s a t i o n of algorithm (8.86)( 8 . 8 7 ) (8.89)(8.9)(8.91): up = 0, pp = 0.

Termination c r i t e r i o n :

Analysis of t h e nwnerical r e s u l t s : With t h e v a l u e of E f i x e d d u r i n g t h e i t e r a t i v e p r o c e s s , t h e r e s u l t s shown i n Table 8.1 were o b t a i n e d . The v a l u e s of u ( 0 . 5 , 0 . 5 ) i n Table 8.1 may be compar-ed with t h e v a l u e 0.413 o b t a i n e d by s o l v i n g t h e problem ( P I ) ( s e e S e c t i o n 3) a s s o c i a t e d w i t h t h e preceding problem (Po) u s i n g t h e method of S e c t i o n 3.4 w i t h h = 1 / 4 0 . Table 8.1 shows t h a t f o r small v a l u e s of E convergence is very slow; t h e r a t e of convergence w a s improved by reducing E andET,

Blasto-plastic torsion of a cylindrical bar

216

(CHAP. 3 )

i n p a r a l l e l , during t h e i t e r a t i v e process: more p r e c i s e l y , t h e p r o c e s s s t a r t s w i t h E = lo-' and cT = lo-' and E and E= a r e t h e n d i v i d e d by 10 when

The improvement t h u s b r o u g h t a b o u t , b o t h i n t e r m s o f t h e r a t e o f convergence and t h e p r e c i s i o n o b t a i n e d , i s c l e a r l y demonstrated by t h e r e s u l t s g i v e n i n Table 8 . 2 .

Table

8. I ( Example 1

u(o.5, 0.5)

Number o f iterations

w1

wz

1.8

1

95

10-210-3 1.8

1

269

1 0 - 3 10-4 1.8

1

E

ET

lo-' lo-'

0.414 Table

853

Computation

t i m e i n secs on C I I 10070

120 s

8.2 (Example 1)

The number o f i t e r a t i o n s i n d i c a t e d i n T a b l e 8 . 2 r e l a t e s t o t h e o v e r a l l p r o c e s s ( i . e . t h e i t e r a t i o n "count" i s n o t r e s e t t o z e r o i n t h e c o u r s e of t h e e x e c u t i o n when convergence o f t h e sub-problems is o b t a i n e d ) . Table 8.3 shows t h e number o f i t e r a t i o n s r e q u i r e d f o r converge n c e , as a f u n c t i o n o f q a n d E , when w 2 = 1. uIo - 0, pp = 0 , ET = 10-4.

(SEC. 8 )

217

Solution of ( P o ) by penalty methods

I

1.6

I

183

I

1055

-1.9

I

197

I

883

I

I

I t may be seen from Table 8 . 3 t h a t t h e number of i t e r a t i o n s r e q u i r e d for convergence i s a r e l a t i v e l y f l a t f u n c t i o n of w, over q u i t e a wide range e i t h e r s i d e of t h e optimal v a l u e of wl.

Example 2 . We consider a g a i n Example 3 of S e c t i o n 7.6, which w a s t r e a t e d earlier i n S e c t i o n 8.2.5, Example 2; w e thus have R = 10, I[ x ]0,1[ and f defined by ( 8 . 4 1 ) . To s o l v e t h e above problem numerically we use, a s i n t h e preceding example, t h e approximation (POJl4o f ( P o ) , with h = 1 / 2 0 , t h e algorithm (8.86), (8.87), (8.89), ( 8 . 9 0 ) , (8.91) being i n i t i a l i s e d by uf = 0,p: = 0. Table 8.4 summarises t h e r e s u l t s obtained with E f i x e d , and Table 8.5 t h o s e obtained by decreasing E and E= i n t h e course of t h e i t e r a t i o n s , as i n t h e preceding example:

u(0.25,0.75)

Number of

iterations

Computation t i m e i n secs o n C I I 10070

0.125

277

489

0.122

1895

336 s

0.088

Table

8.4

(Example 2 )

Elasto-plastic t o r s i o n of a cylindrical bar

218

Table

8.5

(CHAP.

3)

(Example 2 )

The v a r i a t i o n s o f t h e number o f i t e r a t i o n s r e q u i r e d f o r convergence, as a f u n c t i o n o f w1 and E , when w2 = 1, ut = 0 , p t = 0, E~ = a r e i n d i c a t e d i n Table 8.6:

182

1.6

1.

1.9

Table

8.4

I 8.6

427

2 473

I

2835

I

(Example 2 )

Compar i sons

I n view o f t h e preceding r e s u l t s , t h e two p e n a l t y methods cons i d e r e d i n S e c t i o n s 8.2 and 8 . 3 , r e s p e c t i v e l y , appear comparable as f a r as computation time i s concerned; however, t h e formulati o n o f t h e approximate problems i s r e l a t i v e l y complicated i n t h e c a s e o f t h e f i r s t method because of t h e highly nonlinear n a t u r e of t h e p e n a l i s e d problem; t h i s d i f f i c u l t y does n o t e x i s t i n t h e second method, f o r which t h e corresponding p e n a l i s e d problem i s almost linear.

(SEC. 9)

Solution of ( P o ) by d u a l i t y methods

219

By way o f i l l u s t r a t i o n , it i s worth p o i n t i n g o u t t h e u s e by R . TrEmoliPres /1/ o f t h e method of centres with variable trunca t i o n d e s c r i b e d i n Chapter 2, S e c t i o n 3.4; t h i s method, t h e p r i n c i p l e o f which c o n s i s t s o f a p p r o x i m a t i n g t h e s o l u t i o n w h i l s t r e m a i n i n g w i t h i n t h e i n t e r i o r o f t h e domain o f t h e c o n s t r a i n t s , i s more c o s t l y i n c o m p u t a t i o n t i m e s i n c e f o r t h e problem ( P o ) , i n 62 = ]0,1[ x ]0,1[ w i t h f = 1 0 and w i t h t h e same c h a r a c t e r i s t i c s as i n Example 1 o f S e c t i o n s 8 . 2 . 5 , 8.3.5, t h e s o l u t i o n o f t h e approxima t e problem:

Min Ji(uk) V h E Kbh

t a k e s 24 s e e . on a n IBM 360191 i f h = 1/10 and 1 3 9 s e e . i f h = 1 / 2 0 , i . e . c o m p u t a t i o n t i m e s a p p r o x i m a t e l y 20 t i m e s l o n g e r t h a n t h o s e r e q u i r e d when t h e e x t e r i o r p e n a l t y methods o f t h i s S e c t i o n 8 are used. We r e f e r t h e r e a d e r t o T r & m o l i h r e s , loc. c i t . , f o r f u r t h e r d e t a i l s o f t h e a p p r o x i m a t e s o l u t i o n o f ( P o ) by t h i s method.

9.

SOLUTION OF THE APPROXIMATE PROBLEMS RELATING TO ( P o ) BY DUALITY METHODS

9.1

I n t r o d u c t i o n and s y n o p s i s

I n t h i s s e c t i o n we s h a l l s t u d y t h e a p p l i c a t i o n o f t h e duality methods of Chapter 1, S e c t i o n s 3 . 4 , 3 . 5 and o f Chapter 2, S e c t i o n 4 , t o t h e s o l u t i o n o f t h e f i n i t e - d i m e n s i o n a l problems o b t a i n e d by a p p l y i n g t o ( P o ) t h e a p p r o x i m a t i o n methods o f S e c t i o n s 4 and 5. I n o r d e r t o s i m p l i f y t h e p r e s e n t a t i o n , t h e v a r i o u s approaches used w i l l b e d e m o n s t r a t e d for t h e c o n t i n u o u s problem.

9.1.1

A f i r s t formulation

We c o n s i d e r a g a i n t h e formalism o f Chapter 2, S e c t i o n 4 . 2 , we thus define: Example 4 . 2 ; (9.1)

A , = L:(62),

(9.2)

{

: HA(62) --t L’(62) @,(u) = I grad u

I

- 1,

by

Elasto-plastic torsion of a cylindrical bar

220

and t h e Lagrangian 9, : HA(P) x A , (9.3)

9 , ( v , 4) = JW

+R

(CHAP. 3 )

by

+

From t h e r e l a t i o n

we deduce t h a t i f 9,admits a s a d d l e p o i n t { u , p } o n H&?) x A , , hence i n t h i s c a s e , f o r which then u i s t h e solution of ( P ) ; Chapter 2, S e c t i o n 4 . 3 are satisassumptions ( 4 . 1 0 ) , ( 4 . 1 1 ) f i e d , w e c a n s o l v e ( P o )by a p p l y i n g t h e a l g o r i t h m ( 4 . 1 2 ) , , ( 4 . 1 5 ) o f Chapter 2, S e c t i o n 4 . 3 ; namely:

08

...

In

w i t h p" known(€ Al),u"is d e f i n e d as b e i n g t h e element (9.5)

(9.7)

o f H i @ ) which m i n h i s e s

pm> 0 ,

J(u)

+

p" @,(u) dx,

sufficiently s m a l l .

The m i n i m i s a t i o n problem d e f i n e d by ( 9 . 5 ) ( a n d which admits one and o n l y o n e s o l u t i o n ) i s a problem o f m i n i m i s a t i o n o f a nondifferentiable f u n c t i o n a l , v e r y similar t o t h e f l o w problem cons i d e r e d i n Chapter 1, S e c t i o n s 1 . 4 , 3 . 4 , and i n Chapter 2, S e c t i o n 4.2, Example 4 . 1 ; t h e use of t h e algorithm ( 9 . 5 ) , (9.6), (9.7) f o r s o l v i n g ( P o ) w i l l b e c o n s i d e r e d i n Chapter 5, S e c t i o n 8 . 8 9.1.2

A second formulation

We d e f i n e :

and t h e Lagrangian Y 2: H i ( f 2 ) x A2 + R by (9.10)

92(u,

q) = J(u)

From t h e r e l a t i o n

+

221

Solution of ( P o )bg duality methods

(SEC. 9)

we deduce t h a t i f Y z a d m i t s a s a d d l e p o i n t { u , p } on H&2) x A2 , t h e n u i s t h e s o l u t i o n o f (Po). The c o n s i d e r a t i o n s o f Chapter 2, S e c t i o n 4 , l e a d - i n a t o t a l l y formal manner, s i n c e w e a r e no l o n g e r i n t h e Hilbert s e t t i n g o f Chapter 2, S e c t i o n 3.4 ( l ) - t o u s i n g t h e a l g o r i t h m ( 4 . 1 2 ) , , (4.15) o f Chapter 2, S e c t i o n 4 . 3 i n o r d e r t o s o l v e ( P o ) ; t h a t i s :

. ..

[

w i t h p” known ( € A 2 ) , d ‘ i s d e f i n e d as b e i n g t h e element o f H,’(Q) which minimises

(9.13)

pn+’ = SUP (0-P”

(9.14)

pm> 0 ,

+ P n @~(tf))

J(u)

+ iInp” @&)

dx ,

9

sufficiently small.

I t may b e noted t h a t , g i v e n p” o f t h e e l l i p t i c problem

, fl

i s t h e s o l u t i o n i n Hi(l2)

The u s e o f ( 9 . 1 2 ) , ( 9 . 1 3 ) , ( 9 . 1 4 ) f o r s o l v i n g ( P o ) ( i n f a c t t h e i n t e r i o r and e x t e r i o r approximations o f ( P , ) ) w i l l a r i s e i n S e c t i o n s 9 . 2 , 9.3, 9.4. 9.1.3

A third formulation

I n t h e two p r e c e d i n g approaches t h e d i f f i c u l t y o f t h e problem ( i . e . t h e c o n s t r a i n t I grad u 1 Q 1 ) w a s “ e l i m i n a t e d ” ( a t l e a s t , f o r m a l l y ) by a s s o c i a t i n g w i t h it a Lagrange m u l t i p l i e r ( o r a Kuhn-Tucker m u l t i p l i e r ) s a t i s f y i n g a non-negativity c o n d i t i o n , s i n c e t h i s t y p e of c o n s t r a i n t does n o t p r e s e n t any d i f f i c u l t y t h e new approach i s based on t h e f o l l algorithmically s p e a k i n g ; owing v a r i a n t o f t h e c o n s i d e r a t i o n s o f S e c t i o n 8 . 3 . 1 . Once a g a i n we write: (9.16)

A = { q 1 q E L’(l2) x L2(Q),q = (ql, q,), q:(x)

(9.17)

X , = { (u, q) I u E H i @ ) , q E A, q = grad u } ;

(l)

+ q%x)

d 1

a .e. }

And s i n c e , a p a r t from t h e c a s e f = c o n s t a n t , t h e problem of t h e e x i s t e n c e o f a s a d d l e p o i n t f o r S?,, i n H d ( Q ) x A , , seems t o b e open even f o r f e L Z ( Q )( s e e Remark 9 . 2 b e l o w ) .

E l a s t o - p l a s t i c t o r s i o n o f a c y l i n d r i c a l bar

222

(CHAP. 3 )

t h e r e i s t h e n e q u i v a l e n c e between ( P o ) and

S i n c e t h e p r o j e c t i o n L'(62) x L'(62) + A d o e s n o t pose any p r a c t i c a l d i f f i c u l t i e s ( s e e ( 8 . 5 2 ) ) , we s h a l l b e c o n t e n t t o decouple u and q , 2 . e . t o e1imina;te t h e r e l a t i o n q = gradu , by a s s o c i a t i n g w i t h it a Lagrange m u l t i p l i e r ( i n S e c t i o n 8 . 3 we proceeded u s i n g penal-

isation)

(1)

We t h u s d e f i n e : (9.19)

L = L'(l2) x L'(62)

and t h e L a g r a n g i a n p3: HA@) x L x L + R (9.21)

by

Y,(u,q;p)=~

From t h e r e l a t i o n

we deduce t h a t i f Y3admits a s a d d l e p o i n t { u , p ; d } on x A x L, t h e n { u , p } i s t h e s o l u t i o n o f t h e problem (9 . 18 ) e q u i v a l e n t t o (Po); t h e algorithm (4.121, , ( 4 . 1 5 ) o f Chapter 2, S e c t i o n 4.3 t h e n t a k e s t h e form:

...

(9.23)

(9.25)

(l)

{

w i t h 1" known

(€

L ) , we d e f i n e { g , p " } as b e i n g t h e

element o f H,'(Q) x A which minimises Y 3 ( u , q ; A''),

p,, > 0 ,

s u f f i c i e n t l y small.

A method combining b o t h t h e s e methods w i l l b e found i n S e c t i o n 10.

(SEC. 9 )

Solution of ( P o ) b y d u a l i t y methods

With t h e f u n c t i o n k g i v e n i n L , l r a n d f a r e , solutions of

(9.26) (9.27)

- Ad'= 2 f

223

respectively,

+ divA",

p" = PA(A").

The u s e o f ( 9 . 2 3 ) , ( 9 . 2 4 1 , ( 9 . 2 5 ) f o r s o l v i n g ( P o ) w i l l be d i s c u s s e d i n S e c t i o n 9.5.

Remark 9.1. The method o f S e c t i o n 9 . 1 . 3 i s a method o f decomposition-coordination i n t h e s e n s e o f Bensoussan-Lions-Temam /I/, a p p l i e d t o t h e s o l u t i o n of t h e problem (9.18) e q u i v a l e n t t o (Po). 9.1.4

Remarks

Remark 9 . 2 . The problem o f t h e e x i s t e n c e of a s a d d l e p o i n t 9,) on H ; ( Q ) x 4 ( r e s p . H&2) x A2, Hi@) x A x L ) f o r Y , ( r e s p . Y2, seems t o b e open; i n t h i s c a s e it i s p o s s - i b l e , however, by u s i n g ( f o r example) Bensoussan-Lions-Tham /l/, Chapter 2, S e c t i o n 2, s u b s e c t i o n 1, t o prove t h e e x i s t e n c e o f s a d d l e p o i n t s f o r (9.28) (9.29)

(9.30) i n W$m(Q) x A x (Lm(Q))'x (Lm(Q))' where (,) d e n o t e s t h e b i l i n e a r form o f t h e d u a l i t y between Lm(Q) and i t s d u a l (Lm(Q))' w i t h

av

Vi, u Ir = 0

(9.31)

Wd*m(Q)=

(9.32)

(Lm(Q));= { q I q E (Lrn(Q))',( 0, q ) 2 0 vu E L 3 Q ))

E L "(Q),

On t h e o t h e r hand, t h e r e i s no d i f f i c u l t y i n p r o v i n g t h e e x i Y 3), s t e n c e o f s a d d l e p o i n t s f o r t h e r e s t r i c t i o n o f Y l ( r e s p . g2, t o t h e f i n i t e - d i m e n s i o n a l convex sets and s p a c e s a p p e a r i n g i n t h e i n t e r i o r approximations o f ( P o ) defined i n S e c t i o n 4 ( s e e Sections 9.2, 9.3 below); t h e same a p p l i e s f o r t h e e x t e r i o r

Elasto-plastic torsion of a cylindrical bar

224

approximations of Section

(CHAP. 3 )

5 ( s e e Sections 9.4, 9.5 below).

8

Remark 9 . 3 . I n t h e c a s e f = c o n s t a n t (which c o r r e s p o n d s t o t h e p h y s i c a l problem o f S e c t i o n 1 . 2 ) ¶ H . B r e z i s /5/ h a s demonstr a t e d t h e e x i s t e n c e o f P E A , ( = L:(L?)), w i t h p = 0 i n t h e e l a s t i c zone, such t h a t { u , p }, v h e r e u i s t h e s o l u t i o n o f (P,), i s t h e unique s a d d l e - p o i n t o f Y z i n H,'(L?) x A, ( f o r t h e i n t e r p r e t a t i o n o f t h i s r e s u l t i n c o n n e c t i o n w i t h problems i n mechanics, see DuvautLions /l/¶ Chapter 5 , S e c t i o n 6 . 6 ) . 9.2

9.2.1

A p p l i c a t i o n o f t h e d u a l i t y method o f S e c t i o n 9 . 1 . 2 t o t h e s o l u t i o n o f t h e a p p r o x i m a t e problems. Case o f t h e f i n i t e element a p p r o x i m a t i o n o f S e c t i o n 4 . 1 .

On the existence of a saddle point

We s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n 4 . Having d e f i n e d t h e s p a c e V, i n S e c t i o n 4 . 1 . 1 we i n t r o d u c e Lk, a subspace of L m ( 0 ) :

with

(9.34)

OT =

c h a r a c t e r i s t i c function of T ,

and

With t h e L a g r a n g i a n Y Z d e f i n e d by ( 9 . 9 ) , (9 . 10 ) w e t h e n have: P r o p s i t i o n 9.1. { u,,p, } On Vh X A ,

(9.36)

uh

The LagrangianP2 a h i t s a saddle point with:

the solution o f

Min

J(u,J,

01,c KO n V I

and (9.37)

PA( grad u,

1' -

1) = 0

.

Proof. S i n c e t h e s p a c e s Vhand 4 a r e f i n i t e - d i m e n s i o n a l , t h i s r e s u l t f o l l o w s immediately from R o c k a f e l l a r /3/, S e c t i o n 28, Theorem 28.2, 28.3. 8

(SEC.

9)

Solution of ( P o ) by d u a l i t y methods

225

Remark 9.4. S i n c e t h e p a i r { u h , p h ) i sa s a d d l e - p o i n t o f g z o n V, x A,, w e have PZ(th,p#,)6 Uz(Uh,pJ b h E V, s o t h a t , e x p l i c i t l y ,

= 0, w e have, t a k i n g account

and i n p a r t i c u l a r i f

Of

(9.371,

Under t h e c o n d i t i o n s f o r a p p l y i n g Theorem 6.1, S e c t i o n 6.2.1, when h - 0 w e have uh d u s t r o n g l y i n HA(Q)nL w ( f l ) , where u i s w e t h u s have, s i n c e ph 2 0, t h e s o l u t i o n of ( P o ) ;

11 ph lILl(I2) d

(9.40)

vh

'

an e s t i m a t e i n t'(Q) f o r ph.

SoZution of the approximate problem by means of a saddle-point-seeking algorithm ( I ) . Description of the algorithm.

9.2.2.

.. .

We use t h e a l g o r i t h m (4.12), , (4.15) o f Chapter 2, S e c t i o n 4.3, which i n t h i s p a r t i c u l a r c a s e t a k e s t h e form:

(9.41)

i

with of

P;: known

v,,which

(E

Ath) we d e f i n e

6

+-

Jn

minimises J(t+,)

as b e i n g t h e element

fl (I grad uh I' - 1) h,

+ p,(l grad 141' - l ) ) ,

(9.42)

P;"

(9.43)

pm> 0 ,

= SUP ( 0 , ~ ;

s u f f i c i e n t l y small.

The a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) i s a n "approximation" o f a l g o r i t h m ( 9 . 1 2 ) , ( 9 . 1 3 ) , ( 9 . 1 4 ) ; it may b e noted t h a t , given fl , J; is t h e unique s o l u t i o n o f :

i f. (1

(9.44)

+ pi) grad 4.grad uh dX =

v,,

In

fUh

dr

VUh €

v,,

which i s a variational equation in Vh. rn

Remark 9.5. I t i s p o s s i b l e t o w r i t e (9.41) ( a n d / o r ( 9 . 4 4 ) ) and ( 9 . 4 2 ) e x p l i c i t l y i n terms o f t h e parameters qr, T E C , and of

E l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l bar

226

(CHAP. 3 )

t h e "nodal" p a r a m e t e r s ui

= %(Mi) 9

=

G(Mi)

$

0

Mi 6 zh

( s e e S e c t i o n 4 . 1 . 1 , r e l a t i o n ( 4 . 8 ) ) by u s i n g t h e r e l a t i o n s o f Section 4.1.4; w e r e f e r t h e r e a d e r t o B o u r g a t /1/ f o r s u c h a n e x p l i c i t formulation. rn

S o l u t i o n of t h e approximate problem by means of a Convergence o f saddle-point-seeking algorithm ( 11) t h e algorithm.

9.2.3.

.

We have : Proposition 9.2. The sequence (4)" defined ( 9 . 4 2 ) , ( 9 . 4 3 ) converges t o t h e s o l u t i o n uh o f

by algorithm ( 9 . 4 1 ) ,

Proof. Taking a c c o u n t o f P r o p o s i t i o n 9.1 t h i s f o l l o w s from Theorem 4 . 1 o f Chapter 2 , S e c t i o n 4.3, as l o n g as t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d ( s e e r e l a t i o n ( 4 . 1 1 ' ) , Chapter 2, S e c t ion 4.3 ) : (i) (9.45)

eZh: Vh-P Lh

The mapping @zh(t+,)

= I grad

1' -

defined by:

1

i s locally Lipschitz;

(ii)

(9.46)

Vqe &,

Min Uh e v h

J(Uh)

the solution of

:b

+-

q@zh(U) dx

1

remains w i t h i n a bounded region independent of q. P o i n t ( i ) follows immediately s i n c e , vuh E V, I grad uhIz i s quadr a t i c w i t h r e s p e c t t o t h e "nodal" p a r a m e t e r s u, = uh(Mi), and as r e g a r d s ( i i ) ,t h e s o l u t i o n ulg o f ( 9 . 4 6 ) i s t h e t h u s C"; s o l u t i o n i n vh o f t h e v a r i a t i o n a l e q u a t i o n (9.47)

(1 4- q) grad u,.grad

uh dX =

In

VUh E v h9

f U h dX

so t h a t s e t t i n g uh = u, i n ( 9 . 4 7 ) and b e a r i n g i n mind t h a t

(9.48)

(I"

I grad u, IZ

dyZc f Q

II II''(rn

(7

q 3 0,

9

( ' ) C i s t h e PoincarC c o n s t a n t i n . H i ( n ) i . e . t h e r e c i p r o c a l o f t h e s q u a r e r o o t o f t h e s m a l l e s t e i g e n v a l u e o f - A i n Hi@).

(SEC. 9)

Solution of (Po)by d u a l i t y methods

which proves ( i i ) and t h e p r o p o s i t i o n .

227

rn

Remark 9.6. By e x p l i c i t l y w r i t i n g o u t t h e c a l c u l a t i o n s i n t h e proof of Theorem 4 . 1 of Chapter 2, S e c t i o n 4.3, i n t h e p a r t i c u l a r c a s e o f t h e algorithm (9.41), ( 9 . 4 2 ) , ( 9 . 4 3 ) , it may be shown t h a t t h e c o n d i t i o n f o r convergence o f t h e algorithm i s given by

where t h e c o n s t a n t C i s t h e one appearing i n (9.48) and I n p r a c t i c e , t h e e s t i m a t e (9.49) i s very y ( f & = TMin a r # , (Area of 2 ').

pessimistic, a t l e a s t f o r t h e examples considered below. Solution of the approximate probZem by a saddle-point-' seeking aZgorithm (111). PracticaZ considerations on the use of algorithm ( 9 . 4 1 ) , (9.421, ( 9 . 4 3 ) .

9.2.4.

The work i n t h i s s e c t i o n and i n t h e subsequent Section 9.2.5, follows Bourgat 111.

SoZution of ( 9 . 4 4 ) . The determination o f 4 from fi reduces, using ( 9 . 4 4 ) , t o t h e s o l u t i o n o f an (approximate) e l l i p t i c problem which i s of order 2, l i n e a r , and with c o e f f i c i e n t s which vary i n c1 and with n ; t h i s problem, which i s o f D i r i c h l e t type, has been solved by point over-relaxation t a k i n g as v a r i a b l e s t h e nodal values v,, VM&

Termination c r i t e r i a :

$ is i d e n t i f i e d t o

(@,,,.j#,

; we

then d e f i n e (9.50)

D,(n) = Max

I ul -

1,

Mt

c

(9.51)

D,(n) = Y"Ph

lul-ul-ll

c

lull

'

MtcPr

and t a k e f o r algorithm ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) , a termination c r i t e r i . o n of t h e form (9.52)

Di(n) < 8

.

Moreover, a t l e a s t a t t h e s t a r t of t h e i t e r a t i v e process, t h e r e i s ' n o p o i n t i n s o l v i n g t h e D i r i c h l e t sub-problems ( 9 . 4 4 ) f o r t h e over-relaxation algorithm applied with high p r e c i s i o n ; t o problems ( 9 . 4 4 ) , w e t h e r e f o r e use a termination c r i t e r i o n o f

-

228

Elasto-plastic torsion o f a c y l i n d r i c a l bar

t h e type (9.52) with

E

r e p l a c e d by E,

, defined

(CHAP. 3 )

by

Initialisations: I n t h e f o l l o w i n g examples, a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) has been i n i t i a l i s e d w i t h pt = 0 a n d , i n t h e d e t e r m i n a t i o n o f 4 from p ; by s o l v i n g ( 9 . 4 4 ) , t h e o v e r - r e l a x a t i o n a l g o r i t h m h a s been i n i t i a l i s e d w i t h G-'. 9.2.5

Solution of the approximate problem using a saddlepoint-seeking algorithm (IV). Examples

Example 1.

We c o n s i d e r a g a i n Example 1 o f S e c t i o n

7.4, t h a t

is L a = { x I x ~ R ' , x : + ~ < l }and f = l O , t h e notatt h e e x a c t s o l u t i o n b e i n g g i v e n e x p l i c i t l y by ( 7 . 8 1 ) ; i o n and t r i a n g u l a t i o n s used a r e t h e same as t h o s e i n S e c t i o n 7.4 (see F i g u r e s 7 . 3 and 7 . 4 ) . Using t h e i n i t i a l i s a t i o n s i n d i c a t e d i n S e c t i o n 9 . 2 . 4 , we t a k e i n (9.52), i n (9.531, pm=p=l i n ( 9 . 4 2 ) and w = 1 . 5 , where w i s t h e o v e r - r e l a x a t i o n p a r a m e t e r i n t h e s o l u t i o n of (9.44) ( s e e Section 9.2.4). The main r e s u l t s c o n c e r n i n g t h e a p p r o x i m a t e s o l u t i o n o f ( P o ) by means o f t h e above t e c h n i q u e s a r e shown i n T a b l e 9 . 1 ( t h e n o t a t i o n i s t h e same as t h a t i n T a b l e 7 . 1 ) . Table 9.2 shows, f o r t h e c a s e o f t r i a n g u l a t i o n y 2 ,t h e number o f o v e r - r e l a x a t i o n i t e r a t i o n s r e q u i r e d t o s o l v e t h e n t h problem i n ( 9 . 4 4 ) , u s i n g t h e i n i t i a l i s a t i o n s and t e r m i n a t i o n c r i t e r i o n indicated i n S e c t i o n 9.2.4. The r e s u l t s g i v e n i n t h i s s e c t i o n and i n T a b l e Remark 9.7. 7 . 1 o f S e c t i o n 7.4 would a p p e a r t o i n d i c a t e , a t l e a s t , f o r t h e p a r t i c u l a r c a s e c o n s i d e r e d (l), t h a t t h e method o f o v e r - r e l a x a t i o n w i t h p r o j e c t i o n d e s c r i b e d i n S e c t i o n 7 i s more r a p i d t h a n moreover t h e method o f S e c t i o n t h e d u a l i t y method g i v e n above; 7 o f f e r s t h e a d v a n t a g e s o f a s i m p l e r i m p l e m e n t a t i o n ( a n d hence e a s i e r programming) and o f a s m a l l e r s t o r a g e r e q u i r e m e n t ( f o r example, t h e r e i s no need t o s t o r e t h e c o o r d i n a t e s o f pi ) . I t should b e p o i n t e d o u t , however, t h a t i n t h e p h y s i c a l problem o f e l a s t o p l a s t i c t o r s i o n ( i . e . f = c o n s t a n t ) t h e f u n c t i o n p i s of g r e a t p h y s i c a l i n t e r e s t ( s e e Duvaut-Lions 111, C h a p t e r 5 ) and t h i s may j u s t i f y u s i n g a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) s i n c e

(')

T h i s remark i n f a c t i s c o m p l e t e l y g e n e r a l ( s e e b e l o w ) .

(SEC. 9 )

Solution of ( P o ) by d u a l i t y methods

229

t h i s g i v e s , as w e l l as an approximation o f t h e s o l u t i o n u of ( P o ) , an approximation o f t h e f u n c t i o n p .

n

1 2

4

6

8

10

28

24

22

7

6

Number of over-relaxation iterations

Further examples.

S t i l l keeping

f = 10, we have a l s o t r e a t e d

t h e following cases;

Example 2

R

Example 3

R i s t h e e q u i l a t e r a l t r i a n g l e w i t h u n i t side.

Example 4

R i s t h e T-shaped domain shown i n F i g u r e 9.1.

= 10, 1[ x 10, l[.

For t h e t r i a n g u l a t i o n s shown i n F i g u r e s 9.2, 9.3, 9.4, respe c t i v e l y , Table 9 . 3 summarises t h e main r e s u l t s f o r t h e s o l u t i o n of t h e above examples; w e r e f e r t h e r e a d e r t o Bourgat /1/ for

Elasto-plastic t o r s i o n of a c y l i n d r i c a l bar

230

x1

(CHAP. 3 )

t

Fig. 9.1.

Square

Number of v e r t i c e s N = Card(&)

I

225

N' = Card (yh) Pn

I

t

384

I I

432

10-3

I

10-3

169

195

=P 0

I

1.5

I &

r

5 x 10-4 10-3

Comp. time C I1 10070 Table

9.3

f u r t h e r d e t a i l s . I n a d d i t i o n t o t h e t r i a n g u l a t i o n s used, F i g u r e s 9.2, 9.3 and 9.4 a l s o show t h e approximated e l a s t i c and p l a s t i c zones, a t r i a n g l e T b e i n g t a k e n t o l i e i n a p l a s t i c zone when I grad u, I 2 0.995 on T .

Solution of ( P o ) by duality methods

(SEC. 9 )

Fig. 9.2.

Square cross-section : F i n i t e elements ('512 triangles), C = 10. P l a s t i c zones: @ I n i t i a l triangulation:

9.3

231

-

Application o f t h e d u a l i t y method of Section 9.1.2 t o t h e s o l u t i o n o f t h e approximate problems. Case of t h e i n t e r i o r approximation'by eigenfunctions of - A i n HA(Q1

Preliminary remark. I n g e n e r a l t h e eigenfunctions of - A i n H,'(Q) are not known explicitly, and t h i s s e r i o u s l y r e s t r i c t s t h e use of t h e approximation described i n S e c t i o n 4.2; w e have t h e r e f o r e confined our a t t e n t i o n t o a s i n g l e example, corresponding t o Q = ]0,1[ x ]o and f = 10, f o r which t h e above eigenfunctions and t h e corresponding eigenvalues are known e x p l i c i t l y and f o r which approximate s o l u t i o n s f o r problem ( P o ) have been

232

Elasto-plastic torsion o f a cylindrical bar

Fig. 9.3.

Fig. 9.4.

(CHAP. 3 )

F i n i t e - e l e m e n t c a l c u l a t i o n of t h e e l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l bar w i t h t r i a n g u l a r c r o s s section. Number of f i n i t e elements: 384.

T-section:

F i n i t e elements (432 t r i a n g l e s ) , = 10 P l a s t i c zones:

c

I n i t i a l triangulation:

-

(SEC. 9)

S o l u t i o n o f ( P o ) by d u a l i t y methods

o b t a i n e d , by o t h e r methods e a r l i e r i n t h e p r e s e n t volume.

233

rn

Statement of t h e continuous problem and s p e c t r a l decomposition of - A i n H i @ )

9.3.1

We c o n s i d e r t h e problem ( P o ) r e l a t i n g t o R = )O, I[ x ]0,1[ and f = 10, which w a s t r e a t e d e a r l i e r i n S e c t i o n 3 . 5 (making u s e o f t h e e q u i v a l e n c e r e s u l t o f S e c t i o n 2 . 2 ) and i n S e c t i o n s 7.5, 8 . 2 . 5 , 8 . 3 . 5 and 9 . 2 . 5 , by v a r i o u s methods ( s e e a l s o S e c t i o n 9.4 b e l o w ) . For t h e above domain 62, t h e e i g e n f u n c t i o n s o f - A i n H i ( @

are g i v e n by: (9.54)

wpq(xl,x z ) = 2 sin / I

KS,

sin y x.r2 ,

p, q E

N,

and t h e c o r r e s p o n d i n g eigenvalues b y :

(9.55)

,3/

= nZ(p2

+ 92) .

The e i g e n f u n c t i o n s ( 9 . 5 4 ) a r e o r t h o g o n a l i n H;(R) and o r t h o normal i n ~'(62). rn 9.3.2

-

Explicit formulation o f t h e approximate problem

For a s l i g h t l y g r e a t e r d e g r e e o f g e n e r a l i t y , it i s n o t a b s o l u t e l y n e c e s s a r y t o assume f = c o n s t a n t . With M E N , we d e n o t e by V, t h e s u b s p a c e o f H;(n> g e n e r a t e d by t h e wI1 f o r 1 < p , q C M, t h a t is : (9.56)

{

V,=

u ~ u E H A ( R U) ,=

Using t h e n o t a t i o n

w i t h (assuming f

(9.58)

f,

=

C

P,w~,~,ER}

I Sp.q Q M

EL'@)

w&)

( 9 . 5 6 ) , w e have f o r

U E

V,

):

f (4dx .

I n t h e p a r t i c u l a r c a s e i n which

f = c o n s t a n t = C, we.have:

i f p and q a r e odd

(9.58')

fp4 =

0 otherwise

so t h a t i f

VE

V,, rn

Elasto-plastic torsion of a cylindrical bar

234

(CHAP. 3 )

I n view o f t h e symmetries o f t h e p a r t i c u l a r Remark 9.8. problem under c o n s i d e r a t i o n , it i s p o s s i b l e t o t a k e i n ( 9 . 5 7 ' ) only pm with p and q odd, t h e o t h e r c o e f f i c i e n t s being zero; however, w e have d e l i b e r a t e l y chosen not t o t a k e advantage of t h i s s i m p l i f i c a t i o n i n t h e following. For t h e approximation o f K O , we proceed as i n S e c t i o n 4.2.2, Remark 4.5: given M ' i n t e g e r > 0 (which w i l l tend towards i n f i n i t y ) w e put h = 1 / M ' and d e f i n e :

(9.59)

Rk = { Pij I Pij = (ih, jh), i, j E 2 } .

I n t h e p a r t i c u l a r c a s e under c o n s i d e r a t i o n , t h e s e t may be w r i t t e n e x p l i c i t l y i n t h e form

(9.60)

= { PijIP,, = (ih,jh), 1 < i , j <

M' - 1 } ,

KO, t h e approximation

giving, for

(9.61)

= &n

K ~ = { U ( U E V M I&v(Pij)I< ,

1

VPtjEah},

1

t h e c o n d i t i o n 1 grad u(Pij) d 1 being expressed ( w r i t i n g x l i = ih, xZj = j h ) i n t h e e x p l i c i t form

>' +

(9.62)

pppl cosp x x l i sin q I I X ~ ~

Relations ( 9.57) , ( 9.58)-, ( 9.61) , ( 9.62) a l l o w t h e approximate problem t o be expressed e x p l i c i t l y , i n terms o f pm(l d p , q d M ) , very simply: Mia J(u). UEK&

On the existence of a saddle-point

9.3.3

We denote by G, t h e mapping from hand s i d e of (9.6 2 ) and

(9.63)

A2h

=

{ qh Iqh

=

(qIj)prjcl2h,

qij 2

V,

+ R defined by t h e l e f t -

VPtj E a h}

.

It i s n a t u r a l t o a s s o c i a t e with t h e approximate problem t h e approximation of P2 ( s e e S e c t i o n 9 . 1 . 2 ) defined by:

and as t h e proof of P r o p o s i t i o n 9.1 can be adapted without d i f f i c u l t y , 9% admits a saddle-point, ( u M , p h } , on VM x A , with:

(9.65)

uM

the solution of

Mia J(u), UE&

Solution of ( P o )by d u a l i t y methods

(SEC. 9 )

9.3.4

Description of the algorithm and convergence

We use a f i n i t e - d i m e n s i o n a l v a r i a n t of algorithm

(9.14) , namely:

with

(9.67)

235

& known ( € A z L ) , w e

Hi1=

(9.69)

pn > 0 ,

(0, P;I

d e f i n e flu t o be t h e element

minimises Ug(uM,P;),

of V,which

(9.68)

(9.12) , (9.13) ,

+ Pn(G;(U",) - I))

VPi, E 0,

9

sufficiently s m a l l .

The proof of P r o p o s i t i o n 9.2 can be adapted w i t h o u t d i f f i c u l t y , and it may t h u s be shown t h a t t h e sequence (flu). d e f i n e d by (9.67), Min J(u). (9.68) , (9.69) , converges t o t h e s o l u t i o n of AeK8

9.3.5

Practical considerations on the use of algorithm ( 9 . 6 7 ) ,

( 9 . 6 8 ) , (9.69) The terms cospxx,, (resp. cosqnx,,) and sinptxIi (resp. sinqrrx,,) which appear i n algorithm (9.67) , (9.68) , (9.69) through t h e intermedi a r y o f t h e G,, ( s e e ( 9 . 6 2 ) , ( 9 . 6 8 ) ) are obviously c a l c u l a t e d once Ah wN i n t h e c a l c u l Moreover, p u t t i n g U", = and f o r a l l . I*P.q*M

a t i o n of I&( w i t h A known), u s i n g (9.67), t h e vector (A>)15p,qs,, i s t h e s o l u t i o n of a l i n e a r system of equations whose matrix ( t h e c o e f f i c i e n t s of which vary w i t h n ) is symmetric and positived e f i n i t e ( l ) ; t h i s system has been solved by t h e Cholesky method. For t h e i n i t i a l i s a t i o n o f ( 9 . 6 7 ) , ( 9 . 6 8 ) , (9.69), p , O - 0 w a s always used.

9.3.6

Numerical r e s u l t s

Algorithm ( 9 . 6 7 ) , (9.68), (9.69) w a s , applied t o t h e particular case c o r r e s p o n d i n g t o f = 10, and, s i n c e t h e exact s o l u t i o n i s not known, t h e reference s o l u t i o n w a s t a k e n as t h e approximate solution calculated by t h e method of S e c t i o n 3 ( a n d t h u s using t h e equivalence r e s u l t of S e c t i o n 2 . 2 ) w i t h a mesh s i z e of 1 / 4 0 .

Results f o r h = 1/20: For M = 10 ( i . e . 100 eigenfunctions wpr ) , a time of 100 sec w a s required on an IBM 360/91 t o perform 10 i t e r a t i o n s ; taking (l)

And has t h e disadvantage o f being a f u l l m a t r i x .

Elasto-plastic torsion o f a c y l i n d r i c a l bar

236

(CHAP. 3 )

pm independent o f n , i . e . pm= p , t h e o p t i m a l v a l u e o f t h i s parameter i s v e r y c l o s e t o u n i t y ; f o r t h i s v a l u e t h e computed s o l u t i o n d i f f e r e d from t h e r e f e r e n c e s o l u t i o n by o n l y a b o u t ( i n a b s o l u t e v a l u e ) by t h e 5 t h i t e r a t i o n , and t o w i t h i n by t h e For 0.5 < p < 1 . 5 t h e r e f e r e n c e s o l u t i o n i s r e 7th iteration. produced i n l e s s t h a n 1 0 i t e r a t i o n s , t o w i t h i n a t worst.

For M = 5 ( i . e . 25 e i g e n f u n c t i o n s w w ) a t i m e o f 1 6 s e c . w a s r e q u i r e d on a n IBM 360/91 t o perform 20 i t e r a t i o n s ; once a g a i n t a k i n g p m = p t h e o p t i m a l v a l u e remains c l o s e t o u n i t y , and a l i m i t i s r e a c h e d , i n p r a c t i c e , which d i f f e r s from t h e r e f e r e n c e s o l u t i o n by a b o u t i n a b s o l u t e value, i n l e s s t h a n 10 i t e r a t i o n s ; f o r example, a t t h e p o i n t ( 0 . 5 , 0 . 5 ) ( r e s p . 1 / 2 0 , 1 / 2 0 ) o f Q a v a l u e o f 0.427 ( r e s p . 0 . 0 1 9 ) w a s o b t a i n e d i n s t e a d o f t h e 0.413 ( r e s p . 0.029) o f t h e r e f e r e n c e s o l u t i o n . 0 For M = 15, t h e method i s p r a c t i c a l l y u n u s a b l e , r e q u i r i n g a computation t i m e o f s e v e r a l t e n s o f seconds on a n IBM 360/91, f o r a single i t e r a t i o n , and w i t h a v e r y l a r g e s t o r a g e r e q u i r e m e n t .

Results f o r h = 1/10: For M = 10, a t i m e o f 50 s e c . on a n IBM 360/91 w a s r e q u i r e d t o c a r r y o u t 20 i t e r a t i o n s and, w i t h p n = p a g a i n h a v i n g i t s o p t i m a l v a l u e o f a b o u t u n i t y , convergence w a s i n e f f e c t r e a c h e d i n 6 ite r a t i o n s , t h e computed s o l u t i o n d i f f e r i n g by a n a b s o l u t e v a l u e o f about from t h e r e f e r e n c e s o l u t i o n . 9.3.7

Conclusions

T h e , r e s u l t s g i v e n above would a p p e a r t o i n d i c a t e t h a t t h e method o f s o l v i n g t h e problem ( P o ) b a s e d on t h e combined u s e o f approx i m a t i o n by e i g e n f u n c t i o n s o f - A i n H,@) ( s e e S e c t i o n 4 . 2 ) and of a f i n i t e - d i m e n s i o n a l v a r i a n t o f a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) ( s e e S e c t i o n 9 . 3 . 4 ) i s v e r y c o s t l y i n terms o f b o t h c o m p u t a t i o n t i m e and s t o r a g e r e q u i r e m e n t . ( I t s h o u l d a l s o b e remembered t h a t i n g e n e r a l t h e e i g e n f u n c t i o n s o f - A i n Hi@) a r e n o t known explThe above drawbacks are l i n k e d t o t h e f a c t t h a t t h e icitly). a p p r o x i m a t i o n c o n s i d e r e d i s globaZ i n n a t u r e ( t h e e i g e n f u n c t i o n s admit as s u p p o r t ) whereas t h e c o n s t r a i n t I gradu I < 1 i s local; t h i s l e a d s t o f u l l m a t r i c e s o f l a r g e dimension ( v a r y i n g w i t h n ) i n t h e determination of i n (9.67). The above d i f f i c u l t i e s would a l s o b e e n c o u n t e r e d , though admite d l y t o a l e s s e r d e g r e e , i f it were r e q u i r e d t o o b t a i n a n approxb a t e s o l u t i o n o f ( P o ) , u s i n g a n i n t e r i o r a p p r o x i m a t i o n o f H&!) by f i n i t e e l e m e n t s o f o r d e r 3 2 . ( s e e S e c t i o n 4 . 1 . 5 ) .

a

S o l u t i o n of ( P o ) by d u a l i t y methods

237

A p p l i c a t i o n o f t h e d u a l i t y method o f S e c t i o n 9.1.2 t o t h e s o l u t i o n o f t h e a p p r o x i m a t e problems. Case o f t h e e x t e r i o r approximations of S e c t i o n 5.

Synopsis:

We s h a l l c o n f i n e our a t t e n t i o n t o t h e approximation t h e o t h e r c a s e s may be t r e a t e d i n a (see Section 5 ) ; v e r y similar manner and, apart from t h e c a s e m = 1 ( ’ ) , have a l most i d e n t i c a l s t o r a g e r e q u i r e m e n t s . w

Existence of a saddle-point. Description and convergence of the d u a l i t y algorithm

9.4.1

Existence of a saddle-point. We s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n s 5 and 8 . 2 . 3 ; we d e f i n e

thus

and a s s o c i a t e w i t h t h e a p p r o x i m a t e problem (Poh)13 t h e approximati o n Z z h o f Z 2 d e f i n e d by

w i t h ai,, pi, , G , d e f i n e d i n S e c t i o n 8 . 2 . 3 , by ( 8 . 3 3 ) , ( 8 . 3 2 ) , (8.35), respectively. The proof of P r o p o s i t i o n 9 . 1 c a n b e a d a p t e d w i t h o u t d i f f i c u l t y , so t h a t 9 2 h a d m i t s a s a d d l e - p o i n t , { uh,ph } , o n v h x A2h , with:

uh t h e s o l u t i o n of (POhIl3,

(9.73)

P&ij

G;(uh)

-

1) = 0 VMij E s

h

. w

Description o f the algorithm and convergence. We u s e a f i n i t e - d i m e n s i o n a l v a r i a n t o f a l g o r i t h m ( 9 . E ) , (9.13),

(9.14),namely:

{

with

A known (€A2,,), we d e f i n e 4 t o b e t h e element

(9.74)

of

which minimises

(9.75)

P;:

(9.76)

p,,

= max (0, P;, + p,,(Bij G;(G) - 1)) VMi,E s h , > 0 , sufficiently small.

(’)

vh

See S e c t i o n 5.2.

YZh(vh.pi)

Elasto-plastic torsion o f a c y l i n d r i c a l bar

238

(CHAP. 3 )

The proof o f P r o p o s i t i o n 9 . 2 c a n b e a d a p t e d w i t h o u t d i f f i c u l t y ; d e f i n e d by ( 9 . 7 4 ) , ( 9 . 7 5 ) , (9.76) converges t o t h e s o l u t i o n o f (P&3.

it may t h u s b e shown t h a t t h e sequence

(mm

Application

9.4.2

We s h a l l c o n f i n e o u r a t t e n t i o n t o t h e c a s e

=

lo,1[

x ]0,1[ and

f = 10 which has been t r e a t e d i n t h e p r e c e d i n g s e c t i o n s by means o f o t h e r methods ( f o r f u r t h e r examples, and a more d e t a i l e d analys i s o f t h e numerical r e s u l t s , see Bourgat /l/).

Mesh s i z e :

h = 1/20.

Solution of ( 9.74) :

By p o i n t o v e r - r e l a x a t i o n ,

with increas-

ing i , j . By p: = 0 f o r a l g o r i t h m (9.74), (9.751, (9.76) ( w i t h u i l = 0 ) i n t h e s o l u t i o n o f (9.74) by o v e r - r e l -

InitiaZisations: and by axation.

b-’

Termhation c r i t e r i a :

(9.78)

D,(n) = Max

14’ - “.;I

Putting

I,

MtJenh

we t a k e f o r a l g o r i t h m ( 9 . 7 4 ) , ( 9 . 7 5 ) , (9.76) t h e t e r m i n a t i o n c r i t erion: (9.79)

Q 10-3

For t h e s o l u t i o n o f t h e sub-problems (9.74) by o v e r - r e l a x a t i o n , w e u s e a t e r m i n a t i o n c r i t e r i o n o f which t h e s e v e r i t y i n c r e a s e s ’with n , and which has t h e form:

with (9.81)

E,,

= min(En-l, 0.2D,(n

- I)),

E,,

=

lo-”

where t h e i n t e g e r p i n ( 9 . 8 0 ) r e p r e s e n t s t h e number o f over-relaxation iterations.

Numerical r e s u l t s . Table 9.4 summarises t h e main results f o r t h e convergence o f a l g o r i t h m (9.74) , (9.75) , (9.76) :

methods

238.

P. = P w ~

Computation t i m e C I1 10070 'Cornputati o n time IBM 360191 -

Table

9.4

The s o l u t i o n t h u s c a l c u l a t e d coincides t o w i t h i n a n a b s o l u t e value of about LOd5 with t h a t obtained f o r t h e same approximate problem by t h e method o f over-relaxation with p r o j e c t i o n described Figure 9.5 shows t h e r e s t r i c t i o n t o i n S e c t i o n 7.5, Example 2. of t h e e q u i p o t e n t i a l s o f t h e f u n c t i o n p defined i n ]O,f[ x p,2 S e c t i o n 9.1.2, obtained using t h e approximation J$, provided by algorithm ( 9 . 7 4 ) - ( 9 . 7 6 ) . Note t h e c o n t i n u i t y of t h e function p ; t h i s i s i n agreement w i t h results obtained by H . Br6zis 151, G? being convex. Figure 9.6 shows t h e v a r i a t i o n s o f t h e number of i t e r a t i o n s r e q u i r e d f o r convergence, as a f u n c t i o n of p f o r w = 1.5 , and Figure 9.7 shows t h e number of over-relaxation i t e r a t i o n s r e q u i r e d t o s o l v e t h e same sub-problem (9.74) using t h e termination c r i t e r i o n ( 9 . 8 0 ) , (9.81) and with pm= p = 1, w = 1.7.

Fig. 9.5. E q u i p o t e n t i a l s of t h e dual f u n c t i o n p .

Elasto-plastic torsion of a cylindrical bar

240

2 3 PC Variation of no. of iterations as a function ofp ( w = 1 . 5 ) . 1

Fig.9.6.

15,

10.

1

\ 5 .

Y 0

3)

I

I

I

(CHAP.

10

m

30

4

241

Solution of ( P o )by d u a l i t y methods

(SEC. 9 )

A p p l i c a t i o n o f t h e d u a l i t y method o f S e c t i o n 9 . 1 . 3 t o t h e s o l u t i o n o f t h e a p p r o x i m a t e problems

9.5

We s h a l l c o n f i n e o u r a t t e n t i o n t o t h e f i n i t e element approxima t i o n of S e c t i o n 1 and t o t h e e x t e r i o r a p p r o x i m a t i o n s o f (Po) u s i n g t h e a p p r o x i m a t i o n K& of KO ( s e e S e c t i o n 5 . 2 ) .

Case of the f i n i t e eZement approximation of Section 4.1. (I). Formulation of the approximate problem and existence of a saddle-point.

9.5.1

We s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n s 4.1 and 8.3.2; ing:

(9.82)

X O h

=

{ (u, 4) I(u, q ) e x O . U E

vh,

q E'

h

1,

t h e r e i s t h e n equivalence between t h e problem

Min Y

With t h e Lagrangian Proposition 9.3. {uh,ph; Ah}

(9.84)

uh

on

(vh

x

9'3

d e f i n e d by (9.201,

Ph

J(o)

and

EvhnKo

( 9 . 2 1 1 , we have:

The Lagrangian 9, admits a saddle-point, x L h With:

Ah)

the s o l u t i o n of

Min Uh E

(9.85)

putt-

J(uh)

K O n vh

= grad U h

Proof. S i n c e t h e problem is f i n i t e - d i m e n s i o n a l , we may a p p l y 8 R o c k a f e l l a r / 3 / , S e c t i o n 2.8, Theorem 28.2, 28.3.

Case o f the f i n i t e element approximation of Section 4.1. (11). Description of the algorithm and convergence W e u s e a l g o r i t h m (4.12), ... , ( 4 . 1 5 ) of C h a p t e r 2, S e c t i o n 4.3, 9.5.2

E l a s t o p l a s t i c torsion of a cylindrical bar

242

(CHAP. 3 )

which i n t h i s p a r t i c u l a r c a s e t a k e s t h e form:

(9.88)

4"

(9.89)

p.

=

A:

>0,

+ p,(grad 4 - p;), s u f f i c i e n t l y small.

The algorithm ( 9 . 8 7 ) , ( 9 . 8 8 ) , ( 9 . 8 9 ) i s a n "approximation" of algorithm ( 9 . 2 3 ) , ( 9 . 2 4 ) , (9.25).

P r o p o s i t i o n 9.4. I f 0 < a Q pI Q /I< 1 the sequence { G,H} defined by (9.871, (9.881, (9.89) converges t o { u h , p L )with

9.5.3

.

Case of the exterior approximation o f Section 5 . ( I) Formulation of the approximate problem and existence of a saddle-point.

The preliminary remark o f S e c t i o n 8 . 3 . 3 s t i l l h o l d s , i n t h e sense t h a t t h e approximation K& of KO i s t h e only one which i s i d e a l l y s u i t e d t o t h e use o f t h e d u a l i t y method of S e c t i o n 9.1.3. We r e t a i n t h e n o t a t i o n o f S e c t i o n s 5 and 8.3.3, t h a t i s :

We i n t r o d u c e t h e approximation o f

There i s t h e n equivalence between:

xo defined

by

ISEC. 9)

Solution of ( P o )by duality methods

24 3

(9.98)

S i m i l a r l y t h e r e i s equivalence between: (9.99) and

( 9 . loo)

Remark 9.9. The approximation of ( P o ) defined by (9.96) was it may e a s i l y be shown not considered i n S e c t i o n s 5 and 6; t h a t t h e results ( o f convergence, i n p a r t i c u l a r ) demonstrated for t h e o t h e r approximations apply equally well t o problem ( 9 . 9 6 ) .

E l a s t o p l a s t i c torsion o f a c y l i n d r i c a l bar

244

We d e n o t e by Yi t h e f u n c t i o n a l from

vh

x

(CHAP. 3 )

d e f i n e d by

Lh+89

(9.97) i f = 1, and by (9.100) i f l = 2; we a s s o c i a t e w i t h problems ( 9 . 9 8 ) , ( 9 . 1 0 0 ) t h e f u n c t i o n a l s , from ( v h x &) x L, +R, 2'jh and

9'ih , which

a r e a p p r o x i m a t i o n s o f Y3 and a r e d e f i n e d . b y :

'Ji+lj+~

- V i j + ~+ u i + ~ j- u i j 2h

(9.101)

ui

)

- 41i+1/2j+1/2 +

+ 1j+ 1 - ui + 1j + u i j + I - Uij 2h

\

I = 1,2.

Using a v a r i a n t o f P r o p o s i t i o n 9 . 3 i t may b e shown t h a t { d,A,4} on ( v h x A,,) x L,, w i t h :

Yi,,(l=1, 2) admits a s a d d l e - p o i n t , (9.102)

u: t h e s o l u t i o n o f

(9.103)

I

P:i+ 1/2 j+I/*

A

(9.104)

=

+ 1/2 j + 1/2 =

i + 1/2 j + 1/29

(9.96),

u: t h e s o l u t i o n o f

1

4+lj+l-Jil+1 +4+lj-Ufj 2h

4+I j + I - U f + l j + 4 j + l - . i j 2h

vMi+1/2 j +112

(9.99),

ch

9

A+1/2 j+1/21 = P z A i ~+ 112 j+1/21 4 i + I/2 j+ 1/21 ( I = lg2)9

vMi+l/2j+l/2Ezh

where, i n (9.104) Pij i s d e f i n e d i n S e c t i o n 8 . 3 . 3 by ( 8 . 8 2 ) ( 8 . 8 3 )

9.5.4

H

Case o f the e x t e r i o r approximations o f Section 5 . (11). Description of the algorithm and convergence

We u s e a l g o r i t h m ( 4 . 1 2 ) - ( 4 . 1 5 ) o f Chapter 2 , S e c t i o n 4 . 3 , which i n t h i s p a r t i c u l a r c a s e i s e x p r e s s e d ( o m i t t i n g t h e upper index 2 ) :

1)

f o r l = 1, by

I-

w i t h $ known U?+Ij

+

UY-lj

(E Lh)

+ u;j+l + hZ

we d e f i n e U"-]

- 45

=

4~Vh , t h e n

2Jj

+

Ah by:

Solution of ( P o )by d u a l i t y methods

(SEC. 9 ) 2)

24 5

f o r l = 2, by

= 2J;:j

+

VMijeQh

with, f o r

(9.109)

z=

1 or 2

pn > 0 ,

sufficiently s m a l l .

The above a l g o r i t h m i s a n "approximation" o f a l g o r i t h m ( 9 . 2 3 ) , ( 9 . 2 4 ) , ( 9 . 2 5 ) and r e l a t i o n s ( 9 . 1 0 5 ) , ( 2 = 1,2) a r e t h e d i s c r e t e analogues of ( 9 . 2 6 ) . I t f o l l o w s from Chapter 2, S e c t i o n 4.3, Theorem 4 . 1 t h a t t h e sequence { & & } d e f i n e d by ( 9 . 1 0 5 ) t ( 2 = 1,2), (9.106), ( 9 . 1 0 7 ) , ( 9 . 1 0 8 ) i s c o n v e r g e n t , f o r pn s u f f i c i e n t l y s m a l l , t o { u ! , , ~ ! , } .

9.5.5

Case o f the e x t e r i o r approximations o f Section 5 . (111). A nwnericaZ example.

We c o n s i d e r t h e problem (Po) r e l a t i n g t o P = ]0,1[ x ]0,1[ and we c o n f i n e o u r a t t e n t i o n t o f = 10, which w a s t r e a t e d e a r l i e r ; t h e a l g o r i t h m used i s t h u s t h e a p p r o x i m a t i o n d e f i n e d by ( 9 . 9 6 ) ; d e f i n e d by ( 9 . 1 0 5 ) 1 ,

Mesh s i z e :

...

,

h = 1/20.

(9.109):

246

Elasto-plastic torsion of a c y l i n d r i c a l bar

I n i t i a l i s a t i o n of algorithm ( 9 . 1 0 5 ) 1,

.. .

(CHAP.

, (9.109) : A:

3)

= 0.

Determination of 4 from 4 : by p o i n t o v e r - r e l a x a t i o n , t h e a l g o r i t h m b e i n g i n i t i a l i s e d by 4-l With m d e n o t i n g t h e i t e r a t i o n c o u n t f o r t h e o v e r - r e l a x a t i o n , we t a k e as t e r m i n a t i o n c r i t er i o n :

.

c

Mfj E Rh

14" - $ 1

d

El.

Termination c r i t e r i o n f o r algorithm ( 9 . 1 0 5 )

c

. .. , ( 9 . 1 0 9 )

IG+1-lqI<&.

M ~EJRh

Analysis of the numerical r e s u l t s . Table 9.5 shows t h e r e s u l t s r e l a t i n g t o t h e convergence o f , (9.109), f o r an over-relaxation parameter of 1 . 7 5 ( 9 . 1 0 5 ) 1, and w i t h p. = p = 0.75,

. ..

' E N o . of i t e r a t i o n : required f o r Approximate value of u(0.5,0.5) Computation time i n seconds on CII 10070

5 x 10-3

5 x 10-3

10-3

10-4

120

80

0.415

0.415

5 x 10-4

I

10-4

90

0.414

L Table

9.5

A s f a r as t h e p r e c i s i o n o b t a i n e d i s c o n c e r n e d , t h e a p p r o x i m a t e solution thus calculated coincides t o within an absolute value of about w i t h t h o s e o b t a i n e d by t h e o t h e r methods o f t h i s chapter.

10.

DISCUSSION Problems i n e l a s t o - p l a s t i c i t y have g i v e n r i s e t o a v e r y l a r g e

Discussion

(SEC. 10)

247

number of works i n t h e f i e l d o f numerical a n a l y s i s . We r e f e r t h e r e a d e r to h g y r i s /l/, 121, 131, h g y r i s - S c h a r p f Spooner / l / , 121, h g y r i s - S c h a r p f 111, Argyris-Balmer-DoltsinisWillam 111, Nayak-Zienkiewic z / l / , Zienkiewicz-Valliapan-King 111, Zienkiewic z-Valliapan 111, Zienkiewic z-Cormeau /1/ and t o t h e i r respective bibliographies. Likewise t h e problem o f e l a s t o - p l a s t i c t o r s i o n has given r i s e t o numerous t h e o r e t i c a l and numerical i n v e s t i g a t i o n s ; i n the numerical f i e l d , we may c i t e Hodge-Herakovich-Stout 111, Herakovich-Hodge 111, Stout-Hodge /l/, Tchernousko-Banitchouk 111, Bourg a t /l/, Cea-Glowinski-Nedelec 121, Goursat 111, Marrocco 111, Br6zis-Sibony 111, Haugazeau 121, and t h e i r r e s p e c t i v e b i b l i o g r a phies. I n t h e c a s e of t h e t o r s i o n problem, it would appear t h a t t h e f i r s t - o r d e r f i n i t e elements described i n S e c t i o n 4 . 1 a r e t h e b e s t adapted f o r p r a c t i c a l c a l c u l a t i o n s ; t h e s e elements, i n f a c t , allow a b e t t e r approximation of domains with complicated shape than do t h e f i n i t e d i f f e r e n c e methods o f S e c t i o n 5; moreover, t h e low r e g u l a r i t y of t h e s o l u t i o n s , t o g e t h e r ’ w i t h t h e drawbacks pointed o u t i n S e c t i o n s 4 and 6, would appear t o l i m i t t h e scope of f i n i t e elements o f o r d e r 2 or h i g h e r . A s f a r as methods for solving t h e approximate problems a r e concerned, w e would l i k e t o emphasize t h e advantage o f f e r e d by methods of t h e duality t y p e based on t h e u s e of Uzawa‘s algorithm (see S e c t i o n 9 ) , and to p o i n t o u t t h e following v a r i a n t which comt h e n o t a t i o n used i s bines t h e techniques of S e c t i o n s 8 and 9 ; t h e same as t h a t used i n S e c t i o n 8.3, and w e write: (10.1)

L = L’(r2) x L’(i-2)

(10.3)

j ( v , q) =

t h e r e i s t h e n equivalence between ( P o ) and ( ’) :

L e t u be t h e s o l u t i o n of ( P o ) ; then (u,Vu) i s t h e unique s o l u t i o n of ( 1 0 . 4 ) . I t i s n a t u r a l t o a s s o c i a t e with (10.4) t h e Lagrangian 9,obtained by d u a l i s a t i o n of Vu - q = 0 , t h a t i s : 9(u,q;

= j ( u , q)

+

I

p.(Vv

- q ) dx .

With p and q f i x e d , 9 i s l i n e a r i n v and hence non-coercive,

( ’)

I n t h e following, we write : grad v = V v .

248

Elasto-plastic torsion of a c y l i n d r i c a l bar

(CHAP. 3 )

and t h i s i s a d i s a d v a n t a g e i f t h e r e i s a need t o c a l c u l a t e a p o s s i b l e s a d d l e - p o i n t o f Y o n H;(R) x A x L u s i n g U z a w a ' s method; t o remedy t h i s s i t u a t i o n we a p p l y a p r i n c i p l e due t o H e s t e n e s 121, and p e n a l i s e V v - q = 0 , g i v i n g

I t may b e shown t h a t any s a d d l e - p o i n t o f Yz on H ; ( R ) x A x L i s o f t h e form (u. V u ; A) w i t h u t h e s o l u t i o n o f ( P o ) . We a p p l y Uzawa's a l g o r i t h m t o Y&, i . e . (10.5)

AOEL

w i t h 1" known,

given,

#,p", An+1 a r e determined s u c c e s s i v e l y by:

=YEW, P" ;P") d =YE(u, q ;P") (u",p") E Ht(SZ) x A

(10.6)

A"+'

(10.7)

=

A"

+ p.(VUn

- p")

1

Vu, q) E H A @ )

p.

x A

>0

By assuming f o r Ye t h e e x i s t e n c e o f a s a d d l e p o i n t ( u , V u ; A) on H i @ ) x A x L , it may be shown t h a t under t h e c o n d i t i o n 0 < r o d pn d rl <;

(10.8)

2

we have :

lim (u",p") = (u, Vu) strongly i n HA(R) x L n-+m

I t may b e n o t e d t h a t t h e d e t e r m i n a t i o n o f (#,p") , u s i n g (10.61, i s a f i n i t e - d i m e n s i o n a l quadratic programing problem which c a n b e s o l v e d e a s i l y by means o f t h e method o f block over-relaxation with projection d e s c r i b e d and s t u d i e d i n Cea-Glowinski 121. The problem o f t h e e x i s t e n c e o f a s a d d l e - p o i n t f o r Yzo n H,'(R) x A x L seems t o b e open i n t h e g e n e r a l c a s e ; o n t h e o t h e r hand, t h e r e i s no d i f f i c u l t y i n p r o v i n g such a r e s u l t f o r t h e r e s t r i c t i o n o f Yz t o t h e f i n i t e - d i m e n s i o n a l a p p r o x i m a t i o n s o f H ; ( B ) x A x L d e f i n e d ' by means o f a f i n i t e element method such a s , f o r example, t h e f i r s t - o r d e r f i n i t e element method d e s c r i b e d i n S e c t i o n 4 . We r e f e r t o some r e c e n t work by Fortin-Glowinski-Marrocco f o r a more d e t a i l e d s t u d y o f t h e numerical implementation o f ( 1 0 . 5 ) a s t u d y o f o t h e r n o n l i n e a r problems t r e a t e d b y means of (10.7); v a r i a n t s o f ( 1 0 . 5 ) - ( 1 0 . 7 ) may a l s o b e found i n t h i s same a r t icle.

Chapter 4 THERMAL CONTROL PROBLEMS, BOUNDARY UNILATERAL PROBLEMS, AND ELLIPTIC VARIATIONAL INEQUALITIES OF ORDER

4

SYNOPSIS In this chapter we shall consider various problems relating to numerical aspects of the methods and technicalities studied in Chapters 1 and 2. In Sections 1, 2 and 3 we shall study various unilateral elliptic problems of order 2, the unilateral behaviour being related to conditions at the boundary; in Section 4 ire shall study the use of elliptic variational inequalities of order 4 for modelling unilateral problems for the case of thin plates.

PROBLEMS OF THERMAL CONTROL AND OF THE DIFFUSION OF FLUIDS THROUGH SEMI-PERMEABLE WALLS

1.

1.1

Formulation of the problem

A certain number of model problems concerning the thermal control of enclosures R , either by means of air-conditioning applied across the boundary, or by diffusion, lead to the determination of a function 'u(x), X E D which satisfies

- Au= f

in R

(1.1)

pu

(1.2)

au + q(u) = 0

(1.3)

u =0

on

on

r,,

r - rd

or (1.3)'

'an= o

on

r - f d .

In (1.1)-(1.3)', p is a constant 3 0, r,,a part of the boundary and ip : W + W a given increasing function. Following Duvaut-Lions /l/, Chapter 1, the solution of problems (l.l), (1.2), (1.3) and (l.l), (1.2), (1.3)' reduces to'the solution of the minimisation problem:

r of R

UniZateraZ probZems and eZZiptic inequaZities

250

(CHAP. 4 )

with, assuming t h a t f E Lz(62), (1.5)

in

J(u) = -

(1.4), w e

(1.6)

V

=

but + I

I,

grad u 1') dx -

take

{ u I v E H'(Q), u Ir-r, = 0 }

i n t h e case of problem (l.l), (1.2), (1.3) and (1.7)

V

= H'(62)

i n t h e case of problem (l.l), (1.2), ( 1 . 3 ) ' ; i n ( 1 . 5 ) t h e function Y i s t h e p r i m i t i v e of rp which vanishes a t t h e o r i g i n , i . e . (1.8)

"(5)

=

Remark 1.1. I n o r d e r t o h e l p s i m p l i f y t h e d e s c r i p t i o n , it w i l l be assumed i n t h e following t h a t p > 0; t h e r e a d e r may cons u l t Duvaut-Lions /l/y Chapter 1, Section 7.3, for a t h e o r e t i c a l rn study of t h e c a s e p = 0. Remark 1 . 2 . The f u n c t i o n rp considered i n t h i s s e c t i o n w i l l be of t h e following type: if if

t

5/k

- g

if

(<-kg

g

(1.9)

cp(0

=

i

2kg

- kg < 5 < kg

where g and k i n ( 1 . 9 ) are positive c o n s t a n t s . t h u s have

g5-fkgZ

(1.10)

w5) = {

g

-gt-$kg'

if

if if

From (1.8) , we

kg

151 9 k g

5s - k g .

The f u n c t i o n s rp and Y * r e s p e c t i v e l y , a r e shown i n Figures 1.1 and 1 . 2 . Some examples r e l a t i n g t o f u n c t i o n s rp more g e n e r a l t h a n t h a t defined by (1.9), and which may be discontinuous, are given i n Duvaut-Lions /l/, Chapter 1. (See a l s o Section 2 o f t h e present chapter f o r Y(u) = g 1.0 I).

(SEC. 1)

ThermaZ controZ prob Zems

Flg. 1.1.

Fie. 1.2. 1.2

251

Graph o f , d u ) .

Graph of

$(u).

Existence and uniqueness r e s u l t s for problem (1.4), ( 1 . 5 ) .

I n t h e following, I( u (1 w i l l denote t h e standard norm of t h e space H1(f2), t h a t i s

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

252

(CHAP.

4)

We have : Theorem 1.1 Assuming 62 t o be bounded and with Lipschitz boun d a r y r , then the problem (1.4), (1.5) admits one and only one solution.

Proof.

The f u n c t i o n a l J i s s t r i c t l y convex w i t h

-

lim llull + m

J(v) =

+

co ;

s i n c e t h e c o n t i n u i t y i n H'(62) of

is o b v i o u s , i n o r d e r t o a p p l y Theorem 2 . 1 , o f Chapter 1, S e c t i o n 2 . 1 , it i s s u f f i c i e n t t o prove t h e c o n t i n u i t y o f j : HI@)+ W d e f i n e d by

We have :

so t h a t (1.12)

1

I u2

)j(v2) -j(vA d g e

now, w i t h

- u1 I d r Vu2, u1 E L 2 ( r );

d

YO d e n o t i n g t h e t r a c e mapping, w e have

so t h a t t h e r e s u l t f o l l o w s from ( 1 . 1 2 )

- (1.14).

Remark 1 . 3 . The above r e s u l t s t i l l h o l d s i f we assume k = 0 i n (1.9), i n which c a s e j(v)

*g

I,

I v Id r ;

we s h a l l r e t u r n t o t h e above p o i n t i n S e c t i o n 2.

Thermal controZ problems

(SEC. 1)

1.3

253

Approximation by a f i n i t e element method

I n t h i s s e c t i o n w e s h a l l s t u d y t h e a p p r o x i m a t i o n o f problem u s i n g Lagrangian f i n i t e e l e m e n t s o f o r d e r 1 and 2 . To s i m p l i f y t h e d e s c r i p t i o n it w i l l be assumed i n t h e f o l l o w i n g t h a t R i s a poZygonaZ domain ( a n d hence L i p s c h i t z ) .

(1.4), (l.5),

Definition of

Triangulation of 0 .

1.3.1

v h

4, Example

We proceed a s i n Chapter 1, S e c t i o n

4 . 2 and Chapter

we t h u s suppose t h a t F,, i s a finite trianguta3, S e c t i o n 4 . 1 ; tion o f 0 such t h a t (1.15)

T c Z VTEY,,,

U T = C T E T ~

T a x i d T ' E 9 h = - T n T ' = b , o r T and

T have a common s i d e ,

o r T and T and have a common v e r t e x . We p u t : (1.17)

h = l e n g t h of t h e l o n g e s t s i d e of t h e T

E

~

~

and Z h = s e t of t h e v e r t i c e s of (1.18)

F h

ZL = set o f t h e m i d p o i n t s of t h e s i d e s o f t h e t r i a n g l e s

of

sh.

r

I n view o f ( 1 . 1 5 ) all t h e v e r t i c e s o f belong t o r h ; w e s h a l l assume, moreover, t h a t r, i s t h e u n i o n o f s i d e s , the ends of which lie on r , of e l e m e n t s o f F h C l a s s i c a l l y , t h e s p a c e H'(f2) i s approximated by

.

(1.19)

v h

=

I u,, E ff'(0)n co(Z),v h (TE pq

{

VTEy

h

}

i n which u h J T d e n o t e s t h e r e s t r i c t i o n of 0, t o T and Pq t h e space of p o l y n o m i a l s o f d e g r e e < q ; we s h a l l c o n f i n e o u r a t t e n t i o n t o q = 1 and 2. The s p a c e Vh a s s o c i a t e d w i t h t h e problem (l.l), (1.21, ( 1 . 3 ) w i l l t h u s b e d e f i n e d by (1*20)

vh

=

{

Oh

I oh EYhr

Uh I r - r d = O

};

i n t h e c a s e o f problem (l.l), (1.2), (1.3)' w e s h a l l t a k e

254

Unilateral problems and elliptic inequalities

(CHAP. 4)

The n a t u r a l d e g r e e s of freedom f o r t h e above approximations are t h e v a l u e s t a k e n by Oh on ( r e s p . Z,,U.Z;) i f q = 1 (resp. q = 2).

1.3.2

Approximation of j

The f u n c t i o n a l Y(u)dT

j(u) = !rd

i S d e f i n e d on t h e above spaces v h but i s not v e r y p r a c t i c a l t o use; we s h a l l t h e r e f o r e approximate it by j h which i s o b t a i n e d from j by numerical integration, or, which amounts t o t h e same t h i n g , by a n a l y t i c a l i n t e g r a t i o n o f a s u i t a b l y chosen interpolant o f t h e f u n c t i o n P(u,,) : w e s h a l l denote by wh t h e space o f t h e t r a c e s on of t h e f u n c t i o n s on Vh ; given u h € Vh , we "approxh a t e " Y(u& by Yh(u,,) d e f i n e d by

rd

t h e convexity and t h e continuity of j h a r e obvious. With t h e n o t a t i o n shown i n F i g u r e 1 . 3 f o r q = 1 and i n F i g u r e

1 . 4 f o r q = 2 , and summing over t h e s i d e s o f t h e t r i a n g l e s o f

Fig. 1.3.

Fs.1.4.

y h

Thermal control problems

(SEC. 1)

255

The formula (1.24) ( r e s p . ( 1 . 2 5 ) ) i s c l e a r l y r e l a t e d t o t h e

trapezoidal r u l e ( r e s p . Simpson's r u l e ) . Definition and s o l v a b i l i t y of the approximate problems

1.3.3

Using t h e n o t a t i o n d e f i n e d i n S e c t i o n s 1.3.1, 1.3.2, and with W denoting t h e symmetric b i l i n e a r form

a :V x V +

(1.26)

a(u, u) =

Jb

(grad u.grad u

+ puv) dx

w e approximate problem (1.4), (1.5) by

The minimisation problem (1.27) i s e q u i v a l e n t t o t h e v a r i a t ional inequality:

The convexity and t h e c o n t i n u i t y o f jh a l l o w t h e approximate analogue of Theorem 1.1 t o be proved, g i v i n g : Theorem 1.2.

The problem ( 1 . 2 7 ) admits one and only one

solution. I n o r d e r t o s o l v e t h e v a r i o u s approximate problems i n an e f f e c t i v e manner, it i s a p p r o p r i a t e t o e x p r e s s ( 1 . 2 4 ) - or (1.25) e x p l i c i t l y i n terms of t h e v a l u e s t a k e n by uh and uh on if w e s h a l l r e t u r n t o t h e above p o i n t q = 1, or on & u ci i f q = 2; i n S e c t i o n 1.6. To h e l p c l a r i f y t h e study of t h e convergence of uhwhen h -+ 0 , we s h a l l d i s t i n g u i s h between t h e c a s e s q = 1 ( s e e Section 1 . 4 ) and q = 2 ( s e e S e c t i o n 1 . 5 )

1.4

Convergence of t h e approximate s o l u t i o n s ( I ) . The c a s e q = l

Throughout t h i s s e c t i o n , C w i l l be used t o denote v a r i o u s constants.

1.4.1

Two Zemas

For t h e study of t h e convergence, when h

-+

0 , w e s h a l l make

Unilateral problems and elliptic inequalities

256

(CHAP. 4)

u s e o f t h e f o l l o w i n g two lemmas.

We u s e t h e same n o t a t i o n as t h a t i n S e c t i o n 1 . 3 . 2 . Proof. where With M E m i + l , w e have

a=

w i t h t h e f u n c t i o n uh affine and 'p convex, we have s u c c e s s i v e l y ( w r i t i n g ui = u h ( M i ) :

(1.31)

Uh(M)

= (1

- e) Ui + 8 U i + l

Furthermore, w e have Mi+i

(1.33)

j(S) =

C JMl

and from ( 1 . 3 2 ) ,

'p('h)

dr

( 1 . 3 3 ) , we deduce by i n t e g r a t i o n t h a t :

so t h a t from ( 1 . 2 4 )

with, in (1.35 1, C,, independent of k,g, h, s, oh. Proof. L e t u h € Vh ; u s i n g t h e n o t a t i o n shown i n F i g u r e 1 . 5 w e d e f i n e qh yo uh E L 2 ( r ) by: (1 -36)

q h yo

uh

=

1 MisrnZh

mi

Thermal control problems

(SEC. 1)

257

Fig. 1.5.

with (1.37)

W,

= characteristic function of the "arc"

-

Mi-l,2MiMi+llt.

For the function Yh(ua defined by (1.22), it may be shown that

With the notation used in the proof of Lemma 1.1 we have:

258

Unilateral probZems and elliptic inequalities

(CHAP. 4)

and

Moreover, we have:

and l i k e w i s e

-

By i n t e g r a t i o n o v e r MiMi+l we deduce from (1.43),

but s i n c e youh

so t h a t

is

-

a f f i n e o v e r MiMi+, we have

(1.44) t h a t

Thermal control probZcms

(SEC. 1)

259

From t h e d e f i n i t i o n of h ( s e e ( 1 . 1 7 ) ) we deduce from (1.46) that

which may be w r i t t e n :

R e l a t i o n (1.48) i n c o n j u n c t i o n with C independent o f h , s , o h . w i t h ( 1 . 4 0 ) proves (1.35) and hence t h e lemma. 1.4.2

A convergence theorem

The f o l l o w i n g c o n s i d e r a t i o n s a l s o a p p l y i f we assume We have:

k =

0 in

(1.9).

Theorem 1.3. by a constant a.

I f , as h > 0,

-F

we have

0,

the angles o f

y h

are bounded beZow

260

Unilateral probletrns and e l l i p t i e i n e q u a l i t i e s

(1.49)

h u h=u h-0

(CHAP. 4 )

strongZy i n H W )

where u and uh are r e s p e c t i v e l y solutions of t h e problems (1.41, ( 1 . 5 ) and (1.27). Proof. This is a v a r i a n t of t h e proof o f Theorem 4.2 o f We have: Chapter 1, S e c t i o n 4 . 4 . (1 * 50) (1.51)

a(+

vh

a(u, uh

- uh) + jh(vh) - jh(uh) 3

- u) + jbk) - j(u) 2

Lf( vh

I,

f(uh

- uh) dx

VVh E V h 9

- u) dr .

h By adding ( 1 . 5 0 ) and ( 1 . 5 1 ) it may be deduced t h a t V v h ~ Vwe have

With U E V , we denote by r h o t h e p r o j e c t i o n of v on Vh r e l a t i v e t o t h e norm V - r J G 3

( e q u i v a l e n t t o t h e s t a n d a r d norm of H'(C2)

1;

we t h u s have

It i s w e l l known ( s e e , f o r example, Fix-Strang /l/, CiarletR a v i a r t /l/) t h a t under t h e assumptions adopted f o r f h we have (1.54)

limrhv = u h-0

strongly in V ;

p u t t i n g v h = r h u i n ( 1. 50 ) we t h e n deduce from ( 1 . 5 3 ) ,

(1 .55)

a(q,, r, u

- u) = 0

(1.54)

Thermal control problems

(SEC. 1) (1.56)

lim h-0

I*

f(u

261

- rhu)dx = 0 .

Moreover

Property ( 1 . 5 4 ) i m p l i e s t h a t

and (1.57),

(1.58) l e a d t o

Taking account of (1.52) with

vh

= rhu, from

( 1 . 5 5 ) , (1.56),

(1.59), we have l h dub - l(, uh - u) = 0 h-0

which proves t h e s t r o n g convergence.

Remark 1 . 4 .

Theorem 1 . 3

a l s o holds i f i n t h e f u n c t i o n a l J

defined by ( 1 . 5 ) w e r e p l a c e t h e term

5..

dx L i s an element o f t h e dual V' of t h e space V ; ant i n c e r t a i n applications. W

by L ( v ) where t h i s i s import-

Remark 1 . 5 . If we assume t h a t f E L'(f2) , we have f o r t h e s o l u t i o n u of problem ( 1 . 4 1 , (1.5)., a r e g u l a r i t y p r o p e r t y of t h e type (1.60)

uE Y n

.

where a i s a s u i t a b l e r e a l number belonging t o ]0,1] I n view of (1.60), we may expect t o o b t a i n e s t i m a t e s of t h e approximation e r r o r as a f u n c t i o n o f h ; t h i s w i l l form t h e s u b j e c t of Section 1.4.3 below.

m i l a t e r a l problems and e l l i p t i c i n e q u a l i t i e s

262

4)

(CHAP.

Estimates of the approximation e r r o r as a function of h

1.4.3

I n t h i s s e c t i o n we s h a l l g i v e some e s t i m a t e s o f t h e approximat i o n e r r o r which a r e independent o f t h e parameter k i n r e l a t i o n (1.9); t h e s e e s t i m a t e s a r e a l s o v a l i d i f k = 0, i n which c a s e j(v) = g

we have

I,.

I v 1 dT

.

E s t i m a t e s which a r e dependent on k

w i l l not be c o n s i d e r e d h e r e s i n c e t h e p r o c e s s f o r o b t a i n i n g them comes more under t h e c a t e g o r y o f t e c h n i q u e s used f o r s o l v i n g equations t h a n t h o s e used f o r i n e q u a l i t i e s . The r e s u l t s o f t h i s s e c t i o n a r e i n no way i n t e n d e d t o b e o p t i m a l . I n a f i r s t r e a d i n g , t h e r e a d e r may proceed d i r e c t l y t o t h e s t a t e m e n t o f Theorem1.4, where t h e r e s u l t s proved i n t h i s s e c t i o n a r e c o l l e c t e d t o g e t h e r . I n t h e proof o f Theorem 1 . 3 it w a s e s t a b l i s h e d t h a t t h e f u n c t i o n s u and u, , which are r e s p e c t i v e l y s o l u t i o n s of problems ( 1 . 4 ) , (1.5) and (1.27), s a t i s f y ( s e e ( 1 . 5 2 ) ) : (1.61)

a(u#,-u, uh-u)

< a(&,, vh-u)+jh(Vh)-j(u)+

5,

f(u-Uh)

We d e f i n e t h e interpolation o p e r a t o r ( l )

dx

vuh E v h

.

xh : C0@) + *Yh by:

suppose v E C0@) n V ; t h e n , t a k i n g account o f t h e c o n d i t i o n imposed on rr , namely t h a t it i s t h e union o f s i d e s of t r i a n g l e s o f , we i n f a c t have x, U E V, whether Vh be d e f i n e d by ( 1 . 2 0 ) o r g h

(1.21). We s h a l l t a k e uh = x h u i n ( 1 . 6 1 ) ; t h i s i s j u s t i f i e d s i n c e ( s e e Remark 1 . 5 ) we have U E V n H"'(f2) (0 c a < 1) and s i n c e - as f2 i s a two-dimensional open domain w i t h Lipschitz boundary - we have ( i f a > 0 ) H"'(f2) c Co(f2) from a theorem due t o Sobolev ( s e e f o r example, N@as 111). We t h e r e f o r e o b t a i n (I(uh-u, uh-u)

d a(uh, %h u - U ) + j h ( %

or

(l)

"Yh

was d e f i n e d i n

(1.19) .

u)-j(u)+

Themmzz control problems

(SEC. 1)

263

We s h a l l e v a l u a t e t h e term a(u,nhu- u) by n o t i n g t h a t from (1.1) - (1.3)', u i s a s o l u t i o n o f t h e e q u a t i o n :

IU E V . (1.65) we

Taking U = nhu-u i n

(1.68)

-

deduce from

(1.64) that:

I < Co BPII YO nh u IInqr)

U) j(nh U) i f s E [0,11

l h b h

and f i n a l l y , from ( 1 . 9 ) a n d (1.12), (1.13),

To e s t i m a t e a(uh- u,uh- u) it i s t h u s n e c e s s a r y t o e s t i m a t e

II z h u - u Ilnqn)

9

P II YO nh u Ilnqr)

9

II YO nh u - Y O U IILqr)

u s i n g t h e r e g u l a r i t y p r o p e r t y mentioned i n Remark

1 . 5 , t h a t is

Unilateral problems and e l l i p t i c inequalities

264

(CHAP. 4)

U € Y n 0 < a Q 1 ; we shall distinguish between the two cases (i) and (ii) as follows:

( i)

The case Y = { u I u E HI@), u Ird-r = 0 } .

The function u is a solution of the mixed (nonlinear) problem

and from the results we deduce from (1.TO), from 1 cp(u) I < g on concerning the regularity of solutions of mixed problems (see Magsnes-Stampacchia 11 I ) , that: (1.71)

I

(a), SO that y o u ~ H 1 - ' ( r ) , arbitrarily small

H312-8

8

>0

To estimate the right-hand sides of the inequalities (1.67) (1.69)we shall use, once again, interpoZation in Sobolev spaces.

Estimation of

I( ?th u - u I I H 1 ( R ) .

Following Fix-Strang /l/, Ciarlet-Raviart /1/ etc ..., it may be shown that under the assumptions adopted for F h in the statement of Theorem 1.3 we have (1*73)

I

II nh 0 - 0 IItrl(n)d Ch II IIH~(o) VU E H2(Q C independent of h and u .

Likewise if q > 0 we have H1+'(B) c Co(a) and (1.74)

I

1I n h v - v IIH~O)

C(V)II

Iln1+qn)

VU~H'+"(Q)

C(d independent of h and u .

By interpolation between H'+"((n) and H2(Q) we deduce from (1.711, (1.731, (1.74) that:

11 nh (1.75)

-

IlHl(R)

with O1 =

c(q)he' 11 u llH%2-*(R) - 2(& + q )

Q

1 1

1-v

where C(q) is a constant different from that appearing in

(1.74);

Thermal control prob Zems

(SEC. 1)

265

from (1.72), (1.75) we deduce

I( yO(nh u - u) IILqr). boundary r i s Lipschitz and

Estimation of Since t h e

one-dimensional, w e have

H 1 / 2 + r ) ( rc) C o ( f ) i f q > 0 . Suppose U E Co(Td) ; t h e t r a c e of t h e l i n e a r i n t e r p o l a t e nho i s t h e same as t h e i n t e r p o l a t e of t h e t r a c e y o u ; we have, moreover ; (1.77)

11 y o ( 4 D - 0) I)L q r ) c Ch II yo 0 IIHqr) Q Ch II (IBU~(R) Vu E H3''(R)

.

or, a l t e r n a t i v e l y , using (1.72)

I n t h e following w e s h a l l use ( b y way of s i m p l i f i c a t i o n ) t h e same q i n ( 1 . 7 5 ) and (1.79).

Estimation of H II yo

u I(Bqr).

Following (1.71),w e s h a l l t a k e s = 1 - & ; moreover, s i n c e the t r a c e of t h e i n t e r p o l a t e q u is equal t o t h e i n t e r p o l a t e of the t r a c e it may be shown t h a t

II YO n h o IIBi-qr) d C II YO IlB1-w So

that, i n view of (1.72) w e have:

(1.sl)

It yonhu

IIsrl-qr)

Estimation of

d c(&)(Ilf IIrqm + d

II uh - u II,,I(~.

-

(CHAP. 4 )

Unilateral problems and e l l i p t i c inequalities

266

We deduce from ( 1 . 6 6 ) ~ ( 1 . 6 9 ) ,( 1 . 7 6 ) , (1.801, ( 1 . 8 1 ) :

w i t h , i n ( 1 . 8 2 ) , C ( E ) and C ( E , Q ) i n d e p e n d e n t o f h , k , g , f a r b i t r a r i l y small p o s i t i v e q u a n t i t i e s . 8

( i i ) The case

and

E,~I

V = H1(Q).

The f u n c t i o n u i s a s o l u t i o n o f t h e problem

+ cp(u) = 0

(1.83)

an

(1.84)

r = rd it

r d

r-r,

-=Oon If

on

f c l l o w s ( s e e H . B r C z i s 121;

UEH'-*(Q),

E

C h a p t e r l), t h a t

>0

a r b i t r a r i l y s m a l l , and i f G? i s convex w e c a n p u t E = 0 i n ( 1 . 8 4 ) ; moreover :

(1 -85)

I1 u Il112-qm G.C(E)II f I I L ~ o )

w i t h C( E) i n d e p e n d e n t of k , g , f

If rd# (1.86)

r we

have

1 $1

wr)

(

1.

G C g so t h a t , from ( 1 . 8 3 )

u E H3"(Q)

with

(1.87)

II U 1111~qn) Q C(II f I I L + ~ 9) ~ ;

w e deduce from (1.86) t h a t y o u ~ H ' ( T ) a n d , s i n c e t h e f u n c t i o n cp i s L i p s c h i t z ( w i t h c o n s t a n t l / k ) , t h i s i m p l i e s ( s e e , f o r example, (1)

S i n c e e s t i m a t e (1.85) i s i n d e p e n d e n t o f k it s t i l l h o l d s i f

k = 0, i n which c a s e j ( u ) = g present chapter).

1 u IM ( s e e S e c t i o n 2 of t h e

Therma1 contro I problems

(SEC. 1)

267

H. B r e z i s /2/, C h a p t e r 1) t h a t f p ( y o u ) € H ' ( r d ) so t h a t

S i n c e any f u n c t i o n on b e l o n g s t o H''z-a(r), E > 0 imply t h a t U E Hz-K(Q). I t t h i s r e g u l a r i t y property

( so

t h a t j ( u ) = g [dluidr)

r rd

H1(rd) , extended by z e r o i n , a r b i t r a r i l y s m a l l , (1.83) and (1.88) would b e i n t e r e s t i n g t o know whether s t i l l h o l d s i n t h e l i m i t when k = 0

, but

t h i s problem would a p p e a r t o be

open. I n view o f (1.84)-(1.87) it would a p p e a r t h a t i f t h e r e e x i s t s a such t h a t

V = H1(Q)

with

1I u IIRI-qn)

c(ll f IIL'Ca) + 8 )

Q

C independent o f k , g , f. Using ( 1 . 8 9 ) , (1.90), it may b e proved by p r o c e e d i n g as f o r p o i n t ( i ), t h a t

(1.91)

{

11 uh - u Ilfil(a)

< cdll f 11Lqa) + a) + 'dl f IILqn) + 8)

+ C(q)(11 f 1ILqa + 8)'

x hl/z+E

h2(")/('-")

w i t h i n (1.9l),C and C(q) independent o f h, k, g, f, q > 0, arbitrari l y s m a l l ( q = 0 i f a = 1). We summarise t h e p r e c e d i n g r e s u l t s i n :

Let u be the solution of problem (1.41, (1.5) Theorem 1 . 4 . and uk the solution of the corresponding approximate problem (1.27); i f when h + 0 the angZes of y h a r e bounded beZow by a constant EI,, > 0 , wg have the foZZowing estimates for the approximation e @ m r 11 u,, - u llBl(a) ( i ) I f V = { v l u E H l ( Q ) , ulr-r,=O}we have:

ohere i n (1.92), C(&) and C(&,q) are independent of h, k , g , f and

268

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

(CHAP. 4 )

& , q are a r b i t r a r i l y small p o s i t i v e q u a n t i t i e s .

If Y = H'(R) there e x i s t s

(ii)

a, f


1 such that

UE

H1+'(Q)

and we have (1.93)

I

I ur - u IIkqn) Q CdII f IILqm + 8) h + CdII f IILqn) + 8) x

+ C(q)(11 f IILqm + 8)'

x h"'+'

h'"-")'(l-")

where i n ( 1 . 9 3 ) , C and C( 0 ) a r e independent of h, k,g, f and q > 0 i s a r b i t r a r i l y small (q = 0 i f a = 1). Remark 1 . 6 . I f , i n t h e c a s e where Y = { u I V E H'(P), u =0) , we have , U E H"'(R) w i t h 3 Q a Q 1 (which i s n o t t r u e i n g e n e r a l ) w e may r e p l a c e e r r o r e s t i m a t e ( 1 . 9 2 ) by e s t i m a t e ( 1 . 9 3 ) . .&mark

1.7.

I f , i n s t e a d o f (1.27), w e had t a k e n as t h e appro-

ximate problem:

4%.uh - uk) + j(uh) (1.94) uk

In

- Auk) 2 f (oh -

dx

vuh

vh

vh

which i s n o t v e r y p r a c t i c a l from a numerical v i e w p o i n t , t h e n i n s t e a d o f ( 1 . 9 2 ) , ( 1 . 9 3 ) we would have o b t a i n e d , r e s p e c t i v e l y :

(l *95)

11 ub -

fA

lIk'(l2)

d

Z c(&, q) (11 f IILz(n) + 8 ) X

h W

il+8h"

-)

eW% "

II f IlLqm + 8

I n view of ( 1 . 9 2 ) , ( 1 . 9 3 1 , ( 1 . 9 5 ) , (1.96) it would a p p e a r t h a t t h e a d d i t i o n a l e r r o r i n t r o d u c e d by r e p l a c i n g j by j , i s ; we may t h e r e f o r e deduce t h a t i f , from " p r a c t i c a l l y " of t h e p o i n t o f view of r e g u l a r i t y , we have no b e t t e r t h a n U E H " ~ ( Q ) t h e n t h e above e r r o r i s o f t h e same o r d e r o f magnitude a s t h e a p p r o x i m a t i o n e r r o r a s s o c i a t e d w i t h t h e approximate problem ( 1 . 9 4 ) . it would a p p e a r , On t h e o t h e r hand i f we have u ~ H ' + " ( Q ) w i t h a > i n view o f (1.931, ( 1 . 9 6 ) , t h a t t h e p r i n c i p a l p a r t of t h e approxi m a t i o n e r r o r 11 u, - u \lalo i s t h e e r r o r i n t r o d u c e d by r e p l a c i n g i by

O(d)

4

1.5

Convergence o f t h e approximate s o l u t i o n s . (11). The c a s e q = 2.

269

Thermal control problems

(SEC. 1)

Throughout this section, C will be used to denote various constants. A lenitna

1.5.1

We shall use the following variant of Lemma 1.2 for the study of the convergence when h + 0:

where, i n ( 1 . 9 7 ) , C i s independent of k, g , h, S, % Suppose u h € vh Proof. Figure 1.6, and with I we define

, the

notation being that shown in Mi-116 I = I Mi-112 Mi-116 I = 4 I Mt-1 Mt I ;

Fig. 1.6. with mi = characteristic function of Mi-116MJ4i+116

wl-l12 = characteristic function o f

kfj-~&~-11&1-116

It may be shown that for the function that (1.99)

5

r.

$h(uh)

d r = Jr, $(qh

YO

vh) d r

+h(Uk)

defined by (1.22)

m i l a t e r a l problems and e l l i p t i c inequalities

270

so t h a t

(l. loo)

jh("h) =

$(qh YO

Oh)

dr .

Jr

From (1.11)and (1.100) we have

(1.101)

with

(1 .102)

and

(1.103) since

(1.104) we deduce from (1.102),

. .. ,

(1.104) t h a t

so t h a t

or alternatively, writing

T = (a

+ ()/2,

(CHAP.

4)

Thermal contro 2 problems

(SEC. 1)

271

(1.108)

We have

so that, from (1.108)

From (1.112) and from the definition of h we deduce by addition that (1.113)

1)

70 uh

- q h Y O o h llLZ(R) d

11 Y O Oh llHL(T)'

Relation (1.114) in combination with C independent of h, s, uh . U with (1.101) proves (1.97) and hence the lemma. 1.5.2

A convergence theorem

The following result a l s o holds if we assume k = 0 in (1.9).

W e have :

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

27 2

(CHAP.

4)

Theorem 1.5. If, when h -+ 0 , the angles of Yh are bounded below by a constant 8, > 0 , we have

(1.115)

strongly i n H ' ( a )

lim l(h = u h-0

where

u

and

uh

are respectively solutions of the problems (1.4),

(1.5) and (1.27).

With U E V , we denote by r h u t h e p r o j e c t i o n o f u on v h relati v e t o t h e norm u + d m . ( e q u i v a l e n t t o t h e s t a n d a r d norm of H'(l2)) ; w e t h u s have:

Under t h e assumptions adopted f o r g h w e have

(1.118)

strongly i n V .

limrhu = u h+O

P u t t i n g Uh=rhtrin (1.116) we deduce from (1.117), (1.118) t h a t :

(1.119)

a(uh,rhu- u) = 0 r

P r o p e r t y (1.118) i m p l i e s t h a t

I-

lim yo r, u

(1.122)

h-.o v3 E

= you

s t r o n g z y i n H'(l-)

[O, 1/21

and (1.121), ( 1 . 1 2 2 ) l e a d t o :

Therma2 contro 1 problems

(SEC. 1)

273

I t t h e r e f o r e remains t o show t h a t

(1.125)

11 Y O uh l ! H l D ( T ) Q c \I f IIL'G')

'

Taking a c c o u n t o f Lemma 1 . 3 from ( 1 . 1 2 5 ) t h a t : Ijh(uh)

-Auk) I d

cg

& 11 f

-

with s =

i n (1.97)

- we

deduce

IILz(I?)

which gives ( 1 . 1 2 4 ) ; it f o l l o w s from (1.116) w i t h from (1.118)- ( 1 . 1 2 0 ) , (1.123), ( 1 . 1 2 4 ) t h a t

vh

= rhu and

lim a(uh - u, uh - u) = 0

h-0

from which t h e theorem t h e n f o l l o w s .

Remark 1 . 7 .

Theorem 1 . 5 s t i l l h o l d s i f i n t h e f u n c t i o n a l J

d e f i n e d by ( 1 . 5 ) we r e p l a c e t h e term

by

Uv) where

In.

L i s a n element o f t h e d u a l V' of t h e s p a c e V. 1.6

Numerical s o l u t i o n o f t h e a p p r o x i m a t e problems

Synopsis I n t h i s s e c t i o n we s h a l l s t u d y t h e numerical s o l u t i o n o f t h e approximate problems of S e c t i o n 1 . 3 . 3 , namely

by means o f some of t h e o p t i m i s a t i o n methods d i s c u s s e d i n Chapter

2.

(CHAP. 4 )

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

274

F i r s t , it i s a p p r o p r i a t e t o e x p r e s s t h e f u n c t i o n a l s

Jo(uh), j,,(ud e x p l i c i t l y i n t e r m s o f t h e v a l u e s t a k e n by V, on Z h i f q = 1 , or on Z,u Z i i f q = 2 ; t h i s w i l l form t h e s u b j e c t of

S e c t i o n s 1.6.1, 1.6.2.

1.6.1

Formulation of the approximation problem.

(I), q = 1.

In ( 1 . 2 4 ) t h e f u n c t i o n a l j,, w a s e x p r e s s e d e x p l i c i t l y i n t e r m s of t h e v a l u e s t a k e n by u, on (1.127)

jh(uk)

=

Zh n r

, that

is:

4 C I Mi Mi+ i I (Jl(0i) + Jl(ui+ . 1))

In order t o express

J*

f u h d x e x p l i c i t l y , it i s c o n v e n i e n t t o

i n t r o d u c e t h e basis functions o f t h e s p a c e v,,( d e f i n e d by ( 1.19 )) of t h e same t y p e as t h o s e i n Chapter 1, S e c t i o n 4 . 1 , Example 4.2, that is:

d e n o t i n g by (1.129)

ai

t h e s u p p o r t o f wi we t h e n deduce

C

lDfuhdx=

Mi € r h

ui J D , / . i h .

Using t h e n o t a t i o n o f F i g u r e o f t h e t r i a n g l e T, w e have t,

so t h a t

Fig. 1.7.

1 . 7 , d e n o t i n g by A ( T ) t h e measure

27 5

Thermal control problems

(SEC. 1)

F i n a l l y , by u s i n g r e l a t i o n s ( 4 . 1 7 ) - ( 4 . 1 9 ) o f Chapter 3, S e c t i o n

4 . 1 . 4 , w e may prove

so t h a t

R e l a t i o n s (1.1271, ( 1 . 1 2 9 ) - ( 1 . 1 3 1 ) a l l o w t h e approximate problem ( 1 . 1 2 6 ) t o b e w r i t t e n e x p l i c i t l y as a f i n i t e - d i m e n s i o n a l o p t i m i s a t i o n problem.

Formulation o f t h e approximate problem.

1.6.2

(111, q = 2 .

I n ( 1 . 2 5 ) t h e f u n c t i o n a l j , w a s e x p r e s s e d e x p l i c i t l y i n terms on (&vz;)nr , t h a t i s

of t h e v a l u e s t a k e n by

(1.132)

I($(b)

~ h ( u h ) = ~ ~ l M i M i + l

In o r d e r t o write ion 1.6.1, that is

(1.133)

{

+ $(ui+l))*

+4$(vi+l,Z)

5.

we i n t r o d u c e , as i n S e c t -

fv,dxexplicitly,

t h e b a s i s f u n c t i o n s o f t h e s p a c e V hd e f i n e d by ( l . l 9 ) , (wi)M Wi

d e n o t i n g by

IE

(rh U rA)

€v,, ai t h e

W,(Mj) =

6,

s u p p o r t of

V M j € zh

wi

Using t h e n o t a t i o n of F i g u r e we have

, we

U

z;

;

t h e n deduce t h a t

1 . 8 , w i t h A ( T ) = measure of T,

276

h i l a t e r a l problems and e l l i p t i c inequulities

which allows

1.

I"

(CHAP.

4)

I uh ('dx to be written explicitly since

c

luhIzdx=

1nluh12dx.

T€Yh

Fig. 1.8.

Moreover, it may be proved that

(1.136)

i

+

I

u

-

1'

u

2

+ 2(u12 + u 2 3 - % l ) M . l M ,

+ u3 2,

~

3

+

u

*

~

l

-

+ 2(u23 + u31 - u 1 2 ) m 2 1' + + 1 - UI m 3 + o 2 m 1 + o3=2 + 2(u31 + u12 - u 2 3 ) 31' }

which allows

1"

3

~

+ +

+

I grad uh 12dx to be written explicitly since

I grad oh 1' dx =

1

1 1 grad uh 1'

TErh

dx.

Using relations (1.1321, (1.134)-(1.136) the approximate problem (1.126) may be put in the form of a finite-dimensional optimisation problem.

Thermal contro Z prob Zems

(SEC. 1)

277

Solution of the approximate problem by relaxation

1.6.3

Suppose Nh = dim Vh ; h e r e i n a f t e r we s h a l l d e n o t e by a v e c t o r o f W N h , i . e . oh = (ul, u2, ..., uNh) I t w a s seen i n Sections 1.6.1, 1.6.2 t h a t t h e a p p r o x i m a t e problem (1.126) i n f a c t r e d u c e s t o m i n i m i s i n g i n RNh a f u n c t i o n a l d e n o t e d by Jh which s a t i s f i e s t h e following conditions:

.

(i)

Jh i s s t r i c t l y convex

-

(ii)

lim

Jh (Oh)

= -k 00

Ilm I1 + m

( iii)

$ E C1(W).

Jh E C1(WNh)s i n c e

I n view o f t h e s e p r o p e r t i e s , it f o l l o w s from Theorem 1.1 o f Chapter 2 , S e c t i o n 1.1, t h a t t h e convergence o f t h e r e l a x a t i o n a l g o r i t h m a p p l i e d t o t h e m i n i m i s a t i o n o f Jh i n Ph i s a s s u r e d . T h i s a l g o r i t h m i s d e f i n e d as f o l l o w s :

(1.137)

uf = (uy,

w i t h Ir, known, using

(1.138)

aJh

..., u!~)

g i v e n a r b i t r a r i l y i n RNh ;

4" is

d e t e r m i n e d c o o r d i n a t e by c o o r d i n a t e ,

..., 4?,',4+l, @ + l , ..., Gh)= 0 ,

-((ul+',

0Q id

Nh.

aui

Remark 1.8. If t h e v a r i a b l e u, r e l a t e s t o a v e r t e x o f (q = 1) , o r o f Z,, u Z; (q = 2)which d o e s n o t b e l o n g t o rr , e q u a t i o n ( 1 . 1 3 8 ) i s linear.

Z,,

Remark 1.9 If t h e c o n s t a n t k i n (1.9) t a k e s v e r y s m a l l v a l u e s , a l g o r i t h m ( 1 . 1 3 7 ) , ( 1 . 1 3 8 ) i s g u a r a n t e e d a p r i o r i t o perform w e l l . I n f a c t , i n t h e l i m i t when k -+ 0, we have $(r) = g 1 I so that

cI

jh(uh)

=

2g

(l * 1 4 ) jh(h)

=

5 1I

(1*139)

I(I Di I + I v i + 1 I)

Mi

Mi M i +

1

1 (1

01

9

= 1

if

I + 4 I Oi + 111 I -k I ui+1 1)

9

if q = 2 ;

we are t h u s w i t h i n t h e c o n d i t i o n s f o r a p p l y i n g Theorem 1.4 of Chapter 2 , S e c t i o n 1.5, and t h i s a s s u r e s t h e convergence of t h e r e l a x a t i o n a l g o r i t h m - p r o v i d e d , o f c o u r s e , (1.138) (which i s n o t always m e a n i n g f u l ) i s r e p l a c e d by:

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

278

We s h a l l r e t u r n t o t h e above i n S e c t i o n 2 .

(CHAP.

4)

a

Remark 1.10. I n o r d e r t o improve t h e r a t e o f convergence, it can b e u s e f u l t o i n t r o d u c e a n under- o r over-relaxation p a r a m e t e r we t h e n o b t a i n t h e f o l l o w i n g v a r i a n t o f a l g o r i t h m ( 1 . 1 3 7 ) , w; (1.138) : (1.142)

u: g i v e n

(1.143)

-((Ul+’,u;+’,

(1.144)

u;+’

aJk

av*

1.6.4

=

...,tr; Ti, tr;+l”, 4+’, ...) = 0

tr; + oJ(u;+1’* - u;). a

Solution o f the approximate problem by a conjugate gradient method.

I n view o f t h e p r o p e r t i e s o f Jk , it i s also p o s s i b l e t o u s e i n t h e s o l u t i o n of t h e a p p r o x i m a t e problem ( 1 . 1 2 6 ) t h e v a r i a n t ( l ) ( 2 . 2 6 ) - ( 2 . 2 9 ) o f Chapter 2, S e c t i o n 2 . 3 , of t h e conjugate gradient a l g o r i t h m .

1.7

Examples

I n t h e f o l l o w i n g examples t h e approximPreliminary remark. a t e problems a r e a l l s o l v e d by t h e o v e r - r e l a x a t i o n method (1~ 4 2 ) - ( 1 . 1 4 4 ) d e s c r i b e d i n S e c t i o n 1 . 6 . 3 ; a l t h o u g h n u m e r i c a l experiments performed w i t h t h e c o n j u g a t e g r a d i e n t method have g i v e n some , v e r y s a t i s f a c t o r y r e s u l t s , t h e programming o f t h i s method. i s much l e n g t h i e r and t r i c k i e r ( f o r comparable computation t i m e s ) t h a n t h a t o f t h e o v e r - r e l a x a t i o n method; t h e corresponding r e s u l t s w i l l t h e r e f o r e n o t be p r e s e n t e d h e r e . a

1.7.1

Example 1

The problem ( 1 . 4 ) c o n s i d e r e d i s d e f i n e d by:

Geometric data: 62

= ]0,1[ x ]0,1[, fd =

space V : r, = r i m p l i e s V

r.

= H’(62).

Function f:f = 1. Parameter values:

(

p = 1,

Adapted t o non-quadratic

k

=

1, 0.1, 0.01,g = 1 and 0 . 2 .

cases.

279

Thermal control problems

(SEC. 1)

I n i t i a l i s a t i o n of the over-relaxation algorithm: ut

= 0.

Ternination c r i t e r i o n f o r the o v e r - r e l a a t i o n algorithm:

c lul" - $ I c IG"I Nh

(1.145)

*=lNh

QE,

i= 1

with E, =

E,

= if q = 1 ( f i n i t e elements o f o r d e r l), if q = 2 ( f i n i t e elements of order 2 ) .

Triangulation y h : Figures 1.9 and 1.10 show t h e t r i a n g u l a t i o n s employed, r e l a t i n g r e s p e c t i v e l y t o q = 1 and q = 2.

Fig. 1.9. (4

= 1).

Fig. 1-10. (4 = 2).

For 4 = 1 ( r e s P . q = 2 ) w e have 512 ( r e s p . 1 2 8 ) t r i a n g l e s and 289 nodes, and hence Nh = 289.

Analysis of the numerical r e s u l t s . Table 1.1 g i v e s c e r t a i n information concerning t h e s o l u t i o n we have of t h e a.pproximate problems and t h e results obtained; denoted by pl(uJ and p&J t h e following two q u a n t i t i e s :

A(Ud=MaX%(M),MEzh if 4 ' 1 , P2(uJ = Max uh(M),ME

zhn

ME&Uz,' i f 4 = 2 ,

i f q = 1,

M E ( Z h u z ; ) n r if q = 2 .

Unilateral problems and elliptic inequalities

280

(CHAP. 4 )

I

q=l

q=2

q=l

q=2

q=l

q=2

q=l

q=2

---------

1 1.7 1.7 0.256 0.255 0.206 0.206 50 149 0.1 1.7 1.7 0.095 0.094 0.029 0.029 56 53 0 . 0 1 1 . 7 1 . 7 0.073 0.072 0.003 0.003 56 ---------- 30 1 1.9 1.9 0.259 0.258 0.209 0.210 99 76 g = 0.2 0.1 1.9 1.9 0.233 0.232 0.183 0.183 116 185 0.01 1.9 1.9 0.233 0.232 0.183 0.183 125 185 g =

1

Table

1.1

A l l t h e c a l c u l a t i o n s were performed on a C I I 10070 w i t h a mean computation t i m e per iteration of 0.26 s e c . w i t h q = 1, and 0.42 s e e . w i t h q = 2. I n view o f t h e s e r e s u l t s , it would appear t h a t , f o r t h i s t y p e o f problem a t l e a s t , approximation by elements o f o r d e r 1 has t h e advantage over t h a t by elements o f o r d e r 2, s i n c e f o r t h e same p r e c i s i o n and t h e same t o t a l number o f degrees o f freedom, t h e computation times a r e approximately h a l f a s l o n g . F i g u r e s 1.11 and 1 . 1 2 show t h e e q u i p o t e n t i a l s o f t h e approximate s o l u t i o n s f o r g = 0 . 2 w i t h k = 1 and 0.1.

Fig. 1.11.

Fig. 1.12.

(Example 1 : g = 0.2, k = 1).

(Example 2 : g = 0.2, k = 0.1).

Thermal control problems

( S E C . 1)

281

F i g u r e 1.13 shows t h e graphs o f t h e f u n c t i o n s uk(x1,0) for

g = 0 . 2 and w i t h k = 1, 0.1. It may b e s e e n i n F i g u r e 1.13 t h a t t h e f a c t t h a t t h e f u n c t i o n uh(xl,O) p a s s e s beyond t h e c r i t i c a l v a l u e kg does not appear t o have any a d v e r s e r e p e r c u s s i o n s on t h e r e g u l a r i t y o f t h e t r a c e o f the solution. rn

0.10

0.05

Fig.1.13.

1.7.2

1 1

(Example 1:

g = 0.2).

R e p r e s e n t a t i o n o f u,(xl,O).

Example 2 .

The problem

(1.4) i s

d e f i n e d by:

Geometric data :

n = 10, 1[ x 10, 1[

9

rd =

{ (xi, x2) I 0.25 d

XI

d 0.75,

x2

=0).

Space V : V = ( u ( u ~ H ~ ( n ) , u = O on r - r d } . Function f : We have t a k e n f equal t o 10 t i m e s t h e c h a r a c t e r i s t i c f u n c t i o n of t h e s q u a r e ]0.375,0.625[ x ]0.375,0.625[ i . e . f ( x ) = 10 i f 3c belongs t o t h i s s q u a r e , f ( x ) = 0 o t h e r w i s e . Parameter values: We have t a k e n = 0 , which i s p e r m i s s i b l e since

d e f i n e s on V a norm e q u i v a l e n t t o t h e norm induced by H ' ( n ) , and 1, 0 . 2 .

k = 1, 0.1, 0.01, g =

282

Unilateral probZems and eZZiptic inequalities

(CHAP.

4)

Initialisation and termination criterion f o r the over-relaxation algorithm: see Section 1 . 7 . 1 . Triangulation r h : See S e c t i o n 1 . 7 . 1 ; t h e number Nh o f degrees o f freedom i s i n t h i s c a s e equal t o 234. F i g u r e 1 . 1 4 shows a, rr and t h e s u p p o r t o f f .

Fig. 1.14.

(Example 2 )

Analysis of the nwnericaZ results Table 1 . 2 shows some d e t a i l s r e l a t i n g t o t h e numerical r e s u l t s obtained. The mean computation times p e r i t e r a t i o n on a C I I 10070 are e s s e n t i a l l y t h e same a s t h o s e i n Example 1; since t h e values. o b t a i n e d a r e p r a c t i c a l l y t h e same, t h i s example once a g a i n confirms t h e advantage o f working, f o r t h i s type o f problem, w i t h f i n i t e elements o f o r d e r 1.

k

g = l

g = 0.2

1 0.1 0.01 1 0.1 0.01

overrelaxation parameter o

Number of iterations

q= 1

q=2

q= 1

q=2

1.67 1.67 1.67 1.67 1.67 1.67

1.7 1.7 1.7 1.7 1.7 1.7

48

45

42

39

40 48

37 45

42 48

39

Table

1.2

44

Thermal control problems

(SEC. 1)

283

F i g u r e s 1.15, 1 . 1 6 and 1.17 show t h e e q u i p o t e n t i a l s o f t h e approximate s o l u t i o n s f o r g = 1 and k = 1, 0.1, 0.01.

Fig.1.15. (Example 2: g = 1, k = 1).

Fig. 1.16. (Example 2: g = 1, k = 0.1).

Fig. 1.17. (Example 2 : g = 1, k = 0.01).

Geometric data : Q = { (x,, x 2 ) I x: c i r c l e (see Figure 1.18). space Y : Y

=

{ u I U E H'(Q), u

+ xi =

< f },

r, = t h e

lower semi-

o }.

Function f: f i s e q u a l t o 18 t i m e s t h e c h a r a c t e r i s t i c f u n c t i o n of t h e d i s c o f r a d i u s 0.25. Parameter values:

p = 0,

k = 1, k = 0.1, 0.01, g = 1, 0.2.

284

Unilateral problems and elliptic inequalities

(CHAP. 4 )

Fig. 1.18.

Fig. 1.19.

Initialisation and termination criterion for the relaxation 6ee S e c t i o n 1 . 7 .I algorithm: Triangulation r h : t h a t o f F i g u r e 1.19, i . e . 384 t r i a n g l e s , 217 nodes with Nh = 193. Analysis of the numerical results: We have used o n l y approximation by f i n i t e elements o f o r d e r 1 (q = 1 ) ; t h e results o b t a i n e d a r e shown i n Table 1 . 3 .

(SEC. 1)

Thermal control problems

k

g=1

g = 0.2

1 0.1 0.01 1 0.1 0.01

0

28 5

Number of iterations 42

1.76

1.76

35 35 42 48 56

The mean computation time p e r i t e r a t i o n i s about 0.33 s e c . on a C I I 10070; F i g u r e s 1 . 2 0 , 1 . 2 1 , 1 . 2 2 show t h e e q u i p o t e n t i a l s of t h e approximate s o l u t i o n s f o r g = 1 and w i t h k = 1, 0.1, 0.01.

Fig.1.B. (Example 3: g = 1, k = 1).

Fig.1.22. (Example 3:

Fig. 1.21. (Example 3: g = 1, k = 0.1).

g = 1, k = 0.01).

286

Unilateral problems and e l l i p t i c inequazities

(CHAP. 4 )

F i g u r e 1 . 2 3 shows, i n polar coordinates and f o r g = 0 . 2 , k = 1, 0.1, 0.01, t h e g r a p h s o f t h e f u n c t i o n s uh(i,B) f o r O E [ - x , O ] .

-

180

-

-

150

120

- 90 - 60

0 degrees

g = 0.2).

Fig.1.23. (Example 3:

1.7.4

- 30

Example 4 .

Geometric data:

see Section 1.7.3.

Space V : V = H'(O). Function f : f i s e q u a l t o 10 t i m e s t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e d i s c o f r a d i u s 0.25. Parameter values:

= 1, k = 1, 0 . 1 , 0.01, g = 1, 0 . 2 .

I n i t i a l i s a t i o n and termination c r i t e r i o n for the over-relaxation'algorithm: see S e c t i o n 1 . 7 . 1 . Triangulation we have

g h

:

s e e S e c t i o n 1.7.3;

Nh = 217. 8

i n t b c present case

..

Analysis of the numerical r e s u l t s . t h e mean computThe r e s u l t s o b t a i n e d a r e shown i n T a b l e 1 . 4 ; a t i o n t i m e p e r i t e r a t i o n w a s 0.33 s e c on a GI1 10070.

.

F i g u r e s 1 . 2 4 , 1 . 2 5 , 1 . 2 6 show t h e e q u i p o t e n t i a l s o f t h e approxh a t e s o l u t i o n s f o r g = 1 w i t h k = 1, 0.1, 0.01.

Thermal control probZems

(SEC. 1)

k

g=l

g =

0.2

1 0.1 0.01 1 0.1 0.01

1.91

1.91 Table

Fig.1.24.

Number of iterations

u

(Example 4: g = 1, k = 1).

91 96 97 91 111 97

1.4

Fig.1.U. (Example 4: g = 1, k = 0.1).

Fig.1.M. (Example

4: g

= 1, k = 0.01).

Unilateral problems and e Z l i p t i c i n e q m l i t i e s

288

(CHAP.

4)

FRICTION PROBLEMS

2.

2.1

Formulation of t h e problems

A c e r t a i n number o f model problems ( ' ) r e l a t i n g t o t h e d i s p l a cement of a s o l i d body a, w i t h f r i c t i o n on t h e boundary I' o f a, o r o v e r a p a r t rd of t h i s boundary, l e a d t o d e t e r m i n i n g a f u n c t i o n u(x), X E Q , t h e s o l u t i o n o f t h e variational problem

I n ( 2 . 2 ) , w e t a k e f E L2(n),p 2 0, g > 0 ; as r e g a r d s s h a l l confine our a t t e n t i o n t o t h e cases

V

(2.3)

=

{ u I u E H'(P), u = 0 on

v, we

r - r, )

or

V = H'(f2).

(2.4)

Remark 2 . 1 . To h e l p s i m p l i f y t h e d e s c r i p t i o n , we s h a l l we refer t h e reader t o assume i n t h e f o l l o w i n g t h a t p > O ; Duvaut-Lions, loc. c i t y f o r t h e s t u d y o f t h e c a s e p = 0. 2.2

E x i s t e n c e and u n i q u e n e s s r e s u l t s f o r problem ( 2 . 1 ) , ( 2 . 2 ) .

A s f o r Theorem 1.1 of S e c t i o n 1 . 2 , we may prove: Theorem 2 . 1 .

boundary 2.3

Assuming R t o be bounded and with Lipschitz ( 2 . 1 ) , ( 2 . 2 ) admits one and only one solution.

r, problem

R e l a t i o n s h i p w i t h t h e problems o f S e c t i o n 1

We s h a l l show t h a t problem ( 2 . 1 1 , ( 2 . 2 ) may be c o n s i d e r e d as a l i m i t i n g c a s e of problem ( l . h ) , ( 1 . 5 ) o f S e c t i o n 1.1, when t h e t h i s w i l l r e s u l t from: p a r a m e t e r k i n (1.9) t e n d s t o z e r o ; Proposition 2.1.

(I)

Let

u

be the solution of problem (2.1), ( 2 . 2 1 ,

See,Duvaut-Lions /I/, C h a p t e r 3, f o r t h e m o d e l l i n g and s t u d y o f f r i c t i o n problems which a r e much more c o m p l i c a t e d and r e a l i s t i c than those considered i n t h i s chapter.

D i c t i o n problems

(SEC. 2 )

and uk t h a t of probZem (1.4), ( 1 . 5 ) of Section 1.1; limu, = u k-0

Proof: (2.5) (2.6) (2.7)

then

strongly i n V .

We w r i t e :

u(u, u ) =

.

28 9

In

(puu

Av)

=gjrd

ik(u)

=

10

+ grad u.grad u ) d r Idr

/rF(u) dr

where, i n ( 2 . 7 ) , +k is t h e f u n c t i o n d e f i n e d by (1.10) and which was d e n o t e d by $ i n S e c t i o n 1. The s o l u t i o n s u and uk a r e c h a r a c t e r i s e d r e s p e c t i v e l y by

(2.8)

U(U, u

- U) + j ( ~- )j ( ~ )2

so t h a t from ( 2 . 6 ) , ( 2 . 7 ) , we g e t :

From ( 2 . 1 0 )

, (2.12)

we deduce

so t h a t (2.13)

l h a(uk k-0

since u uced by

- u, uk - u) = 0 ;

defines a norm on V equivalent t o t h a t ind+ m HI@), ( 2 . 1 3 ) i m p l i e s t h e s t r o n g convergence o f uk. rn

(CHAP. 4)

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

290

Remark 2 . 2 . The above r e s u l t i s fundamental f o r what f o l l o w s since-being a l s o t r u e f o r t h e approximate problems (for f i x e d h ) - it i m p l i e s t h a t t h e a p p r o x i m a t i o n e r r o r e s t i m a t e s , independent of k , o b t a i n e d i n S e c t i o n 1, a r e also v a l i d f o r problem ( 2 . 1 ) , ( 2 . 2 ) Remark 2 . 3 . Problem (1.4), ( 1 . 5 ) may b e c o n s i d e r e d as b e i n g o b t a i n e d from problem (2.1), ( 2 . 2 ) by regularisation o f t h e non-

differentiable t e r m

2.4

j(u) = g

1,

I u I dr.

Dependence o f t h e s o l u t i o n on g

We s h a l l , f o r t h e moment, d e n o t e by ua t h e s o l u t i o n o f ( 2 . 1 ) , (2.2); r e l a t i n g t o t h e p r o p e r t i e s o f t h e mapping g + u,, , we have :

We have :

Proposition 2.2.

The mapping g

( i)

( ii)

The mapping

(iii)

The mapping

u,, i s Lipschitz from

-1, g

I U~ 1

+ u(u,,

R,

+

V,

i s Lipschitz decreasing.

u,,) i s convex decreasing.

These p r o p e r t i e s f o l l o w from t h e more g e n e r a l r e s u l t s Proof. o f Lemma 2 . 1 below ( t h i s would a l s o a p p l y t o Chapter 5 , S e c t i o n 2.2, b u t , i n t h i s c a s e , we have p r e f e r r e d t o a r g u e d i r e c t l y ) . We u s e , i n p a r t , t h e same formalism as t h a t employed i n t h e precedin s e c t i o n s . /.tx $a.

Lkqwia 2.1.

Suppose we have:

I r i

- V a r e a l Hilbert space, - 0. : V x V + W a coercive, symmetric, continuous, b i l i n e a r fom - L : V-+ R, -

& 'V

linear continuous a seminorm on V, continuous on V,

*m-.

- g a scalar 2 0 ;

we shall denote by J the functional from V - r Rdefined by

J(4=

u)

+ gj(u) - L(v) .

The problem

then admits one and only one s o l u t i o n and the mapping possesses the following properties:

g+ua

Friction problems

(SEC. 2 ) ( i ) the mapping

g + ug i s Lipschitz from

291

V, ( i i ) the mapping g +j(ug) i s Lipschitz decreasing, ( iii) the mapping g + dug,u,) i s convex decreasing. R,

+

Proof. I n view o f t h e p r o p e r t i e s o f t h e form a , we may always assume t h a t t h e H i l b e r t s t r u c t u r e o f V i s d e f i n e d by t h e i n n e r product u,u =a(u,u), s o t h a t f o r t h e a s s o c i a t e d norm, we have ; moreover t h e r e e x i s t s ( R i e s z ’ s Theorem) f c V , IIOI( = unique, such t h a t L(u) = u) V U E V, which g i v e s

&

J(u) =

+ I1

u,

u

+ gi(u) - ( A 4.

112

Finally t h e continuity of j implies (2.15)

i(u) Q l i t (I u It

VUE

v

with

VfO

.

Proof vf ( i ) L e t gl,gz E R,, u1 = us,,u2 = uI, b e t h e correspondi n g s o l u t i o n s o f ( 2 . 1 4 ) ; we have (2.16)

(u19 0

- 4)+ g1(i(u) -i(U1)) 2 ( A 0 - UI)

and hence ( i ) .

rn

Proof of ( i t ) . (2.19),

We have, s i n c e j i s a seminorm, and from ( 2 . 1 5 1 ,

UniZateral problems and e l l i p t i c inequazities

292

Proof of ( i i i ) . We s h a l l f i r s t p r o v e t h e c o n v e x i t y ;

(CHAP.

4)

ug sat-

isfies

(2.20)

(u,, u - u,) Ue€ V ,

+ gj(u) - gj(tr,) 2 (Lu - ug)

s o t h a t by s u c c e s s i v e l y p u t t i n g

(2.21)

II u# (I2

u = 0 and

Vu E V

u = 2 u g i n ( 2 . 2 0 ) we g e t

+ 5u(Ug) = (LU J .

From ( 2 . 2 1 ) it f o l l o w s t h a t

(2.22)

J(UJ = - f 11 ug 112.

Now

J(u,) = min Cf II V € V

11'

+ d o ) - (Lu)I,

so t h a t t h e f u n c t i o n g + J ( u , ) t h u s a p p e a r s as t h e lower envelope o f a f a m i l y o f a f f i n e f u n c t i o n s (which a r e t h u s concave), and i s t h e r e f o r e i t s e l f com(Zz)e; t h i s r e s u l t i n combination w i t h ( 2 . 2 2 ) means t h a t t h e f u n c t i o n g -P 11 ue 11' i s convex. I n o r d e r t o prove t h a t g-, 11 ug )I2 is d e c r e a s i n g , we s h a l l u s e a p r o c e d u r e which i n f a c t w i l l a l s o d e m o n s t r a t e a number o f o t h e r With p r o p e r t i e s of t h e s o l u t i o n u o .

V,

=

Ker(j) = { u I U

E

V,j(u) = 0 } ,

VI i s a cZosed subspace o f V ; V: , so t h a t (2.23)

we s h a l l d e n o t e by V , t h e subspace

V = V k @ Vz

and t h e c o r r e s p o n d i n g d e c o m p o s i t i o n s o f u , J u , a r e

(2.24)

u =

01

+ uZ,

f = f i

+fz

9

+ ugz.

U, = ~ 9 1

I t may b e shown t h a t

(2.25)

j(u) =j(uz)

VUE V

and t h a t j ' i s a norm on v,. From (2.23)-( 2 . 2 5 ) , we t h e n have

and from (2.261, ( 2 . 2 7 ) w e deduce t h a t u,,,

is the solution of

Friction problems

(SEC. 2 )

(2.28)

Min { f II 1) 11’ - ( f ~4,1 U€V,

293

.

so t h a t

S i m i l a r l y , u12 i s t h e s o l u t i o n o f t h e problem

(2.29)’

Min { t II 0 11’ UEV*

+ ai(4 - (fzt 4 1

and w e t h u s have ( c f . ( 2 . 2 1 ) ) :

NOW

(2.31)

(2.30) implies

u,z 1 I S I I f z II

so t h a t (2.32)

i.(u,d

Q

2

;II fz II’.

From ( 2 . 3 1 ) , ( 2 . 3 2 ) , we t h e n deduce

(2.33)

limu,, = 0

weakZy i n V

0-W

and ( 2 . 3 0 ) , (2.33) imply

(2.34)

limu12 = 0 s t r o n g t y i n V .

We have (1 uIz /I2= 11 uI 11’- 11 fi 11’ , so t h a t t h e f u n c t i o n g + 11 uI211’ is t h e r e f o r e convex; moreover

(2.35)

lim 11 uI2 11’ = 0 ;

I--

11 uIz 11’ 3 0 vg , t h e convexity of taking account of (2.35) and of t h e f u n c t i o n g + 11 uIz 11’ implies t h a t it i s decreasing and hence t h a t g + )I uI 11’ i s decreasing, s i n c e I1 u, 11’

=

II fi 11’

+ II U,’

11’.

Proposition 2.2 may be deduced immediately from Lemma 2 . 1 by noting t h a t t h e mapping v

+

I 0 Idr

UnilateraZ probZems and e Z l i p t i c i n e q u a z i t i e s

294

(CHAP.

4)

s a t i s f i e s t h e p r o p e r t i e s r e q u i r e d of j i n t h e s t a t e m e n t o f t h e lemma. H Let Vi = { u I u E V,j ( u ) = 0 ) and l e t ti b e t h e s o l u t i o n o f

with

Jo(v) = + U ( U , u)

- L(u) ;

It may be seen t h a t by proving p o i n t ( T i ? ) o f Lemma 2 . 1 , w e have a l s o proved: Lemma 2 . 2 .

Ve have:

lim ue = ti strongZy in V .

H

#++a3

From t h i s we deduce: P r o p o s i t i o n 2.3. li t h a t of

Let

ue

be t h e s o l u t i o n o f (2.1), ( 2 . 2 ) , and

Iw = { v I U E v,v lrd = 0 } ; then lim ug = ti

s t r o n g l y in V .

H

#-+m

Remark 2 . 4 . The s o l u t i o n ti of ( 2 . 3 7 ) i s a s o l u t i o n o f t h e D i r i c h l e t problem

{

-Ali+pG==f liIr=O

i f V i s d e f i n e d by ( 2 . 3 ) , and of t h e mixed problem

-Ati+pG==f li

[~ 2.5

r, = ono r - r ,

=o

on

Duality p r o p e r t i e s

We denote by A t h e c l o s e d convex s u b s e t o f

Lz(rJ d e f i n e d by

(SEC. 2) (2.38)

Friction problems.

295

A = { q l q ~ L ~ ( r d ) , I q ( X ) J d la . e . on r d } ;

we t h e n have:

The solution u of problem ( 2 . 1 ) is characterised Theorem 2.2 by the existence of P E A , unique, such that

[ In

(2*39)

(grad u.grad u+puu) dx+g

1I

pu uE=V 1 u 1 a . e . on PEA.

I, I pv d r =

fudx

Vu E V ,

r d

Theorem 2.2 may b e proved by u s i n g a v a r i a n t o f t h e Proof. proof ( b a s e d on t h e Hahn-Banach Theorem) o f Theorem 3.3 o f Chapter however, we p r e f e r t o g i v e a more c o n s t r u c t i v e 1, S e c t i o n 3.4; p r o o f , based (amongst o t h e r t h i n g s ) on t h e regularisation p r o p e r t y Using t h e same formalism a s t h a t employof P r o p o s i t i o n 2.1. (1). ed i n S e c t i o n 2.3, w e denote by fp, t h e d e r i v a t i v e o f + k , so t h a t (cf. (1.9)):

(2.40)

rp,(t) =

g

if

t/k

if

- g

if

i

takg - kg < t d kg <<-kg;

t h e s o l u t i o n u, o f problem ( 1 . 4 ) , ( 1 . 5 ) of S e c t i o n 1.1 i s t h u s a l s o t h e s o l u t i o n of t h e n o n l i n e a r v a r i a t i o n a l e q u a t i o n

We d e f i n e pk by (2.41) t h a t :

pk = fpk(ur)/g

j

it t h e n f o l l o w s from ( 2 . 4 0 ) ,

and

L 2 ( f J , we may S i n c e t h e convex set A i s weakly compact i n hence deduce t h e e x i s t e n c e of P E A and of an e x t r a c t e d subsequence, a l s o denoted by p k , such t h a t

( I ) This p r o c e s s w i l l be t a k e n up a g a i n i n Chapter 5, Section 6.1.4.

296

UrziZateraZ probZems and eZZiptic ineqwzZities

(2.44)

pr + p weakly i n Lz(rd) ( )

(CHAP. 4 )

Taking a c c o u n t o f P r o p o s i t i o n 2 . 1 and o f ( 2 . 4 4 ) , we may t h e n deduce i n t h e l i m i t i n ( 2 . 4 3 ) t h a t

(2.45)

a(u, u )

+g

dT =

dx Vu E V

It f o l l o w s from t h e p r o o f o f Lemma 2 . 1 t h a t a@, u)

(2.46)

+

gird I

u Id r = lnf.d..

Moreover, p u t t i n g u = u i n ( 2 . 4 5 1 , we may t h e n deduce t h a t :

u(u,u) + . s , P u d r

=

J fud., n

so t h a t with (2.46) :

which may b e w r i t t e n

(2.47)

[r(lu[ -pu)dr=O.

But p E A i m p l i e s PuGlpulGlUl

so t h a t (2.48)

Iu) - p u > 0

a . e . on

r,.

I t t h e n f o l l o w s from ( 2 . 4 7 ) , ( 2 . 4 8 ) t h a t l u l = p u a . e . on f d . We p r o v e t h e u n i q u e n e s s by reductio ad absurdm; suppose t h a t t h e r e e x i s t s p l , p z E A s a t i s f y i n g ( 2 . 3 9 ) , so t h a t

(2.49)

(pZ- p l ) u d f = 0 V U E V r

and, s i n c e t h e s p a c e o f traces o f t h e f u n c t i o n s o f V o n f d i s dense i n L Z ( f d ) , ( 2 . 4 9 ) i m p l i e s p z = p l , t h u s p r o v i n g t h e uniqueness. Conversely i f u obeys ( 2 . 3 9 ) , t h e n we can e a s i l y p r o v e t h a t u i s t h e s o l u t i o n o f problem ( 2 . 1 ) .

Remark 2 . 5 . (2.1) s a t i s f i e s )

I t f o l l o w s from ( 2 . 3 9 ) t h a t t h e s o l u t i o n u o f

I n f a c t we have pk + p w e a k l p i n

L"(f3.

Friction problems,

(SEC. 2 )

I

2-97

in 0

-Au+pu=f

a + p = O on

r d

au u=o onr-rd

if V i s d e f i n e d by ( 2 . 3 ) and -Au+pu=f -+p=Oon

r d

r-rd

-=Oon

2.6

in 0

Approximation by f i n i t e e l e m e n t s

Formulation of the apbrozimate problem

2.6.1

We assume, a s i n S e c t i o n 1 . 3 , t h a t n i s p o l y g o n a l ; t h i s t h e n a l l o w s yi and v h t o be d e f i n e d as i n S e c t i o n 1 . 3 ; i n p a r t i c u l a r , i s the union of sides, the ends we r e t a i n t h e assumption t h a t

o f which i i e on

r,

rd

of elements of

r h 'I.

A s far as t h e a p p r o x i m a t i o n o f j ( u ) = g

s,

I u I fl i s

concerned, w e

u s e t h e l i m i t s (I), when k - 0 , o f t h e f u n c t i o n a l s d e f i n e d i n S e c t i o n 1 . 3 by (1.24), (1.25), g i v i n g :

f cI

I(I

I + I uh(Mi+l) 1)-

(2*50)

jh(ub =

(2.51)

j h ( u b = i Z I M ~ M ~ + ~ I I I+ ~ 4 1~ ~ h( (W ~ i +I 1 , 2 ) + 1

Mi

uh('i)

if 4 =

1

+ I u h ( M i + l ) I). if

q = 2,

t h e summations e x t e n d i n g o v e r t h e s i d e s o f t h e t r i a n g l e s of r h whose u n i o n c o n s t i t u t e s r d We t h e n a p p r o x i m a t e problem ( 2 . 1 ) by t h e f i n i t e - d i m e n s i o n a l problem d e f i n e d ( i n t h e form o f a v a r i a t i o n a l i n e q u a l i t y ) by

.

(2.52)

i

uh E

vh

.

We immediately have:

(1)

These are g i v e n i n e x p l i c i t form i n S e c t i o n 1.6.3, Remark

1 . 9 , r e l a t i o n s (1.1391, (1.140).

298

UniZateraZ problems and e l l i p t i c i n e q u u l i t i e s

(CHAP.

4)

Theorem 2.3. The approximate problem (2.52) admits one and only one solution.

Convergence o f the approximate solutions

2.6.2

Using p r o o f s i d e n t i c a l t o t h a t o f Theorem 1.3 of S e c t i o n 1.4.2

( q = 1 ) and o f Theorem 1.5 o f S e c t i o n 1.5.2 ( q = 2) may prove, f o r q = 1, 2: Theorem 2.4. If, when h -+ 0 , the angles o f below by a constant e o > 0 , we have

limuh = u h+O

where u and

uh

r h

are bounded

strongZy i n W ( n )

are respectively solutions o f problems (2.1) and

(2.52).

Estimates o f the approximation error.

2.6.3

For q = 1, t h e e s t i m a t e s o f t h e approximation e r r o r o b t a i n e d i n S e c t i o n 1.4.3, which a r e independent of k , a l s o hold f o r t h e approximate s o l u t i o n s o f pmblem (2.1) d e f i n e d by (2.52).

2.7

Numerical s o l u t i o n o f t h e approximate problems

Synopsis We s h a l l s t u d y t h e s o l u t i o n o f t h e approximate problem (2.52) by t h e r e l a x a t i o n method c o n s i d e r e d i n S e c t i o n 1.6.3 and by a d u a l i t y a l g o r i t h m o f Uzawa t y p e ( s e e Chapter 2, S e c t i o n 4.3).

2.7.1

Formulation o f the approximate probZems

I n o r d e r t o w r i t e t h e approximate problem (2.52) e x p l i c i t l y , we s h a l l u s e t h e e x p r e s s i o n f o r j h g i v e n i n ( 2 . 5 0 ) ( r e s p . (2.51)) and t h e formulas of S e c t i o n 1 . 6 . 1 ( r e s p . ( 1 . 6 . 2 ) ) i f q = 1 ( r e s p . q = 2).

2.7.2

Solution o f the approximate problem by relaxation

I n view of Remark 1 . 9 o f S e c t i o n 1.6.3, t h e c o n s i d e r a t i o n s o f S e c t i o n 1.6.3 r e l a t i n g t o t h e approximate problem (1.126) a l s o hold f o r t h e approximate problem (2.52)which i s e q u i v a l e n t t o t h e f i n i t e - d i m e n s i o n a l o p t i m i s a t i o n problem:

Friction pro b lems

(SEC. 2 )

299

SoZution o f the approximate probZem by a duality method

2.7.3

We s h a l l f i r s t study t h e continuous case; t h e problem ( 2 . 1 ) , (2.2) f a l l s i n t o t h e category of Chapter 2, Section 4 i f we take:

I n f a c t , we then have

Jrd

and using r e l a t i o n (2.39) of t h e statement of Theorem 2.2 we may prove :

The LagrangianY adnrits a unique saddle point Theorem 2.5. { u , p } on V x A, where u i s the solution o f ( 2 . 1 ) , ( 2 . 2 ) and p i s reZated t o u by ( 2 . 3 9 ) . We denote ulrd by y d u ; i n view o f Theorem 2 . 5 , problem ( 2 . 1 ) , ( 2 . 2 ) may be solved by t h e algorithm (4.12)-(4.15) of Chapter 2, Section 4.3, which i n t h i s p a r t i c u l a r c a s e t a k e s t h e form: (2.55)

po E A

(2.56)

U ( f f , p n )d Y ( u , p " )

(2.57)

P t 1= PA@I + P n n d p ) , P n > 0 .

chosen a r b i t r a r i l y ( f o r example, zero)

V U E V,ff E V

Writing (2.56) e x p l i c i t l y , w e o b t a i n

(2.58)

1 -Aff+M=f

1

+ gpn = 0 on

inB r d

with

(2.59)

ff

=0

on

r - rd i f

V i s given by ( 2 . 3 ) ,

(2.60)

=o an

on

r-

v = H ~ ( B ;)

r d

if

moreover

(2.61)

PA(q) = sup (- 1, inf (1,q)) Vq E L2(rd)

which allows (2.57) t o be w r i t t e n e x p l i c i t l y .

Unilateral problems and e l l i p t i c inequalities

300

Since

yd

i s a l i n e a r c o n t i n u o u s mapping from u

11 Yd u lILz(r.+) < 11 ?d 11 11

(2.62)

4.1 o f

it t h e n f o l l o w s from Theorem (2.63)

, we

+ L2(fd)

4)

have

;

VuE

IIY

(CHAP.

Chapter 2, S e c t i o n 4.3,

that:

strongly i n V

u"+u

with t h e condition (l):

(2.64)

2

O
g2 11 Yd

(I2 '

m

The approximate problem ( 2 . 5 2 ) i s s o l v e d i n a s i m i l a r manner u s i n g t h e d i s c r e t e a n a l o g u e o f a l g o r i t h m ( 2 . 5 5 ) - ( 2.57) : We w r i t e ph = { pi }i w i t h M i € Zh n f, i f q , = 1 or w i t h M i € (Zh n ZJ n rd i f q = 2, pi E R Vi ; w e t h e n d e f i n e t h e approxi m a t i o n Ahof A by fh={phI

I P i l G 1 vi},

and t h e n t h e a p p r o x i m a t i o n 9 , , o f Y by yh(uh, ph) = JO(uh) +

(2.65)

$

yh(uh, ph) = JO(uh)

+

+$

7I

9

Mi Mi+I

Pi+l/2uh(Mi+1/2)+

+

1 (pi

if 4 =

-k pi+l uh('i+l))

(2.66)

I Mi

k+l

I (Pi uh(Mi) +

uh(Mi+l))

if

4 =2.

The L a n g r a n g i a n g h admits a u n i q u e s a d d l e p o i n t { uh, A h } on , w i t h uh t h e s o l u t i o n o f t h e approximate problem ( 2.52) ; t h e d i s c r e t e analogue of (2.55)-(2.57) i s then

V, x Ah

(2.67)

AiEAh

(2.68)

yh(&

4)2 gh(uh, A:)

vuh

E Vh,

4E v h

It may b e shown t h a t under t h e c o n d i t i o n ( l)

YdE

By t a k i n g

( p o 2 + I grad u

12) dx

0 < a.

< pn d a,, a1

as norm on V, t h e norm o f

U ( V ;Lz(fd)) c o r r e s p o n d i n g t o t h i s c h o i c e .

Friction problems

(SEC. 2)

S u f f i c i e n t l y s m a l l , w e have lim (2.52). n-+m

2.8

301

4 = uh, u, i s

t h e s o l u t i o n of

An example

We s h a l l now apply t h e above methods o f approximation and s o l u t i o n t o t h e example d e f i n e d i n S e c t i o n 2 . 8 . 1 below. 2.8.1

The continuous problem

The problem ( 2 . 1 ) , ( 2 . 2 ) c o n s i d e r e d i s d e f i n e d by:

Geometrical data:

n = ]0,1[ x ]0,1[

r d = { ( X l r X z ) i o ~Q X Ii , x z = o } u ~ ( x , , x , ~ ~Qo i~, xx, =, i )

Space V : V = { U ~ U E H ’ ( P ) ,u = O

on

r-rd}.

Function f : f(x)=IO i f XE]O,)[X]O, I[ , f(x)=-lO

Parmeter values: The c o n d i t i o n

p = 0;

u = 0 on

if

g = 0 . 2 , 0.5, 1,

r - rd i m p l i e s

XEH. I[x]O, I[. 1 . 5 , 2.

that u +

(1”

I grad u I’dx)

1P

d e f i n e s on V a norm which i s e q u i v a l e n t t o t h a t induced by H ’ ( n ) , s o t h a t it i s p e r m i s s i b l e t o t a k e p = 0 i n ( 2 . 2 ) . 2.8.2

The approximate problem

The c a s e q = 1 ( r e s p . q = 2 ) corresponds t o approximation by f i n i t e elements o f o r d e r one ( r e s p . o f o r d e r two); f o r t h e approximate problem d e f i n e d by ( 2 . 5 2 ) we have used t h e t r i a n g u l a t i o n Yhshown i n F i g u r e 1 . 9 ( r e s p . 1.10) i f q = 1 ( r e s p . q = 2 ) , c o r r esponding t o 512 t r i a n g l e s ( r e s p . 128 t r i a n g l e s ) , 289 nodes and 255 d e g r e e s o f freedom. For q = 1 ( r e s p . q = 2 ) t h e approximate problem may b e w r i t t e n i n e x p l i c i t form u s i n g t h e r e l a t i o n s o f ’ S e c t i o n 1.6.1 ( r e s p . 1 . 6 . 2 ) and of ( 2 . 5 0 ) ( r e s p . ( 2 . 5 1 ) ) . 2.8.3

Solution by over-relaxation

The approximate problem ( 2 . 5 2 ) i s e q u i v a l e n t t o t h e f i n i t e dimensional minimisation problem

with

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

302

1

3

=

Jh(vh)

In

I grad v h 1' dX + j h ( u h ) -

t o s o l v e ( 2 . 7 0 ) , we have used ( ' )

(9,..., ugh) chosen

(2.71)

ui =

With using

U;:

(2.72)

Jh(u;+

< Jh(u?+', @+' = @

(2.73)

4" , c o o r d i n a t e

)

-

dX ;

t h e following algorithm:

#+"',@+ 1, ...) < ... #?:, vi, @ + I , ...)

+ o(@+"'

fuh

4)

i n WNh .

known, we determine

', ..., @?:,

I,

(CHAP.

VviE

by c o o r d i n a t e ,

R

@I.

I n t h e example c o n s i d e r e d , t h e a l g o r i t h m ( 2 . 7 1 ) - ( 2 . 7 3 ) w a s i n a l l c a s e s i n i t i a l i s e d by u t = o , t h e termination c r i t e r i o n being d e f i n e d by i

GIG" I

G 10-5

i

Analysis of the numerical r e s u l t s Table 2.1 shows c e r t a i n d e t a i l s r e l a t i n g t o t h e s o l u t i o n o f t h e approximate problems and t o t h e r e s u l t s o b t a i n e d ; we have denoted by pl(uh) and p2(u,J t h e q u a n t i t i e s d e f i n e d by i f q = 1, M E Z , , U Z ; i f q = 2 ,

pl(Uh)=MaxUh(M),

M E Z h

pZ(uh)=MaXuh(M),

M E Z h n r i f q = 1,

if

9

0.2 0.5 1 1.5 2

q=2.

o v e r r e l a x a ti o n parameter o

-

ME(z,,u.z;)~~

Number of iterations

~ l ( ~ h )

q=l

q=2

q=l

q=2

q=l

q=2

q=l

q=2

1.74 1.74 1.74 1.74 1.74

1.75 1.75 1.75 1.75 1.75

0.309, 0.304 0.295 2.286 0.284 Table

0.309 0.304 0.295 0.287 0.284

0.216 0.221 0.129

0.275 0.219 0.126 0.038 0

26 26 26 25 27

40 30 33 35 31

0.040

0

2.1

The n o t a t i o n i s t h e same as t h a t used i n S e c t i o n 1 . 6 . 3 .

F r i c t i o n problems

(SEC. 2)

303

All the Computations were performed on a C I I 10070, the mean computation time per iteration being 0.2 sec. for q = 1, and 0.4 sec. for q = 2. Figure 2.1 (resp. 2.2) shows the function x1 uh(xl,$)(resp. x1 -+ u&,, 0)) calculated for q = 1. Figures 2.3, 2.4 show the equipotentials of the computed solutions (with q = l), corresponding respectively to g = 0.2 and g = 1.5. -+

Remark 2.6. If, all other things being equal, we solve the approximate problem (1.27) of Section 1.3.3, the computed solutions are practically coincident with those of problem (2.52) when kg ; this gives an experimental confirmation of the results of Section 2.3.

t

Fig. 2.1.

2.8.4

Representation of

u (xl, 1/2) (q = 1).

S o l u t i o n by a d u a l i t y method

We shall now solve the approximate problem defined in Sections 2.8.2, using the duality algorithm (2.67)-(2.69) and confining our attention to the case q = I; the conditions used for the computation are as follows:

2.8.1,

! 304

Unilateral problems and e l l i p t i c i n e q u u l i t i e s

(CHAP.

0.3

0.2 0.1

0.0 0 - 0.1 - 0.2

- 0.3 Fig. 2.2. Representation of u (xl,0) (q

Fig. 2.3. Equipotentials of u (g = 0.2).

= 1).

c: ;r:

Fig. 2.4.

Equipotentials of

u (g = 1.5).

4)

Friction problems

(SEC. 2 )

I n i t i a Z i s a t i o n o f (2.67)-(2.69): 1;

305

= 0.

Solution of (2.68); by point over-relaxation using an optimal parameter and initialising with 4-l. Choice of

p , ,:

this was taken as pn = p independent of n

Termination c r i t e r i o n f o r ( 2.67)-(2.69):

The results thus calculated coincide with those obtained using the over-relaxation method described in Section 2.8.3. Table 2.2 shows the number of iterations required for convergence, as a function of the parameter p.

Remark 2.7. For g = 0.2, 0.5 and 1 it can be seen that l & l is everywhere equal to unity; it therefore follows that algorithm (2.67)-(2.69)in fact converges for any positive value of p ; on the other hand, for g = 1.5 and 2, l L h l takes values strictly less than unity and the algorithm diverges if p is too large. Remark 2.8.

For the case of the example in Section 2.8.1 we

can determine a lower bound for the quantity

2 ,I *d aPPeariny.

g2

1,2

in the conditions for convergence of the duality alcorithm (2.55)(2.57); we recall (see Section 2.7.3, equation (2.64)) that (2.55)-( 2.57) converges for

Table

2.2

h i l a t e r a l problems and e l l i p t i c i n e q u a l i t i e s

306

We have V = { u I u E H1(P),u = 0 on

as norm over V . The eigenfunctions of

- A u = b on

u=O

r - r,}, so

(CHAP.

that we can take

- A in V, i.e. the solutions of

in P

r-r,

namely

umo=

Jzsin mnxl, rn 2 1

urn, = 2 sin mnxl cos nnx,,

m, n 2 1 ,

form a basis for V (orthonormal in L.'(P)), so that V u c V : u(x,, x,)=

1 umo f l s i n

mnx,

mb 1

+

c

u,, 2 sin mnxl cos nnx,

m.nb 1

From this we deduce that: (2.75)

(1 v 11

=

I grad v l2 dx = nzm(l;

m2 V T O +

c

m,n3 1

For the traces on

so that

which implies

r d ,we

have

(m2+ nZ)k

.

4)

Friction prob Zems

(SEC. 2 )

(:

307

:>

= 4 7 + -IIV1l2.

We t h u s have

so t h a t t h e convergence o f a l g o r i t h m ( 2 . 5 5 ) - ( 2 . 5 7 ) i s a s s u r e d i f

Noting t h a t 3 x 2 / ( 6 + R 2 ) N 1.9 , we may t h e n deduce, by l o o k i n g a t Table 2.2, t h a t ( 2 . 7 6 ) g i v e s a p e s s i m i s t i c e s t i m a t e o f t h e convergence i n t e r v a l f o r a l g o r i t h m ( 2 . 6 7 ) - ( 2 . 6 9 ) . 2.8.5

Various remarks

A s p o i n t e d o u t e a r l i e r i n Remark 5.10 o f Chapter 2 , S e c t i o n

5.7, t h e o v e r - r e l a x a t i o n a l g o r i t h m i s more economical t h a n t h e d u a l i t y a l g o r i t h m ( I ) , b o t h i n t e r m s o f computation t i m e and s t o r a g e r e q u i r e m e n t . However, t h e d u a l i t y a l g o r i t h m h a s t h e advantage o f a l s o g i v i n g t h e a p p r o x i m a t i o n o f t h e ' m u l t i p l i e r ' p and a l l o w s ( 2 . 1 ) , ( 2 . 2 ) t o b e s o l v e d e a s i l y , a t t h e c o s t o f t r i v i a l m o d i f i c a t i o n s i f a program f o r s o l v i n g e l l i p t i c problems of mixed t y p e i s a v a i l a b l e ; i n f a c t t h e e s s e n t i a l s t a g e i n t h e implementation o f a l g o r i t h m ( 2 . 6 7 ) - ( 2 . 6 9 ) i s c l e a r l y t h e s o l u t i o n H of problem ( 2 . 6 8 ) . A s far as t h e t y p e o f f i n i t e e l e m e n t s u s e d i s concerned, it would a p p e a r t h a t f o r a g i v e n d e g r e e o f p r e c i s i o n t h e u s e o f e l e ments o f o r d e r one ( q = 1) i s much more economical t h a n t h a t of elements o f o r d e r tuo ( q = 2 ) ; t h e a d v a n t a g e o f e l e m e n t s o f o r d e r one becomes even g r e a t e r if t h e p r e s e n c e of curved boundari e s i n t r o d u c e s t h e need f o r isoparametric f i n i t e elements o f order two. ( 2 ) . I n f a c t , i n o u r o p i n i o n , t h e f o r e g o i n g remark a p p l i e s f o r e l l i p t i c problems of order two whenever t h e s o l u t i o n s a r e o f ZOW regularity ( s a y when t h e r e g u l a r i t y i s no b e t t e r t h a n H ' ) . H (l)

(')

For s o l v i n g (2.1), ( 2 . 2 ) a t l e a s t See C i a r l e t - R a v i a r t / 2 / f o r t h e d e f i n i t i o n and t h e o r e t i c a l s t u d y o f curved f i n i t e e l e m e n t s o f i s o p a r a m e t r i c t y p e .

Unilateral problems and e l l i p t i c inequalities

308

3.

(CHAP. 4 )

A PROBLEM W I T H UNILATERAL CONSTRAINTS AT TiE BOUNDARY

3.1

Synopsis

I n t h i s s e c t i o n we s h a l l c o n s i d e r t h e n u m e r i c a l a n a l y s i s o f t h e problem

grad u.grad (u - u) dx

+p

f ( ~ u)dx V U E K

(3.1)

w i t h f E L 2 ( Q ) and

(3.2)

K = ( U ~ U E H ~ ( Q ) , U >aO. e . on

r}.

A v a r i a n t o f t h i s problem ( w i t h p = 0 ) w a s c o n s i d e r e d e a r l i e r (from a t h e o r e t i c a l v i e w p o i n t ) i n Chapter 1, S e c t i o n s 1.1 and 2.1, t o g e t h e r w i t h a n i n d i c a t i o n o f t h e c o r r e s p o n d i n g p h y s i c a l background. 3.2

E x i s t e n c e a n d u n i q u e n e s s r e s u l t s for problem ( 3 . 1 ) . ( 3 . 2 1

We s h a l l suppose i n t h e f o l l o w i n g t h a t p > 0 ; it may b e n o t e d t h a t t h e r e i s e q u i v a l e n c e between (3.1), ( 3 . 2 ) and

Min J,(u) , (3.3)

s i n c e t h e f u n c t i o n J , i s c o n t i n u o u s and s t r i c t l y convex i n H'(Q) with

lim

II lJIl-+-

J,(u) =

+ co ,

and s i n c e K i s convex a n d c l o s e d i n H'(Q), Theorem 2 . 1 o f Chapter 1, S e c t i o n 2.1, i s a p p l i c a b l e , so t h a t t h e e x i s t e n c e and uniquen e s s of a s o l u t i o n f o r ( 3 . 3 ) (and, by e q u i v a l e n c e , f o r ( 3 . 1 ) , ( 3 . 2 ) ) , t h e n follow. 3.3

Regularity r e s u l t s

If t h e boundary r o f l-2 i s s u f f i c i e n t l y r e g u l a r , it may b e shown, ( s e e Lions /l/, Chapter 2 , S e c t i o n 8 and H . B r e z i s /2/) t h a t t h e s o l u t i o n u of ( 3 . 1 ) , ( 3 . 2 ) b e l o n g s t o K n H2(G?)with

(SEC. 3 )

3.4

Problem with unilateral constraints

309

Duality results

F i r s t , w e s h a l l prove:

The s o l u t i o n u of ( 3 . 1 ) , ( 3 . 2 ) i s character-

P r o p o s i t i o n 3.1.

ised by

a

in

(-Au+pu=j

Proof. We w r i t e : a(u, v ) =

I In

(grad u.grad v

+ p v ) dx .

A s K i s a convex cone w i t h apex 0, t h e s o l u t i o n u o f (3.1), ( 3 . 2 ) i s i n f a c t c h a r a c t e r i s e d by

[ a(u, v ) 3

we have implies

9(Q)cK

f v dx Vv E K

,

so t h a t t h e f i r s t r e l a t i o n i n ( 3 . 5 ) t h e n

so t h a t (3.6)

a(u, 0) =

and ( 3 . 6 ) l e a d s t o

(3.7)

-Au+p=f

inff.

A f t e r m u l t i p l i c a t i o n by V E K , and i n t e g r a t i o n by p a r t s , we deduce from ( 3 . 5 ) , ( 3 . 7 ) t h a t

(3.8)

lrvgdT.=o(u,u)-

I,

fvdx>O

VUEK;

r e l a t i o n (3.8) then i m p l i e s au/an>O a.e. on v . = u i n ( 3 . 8 ) , we hence deduce t h a t

r,

and p u t t i n g

310

UniZateraZ probZems and eZZiptic i n e q u a z i t i e s

(CHAP. 4 )

now u 2 0, aulan 2 0 a . e . s o t h a t u(au/an) = 0 a . e . on r, which completes t h e proof o f ( 3 . 4 ) . H Conversely, s t a r t i n g from ( 3 . 4 ) , t h e c h a r a c t e r i s a t i o n ( 3 . 5 ) may be r e c o v e r e d w i t h o u t d i f f i c u l t y . H We d e n o t e by g t h e Lagrangian d e f i n e d by g ( o , q ) = J ~ ( u)

1:

qo dT

and by A t h e p o s i t i v e cone o f

L2(r), that is,

r};

A = { q l q ~ L ~ ( r ) , q > o a . eon .

n o t i n g t h a t u E H'(61) i m p l i e s aulan E H ' / ' ( r ) c L 2 ( r ) we may t h e n prove (I), s t a r t i n g from P r o p o s i t i o n 3.1: Theorem 3 . 1 . Suppose u i s the soZution o f ( 3 . 1 ) , ( 3 . 2 ) ; then the Lagrangian Y admits { u, duldn } as unique saddZe p o i n t on

H'(61) x A .

H

The above r e s u l t s w i l l be u s e d i n . S e c t i o n 3 . 7 i n p r o v i n g t h e convergence o f a d u a l i t y a l g o r i t h m which i s a v a r i a n t o f t h a t described i n S e c t i o n 2.7.3. H 3.5

Approximation by f i n i t e e l e m e n t s o f o r d e r one a n d two

FormuZation o f the approximate probZem

3.5.1

We assume, a s i n S e c t i o n 1 . 3 , . t h a t 61 i s p o l y g o n a l ; this a l l o w s r h t o b e t a k e n as i n S e c t i o n 1 . 3 . 1 and t h e a p p r o x i m a t i o n v h o f t h e s p a c e H'(61) t o b e d e f i n e d by (1.19). We s h a l l approxi m a t e K by Kh, a c l o s e d convex s u b s e t o f V h d e f i n e d by

(3.9)'

Kh=Kn Vh={uhIuhEVh,Uh(P)>O

(3.912

Kh={

uh

I u h E V h, uh(P) > 0

VPEZhnr}

VPE(Z~UZ;)~T}

if

q=l

if q = 2 .

I t i s worth n o t i n g t h a t i f q = 2, we have Kh 9 K . The problem ( 3 . 1 ) , ( 3 , 2 ) , i s t h e n approximated by: (3.10)

(l)

i

a(uh, uh

- uh) >

I,

f(Uh

- uh) dx

vuh E

Kh

Kh

The proof o f Theorem 3 . 1 i s a d i r e c t v a r i a n t o f t h a t o f Theorem 5.2 of Chapter 2, S e c t i o n 5.6.

Problem with u n i l a t e r a l constraints

(SEC. 3 )

311

which a d m i t s one and o n l y one s o l u t i o n .

Convergence of the approximate solutions

3.5.2

We s h a l l now p r o v e :

We have

Lemma 3 . 1 .

C m ( E ) n K =K .

(3.11)

Proof. (3.12)

If

U E

H ' ( B ) t h e n u = u'

K = { u I u E H ' ( B ) , u-

E

- u- w i t h u ' , u - E H ' ( B ) and

Hi(B)} .

-

I n view o f ( 3 . 1 2 ) and s i n c e g(62)= H { ( B ) , it i s s u f f i c i e n t , i n o r d e r t o prove ( 3 . 1 1 ) , t o show t h a t V U E H ' ( O ) , u 2 0 a . e . on 61, it i s p o s s i b l e t o f i n d U , E Cm(a), u, > 0 on 61 such t h a t lim u, = u i n .-+m

H1(62)(strongly). If U E H'(61), u 2 0 a . e . t h e n u admits a n e x t e n s i o n b i n t o H'(R2) such t h a t b 2 0 a . e . o n W 2 ; i n f a c t i f 5 i s an a r b i t r a r y e x t e n s i o n ( l ) o f u i n t o H ' ( R 2 ) , u > 0 a . e . on 61 i m p l i e s t h a t 5 = I GI i s a l s o a n e x t e n s i o n o f u i n t o H'(W2). L e t (p,,), be a reguzarising sequence and l e t iJm = v ' z p , ; t h e n lim 5, = n-+m

i n H1(Rz) ( s t r o n g l y ) ; moreover, 5, &(x) =

)

p,(y) $x

E

Cm(W2)and

- y ) dy 2 0

Vx E R2

R'

s i n c e p , > O and 5 3 0 a . e . on W2. Let U, b e t h e r e s t r i c t c o n o f 3, t o 62; i t f o l l o w s from t h e prope r t i e s of &, t h a t U , E Cm(61),.u, > 0 o n 62 w i t h lim u, = u i n H'(61) m-+m (strongly). I n view o f t h i s lemma we have: Theorem 3 . 2 . If, when h + 0, the angZes of T,,are bounded below by a constant eo > 0, we have

lhl(h = u h+O

with u and

uh

strongly i n

H'(61)

solutions of ( 3 . 1 ) , ( 3 . 2 ) and

(?,.lo),

respectively.

Proof of Theorem 3 . 2 f o r q = 1. We a p p l y Theorem 4 . 2 o f Chait i s t h u s p t e r 1, S e c t i o n 4 . 3 , t a k i n g a c c o u n t o f Remark 4 . 1 ; necessary t o v e r i f y t h a t :

(I)

r

Since i s L i p s c h i t z , such a n e x t e n s i o n e x i s t s ( s e e , f o r example, N6fas /l/).

312

UnilateraZ problems and e l l i p t i c i n e q u a l i t i e s

(i) Vv E 1, = K, we can f i n d i n HI@);

(3.13)

(ii) i f v h e K k ,

Oh

4)

with vh + v s t r o n g l y

VkE

weakly i n H ' ( Q ) , t h e n

+v

(CHAP.

VEK

I n view of t h e i n c l u s i o n Kh c K p o i n t ( T i ) follows T F e d i a t e l y . To prove l i ) we can t a k e , following Lemma 3.1, x = Cm(62)nK; l e t n, : CO(62)+ Vhbe t h e i n t e r p o l a t i o n o p e r a t o r defined by (1.621, S e c t i o n 1 . 4 . 3 . If v E X t h e n

under t h e c o n d i t i o n on t h e a n g l e s of F h i n t h e statement of Theore m 3.1, we have, moreover, lim nhu = v i n H'(62) ( s t r o n g l y ) so t h a t t o h+O

prove (i) it is t h u s s u f f i c i e n t t o t a k e

x

=

~ ~ (n K5 ) and

= nhv.

Once a g a i n it i s s u f f i c i e n t Proof of Theorem 3.2 f o r q = 2. t o v e r i f y ( 3 . 1 3 ) ; p o i n t ( i ) i s t r e a t e d e x a c t l y a s f o r q = 1; it t h u s remains t o be shown t h a t ( t i ) holds. L e t p E Co(r); using t h e same n o t a t i o n as i n S e c t i o n 1 . 4 . 1 , E g u r e 1.5, w e d e f i n e t h e f u n c t i o n p,,, d i s t r i b u t e d over r, by ( 3 * 14)

ph(p) =

with (3.15)

c

dMi+1/2)

xi+l12(p)

i

= c h a r a c t e r i s t i c f u n c t i o n of

limp, = p uniformZy h+O

V

~

co(r) E and

-

MiMi+, ; @,

2

o

we have

if p 2

o

Since Simpson's r u l e i s e x a c t f o r polynomials of degree w e have

Jr (Ph

dr 2 0

QVh E

Kh , v p E

<

co(r),rp 2 0 .

If + v weakZy i n H1(G) t h e n y o v h -+ y o u strongZy i n L 2 ( r ) so t h a t from ( 3 . 1 5 ) w e have:.

3,

Problem with u n i l a t e r a l constraints

(SEC. 3)

From (3.18) we have u 2 0 on

3.6

r,

313

which completes t h e p r o o f .

Numerical s o l u t i o n o f t h e approximate problems

Synopsis. We s h a l l now s t u d y t h e s o l u t i o n of t h e approximate problem

(3.10) by means of t h e method o f p o i n t o v e r - r e l a x a t i o n w i t h p r o j e c t i o n d i s c u s s e d i n Chapter 2, S e c t i o n 1 . 4 , and by means o f a d u a l i t y a l g o r i t h m o f Uzawa t y p e , ( s e e Chapter 2, S e c t i o n 4.3).

3.6.1

Formulation o f the approximate problems.

To w r i t e t h e approximate problem (3.10) i n e x p l i c i t form, we s h a l l u s e (3.9)l ( r e s p . (3.9)2) and t h e formulas o f S e c t i o n 1.6.1 ( r e s p . 1.6.2) i f q = 1 ( r e s p . q = 2 ) .

3.6.2

Solution o f the approximate problem by over-relaxation with projection

The approximate problem (3.10) i s e q u i v a l e n t t o t h e finite-dime n s i o n a l problem Vh

E Kh

(3.19)

(1 gad uh 1’ + Pu:)

-

In

full

dr.

I n view o f t h e n a t u r e of & ( s e e (3.9)] i f 9 = 1, (3.9)2 i f . q = 2 ) we may apply t h e method o f p o i n t o v e r - r e l a x a t i o n with proj e c t i o n , d i s c u s s e d i n Chapter 2, S e c t i o n 1 . 4 , t o t h e s o l u t i o n o f (3.10); t h e convergence of t h i s method i s a s s u r e d VUE]O,~[.

3.6.3

Solution o f the approximate probZem by the duality method

F i r s t , w e s h a l l c o n s i d e r t h e continuous c a s e ; problem (3.1), (3.2) f a l l s i n t o t h e c a t e g o r y of t h e problems d i s c u s s e d i n Chapter 2, S e c t i o n 4 , i f we t a k e

Jo(u) =

:In

(I grad LJ I’

M = Y = H’(B), L

+ pu’)

dx

-

In

fu dx , U ( u , q)

= L Z ( r ) , @(u) =

- you, A

=

J,,(u) -

= Lt(l-).

UniZateraZ problems and elziptic inequalities

314

(CHAP. 4 )

I n view of Theorem 3 . 1 ( s e e S e c t i o n 3 . 4 ) we may a p p l y a l g o r i t h m ( 4 . 1 2 ) - ( 4 . 1 5 ) o f Chapter 2, S e c t i o n 4 . 3 t o t h e s o l u t i o n o f ( 3 . 1 ) , ( 3 . 2 ) ; i n t h i s p a r t i c u l a r c a s e t h i s t a k e s t h e form:

(3.20)

~ O E A chosen a r b i t r a r i l y ( f o r example, z e r o )

(3.21)

Y ( u " , p n ) < U(o,p") v o E H ' ( Q ) , u" E

(3.22)

p"+'

=

v

PA@" - p. yo u") , pm> 0 .

W r i t i n g ( 3 . 2 1 ) e x p l i c i t l y , we o b t a i n t h e Neumann problem -Au"+pn=

(3.23) on

[ : = A n

f

i n 51

r;

moreover

(3.24)

vq E L 2 ( r )

(PAq)) = 4'

which a l l o w s ( 3 . 2 2 ) t o b e p u t i n e x p l i c i t form. S i n c e yo i s a l i n e a r c o n t i n u o u s mapping from H ' ( Q ) + L ' ( ~ ) we have

vo

E

H'(Q) ;

from t h e theorem i n Chapter 2, S e c t i o n 4 . 3 , we may t h e n deduce that

(3.26)

u"

+

u

strongZy i n H ' ( Q )

under t h e c o n d i t i o n

(3.27)

0 < a.

< pn < a l

2 <7 .

II Yo II

The a p p r o x i m a t e problem ( 3 . 1 0 ) i s s o l v e d i n a similar manner u s i n g t h e d i s c r e t e a n a l o g u e o f a l g o r i t h m ( 3 . 2 0 ) -( 3 . 2 2 ) : We w r i t e

ph = { pi}i

MiE&uzh)

with

if q = 2 ;

MisCh

if

q = 1,

Problem w i t h u n i l a t e r a l c c n s t r a i n t s

(SEC. 3 )

315

The L a g r a n g i a n Y h a d m i t s a u n i q u e s a d d l e p o i n t ( 1 ) {u,,,A,+} on t h e d i s c r e t e analogue Vh x Ah where u,, i s t h e s o l u t i o n or^ ( 3 . 1 0 ) ; of ( 3 . 2 0 ) - ( 3.22) i s t h e n

I t may b e shown t h a t under t h e c o n d i t i o n 0 < a. < p. d a1 , a1 s u f f i c i e n t l y s m a l l , we have lim ul; = uh , uh b e i n g t h e s o l u t i o n of n-+m (3.10).

3.7

An example

3.7.1

FormuZation o f t h e continuous problem

Let B = ]0,1[ x ]0,1[ and r, V,f b e as i n S e c t i o n 2.8.1; we have a p p l i e d t h e above methods o f approximation and s o l u t i o n t o t h e f o l l o w i n g v a r i a n t ( 2 , of problem ( 3.1) , ( 3 . 2 ) :

MinJ,(u)

K= {

(3.34) 3.7.2

U ( U E V, u

20

a . e . on

The approximate prob Zem

We proceed as i n S e c t i o n 2.8.2, 3.7.3

r,,}. and u s e t h e same t r i a n g u l a t i o n s .

S o l u t i o n by over-relaxation

The a l g o r i t h m o f p o i n t o v e r - r e l a x a t i o n w i t h p r o j e c t i o n d i s c u s s ed i n Chapter 2, S e c t i o n 1 . 4 , w a s used under t h e f o l l o w i n g condi t i o n s ( f o r q = 1 and 2 ) :

Initialisation:

(l) (2)

u: = 0.

i h i s a Xuhn-Tucker v e c t o r f o r ( 3 . 1 9 ) We t h u s have p = 0.

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

316

(CHAP.

4)

Termination c r i t e r i o n :

CI.;+l - 51 Value of w : w = 1.74. A n a l y s i s of the nwnerical r e s u l t s : For q = 1, convergence i s a t t a i n e d i n 27 i t e r a t i o n s and for q = 2 i n 42 i t e r a t i o n s . The mean time p e r i t e r a t i o n ( o n a C I I 10070) i s 0 . 2 s e c . for

q = 1, and 0.4 s e c . for q = 2. F i g u r e 3 . 1 shows t h e graphs o f t h e f u n c t i o n s x1 + uh(xl, 0)and Uh(x1,j) and F i g u r e 3 . 2 t h e e q u i p o t e n t i a l s of t h e approximate s o l u t i o n u,,. x,

+

0.4

j

1

- 0.31

Fig. 3.1.

3.7.4

Fi.3.2.

Solution by a d u a l i t y method

We s h a l l c o n f i n e o u r a t t e n t i o n t o q = 1; from a p r a c t i c a l p o i n t o f view, a l g o r i t h m ( 3 . 3 0 ) - ( 3 . 3 2 ) was a p p l i e d u s i n g o n l y t h e M,EZhnr, i n ( 3 . 3 2 ) . I t may b e s e e n t h a t , under t h e same opera t i n g c o n d i t i o n s as t h o s e i n S e c t i o n 2.8.4, t h e computed r e s u l t s c o i n c i d e w i t h t h o s e o b t a i n e d i n S e c t i o n 3.7.3 u s i n g o v e r - r e l a x a t ion with projection; Table 3.1 shows t h e number of i t e r a t i o n s r e q u i r e d f o r convergence, a s a f u n c t i o n of t h e parameter p.

(SEC.

4)

317

Fourth-order variationu l i n e q u a l i t i e s

2 Number of i t e r a t i o n s

Table

4

6

16

12

--17

3.1

The remarks i n S e c t i o n 2.8.5 r e g a r d i n g t h e r e l a t i v e m e r i t s o f t h e v a r i o u s methods o f approximation and numerical s o l u t i o n a l s o hold f o r t h e problem c o n s i d e r e d i n t h i s S e c t i o n 3.

4.

NUMERICAL ANALYSIS OF VARIATIONAL INEQUALITIES OF ORDER

4.1

4

Synopsis

I n S e c t i o n 4 we s h a l l d e s c r i b e a c e r t a i n number o f s p e c i f i c The numerical procedures f o r v a r i a t i o n a l problems o f o r d e r 4 . s i z e and complexity o f t h e s u b j e c t would j u s t i f y c o n s i d e r a b l y more e x t e n s i v e d i s c u s s i o n s t h a n t h e r e l a t i v e l y b r i e f o u t l i n e given here; we t h e r e f o r e r e f e r t h e r e a d e r f i r s t l y t o Duvaut-Lions /1/ Chapter 4 , and 141, / 5 / f o r t h e t h e o r e t i c a l s t u d y and t h e p h y s i c a l background t o v a r i a t i o n a l i n e q u a l i t i e s o f o r d e r 4, and secondly ( ’) t o Begis-Glowinski 111, Ciarlet-Glowinski 111, 121, C i a r l e t R a v i a r t 131, Glowinski / 6 / f o r f u r t h e r d e t a i l s r e g a r d i n g t h e f i n i t e element methods and i t e r a t i v e t e c h n i q u e s d e s c r i b e d b r i e f l y in t h i s section.

4.2

A n i t e r a t i v e method f o r s o l v i n g c e r t a i n v a r i a t i o n a l problems o f o r d e r 4

I n t h i s s e c t i o n we s h a l l show t h a t c e r t a i n v a r i a t i o n a l problems of o r d e r 4 may be reduced, by means o f a n i t e r a t i v e p r o c e s s , t o t h e s o l u t i o n o f a sequence o f second-order problems. 4.2.1

Reduction of the Dirichlet probZem f o r Az t o a sequence o f second-order problems

Let f2c @ b e a bounded open domain w i t h a regular boundary t h e (homogeneous) D i r i c h l e t problem f o r A’ i s d e f i n e d by

( A % =f

(’1

i n f2

And t o t h e i r r e s p e c t i v e b i b l i o g r a p h i e s .

r;

U n i l a t e r a l problems and e l z i p t i c i n e q u a l i t i e s

318

(CHAP.

4)

Functional framework and n o t a t i o n

=

{

u

I V E H’(Q), u I r

=

-

s i n c e t h e domain Q i s bounded and mapping

r

i s sufficiently regular the

d e f i n e s on V a norm which i s e q u i v a l e n t t o t h a t induced by H Z ( a ) , We s h a l l d e n o t e by yo a n d y, t h e two t r a c e mappings d e f i n e d 0 = ao/an lr. by y o u = u ir.

VariatiationaZ formulation of probZem (4.1). ueness and r e g u l a r i t y r e s u l t s .

E x i s t e n c e , uniq-

L e t us assume ( f o r t h e moment) t h a t f E H - ’ ( Q ) (H-2(Q): t o p o l o g i c a l d u a l o f H i @ ) ) ; it i s w e l l known t h a t ( 4 . l ) admits one and o n l y o n e s o l u t i o n i n Hi(Q) ; t h i s solution i s also the unique s o l u t i o n o f t h e v a r i a t i o n a l e q u a t i o n : r

(4.2)

I

r

AuAudx

=

( J u )

VUEH:(Q)

JD

u E H&?)

as w e l l a s o f t h e o p t i m i s a t i o n problem

I n ( 4 . 2 ) and ( 4 . 3 ) , (,) r e p r e s e n t s t h e b i l i n e a r form o f t h e d u a l i t y between H - ’ ( Q ) and Ht(62). A s f a r as r e g u l a r i t y i s c o n c e r n e d , we s h a l l c o n f i n e o u r a t t e n t i o n t o t h e f o l l o w i n g r e s u l t ( f o r which we r e f e r t h e r e a d e r t o Lions-MagPnes /1/): i f f~ L ’ ( 9 ) ( ’)

we have u E H4(Q)n H t ( 0 ) .

Description of t h e i t e r a t i v e method. We assume i n t h e f o l l o w i n g t h a t f~ L’(Q) and w e c o n s i d e r t h e

algorithm d e f i n e d by

(SEC. 4 )

(4.4)

Fourth-order varia t i o n a 1 inequa 1i t i e s

319

chosen a r b i t r a r i l y i n L 2 ( r ) (1' = 0 f o r example),

'1

t h e n , having c a l c u l a t e d R E L 2 ( r ) by means o f

, we

s u c c e s s i v e l y determine f

f V~ a n d

A2u" = f

(4.5)

i

you" = 0

y o Au" = - 1"

(4.6)

1""

=1"+P,Ylff,

P"'0.

Convergence of a l g o r i thm ( 4 .4) - ( 4 .6 1 We have :

The sequence (~r), defined by ( 4 . 4 ) - ( 4.6) converges Theorem 4 . 1 . strongly i n V t o t h e s o l u t i o n u of ( 4 . 1 ) , under the condition (4.7)

O < ro Q p,,

< rl

c2 4

with (4.8)

II

inf

uo =

vEH=nHd

IILZ(0)

1 %I( av

L w

'

We f o l l o w Glowinski / 6 / ; l e t u be t h e s o l u t i o n of Proof. ( 4 . 1 ) ; we have u E H4(Q)n H i @-) s o t h a t yo h E H 3 " ( f ) c L 2 ( r ) ; we We have ( s i n c e y1 u = 0) put 1 = - yo Au and ii" = u" - u, 1" = 1" - 1

.

A2u = f you = 0

(4.9)

yo

AU

= -

1

and (4.10)

1 = rl

+ pRylu

so t h a t by s u b t r a c t i n g t h e s e e q u a t i o n s from ( 4 . 5 ) , ( 4 . 6 ) , w e have

A2P = 0

y0v = 0

(4.11)

y0AP= (4.12)

-

2""

-p

=P+pnylfi".

With p ~ L ' ( f ), we write from ( 4 . 1 2 ) t h a t

l p l = (IpI(Lz(,.); we may t h e n deduce

Unilateral problems and e l l i p t i c inequalities

320

(CHAP. 4 )

Using Green's formula

(4.14)

J"(UAW -

W A V ) ~=~ Jr

( 2 w -):; u--

dr,

and ( 4 . 1 1 ) , (4.13), w e have O = S . Ls2ij"Pdx== JnlAij"lzdx -[ryoAij"ylCdl.

and t h e n

I ;i.I' - I ;i.+ I'' = 2 P. II 0II&)

-P I 71I

Itqr,

9

s o t h a t from t h e d e f i n i t i o n o f a. we have:

(4.15) Under c o n d i t i o n ( 4 . 7 ) , we deduce from ( 4 . 1 5 ) t h a t t h e sequence I i s _decreasing and convergent, so t h a t w e have 12' 1' - I An+' 1' + 0 when n + m and hence A 2 + 0 strongly i n L2(@ and t h i s proves t h e theorem. rn

I

+

Remark 4.1. Under c o n d i t i o n ( 4 . 7 ) , we deduce from (4'.15) t h a t R i s bounded i n L Z ( r ) , so t h a t , t a k i n g account of ( 4 . 5 ) , u" i s bounded i n H5/'(D), which t o g e t h e r w i t h Theorem 4 . 1 l e a d s t o u"+u strongly i n Ha@) vs < $. rn Remark 4.2.

I

I n t h e c a s e o f t h e inhomogeneous problem, namely

A% = f YOU

= 91 >

i n 62 Yl

= g,

w e can, i f g1 and g, a r e s u f f i c i e n t l y r e g u l a r , u s e t h e f o l l o w i n g g e n e r a l i s a t i o n o f a l g o r i t h m (4.4)-(4 . 6 ) :

(4.16)

lo chosen a r b i t r a r i l y i n L 2 ( r )

A%" = f

(4.17)

You" = 81 Y,,A~" = -

an

(SEC. 4 )

Fourth-order variationa 1 i n e q u a l i t i e s

3 21

where t h e convergence o f t h i s a l g o r i t h m i s proved i n e x a c t l y t h e same way, and t h e c o n d i t i o n on pn i s s t i l l ( 4 . 7 ) . L e t X,EO a n d l e t 6(xo) b e t h e D i r a c measure o f Remark 4 . 3 . i f N = 2 and s > 1 we have H'(f2) c Co(f2) w i t h continuous i n j e c t i o n , so t h a t 6(xo) i s t h u s l i n e a r c o n t i n u o u s o v e r H'(Q i f s > 1. It t h e n f o l l o w s ( s e e Lions-MagPnes /l/) t h a t i f f = 6 ( x , ) i n ( 4 . 1 ) ( ' ) , we have U E Hi(f2) n H'(O), Vs < 3, s o t h a t yo Auc L'(r), which a l l o w s - s i n c e Theorem 4 . 1 s t i l l h o l d s - a l g o r i t h m ( 4 . 4 ) (4.6) t o be applied t o t h e solution of ( 4 . 1 ) . x,;

Remark 4 . 4 . From (4.5) ( a n d t h i s a l s o h o l d s f o r ( 4 . 1 6 ) - ( 4 . 1 8 ) ) t h e d e t e r m i n a t i o n o f u" from Rn f a c t o r i s e s i n t o two D i r i c h l e t problems for - A , t h a t i s :

{ - w =R"f

(4.19)

-Ad'=

i n f2

p"

i n f2

yo u" = 0 .

yop" =

T h i s remark i s fundamental s i n c e it shows t h a t t h e s o l u t i o n o f

( 4 . 1 ) may be r e d u c e d t o t h a t o f a sequence o f D i r i c h l e t problems for - A . I t i s t h u s n a t u r a l t o a p p r o x i m a t e u" and p" i n H1(f2), and t h i s l e a d s t o a p p r o x i m a t i o n s f o r P w h i c h a r e e x t e r i o r ; we s h a l l r e t urn t o t h i s p o i n t i n S e c t i o n s 4.3, 4.4 and 4 . 5 .

Interpretation of algorithm ( 4 . 4 ) - ( 4 . 6 ) .

Duality r e s u l t s .

We s h a l l r e t a i n t h e a s s u m p t i o n f € L 2 ( f 2 ) and t h e n o t a t i o n w e d e f i n e 2' : V x L '(r ) + W , t h e Lugrangian a s s o c ; i a t e d w i t h problem ( 4 . 3 ) , as f o l l o w s :

1 = - yoAu

U(U, p)

(4.20)

=

:I

-

I AU 1' dx

-

I + I, fi

dx

py1

udr.

We may t h e n prove ( s e e Glowinski, loc. c i t . Theorem 4 . 2 .

on

vx

The pair

,

Section 1 . 4 ) :

(u,l) i s the unique saddle point o f 2'

Lz(r).

I n view o f Theorem 4 . 2 , w e may a p p l y t o Y t h e a l g o r i t h m ( 4 . 1 2 ) -

(4.13) o f C h a p t e r 2, S e c t i o n 4 . 3 , w i t h V = H'(l2) n H i @ ) , M = V, L = L 2 ( r ) A = L, @ = yl; t h e a l g o r i t h m t h u s o b t a i n e d

t u r n s o u t t o b e t h e same a l g o r i t h m as ( 4 . 4 ) - ( 4 . 6 ) . A variant of algorithm ( 4 . 4 ) - ( 4 . 6 ) . If

('1

U E H'(i2)

, we

have

aulan E H 1 I 2 ( r ) , s o t h a t it i s n a t u r a l t o

This i s i m p o r t a n t i n c e r t a i n a p p l i c a t i o n s .

UniZateraZ probZerns and eZZiptic inequuZities

322

extend t o

(4.21)

(CHAP. 4 )

V x H-'12(T) t h e Lagrangian 2 ' d e f i n e d b y ( 4 . 2 0 ) so t h a t

P(o,p) =

f~dx

+ ( p,

71

u):

i n ( 4 . 2 1 ) , ( , ) d e n o t e s t h e b i l i n e a r form o f t h e d u a l i t y between and H'12(r) ( l )

.

H-'I2(T)

With t h e o p e r a t o r S : H'/'(T) + H - ' / ' ( T ) a d u a l i t y o p e r a t o r ( 2 ) , we c o n s i d e r t h e f o l l o w i n g v a r i a n t o f a l g o r i t h m ( 4 . 4 ) - ( 4 . 6 ) :

t h e n , h a v i n g c a l c u l a t e d I " , w e s u c c e s s i v e l y d e t e r m i n e U"E V and A"+' E H - ' / ' ( r ) by means o f :

A'u" = f (4.23)

youn = 0 yo A d = -

I"

Using a v a r i a n t o f t h e p r o o f o f Theorem

4 . 1 we may prove:

Theorem 4.3. The sequence (u", I"), defined by ( 4.22)-( 4.24) converges strongZy i n V x H-'12(r)to (u, - yoAu), where u i s the soZution o f ( 4 . 1 ) , under the condition:

(4.25)

0 < ro < pn < rl < 2 4

with

Remark 4 . 5 . have :

We may p r o v e t h a t under c o n d i t i o n ( 4 . 2 5 ) , w e

(SEC. 4)

Fowlth-order v a r i a t i o n a l i n e q u a l i t i e s

wit.h ( l ) 0 < K < 1 ; the convergence of (4.22)-(4.24) first order.

323

is thus of

Remark 4.6. To prove the convergence of (4.22)-(4.24), it is sufficient to suppose that y o A U EH - ’ ‘ ’ ( T ) ; we may then weaken quite considerably the condition f E L Z ( f i ) , and Remark 4.3 h o l d s a fortiori. Remark 4.7. The practical utility of (4.22)-(4.24) is restricted by the difficult numerical use of H-”’(T), H’’’(r), S (however, see Glowinski 171, /a/. 8 Generalisation t o variationaZ i n c q u a z i t i e s

4.2.2

We shall use three simple examples to demonstrate the possibility of generalising algorithm (4.4)-(4.6) to the solution of varIn the following, a is a iational inequalities of order 4. bounded open domain of W N , with regular boundary r .

Example 1.

Statement of the problem.

Let K be the closed convex subset of ined by

K = { u I u E V ,y , u > O and let f E L’(62) .

V = H’(62)

n HA@)

, def-

a.e. }

The variational inequality (of order

4)

f ( -~ u ) d x V U E K UEK admits one and only one solution, which is also the solution of the minimisation problem J~(u)Q J~(u) V U E K (4.30) U EK

{

Regularity and d u a l i t y r e s u l t s Following the ideas of Lions 111, Chapter 2, Section 8.7.2, it may be proved that under the condition f EL’@), we have (l)

K depends, amongst other things, on ro and r ,

.

LFniZateraZ probZems and e Z Z i p t i c i n e q u a l i t i e s

324

, which

U E H'(61) n H&?)

yo

A24

(CHAP.

4)

implies

E Hl'Z(l-) c LZ(61).

From this regularity result, we deduce that u is characterised by (4.31)

Azu = f in 61,

{

U E

H2(61), u

yo(Au) 2 0 , y1 u 2 0 ,

=

0

on

yo(Au) y1 u = 0 on

r r.

L e t 9 : V x Lz(r)+ R be defined by (4.32)

P) = J ~ ( v ) Jr

=YP(U,

w1v d r

and

from the characterisation (4.31), we may deduce: Theorem 4.4. Suppose t h a t u i s t h e s o l u t i o n o f (4.29), then 9 admits (u, yoAu) a s a m i q u e saddle p o i n t on V x A . a

SoZution of (4.29) by a d u a Z i t y aZgorithm. In view of Theorem 4.4, we may apply to 9 t h e duality algorithm (4.12)-(4.13) of Chapter 2, Section 4.3; we then obtain the following variant of algorithm (4.4)-(4.6): (4.33)

A0

(4.34)

i

chosen arbitrarily in A (for example, zero) A%"

=

f

you" = 0

y o A ~ P=

(4.35)

P+l

an,

= P,t(J" - pn

ff)=

(an - Pn

d)'

9

Pn > 0 .

The conditions for convergence of (4.33)-(4.35) are exactly those of (4.4)-(4.6), i.e. conditions (4.7); Remarks 4.1, 4.3 and 4.4 still hold. a ExqZe 2

Statement of t h e probZem Let V = H'(61) n Hi(61), variational inequality

g

constant > 0 and

f €LZ(61); the

(SEC.

4)

Fourth-order variational iniqua2itie.s

- U) dx

Au A(u

+ g Jr

12

ldr - g j r

I

32 5

dT 2

(4.36)

f(u-U)dx

VUEV

U E V

admits one and only one s o l u t i o n , which i s a l s o t h e s o l u t i o n of t h e minimisation problem:

with

Lhcaliby and regularity r e s u l t s .

Let A 6 e t h e closed convex subset of L z ( r ) defined by: (4.38)

A ' = { p l p c L 2 ( r ) ,I p ( x ) I d 1

r};

a.e. on

proceeding as i n Chapter 1, S e c t i o n 1 . 3 ( o r a l t e r n a t i v e l y by r e g u l a r i s a t i o n as i n Chapter 4 , S e c t i o n . 2 . 3 and Chapter 5 , Section 6.1.4) we may prove t h e e x i s t e n c e of A E A such t h a t t h e s o l u t ion u of (4.36) i s c h a r a c t e r i s e d by

(4.39)

1 t

With (4.40)

A2u = f you = 0 yo Au =

i n GI

- grt

l y l u = l y l u l a.e.

Y : V x L 2 ( r )+ W defined by

W, p) =

;In

I

l2 d.x

it follows from (4.39) t h a t

point of. P on V x A . w UE

Jb.

dx

( --yoAu : u,

I, 1

+g

J I

dr

I

i s t h e unique saddle

regularity of u i s concerned, w e may deduce from i n f a c t , by analogy with t h e r e s u l t s V n H5/2(Q) j

As far as t h e

(4.39) t h a t

-

326

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

(CHAP.

4)

of Br6zis /2/ for problems of order 2, it is reasonable to conjecture that U E V n H3(SZ). rn

S o l u t i o n o f (4.36) by a d u a l i t y algorithm. In view of the above saddle-point result, we may apply to 2 the duality algorithm (b.12)-(b.13) of Chapter 2, Section 4.3; we then obtain the following variant of algorithm ( b . b ) -( 4.6): (4.41)

L'EA

i

A%"

(4.42)

chosen arbitrarily (for example, zero) =f

yod =0 yo Au" = -

(4.43)

A"+'

=

gl"

J'"(2+ P n 971 u")

Pn

3

>0

with

-

PA@) = sup (inf (1, p),

1) V p E L Z ( r ) .

The conditions for convergence can be defined precisely by (4.44)

0 c ro

and Remarks

<

< rl

< (2/g2) u6,

4.1, 4.3, 4.4

rn

still hold.

Example 3 Statement o f t h e problem. Suppose $ E Lm(SZ) and

i"

AU A(u -

(4.45)

U)

dx 2

f E L2(SZ) ; the variational inequality

J', f

(V

-

U ) dx

V UE K

U E K = { U ~ U E H ~ ( uS >Z $) , a.e. }

admits one and only one solution ( l ) which is also the solution of the minimisation problem

with Jo(u) =

;Ifl

I Au I'dx

-1'

( l ) Provided we assume that

fudx.

rn

K # 0

, which

R

we do in the following.

Four th-order va t i a t iona 1 inequa 1i t i e s

(SEC. 4)

327

Remark 4.8. If N = 2 , t h e i n e q u a l i t y ( 4 . 4 5 ) models t h e small v e r t i c a l d e f o r m a t i o n s o f a h o r i z o n t a l , e l a s t i c t h i n p l a t e , which i s r i g i d l y clamped around i t s p e r i m e t e r r , under t h e i n f l u e n c e o f a v e r t i c a l force f i e l d of density f , t h e p l a t e being constrained t o remain above a n o b s t a c l e whose h e i g h t i s g i v e n by $. 8 h a l i t y results. If N = 2 , as w i l l be assumed h e r e , we have H 2 ( 9 ) c Co(a) w i t h uE Hi(9) we have u JI E Lm(62); l e t (Lm(9))' be t h e t o p o l o g i c a l d u a l o f Lm(Q), <, $ t h e b i l i n e a r form o f t h e d u a l i t y between (L"(9))' and L m ( 9 ) , a n d A t h e p o s i t i v e cone o f (L"(9))' i . e .

-

continuous i n j e c t i o n , so t h a t i f

(4.47)

A = { p I p ~ ( L " ( 6 2 ) ) ' , < p , o $ > O t / u ~ L ~ ( d ul )> , O

Y :Hi@)

We t h e n d e f i n e (4.48)

Y ( up,)

x (L"(9))' --*

+ < p, $ - u $

J~(u)

=

a . e . }.

R by

;

we assume ( b y way o f s i m p l i f i c a t i o n ) t h a t $ i s n e g a t i v e everywhere i n a ne
Remark 4.9. fEH-'(62).

The above s a d d l e - p o i n t r e s u l t a l s o h o l d s if

8

S o l u t i o n of ( 4 . 4 5 ) by means 'of d u a l i t y algorithms. I n the infinite-dimensional c a s e , Example 3 l i e s o u t s i d e t h e Hilbert framework i n which t h e d u a l i t y a l g o r i t h m s a l l o w i n g t h e s a d d l e - p o i n t s o f a c e r t a i n c l a s s o f Lagrangians t o b e c a l c u l a t e d This i s not t h e case f o r were s t u d i e d i n Chapter 2, S e c t i o n 4 . t h e finite-dimensional problems a p p r o x i m a t i n g ( 4 . 4 5 ) ; t o solve t h e s e problems, we s h a l l u s e a n a p p r o x i m a t e form o f t h e f o l l o w i n g a l g o r i t h m , which i n general i s n o t meaningful i n the i n f i n i t e -

dimensional case: (4.49)

'1 3 0

chosen a r b i t r a r i l y ;

A",

having c a l c u l a t e d

(4.50)

{

A2d = f you" =

+ 1"

y1

we c a l c u l a t e d and An+' by means of i n 62

u" = 0

UniZateraZ probZems and eZZiptic inequazities

328

(CHAP. 4 )

The biharmonic problem ( 4 .SO) may i t s e l f by solved by means We s h a l l meet of t h e d u a l i t y method s t u d i e d i n S e c t i o n 4 . 2 . 1 . i n Secion 4.7 a finite-dimensional v a r i a n t of (4.49)-( 4.51) appl i e d t o t h e s o l u t i o n o f a problem of t h e type . ( 4 . 4 5 ) .

4.3

A new v a r i a t i o n a l formulation o f t h e D i r i c h l e t problem

f o r A2,

.

synopsis. The algorithm ( 4 . 4 ) - ( 4.6) expZicitZy involves y1 u = a U / h which can p r e s e n t numerical d i f f i c u l t i e s when a p p l i e d t o t h e approximate problems; moreover ( s e e Remark 4 . 4 ) it would appear d e s i r a b l e t o be a b l e t o use approximations which are i n t e r i o r f o r problems of To t h i s end we o r d e r two but exterior f o r t h o s e of o r d e r fo ur. s h a l l g i v e f o r problem ( 4 . 2 ) a n e q u i v a l e n t v a r i a t i o n a l formulati o n which involves only t h e spaces L2(f2),H'(f2) and H,'
In

grad u.grad p dx =

In

qp dx Vp E H 1 ( f 2 ) } ;

we then have: Theorem 4.5. There i s equivaZence between probZems ( 4.2) ( 4.3) and the prob Zem

(l)

,

The c a s e f €H-'(f2),s > 1 would p r e s e n t a d d i t i o n a l t e c h n i c a l d i f f i c u l t i e s and t h e r e f o r e w i l l n o t be considered i n t h e r a t h e r b r i e f o u t l i n e given i n t h i s s e c t i o n .

(SEC.

(4.55)

4)

Fourth-order v a r i a t i o n a l i n e q u a l i t i e s

1,

grad u.grad p dx =

Taking (4.56)

329

I',

p p dx Vp E HL(Q)

p e H i ( Q ) i n ( 4 . 5 5 ) , we t h e n deduce

-Au=p

i n 62

so t h a t , i f r i s s u f f i c i e n t l y r e g u l a r , From ( 4 . 5 5 ) , ( 4 . 5 6 ) we have

1,

grad u.grad p dx =

-

J-,

u E H Z ( Q )n HG(Q).

Aup dx Vp E H'(Q)

s o t h a t by a p p l y i n g G r e e n ' s formula

-

J, Aup dx + lr$

p dT =

-

I,

Aup dx V p E H'(Q) ;

we t h u s o b t a i n

which, w i t h U E Hz(Q) n H i @ ) , l e a d s t o U E Ht(62). With U E H ; ( Q ) , it may b e shown t h a t (u, - AU)EW ; it t h e n f o l l o w s from ( 4 . 5 4 ) and from t h e p r o p e r t i e s of u d e f i n e d above that

and u i s t h u s t h e s o l u t i o n o f ( 4 . 2 ) , ( 4 . 3 ) . Conversely, if u i s t h e s o l u t i o n o f ( 4 . 2 ) , ( 4 . 3 ) it may b e shown t h a t (u, - Au) s a t i s f i e s t h e n e c e s s a r y and s u f f i c i e n t c o n d i t ions ( 4 . 5 4 ) , ( 4 . 5 5 ) .

Remark 4.10.

I n t h e c a s e o f t h e problem

A% = f

in Q

(4.58) YOU

= 91 ,

71

u = 9 2 ; g,, g2 s u f f i c i e n t l y r e g u l a r

it i s n e c e s s a r y t o r e p l a c e t h e s p a c e W, s e e ( 4 . 5 2 1 , by

Unilateral probZems and e l l i p t i c i n e q w z l i t i e s

330

(CHAP.

4)

where g denotes the pair { g,,g2 }. There is then equivalence between ( 4 . 5 8 ) and

4.4

An iterative method for the solution of the Dirichlet problem for A ' based on the variational formulation given in Section 4.3.

We shall give in explicit form a variant of algorithm ( 4 . 4 ) ( 4 . 6 ) associated with the equivalent variational formulation ( 4 . 5 3 ) it is convenient, and reasonable enough, of problem ( 4 . 2 ) - ( 4 . 3 ) ; to interpret ( 4 . 5 3 ) as an optimal control problem in order to demonstrate the following algorithm. Let A b e a complement of HA(S2)

in

H'(Q)

; we thus have

H'(S2) = HA(S2) @ "4

and there is equivalence between ( 4 . 5 3 ) and the following optimal control problem with s t a t e c o n s t r a i n t s :

the s t a t e variable u being a function of the control variable q through the intermediary of the s t a t e equation

iI"

grad u . grad p dx

(4.62)

u E Hi(i2) :

=

In

qp dx Vp E HA@)

q'and u must also satisfy the s t a t e constraints

(SEC.

(4.63)

4)

Fourth-order v a r i a t i o n a l i n e q u a l i t i e s

Jn grad v.grad p dx =

331

In

q p dx ~p E M

Let u be the solution of (4.2), (4.3); problem (4.61)-(4.63) is equivalent to (4.53) and thus to ( 4 . 2 ) , (4.3) and therefore admits p = - h as its unique solution, u being the corresponding state function. A Lagrangian associated with (4.61)-(4.63) is 9 : L 2 ( Q ) x M + IW (in which u is a function of q through the intermediary o f (4.62)) defined by

(4.64)

grad v.grad p dx

Y(q,p) =

-

We shall now prove: Theorem 4.6. Let u be t h e s o l u t i o n of (4.21, (4.3) and p = - Au ; we assume t h a t U E H 3 ( Q ) n H i @ ) ( ; then 2’ admits { p , p 2 } as unique saddle p o i n t on L 2 ( Q ) x .& , w i t h p 2 defined by P

= P1

+ Pl1

PI E H m , P 2 E A .

It follows’from Theorem 4.6 that we may apply to 9 the algorithm (4.12), (4.13) of Chapter 2, Section 4.3, that is: (4.65)

l oE A

(4.66)

Y(p”, A”) < Y ( q ,A”) Vq E L2(L?), p” E L2(Q)

(4.67)

I

chosen arbitrarily

L I ( ~ “ + ~ p, ) = b ( P , p ) + p , VPE A , p”

A”+’ E A ,

grad u“.grad p dx-

>0;

in (4.67),’ b is a bilinear form which is continuous, symmetric and coercive on A x A. The strong convergence of p” to p is assured if 0 c a. < p. < a l , a1 sufficiently small. We shall now write problem (4.66)explicitly; in view of (4.64), (4.66) has the form of an unconstrained optimal control problem, which admits (see Lions/2/, Chap. 2) a unique solution p” characterised by the existence of an adjoint state y ” e H : ( f 2 ) such that (l)

If

r

is regular, then this holds for f ~ H - ’ ( a ) .

Unilateral problems and e l l i p t i c inequalities

332

grad u".grad p dx

(CHAP.

4)

p" p dx Vp E H,'(Q)

5

u" E H&Q)

(4.68)

(4.69)

p" = u"

+ I".

In view of (4.69)we may replace (4.68), (4.69) by gradp".gradpdx = p"

fpdx

- I" E H ; ( Q ) ,

which gives the explicit form of gradp".grad u dx = (4.70)

[ p" - I" E H;(Q)

/In

grad u".grad u dx =

(4.71)

[

Vp~Hi(f2)

u"~H&2);

I.

(4.66):

fu dx Vu E H & 2 )

1.

p" u dx V u E H,'(Q)

the determination of u"thus reduces to the solution of two Dirichlet problems for - A , and it may be noted that in view of (4.71) the strong convergence, in L 2 ( Q ) , of JP to - Au implies the strong convergence of u" to u in HZ(Q).

4.5

An approximation of problem (4.2), (4.3) by mixed finite

elements

4.5.1

Synopsis

Following Ciarlet-Raviart /3/, we shall now approximate problem (4.2), (4.3), via the equivalent formulation (4.53) , by a finite element method which is i n t e r i o r in H'(Q) and exterior in H2(Q) (see Remark 4.4). 4.5.2

Approxkztion of

L2(Q).H'(f.2). H&2), A

By way of simplification, we shall assume that D is a polygonal

(SEC.

4)

Fourth-order VariationuZ ineqwzzities

333

w i t h f h a t r i a n g u l a t i o n of La which s a t i s f i e s t h e domain of R'; assumptions o f Section 1 . 3 of t h i s c h a p t e r ( t h e n o t a t i o n of which i s r e t a i n e d h e r e ) and w i t h P k denoting t h e space o f polynomials with two v a r i a b l e s of degree 6 k , w e approximate L'(Q) and H 1 ( Q ) by v h

then H&2)

=

{

0,

1 0, E co((;iji>, 0, IT E Pk VTE f l h }

by

voh = {

uh

I l)h E

vh, 0,

= 0 on

r}.

For A h , we may t a k e any a r b i t r a r y complement o f V,in t h e simplest i s t h a t defined by = 0 VTEflh,a T n r = 0 ) = { l)h I 1 ) h E v h , 0,

vh,

but

and t h i s is t h e only one which w i l l be considered in t h e following.

Approccimation of (4.5;)

4.5.3

L e t W,be t h e approximation of W ( s e e ( 4 . 5 2 ) ) defined by (uh,

qh)

I (Uh, q h ) e v,

x

vh,

grad %.grad p h dx =

We approximate (4.53) by

F i n i t e element approximations o f t h i s type a r e s a i d t o be mixed, s e e C i a r l e t - R a v i a r t /3/.

4.5.4

Convergence resuzts (k 3 2)

We suppose t h a t t h e angles of fhaare bounded below, UnifOYdy t?,>O , and a l s o t h a t T h s a t i ' s f i e s t h e following asswnp t i o n ( t h e so-called inverse hypothesis): Max h(T) h(T) 6 T v f h , r independent of h , (4.73)

in h , by

iz TC

J i

with, i n ( 4 . 7 3 ) , h(T) = l e n g t h of t h e l o n g e s t s i d e of T . It i s shown i n C i a r l e t - R a v i a r t , c i t . , t h a t using t h e above assumpt i o n s we have

&.

(4.74)

II uh - IIBI(o)

+ 1 (-

AU)

- ph IIL'(a) 6 C II 11 lIW+l(n)p-'

Unilateral problems and e l l i p t i c i n e q u u l i t i e s

334

with, i n

( 4 . 7 4 ) , C independent o f h and

(CHAP.

4)

u.

Remark 4.11. When u # H'''(61) we have, when h + 0, and u s i n g t h e above assumptions on Fh,strong convergence o f u h t o u i n H'(61) and o f ph t o -Au i n L'(Q) ; however, we do not have any e s t imate o f t h e approximation e r r o r s in t h e s e s p a c e s .

4.5.5

The case k = 1

I f k = 1, we do n o t i n general know ( 3 ) whether {uh,Ph}Converges A s u f f i c i e n t c o n d i t i o n f o r having {uh,Ph}+ ( u , -Au} t o {u, -Au}. s t r o n g l y i n H'(61) X L'(Q)is t o assume t h a t T h i s g e n e r a t e d by 3 f a r i l i e s of parallel s t r a i g h t l i n e s ( e x c e p t p o s s i b l y i n a neighbourhood o f r t h e measure o f which t e n d s t o z e r o ) . Moreover i f , f o r k = 1, we r e p l a c e i n t h e d e f i n i t i o n o f Whthe condition

(4.75)

-

grad o h grad ph dx

=

by ( l ) ( s e e F i g u r e 4 . 1 )

(4.76)

Fig. 4.1. which i s o b t a i n e d by numerical integration o f t h e right-hand s i d e of ( 4 . 7 5 ) , we g e t , i f 61 i s a r e c t a n g l e ( 2 ) and f o r t h e t r i a n g u l a t i o n shown i n F i g u r e 4.2, t h e c l a s s i c a l 13-point scheme f o r t h e d i s c r e t i s a t i o n o f t h e D i r i c h l e t problem f o r A'. (l)

A ( T ) = a r e a o f T.

(2)

This a l s o a p p l i e s i f t h e coordinate axes;

r

i s t h e union o f segments p a r a l l e l t o i n c i d e n t a l l y , t h e same numerical i n t -

e g r a t i o n formula i s needed t o e v a l u a t e (3)

i[* 1 %

i'&

in (4.72)-

I n f a c t it has been shown, subsequent t o t h e p u b l i c a t i o n o f t h e o r i g i n a l French v e r s i o n of t h e p r e s e n t volume, t h a t convergence can be proved even i n t h e c a s e k = 1 (seeAppendix4).

(SEC. 4)

Fourth-order variationa I inequa I i t i e s

335

Fig. 4.2.

4.6

An iterative method for the solution of the approximate Generalisation to inequalities of problem (4.72). order 4.

4.6.1

Description of the aZgorithm

The approach used in Section 4.4, based on optimal control, also holds for the finite-dimensional problem (4.72); in order to solve the latter problem we may therefore use the following algorithm ( 1) : (4.77)

A,"EY/~;,

chosen arbitrarily,

then, having calculated

il.

grad p;.grad

(4.78)

Pf

- If

{I.

Vh

A f E d h ,

dx =

( "The

4

fVh

dX

VVh E

4,If+'

by

vo,

vOh

grad uf .grad 0, dx

(4.79)

I,

we calculate pi,

=

pf Oh dx

v0h E

voh

vOh

finite-dimensional variant of (4.65), (4.'(O) ,(

4.71).(4.67).

Unilateral problems and e l l i p t i c inequalities

336

where, in (4.80), Ph : A, x Ah-+ R metric and coercive.

(CHAP. 4)

is bilinear, continuous, sym-

Remark 4.12. If k = 1 we may, of course, replace the righthand side of (4.79), (4.80) by the term obtained if (4.76) is used instead of (4.75).

4.6.2

Convergence o f algorithm (4.77)-(4.80)

It may be shown, see Ciarlet-Glowinski /l/, that under the condition 0 < a. Q P. Q a l , a1 sufficiently small, we have

lim { $ipi, nli" 1 = { uh,ph, 1

n++m

where { uh,ph}isthe solution of (4.72) and & € A h is a component of in the decomposition v h = V, @ A h . In the following, we shall denote by 2 4 the upper bound of the al values which give convergence of algorithm (4.77)-(4.80).

4.6.3

Remarks on the choice of

The bilinear forms flhdefinedbelow possess the properties required in Section 4.6.1: (4*81)

Ph)

=

(4.82)

8t(h*Ph)

=

Ah.*ad

Ir

Ph

dX

b Ph dr .

These forms have the disadvantage that, in the calculation of by means of (4.80), a linear system ( l ) must be solved at each iteration of algorithm (4.77)-(4.80). To remedy this situation we may profitably use a numedeal! integration method to obtain a form awhose matrix is diagonul with respect to the degrees of freedom defining the finite-element approximation considered; in particular, this will be the case if is approximated by

.A:+'

(')

It must be noted, however, that the matrix of this system is symmetric, positive-definite, and of relatively small bandwidth; consequently it is easy to employ a Cholesky factorisation which is performed once and for all.

(SEC. 4 )

Foi*rth-order v a r i a t i o n a l i n e q u a l i t i e s

337

t h e n o t a t i o n u s e d i n ( 4 . 8 3 ) , ( 4 . 8 4 ) i s t h e same a s t h a t employed i n Section 1.3.2 of t h i s chapter.

4.6.4

Behaviour of

q, when

h

+

0

L e t a o b e t h e c o n s t a n t d e f i n e d by ( 4 . 8 ) ; w e suppose t h a t t h e form bh used i s t h a t d e f i n e d by ( 4 . 8 2 ) . Using t h e assumptions made i n S e c t i o n 4 . 5 . 4 f o r T h i t may t h e n b e shown, see C i a r l e t - G l o w i n s k i /l/, t h a t Vk 2 1, w e have

lim oh = uo .

(4.85)

h-0

The r e s u l t ( 4 . 8 5 ) a l s o h o l d s i f we approximate k = 1, and by ( 4 . 8 4 ) i f k = 2 .

by (4.83) i f

The r e s u l t ( 4 . 8 5 ) a l l o w s a p r a c t i c a l estimate o f t h e q u a n t i t y a i s a n open domain f o r which it i s possi b l e t o d e t e r m i n e , or a t l e a s t t o estimate, go ( s e e J . Smith /l/). oh t o be o b t a i n e d when

4.6.5

Application t o the s o l u t i o n o f v a r i a t i o n a l i n e q u a l i t i e s of order 4 .

We s h a l l now c o n s i d e r t h e i n e q u a l i t i e s o f Examples 1 and 2 o f S e c t i o n 4.2.2; w e assume i n t h e f o l l o w i n g , c o n f i n i n g our a t t e n t i o n t o k = 1 and 2 , t h a t t h e forms Ph used a r e t h o s e d e f i n e d by (4.83), (4.84). Furthermore, t h e l i n e a r mapping A,,-+R d e f i n e d by

(l)

Indeed, t h i s a l s o h o l d s when k = 1, i f w e u s e of ( 4 . 7 5 ) .

(4.76) instead

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

338

(CHAP.

4)

Approximate solution of inequality (4.29) We "approximate" A = L : ( f ) by Ah={phIphEAh,ph(P)aO

V P E Z h n f if k = l ,

V P E ( Z h u Z C I ; ) n r if k = 2 } ;

in order to solve inequality (4.29) in an approximate manner, we may then use the following algorithm (a finite-dimensional variant of algorithm ( 4.33)-( 4.35)) : (4.88)

A:

(4.89)

having calculated A:E Ah we determine p;,

E Ah,

chosen arbitrarily

4

by

(4.78), (4.791, then A;"

by

with p,,

7

0.

.

Approximate s o l u t i o n o f inequality (4.36) We "approximate" A ( see 4.38)) by

and then, to solve (4.36) 'in an approximate manner, we use the following finite-dimensional variant of algorithm (4.41)-(4.43) : (4.91) (4.92)

Ah, chosen arbitrarily

having calculated A:E A, we determine p:, d by (4.78) (4.791, then A;+' by

(SEC. 4 )

Fourth-order v a r i a t i o n i n e q u a l i t i e s

w i t h pn > 0 and P I - l.+,,(a)= max ( - 1, min (a, 1))

339

Va E R.

Convergence of a Zgori thms ( 4 .88 ) - ( 4 .go) and ( 4 .91)- ( 4 .93 ) The c o n s t a n t a,, b e i n g t h e one a p p e a r i n g i n S e c t i o n 4 . 6 . 2 , may show t h a t under t h e c o n d i t i o n s

(4.94)

0 < No

< p. < a ,

< 2 a:

(4.95)

0 < a0

< pn < a,

< 7 0 ;f o r problem ( 4 . 3 6 ) ,

f o r problem ( 4.29)

. we

,

2 9

t h e r e i s convergence o f t h e above a l g o r i t h m s t o a p a i r { u , , p , , } which converges s t r o n g l y i n H I @ ) x L 2 ( R ) , when h + 0 ( t h e hypotheses on Y,,being t h o s e of S e c t i o n 4 . 5 . 4 ) t o { u, - A u } , where u i s t h e s o l u t i o n o f t h e c o r r e s p o n d i n g c o n t i n u o u s problem.

Remark 4.13. The numerical s o l u t i o n of problem ( 4 . 4 5 ) i n Example 3, S e c t i o n 4 . 2 . 2 , does n o t p r e s e n t any s e r i o u s p r a c t i c a l d i f f i c u l t i e s ; i n f a c t t h e s o l u t i o n of problem ( 4 . 4 5 ) , s u i t a b l y approximated, may be r e d u c e d t o t h a t o f a sequence o f problems o f t h e t y p e ( 4 . 7 2 ) , which may t h e n be s o l v e d by means o f a l g o r i t h n i ( 4 . 7 7 ) - ( 4 . 8 0 ) ( s e e S e c t i o n 4.7 b e l o w ) . W

4.7

Example and n u m e r i c a l a p p l i c a t i o n

Synopsis I n o r d e r t o i l l u s t r a t e t h e a p p r o x i m a t i o n and s o l u t i o n methods of S e c t i o n 4 , we have chosen a problem which f a l l s i n t o t h e f o r f u r t h e r fourth-order c a t e g o r y o f Example 3 , S e c t i o n 4 . 2 . 2 ; W problems t r e a t e d by a n a l o g o u s methods, s e e Bourgat / 2 / .

4.7.1

Formulation o f the problem

Let R and E be two open bounded domains o f R2, such t h a t

Bc R ;

Unilateral problems and e l l i p t i c inequalities

340

(CHAP. 4 )

we denote by K t h e closed convex subset of Hi(61) defined by

K = ( U ~ U E H ~ ( ~ ~ ) , U (V X X E) E>}C Under t h e s e c o n d i t i o n s t h e problem (4.96)

admits one and only one s o l u t i o n , and t h e r e i s equivalence between ( 4 . 9 6 ) and t h e following fourth-order i n e q u a l i t y : (4.97) @ A. : : ! {

-

4

2 0 vu E K

Approximation of problem ( 4 . 9 6 ) , ( 4 . 9 7 )

4.7.2

t o approximWe assume i n t h e following t h a t 61 i s polygonal; a t e ( 4 . 9 6 ) , (4.97) we s h a l l use t h e f i n i t e element approximation of S e c t i o n 4 . 5 , with f i n i t e elements of o r d e r 1; t h i s i s perm i s s i b l e i f ( s e e S e c t i o n 4 . 5 . 5 ) t h e t r i a n g u l a t i o n F h u s e d i s gene r a t e d (except p o s s i b l y i n t h e neighbourhood of r) by t h r e e fami l i e s of pa ral le l st raigh t l i n e s . With t h e space K d e f i n e d by

K

=

{ (4, qh) I (uh, qh) E VOh x

vh

9

(vh, q h ) s a t i s f i e s

(4*76) }

9

w e denote by Zh t h e s e t of t h e v e r t i c e s of F h a n d we approximate, r e s p e c t i v e l y , K and ( 4 . 9 6 ) , ( 4 . 9 7 ) by (4-98)

Kh

=

{ (bqh) I (uh, q h ) E &, uh(P) >

c

VP E E n Zh }

The approximate problem (4.99) admits one and only one s o l u t ion, (uh,ph). 4.7.3.

I t e r a t i v e methods of solution

A f i r s t method of solu'tion. This i s a d i s c r e t i s e d v e r s i o n o f algorithm ( 4 . 4 9 ) - ( 4 . 5 1 ) ; with Mhdenoting t h e number o f elements of &, n E , we d e f i n e g,, : Wh x R M h + R , t h e Lagrangian a s s o c i a t e d with (4.99) , by

(SEC. 4 )

Fourth-order variational i n e q u a l i t i e s

341

where, i n (4.100), T ( P ) = sum of t h e a r e a s of t h e t r i a n g l e s on r h S i n c e t h e Lagrangian 5f'hadmits which admit P as common v e r t e x . W, x RY , namely (Uh,Ph,Yk) where ( u h , p ~i s t h e a saddle-point on s o l u t i o n o f ( 4 . 9 9 ) , t h e n i n o r d e r t o s o l v e t h i s l a t t e r problem we may use t h e f o l l o w i n g "approximation" o f a l g o r i t h m ( 4.49)-( 4.51) : (4.101)

y: E

RY,

chosen a r b i t r a r i l y

then having c a l c u l a t e d

A , we determine (4,m and A+' by

Under t h e c o n d i t i o n 0 < a. d p. d a1, a1 s u f f i c i e n t l y s m a l l , we It may be have convergence o f t h e sequence ( & p a t o (uh,ph) noted t h a t problem ( 4 . 1 0 2 ) i s o f t h e t y p e c o n s i d e r e d i n S e c t i o n 4.6.1 and hence may be s o l v e d u s i n g a l g o r i t h m ( 4 . 7 7 ) - ( 4 . 8 0 ) .

.

A second method o f solution. By combining a l g o r i t h m s (4.101)-( 4.103) and ( 4 . 7 7 ) - ( 4 . 8 0 ) we o b t a i n ( u s i n g t h e n o t a t i o n o f S e c t i o n s 4 . 5 . 5 and 4.6.5) t h e f o l l owing a l g o r i t h m : (4.104)

(A:, y:)

E

Ahx

Wy,chosen

then having c a l c u l a t e d

arbitrarily

(1i,m we

determine ( G , p D and (L;+',X+') by

(4.105)

(4.106)

"

(4.107)

(4.108)

It may b e shown t h a t under t h e c o n d i t i o n s 0 < a: d p: Q a:, i = 1,2 , w i t h a: s u f f i c i e n t l y s m a l l , we have convergence o f

342

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

(CHAP.

4)

(G,& t o (u,,,ph) ; t h e problem o f t h e d e t e r m i n a t i o n o f t h e sequence (p:,p:)" which g i v e s ( 4 . 1 0 4 ) - ( 4 . 1 0 8 ) t h e o p t i m a l r a t e o f convergence r e q u i r e s f u r t h e r a t t e n t i o n . 4.7.4

An a p p l i c a t i o n

Geometric d a t a : d i s c w i t h c e n t r e ( 0 . 5 , 0 . 5 ) and r a d i u s 0.25,

61=]0,~l[x]O,l[, E : c=0.062 5.

Triangulation of 61: The t r i a n g u l a t i o n used i s t h a t o f F i g u r e 1 . 9 i n S e c t i o n 1 . 7 . 1 o f t h i s c h a p t e r , comprising 800 t r i a n g l e s , 4 4 1 nodes and 361 i n t e r i o r nodes. I t may b e noted t h a t y h i s indeed g e n e r a t e d by t h r e e f a m i l i e s o f p a r a l l e l s t r a i g h t l i n e s . I t may a l s o b e noted t h a t

3 Z(P)= 400

VP.

Sol u t i o n algorithm : We use ( 4 . 1 0 4 ) - ( 4 . 1 0 8 ) .

I n i t i a l i s a t i o n of algorithm ( 4 . 1 0 4 ) - ( 4.108) A:

= O,y,o(P) =

:

100 v P € E n z h .

S o l u t i o n of the D i r i c h l e t problems ( 4 . 1 0 5 ) , ( 4 . 1 0 6 ) : B y t h e Cholesky method.

Parameters

pt, p i :

These were t a k e n a s p; = p1 = 1.25, pt = p2 = 750 ; it may b e noted t h a t t h e o r d e r s o f magnitude o f t h e s e two parameters a r e very d i f f e r e n t .

Analysis of the r e s u l t s . Table 4 . 1 shows t h e v a r i a t i o n s of

(SEC. 4 )

Fourth-order v a r i a t i o n a 1 i n e q u a l i t i e s

as a f u n c t i o n o f t h e number o f i t e r a t i o n s ; R;h

343

t h e s i g n i f i c a n c e of

i s t h a t t h i s q u a n t i t y i s an approximation o f

sI:,

dr

, and

t h i s t h e r e f o r e a l l o w s us t o a s c e r t a i n whether t h e c o n d i t i o n

a U / a n = O h a s been p r o p e r l y s a t i s f i e d ( a p p r o x i m a t e l y a t l e a s t ) .

Number o f i t e r a t i o n s 50 100 200 300 400 500 600 700 750

rh

9x10-’ 1.3 x ~ O - ~ 1.7xlO-’ 1.3 xl0 1.2x10-6 9.7~10-~ 7.7 x106.25 xl0 5.6~10-~

Table

R;h

1 . m 0- 2 7.3x10-’ 1.2x10 - 3 1 . 5 10~ 1.07 x108.&10-’ 6.9~105.6~10-~ 5x10-’

’

4.1

The computation t i m e i s 2 min. 40 s e e . f o r 100 i t e r a t i o n s on a C I I 10070, t h e c a l c u l a t i o n s b e i n g performed i n double p r e c i s i o n . F i g u r e 4 shows t h e e q u i p o t e n t i a l s o f t h e computed s o l u t i o n uh, and F i g u r e 4 . 4 r e p r e s e n t s t h e f u n c t i o n x1 + Uh(Xlr 0.5).

Remark 4 . 1 4 . With t h e domain 62 s u p p o r t i n g a n e l a s t i c t h i n p l a t e , it a p p e a r s - and t h i s c o u l d b e p r e d i c t e d t h e o r e t i c a l l y t h a t t h e zone of c o n t a c t between t h e p l a t e and t h e o b s t a c l e ( i . e . t h e c y l i n d e r o f b a s e E and o f h e i g h t C ) , i s l i m i t e d t o a c i r c l e . o f r a d i u s 0.25 and h e i g h t C. I t t h e n f o l l o w s t h a t t h e m u l t i p l i e r yh h a s no l i m i t i n L2(62)when h + O ; t h i s i s thus t y p i c a l of a s i t u a t i o n i n which t h e a l g o r i t h m ( 4 . 4 9 ) - ( 4 . 5 1 ) i s n o t meaningful. I t may a l s o b e _ n o t e d t h a t t h e m u l t i p l i e r yh is z e r o e x c e p t a t t h e p o i n t s o f & , n E n e i g h b o u r i n g t h e c i r c l e aE, t h e boundary o f E ; a t t h e s e p o i n t s , yh t a k e s v a l u e s which become p r o g r e s s i v e l y l a r g e r as t h e v a l u e o f h d e c r e a s e s ( o f t h e o r d e r o f 1000 f o r t h e t r i a n g ulation Fhused). rn

Remark 4.15. It i s e v i d e n t t h a t f o r t h e shape o f B a n d t h e t r i a n g u l a t i o n Y h u s e d , t h e a p p r o x i m a t e problem ( 4 . 9 9 ) r e l a t i n g t o t h e example i n S e c t i o n 4.7.4 c o u l d have been d e s c r i b e d i n terms (1)

I n f a c t t h e l i m i t o f yh i s n o t a f u n c t i o n .

344

h i l a t e r a l probZems and e l l i p t i c i n e q u a l i t i e s

Mg.4.3.

Equipotentials of the function u,.

Fig. 4.4.

Graph of the function sr,(xl,O.S).

(CHAP.

4)

of a f i n i t e di ff erence approximation; we have not pursued this viewpoint, however, since'the application of finite difference methods for problems of order 4 is i n pra c tic e limited to domains I) whose boundaries consist of segments p a r a l l e l t o the coordinate a xe s; the finite element approximations studied in Section 4 do 8 not suffer from this limitation.

(SEC.

5.

4)

Fourth-order variational i n e q u a l i t i e s

34 5

DISCUSSION

S e c t i o n s 1, 2 , 3 a r e g i v e n f o r r e a s o n s o f methodology; i n f a c t t h e t 6 c h n i q u e s employed a r e o f a g e n e r a l i n t e r e s t and a r e t h e ones used i n a number o f i n t e r e s t i n g a p p l i e d problems. For exampl e , t h e r e a d e r may c o n s u l t Fr6mond 121. The methods o f S e c t i o n 4 seem t o b e w e l l s u i t e d t o t h e s o l u t i o n o f c e r t a i n problems i n F l u i d Mechanics. I n f a c t , u s i n g t h e n o t a t i o n o f S e c t i o n 4 . 3 , t h e f u n c t i o n s u and p (p = - Au) a p p e a r r e s p e c t i v e l y as t h e stream function ( ’ ) a n d t h e v o r t i c i t y ( 2 , when, f o r example, S t o k e s and Navier-Stokes flows a r e b e i n g modelled by t h e finite-element p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f o r d e r 4; method o f S e c t i o n 4 . 5 - which f o l l o w s C i a r l e t - R a v i a r t / 3 / - posse s s e s , amongst o t h e r a d v a n t a g e s , t h a t o f g i v i n g a continuousapprox i m a t i o n o f t h e s t r e a m f u n c t i o n and o f t h e v o r t i c i t y . Moreover, it r e s u l t s from S e c t i o n 4 . 6 - which f o l l o w s C i a r l e t - G l o w i n s k i 111, /2/ - t h a t i n o r d e r t o s o l v e t h e D i r i c h l e t problem f o r A’, it is s u f f i c i e n t t o have a v a i l a b l e a f i n i t e - e l e m e n t program f o r t h e D i r i c h l e t problem f o r - A ( i n f a c t , t h e a p p r o x i m a t i o n d e s c r i b e d i n Some s t u d i e s o f i t e r Section 4 . 5 w a s constructed t o t h i s end). a t i v e a l g o r i t h m s o f t h e t y p e c o n s i d e r e d i n S e c t i o n 4 may b e found, f o r example ( 3 ) , i n Smith 111, 121, 131, B o s s a v i t 111, E h r l i c h 111; however, t h e s e methods a r e s u i t a b l e o n l y f o r f i n i t e difference a p p r o x i m a t i o n s , and f o r domains whose b o u n d a r i e s a r e formed by t h e union o f segments p a r a l l e l t o t h e c o o r d i n a t e a x e s . The a p p l i c a t i o n o f t h e s e methods, i n c o n j u n c t i o n w i t h some a p p r o x i m a t i o n s by f i n i t e elements, a p p e a r s i n Glowinski 161, where a n i n t e r p r e t a t i o n i s a l s o g i v e n o f t h e a l g o r i t h m s mentioned above as b e i n g algori-thms t h i s l a t t e r p o i n t o f view f o r t h e c a l c u l a t i o n of s a d d l e - p o i n t s ; i s u s e d s y s t e m a t i c a l l y i n C i a r l e t - G l o w i n s k i 111, 121 and i n S e c t i o n 4 o f t h e p r e s e n t c h a p t e r , where t h e p o s s i b i l i t y o f combining t h e s e i t e r a t i v e methods w i t h p r a c t i c a l l y any f i n i t e element approximation ( e x c e p t e l e m e n t s o f o r d e r 1; s e e S e c t i o n 4.5.5) ( 4 ) i s demonstrated.

~~

(’) (2)

(3) (4)

U s u a l l y d e n o t e d by Y . U s u a l l y d e n o t e d by w. We r e f e r t o Glowinski 161 and C i a r l e t - G l o w i n s k i 111, 121 f o r a d e t a i l e d b i b l i o g r a p h y on t h i s s u b j e c t . Subsequent t o t h e p u b l i c a t i o n o f t h e o r i g i n a l French v e r s i o n o f t h e p r e s e n t volume, it h a s been shown t h a t even t h i s l i m i t a t i o n does n o t a p p l y ( s e e Appendix 4 ) .

This Page Intentionally Left Blank

Chapter 5 N U M E R I C A L ANALYSIS OF T H E STEADY FLOW OF A B I N G H A M FLUID I N A C Y L I N D R I C A L

DUCT

INTRODUCTION

I n t h i s c h a p t e r we s h a l l i n v e s t i g a t e t h e a p p l i c a t i o n o f t h e r e s u l t s and methods of C h a p t e r s 1 and 2 t o t h e n u m e r i c a l a n a l y s i s 8 o f t h e f l o w problem c o n s i d e r e d i n Chapter 1, S e c t i o n 1 . 4 .

1. STATEMENT O F THE CONTINUOUS PROBLEM. SYNOPSIS

PHYSICAL MOTIVATION.

1.1 S t a t e m e n t o f t h e c o n t i n u o u s problem

The n o t a t i o n i s t h a t of Chapter 1, S e c t i o n 1 . 4 ; a i s an open r . On Hd(C2) we d e f i n e :

bounded domain o f R2 w i t h boundary

I,

I grad u I dx ,

(1.3)

j(v) =

(1.4)

J8(u) = Jo(v) + g j ( u ) , g constant 2 0 ,

and we c o n s i d e r t h e v a r i a t i o n a l problem: (1.5)

( P o ) : min J,(v). v E H$?)

S i n c e t h e f u n c t i o n a l Je i s c o n t i n u o u s and s t r i c t l y convex on H;(Q), w i t h

lim IIuIIH;(,)'

J,(u) +m

=

+ 03

,

Theorem 2 . 1 o f Chapter 1, S e c t i o n 2 . 1 a p p l i e s , g i v i n g t h e e x i s t e n c e and u n i q u e n e s s o f a s o l u t i o n o f (Po)l. We s h a l l denote t h i s s o l u t i o n by ue ; ug i s a l s o t h e unique s o l u t i o n o f the variational inequality :

e q u i v a l e n t t o (Po). 8

Numerical analysis of Bingham fluid flow

348

1.2.

(CHAP.

5)

Physical motivation

I n Duvaut-Lions /l/, Chapter 6, f o l l o w i n g t h e d e f i n i t i o n of a Bingham f l u i d , it i s shown t h a t i n t h e s p e c i a l c a s e of s t e a d y laminar flow i n a c y l i n d r i c a l duct of c r o s s - s e c t i o n t h e flow v e l o c i t y i s t h e s o l u t i o n o f ( l . 5 ) , (1.6), where f i s t h e l i n e a r p r e s s u r e drop ( c o n s t a n t i n p r a c t i c e ) , g i s a p o s i t i v e c o n s t a n t r e p r e s e n t i n g a s t r e s s t h r e s h o l d below which t h e f l u i d behaves as a r i g i d material and above which it behaves a s an incompressible f l u i d w i t h v i s c o s i t y p ; h e r e a f t e r we s h a l l t a k e p = 1. For t h e numerical a n a l y s i s of more g e n e r a l Bingham flows t h e r e a d e r i s r e f e r r e d t o BCgis /l/, F o r t i n /1/ and Chapter 6 of t h e p r e s e n t volume.

1.3.

Synopsis

I n S e c t i o n 2 we s h a l l g i v e some o f t h e p r o p e r t i e s o f t h e s o l u t i o n o f problem ( P o ) ; from t h e Numerical Analysis p o i n t o f view t h e major d i f f i c u l t y o f t h e problem c o n s i d e r e d i s t h e nondifferentiability of t h e f u n c t i o n a l J # ; i n S e c t i o n s 6 and 7 we s h a l l d e s c r i b e t e c h n i q u e s o f regularisation ( c f . Chapter 1, S e c t i o n 3.3) and duality ( c f . Chapter 1, S e c t i o n 3.4 and Chapter 2 ¶ S e c t i o n 4 ) which a l l o w us t o s o l v e (Po) ( l ) , or, more p r e c i s e l y , approximations o f (Poh t h e s e approximations a r e i n v e s t i g a t e d i n S e c t i o n s 3, 4 and 5 . I n S e c t i o n 8 we s h a l l apply t h e r e s u l t s o f t h e p r e c e d i n g s e c t i o n s t o t h e s o l u t i o n of t h e e l a s t o - p l a s t i c t o r s i o n problem i n v e s t i g a t e d i n Chapter 4 , u s i n g t h e d u a l f o r m u l a t i o n o f Chapter 3, S e c t i o n 9.1.1. 2. SOME PROPERTIES 2.1.

OF THE SOLUTION

OF THE CONTINUOUS PROBLEM

R e g u l a r i t y results

n

If i s a bounded open domain w i t h a s u f f i c i e n t l y regular boundary r t h e n it i s shown i n H. B r C z i s /6/ t h a t f E L 2 ( 9 ) i m p l i e s , for t h e s o l u t i o n ue of ( P o ) : ue E

(2.1)

H 2 ( a )n H J ( Q ) .

Remark 2.1. We s h a l l see i n S e c t i o n 2 . 3 t h a t t h e above p r o p e r t y i s o p t i m a l , at le,ast i n Sobolev spaces o f whole number order. 2.2.

Dependence of t h e s o l u t i o n on g

It i s shown i n Duvaut-Lions

('1

/1/¶Chapter 6, S e c t i o n 8.3 t h a t :

See a l s o S e c t i o n 9 f o r an a l g o r i t h m o f t h e augmented Lagrang i a n t y p e ( c f . Chapter 2 , S e c t i o n 6 ) .

Statement of continuous problem

(SEC. 1)

Theorem 2.1. The function from R+ -+ H,'(B). Moreover:

(2.2)

u, = 0 f o r

g

g -+ u,

349

i s Lipschitz continuous

2 go = criticaZ threshoZd depending on B k? J

and (2.3)

u& (resp. the function { decreasing (zero for 2 g -+ u(u,,

g

go).

g -+j(u,) )

i s continuous

8

I n t h e s p e c i a l c a s e f = c o n s t . (which i s o f i n t e r e s t i n p r a c t i c a l a p p l i c a t i o n s ) , ( 2 . 2 ) can be rendered p r e c i s e by showing ( s e e Glowinski 121) t h a t : (2.4)

go =

Ylfl,

Theorem 2 . 2

The mapping

g -+ u(u,, u&

is convex from R + + R+.

The f u n c t i o n u , i s t h e s o l u t i o n o f ( 1 . 6 ) ; by t a k i n g a n d u = 2 u , i n (1.61,w e t h e n deduce:

Proof. u=O

(2.6)

4u,, u,) + ai(u,) =

whence

I"

fu, dx

t h e mapping g+J,(u,), t h e lower envelope of a f a m i l y o f a f f i n e l i n e a r f u n c t i o n s of g , i s t h u s concave, s o t h a t g + a(u,, u& i s convex from ( 2 . 7 ) ( 1 ) .

Remark 2.2. Other p r o p e r t i e s o f t h e s o l u t i o n of (Po) w i l l be found i n Puvaut-Lions, loc. c i t . ; i n p a r t i c u l a r i f f 2 0 on B we have u,>O on B. 8 Remark 2.3.

We s h a l l apply t h e d u a l i t y t h e o r y due t o

( ' ) This proof does not use t h e s p e c i a l s t r u c t u r e o f U(U,U) and j(u).

Numerical analysis of Bingham f l u i d f l o w

3 50

(CHAP.

R o c k a f e l l a r /1/ t o t h e p r e s e n t problem (we a l s o r e f e r t o t h e account by Ekeland-T6mam /l/) We p u t ( 1 ) :

In

I grad u I dx - ( A u) ,

(2.9)

F(u)

(2.11)

nu= gradu

=g

(2)

B E H&2)

=

V

A E Y ( V ; Y).

The problem i s e q u i v a l e n t t o

(2.12)

inf [ f l u ) U

+ G(Au)] ;

we then have

(2.13)

inf [ f l u )

”

+ G(Au)] = sup [- F*(A* q) - G * ( -

q)],

@

where F* and G* a r e t h e c o n j u g a t e f u n c t i o n s of is t h e a d j o i n t o f A. Thus

A*q =

F and

G and A*

- divq

and

(2.15)

F*(A*q) = sup v

[-

Inudivqdx - g

lgradu Idx

1

+ cfi u) .

If w i s t h e s o l u t i o n o f

(2.16)

-Aw=f,

w~,=O,

then

u,u) = (grad w,

grad u )

and t h u s , i f we p u t : p =q

,&

+ gradw,

n > 2. A should not be confused w i t h t h e convex s e t denoted l a t e r on by t h e same term.

( l ) This a l s o h o l d s f o r ( 2 ) This

5)

Properties of so Zution of ( P o )

(SEC. 2 ) we have :

J*

F * ( A * q ) = s u" p [ b p . g r a d u d x - g

351

1

Igraduldx .

From this we deduce t h a t

IJ

[ O if F*(A* q) =

I+ a,

F*(A* q) =

p . g r a d u d x 1 < g J D Igraduldx

VU,

D

otherwise

{ + if 0

00

pEgK otherwise.

I f the dimension i s 1 w e have: K = { p , IP(X)I
4 Ip

- grad w

1' ,

I

I

= norm(L2(9))".

PEEK

Consequently

(2.18)

a(u,, u,) =

inf I p - grad w I' , p

E gK.

P

If n

=

1 and i f K

=

{ p Ip(x) I

<1}

we deduce e x p l i c i t l y

4u,, u,) =

where

dw h = - belongs to H'(l2) dx We t h e n deduce that

c C0@)

, from (2.161, if f ~ L ' ( n ) .

(CHAP. 5)

Numerical analysis of Binghwn f l u i d f l o w

352

I

I

I f t h e set on which h(x) = c h a s measure zero Vc, we t h e n have

d2

g = gc .

a(u,, u,) = 0 i f &I

--

We deduce a@,, u ~ ) and

d3

u,) =

-a@,,

413

m(g

- gc)3 i n t h e neighbourhood of 1

- 2 lim - measure { x

g

E

c Ih I

d g+E

}

gc(m 2 0)

d 0.

I n t h e g e n e r a l case ( 2 . 1 8 ) ,

L e t PK = orthogonal p r o j e c t i o n on K i n (L’(L2))”. We have: (2.19)

a@,, u,) = 8’

I Pdhlg) - h/g IZ

*

Duality relation. L e t q o be t h e s o l u t i o n of t h e dual problem, so t h a t go = PO - grad w ,

Po = g

p K grad ( T w > .

We have :

r;w).

ThUS

(2.20)

grad u = grad w

2.3.

- gP, -

Some p a r t i c u l a r cases f o r which t h e s o l u t i o n i s known

A f i r s t exwnple

2.3.1

We t a k e Q = p, 1[ and f = c > 0; t h i s corresponds t o a Bingham flow between two p l a n e s u n i t d i s t a n c e a p a r t . Problem (Po) i s t h u s defined by: (2.21) u

with

min

qcn,

[; j;

u’ = dvldx.

V”dx

+

gI;

,v’ldx -

cj;“dx],

(SEC. 2 )

Properties of solution of ( P o )

353

The s o l u t i o n of ( 2 . 2 1 ) i s g i v e n by: (2.22)

ug = 0

and i f g < c / 2

i

if

by: C

u,(x) = - x ( l 2

(2.23)

g 2 go = c / 2 ,

1 2

- x ) - gx

if 0 < x < - - 8

-1 - c-g < x 2 1 x) i f f <x if

U,(X)

C = -x(l

2

- x)

- g(1 -

+

c'

g < 2-1 + -, c

<1.

If can b e v e r i f i e d t h a t f o r g < c/2 we have ug E H i ( 0 , l ) n H2(0, 1) but t h a t u,# H3(0, 1 ) . F i g u r e 2 . 1 shows t h e s o l u t i o n o f ( 2 . 2 1 ) c o r r e s p o n d i n g t o g = l and c = 10 (we have f - g/c = 0.4 )

.

0.8

1

c X

Fig. 2.1.

2.3.2

A second example

We t a k e s1 = { x I x = (x,, x2), x: (Po) i s g i v e n by: (2.24)

ug = 0

and i f g < cR/2

(2.25)

+ x i < R 2 }, f = c > 0 ; t h e

i f g 3 go = cR/2, by:

1

+ R') - 2 9

if R ' < r < R ,

1

if 0

< r < R',

s o l u t i o n of

Numerical analysis of Bingham fluid flow

354

(CHAP.

5)

with

(2.26)

r =

,/=R‘ ,= 2 g / c .

Remark on the rum-uniqueness of the s o l u t i o n of the dual problem. It can e a s i l y b e shown t h a t i n t h e c a s e o f t h e example o f S e c t i o n 2 . 3 . 1 t h e d u a l problem

admits one and o n l y one s o l u t i o n , t h e r e a s o n f o r t h i s b e i n g obvious ! I n t h e c a s e o f t h e example o f S e c t i o n 2 . 3 . 2 , w e s h a l l s e e t h a t t h e d u a l problem admits an i n f i n i t e number o f s o l u t i o n s : Consider a system of pozar coordinates i n which w e u s e t h e n o t a t i o n p = { p,, } ; we t h e n h a v e :

(2.28)

a divp = -i -(rp,) r a.

+ -r1 -.ae

Using ( 2 . 2 8 ) and t h e c h a r a c t e r i s a t i o n :

[

- Au

- gdiv1= c ,

1.grad u = I grad u I , where u ( r e s p . 1) i s t h e s o l u t i o n o f it can be shown t h a t 1, d e f i n e d by:

-1,

(2.30)

1,

(2.31)

1, = - r / R ’ ,

=

1,

=

(Po) (or t h e d u a l p r o b l e m ) ,

0 i f R’ < r c R ,

1, = b(r) if 0 < r c R ’ ,

with:

(2.32)

BeLOD(O,R’),

l B ( r ) l Q , / m

a.e.

on ( O , R ’ ) ,

i s a s o l u t i o n of t h e dual problem which t h u s admits an i n f i n i t e number of s o l u t i o n s , b u t o n l y one r a d i a l l y symmetric s o l u t i o n , o b t a i n e d by t a k i n g b = 0 i n ( 2 . 3 1 ) .

Remark 2.4. Miasnikov ring.

O t h e r e x a c t s o l u t i o n s may b e found i n Mosolovp a r t i c u l a r for t h e c a s e where 61 i s a c i r c u l a r

111, i n

Approximation o f ( P O ) by f i n i t e elements

(SEC. 3 )

355

Remark 2 . 5 . We can now complete Remark 2.1; i n t h e two examples above, i n which t h e d a t a i s v e r y r e g u l a r , we have, f o r sufficiently s m a l l g, ugE

c (SZ) n~ ' ( n 0H ) ~ ( o but )

ug4

c2(ii), ug4 ~

~ ( . 0 )

INTERIOR APPROXIMATION OF ( P o ) BY A FINITE ELEMENT METHOD

3.

I n t h e following s e c t i o n s , unless s t a t e d otherwise, t h e s o l u t i o n o f (Po) w i l l simply by d e n o t e d by u .

Synopsis. I n t h i s s e c t i o n we s h a l l d e f i n e a method o f approximating (PO) using f i r s t - o r d e r f i n i t e e l e m e n t s , i . e . u s i n g c o n t i n u o u s f u n c t i o n s which a r e p i e c e w i s e l i n e a r and a f f i n e on a t r i a n g u l a t i o n o f on 6 2 ; t h e convergence of t h i s approximation w i l l be i n v e s t i g a t e d i n S e c t i o n 5 and n u m e r i c a l methods f o r s o l v i n g t h e approximate problem w i l l be g i v e n i n S e c t i o n s 6 and 7.

a

3.1.

Triangulation of 0 .

Definition of

v,.

We p r o c e e d as i n Chapter 3, S e c t i o n 4 . 1 . 1 , whose n o t a t i o n we r e t a i n ; we can t h u s c o n s t r u c t an i n t e r i o r approximation o f HA(0) - i n t h e s e n s e o f Chapter 1, S e c t i o n 4 - u s i n g a f a m i l y v h of subspaces of Hi@). 3.2.

D e f i n i t i o n of t h e approximate problem

We d e f i n e t h e approximate problem (Po,,) by: (3.1)

(POh) min Jg(Uh) . Vh E v h

3.3.

S o l v a b i l i t y of t h e approximate problem

S i n c e t h e f u n c t i o n a l J,, i s c o n t i n u o u s and s t r i c t l y convex on vh, w i t h lim Jg(uh)= co, it r e s u l t s from Chapter 1, S e c t i o n IIvh

11

-

+

+m

2 . 1 , t h a t (PO,,) admits one and o n l y one s o l u t i o n , uh s a y . approximate s o l u t i o n uh i s o b v i o u s l y c h a r a c t e r i s e d by:

3.4.

The

E x p l i c i t f o r m u l a t i o n o f t h e approximate problem

The f o r m u l a s of Chapter 3, S e c t i o n 4.1.4 a l l o w u s t o o e x p r e s s Jg(uh) as a f u n c t i o n of t h e n o d a l v a l u e s vM = uh(M),M E Z h ; t h e

(CHAP. 5 )

Numerical analysis of Bingham f l u i d flow

356

approximate problem ( 3 . 1 ) then comes down t o t h e minimisation of 0 a n o n d i f f e r e n t i a b l e f u n c t i o n a l on WN (N = dim vh = card (&))-

3.5.

On t h e use of f i n i t e elements of o r d e r g r e a t e r t h a n 1

I n t h e case of an approximation by f i r s t - o r d e r f i n i t e elements, f o r which t h e g r a d i e n t of oh i s c o n s t a n t over each t r i a n g l e of r h

,

t h e r e i s no d i f f i c u l t y i n e x p r e s s i n g j(u,,) =

I,

I grad vh I dx

as an e x p l i c i t f u n c t i o n of t h e nodal values ;,0 on t h e o t h e r hand, such a c a l c u l a t i o n i s impossible i f we use an approximation by f i n i t e elements of o r d e r k, k 2 2 ; it i s t h e n necessary t o use an approximation of t h i s i n t e g r a l ; of all t h e p o s s i b l e choices, we c i t e t h e following i n which A(T)= area of T, Gr= c e n t r e of g r a v i t y of T, i n which t h e n o t a t i o n i s as shown i n Figure 3.1: MI T

Fig. 3.1.

I n S e c t i o n 7 w e s h a l l meet an example taken from F o r t i n 111, i n which t h e s o l u t i o n o f , (Po) i s approximated by a f i n i t e element method of o r d e r 2 , t h e i n t e g r a l

b

I grad v I dx being e v a l u a t e d using

( 3 . 3 ) and ( 3 . 5 ) . I n c i d e n t a l l y , it may be noted t h a t because of t h e r e l a t i v e l y low r e g u l a r i t y of t h e s o l u t i o n , t h e advantage of u s i n g f i n i t e elements of o r d e r g r e a t e r t h a n one, f o r an equal number of degrees of freedom, is not c l e a r l y e s t a b l i s h e d .

(SEC. 4 )

4.

Exterior approximation of ( P o )

357

E X T E R I O R A P P R O X I M A T I O N S OF ( P o l

We r e t a i n t h e d e f i n i t i o n s and t h e formalism of Chapter 3 ¶ S e c t i o n s 3 . 2 . 1 and 5.

4.1.

Approximation o f Jo

We approximate

d e f i n e d by: (4.1)

JOh(vh)

=

1 2 k= 1

In

5,

I 61, q h v h I2 dx -

f q h Oh

dX

9

t h e e x p l i c i t form of ( 4 . 1 ) b e i n g g i v e n by:

[(

- M cI I E ~ ~ h2

=

JOh(uh)

+

(4.2)

~ i + l j- v i j ) 2

( O i l + Ih-

1

vij)2

fil"lj;

+

(

vpq

+

(ui-ijh-

Vij)'

+

h vij- Ih-

=

M U E nh

0

- h2

if

Mpq#ah,

with: (4.3)

L j

'I

=p

f (x) dx

V M i j

Enh

.

iJjeCM,],

The s e t s

nh,zhwere

d e f i n e d i n Chapter 3 ¶ S e c t i o n 3.2.1.

Remark 4.1. We c o u l d a l s o use t h e e x t e r i o r approximation o f J o g i v e n i n Chapter 3, S e c t i o n 5.1.2, e q u a t i o n ( 5 . 4 ) , b u t s i n c e t h i s approximation h a s n o t been u s e d i n any n u m e r i c a l c a l c u l a t i o n we r e s t r i c t our a t t e n t i o n i n t h e f o l l o w i n g t o ( 4 . 2 ) . 4.2.

j;

E x t e r i o r a p p r o x i m a t i o n s o f jl

W e s h a l l g i v e below f o u r p o s s i b l e e x t e r i o r a p p r o x i m a t i o n s f o r t h e most n a t u r a l approximation i s c l e a r l y :

(4.4)

ji(uh) =

%/I 61 q h vh 1' + I a2 q h uh l2 dx

1

which can be e x p r e s s e d i n t e r m s of t h e vlj by:

NwnericaZ a n a l y s i s of Bingham f l u i d f l o w

358

+( +( +(

(4.5)

I

Upp

(CHAP. 5 )

u i - l j - uij h ui- - uij h ui+lj

=0

- uij

h

if

Mpq#Qh.

It r e s u l t s from ( 4 . 5 ) t h a t ji(u3 i s approximately t h e sum of 4card (62,)nondifferentiable terms; t o reduce t h i s q u a n t i t y t o about card (ah), we can use t h e following approximations:

f

j:(vh)=

(4.6)

+

2

( 4 . 6 ) , ( 4 . 7 ) and ( 4 . 8 ) can r e a d i l y be expressed i n terms of t h e ui,, giving:

(4.9)

(SEC.

4)

Exterior approximation bf ( P o )

I

u,=o

Remark 4.2.

I

359

if Mpq$62h. Approximation ( 4 . 1 0 ) can be r e p l a c e d by

with: i f M i j $ Q hw i t h two n e i g h b u r s uij = 1 i f M,E Oh, uij = $12 i n ah, uij = Jz/2 i f M i j $Qh w i t h a s i n g l e neighbour i n ah

.

I n fact t h i s corresponds t o t h e s i t u a t i o n portrayed i n Chapter 3 , S e c t i o n 5.2 by F i g u r e s 5 . 3 , 5 . 4 . 4.3.

Formulation o f t h e approximate problem

4.4.

S o l v a b i l i t y o f t h e approximate problem

We can a p p l y t h e r e s u l t s o f Chapter 1, S e c t i o n 2 . 1 , by n o t i n g t h a t t h e mapping

d e f i n e s a norm on V h ; w e t h e r e f o r e h a v e - i n view o f t h e c o n t i n u i t y and c o n v e x i t y o f j,,! (I = 1, 2, 3, 4) - t h e e x i s t e n c e and uniqueiless of a s o l u t i o n o f (PO,,),, I = 1,2,3;4.

5.

CONVERGENCE OF THE INTERIOR AND EXTERIOR APPROXIMATIONS

Synopsis. I n t h i s s e c t i o n w e s h a l l p r o v e t h e convergence o f t h e approximation c o n s i d e r e d i n S e c t i o n s 3 and 4 ; we s h a l l make u s e o f t h e methods o f Chapter 1, S e c t i o n 4 f o r t h e i n t e r i o r approximations and Chapter 1, S e c t i o n 5 f o r t h e e x t e r i o r a p p r o x i m a t i o n s . The convergence p r o o f s f o r t h e above a p p r o x i m a t i o n s a r e based

Nwnerical analysis of Bingham f l u i d flow

360

e s s e n t i a l l y on t h e d e n s i t y of 9(62)i n HA(i2).

(CHAP. 5 )

W

Conveqence of t h e f i n i t e element method of S e c t i o n 3.

5.1.

We s h a l l i n v e s t i g a t e t h e convergence of t h e f i r s t - o r d e r f i n i t e element approximation of Section 3 , u s i n g t h e r e s u l t s o f Chapter 1, S e c t i o n s 4.3 and 4 . 4 ; t h e n o t a t i o n i s t h a t of S e c t i o n 3 and Chapter 3, S e c t i o n 6.2.1. We have :

Theorem 5.1. We assume t h a t i f 0 denotes an angle o f T, T c F k , there e x i s t s e0 > 0 such that 0 2 Oo V T € Y h ,Vh. We a l s o assume t h a t ?.f - f.?h + 0 ( i n the sense t h a t , f o r a l l compact s e t s E of 62 we have E c f.?h f o r h s u f f i c i e n t l y small). Let u, ( r e s p . u ) be the solution o f (Pod ( r e s p . (Po) ). Then uh + u strongly i n H&?)

Pro0f. 1) We put (5. l)

uh)

Vh

= O i n relation

+ d(%)d

( 3 . 2 ) of S e c t i o n 3.3, g i v i n g :

fuh d-X

from which w e deduce, s i n c e j(uJ 2 0

where, i n (5.2), C,,= l/&, l o b e i n g t h e smallest eigenvalue of H;(I)). 2.) It r e s u l t s from t h e proof of Theorem 6.1, Chapter 3, Section 6.2.1, and from t h e d e n s i t y of g(f.?) i n H;(f.?), t h a t V v c H i ( @ w e can c o n s t r u c t a sequence r,v such t h a t

- Ain

(5.3)

{

rhV€

vh

r,v + v

Vh, s t r o n g l y i n H;(n).

If u i s t h e s o l u t i o n of

1.1) :

( P o ) , w e t h e n have ( s e e (1.61, Section

(SEC. 5 )

Convergence of approximations

and hence (uhlh i s s t r o n g l y c o n v e r g e n t i n H&2) r,,u + u s t r o n g l y i n Hi(62) .

Remark 5.1.

361

s i n c e , from (5.31,

A f a i r l y g e n e r a l d e f i n i t i o n o f t h e convergence

of Qh t o 62 w a s g i v e n i n Chapter 3 , S e c t i o n 6.2.1; u n d e r more

r e s t r i c t i v e c o n d i t i o n s - which a r e a l m o s t always s a t i s f i e d i n p r a c t i c a l a p p l i c a t i o n s - and f o r which we r e f e r f o r example t o C i a r l e t - R a v i a r t /1/, we can c o n s t r u c t rh : H ~ ( S Z+) vh such t h a t ( )

i

Vu E HG(Q) n H2(62) w e have

11 rh 1 rh

(5.8)

-v -

llH&n)

< c,(v)h'" .

c2(u)

IIL'(f2)

,

I n p a r t i c u l a r ( 5 . 8 ) i s s a t i s f i e d i f we d e f i n e

{

(5.9)

r h u = p r o j e c t i o n of u on f o r t h e norm u

rh by:

vh

--*

f o r which c a s e it i s c l e a r t h a t

(5.10)

a@,, rh u

- u) = 0

V u E H,'(62), Vuh E Vh .

I n view o f ( 5 . 6 ) , ( 5 . 8 ) and ( 5 . 1 0 ) w e t h e n have

1I uh - u llj2,;(") d agh'"

(5.11) 5.2.

+ fib,

a and

p

independent o f g and h.

Convergence o f t h e e x t e r i o r a p p r o x i m a t i o n s

I n t h i s s e c t i o n we s h a l l i n v e s t i g a t e t h e convergence o f t h e e x t e r i o r a p p r o x i m a t i o n s d e f i n e d 'in S e c t i o n 4 , u s i n g t h e r e s u l t s of Chapter 1, S e c t i o n 5; t h e n o t a t i o n i s t h a t of S e c t i o n . 4 .

5.2.1.

Reduction o f (P,,J, t o a v a r i a t i o n a l i n e q u a l i t y

The approximate problem (Poh),,I = 1,2, 3,4 i s e q u i v a l e n t t o t h e variational inequality:

( )

We r e c a l l t h a t

h = max Area(T). T E J h

Nwnerical analysis of Bingham f l u i d flow

362

(CHAP. 5 )

with

With t h e function f we a s s o c i a t e L : H;(Q) + W defined by:

L(u)=Injudx.

(5.14) 5.2.2.

Exterior approximations f o r H;(O), a, j , L

1) Exterior approximation f o r

H,’(Q), a, L

Since t h e b i l i n e a r form ah i s i d e n t i c a l t o t h a t denoted by a: i n Chapter 3, S e c t i o n 6.3.1, w e r e f e r t o Chapter 3 , S e c t i o n 6.3.2 where it was proved t h a t vh, ah, uh

+

In

f q h uh

dx

c o n s t i t u t e e x t e r i o r approximations of H;(Q), a, L, i n t h e sense of Chapter 1, S e c t i o n 5 . 2)

respectively,

Exterior approximations o f j

We ‘have:

The functionals j i I Proposition 5.1. e x t e r i o r approximations o f j . Proof.

= 1,

2, 3,4, c o n s t i t u t e

We have t o show ( s e e Chapter 1, Section 5.2) t h a t

V e r i f i c a t i o n of (5.15), (5.16) i s immediate f o r j i ; moreover, by using t h e n o t a t i o n of Chapter 3, S e c t i o n 6.3.2 w e have:

(SEC. 5 )

Convergence of approximations

363

(5.17)

(5.18)

(5.19)

+I

7 1 ( h / Z ) 6 2 q h Oh

+ ?I(-h/2)

6 2 q h Oh

2

1’

dx,

It i s t h e n a l m o s t o b v i o u s t h a t p r o p e r t i e s (5.151, ( 5 . 1 6 ) r e s u l t from t h e t r a n s l a t i o n p r o p e r t i e s d e s c r i b e d i n Chapter 3 , Section 6.3.2 i n t h e p r o o f of P r o p o s i t i o n 6 . 1 ( s e e ( 6 . 7 2 ) , (6.73)).

A strong convergence r e s u l t

5.2.3.

I n view o f S e c t i o n s 5 . 2 . 1 and 5.2.2 we can a p p l y Theorem 5.2 of Chapter 1, S e c t i o n 5.4 t o t h e problems (POk),, I = 1,2,3,4, and more p r e c i s e l y t o t h e e q u i v a l e n t f o r m u l a t i o n s ( 5 . 1 2 ) . Hence:

Let

Theorem 5.2.

( P o ) ; when h + O

of

(5.20)

I

(Jh

JOh(&

ul, a l ( J h +

u: be the soZution of (POh),, we have:

u:. 6 2 q h

JO(u)

uL }

+

{

u,-

-

u

t h e solution

strangZy i n (L2(Q))’

9

jk4> -,i(u) Vf = 1 , 2 , 3 , 4 .

6.

IGTHODS

OF SOLUTION BY REGULARISING

j

Synopsis. I n t h i s s e c t i o n we s h a l l make u s e o f t h e r e g u h - i s a t i o n t e c h n i q u e s i n t r o d u c e d i n Chapter 1, S e c t i o n 3.3. I n S e c t i o n 6.1 w e s h a l l c o n s i d e r t h e r e g u l a r i s a t i o n of t h e c o n t i n u o u s problem (Po) and i n v e s t i g a t e t h e a p p r o x i m a t i o n e r r o r t h u s c r e a t e d as a f u n c t i o n o f t h e r e g u l a r i s a t i o n parameter E. I n S e c t i o n s 6.2 and 6.3 r e s p e c t i v e l y , we s h a l l i n v e s t i g a t e t h e r e g u l a r i s a t i o n

364

(CHAP. 5 )

Numerical analysis of Bingham f l u i d flow

of t h e approximate problems defined i n Sections 3 and 4 , and t h e numerical s o l u t i o n methods f o r t h e r e g u l a r i s e d approximate problems; t h e s e techniques w i l l be a p p l i e d t o s e v e r a l examples i n Section 6.4. Regularisation of t h e continuous problem (PO)

6.1.

6.1.1

Formulation of the regularised problem

There a r e various ways of r e g u l a r i s i n g t h e n o n d i f f e r e n t i a b l e term

I,

I grad u 1 dx, but we s h a l l r e s t r i c t our a t t e n t i o n t o t h e two

approximations defined below. We consider t h e two numerical f u n c t i o n s 2 0 , and of c l a s s C’ on R, defined by:

4’.(7)

(6.1)

=

d K z-

41Kand

4zc, convex,

E ,

With t h e s e we a s s o c i a t e (pf) , approximations which are obtained by r e g u l a r i s i n g (Po) and defined by:

6.1.2

Solvability of the regularised problems. Convergence and estimation of the regularisation error. j l s ( u ) = J ’ a Q k ( l g r a d u l ) d x , I = 1,2;

We w r i t e

we then have:

Proposition 6.1. The approximate problems (Pn,I = 1, 2, admit 4 , characterised b y : one and only one solution, (6.4)

{ U- ’A.4= +o mgjA(u’3r ,

(with j ;

(6.5)

=f m P ,

=

gradient o f j . ) and by :

44, u-u’3+gjk(U)-~h,(U’.i,) 4 E H,(P) .

2 J’,f(U-d!

dX

+

Vv E HA(Q) 9

Proof. Since t h e f u n c t i o n a l s u + Jo(u) &(u) convex, continuous and d i f f e r e n t i a b l e with lim

[J&)

11 I-+-

+ gj.(u)]

=

+ 03,

are s t r i c t l y

(SEC.6 )

36 5

S o l u t i o n by r e g u l a r i s a t i o n of j

t h i s r e s u l t s from Chapter 1, S e c t i o n s 2 and 3 . 3 , and from t h e g e n e r a l t h e o r y o f monotone o p e r a t o r s ( s e e , e.g. Lions /1/ I . The r e g u l a r i s a t i o n p r o c e s s d e f i n e d above i s j u s t i f i e d by:

6.1.

Theorem we

have, f o r

If

(I 4 - u

(6.6)

Proof. (6.7)

4 is

4)

I( 4

-u

II

{ c1

= J2

In

.fu&

+

.

dx,

- U) d

(i(4 - ilc(4>>l .

jcl,

which p r o v e s

g measureja)

C2 = J2 g(1

Remark 6.1.

6.1.3.

E

d 2 gyI measure ( 0 )E ,

with y1 = 1, y2 = (1 - 2/7t),

form u +

- silC(4) 2 fnf(u -

- j(u))

whence, by d e f i n i t i o n o f

(6.12)

t h a t of (Po),

( 1 . 6 ) and ( 6 . 5 ) ) :

- u I(ib(n)= U(U~- U, 4

d g[(i&)

(6.11)

u

4 - u) + sA4) - si(4 2 Inf(d- 4 d.x

&f. - + si&) (6.8) so t h a t , by a d d i t i o n , (6.9)

; C, independent of

6 C,

We have ( s e e

a(u,

t h e s o l u t i o n of (Pi)and

1 = 1,2 :

- 2/n)

(6.6) with

,

measure ( 0 ) .

The above p r o o f h o l d s i f t h e c o n t i n u o u s l i n e a r

i s r e p l a c e d by

LE H - ' ( P ) .

Estimates of t h e r e g u l a r i s a t i o n e r r o r i n the onedimensional case.

It i s r e a s o n a b l e t o suppose t h a t t h e e s t i m a t e ( 6 . 6 ) o f t h e I n t h i s s e c t i o n we s h a l l r e g u l a r i s a t i o n e r r o r i s not optimal. t h e r e f o r e g i v e an e s t i m a t e which i s more p r e c i s e t h a n ( 6 . 6 ) f o r t h e p a r t i c u l a r one-dimensional c a s e c o n s i d e r e d e a r l i e r i n S e c t i o n 2.3.1; we s h a l l r e s t r i c t our a t t e n t i o n t o r e g u l a r i s a t i o n by t h e f u n c t i o n 41a.

Numerical analysis of Bingham f l u i d flow

366

(CHAP.

5)

The problem (Po) i s d e f i n e d by:

(6.13)

(Po)v Emill Hb(0. I ) [ ~ f ~ i i , ' d x + g l 0' l i l d x - c f ~ " d x ] ,

and t h e r e g u l a r i s e d problem (6.14)

(P,) min v E H&O. I )

[i 1;I

u'('dx

(Pb by

+ 81:

, / m d x

-c

u&].

Problems (Po) and (Pb admit unique s o l u t i o n s , denoted i n t h e following by uo and u, r e s p e c t i v e l y . We s h a l l now prove: Theorem 6.2.

If

c>2g>O

(6.15)

II uo - u, IIp(0.l) d

(6.16)

11 uo - u,

llHA(O,])

we have:

C1E, Cl independent of

E ,

CZ independent of

d Cz E =/,E,

8 .

Proof. This i s an e x e r c i s e i n c a l c u l u s ; by v i r t u e of symmetry w e may r e s t r i c t our a t t e n t i o n t o t h e i n t e r v a l . 0 d x d f We p u t po = uh and p, = 4 ; on t h e one hand w e have ( s e e S e c t i o n 2.3.1).

and on t h e o t h e r hand

poand p, are shown g r a p h i c a l l y i n F i g u r e

1

Fig. 6.1.

6.1.

(SEC. 6 )

367

S o l u t i o n by r e g u l a r i s a t i o n of j

Proof of (6.15). Twice t h e area o f t h e h a t c h e d r e g i o n i n Figure 6 . 1 i s e q u a l t o \Iu:. Hence, i n view o f (6.171,

(6.18):

(6.19)

I1 u: - 4 IlL1(0.1)

Q

'' -

10'"

1 - J 7 ) Pd p = 3 c . E2 p

(

From ( 6.19) and from u,(x) - uo(x) =

which p r o v e s

["(ul(<) - ub(<))d(

Jo

(6.15)and r e n d e r s it p r e c i s e .

Proof of (6.16).

C

+

Since t h e solution

w e deduce

1

u, o f ( 6 . 1 4 ) i s a

solution of:

.it i s t h u s C " ; w e deduce from e q u a t i o n ( 6. 1 8) t h a t

and

so t h a t

The f a m i l y ( p 3 , i s t h u s equi-Lipschitzian; i n view o f ( 6 . 2 2 ) we can a p p l y A s c o l i ' s theorem t o (pz)e Hence:

.

(6.25)

pc + p o u n i f o r m l y ,

In t h e following we s h a l l w r i t e

pc = p,(i - g/c);

w e t h u s have

368

Numerical a n a l y s i s of Bingham fluid flow

(CHAP. 5 )

and hence BE2

(6.28)

pc = (pc

When

+ Jm) J E T

.

pc + 0 and p c / , / p p

E + 0,

+

1; w e t h u s have

and hence

We have

we put

Jo

Estimation o f On

(6.30)

II

[i- g/c, $1 we II =

1”’ 1/2

- e/c

have po(x) = 0

p f ( x )dx

and hence

.

We put

(6.31)

pc = c s h 4 ,

which g i v e s

(6.32)

E

sh 4

+

th 4 = c($ - x ) ,

and

An elementary c a l c u l a t i o n shows t h a t

(6.34)

I , = -E2- ( ~ ~ h ’ 4 , + q 4 ~ - g t h q 5 . )

.

We deduce from ( 6 . 2 9 ) , ( 6 . 3 3 ) , (6.34) t h a t

e/pc + 0

(SEC. 6 )

Solution by regularisation of j

Estimation of

I,.

On [0, i - g/c] we have p(x) and hence

(6.36)

369

+ g th 4 = c() - x )

and p o ( x )

+ g = c(i - x),

p(x) - po(x) = g(l - th 4) = g(e-*/ch 4) ,

and, i n view of ( 6 . 3 2 )

P u t t i n g t = e*, and p e r f o r m i n g t h e n e c e s s a r y m a n i p u l a t i o n s we obtain :

I

+ a e-2*

(6.38)

*c

ch 4 =

(e ch 4

">

+ ch2 4

d4 =

2 c(e-*c - Arctg e-*.)

89 1 +3 (eZ*c + 1)" :

and u s i n g t h e f a c t t h a t e*= 'Y *,-'I3

which g i v e s

(6.40)

I2 d zgc 2 [ 1

+ a(&)]

lha(&) = 0 . 8

with

r-0

From ( 6 . 3 5 ) and ( 6 . 4 0 ) w e immediately deduce t h a t

(6.41)

11 uz - uO

IlH&O.l) %

J2g/3c

which p r o v e s Theorem 6.2.

Remark 6 . 2 .

&

Jx& 1

8

Consider t h e problem

r e g u l a r i s e d by

If P i s a d i s c o f r a d i u s R, it can be proved by p r o c e e d i n g as in t h e p r o o f o f Theorem 6 . 2 t h a t

(CHAP. 5 )

Numerical analysis of Binghwn f l u i d flow

370

(6.42)

11 u, - uo [ILao(")

(6.43)

11 u, - uo I H&",

< CI&,C , independent < C2E d-,

of

E

,

C2independent o f

E

,

(Po) ( r e s p . ( P e ) ) .

rn

where uo ( r e s p .

u, ) i s t h e s o l u t i o n o f

Remark 6.3. with f E H - ' ( P ) :

S i n c e t h e domain 0 i s a r b i t r a r y i n @ we c o n s i d e r ,

If uo and u, are t h e r e s p e c t i v e s o l u t i o n s of ( 6 . 4 4 ) and ( 6 . 4 5 ) , we c o n j e c t u r e on t h e b a s i s o f p a r t i c u l a r e x p l i c i t c a s e s t h a t :

(6.46)

1I u,

6.1.4.

- u0 IIH;(0)

< C E " ~C, independent

of

E

. rn

Application of the regularisation process fu obtaining duality results.

We s h a l l u s e t h e above r e g u l a r i s a t i o n p r o c e s s t o g i v e an elementary d e m o n s t r a t i o n o f t h e ( " d u a l i t y " t y p e ) c h a r a c t e r i s a t i o n (3.46) - ( 3 . 5 1 ) of Chapter 1, S e c t i o n 3.4; we assume i n t h e following t h a t ftz H - ' ( 0 ) . We f o r m u l a t e ( P d i n t e r m s of t h e e q u i v a l e n t v a r i a t i o n a l inequality

and t h e c o r r e s p o n d i n g regularised problem by

(6.48)

I

+ I grad v I' dx -

,/e2

2 (f, u - u, ) Vv E H;(G?),

By p u t t i n g that

(6.49)

v = 0 and v = 2 u

i n (6.47) it can r e a d i l y be shown

u); a(u, U) + d(u) = (f,

moreover t h e s o l u t i o n u, o f (6.48) i s c h a r a c t e r i s e d by

.

grad u, grad u

(6.50)

JE'

+ I grad u,

dx= (2

( X U ) VVEH;(Q),

(SEC. 6 )

Solution by regularisation of j

L e t A = { P I P = ( P ~PJ , E L2(Q) x LZ(Q),d ( x ) w e d e f i n e A,EA by

(6.51)

1, =

+ P%X)

d 1 a.e} ;

grad u,

+ I grad u, 1'

'

There t h e n e x i s t s ~ E and A an e x t r a c t e d subsequence, a g a i n denoted by (A& , such t h a t

(6.52)

1, + 1 weakly" i n Lm(Q) x Lm(Q).

From (6.48) w e have

(6.53)

a(u,,u) +gInl,.gradudx = ( L o ) V U E H & ~ ) ,

(6.54)

due, uK) + g

In

',.grad

' K

dx

=

(A

' K

)9

and because of t h e strong convergence o f u, t o u ( s e e Theorem 6.1) w e have, i n t h e l i m i t i n ( 6 . 5 3 1 , ( 6 . 5 4 ) :

(Lu),

(6.55)

o(u,u)+gInA.gradudx=

(6.56)

o(u,u)+gInA.gradudx= ( L u ) .

From ( 6 . 4 9 ) , ( 6 . 5 6 ) , w e deduce t h a t :

(6.57)

InA.gradudx=

I,

(graduldx.

We a l s o have t h e e q u i v a l e n c e

1.grad u d x = I I grad u I dx] (6.58)

{

A.grad u= Igrad u 1 AEA

which, w i t h ( 6 - . 5 5 ) , g i v e s

- Au - g d i v 1 = f , UIr = 0 ,

(6'60)

1.grad u = I grad u I , {LEA,

i. e. t h e c h a r a c t e r i s a t i o n r e q u i r e d .

6.2. R e g u l a r i s a t i o n o f t h e approximate problems 6.2.1.

Case of the interior approximation of Section 3

The n o t a t i o n is t h a t o f S e c t i o n s 3 and 6.1.1,

6.1.2;

the

(CHAP. 5)

Numerical a n a l y s i s o f Bingham f l u i d f l o w

372

r e g u l a r i s e d problem c o r r e s p o n d i n g t o t h e approximate problem ( 3 . 1 ) o f S e c t i o n 3.2 i s d e f i n e d by

1

a(d!, vh - d h ) + ddUh) -adUL)2

(6.61) with

E vh

1= 1,2.

I,f

( h- &)

voh E

vh

9

9

I t i s simple t o prove:

Theorem 6.3. Problem (6.61) admits one and o n l y one s o l u t i o n , E + 0 we have, V l = 1 , 2

and when (6.62)

d!

+

u,,

where uh is t h e s o l u t i o n o f t h e approximate problem (Pd ( s e e (3.11, S e c t i o n 3 . 2 ) . Remark 6.4.

The f o l l o w i n g convergence r e s u l t s can be r e a d i l y

proved :

(6.63) (6.64)

{

lim UL = 4 strongly in H @ ) , e:re

u’, i s t h e s o l u t i o n o f (Pc’) ( s e e (6.3) S e c t i o n 6 . 1 . 1 ) ,

lim d! = u strongly in H&2), u b e i n g t h e s o l u t i o n o f (Po).

r.h-0

I n f a c t , under t h e c o n d i t i o n s f o r which t h e estimate ( 5 . 1 1 ) was o b t a i n e d ( s e e S e c t i o n 5 . 1 , Remark 5 . 1 ) it can be shown t h a t i f u i s t h e s o l u t i o n o f (PO),

(6.65)

{ I1 p,- G

u Ili:(n, Q ah + g(Bh”z + 1’1 4 , a, yl independent o f h and E ; a, j3 independent o f 1 .

From t h e p o i n t o f view o f t h e nwnerical soh.4tion o f (6.611, we s h a l l i n f a c t u s e t h e e q u i v a l e n t f o r m u l a t i o n , g i v e n i n v a r i a t i o n a l form by

t h e c a s e 1 = 2 b e i n g a l i t t l e more c o m p l i c a t e d t o w r i t e explicitly. There i s i n p r i n c i p l e no d i f f i c u l t y i n e x p r e s s i n g ( 6 . 6 6 ) , u s i n g t h e f o r m u l a s o f Chapter 3, S e c t i o n 4.1.4 i n t h e form o f a system o f N( l ) n o n l i n e a r e q u a t i o n s whose unknowns a r e t h e v a l u e s ( ) N = dim v h = card (&).

(SEC. 6 )

Solution by regularisation of j

of dd a t t h e nodes ME&, of t h e t r i a n g u l a t i o n

373

g h .

Case of the e x t e r i o r approximation of Section 4

6.2.2.

The n o t a t i o n i s t h a t o f S e c t i o n s 4 and 6.1 .1, 6 . 1 . 2 ; the approximations a r e d e f i n e d by r e g u l a r i s i n g t h e f u n c t i o n a l s A k = 1, 2, 3, 4, o f S e c t i o n 4 . 2 u s i n g , r e s p e c t i v e l y

(6.68)

\.

\

(6.69)

(6.70)

(6.71)

with urn = 0 i f M r n $ Q h ; i n (6.70) we t a k e ui, = 1 i f 3 i s given by (4.10), or t h e v a l u e s g i v e n i n ( 4 . 1 0 ’ ) i f j i , g i v e n by t h i s l a t t e r equation, i s r e g u l a r i s e d . The r e g u l a r i s e d approximate problems o f ( 4 . 1 2 ) c o r r e s p o n d i n g t o (6.68) ( 6 . 7 1 ) a r e g i v e n by

...

It i s s i m p l e t o p r o v e :

Numerical a n a l y s i s of Bingham f l u i d flow

374

(CHAP.

5)

Theorem 6.4.

&,

Problem ( 6 . 7 2 ) a h i t s one and o n l y one s o l u t i o n , and when E - 0 we have,Vl= 1,2,

4-4,

(6.73)

where 4 i s the s o l u t i o n of the approximate problem (Po& defined by ( 4 . 1 2 ) i n Section 4.3. The problems ( 6 . 7 2 ) a r e e q u i v a l e n t t o t h e system of N ( ' ) nonl i n e a r e q u a t i o n s i n Nunknowns g i v e n by

w i t h I& = (uI,)Y,,~~~. I t i s c l e a r t h a t t h e n u m e r i c a l s o l u t i o n of ( 6 . 7 2 ) w i l l be c a r r i e d o u t on t h e e q u i v a l e n t f o r m u l a t i o n ( 6 . 7 4 ) .

Remark 6.5. For k = 1,2,4 ( r e s p . k = 3 ) system ( 6 . 7 4 ) corresponds t o a d i s c r e t i s a t i o n of t h e r e g u l a r i s e d continuous problem ( 6 . 4 ) by a 9-point ( r e s p . 13-point) f i n i t e d i f f e r e n c e scheme; s e e Chapter 3, S e c t i o n 8 . 2 . 5 , F i g u r e 8.2 ( r e s p . S e c t i o n 8 . 2 . 4 , F i g u r e 8.1) for t h e l o c a t i o n o f t h e p o i n t s i n q u e s t i o n . 6.3.

S o l u t i o n o f t h e r e m l a r i s e d approximate problem ( I ) . Method of p o i n t o v e r - r e l a x a t i o n

Description and convergence of t h e method

6.3.1.

The v a r i o u s r e g u l a r i s e d approximate problems d e f i n e d i n S e c t i o n 6 . 2 are of t h e f o l l o w i n g t y p e ( w e u s e t h e n o t a t i o n o f Chapter 2 , S e c t i o n 1):

min J(u) ,

(6.75)

V€RN

with J :

W N + W c l a s s C', s t r o n g l y convex lim J(u) = + 00 ; '

( 2 ) and s a t i s f y i n g

llVIl'+~

t h u s f o r t h e a c t u a l s o l u t i o n of ( 6 . 7 5 ) we c a n u s e t h e r e l a x a t i o n a l g o r i t h m d e s c r i b e d i n Chapter 2, S e c t i o n 1.1, whose convergence f o l l o w s as a r e s u l t o f Theorem 1.1 o f t h e same s e c t i o n ; t h a t i s :

(6.76)

uo = { uy,

...,u$ } g i v e n ;

w i t h 't known, we c a l c u l a t e u"", c o o r d i n a t e by c o o r d i n a t e , u s i n g

(SEC. 6)

SoZuti.on by r e g u l a r i s a t i o n o f j

{

(6.77)

G'', @+'. ...) i = 1 ,..., N .

J(ul+', ..., G?:, VuiER,

37 5

< J W ' , ..., G?:, vi, G+',...I

There i s a w e l l known e q u i v a l e n c e between ( 6 . 7 7 ) and

aJ

(6.78)

..., @?:,@+',@+I,

-((u;+',

80,

...) = 0 .

We can s e e k t o improve t h e speed o f convergence o f t h e a l g o r i t h m by i n t r o d u c i n g an under- or o v e r - r e l a x a t i o n p a r a m e t e r ( f o r t h e c a s e o f l i n e a r e q u a t i o n s s e e Chapter 2 , S e c t i o n 1.4), g i v i n g t h e f o l l o w i n g two p o s s i b l e v a r i a n t s of (6.761, ( 6 . 7 7 ) :

First algorithm. uo = { uy,

(6.79)

..., u i } g i v e n

with u" known, w e c a l c u l a t e

aJ aoi

c o o r d i n a t e by c o o r d i n a t e , u s i n g

..., ul+' - +' ,-1.4 ,G+l , . . . ) = 0 ,

(6.80)

-(u:+',

(6.81)

#+' = @

with

ff",

+ fl+' - @),

i = 1, 2, ..., N.

Second aZgor it hm. With ff known, we c a l c u l a t e u " + ' , c o o r d i n a t e by c o o r d i n a t e , u s i n g

..., ,@ ;: (6.82) =

@+I,#+',

aJ (1 - w)-(u;+', avi

...) =

..., q?;,q,q+', ...),

w i t h i = , l , 2, ..., N. I n t h e examples s o l v e d n u m e r i c a l l y by r e g u l a r i s a t i o n and overr e l a x a t i o n we have u s e d t h e r e g u l a r i s a t i o n o f

jn,/&'+

(gradul'dx

.

J"

Igradoldx

by

Under t h e s e c o n d i t i o n s t h e f u n c t i o n a l J o f

t h e c o r r e s p o n d i n g problem (6.75) is C" and i t s Hessian m a t r i x a t t h e p o i n t v, J"(u) i s p o s i t i v e d e f i n i t e V U E W ~ ; from S c h e c h t e r 111, 121, 131, t h e s e two p r o p e r t i e s a r e s u f f i c i e n t t o e n s u r e t h e convergence o f ( 6 . 7 9 ) , ( 6 . 8 0 ) ~(6.81) u n d e r t h e c o n d i t i o n :

(6.83)

0 <w


where i n p r a c t i c e t h e bound /I i s g e n e r a l l y d i f f i c u l t t o d e t e r m i n e explicitly. To t h e b e s t of our knowledge, a l g o r i t h m ( 6 . 7 9 ) , (6.82) does n o t a p p e a r t o have been s u b j e c t t o any t h e o r e t i c a l i n v e s t i g a t i o n , a l t h o u g h i n numerous c i r c u m s t a n c e s it h a s proved

(CHAP. 5 )

Numerical analysis of Bingham fluid f l o w

376

more "robust" t h a n

( 6 . 7 9 ) , ( 6 . 8 0 ) , (6.81).

8

Remark 6.6. I n t h e examples c o n s i d e r e d ( s e e S e c t i o n 6.3.2 below) t h e n o n l i n e a r e q u a t i o n s (6.80) and ( 6 . 8 2 ) were s o l v e d by Newton's method; t h a t i s , i n t h e c a s e o f t h e c a l c u l a t i o n o f %+' from ( 6 . 8 0 ) : (6.84)

g".'

=

4, aJk

-(uy+',

(6.85)

Si;+'."'+'

= $+'*"'

-

..., U ; : ; , q + 1 * m ,

a2Jk

-(4+',..., U;_+;,i$+l+l.m, av:

U;+', ...) U;+', ...)

S i m i l a r r e l a t i o n s would be o b t a i n e d f o r t h e c a l c u l a t i o n o f from ( 6 . 8 2 ) . On t h e s u b j e c t of Newton's method, it should be p o i n t e d out t h a t i n S c h e c h t e r , ~ O C . c i t . , it i s shown i n connection w i t h a l g o r i t h m ( 6 . 7 9 ) , (6.801, (6.81) t h a t under t h e assumptions J E Cm(p), J"(v) p o s i t i v e d e f i n i t e Vve p,o " s u i t a b l y chosen", o n l y a s i n g l e i t e r a t i o n o f Newton's method need be c a r r i e d out i n t h e c a l c u l a t i o n o f @+', and t h u s , u s i n g t h e n o t a t i o n o f ( 6 . 8 4 ) , ( 6 . 8 5 ) , w e can d e f i n e 6" by

4 ' '

(6.86)

U;" = U;

+ CL@'+'*'- 0 ,

and s t i l l have convergence o f t h e algorithm.

8

Remark 6.7. I f , i n t h e a l g o r i t h m s ( 6 . 7 9 ) , (6.801, (6.81) and ( 6 . 7 9 ) , (6.821, only one Newton i t e r a t i o n i s c a r r i e d o u t i n t h e c a l c u l a t i o n o f *+',then cide. 8 6.3.2.

t h e new a l g o r i t h m s t h u s o b t a i n e d coin- .

Applications.

For B w e t a k e t h e d i s c o f r a d i u s 4 and f = c o n s t = C ; t h e s o l u t i o n o f (Po) i s t h e n given by e q u a t i o n s ( 2 . 1 9 ) ,. (2.21) i n S e c t i o n 2.3.2. The approximation used i s t h e i n t e r i o r approximation, u s i n g f i r s t - o r d e r f i n i t e elements, of S e c t i o n 3 ; F i g u r e 6.2 shows t h e corresponding t r i a n g u l a t i o n f k: t h e r e are 1024 t r i a n g l e s , 545 nodes and 481 i n t e r i o r nodes, t h u s g i v i n g an approximate problem w i t h 481 unknowns.

..

(see. 6)

S o l u t i o n by regularisation of j

377

Fig. 6.2.

a(&,,

0,

- )24:

-g (6.87)

<

In

JE’

2 c

(oh

+g

JE’

-t I grad

+ I grad u; - .A) dx

1’ dx

-

1’ dx 2

vvh

E

vh,

The r e s u l t s o f S e c t i o n 6 . 3 . 1 c a n be a p p l i e d t o problem (6.88) which, i n p a r t i c u l a r , comes w i t h i n t h e scope o f Remark 6.6; we can t h u s s o l v e (6.88) by n o n l i n e a r o v e r - r e l a x a t i o n , c a r r y i n g o u t ME,^^). o n l y a s i n g l e Newton i t e r a t i o n t o d e t e r m i n e ( ’ ) The method d e s c r i b e d above w a s a p p l i e d u s i n g t h e f o l l o w i n g parameter values

(’)

The suffix

E

has b e e n omitted.

378

Numerical anaZgsis of Bingham f l u i d f l o t ,

(6.89)

C = 10,

g = 1

and

10,

c =

(CHAP. 5 )

lo-*.

We i n t r o d u c e t h e " r e s i d u a l " R(n) d e f i n e d by:

1-l f l M + ' - f l l (6.90)

YErh

R(n) =

With t h e n o n l i n e a r o v e r - r e l a x a t i o n a l g o r i t h m - u s i n g one Newton i t e r a t i o n - i n i t i a l i s e d by

(6.91)

U: =

0,

t h e f o l l o w i n g phenomena are observed:

-

Each i t e r a t i o n t a k e s about 2 s e c on a C I I 10070. For g = 1 , it appears t h a t t h e optimum v a l u e o f w i s o f t h e o r d e r of 1 . 4 and t h a t f o r t h i s v a l u e o f w t h e q u a n t i t y R(n) i s of t h e o r d e r of a f t e r 500 i t e r a t i o n s ; t h i s t h e r e f o r e l e a d s t o a computation t i m e which i n our opinion i s t o o l o n g , p a r t i c u l a r l y when compared w i t h t h o s e o b t a i n e d w i t h t h e methods desc r i b e d i n subsequent s e c t i o n s ; t h e approximation error r e l a t i v e t o t h e e x a c t s o l u t i o n i s a c c e p t a b l e s i n c e , f o r n = 500, we have

where, i n ( 6 . 9 2 ) , u denotes t h e s o l u t i o n o f (Po). - For g = 10, t h e computation t i m e s r e q u i r e d t o g i v e an a c c e p t a b l e approximation t o t h e e x a c t s o l u t i o n a r e f a r t o o l o n g f o r t h e method used t o be regarded as a p r o d u c t i o n t o o l .

6.4.

S o l u t i o n of t h e r e a u l a r i s e d approximate problems (11). Gradient method w i t h a u x i l i a r y o p e r a t o r .

6.4.1.

Description and convergence o f t h e method

I t w a s shown i n Chapter 2, S e c t i o n 2 . 1 - r e f e r r i n g t o C6a / 2 / f o r t h e p r o o f s - t h a t problems o f t h e t y p e (6.75) can be s o l v e d by means of t h e a l g o r i t h m : (6.93)

uo

(6.94)

u"+' = u"

(6.95)

pm> 0 s u i t a b l y chosen.

given,

- P.

J'(u") 9

A u s e f u l v a r i a n t o f t h i s a l g o r i t h m i s as f o l l o w s : s i n c e S E ~ ( P7;)i s symmetric and p o s i t i v e d e f i n i t e , t h e a l g o r i t h m (6.931, ( 6 . 9 4 ) , (6.95) can be g e n e r a l i s e d by determining P+'

(SEC. 6)

S o l u t i o n by r e g u l a r i s a t i o n o f j

379

using

(6.96)

U"+' = U"

- p,, S-1 ~ ' ( g ) ,

t h e c a s e ( 6 . 9 3 ) , ( 6 . 9 4 ) , (6.95) t h e n c o r r e s p o n d s t o S = I ; w i t h a s u i t a b l e c h o i c e o f s ( o n t h i s p o i n t , s e e S e c t i o n 6.4.2 below) we can c o n s i d e r a b l y a c c e l e r a t e t h e convergence compared w i t h t h e c a s e S = I, a t l e a s t when J'(u) i s t h e d i s c r e t i s e d form o f an unbounded o p e r a t o r which may be l i n e a r o r n o n l i n e a r ; such i s t h e c a s e f o r t h e examples c o n s i d e r e d i n t h i s book. Regarding t h e convergence o f a l g o r i t h m ( 6 . 9 3 ) , ( 6 . 9 6 ) , ( 6 . 9 5 ) , we have : Theorem 6.5.

Under t h e asswnptions

- J'(v,), v2 - vl) 2 a

It 11' - u1 [ I 2 ,

(6.97)

(J'(02)

(6.98)

1) J ' ( o z ) - J'(oJ I( d B (1 v2 - u1 (1 , B

a

>0,

independent of

v1 ,

U*EW,

and under t h e c o n d i t i o n (6.99)

0 < rl d pn d r2 < 2 aa/B2,

where Q i s t h e s m a l l e s t eigenvalue of S, we have, V u o ~ R N , convergence t o t h e s o l u t i o n u of ( 6 . 7 5 ) ( l ) o f the sequence U" defined b y ( 6 . 9 3 ) , ( 6 . 9 6 ) . Proof. T h i s i s a v a r i a n t o f t h e s t a n d a r d p r o o f o f t h e convergence o f t h e g r a d i e n t method; we p u t 111 u (I(= (Sv, u)'''. S i n c e t h e s o l u t i o n u o f (6.75) s a t i s f i e s

(6.100)

u

=

u - p.

S-I

J'(u) V ~ I ,

and, by s u b t r a c t i n g t h i s f r o m (6.96) ( w r i t i n g ii" = u"

(6.101)

i?'+

=

-u),

we have

2' - pn S-'(J'(d') - J ' ( u ) ) ,

and hence, by s c a l a r s q u a r e s ,

(6.102)

((1 Z + 'I([' = (11 ii" (I(' - 2 ~ , ( J ' ( u " ) J'(u), i") +

+ pi(S-l(J'(u") - J'(u)), J'(U") - J'(u)) ,

o r , i n view o f ( 6 . 9 7 ) , (6.98) and u s i n g t h e f a c t t h a t we have

1) S-I 11

=

l/a,

( l ) T h i s "u" h a s n o t h i n g t o do ( a t l e a s t f o r t h e moment) w i t h t h e s o l u t i o n o f (Po) ( d e f i n e d by (1.5)).

380

(6- 103)

Numerical a n a l y s i s of Bingham f l u i d f l m

III p 1112 - III

5)

(CHAP.

1112 2 pn(2 a - pn B2 / g) II En 112 ,

and hence convergence of g u n d e r t h e condition (6 .99) .

rn

Remark 6.8. Using t h e conventional n o t a t i o n p(S) = 11 SII it i s easy t o show, s t i l l using ( 6 . 9 7 ) , ( 6 . 9 8 ) , ( 6 . ~ 1 3 ) ~

giving t h e geometric convergence of condition

111 u" - u'III

t o zero under t h e

which i s more r e s t r i c t i v e than (6.99) (except when S = Z ).

rn

Remark 6.9. An i n v e s t i g a t i o n of t h e convergence of algorithms of t h e t y p e ( 6 . 9 3 ) , (6.96) can be found i n Brezis-Sibony /2/ and Sibony 111, under assumptions which a r e somewhat weaker t h a n t h o s e of Theorem 6.5, with a p p l i c a t i o n t o t h e s o l u t i o n of nonrn l i n e a r boundary-value problems. Remark 6.10. The r e a d e r may check t h a t algorithm (3.20) of Chapter 1, Section 3.1, Remark 3.2 i s an infinite-dimensional v a r i a n t (given i n v a r i a t i o n a l form) of algorithm ( 6 . 9 3 ) , (6.96). The g r a d i e n t of t h e r e g u l a r i s e d approximate f u n c t i o n a l s of Section 6.2 c l e a r l y s a t i s f i e s t h e assumptions f o r t h e a p p l i c a t i o n of Theorem 6.5, t h e L i p s c h i t z constant B being o f t h e form (6.106)

8 = 2B ( 1 + W E ) ) , &

6.4.2.

limO(E)=O.

rn

C-0

An example o f a p p l i c a t i o n

We consider t h e p a r t i c u l a r

62=]0,1[ x ]0,1[,

(Po) problems defined by

p= 1,f=

l O , g = 1 and 1.6.

Type o f d i s c r e t i s a t i o n : By f i n i t e d i f f e r e n c e s ( e x t e r i o r approximation) t h e n o n d i f f e r e n t i a b l e term being approximated by t h e v a r i a n t ( 4 . 1 0 ' ) of 4 and by 1: (see equation ( 4 . 1 1 ) ) . Type of r e g u l a r i s a t i o n : Mesh s i z e :

By t h e f u n c t i o n (6.2)).

(')

(see relations

h = 1/20.

( l ) This approximation of 7 + 1 7 1, although more complicated than t h a t using 7 + ,/-, i s widely used ( a t l e a s t , i t s f i r s t and second d e r i v a t i v e s ) f o r t h e t r a n s i e n t a n a l y s i s of e l e c t r i c a l networks, s i n c e it i s known how t o produce it physically.

S o l u t i o n by r e g u l a r i s a t i o n of 3'

(SEC. 6 )

Regularisation parameter :

E

381

=4~10-~.

Implementation o f the algorithm ( 6 . 9 3 ) , ( 6 . 9 6 ) : I n view o f t h e e x t e r i o r a p p r o x i m a t i o n s used and w i t h t h e corresponding n o t a t i o n , a l g o r i t h m ( 6 . 9 3 ) , (6.96) a p p l i e d t o t h e s o l u t i o n o f t h e r e g u l a r i s e d approximate problems t a k e s t h e form (6.107)

{ " given

U;+' = U; - pn Fh(G), where Fh i s t h e o p e r a t o r from vh*vh d e f i n e d by t h e l e f t - h a n d s i d e o f ( 6 . 7 4 ) ( w i t h 1 = 2 and k = 3, 4 ) and s h t h e d i s c r e t i s e d We i n i t i a l l y t o o k form o f a s u i t a b l y chosen e l l i p t i c o p e r a t o r . S = S, = I b u t w i t h such a c h o i c e t h e convergence proved t o be far t o o slow; we t h e r e f o r e t o o k , as s u g g e s t e d by Godunov-Prokopov /1/ i n c o n n e c t i o n w i t h l i n e a r e l l i p t i c problems,

t h e boundary c o n d i t i o n s b e i n g o f D i r i c h l e t t y p e , homogeneous w i t h respect t o each f a c t o r . Another s u i t a b l e c h o i c e i s S3 = - A f o r homogeneous D i r i c h l e t c o n d i t i o n s . For a g i v e n t e r m i n a t i o n c r i t e r i o n ( s e e (6.108) below) t h e two methods a r e l a r g e l y e q u i v a l e n t i n t e r m s o f t h e number o f i t e r a t i o n s ; however, t h e p o s s i b i l i t y , w i t h Sz , o f s o l v i n g t h e a u x i l i a r y problem at each i t e r a t i o n by d i r e c t methods o f t h e Gauss o r Choleski t y p e ( i n view of t h e tridiagonal s t r u c t u r e o f t h e d i s c r e t i s e d forms o f I - a2/ax:, I - a2/ax:) l e a d s f o r t h e examples c o n s i d e r e d t o a method about t e n times as f a s t , i n t e r m s of computation t i m e , as a method u s i n g S3 = - A w i t h s o l u t i o n o f t h e a u x i l i a r y D i r i c h l e t problems by p o i n t o v e r - r e l a x a t i o n w i t h o p t i m a l p a r a m e t e r .

Numerical r e s u l t s . With t h e termination c r i t e r i o n f o r t h e i t e r a t i o n i n t h e form: (6.108)

I&" -

R" =

I<

MIJE%

and w i t h ui g i v e n by t h e r e s t r i c t i o n o f t h e f u n c t i o n xY(1 - x ) x (1 - y ) to f&, , t h e r e s u l t s r e l a t i n g t o t h e convergence o f a l g o r i t h m (6.107) u s i n g s = Sz a r e summarised i n T a b l e 6 . 1 overleaf. The IBM 360191 machine t i m e u s e d i s o f t h e o r d e r o f 3 s e e . f o r each problem. With r e g a r d t o t h e a c c u r a c y o b t a i n e d , we can s t a t e t h a t t h e r e i s coincidence t o within a r e l a t i v e accuracy of at l e a s t compared w i t h t h e r e s u l t s o b t a i n e d f o r t h e same problems u s i n g t h e d u a l i t y methods of S e c t i o n 7 below; f o r f u r t h e r d e t a i l s we r e f e r t o Goursat 121.

Nwnerical analysis of Bingham f l u i d flow

382

(CHAP. 5 )

Nunber of i t e r a t i o n s g

1

jl:

P.

required f o r convergence

ji

134

ji

S t a r t i n g w i t h po = 1 retaining p. = 1 u n t i l R" c e a s e s t o d e c r e a s e , and t h e n taking:

i:

p. = 1/20

210

Jk*

1.6

Table

7.

164 133

6.1

DUALITY METHODS

Synopsis. I n t h i s s e c t i o n we s h a l l i n v e s t i g a t e t h e a p p l i c a t i o n o f t h e d u a l i t y results and a l g o r i t h m s of Chapter 1, S e c t i o n s 3.4, 3.5 and Chapter 2 , S e c t i o n 4. F i r s t , t h e p r i n c i p l e s o f t h e method f o r t h e continuous problem (1.5) are reviewed i n S e c t i o n 7.1, and t h e n v a r i a n t s o f t h e continuous c a s e w i l l be employed, i n S e c t i o n s 7 . 1 and 7.2 r e s p e c t i v e l y , t o s o l v e t h e approximate problems d e f i n e d i n S e c t i o n 4 ( e x t e r i o r approximations) and S e c t i o n 3 ( f i n i t e element approximations).

7.1.

A p p l i c a t i o n t o t h e s o l u t i o n o f t h e continuous problem

It results from Chapter 1, S e c t i o n 3.4 t h a t t h e s o l u t i o n u of

( 1 . 5 ) i s c h a r a c t e r i s e d by t h e e x i s t e n c e of 1 ~ A = { p l p ~ L ~x( L9 2)( 9 ) , I p ( x ) l g1a.e.J such t h a t (7.1)

1.grad u = I grad u 1 ,

- AU - g d i v 1 = f U = O

in

n,

~nr.

It f o l l o w s immediately from (7.1), ( 7 . 2 ) t h a t { u , l } i s a saddZe point on H i ( 9 ) x A o f t h e Lagrangian 9 : H i @ ) x (L2(Q))'+ R d e f i n e d by (7.3)

. i r'

-

1 ..' 9(1),p) = p ( u . u )

- V,u)+ g

1"

p.gradudx.

We can u s e (as has a l r e a d y been mentioned i n Chapter 2 , S e c t i o n

4.4, Remark 4.9) t h e d u a l i t y a l g o r i t h m ( 4 . 1 2 ) , ( 4 . 1 3 ) o f Chapter 2, S e c t i o n 4.3 w i t h A d e f i n e d as above and (7.4)

v = Hi&?),

M =

v,

(SEC. 7 )

h a l i t y methods

L

(7.5) (7.6)

=

383

L2(Q) x L 2 ( 0 ) ,

4 : V+L

d e f i n e d by

4v

= grad v

With r e g a r d t o t h e o p e r a t o r P A : L

+A

. t h i s i s d e f i n e d by

The e x p l i c i t form o f t h e a l g o r i t h m mentioned above i s t h e n given by:

'1 g i v e n ( a r b i t r a r i l y ) ;

(7.8)

with k known (7.9)

{

(7.10)

1""

( € A ) , we d e f i n e

-Au"-gdiv1"=f u " = O on r , =

PA@"

u", and t h e n 1""

by

bQ,

+ p,, grad u") .

The f o l l o w i n g p r o p o s i t i o n can b e deduced from t h e p r o o f of Theorem 4 . 1 o f Chapter 2, S e c t i o n 4.3:

7.1.

Proposition (7.11)

0 < a.

< p. < a,

Under the condition < 2/g,

f o r the sequence u" defined by ( 7 . 8 ) , ( 7 . 9 ) , (7.10)we have (7.12)

where

strongly i n H&2)

u" + u u

i s the s o l u t i o n of ( 1 . 5 ) .

7.2.

7.2.1.

A p p l i c a t i o n t o t h e s o l u t i o n o f t h e approximate problems (I). Case o f t h e e x t e r i o r a p p r o x i m a t i o n s o f S e c t i o n 4.

Duality algorithm f o r the approximate problems of Section 4 .

We s h a l l i n f a c t c o n f i n e o u r a t t e n t i o n t o t h e approximate problem ( 4 . 1 2 ) ( s e e S e c t i o n 4 . 3 ) r e l a t i n g t o ( d e f i n e d by ( 4 . 1 1 ) ) , s i n c e - as can be r e a d i l y v e r i f i e d - o f a l l t h e approximate problems ( 4 . 1 2 ) t h i s i s t h e one which p o s s e s s e s t h e s i m p l e s t d u a l problem (I). With t h e n o t a t i o n o f S e c t i o n 4 and o f Chapter 3, S e c t i o n 5.1.2, we d e f i n e t h e s p a c e Lh by ~

(')

See t h e a r t i c l e by C6a-Glowinski /1/ on t h i s s u b j e c t .

384

Numerical analysis of Bingham f l u i d flow

(CHAP. 5 )

and t h e closed convex s e t Ahof Lh by (7.14)

{C(hI~hELh,I~:+1/2j+l/212

Ah=

+

Id+l/Zj+1/2

1' 6

+

vMi+l/2j+1/2Ezh}?

and, f i n a l l y , t h e mapping G,, : Vh + L h ( t h e d i s c r e t e analogue of grad) by

uh

Under t h e s e c o n d i t i o n s , it can be shown t h a t t h e s o l u t i o n ( l ) of (Po& i s c h a r a c t e r i s e d by t h e e x i s t e n c e of 1 h . k E A h such t h a t "i+lj

I-

+ 4 - l j + " i j + l + uij-1 - 4"ij h2

=Aj

+

We can t h u s apply t h e r e s u l t s of Chapter 2 , Section 4 t o t h i s finite-dimensional s i t u a t i o n , g i v i n g t h e d i s c r e t e analogue of algorithm ( 7 . 8 ) , ( 7 . 9 ) , (7.10), t h a t i s :

(7.19)

(I)

4

given;

Henceforward t h e s u p e r s c r i p t

4

w i l l be omitted.

(SEC. 7 )

with

4

Duality methods

known

(€Ah)

, we

define

4 and t h e n

I-

(7.20)

=Aj

by

+

I+

I+

(7.21) u s i n g t h e n o t a t i o n of Chapter 3 , S e c t i o n 8.3.3, Remark 8.10. S i n c e Theorem 4 . 1 of Chapter 3, S e c t i o n 4 . 3 can a l s o be a p p l i e d h e r e , we have convergence of t h e sequence 6 , d e f i n e d by ( 7 . 1 9 ) , ( 7 . 2 0 ) , ( 7 . 2 1 ) , t o t h e s o l u t i o n uh o f (POh)4 f o r 0 < aoh< pn < alh, a],, s u f f i c i e n t l y s m a l l .

8

Remark 7.1. E q u a t i o n s (7.16)~( 7 . 1 7 ) , ( 7 . 2 0 ) , ( 7 . 2 1 ) cons t i t u t e " n a t u r a l " d i s c r e t i s a t i o n s of t h e c o r r e s p o n d i n g c o n t i n uous e q u a t i o n s ( 7 . 2 ) , ( 7 . 1 ) , ( 7 . 9 ) , (7.10). 8 7.2.2.

Solution of a particular case for which the exact solution i s known.

Construction of the t e s t problem. The e x a c t s o l u t i o n of (Po) when f = c o n s t = C and 0 i s a d i s c of r a d i u s R was g i v e n i n S e c t i o n 2.3.2. Let u b.e t h e r e s t r i c t i o n t o 0 =]0,1[ x ] 0 , 1 [ o f t h e s o l u t i o n o f (Po) r e l a t i n g t o t h e d i s c o f r a d i u s R = G I 2 and c e n t r e ( 0 . 5 , 0 . 5 ) f o r p = 1, C = 10, g = 1 . 2 5 = 514 ( h e n c e R ' = $ , s e e F i g u r e 7 . 1 ) . The e s s e n c e o f what h a s a l r e a d y been s a i d i n t h e p r e s e n t c h a p t e r a p p l i e s t o t h e (non-homogeneous) v a r i a t i o n a l problem

.

I

H = { u 1 U E H1(62), u Ir = u

lr},

whose s o l u t i o n i s c l e a r l y u ; i n p a r t i c u l a r , ( 7 . 2 2 ) can be s o l v e d a p p r o x i m a t e l y u s i n g a l g o r i t h m (7.19) , ( 7 . 2 0 ) , (7.21); it i s s u f f i c i e n t t o r e p l a c e i n ( 7 . 2 0 ) t h e c o n d i t i o n u, = 0 i f ( i n which c a s e M, t h e n b e l o n g s t o f ) by t h e c o n d i t i o n M,$62,

g , = u(M,)

if M,E

r.

8

386

Numerical analysis of Bingham f l u i d f l o w

(CHAP.

5)

Fig. 7.1.

Solution of the approximate Dirichlet problems ( 7 . 2 0 ) . This i s c a r r i e d out u s i n g p o i n t o v e r - r e l a x a t i o n w i t h o p t i m a l parameter; i f we denote by t h e sequence of i t e r a t e s given by t h e above o v e r - r e l a x a t i o n a l g o r i t h m , t h e c a l c u l a t i o n o f 111: i s i n i t i a l i s e d by

em

(7.23)

eo= u"-' h

9

t h e t e r m i n a t i o n c r i t e r i o n used b e i n g (7.24)

c

Igjm - q

-

1

Id

10-4.

Mi16nh

I n i t i a l i s a t i o n of algorithm (7.19), ( 7 . 2 0 ) , (7.21): (7.25)

1;

= 0.

Termination c r i t e r i o n f o r algorithm (7.19), (7.20), (7.21): (7.26)

R" =

c

I

- #i' I d

3~10-~.

MII 6 Qh

Choice of

pm i We t a k e

p,, = p

=

1/g = j .

Analysis of the numerical r e s u l t s : We use t h e n o t a t i o n

(SEC. 7 )

h a l i t y methods

.

387

where u i s t h e e x a c t s o l u t i o n o f ( 7 . 2 2 ) ; r e s u l t s r e l a t i n g t o t h e convergence and accuracy o b t a i n e d are shown i n Table 7.1.

1 10

1 -

20

i-6

25

25

36

RN

2.9 XI 0-4

2.6x10-*

2.~(10-~

EN

5.3x10-’

2.1 X I O - 2

2.7x 1 0-

0.35 s

1.53 s

9.62 s

h Number of iterations N required f o r convergence

Computation time on IBM 360/91

Table

Remark 7 . 2 .

For h = 1/20, max Aft, E

1

’

7.1

114;’

- u(Mij)

I

5

lo-*.

Oh

A more d e t a i l e d a n a l y s i s of t h e convergence of a l g o r i t h m

( 7 . 1 9 ) , (7.20),( 7 . 2 1 ) and of t h e numerical r e s u l t s o b t a i n e d , i n c l u d i n g t h e dependence of t h e s o l u t i o n s on g , may be found i n S e c t i o n 7.2.3. below. 7.2.3.

Other examples (I)

We s h a l l c o n s i d e r examples whose e x a c t s o l u t i o n i s n o t known (l).

D e f i n i t i o n of the domain and vaZues of the parameters. We t a k e D = 10, I[ x )o, I[, p = I , f = 10; we s h a l l t a k e s e v e r a l v a l u e s for g , w i t h t h e a i m o f i n v e s t i g a t i n g t h e dependence of t h e s o l u t i o n s w i t h r e s p e c t t o t h i s parameter.

Mesh s i z e :

h

=

1/20.

SoZution o f the approximate DirichZet probzems ( 7 . 2 0 ) : By p o i n t o v e r - r e l a x a t i o n w i t h o p t i m a l parameter.

(’1

Except when g i s s u f f i c i e n t l y l a r g e , i n which c a s e t h e s o l u t i o n is e x a c t l y z e r o .

Numerical a n a l y s i s of Binghm f l u i d f l o w

388

(CHAP.

5)

I n i t i a l i s a t i o n and termination c r i t e r i a of t h e algorithms : Those of t h e example i n S e c t i o n 7.2.2.

Choice of p , , : We t a k e p,, = p . Analysis o f the nwnericaZ r e s u l t s : F i g u r e 7 . 2 shows t h e number of i t e r a t i o n s r e q u i r e d f o r convergence, as a f u n c t i o n o f p, when g = 1.

Fi.7.2.

V a r i a t i o n o f t h e number of i t e r a t i o n s r e q u i r e d f o r convergence, as a f u n c t i o n o f p, ( g = 1).

For p = 2/g = 2 t h e r e i s s t i l l convergence, i n 136 i t e r a t i o n s , b u t f o r p = 2 . 1 , divergence; t h e r e s t r i c t i o n (7.11), which r e l a t e s t o t h e continuous problem, i s t h u s very r e a l i s t i c f o r For p = 1.1 (which roughly c o r r e s t h e approximate problem (P& ponds t o t h e o p t i m a l v a l u e f o r p ) t h e convergence o f a l g o r i t h m (7.19), ( 7 . 2 0 ) , ( 7 . 2 1 ) i s a t t a i n e d i n 1 2 i t e r a t i o n s , g i v i n g an e x e c u t i o n t i m e o f 1 s e c . on an IBM 360/91. F i g u r e 7.3 shows t h e number of o v e r - r e l a x a t i o n i t e r a t i o n s r e q u i r e d t o s o l v e t h e approximate D i r i c h l e t problems ( 7 . 2 0 ) with g = 1, p = 1, u s i n g t h e i n i t i a l i s a t i o n ( 7 . 2 3 ) and t h e t e r m i n a t i o n c r i t e r i o n (7.24).

Duality methods

I

1 2

3

I

I

I

389

I

I

.

4 5 6 7 8 910111213%

Fig. 7.3. . V a r i a t i o n of t h e number of o v e r - r e l a x a t i o n i t e r a t i o n s r e q u i r e d t o s c l v e t h e approximate D i r i c h l e t problems ' ( 7 . 2 0 ) ( g = 1, p = 1). F i g u r e s 7.4, 7 . 5 s h G w t h e f l u i d zones ( ' ) and r i g i d zones ( 2 ) ( h a t c h e d ) c o r r e s p o n d i n g t o g = 1 and g = 1.8 r p s p e c t i v e l y . F i g u r e 7 . 6 r e p r e s e n t s t h e f u n c t i o n x1 + u(xl,$) f o r p = 1, g = 1, f = 10.

p = 1, g = l,f = 10, urn,,= 0.291.

Fig. 7.4.

( ) i.e. ( 2 ) i.e.

I grad u I > 0. 1 grad u 1 = 0.

Fig. 7.5.

p = 1, g = 1.8,f = 10, urnax=0.080.

390

Numerical a n a l y s i s of Bingham f l u i d f l o w

(CHAP.

5)

XI

Fig. 7.6.

Representation of

u(x,, 1/2), p = 1, g = 1, f = 10.

We c o n s i d e r it a p p r o p r i a t e t o show, i n F i g u r e 7.7, t h e d e c r e a s e )I ug’pItrhn,as a f u n c t i o n o f g ; i n t h e c a s e when 61 i s a d i s c and f i s a c o n s t a n t it i s easy t o deduce from e q u a t i o n s ( 2 . 2 0 ) o f w e have i s p r o p o r t i o n a l t o I g - gc 1’; S e c t i o n 2.3.2 t h a t 11 ug I&, t h e r e f o r e p l o t t e d g + I( ug lI2irn , u s i n g c a l c u l a t i o n s c a r r i e d out f o r a f a i r l y l a r g e number o f v a l u e s o f g ; s i n c e t h e graph o b t a i n e d i s a s t r a i g h t l i n e it appears t h a t i n t h e c a s e when 61 again v a r i e s a s i s a s q u a r e , and f i s a c o n s t a n t , 11 ug ( a p a r t from approximation and c a l c u l a t i o n e r r o r s ) . ‘p g - gc’1’

of

7.2.4. Other examples (11). We once a g a i n c o n s i d e r an example f o r which t h e e x a c t s o l u t i o n i s n o t known ( a t least f o r g s u f f i c i e n t l y s m a l l ) .

g

Fig. 7.7.

R e p r e s e n t a t i o n of

g

+

11 u,, l $&.

D e f i n i t i o n o f the domain and values of t h e parameters: For 0 w e t a k e t h e h a l f - d i s c o f r a d i u s $, p = 1, f = 10, g = 0.75. Mesh s i z e :

h = 1/20.

Analysis of the numerical r e s u l t s : W e u s e a l g o r i t h m ( 7 . 1 9 ) , ( 7 . 2 O ) , ( 7 . 2 1 ) as i n t h e p r e v i o u s examples, which g i v e s t h e r e s u l t s shown i n F i g u r e 7 . 8 for t h e f l u i d and r i g i d ( h a t c h e d ) zones.

S e m i - c i r c u l a r d u c t (p = 1, f = 10,g = 0.75).

Fig. 7.8. 7.3.

7.3.1.

A p p l i c a t i o n t o t h e s o l u t i o n o f t h e approximate problems (11). The c a s e o f f i n i t e element a p p r o x i m a t i o n s .

The case o f the f i n i t e element approximation o f Section 3.

Description and convergence o f the d u a l i t y algorithm. We u s e t h e n o t a t i o n o f S e c t i o n 3 and o f Chapter 3 , S e c t i o n 8.3.2; t h e s o l u t i o n uh o f ( 3 . 1 ) ( a n d o f t h e e q u i v a l e n t problem ( 3 . 2 ) ) i s c h a r a c t e r i s e d by t h e e x i s t e n c e o f & € A h such t h a t a(uh, Oh)

+g

In

&.grad 0,

(7.28) uh

E

vh

9

which ' a r e approximate a n a l o g u e s of (7.1), (7.2) ; t h e p a i r { uhr } i s a s a d d l e p o i n t i n V , x Ahof t h e Lagrangian .Yk d e f i n e d by :

Numerical analysis

392

(7*30)

ph) =

9h(uh,

1

dub, uh) -k g

I

Oj?

Bingham f l u i d flow

&.grad

uh

dx -

J',

fUh

(CHAP.

5)

dx .

We can t h u s apply t o t h i s i n f i n i t e - d i m e n s i o n a l s i t u a t i o n t h e r e s u l t s of Chapter 2 , Section 4 , which y i e l d t h e approximate analogue of algorithm ( 7 . 8 ) , (7.9), (7.10) given by:

A:

(7.31)

given;

with 2; known (E .4,)

i

(7.32)

4 E

A;+'

(7.33)

, we

define

4

and t h e n 2;"

by

vh,

= PAh(A;

+ p,, grad 4),

with, w e r e c a l l , PA, : L . h + A h

d e f i n e d by

(7.34) Since Theorem 4 . 1 of Chapter 2, S e c t i o n 4.3 may a l s o be applied h e r e , t h e convergence of t h e sequence 4 defined by (7.31), ( 7 . 3 2 ) , ( 7 . 3 3 ) t o t h e s o l u t i o n u h o f problem ( 3 . 1 ) can t h u s be deduced f o r 0 < a. Q p,, d a1 < 2/g,

(7.35)

as i n t h e c a s e of t h e continuous problem.

Remark 7 . 2 . It i s easy t o express equations (7.281, ( 7 . 2 9 1 , and ( 7 . 3 2 ) , ( 7 . 3 3 ) i n terms o f t h e nodal v a l u e s U, =vh(M), using t h e formulas of Chapter 3, S e c t i o n 4.1.4. 7.3.2.

The case o f the f i n i t e element approximation of Section 3. ( 1 1 ) . Examples ( l )

Solution o f t h e approximate DirichZet problems ( 7 . 3 2 ) . By p o i n t o v e r - r e l a x a t i s n w i t h w = 1 . 5 , f o r a l l t h e examples considered; t h e c a l c u l a t i o n of 4 i s i n i t i a l i s e d by up = 4-l t h e s u p e r s c r i p t p being used t o denote t h e over-relaxation i t e r a t i o n s , with uf" f o r t h e p t h i t e r a t e .

(')

The numerical t r e a t m e n t of t h e examples below i s due t o J F. Bourgat.

.

(SEC. 7 )

D u a l i t y methods

393

Termination c r i t e r i a : We d e f i n e D,(n) by

and D 3 P ) by

(7.37)

D;(p)

= M

c€1, I uRi" -

uX;-l*n

I/

c

IuiYl.

HE&

For a l l t h e examples t r e a t e d below we t h e n t a k e as termination c r i t e r i a (7.38)

D , ( ~ )Q 1 0 - ~

f o r a l g o r i t h m ( 7 . 3 1 ) , ( 7 . 3 2 ) , ( 7 . 3 3 ) and

with

f o r t h e over-relaxation algorithm.

I n i t i n l i s a t i o n of algorithm ( 7 . 3 1 1 , ( 7 . 3 2 1 , ( 7 . 3 3 ) : 1; = 0. Values o f p.: considered

.

We t a k e pn = p = 0.7 f o r a l l t h e examples

Example 1. We t a k e an example f o r which t h e e x a c t s o l u t i o n i s known and g i v e n by formulas ( 2 . 2 5 ) of S e c t i o n 2 . 3 . 2 , namely: P i s t h e d i s c of r a d i u s $, p = 1, f = 10, g = 1.25, g i v i n g R' = f . With t h e t r i a n g u l a t i o n shown i n F i g u r e 7 . 9 , c o n s i s t i n g o f 512 t r i a n g l e s and 225 i n t e r i o r nodes, convergence o f a l g o r i t h m ( 7 . 3 1 ) , ( 7 . 3 2 ) , ( 7 . 3 3 ) i s r e a c h e d i n 10 i t e r a t i o n s under t h e above c o n d i t i o n s , g i v i n g a computation t i m e o f 90 s e c . on a C I I 10070; i f u i s t h e e x a c t s o l u t i o n we have Max I u i 0 ( M ) - u ( M ) I = 6 ~ 1 0 - ~ . ME&

F i g u r e 7.9 shows t h e f l u i d and r i g i d ( h a t c h e d ) zones t h u s c a l c u l at e d.

Example 2. We t a k e 62 = 10, l.[ x ]0,1[ , p = 1, f = 10, g = 1. Using t h e t r i a n g u l a t i o n shown i n F i g u r e 7.10, c o n s i s t i n g of 512 t r i a n g l e s , convergence i s r e a c h e d i n 14 i t e r a t i o n s , g i v i n g a computation t i m e of 2 min 30 s e e on a CII 10070.

Numerical analysis of Bingham f l u i d flow

394

(CHAP.

5)

F i g u r e 7.10 a l s o shows t h e c a l c u l a t e d f l u i d and r i g i d ( h a t c h e d ) zones. We t a k e 61 t o be t h e h a l f - d i s c o f r a d i u s t , p = 1, With t h e t r i a n g u l a t i o n shown i n F i g u r e 7.11, g = 0.75. 10, c o n s i s t i n g of 256 t r i a n g l e s , convergence i s reached i n 2 3 i t e r a t i o n s , g i v i n g a computation t i m e of 1 min. 10 s e c on a C I I 10070. F i g u r e 7.11 shows t h e c a l c u l a t e d f l u i d and r i g i d ( h a t c h e d ) zones.

Example 3.

f=

7.3.3.

The case of approximation by second-order Lagrcmgian f i n i t e elements.

General considerations. The f o l l o w i n g c o n s i d e r a t i o n s a r e t a k e n from F o r t i n 111. For an approximation of HJ(61) given by a f a m i l y o f subspaces formed from continuous f u n c t i o n s , and whose r e s t r i c t i o n t o t h e t r i a n g l e s T o f a t r i a n g u l a t i o n 9'-,, o f B i s o f degree d 2 , it has a l r e a d y been p o i n t e d out i n S e c t i o n 3.5 t h a t t h e u s e o f t h i s t y p e of f i n i t e element i n t h e s o l u t i o n o f problem (1.5) poses c e r t a i n d i f f i c u l t i e s connected w i t h t h e approximation o f t h e

v,

n o n d i f f e r e n t i a b l e term

B Jn I grad I I3.c

The approximation used b e i n g o f Lagrangian t y p e i n which t h e

Duality methods

Fig.7.10.

p = I,f

Fig.7.11. p = I , f =

= 10,g = 1.

IO,g=O.75.

d e g r e e s o f freedom a r e t h e v a l u e s t a k e n by oh a t t h e p o i n t s MiT,rniT(i= l , 2 , 3 ) o f F i g u r e 3 . 1 o f S e c t i o n 3.5, w e s h a l l u s e numerical i n t e g r a t i o n f o r m u l a s t o approximate t h e f u n c t i o n a l

IgraduIdx J,(u) =

; w e approximate t h e f u n c t i o n a l

-21 a(v, u)

f i r s t by

and s e c o n d l y by

+ d(V)

-

I,

f o dx ,

395

Nwnerical a n a l y s i s o f Bingham f l u i d f l o w

396

5)

(CHAP.

with G, = c e n t r e of g r a v i t y of T and A ( T ) = a r e a of T. It can be shown without d i f f i c u l t y t h a t t h e approximate problems :

admit, r e s p e c t i v e l y , one and only one s o l u t i o n u i , u i say, and t h a t when h - 0 as i n t h e statement of Theorem 5 . 1 of Section we have (7.45)

s t r o n g l y i n H&2)

ui -* u

5.1,

(i = 1,2),

where u i s t h e s o l u t i o n of (Po) ( l ) . Given t h e Lagrangians 9:, 9;defined by

w2,

it can be proved t h a t t h e following with p,, pi, (i = 1,2,3) E finite-dimensional v a r i a n t s of algorithm (7.81, ( 7 . 9 1 , (7.10) define sequences up (i = 1, 2) which converge, r e s p e c t i v e l y , t o t h e s o l u t i o n ui of (7.43), (7.44):

Case i = 1: (7.48)

{: 1

given;

with nl, known, w e determine ub)+g

(7.49)

1 TEYh

4

and

by

A(T) L”,Pad uh(GT)= ,

( ’ 1 This l a s t p o i n t i s a r e l a t i v e l y immediate consequence of g e n e r a l r e s u l t s on t h e convergence o f f i n i t e element approximat i o n s ( s e e f o r example Fix-Strang /l/, Ciarlet-Raviart /l/].

397

(7.50)

';A

(7.51)

0 < aAh < p.

=

%(A;

+ p. A ( T ) grad U;:(Gr)), < aih,

T EY h ,

mih s u f f i c i e n t l y s m a l l .

Case i = 2:

Y

We r e c a l l t h a t if y = ( y l , y 2 )R2 ~ w e have Pa(y) =

max (1,

JTX) '

Application t o an exuniple for which the exact s o l u t i o n i s known. We c o n s i d e r a g a i n t h e (nonhomogeneous) example o f S e c t i o n w e s h a l l n o t go i n t o t h e d e t a i l s of t h e obvious 7.2.2; m o d i f i c a t i o n s which need to be a p p l i e d t o ( T . h 9 ) , (7.53), n o r i n t o t h o s e which r e l a t e t o t h e u s e o f t h e d u a l i t y a l g o r i t h m s given e a r l i e r . We s h a l l simply s t a t e t h a t w i t h a t r i a n g u l a t i o n r h of 1 2 8 t r i a n g l e s w e o b t a i n approximate s o l u t i o n s ui such that

t h e s e maximum e r r o r s b e i n g o b t a i n e d i n t h e neighbourhood of t h e i n t e r f a c e between t h e r i g i d and f l u i d zones (where d e r i v a t i v e s of o r d e r 3 2 are d i s c o n t i n u o u s ) .

Nwnerical analysis o f Bingham f l u i d flow

398

8.

(CHAP. 5)

APPLICATION TO THE SOLUTION OF THE ELASTO-PLASTIC TORSION PROBLEM OF CHAPTER 3 .

8.1.

Synopsis

A d u a l i t y a l g o r i t h m was formulated i n Chapter 3, S e c t i o n 9.1.k e q u a t i o n s ( 9 . 5 ) , ( 9 . 6 ) , ( 9 . 7 ) , which a l l o w s t h e s o l u t i o n of t h e e l a s t o - p l a s t i c t o r s i o n problem o f Chapter 3 t o be o b t a i n e d . The a p p l i c a t i o n of t h i s a l g o r i t h m e n t a i l s , among o t h e r t h i n g s , t h e minimisation of a n o n d i f f e r e n t i a b l e f u n c t i o n a l of a t y p e very similar t o t h a t c o n s i d e r e d i n Chapter 5 , S e c t i o n s 1 t o 7; we s h a l l t h e r e f o r e u s e a v a r i a n t o f a l g o r i t h m ( 7 . 8 ) , ( 7 . 9 ) , (7.10) o f S e c t i o n 7 . 1 t o s o l v e t h e above n o n d i f f e r e n t i a b l e problem. 8.2.

Reformulation o f a l g o r i t h m ( 9 . 5 ) , ( 9 . 6 ) , ( 9 . 7 ) o f Chapter 3, S e c t i o n 9.1.1.

For r e a s o n s of c l a r i t y we s h a l l r e f o r m u l a t e a l g o r i t h m ( 9 . 5 ) , ( 9 . 6 ) , ( 9 . 7 ) of Chapter 3 , S e c t i o n 9.1.1, u s i n g t h e same formalism; (8.1)

we thus define A , = L:(G?),

4, : H&2) + LZ(G?) by 41(4= I grad D I - 1 , and t h e Lagrangiaa Y1 : Hi(l2) x A ,

(8.3)

a,(& 4) = -* 4 v . 4 -

I,

+

f v dx

W by

+

Q

q41(4 dx *

From t h e r e l a t i o n

w e deduce t h a t i f Yladmits a s a d d l e p o i n t { u , p } on u i s t h e s o l u t i o n o f t h e probleni (Po) :

(8.6)

H i @ ) x A,,

K o = { o l u ~ H ~ ( G ?I gr ) , ad vlG 1 a.e.1

Thus, i n t h e c a s e where t h e assumptions (4.10), ( 4 . 1 1 ) of Chapter 2, S e c t i o n 4.3 are s a t i s f i e d , w e can s o l v e ( P o ) by applying algorithm (4.12) ,(4.15) of Chapter 2 , S e c t i o n 4.3, t h a t is:

,. . .

(SEC. 8 )

A p p l i e a t i o n t o e las to-p las t i c torsion

w i t h @' known (en,), u"

i s defined as b e i n g t h e element

o f Hh(S2) which m i n i m i s e s

(8.7) (8.8)

pn+l= sup(0,p" +

(8.9)

p">o

8.3.

399

P"&,(U")),

sulficiently small.

Approximate implementation o f a l g o r i t h m ( 8 . 7 ) , ( 8 . 8 ) ,

(8.9). We s h a l l r e s t r i c t our a t t e n t i o n t o t h e s o l u t i o n o f t h e approximate problem (Po,),, of Chapter 3, S e c t i o n 5 . 3 ; w e s h a l l use t h e n o t a t i o n of Chapter 3, S e c t i o n 5 and Chapter 5, S e c t i o n 7.2.

Existence o f a saddle point. Description and c o n v e r gence o f the duality algorithm

8.3.1.

Existence of a saddZe point. We d e f i n e (8'10)

Alh

=

{ q h I qh

vMi+

=

1/2 j + 1/2

q i + 1 / 2 j + 1 / 2 } M 1 + t / ~ ~ + r / ~P ~ Pqpi +, 1 / 2 j + 1 / 2 zh

1

9

and we a s s o c i a t e w i t h t h e approximate problem i m a t i o n 9 1 h o f 9, d e f i n e d by 9lh(uh,

(8.11) X

qh) = JOh(uh)

2

+ h2

(JI G/+ 1/2j+1,2(%)

(PoJ14

t h e approx-

qi+1/21+1/2

M ~ + I / Z J + fI /& Z

I2

i- I G+:

112 j+ 1/2('h)

I2 - '1

S i n c e t h e proof o f P r o p o s i t i o n 9 . 1 o f Chapter 3, S e c t i o n admits a s a d d l e 9 . 2 . 1 can b e a d a p t e d w i t h o u t d i f f i c u l t y , p o i n t , { u,,p, }, on Vh x Alh w i t h

Description o f the algorithm m d convergence. W e use a finite-dimensional v a r i a n t o f algorithm ( 8 . 7 ) , ( 8 . 8 ) ,

( 8 .9 ) : with

fl known

(en,,,), w e d e f i n e

V, which m i n i m i s e s

91h(Uhr&),

U;:

as t h e element o f

Numerical analysis of Binghm fluid fZow

400

(CHAP.

5)

pln > 0 , s u f f i c i e n t l y s m a l l .

(8.15)

I t r e s u l t s from Theorem 4 . 1 of Chapter 2 , S e c t i o n 4.3 t h a t t h e sequence (@,, d e f i n e d by ( 8 . 1 3 ) , ( 8 . 1 4 ) , (8.15) converges t o t h e s o l u t i o n of (P0Jl4. w

There i s no d i f f i c u l t y i n a d a p t i n g t h e c o n s i d e r a t i o n s o f S e c t i o n 7.2.1 t o t h e s o l u t i o n of t h e n o n d i f f e r e n t i a b l e problem (8.13), and we t h e n o b t a i n t h e f o l l o w i n g v a r i a n t o f a l g o r i t h m (7.19), ( 7 . 2 O ) , ( 7 . 2 1 ) o f S e c t i o n 7.2.1 f o r c a l c u l a t i n g 4 from

- @;";j+

@?I,+

4: + qj!! 1

1-4

qj"-

h2

(8.17)

2+A;?;zj-l/z--~;Yiz,h

(n::;z,+l/2-A;?;z,+l/2

112

4

+

2n m AZn m 2n,m A i + * 1 / 2 j + 112i + * l / Z j - 1/2 +Li- I / ? , + 1/2-J-i-

2h

+

1 / 2 j - 112

2w

)

o f Chapter Chapter 22,, SSeeccttiioonn 44..33 aallssoo aapppplliieess hheerree,, Theorem 44..11 of SSiinnccee Theorem $" ddeeffiinneedd by by (8.16), (8.16),((8.17), (8.18) converges c o n v e r g e s ttoo sequence $" tthhee sequence 8 . 1 7 ) , (8.18) U;: of o f (8.13) (8.13) ffoorr p;,,>O &>O tthhee ssoolluuttiioonn U;: ssuuffffiicciieennt tl lyy ssmmaal ll l. .

401

Application t o e l a s t o - p l a s t i c t o r s i o n

(SEC. 8 )

8.4. A p p l i c a t i o n t o a n example

(')

We c o n s i d e r a g a i n t h e b a s i c example o r Chapter 3 , namely 10, l[,f = 10.

ra = p, 1[ x

Mesh s i z e : h

= 1/20.

Solution of the p p r o x i m a t e D i r i c h l e t problems (8.17): By p o i n t o v e r - r e l a x a t i o n w i t h o p t i m a l p a r a m e t e r , i n i t i a l k i n g u s i n g a t e r m i n a t i o n c r i t e r i o n o f t h e t y p e (7.24). with @'"-I,

I n i t i a l i s a t i o n of algorithm (8.16),(8.17), (8.18): (8.20)

' ; 1

=

A;-'

Termination c r i t e r i o n for algorithm (8.16), (8.17), (8.18):

1

(8.21)

I Gj'" - u?!'"-1 1 G 10-3. ,J

Mij E nh

I n i t i a l i s a t i o n of algorithm (8.13), (8.14),(8.15): (8.22)

p: = 0 .

Termination c r i t e r i o n for algorithm (8.13), (8.141, (8.15): (8.23)

Choice o f

pI. and pl,,

:pin =

p1 = 1,

p;," = p2 = 0.1.

Numerical resu 1 t s : Convergence w a s a t t a i n e d i n 12.57 s e c on an IBM 360191, t h e t o t a l number of o v e r - r e l a x a t i o n i t e r a t i o n s c a r r i e d o u t b e i n g

794. 8.5. A v a r i a n t of a l g o r i t h m (8.13), (8.14), (8.15) A s t h e computation time for t h e example i n S e c t i o n 8.4 appears p r o h i b i t i v e , we have a l s o used t h e f o l l o w i n g a l g o r i t h m : (8.24)

with

(8.25)

(I)

(u,",p,",) :1

u;,p;,

r

given;

known, we c a l c u l a t e p;", At+', ~ Z i ' / 2j +

112

4"

by

=max (0, P;+ 112 j+112 +

(JI ~ : + 1 / 2 j + l / t (I'+I ~) VMi+L / Z j + 1/2 6 zb

+ ~ l n

1'-1))

~ ? + 1 / 2 j + 1 / ~ ~ : )

3

9

T h i s numerical implementation i s due t o M. Goursat.

Numerical analysis of Bingham f l u i d flow

402

#(A: (8.26)

j+l/29

(CHAP.

5)

2n+ 1 4, 1/2,+1/2)=

= ~ p ~ : ~ , * , * , , ~ (j +4 l~/ Zl +/ P 2zn G l / Z j + I / z ( a

221/2 j + 112 +

G i ' +

~ z n

112 j + 1/2(G))9

V M i + 112 j + 112

E

zh

;

and f i n a l l y determine G+' by u s i n g only one o v e r - r e l a x a t i o n i t e r a t i o n , s t a r t i n g from 4 , i n t h e s o l u t i o n of t h e l i n e a r system ui

+ 1j + u i j + I + u i -

1j

h2

+ uij- 1 - 4

uij

-

pzn = p2 = 0.2 , convergence i s a t t a i n e d i n 140 with p l n = p1 = 0.4, i t e r a t i o n s o r 3.9 s e c . e x e c u t i o n time on an IBM 360191.

Remark 8.1. The d u a l i t y method developed i n t h i s s e c t i o n w i t h a view t o s o l v i n g t h e e l a s t o - p l a s t i c t o r s i o n problem o f Chapter 3, appears t o have a performance i n f e r i o r t o t h o s e of t h e o t h e r d u a l i t y methods considered i n Chapter 3, S e c t i o n 9; i n p a r t i c u l a r t h i s method has a g r e a t e r c o r e s t o r a g e requirement The only advantage i s t h a t t h e t h a n t h e method of S e c t i o n 9.4. m a t r i x o f t h e approximate D i r i c h l e t problem t o be s o l v e d at each i t e r a t i o n i s independent o f t h e i t e r a t i o n under c o n s i d e r a t i o n ; t h i s i s not t h e c a s e w i t h a l g o r i t h m ( 9 . 7 4 ) , ( 9 . 7 5 ) , (9.76) o f Chapter 3, S e c t i o n 9.4.1 i n which t h e m a t r i x o f t h e approximate D i r i c h l e t problem (9.74) depends on p:.

9.

DISCUSSION

S e c t i o n s 1 t o 7 of t h i s c h a p t e r r e p r e s e n t an e x t e n s i o n of t h e paper of C6a-Glowinski /1/ i n which t h e a u t h o r s r e s t r i c t e d t h e i r a t t e n t i o n t o f i n i t e d i f f e r e n c e approximations. The r e s u l t s o f t h i s c h a p t e r would appear t o i n d i c a t e t h a t t h e method b e s t s u i t e d t o t h e s o l u t i o n o f t h e v a r i a t i o n a l problem (Po) d e f i n e d by (1.5) f o r domainsf2of g e n e r a l shape, c o n s i s t s o f t h e combined use o f t h e f i r s t order f i n i t e element approximation of S e c t i o n 3 and t h e v a r i a n t ( 7 . 3 1 ) , ( 7 . 3 2 ) , ( 7 . 3 3 ) ( s e e S e c t i o n 7 . 3 . 1 ) o f t h e duality a l g o r i t h m ( 7 . 8 ) , ( 7 . 9 ) , (7.10) o f S e c t i o n 7.1. However, we c o n s i d e r e d it a p p r o p r i a t e t o dwell at some l e n g t h ( i n S e c t i o n 6 ) on regularisation, i n s o f a r as

-

t h i s t y p e of method seems a t l e a s t at p r e s e n t - t h e most s u i t a b l e f o r t h e n u m e r i c a l s o l u t i o n o f c e r t a i n optimal control problems i n which t h e u s u a l s t a t e equation i s r e p l a c e d by a Variational i n e q u a l i t y i n v o l v i n g a n o n d i f f e r e n t i a b l e Term; i n d e e d r e g u l a r i s a t i o n a l l o w s t h e above o p t i m a l c o n t r o l problem t o be approximated by a problem i n which t h e " s t a t e i n e q u a l i t y " i s r e F l a c e d by a n o n l i n e a r s t a t e e q u a t i o n for which it i s p o s s i b l e t o s t a t e e x p l i c i t l y c o n d i t i o n s n e c e s s a r y f o r o p t i m a l i t y ; on t h i s s u b j e c t , s e e Yvon /l/, /2/. With r e g a r d t o r e g u l a r i s a t i o n , we n o t e t h a t i n h i s p r o o f o f t h e H 2 r e g u l a r i t y o f t h e s o l u t i o n of (PO) (mentioned i n S e c t i o n 2.1), B r e z i s / 6 / u s e s - among o t h e r t h i n g s - a r e g u l a r i s a t i o n of t h e n o n d i f f e r e n t i a b l e term

jflI

grad 0 I dx.

With r e g a r d t o d u a l i t y methods, we s h o u l d mention i n conn e c t i o n w i t h t h e s o l u t i o n o f problem ( 1 . 5 ) t h a t t h e u s e o f a finite-dimensional v a r i a n t of t h e d u a l i t y algorithm ( ) :

(9. I )

' 1

(9.2)

#"+I

(9.3)

g i v e n ( € A ) , uo g i v e n (E H;(O)),

u" - p1 S-'(- Au" - g div 1" - f), p , > 0 , 1"" = PA@" + p2 grad u"") , p2 > 0 , =

t a k i n g S d e f i n e d by (9.4)

s=

(I - ;12) (I

-

2) '

w i t h homogeneous D i r i c h l e t boundary c o n d i t i o n s on r, d i d n o t l e a d ( 2 , t o a computation t i m e l e s s t h a n t h o s e o b t a i n e d w i t h t h e f i n i t e - d i m e n s i o n a l v a r i a n t s o f a l g o r i t h m (7. ( 7 . 9 ) , (7.10)of S e c t i o n 7.1. H TO conclude t h i s c h a p t e r , we would l i k e t o mention a n o t h e r a l g o r i t h m f o r s o l v i n g (1.5); t h i s i s a v a r i a n t o f a l g o r i t h m ( 1 0 . 5 ) - ( 1 0 . 7 ) o f Chapter 3, S e c t i o n 1 0 , t h e n o t a t i o n o f which we s h a l l use here. We t h u s w r i t e :

a),

L = L2(R) x LZ(s2)

t h e r e i s t h e n e q u i v a l e n c e between (Po) and

('1

Of Arrow-Hurwicz t y p e (see C h a p t e r 2 , S e c t i o n 4 . 4 ) ( 2, For r e c t a n g u l a r domains and f i n i t e - d i f f e r e n c e approximations, which a r e t h e o n l y p r a c t i c a l c a s e s f o r which it i s p o s s i b l e (and worthwhile) t o apply algorithm ( 9 . 1 ) , ( 9 . 2 ) , (9.3) with s d e f i n e d by ( 9 . 4 ) .

404

Numerical a n a l y s i s o f Bingham f l u i d flow

(CHAP.

5)

L e t u be t h e s o l u t i o n o f ( P o ) . Then (u;Vu) i s t h e unique s o l u t i o n o f ( 9 . 5 ) ; a Lagrangian n a t u r a l l y a s s o c i a t e d w i t h ( 9 . 5 )

is U(V,

9 ; P ) = AV, d

+

J', p . W

- 4) dx.

With p and q f i x e d , s i n c e piis non-coercive i n u, w e s h a l l a g a i n p e n a l i s e Vu q = 0 , giving

-

It can be shown t h a t admits a x L x L, of t h e form (u,Vu;A) (Po) and 1 t h e s o l u t i o n o f t h e d u a l Chapter 1, S e c t i o n 3.5. We s h a l l a p p l y U z a w a ' s a l g o r i t h m

u n i q u e s a d d l e p o i n t on where u i s t h e s o l u t i o n of problem o f (Po) d e s c r i b e d i n

Hi@)

(9.6)

A0€L

t o pC,g i v i n g

given;

w i t h An known, we s u c c e s s i v e l y d e t e r m i n e u",p",An+' by

(9.7)

{ YCW, P")

(9.8)

An+' = 1 . + P.(Vu"

p" ;p") d

(u",

Ye(V,

E Hi@) x

q ;P 9

V(U,

4) 6 H i m x L

L

- p") ,

P. > 0 .

I n view o f t h e e x i s t e n c e o f a s a d d l e p o i n t o f Pe on H i @ ) x L x L, under t h e c o n d i t i o n

(9.9)

0 < ro d Pn d ri < 2 / E

w e have

lim (u",~') = (u,Vu) ( s t r o n g ) i n

x L

n-+m

It may be n o t e d t h a t t h e d e t e r m i n a t i o n o f (u",p")u s i n g ( 9 . 7 ) i s an i n f i n i t e - d i m e n s i o n a l problem o f t h e m i n i m i s a t i o n o f a

nondifferentiable functional;

however, b e c a u s e of t h e v e r y s p e c i a l s t r u c t u r e of t h i s f u n c t i o n a l , we can u s e t h e r e l a x a t i o n methods d e s c r i b e d and s t u d i e d i n C6a-Glowinski 121 t o s o l v e ( 9 . 7 ) . W e r e f e r t h e r e a d e r t o Glowinski-Marrocco /3/ f o r a d e t a i l e d s t u d y of t h e a p p l i c a t i o n o f (9.6)-(9.8) t o a n a p p r o x i m a t i o n of (Po) u s i n g f i r s t - o r d e r f i n i t e e l e m e n t s .

Chapter 6 GENERAL METHODS FOR THE APPROXIMATION AND SOLUTION

OF TIME-DEPENDENT VARIATIONAL INEQUALITIES

INTRODUCTION In this chapter we study time-dependent problems. We shall consider the general subject of variational i n e q u a l i t i e s , exarnining the appro&mation and s o h t i o n of these inequalities in some detail. Three fundamental types of inequalities will be investigated: parabolic inequalities of type I, parabolic inequalities of type I1 and time-dependent inequalities of the second order in time. Each of these types of problem will be considered separately, the emphasis being laid on concrete examples which bring out the nature of the general formulation used. This general formulation will enable us to make an in-depth study of the finite difference approximation ( l ) and then for each type of inequality we shall investigate the numerical solution of several typical problems. To help identify the nature of the problems in question, let us say that it is natural from both the physical and mathematical viewpoints to extend the general formalism of steady-state inequalities : (Au - J o

- u) + j ( o )

- j ( u ) 2 0 Vo E K,u E K ,

to the case in which the solution evolves in time. an examination of problems of the following types:

This leads to

- Parabolic i n e q u a l i t i e s o f type I: find a function r-)u(r) for r E [0, T I , with values in K c V such that

Vo E K,u(r) E K,u(0) = u,,

given.

- Parabolic i n e q u a l i t i e s o f type 11: find a function t r E [0, TI , with values in V such that

+

u(r)

for

(’)

All of what follows is valid (and is somewhat simpler) for i n t e r i o r approximations of the “finite element” type; moreover, it is an interior method which we shall use in Section 11.

Time-dependent variational inequalities

406

( CHAP.

6)

- Hyperbolic inequalities (or w e l l posed i n t h e P e t r o w s k i t E [ O , T ] , with values i n V sense): find a function t+u(t) f o r such t h a t

1.

BACKGROUND

Following on from t h e i n i t i a l c o n c e p t s o u t l i n e d i n C h a p t e r 1, S e c t i o n 1.1, we g i v e h e r e a b r i e f r e v i e w o f t h e p r i n c i p a l c o n c e p t s o f f u n c t i o n a l a n a l y s i s which w i l l b e o f u s e l a t e r . The fundamenta l a s p e c t s may b e found, f o r example, i n Duvaut-Lions /l/, Chapter 1. More advanced developments are g i v e n i n Lions-MagPnes /l/, Schwartz /1/ and Sobolev /1/.

1.1

Spaces o f v e c t o r - v a l u e d

d i s t r i b u t i o n s and f u n c t i o n s

+

Given a n i n t e r v a l [O, T ] c R ( i n g e n e r a l T < 00) a n d a Banach s p a c e X w i t h norm (I. IIx , we d e n o t e by Lp(O,T ; X ) t h e s p a c e o f ( c l a s s e s o f ) f u n c t i o n s t + f ( t ) which a r e measurable from [0, TI -+ ( f o r t h e measure dt) s u c h t h a t

The s p a c e s Lp(o,T ; X ) a r e Banach s p a c e s f o r t h e f i r s t norm i f p # 00, , and f o r t h e second norm i f p = + 00. If X i s a H i l b e r t s p a c e equipped w i t h a n i n n e r p r o d u c t (,)x t h e n t h e s p a c e L2(0, T ; X ) i s a l s o a H i l b e r t s p a c e f o r t h e i n n e r

+

product

X

Background

(SEC. 1)

407

We d e n o t e by 9 ( ] 0 , T [ ; X ) t h e s p a c e o f i n d e f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s w i t h compact s u p p o r t i n p , T [ and v a l u e s i n X, and by LY(]O, T [ ; X )t h e s p a c e o f d i s t r i b u t i o n s on 10, r[ w i t h v a l u e s i n X , d e f i n e d by

WP,T [ ; X ) = U ( 9 0 0 , Ti) ; X ) , where i n a g e n e r a l manner Y ( M ; N ) d e n o t e s t h e s p a c e o f l i n e a r c o n t i n u o u s mappings from M h N . I n p a r t i c u l a r we can a s s o c i a t e w i t h f E Lp(O,T ; X ) a d i s t r i b u t i o n 7 : 9(]0,T [ )+ X d e f i n e d by

m

=

p(r)

W) dt

4 E 9(10, Ti) .

>

0

4+f(#) i s i n d e e d l i n e a r c o n t i n u o u s from Moreover, f-f i s a n i n j e c t i o n and it is appropr i a t e t o i d e n t i f y f and f Also f + O i n Lp(O,T ; X ) -?+ 0 i n T(4) --+ 0, V 4 E 9(]0,T[). We t h u s have, a l g e b r a i c 9’(]0, T [ ;X ) , i e a l l y and t o p o l o g i c a l l y The mapping

9(]0, T [ )+ X .

..

LP(0, T ; X ) c LY(]O, T [ ; X ) .

For

Y E B’(]O,T [ ;X ) , we

d e f i n e d’ffldt = f ( k ) by

Pk)(4)= (- Okf(4(lr))v4 E q10, TI), and hence, i n p a r t i c u l a r ,

1.2

ftkJ

E

9’(]0, T [ ;X ) , Vk. rn

Functional s e t t i n g

Whereas i n steady-state problems we have used one H i l b e r t s p a c e V ( o r , i n c e r t a i n c a s e s , one Banach s p a c e ) , i n t h e c a s e o f time-dependent problems w e must make u s e o f two HiZbert spaces ( a n d i n c e r t a i n c a s e s , two Banach s p a c e s ; however, we s h a l l r e s t r i c t our a t t e n t i o n t o t h e c a s e of H i l b e r t sp aces ). Thus l e t V and H be two H i l b e r t s p a c e s w i t h : (1 ’ 1)

V c H,

t h e i n j e c t i o n o f V i n H i s c o n t i n u o u s , V dense i n H .

We d e n o t e by ((J) ( r e s p . (,)) t h e i n n e r p r o d u c t i n V ( r e s p . and by 1) )I ( r e s p . 1 I ) t h e c o r r e s p o n d i n g norm. We t h u s have (1.2)

1 u I < c I1 u II

VUE

v.

We i d e n t i f y H w i t h i t s dual. Then, i n t h e i d e n t i f i c a t i o n c o m p a t i b l e w i t h t h e above, we have: (1.3)

V c H c V’

( V = dual of V ) .

H)

Time-dependent variational inequalities

408

( CHAP.

6)

We equip V' w i t h t h e "dual" norm

II w II+ = sup (w, u ) , w i t h V€V

11 t' 11

= 1

.

We d e f i n e t h e space W ( 0 , T ) by

du

u I u E L2(0, T ; V),u' = -E L2(0,T ;V') dr With t h e i n n e r product

W(0, r ) i s a H i l b e r t space. Every V E W(0, T ) i s , a f t e r p o s s i b l e m o d i f i c a t i o n on a s e t of measure z e r o , continuous from [0, TI-, H. We can t h e n d e f i n e a c l o s e d a f f i n e subspace of W(0,T) by Wo(O, T ) = { u I u E W(0, T ) , 40) = uo,

uo given i n

H}.

INTRODUCTION TO PARABOLIC TIME-DEPENDENT INEQUALITIES OF TYPE I

2.

I n t h i s s e c t i o n we s h a l l i n t r o d u c e by means o f two examples t h e g e n e r a l formalism o f p a r a b o l i c time-dependent i n e q u a l i t i e s o f type I. 2.1

Examples o f p a r a b o l i c i n e q u a l i t i e s o f t y p e I

Example I

2.1.1

I n t h i s f i r s t example we a g a i n c o n s i d e r t h e d i f f u s i o n problem given as a f i r s t example i n Chapter 1, S e c t i o n 1.1. Here, however, we s h a l l c o n s i d e r t h e g e n e r a l c a s e i n which w e seek t o c a l c u l a t e t h e evolution of t h e f l u i d p r e s s u r e when a p r e s s u r e d i f f e r e n c e i s a p p l i e d a t t h e o u t e r s u r f a c e o f t h e t h i n semi-permeable w a l l . Thus, i n s t e a d o f a t t e m p t i n g t o f i n d what t h e p r e s s u r e w i l l b e when e q u i l i b r i u m i s reached, w e seek t o determine t h e e v o l u t i o n o f t h e p r e s s u r e a t each p o i n t o f t h e e n c l o s u r e 9, s t a r t i n g from the i n i t i a l instmrt a t which a p r e s s u r e d i f f e r e n c e i s e s t a b l i s h e d a t t h e u n t i l a given i n s t a n t T ( i f T i s s u f f i c i e n t l y boundary r o f l a r g e , t h e p r e s s u r e must evolve towards t h e s t e a d y - s t a t e s o l u t i o n ) . This problem l e a d s t o t h e f o l l o w i n g formalism: f i n d a f u n c t i o n 4 x . r ) f o r r E [ O , TI, X E B such t h a t

n,

(2.1)

au

ar - Au

=f,

f o r (x, r ) E Q = p, T [ x 9 ,

w i t h the boundary conditions ( h ( x ) b e i n g t h e e x t e r n a l p r e s s u r e applied a t x c r ) :

(SEC. 2 )

(2.2)

Parabolic inequalities of type 1

824 u(x,t) > h(x)=s-(x,r) an

= 0,

409

x E r , t fixed

( w i t h t h e i n t e r n a l pressure higher than t h e ex t e r n al pressure, t h e semi-permeable w a l l p r e v e n t s any exchange o f f l u i d ) ,

(2.3)

U(X,t ) G h(x) =.

au (x, I ) 2 0 ,

xE

r,

(with t h e e x t e r n a l pressure higher than t h e i n t e r n a l pressure, t h e semi-permeable w a l l a l l o w s a p o s i t i v e f l o w from t h e o u t s i d e towards t h e i n s i d e o f Q; t h u s by v i r t u e o f c o n t i n u i t y , t h e o n l y p o s s i b l e c a s e i s u(x, t ) = h(x) ) The initial condition f o r t h e problem i s :

.

(2.4)

u(x, 0 ) = uo(x) = g i v e n ,

X E

l2 (uo 2 h ) ,

( i t is c l e a r t h a t t h e e v o l u t i o n o f t h e p r e s s u r e i s dependent on the pressure a t the i n i t i a l i n s t a n t ) . We s h a l l pose t h i s problem i n t h e form o f a time-dependent variational inequality. To t h i s end we i n t r o d u c e

,

(2.5)

V

(2.6)

u(u, u ) =

(2.7)

(hd =

(2.8)

K = {uluEV,u2honr},

= H'(R)

H

= Lz(12),

I, 1,

grad u grad u dx ,

fgdx,

and d e n o t e by u(f)the f u n c t i o n problem :

x+u(x,t);

we t h e n c o n s i d e r t h e

f i n d u ( t ) ~ K ,t t r a v e r s i n g t h e i n t e r v a l [0, TI, such t h a t

Applying G r e e n ' s formula i n ( 2 . 9 ) we o b t a i n

AU - J

D

- U)

+Ir

$(u

- U) dr 2 0 , Vu E K , u E K .

Using t h i s i n e q u a l i t y it may r e a d i l y b e shown t h a t t h e problems (2.1) - ( 2 . 4 ) a n d ( 2 . 8 ) , ( 2 . 9 ) a r e e q u i v a l e n t . An e q u i v a l e n t formalism i s as f o l l o w s :

Time-dependent variational inequalities

410

(CHAP. 6 )

We p u t :

and c o n s i d e r t h e problem d e r i v e d from ( 2 . 9 ) by i n t e g r a t i o n :

I (2*12)

I 1;

find u e X 0

( d ,u

- u) dr

such t h a t

+

I t can be shown without d i f f i c u l t y t h a t ( 2 . 1 2 ) i m p l i e s ( 2 . 9 ) . 2.1.2

Example I1

I n t h e p r e v i o u s example we i n t r o d u c e d a time-dependent inequali t y r e l a t i n g t o a convex s e t . I n t h e example which f o l l o w s we s h a l l i n t r o d u c e a time-dependent i n e q u a l i t y which i n v o l v e s a nondifferentiable term. This i n e q u a l i t y w i l l b e o b t a i n e d from t h e time-dependent f o r m u l a t i o n o f a thermal control problem ( c f . Duvaut-Lions 111, Chapter 1, 2 . 3 . 1 ) . . The n o t a t i o n w i l l be a s f o l l o w s : Q = domain i n s i d e t h e enclosure, f =boundary o f Q, u(x, 1) = temperature a t x E Q a t t h e i n s t a n t 1. A i r - c o n d i t i o n i n g u n i t s a r e p r e s e n t whose f u n c t i o n i s t o i n j e c t h e a t f l u x e s a c r o s s f when t h e temperature u(x,t), x e f , l i e s o u t s i d e an i n t e r v a l [hl, h,]. When u(x, t ) 9 [hl, h,l,

x Ef ,

w e assume t h a t we can produce a h e a t f l u x whose v a l u e i s p r o p o r t i o n a l t o t h e d i f f e r e n c e between u(x,t), x e f and t h e n e a r e r o f t h e two numbers hl, h, The problem is formulated as f o l l o w s : f i n d u(x,t) f o r r e [0, TI, x e Q such t h a t :

.

(2.13)

au - Au = f at

for ( x , t ) e Q = Q x ] O , T [ ,

w i t h t h e boundary conditions

where &u) i s a f u n c t i o n , which i s n o t d i f f e r e n t i a b l e everywhere, d e f i n e d by:

Parabolic i n e q u a l i t i e s of t y p e 1

(SEC. 2 )

gl(A - h , ) (2.15)

&A)

=

gz(l

- h2)

if

1
if

h,

if

A 2 hz,

411

< A < hz ,

with 91<0<92,

hIdh2,

and the i n i t i a l c o n d i t i o n (2.16)

u(x, 0) = uo(x) = given initial temperature, x E a

.

This problem can be written in the form of a time-dependent To this end we introduce (2.5), (2.61, (2.7) and

variational inequality.

and we consider the problem

I (2.18)

find u(l) E V, I traversing the interval [0, TI, such that

(g,

u - u)

+ a(u, v - u) - ( J u - u) + j ( u ) - j ( u ) 2 0 ,

V U E V , with

u(O)=u,.

Applying Green's formula in (2.18) we obtain (2.19)

(g -

du

-Lo - u )

+

Ir

g ( u - u)dT+

Using this inequality it is possible to show that the problems (2.1?,)-(2.16) and (2.17), (2.18)are equivalent, these problems themselves being equivalent to the following: find

IoT

U E

Wo(O,T ) such that

(u', u - U) dr

(2.20)

+ 2.2

+

a(u, u

- u) dr

-

JOT

(J u

- u) dr +

[ j ( u ) - j ( u ) ] dr 2 0 , Vu E W(0, T).

Abstract formulation

Throughout the rest of the discussion of parabolic inequalities

Time-dependent variationa2 inequalities

412

( CHAP. 6 )

o f t y p e I , we s h a l l u s e t h e f o l l o w i n g s e t t i n g and n o t a t i o n :

(2.21)

V,V',H,ll lI,((J),l

(2.22)

K,

(2.23)

uo E K,

(2.24)

A

(2.25)

(Au, u) 2 a 11 u 1 1 2 ,

I,(,)

d e f i n e d as i n (1.11, ( 1 . 2 1 , ( 1 . 3 )

non-empty convex s e t o f V

E

given i n i t i a l condition

U ( V ; V') w i t h

j : K+R,#

ri c o n s t a n t

f a o n K, 1 . s . c . on K a n d i n t e g r a b l e

(2.27) f o r a l l u E L2(0, T ; K) i.e.

(2.28)

>0.

lIoT I j(u)dt

<

+

00,

f E L 2 ( 0 , T ; V'), ;e = { u l v ~ W ( 0 , T ) , u ( t ) ~ K ,a.e. = { u I u E Wo(O, T ) , u ( t ) E K, a.e.

(2.29)

.fo

rE[O,r]}, rE[O,

r]}.

The g e n e r a l problem i n v e s t i g a t e d i s

I (2.30)

1 J:

f i n d U ' E X ~such t h a t

(u'+Au-f, u-u) d t +

JOT

[j(u)-j(u)] dt 2 0 , Vu E X ,

or i n a n e q u i v a l e n t l o c a l ( o r i n s t a n t a n e o u s ) form: f i n d u E T o such t h a t (u'+Au-S, u-u)+j(u)-j(u) 2 0 , VUE K,

a.e.

r E [O, TI

A s o l u t i o n u o f ( 2 . 3 0 ) or ( 2 . 3 1 ) , when it e x i s t s , i s c a l l e d a

strong s o l u t i o n . The f o l l o w i n g r e s u l t s can b e proved:

Under the conditions ( 2 . 2 1 ) , . . . ,( 2.29) and under Theorem 2.1. the supplementary asswnptions = %E L2(0, T ; V') ,

(2.32)

J'

(2.33)

j(u) = 0 ,

(2.34)

f ( 0 ) - AuOEH,

413

Parabolic i n e q u a l i t i e s of type 1

(SEC. 2)

the problem (2.31) ( o r (2.30)) admits a unique solution over t h i s s o l u t i o n s a t i s f i e s u, u' E L2(0, T ; V) n Lm(O,T ; H).

u.

More-

Remark 2.1. The results of the above theorem are still valid when J(u) f Oprovided assumption ( 2.34) is replaced by an assumption See Duvaut-Lions of the same type involving the value of j(uo) . 111, Chapter 1, Section 5.2 and Br6zis 111, 121. Remark 2.2. The numerical methods described below allow us to prove this theorem. Remark 2.3. Because of the assumptions necessary for the existence of the solution of (2.30) or (2.31), we are led to consider a "weak" formulation of problems (2.30), (2.31). To this end, we define X , = { U ~ U E L ~ ( O , T ; V ) , ~ ( a.e. ~)EK ~ E, [ O , T ] } , and consider the problem

I

find UEX'

such that

(d+AU--f,v-u)df+

[ j ( ~ ) - j ( ~ ) ] d t2 0 , V U C X o .

It can be shown without difficulty that any solution of (2.30) is a solution of (2.35). However the converse is not t r u e , unless supplementary assumptions are made (for example (2.32)-(2.34)). A solution u of (2.35) is termed a weak solution of the problem. We have: Theorem 2.2. With conditions (2.21), (2.35) possesses a unique weak solution.

...,(2.29) the

problem

Remark 2.4. When j ( u ) is differentiable everywhere - and convex - with gradient j'(u), then in the inequalities (2.30) and (2.35) we can replace the term

IoT

Ci(4 - A41 dt

by

and in (2.31) the term

Remark 2.5.

We assume here that A, j do not depend on For problems in which K depends on 1 , see Br6zis /4/, Attouch and Damlamian 111, J. Moreau / h / and J.C. Peralba 111. This assumption is made solely to avoid encumbering the investigation with an excessive number of assumptions. f E [ O , T]

.

414

Time-dependent variational i n e q u a l i t i e s

( CHAP.

6)

When K = V and when j(u) is differentiable and Remark 2 . 6 . convex, the inequalities reduce t o equations; for example ( 2 . 3 1 ) becomes find u ' + Au

U E

Wo(O, T ) such that

- f +j'(u)

= O , X E ~ , a.e. t € [ O ,

TI.

3. APPROXIMATION OF PARABOLIC INEQUALITIES OF TYPE I Synopsis. In this section we shall introduce formalisms for the nwnerica2 solution of the inequality (2.30). For this we shall define a family of inequalities "close" to (3.1)

loT@' +

Au - X u

- u)dt +

vu E x , u E xo ,

16

[j(u)

-j(u)]dt 2 0 ,

or to the equivalent instantaneous inequality: (3.2)

(Ir+Au-f,u-u)+j(u)-j(u),O, V U E K , U Q ; Y ~a.e. , te[O,T],

and for which it will be possible to calculate the solutions more easily. The solutions of these inequalities will be sought in spaces L2(0,T ;vh), v h being finite-dimensional. Denoting one of these solutions by uhwe shall show that when the dimension of the spaces vhtends to infinity we have uh 4 u weakly in L2(0, T;V)

where u is the "weak1'solution of (3.3)

[I(u'

+ Au - X u

- u)dt

VU€JYo,UE xf,

+

I:

[j(u) -j(u)]dt

20,

and, in the-casewhen (3.1) possesses a unique strong solution u we shall show that uh + u strongly in L2(0,T;V)

In general, the approximation of time-dependent problems consists of

-

a spatial discretisation (discretisation of a ) ;

(SEC. 3 )

Approximation of inequalities of type I

- a time discretisation (discretisation of

415

[0, T] ) ;

- an approximation of the space V by a space vhof functions allowing each element of V t o be suitably approximated, and the investigation of the way in which Vhapproximates H ( ' ) ;

K c V b y a convex set Kh of Vhsuch that with each element of K we can associate a sufficiently "close" element taken in Kh ;

- an approximation of the convex set

- an approximation of the inequality (3.1) by an inequality which is compatible with the other characteristics of the approximation. In the following section we shall examine these points and give the fundamental assumptions leading to the formulation of problems which are nwnericaZZy close to the general problem (3.2). Fundamental assumptions for the approximation

3.1

As in Chapter 1, Section 5, we shall first "axiomatise" the approximation assumptions. For the moment we shall restrict our attention to exterior approximations (see Chapter 1, Section 5.2); the theory which follows is valid (and in fact even a little simpler) in the case of interior approximations (of the finite element type). Approximation of V and H

3.1.1

We consider V,H , 11 11, I 1 as in Section 1.2 of the present chapter, and then introduce a simultaneous exterior approximation of V a n d H, using the following:

[,I0

(3.4)

Cg,

(3.5)

F = H x Cg equipped with the inner product and the product norm ( 2 )

(3.6)

n : F + H , first canonical projection

(3.7) (3.8)

i Z : F -P Cg, second canonical projection u : V + F, extension isomorphism from v onto UV c F with:

(3.9)

nu

(3.10)

h,

'( '1 ( 2 )

a Hilbert space, with inner product norm Ie.9

=

and with

identity

parameter which tends to zero.

This is one of the elements which are new compared with the steady-state case. For simplification we shall not introduce the space the "extension" of H (see C6a /3/, Raviart /l/).

416

( CHAP. 6 )

Time-dependent variational i n e q m Z i t i e s

(3.11)

v,,,

a H i l b e r t space with i n n e r product ((,))h a n d norm ( i n g e n e r a l t h i s s p a c e w i l l be o f f i n i t e II Ilk = II dimension N(h) w i t h N(h) + 03, when h + 0 ) .

+

a l i n e a r o p e r a t o r from Vh+ F c a l l e d t h e e x t e n s i o n o f Vh t o F, s a t i s f y i n g a s t a b i l i t y c o n d i t i o n :

(3.12)

h ,

(3.13)

11 p h t+, IIF

Q

C )I o h

C a c o n s t a n t i n d e p e n d e n t o f h, vuh E

v h

,

and a convergence c o n d i t i o n ( s e e ( 5 . 2 ’ 0 , C h a p t e r 1 ) : (3.14)

Vo E V ,

with

3uhE v h

P h oh +

m strongly i n

F a n d 11 uh [Ih

Q

c.

I n a d d i t i o n w e have (3.15)

qh, a l i n e a r o p e r a t o r from q h = nPh.

v h +

H,

i n j e c t i v e such t h a t

We d e f i n e a second inner product and a second norm on

I uh

and

(uh, wh)h

Ih

9

which “approximate” t h e i n n e r p r o d u c t (3.16)

I

(uh? Wk)h

=

I uh

I q h uh I

Ih

=

( q h uh, q h wh)

and we assume t h a t (3.17)

1 o h lh

Q

C (1

(,) a n d t h e norm

II

on H, by

9

9

VUhE v h

oh llh

v h ,

:

( i n t h e same way t h a t

I I d C I1 I1 1 ;

and t h a t

The d e f i n i t i o n o f v h a n d t h e a p p r o x i m a t i o n i f i e d a t t h e time of a p p l i c a t i o n .

Typical example.

vh

of

w i l l b e spec-

I n t h e s e t t i n g o f Chapter 1, S e c t i o n 5.1,

V = H;(Q), H = L2(Q),@ = (L’(Q))”,

f = {fo, 4 } E H

a h

x @ : nf = fo,

i s g i v e n by ( 5 . 1 0 ) , Chapter 1:

Zf = 4

;

Approximation of i n e q u a l i t i e s of t y p e I

(SEC. 3 )

417

It can then be verified that we have (3.18) with: L

S(h) = -

Ihl'

3.1.2

8

D i s c r e t i s a t i o n of time and of the d i f f e r e n t i a l operator

We divide [O, into N = N ( k ) intervals of length k, where k tends to zero as N ( k ) + 00. We seek an "approximation" to the solution of the inequality (3.1) in the class of step functions on [ O , T ] with values in v,+. Let U ~ V,, E i = 0, ..., N and let &t) be the characteristic function of [ik, ( i 1) k [ . We put ( ] )

+

+

N

On the space of these functions we approximate the operator

o r by

Remark 3.1. We could also approximate u + u' by discretisation operators of second order, third order, etc... To second order we would put:

3.1.3

Approximation of S a n d of the i n i t i a l condition

We adopt take Khas a We shall for example

( 2 . 2 2 ) , ( 2 . 2 9 ) for the definition of K and X . We closed convex set of V,. make use of the following easily-proved results, see Tr6moliDres 141:

Lemma 3.1 (consistency). we a s s m e that (Chapter 1, ( 5 . 3 3 ) ) .

Let

('1

0k.k

w i t h h, k

+ 0,

(K,,K} i s c o n s i s t e n t

be a sequence of f u n c t i o n s such t h a t :

If necessary, we may define uk&) from 0 to N + 1.

over

[o, T + k ]

by summing

Time-dependent v a r i a t i o n a l i n e q u a l i t i e s

L2(0, T ; K )

1

6)

We asswne t h a t { Kh,K ) i s convergent

Lemma 3.2 (convergence). (Chapter 1, (5.34)). Then: VU E

( CHAP.

3uh,k E

L '(0, T ; K&

(where h i s not

connected w i t h k ) such t h a t q h Uh,k -, u

s t r o n g l y i n L'(0, T ; H )

strong19 i n L'(0, T ; F) and i f i n a d d i t i o n u' E L2(0,T ; V ' ) t h e n

p h 0h.k

q h 6 u h . k -, u'

--t

uu

s t r o n g l y i n L'(0, T ; V').

Moreover we assume that the given initial condition u , ~ K cV can be approximated by a sequence uOshE Kh(h + 0) such that (3.21)

P h U 0 . h -, uu0

11 U 0 . h

(3.22) 3.1.4

< C,'

strongly in F

C constant, independent of k .

Approximation o f

a(u, u)

Given a(u, u) = (Au, u) as in ( 2.24)-( 2.26) we take = ( A h l(h, uh)h satisfying assumptions ( 5 - 2 8 ) - ( 5-31) of Chapter 1, and Ah E 9( v h ; Vh). The following results can be verified without difficulty: a h ( u h , Uh)

Lemma 3.3 (consistency). p h Uh,k -, uu

Given uhk(t)6 Kb a.e.

t E

[0, TJ

such that

weakly i n L2(0, T ; F)

then i f t h e p a i r { a h ( A a ( , ) } i s c o n s i s t e n t (Chapter 1 , (5.31)) we have :

(SEC. 3 )

Approximation o f inequalities of type I

Lemma 3.4 ( convergence).

3.1.5

Approximation of

Given

u E L’(0, T ; K), w E L’(0, T ; K )

419 and

j(u)

With j(u) defined as in (2.27)we take a family of approximations of j by convex 1.s.c. functions j h which are integrable for all u h E L 2 ( 0 , T ; K h ) , and which satisfy (5.35), (5.36) of Chapter 1. The fdllowing results can then be proved without difficulty:

i.

Time-dependent variationa2 i n e q u a l i t i e s

420

3.1.6

( CHAP.

6)

Approximation of f ( t )

We take f ~ L ' ( 0 T , ; V ? and a family of functions h , k ~ L Z ( T; O , Vh) such that

l+h,k(t)constant in (3.26)

[ik, (i

+ 1) k] ,

11 &(I) 1;

i

=

t over each interval

0, ..., N ,with

< C , constant independent of

h and k ,

N

where x: is the characteristic function of [ik, (i

+ I ) k [ with

and fh(t) approximates f ( t ) as in Chapter 1, Section 5.2; alternatively, iff is continuous and belongs to %'(O,T;H), we can take (3.27 ' )

fi = f,(ik);

we can also take fi = f ( ( i - 1)k) o r f ( ( i - ))k); illustrations which follow will be given with

fi = h ( i k ) . 3.2

the numerical

m

Approximation schemes for parabolic inequalities of type I

In this section we shall give the principal ideas behind the approximation of the general parabolic inequality given in (3.2). We recall this inequality: find (U'

U E Xo

such that, a.e. t E (0, TI,

+ AU - f, u - U) + j ( ~-) j ( ~2) 0 ,

VU E K .

Approarimation of i n e q u a l i t i e s of type I

(SEC. 3 )

4 21

N

find

Uh,k

1 ULXi

=

such t h a t

i=O

(3.29)

u: = u ~ ,a~n d, f o r a l l (zuh.k vuh

+ Ah

E Kh

find

9

uh.k

- fh,k,

uh.k(f) E

u:,

uh 9

f E

[k, r] such t h a t

- Uh.bh + jdUh)- jh(uh.k) 2 [o, TI .

vt E

i = 0, 1, ..., N , w i t h u: = UO,h,

such t h a t f o r i = 0, ..., N ,

T h i s scheme i s termed f u l l y " i m p l i c i t " b e c a u s e of t h e t e r m wliich a p p e a r s i n t h e i n e q u a l i t y . This term o f t e n gives r i s e t o a somewhat i n t r i c a t e n u m e r i c a l s o l u t i o n a n d f o r c e r t a i n problems '(as we s h a l l s e e i n t h e n u m e r i c a l a p p l i c a t i o n s ) it can be u s e f u l t o r e p l a c e it by t h e term A k u L , i n which c a s e we t a l k o f a n " e x p l i c i t " scheme ( I ) . We n o t e t h a t ( 3 . 3 0 ) c a n b e w r i t t e n : AhU;"

(3.31)

((Ilk 4- Ah)

Ur'

- dJk - f;",

+ jh(uh) - jh(up ') 2 0

- ur')h

+

and, w i t h t h e r e s e r v a t i o n t h a t A h b e symmetric, assuming ui t o b e known (which i s t r u e f o r u i ) , we o b t a i n as a s o l u t i o n o f t h e minimisation problem:

ur'

With t h e a i m o f simplifyingasornewhat t h e i n v e s t i g a t i o n o f t h e i m p l i c i t a n d e x p l i c i t schemes we s h a l l assume t h a t j = O , and i n o r d e r t o a v o i d h a v i n g t o r e p e a t v e r y s i m i l a r p r o o f s we i n t r o d u c e "scheme 1" :

(')

I n f a c t t h i s i s a misnomer: t h e c a l c u l a t i o n o f assumes t h e s o l u t i o n o f a v a r i a t i o n a l inequality with respect t o t h e i d e n t i t y o p e r a t o r i n 4 , i . e . t h e c a l c u l a t i o n of t h e projSee a l s o (3.34). e c t i o n PI., on & i n t h e s e n s e o f (,&

.

(CHAP. 6 )

The-dependent variationul inequalities

422

where, i n ( 3.33) 8 i s fixed i n [0,1]. For 8 = I(resp. 8 = 0 ) t h i s i s an i m p l i c i t scheme ( r e s p . e x p l i c i t ) . With 8 = $ and i n t h e c a s e when K,, = v h , t h i s i s t h e c l a s s i c a l so-called Crank-Nicholson scheme. Another category o f scheme i s given by scheme I1 Remark 3.2. (which i s a "decoupled" scheme, as opposed t o scheme I which i s not decoupled)

.

I

N

find :

=

<'I2

i=o U:

=

- u:

k u;"

N

and ukh =

u i x i with i- 0

such t h a t

UO,h,

C ' / 2

(3.34)

xi

+ Ah(8- 4 + Bc I& 'I1 + 8+ 4+')- fi+'

= 0,

and

= PKk[4+1'2] ( p r o j e c t i o n on K h i n t h e sense of t h e

norm (&,) with 8-,P,8+ 2 0 , 8-

+ 6" + 8+ = 1 .

I t can be v e r i f i e d t h a t t h i s scheme i n c l u d e s scheme I (nondecoupled) a s a s p e c i a l c a s e ( p u t 6" = 0 and it can be seen t h a t 'm i f YhA s a t i s f i e s ( 3 . 3 4 ) t h e n it a l s o s a t i s f i e s ( 3 . 3 3 ) )

.

3.3

Convergence of t h e approximate i n e q u a l i t i e s

I n t h i s s e c t i o n w e s h a l l i n v e s t i g a t e t h e complete approximation o f scheme I ( f o r t h e c a s e i n which j $ 0 and f o r t h e i n v e s t i g a t i o n of ( 3 . 3 4 ) , w e r e f e r t h e r e a d e r t o Tr6moliSres / 4 / ) . For convenience we r e c a l l t h e "continuous" i n e q u a l i t y and t h e "discrete" inequality:

(3.35)

I

(u'+Au-f,o-u)20, VVEK, U E X O ,'

a.e. ~ E [ O , T ] ,

20,

+AhUr'-fkl+',Vh-U~'

(3.36)

>h

vvhEKh,

C' E K,, ,

i = 0, 1, ..., N , with u: = uo,h and

Approximation of i n e q u u l i t i e s of type I

(SEC. 3 )

423

We s h a l l show t h a t f o r s u i t a b l e v a l u e s of h and k, t h e s o l u t i o n N

of ( 3 . 3 6 ) can be considered a s r e a l i s i n g a n approximation of t h e s o l u t i o n .I( of t h e i n e q u a l i t y ( 3 . 3 5 ) The proof t a k e s p l a c e i n f o u r s t a g e s :

.

-

d e f i n i t i o n of t h e "weak" d i s c r e t e i n e q u a l i t y ,

- i n v e s t i g a t i o n of t h e s t a b i l i t y ,

- weak convergence, - s t r o n g convergence. F i r s t we note t h a t t h e e x i s t e n c e and uniqueness of ux" a r e immediate consequences o f t h e c o e r c i v i t y o f t h e o p e r a t o r ( I n the ( s e e Chapter 1, Theorems 2 . 1 and 2 . 2 ) (f/k+OA,,), VBc [0,1] c a s e when O = O t h e c o e r c i v i t y must be considered with t h e norm 1 Ih 3 l/s(h) 11 IIhr s(h)E 10, + a[, Vh.1

.

3.3.1

- Definition of the "weak" discrete inequality

We havC : N

Lemma 3.5

Let

0h.k

=

1 u; x:, i=o

N

uh,k

=

C u: xi.

i=o

Then

Time-dependent variational i n e q u a l i t i e s

424

(CHAP. 6 )

and use t h e c l a s s i c a l r e l a t i o n (3.41)

(U

- b , ~ =) f I u 1' - f I b 1'

By summing ( 3 . 3 8 ) from

+ 4 1 u - b 1'.

i = 0 to N

-

1 we obtain

and hence w e deduce (3.40) by s u b s t i t u t i n g t h i s upper bound i n (3.39).

Remark 3 . 3 . ( 3.39) t h a t (3 * 42)

5

(&.k

I n t h e c a s e i n which V ~ , ~ ( O=) uhc(0) w e deduce from

+ Ah 4 . k - h . k ,

h.k

- uh.k)h

dt

2

.

StabiZity investigation

3.3.2

We s h a l l show t h a t t h e s o l u t i o n ukk defined by ( 3 . 3 7 ) as t h e (unique) s o l u t i o n of (3.36) remains, with h and k p o s s i b l y s u b j e c t t o s t a b i l i t y c o n d i t i o n s , w i t h i n bounded s e t s . ( I n o r d e r t o f a c i l With u = 0 i n (3.36) i t a t e t h e proof w e s h a l l assume t h a t O E K ) and o m i t t i n g t h e s u f f i x h f o r t h e time being, w e o b t a i n :

.

1

.

-(U'+l

k

- u',d + 1 ) + U(Ui+@, U i + l ) < y . i + l ,

By using r e l a t i o n ( 3 . 4 1 ) w e o b t a i n

and

Ui+l).

Approximation of i n e q u a l i t i e s of type I

(SEC. 3 ) so that

(3.43) gives

425

(l)

I u'+' I2 - I u' I2 + 1 u'+' - u' 12

+ 2 k[ea(u'+') + (1 - e) a(u')] Q

2k[BCfi+l.ui+l) + (1 - e ) c / i + l , u i ) + + ( I - e)cfi+',u'+i - 4 3 - 2 4 1 - e)a(u',u'+i - uy

Q

However, from the classical relation (3.43')

1

2(a,b)<-(aI2 &

+E(bI2,

VE>O,

we have the upper bounds

Ui+',u'+')

Q

I( f"' (I* (1 ui+' (I Q

1 Q -1 I f"' 1: 2a

+ z1 a ( u ' + ' ) ,

(f'+',u'+' - u') Q 1I f"' [I* 11 u'+'

(6

= a),

- u' (1

Q (E

2E

> 0 arbitrary),

and substituting in (3.43) we obtain VE > 0 : (3.44)

kS(h)z(l-e)

Iu'+lI2--lu'lZ+

x Q

1 u*+1- u' 12 + (1 - e)ka(u') + eka(u'+') Ck II f'+' 1.:

Q

1

x

We then introduce the following assumption, termed the s t a b i l i t y assumption , (3.45)

1

2 c2 - -kS(h)Z(l a

-

e) >/ /3 > 0 ,

Vh,k.

(The assumption is clearly redwidant when 8 = 1 ) . ( 3.44) then gives (reintroducing the suffix h )

Time-dependent variational i n e q m l i t i e s

426

With assumption ( 3 . 2 6 ) on to n
I l 4 + I ;1

+p

I

- u:,;1

i=o

We hence deduce t h a t ,

(3.46) (3.47)

h.Lwe

Vh, k

+

s a t i s f y i n g (3.45) :

L'(0, T ; F),

qhuo remains i n a bounded s e t o f

Lm(O,T ; H ) ,

N-

1

I ur'

-

4 :1 < C ,

6)

o b t a i n , by summing from i = O

ph uh,&remains i n a bounded s e t of

(3.48)

(CHAP.

fixed constant.

8

i=O

3.3.3

Weak convergence

Let u e g and uhL be as i n Lemma 3.2. (3.36) we o b t a i n

By s u b s t i t u t i n g

into

where we r e c a l l t h a t :

However, from (3.461, ( 3 . 4 7 ) and Lemma 3.1 ( c o n s i s t e n c y ) w e can e x t r a c t from u ~ a, ~ subsequence a g a i n denoted by u,, such t h a t qr 11h.k

4

u

ph 4 . k + au

weakly" i n LQ(O,T ; H ) weakly i n L2(0, T ; F)

with u c K . Proceeding t o t h e l i m i t i n ( 3 . 4 9 ) we o b t a i n :

Approximation o f inequalities o f type I

(SEC. 3 )

It remains to show that the right-hand side is 3 0 . We have the upper bound:

By using (3.461, (3.48) and the convergence assumption (redundant when 0 = 1 ) :

we obtain

and hence IOT(u'

+ Au - A U - u ) d t 3 0 ,

VUE Xo,

and u is indeed a "weak" solution ( l ) . 3.3.4

U E

4,

w

Strong convergence

We shall now show that u ~ "approximatesn , ~ the (strong) solution of the inequality (3.35). To this end we assume first that the existence and uniqueness conditions for this solution are realised. W e then know (see Remark 2.2) that the inequalities: (I)

We note that, in contrast to the case of equations, s t a b i l i t y

does not necessarily imply convergence.

Time-dependent variational inequalities

428

(3.52)

( CHAP.

6)

~T(ut+A~-/.u-u)dtZOV . ue X , u e X 0 . 0

(3.53)

l:(uf

+ Au - f , u

- u)dt 2

0, Vue Xo,

U E $1,

are e q u i v a l e n t , t h a t i s , t h a t t h e s o l u t i o n u e x , o f ( 3 . 5 3 ) i s i n We t h e r e f o r e know f a c t such t h a t u e Xo and u s a t i s f i e s ( 3 . 5 2 ) . t h a t phuhekt e n d s weakly towards bu i n L’(0, T ;F) w i t h u a strong I n f a c t it may b e shown t h a t p h ~ h , tke n d s s o h t i o n of ( 3 . 5 2 ) . For t h i s w e c o n s i d e r a seqstrongly towards bu i n L2(0, T ;F) uence E,,, which approximates u as i n Lemma 3 . 2 and w e s h a l l e v a l uate

.

and show t h a t t h i s q u a n t i t y t e n d s t o z e r o . From Lemma 3 . 5 , t a k i n g account o f Remark 3.3 and p u t t i n g uhSk= 2h,k i n i n e q u a l i t y ( 3 . 4 2 ) , we o b t a i n

However (3

.Now

(SEC. 3)

Approximation of inequaZities of type I

429

o r a l t e r n a t i v e l y , using (3.57),

We s h a l l now show t h a t

t e n d s towards z e r o .

We o b t a i n

and hence

By u s i n g ( 3 . 4 8 ) , t o g e t h e r w i t h t h e f a c t t h a t P h lUhL and ph uhsk remain bounded i n L z ( O , T ; F ) , a n d by imposing t h e convergence condi t i o n ( 3 . 5 1 ) , we o b t a i n zh,k

Consequently JhT (J2h,k

. xh,k +

+ Ah

0 if

Gh.k

- h.k,'h.k

- 'h.k)h

dl

-+

9

Time-dependent variationa l inequa Z i t i e s

430

(CHAP. 6)

which follows immediately by v i r t u e of ( 3 . 4 6 ) , ( 3 . 4 7 ) , t h e properti e s given i n Lemma 3.2 f o r 2h.k , t h e r e s u l t s of Lemma 3.4 f o r Ahiih,k and t h e assumptions ( 3 . 2 6 ) , ( 3 . 2 7 ) on fk. Then x,.,+o and we can deduce from t h e uniform c o e r c i v i t y o f A h i n h t h a t Ph(uh,k

- r?k.&

0

+

9

and t h u s s t r o n g l y i n L’(0, T ; F ) .

ph uh,k+ uu

The s e t o f r e s u l t s obtained enables us t o s t a t e : Theorem 3 .l. For asswnption (1 - 0) kS(h)’

11, h, k + 0, and with the supplementary (redundant i f 8 = 1 ) ,

O E [O, +0

N

the solutions

uh,k

=

1 ui x:

of scheme I (3.33) s a t i s f y :

i=O

q h Uh.k

+

p h uh,k +

u

weakly * i n L“(0, T ; H )

cu

weakly i n

L’(0, T ; F )

where u is the unique (weak) solution of . ( 3 . 5 3 ) . When t h i s solution is the unique strong solution o f (3.52) we have P h uh,k

+

UU

Strongly i n L2(o, T ; F ) .

8

Remark 3.4. When h and k a r e a b l e t o tend t o zero without it being necessary t h a t kS(h)’ + 0 we say t h a t t h e scheme i s unconditThis i s t h u s t h e c a s e h e r e f o r ionally s t a b l e and convergent. e = 1. I n t h e o t h e r c a s e s (0 Q 8 < 1) t h e scheme i s s a i d t o be conditionally s t a b l e and convergent. 8 I t i s shown i n Tr6moliEres 141 t h a t scheme I1 Remark 3.5. given i n Remark 3.2 i s conditionally s t a b l e and convergent u n l e s s

(3.61)

e+ = 1 ,

e-

= 0,

e*

=0,

and unless

(3.62)

8+ 3 4,

8- = 0 ,

and

Kh = V,, K = V

I n t h i s l a t t e r c a s e t h e problem t o be solved, i f j i s d i f f e r e n t i a b l e , i s a v a r i a t i o n a l equation. Under assumptions (3.61) o r ( 3 . 6 2 ) , scheme I1 i s unconditionally s t a b l e and convergent. I n p a r t i c u l a r w e recover t h e r e s u l t t h a t Crank-Nicholson schemes a r e unconditionally stable and convergent ( s e e Raviart 111, T6mam /3/). 8

Remark 3.6. The r e s u l t s o f S e c t i o n 3.3.3 e s t a b l i s h t h e e x i s t ence of a weak s o l u t i o n . 8

(SEC. 4)

4.

Numerical solution of inequalities of type I

431

NUMERICAL SOLUTION OF SOME PARABOLIC INEQUALITIES OF TYPE I

In this section we propose to solve three examples o f parabolic inequalities of type I. These examples will allow us to assess the relative merits of explicit schemes (scheme I, with O = 0), schemes which are "implicit in projection" (scheme I, with 0 = I ) , and schemes which are "implicit in equation" (scheme 11, with 0* = 1, 0- = 0' = 0 1 . At the same time we shall investigate the sensitivity of the schemes to the quantities h,kand the ratio klh' 5 kS(h)'. We shall carry out this investigation on: - A model problem using the convex sets studied in Chapter

3;

- A friction problem;

- A problem of the deformation o f a membrane. 4.1

A model problem

Consider the parabolic inequality (4.1)

(u'-Au-f,u-u)>O,

V U E K , ue.XO,

a.e. t E [ o , T ] ,

with:

.%o = { u I u E Wo(O,T), u(t) E K , a . e . t E [0, TI }

,

where d(x,r) is the Euclidian distance from X E Q to the boundary o f 61. We take a = ]0,1[ and the time interval [0,1] It can be shown that for

.

f(x,

f) =

i

100 i f x < t / 2 , o r i f x > l - t / 2 , - [l/(t - l)'] (x' - x - t(t - 2)/4) - 2/(t if t/2 < x < 1 - t/2,

and the initial condition uo(x) = x(l - x) , the problem has the exact solution (see Figure

u(x, t) =

i

- 1)

4.1):

x if O < x < t / 2 ,

+

[l/(t - I)] [x2 - x t2/4] i f t/2 < x < 1 1 - x if l - t / 2 < x < l .

- t/2,

r

Time-dependent variationaZ inequazities

432

Flg. 4.1.

(CHAP.

6)

Evolution of the solution of the elasto-plastic problem.

We discretise 8 = lo, 1( into M + 1 intervals of length h ’ and the time interval [0, TI= 1 into N intervals of length k. We shall seek the discrete solution in the following form ( 6 being the characteristic function of the element w:(ih)) :

c u; ei,

Y+l

Uh.l;U)

=

3

t E [nk,(n

I=O

+ 1) k[

9

and we shall evaluate the quality of the solution found at the instant nk by means of ( * ) :

We take up = ih(1

(4.2)

- ih),

fl = f ( i h , n k ) ,

+ 1, +1

i = 0, ..., M i = 0,...,

and

n = 0,..., N ,

d,=min{ih,M+l-ih}.

All the results relating to problem (4.1)were obtained on a BULL-GE-265 and the times are quoted in sixths of a second, including compilation time.

(*ITmnsZator ‘ s note:

The original French terminology, retained here, means ”deviation”.

(SEC. 4)

4.1.1

Nwnerical solution of i n e q u a l i t i e s of type I

433

Investigation of the e x p l i c i t scheme

The scheme c a n b e w r i t t e n , f o r n = 0,

..., N - 1

i = l , ..., M-1 ,

&+'=max

(-dM,

min

and c l e a r l y , w i t h t h e i n i t i a l c o n d i t i o n ( 4 . 2 ) a n d , f o r a l l n , Po =

=

0 .

The r e s u l t s a r e g i v e n i n T a b l e 4 . 1 .

(4.3)

k

G+'P - h2 (q:j/'

- 2 fl+'I2

+ G?;") = @ + kfl+' ,

i = 1, ..., M , (4.4)

u?+' = m a ~ ( - d ~ , r n i n ( @ + ' / ~ , d ~i )=) , 1 ,..., M ,

w i t h $ g i v e n by ( 4 . 2 ) and, for a l l n,u",=u",+,

=O.

For each n, n = 0, 1, ... we have t o s o l v e t h e system o f e q u a t i o n s ( 4 . 3 ) , which g i v e s i = 1,...., M and we t h e n o b t a i n # + I , i = 1, ...,M using ( 4 . 4 ) . System ( 4 . 3 ) w a s s o l v e d by t h e Gauss-Seidel and The common t e r m i n a t i o n c r i t e r i o n i n c o n j u g a t e g r a d i e n t methods. t h e s e a r c h f o r t h e s o l u t i o n U"+1/2 a t t h e i n s t a n t t = nk by each of t h e s e two methods i s

with

J;(u") =

(I + 2;)

$ - 2fl-l k

k

-

- kfl

where u n - ' i s t h e s o l u t i o n found a t t h e i n s t a n t t = ( t i - I ) & . The c a l c u l a t i o n s w e r e c a r r i e d o u t on a BULL-GE-265, t h e computa t i o n times b e i n g quoted i n s i x t h s o f a second.

1

Values o f h-2 x ECART2 : (exact solution - calculated solution) for v a r i o u s values of t E 10, I]

Characteristics h

k

OOO

t =

k

0.2

0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

I

Solution time

0

0.003 3 0.002 1

0.0048

0.001 8 0.006 5 0.001 0 0.010 8 0.000 3 0.022 8

0

548

1/m

0

0.0035

0.0023

0.0051

0.0020

0.011 0 0.000 4 0.023 2

0

244

1/500

0

0.0036

0.0024

0.005 2 0.002 I

0.007 0 0.001 2 0.01 I 2 0.000 5 0.023 5

0

207

0.0038

0.0026

0.0055

0.0023

0.007 3 0.001 3 0.011 4 O.OOO6

0.004 2 0.003 3

0.0062

0.0030

0.0080 0.001 8 0.0123

1/1

0.006 9 0.001 I

0.023 9

0

170

0.001 0 0.0250

0

128

0.008 5 0.002 2 0.012 8 0.001 2 0.025 0

0

115

1/400

0

1/250

0.000 I

1/200

0.000 1 0.0046

0.003 8

0.0067

0.0034

1/150

0.000 3 0.0058

0.091 6

28.9

1004

35381

lilOO

0.0006 0.1126

84054

5x10'

1010

0.0014

0.0014

1/10

Instability

135

1/1000

0

0.0010

1/800

0

0.001 1 0.001 5

0

0.5630

3500

-

-

-

O.OOO8

3632

-

-

-

-

0.001 3

106

107

lop

10'0

10'2

101

1013

1016

1018

IO*~

1024

95

0.001 2 0.0010

0.0007

0.0004

0.0003

0

617

0.001 6 0.001 5 0.001 4 0.001 I

0.0008

0.0004

0.0003

0

580

-

-

-

-

-

-

-

-

-

-

1/20

Instability 1/100

Table

4.1.

Explicit scheme.

-

-

(SEC. 4)

Numerical solution o f i n e q u a l i t i e s of type I

435

For the Gauss-Seidel method, the values of the “optimal“ parameter wOp are given in Table 4.2 as a function of h and k ( l ) . Figure 4.2 shows the variation of wopcas a function of h and k .

wopt. theo.

Computation time

1/10

1/6W 11400 1 /200 1/150 1/100 1/50 1/10 115 1 12

1.014 58 1.026 54 1.06403 1.087 32 1.127 86 1.21296 1.40481 1.45676 1.496 33

54 52 53 57 73 53 56 55 58

1/20

1/800 1/100 1/10 115

1,06990 1.35260 1.639 48 1.679 12

91 108 112 113

1 /30

1/100 1/10

1.499 55 1 .I4324

109 137

h

k

Table

4.2

The results obtained by the above two methods are recorded in In this table, the figures in the upper triangles Table 4.3. give the numbers of iterations required by the conjugate gradient method to obtain the solution at the corresponding instant t = nk. The figures in the lower triangles give the numbers of iterations required by the Gauss-Seidel method used with optimum parameter (coopt. thee.: see Table 4.2) to obtain the solution at the corresponding instant t = n k . The figures in the last column of the table give, in the upper triangles, the total solution time using the conjugate gradient method, and in the lower triangles, the total solution time using the Gauss-Seidel method.

17

4.1.3

Investigation of the “ i m p l i c i t i n inequality” scheme.

The scheme can be written, for n = 0,

(’1

..., N,

This parameter was obtained by applying the iterative method of Varga /1/ to the unconstrained problem. (See also TrgmoliZres / 4 / and Bourgat-TrCrnoli?res /l/).

F-

w

cn

Conj u g a t e gradient method Gauss-Seidel method with Wept. theo.

v

r.

m

v,

Table

4.3.

"Implicit in equation" scheme.

(See text for explanation of table).

h

(SEC. 4)

Numerical sozution of inequalities of type I

_lo -100 - 200

400

600

800

437

lo00

mg.S.2.

k

? (l4+1-l4--(l4:;-2 h2

u;+'+u;_+;)-kf;"+',ui-u;+l

is1

)

20,

..., uN) with I ui I Q d i , i = 1, ..., M , and rf+' = (u;+', ..., 4 ' ' )such that I fl+' I Q d,, i = 1, ..., M ,

Vu = (ul,

with up given by (4.2) and, for all n, This scheme can be solved at the instant over

t = nk

4 = t&+l

=

0

by minimising

K = { U I~u i I Q d i , i = O , ..., M + 1 }

the function J(u) which has the gradient J'(u) = { J;(u), ..., &(u) } with:

438

Time-dependent variationa2 i n e q u a l i t i e s

(CHAP. 6 )

This minimisation was carried out using the Gauss-Seidel method in projection, with the "optimal" parameter given by Table 4.2 The termination criterion in the search for the solution at (I). the instant t = nk is given by:

with

The results are given in Table 4.4.

/I

k 1 0 - 5 0.003 I 0.001 3 0.0044 0.0012 a0063 0.0006 aoioo 0.0001 0.022 lo-* 0.003 2 0.001 2 0.0046 0.001 1 0.006 5 0.0006 0.0106 0.0002 0.022 lo-' 0.0038 0.0004 0.0055 0.0007 0.0079 0.0006 0.0121 0.0003 0.024

0 0

lo-' 0.0003 0.0005 0.0005 0.0006 0.0005 0.0004 0.0003 0.0701 lo-* 0.0008 0.0010 0.0010 0.0008 0.0006 0.0004 0.0008 0.0001

0

lo-'

lo-' 0.000 5 0.0002 0.000 7 0.0003 0.001 0 0.0003 0.001 8 0.000 I 0.004 3 lo-' 0.0005 0.001 1 0.0005 0.0009 0.0009 0.0003 0.0021 0.0001 0.0046

Table

4.4.

o

0

0 0

"Implicit in inequality" scheme.

Remark 4.1. The "implicit in inequality" scheme has also been solved by a special method, adapted from the conjugate gradient method and using truncation. This method (see Tr6molikes /4/ gives slightly better results than the Gauss-Seidel method; (1)

By "optimal" parameter we mean the theoretical optimum value of the parameter o for the unconstrained case. In fact in the case of inequalities this parameter is in no way optimal. Here we should take the a,,,,, of the Dirichlet problem defined on the domain enclosed by the free boundary, 2.e. the zone in which I u I < d (see Chapter 3, Remark 3.7).

4)

(SEC.

Nwnerical s o l u t i o n of i n e q u a l i t i e s of type I

however it is more awkward to program.

4.1.4

439

rn

Conclusions of 4.1

With regard to the model problem considered here, we have the following conclusions:

E x p l i c i t scheme.

1.

- very simple programming, - importance of a good choice of k and h : k/h2 < const.

- i,

- divergence and overflow if k is too large, - thus, in the absence of the evaluation of the stability c o n s t a n t , this method necessitates "numerical trial and error". "Implicit i n equation" scheme.

2.

- fairly simple programming (when the "projection" does not cause difficulty), - for a given choice of k and h , the scheme appears to be l e s s

accurate and h a l f a s f a s t as the explicit scheme, - the danger of divergence appears to be reduced.

"Implicit i n i n e q u a l i t y " scheme.

3.

- more awkward programming, - very high accuracy ( l ) , and, for a given choice of h and k, the scheme is faster than that above, and, f o r the same rn

p r e c i s i o n , f a s t e r than t h e e x p l i c i t scheme. 4.2

A model problem of time-dependent friction

For t E [0, T], we consider the inequality (4.5)

(u', u

- u)

+ a(u, u - u) + j ( u ) - j(u) 3 ( J u - u) ,

VUEV,

u€Vo,

with

(

Observed experimentally.

Time-dependent variational inequalities

(CHAP. 6 )

O=]O,l[,T = 1 , Y={uluEH'(O), u(x=O=.O)r Yo= { u I u E W(0, T),do) = u0 given E V } ,

1

a(u,v)=

I,

gradugradudx,

= g I u Ix=l , g = 10 I r($ - 1) 1, f = 20 x($ - x) ($ - 2 t ) + 40 t($ - r) .

AV)

This problem is also equivalent to

I

u f - b = f

in O ,

With the above properties and with u o = O the inequality has the exact solution u = 20x(3

- x) t (3

- 1).

This solution is depicted in Figure 4.3 below. This problem was solved by the explicit scheme and by the "implicit in inequality" scheme. The solution of the implicit scheme was carried out for each time iteration by a dual method (see Chapter 2 ) , in view of the nondifferentiable term due to

A4

For the spatial discretisation over ]0,1[we seek uh in the form

where

is the characteristic function of the interval - h/2,1[, with

- 1 and @f is that of [l - 4) h, (i + 4) h[ for i = l?..,M Mthe integer M = l/h. [(i

Befol'e discussing the numerical results, we shall briefly indicate how these schemes can be obtained.

4.2.1

Obtaining the implicit scheme

For convenience we shall first consider the case of semi-disCrete approximations (discretisation in time but not in space). At the time t = nk ( k = 1/N , integer 'N ), with n = 0,1, ..., N, the inequality can be written:

(SEC.

4)

Numerical s o l u t i o n o f i n e q u a l i t i e s o f type I

441

t = 0.3

0

0.5

1

c

- 3Fig.4.3.

V a r i a t i o n of t h e e x a c t s o l u t i o n o f t h e time-dependent f r i c t i o n problem.

442

(CHAP. 6 )

Timedependent variationa Z ineqwz 2 i t i e s

+Jngradun".grad(u-u"+l)dx-

.u-ff+'>

(4.6)

- ( f " + ' , U - u " + l ) +i""(u) -j"+yu"+l)a 0, VUEV, ~"+'EVu , o = u o given.

It is easy to see that this inequality is equivalent to the problem

1 u 1' dx (4.7)

- k(f"",

u)

- (u, u")

+

+ kj"+'(u),

or, because of the special form of j(u) ent to solving

(see Chapter

4),

equival-

We then discretise i n space, so that (dividing by h , noting that the measure o f the last interval is h / 2 , and putting G+'=O):

(4.9)

We then write the conditions for optimality: and p such that:

u;"-u; (4.10)

k

1

- -(u;+': - 2 G + I + u ? + : ) - f l + l hz

i = 1, ..., M (4.11)

&+I

- ffM

k

+ h22

-(&+I

find

= 0,

- 1, -

U&+_l])

- f;"

2 + -pg"+I h

=0,

4",...,u"L'

(SEC.

4)

Numerical solution of i n e q u a l i t i e s o f type I

443

It is then possible to refer to a semi-implicit dual method to define the following algorithm which, for U;: known (recall that is given) , allows us to determine U;:" as follows (we deuf = 0 note by g"(" the ith iterate in the search for u ; " ) : - take any arbitrary

PO,

suitable interval p > O

for example p o ; then

- solve (4.10), (4.11) for - generate p' = P [ p o on [O, 11 Y

+ pg""

ffy+'(oq

- solve (4.10), (4.11), for p - generate

p = po

= pl

=

sign (ffy) x 1 , and a

, which

defines

fly+'('),

where P is the projection (which defines

p z = ~ [ p +' pg"+' fly+'(')], etc

G+'()',)

.

A multiplicity of variants of these schemes may be defined: for example, in (4.9) we can replace

this quantity corresponding to the square of the "quadratic" approximation of the derivative at x = 1 . 4.2.2

Obtaining the f u l l y e x p l i c i t scheme

Although this has not been justified theoretically, we can replace (4.10), (4.11)y(4.12) by

G+'- G -j-++11

(4.13)

-2g+G-')-fl+I

k

i = 1,

(4.15)

..., M - 1 ,

p" = sign (ffM) x 1

4.2.3

=o,

.

Numerical r e s u l t s

With the aim of assessing the merits of these schemes and of the approximation, we define the following quantities:

- the discrete normal derivative of the exact solution at x = l

and t = n k ,

Time-dependen t varia tiona 2 inequalities

444

V,G

I F 1

=

~(1,nk)- 41

h

(CHAP.

6)

- h,nk) ,

- the discrete normal derivative of the approximate solution at x = 1 and at t = n k , - then V= I=

U

=

max ....N = Ilk

1.

max n= 1.

....N

IVN-Vkunklx=l,

1 u(ih, nk) - u,(ih, nk) 1 ,

I = 1.....M

- and finally the difference between the exact solution and the calculated solution at t = nk ,

The times The two schemes were investigated on a BULL-GE-265. are quoted in sixths of a second, including compilation time. Table 4.5 gives the results of the explicit scheme and Table 4.6 those of the implicit scheme. For this scheme equations (4.10), (4.11) were solved by the Gauss-Seidel method with the values oop(.cxp, of the over-relaxation parameter obtained experimentally (see Table 4.7) and the parameter p fixed at 0.1. Table 4.8 shows Uand A in terms of h and k for the explicit and implicit schemes.

4.2.4

Conclusions of 4.2

With regard to the model problem of time-dependent friction considered here, we have:

1.

Fully e x p l i c i t scheme

-

danger of instability

2.

Implicit. scheme

stability appears to correspond to k/h2 Q 112.

- high accuracy, practically independent of h -

and k (but the exact solution is "quadratic"), f o r the same precision, much slower than the explicit scheme.

ECARTl

h

k

r=k

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

The

~

0.004 18 106 lo*] 1016 1021 IO~J 1030 1035 ... ... 1/500 0.OOO 1 0.001 5 0.001 2 O.OOO5 0.001 4 0.0033 0.0056 0.0083 0.011 1 0.0142 0.0174 111 OOO O.OOO045 O.OOO78 O.OOO64 O.OOO25 O.OOO70 0.001 67 0.00284 0.004 15 0.00558 0.007 10 0.00869 1/2 OOO O.OOOO11 O.OOO39 O.OOO32 0.OOO 12 O.OOO35 O.OOO83 0.001 42 0.00207 0.00276 0.00353 0.00434

145 247 463

0.006 5 lo* 1020 1032 ... 1013 1030 ... 1/500 0.0026 U20 1/1 OOO O.OOOO66 O.OOO77 O.OOO64 O.OOO24 O.OOO68 0.001 64 0.00280 0.004 10 0.00552 0.00702 0.00859 1/2 OOO O.OOOO16 O.OOO38 O.OOO32 O.OOO12 O.OOO34 O.OOO82 0.001 40 0.00205 0.00276 0.00351 0.00425

427 802

1/100

Ui0

1/100

]/I00 1/500

...

...

... ...

1/1 OOO 0.000081

1/2 OOO O.OOOO20 O.OOO38 O.OOO32 0,OOO 11 O.OOO33 O.OOO82 0.001 39 0.00204 0.002 74 0.00349 0.00427

Table

4.5.

Time-dependent

friction:

e x p l i c i t scheme

1548

ECARTZ h

k

t = k o . l

0.3

0.2

0.5

0.4

0.6

0.8

0.75 (')

0.9

1.0

Time ~

1/100 ~ O % I O - ~ 81MO-* 1 4 ~ 1 0 - ~19M0-3 26x10-' 2 8 ~ l O - ~31x10-' 3 2 ~ 1 0 - ~3 3 ~ 1 0 - ~3 4 ~ 1 0 - ~ 210 1/10 1/500 36X10-6 16~10-* 29x10-' 39xlO-* 47x10-* 53x10-* 58x10-* 63x10-* 64X1O-* 67x10-* 68x10-* 506 1/1 OOO 91XlO-' 81X10-5 l&lO-* 19xlO-* 23XlO-* 26x10-* 29XlO-* 31x10-* 32x10-* 3k10-* 35x10-* 840 1/100 ,88x10-5 79x10-* 14X10-3 19x10-3 2 2 ~ 1 02~5 ~ 1 0 - 2~ 8 ~ 1 0 - 3ot10-3 3 1 ~ 1 0 - ~3at10-3 3 3 ~ 1 0 - ~ 586 ~ 1/20 1/500 36X10-6 l6xlO-* 2@lO-* 38x10-* *lo-* 52X10-* 5&10-* 61X10-* 63X10-* 6%10-* 67%10-* 1349 ~ l&lO-* 19xlO-* 2?xlO-* 26x10-* 28xlO-* 3(k10-* 31x10-* 3&10-* 33x10-* 2 029 1/1 OOO ~ O X I O - 80X10-5 1/100 82x10-' 78x10-* 14x10-3 18xlO-' 22X10-3 2 5 ~ 1 0 - ~28x10-' 3(k10-' 3 1 ~ 1 0 - ~3 2 ~ 1 0 - ~3 3 ~ 1 0 - ~1183 1/30 1/500 3 6 ~ l O - ~15~10-* 28x10-* 38xlO-* 45x10-* 51x10-* 56x10-* 61x10-* 62xlO-* 6&lO-* 66XlO-* 2786 1/1 OOO S O X ~ O - ~80x10-* 19x10-* 19xlO-* 2klO-* 25x10-* 28xlO-* 3ot10-* 31x10-* 3&10-* 33x10-* 4 333

Table (l)

4.6.

Time-dependent

friction:

i m p l i c i t scheme

For t h i s v a l u e of t t h e s o l u t i o n i s zero and j ( v ) i s " n o n d i f f e r e n t i a b l e " .

h

k "0pl.crp

iterations

I

1/10

1/20

~

1

1 1.4

/

5 1.1 6

0

1/30

~

~~

1/1 0 OOO

1/100

1/500

1/1 OOO

1/100

1/500

1/1 OOO

1.1

1.45 20

1.15 9

1.1 7

1.5 28

1.25 13

1.12 10

5

Table

4.7.

h

a

v

(SEC. 4 )

NwnericaZ sozution o f i n e q u a l i t i e s of type I

1 h

E x p l i c i t scheme

U

k

Divergence and overflow

1/100 1/500 1/1 000 1/2 000

1/1°

2x10-3 1.5A0-3

1/1 000 1/2 000

4.3

3x10-’ 2x10-3

1/100 1/500 1/1 OOO 1/2 OOO

Table 4.8.

10-2 6~10-~ 5~10-~

Divergence and overflow

1/100 1/50

1/30

A

5x10-’ 2x10-3

447

I m p l i c i t scheme

U

A

8~10-~ 2~lO-~ 2 ~ 1 0 - ~ 4x10-5 10-3 2x10-5

not considered 8x10-3 2x104

10-4 2x10-5

10-3

10-5

not considered

Divergence and overflow

8 ~ 1 0 - ~ 5x10-’ 2x104 10- 5 10-3 5x10-6

10-3

n o t considered

2x10-3

R e l a t i v e accuracy of t h e schemes.

A model problem of t h e deformation o f a membrane

We consider a membrane f i x e d over a h o r i z o n t a l r e c t a n g u l a r frame under a constant t e n s i o n F and s u b j e c t t o a ( p o s i t i v e or negative) load q(x,t) The deformations of t h i s membrane a r e r e s t r i c t e d by a f i x e d h o r i z o n t a l plane s e t a t a d i s t a n c e u from t h e frame. We a r e r e q u i r e d t o determine t h e d e f l e c t i o n u(x, 1) o f t h e membrane. Let Q be t h e domain bounded by t h e frame and Q,the region i n which t h e membrane i s i n c o n t a c t with t h e h o r i z o n t a l plane. The problem can be expressed i n t h e form: f o r ~E[O,r] with u(x, 0) = uo(x) given, f i n d u(x, t ) such t h a t

.

I

u(x, t )

=u

on 0,.

448

Time-dependent variational i n e q w d i t i e s

(CHAP. 6)

With f = q/F, a = 0 ( l ) , we arrive at the following problem for t traversing the interval [O,T] , solve the inequality:

For this application we shall take R = ]0,1[ x )0,1[ We put a(r) = f + t sin (2 nt) ,

(2 ) :

and T = 1.

and for the exact solution we take (putting x1 = x, xz = y ) :

with the initial condition:

Corresponding to this solution we have

f

(4.17)

Figure

Y = f,

%

XE

={

11

- AU, x

Q 0

S a(t),

arbitrary ,x > a(t)

4.4 shows a cross-section of the exact solution for In this figure t = 0,0.1,0.2, ..., 1

[O, 11 for the instants

.

a double arrow CI represents the zones of R where u = 0 (i.e. the zones where the constraints are active, for which we say the convex set K is "saturated"). The spatial discretisation is the standard discretisation of (see Chapter 3, Section 3 ) H;(CI) h = h, = h, = 1/20 or 1/41,

(thus with Nh = l / h

- 1,

we have Ni variables) ,

(')

Here we have arbitrarily assumed that the membrane rests on the horizontal plane.

(2)

A time-dependent variant of the model problem considered in Chapter 2, Section 5.

(SEC.

4)

Numerical solution of i n e q u a l i t i e s o f type I

u ~ =, 1 ~ u,(M)@',

449

i s t h e c h a r a c t e r i s t i c function

where

M En k

o f t h e panel v h

G:(M),

= s p a c e o f s t e p f u n c t i o n s on

oh,

z e r o a t t h e edge,

a n d c o n s t a n t on t h e p a n e l s G:(M), M E oh, Kh

=

1 uh E

{

vh, vh

2 0 on

oh } .

For t h e t i m e d i s c r e t i s a t i o n we t a k e

k

= S X ~ O - ~ ,lo-',

~xIO-',

The approximation o f f i s , f o r

f&

=

c

5xlO-' r E

[&,(PI + 1) k[,

f(M.(n + 1) k) 0,M .

MEDh

Moreover, s i n c e f o r x > a(t) t h e c h o i c e o f f < 0 i s a r b i t r a r y , we c a r r i e d o u t t e s t s w i t h f = 0, f = - %x - a(t)), f = 4(x - Or(t)), f = - lOO(x - a(r)) f o r x > a ( t ) . For a l l t h e schemes used, t h e numerical r e s u l t s are p r a c t i c a l l y i d e n t i c a l ( s e e Tr6moli8res /4/). The numerical r e s u l t s g i v e n below r e l a t e t o t h e c a s e f=-~x-a(t)). The approximation o f A i s g i v e n by

-

Ah

=

c

uh(x

+ hzi) - 2 uh(x) + uh(x - hzi)

i = 1.2

4.3.1

h2

Approximation schemes

The schemes i n v e s t i g a t e d a r e as f o l l o w s :

Time-dependent variationa l i n e q u a l i t i e s

(CHAP.

Parabolic inequality Cross-sections of t h e exact solution, f o r y = f , a t t h e instants t = o , o . 1 , 0 . 2 )...) 1 . (u'

- Al4 - J v - u) 2 0, VVEK,UE X

I

t = 0.1, = 0.4

I +saturatim

t = 0.7 t = 0.8

t

4

4

zone of K at

X

t = o

*

0.2 t = 0.3

I =

t =

4

t = 0.6 t

*

= 0.7

t = 0.8

4

Fig. 4.4.

c ~

0.4

t = 0.5

4

-

*

t = 0.1

4

+

t+

rn

+

+ c

6)

4)

(SEC.

Numerical solution of i n e q u a l i t i e s o f type I

451

5 3 - Crank-Nicholson scheme ( l )

Sb

- Fully implicit scheme

4.3.2

Numerical r e s u l t s

Except i n t h e c a s e o f t h e e x p l i c i t scheme, t h e c a l c u l a t i o n of i s c a r r i e d o u t by t h e Gauss-Seidel method w i t h t r u n c a t i o n ( 4 2 0 ) ; t h e r e l a x a t i o n p a r a m e t e r w b e i n g chosen i n a c c o r d a n c e w i t h t h e ,formula ( s e e Varga /l/) ( 2 )

4"

3 L

w =

nh2/2)4 h2 +

(l - [ ( l i +

2h2/k2

>'I

I"

'

The t e r m i n a t i o n c r i t e r i o n u s e d i n t h e c a l c u l a t i o n o f Gauss-Seidel method w i t h p r o j e c t i o n i s EO

=

c

1 d+l (rn+ l)(M-) u;+""'(M) I Q

4"

by t h e

10-6,

ME&

th where 4+'('") i s t h e mth i t e r a t i o n v e c t o r a n d 4+1(m+1) t h e ( m + 1) (One i t e r a t i o n c o r r e s p o n d s t o a complete sweep o v e r t h e c o o r d i n ates). The r e l a t i v e a c c u r a c y o f t h e schemes w i l l be measured a t each i n s t a n t t = n k i n terms of t h e q u a n t i t y

('1

A v a r i a n t o f t h i s scheme i s g i v e n by:

I (2)

.h"

=

uO,h

T h i s i s t h e o p t i m a l p a r a m e t e r i n t h e absence o f c o n s t r a i n t s .

.

Time-dependent variational inequalities

ECART =

(CHAP. 6 )

1 I G ( M ) - u(M ;nk) I MEQk

where U;: i s t h e s o l u t i o n c a l c u l a t e d a t t = nk and u ( M ; n k ) i s t h e exact solution a t t = nk , a t t h e p o i n t M . With scheme S1 divergence w a s observed f o r h = 1/20 and k = d u r i n g t h e i n i t i a l t h e - i t e r a t i o n s ( e r r o n e o u s r e s u l t s and overflow before t = 100 k = 0.1 , w i t h T = 1). A second t e s t w i t h h = 1/20, k = S x lo-* gave a c c u r a t e r e s u l t s . For t h e S2 scheme, s e v e r a l t e s t s showed t h a t w i t h h = 1/20 , t h e b e s t time-step appeared t o b e k = s i m i l a r l y f o r S3 and S4. Table 4.9 shows t h e b e s t r e s u l t s f o r t h e s e f o u r schemes.

Conclusions of 4.3

4.3.3

For t h e s o l u t i o n of t h e i n e q u a l i t y d i s t r i b u t e d over we have:

(4.16) w i t h c o n s t r a i n t s

E x p l i c i t scheme

1.

-

danger of divergence and overflow,

-

an a c c u r a t e s t a b i l i t y e s t i m a t e i s k/h2 < i, f o r t h e same p r e c i s i o n , twice as f a s t as t h e o t h e r schemes.

ECART a c t e r k t i c s r = O . l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 S1

s2 S3 s4

h= 20 Ik=0.5)~10-~0.28 0.29 1 h = -20 k=10-3 0.27 0.43

Table 2.

-

4.9.

1

11 The

0.51 0.78 0.74 0.38 0.12 0.04 0.39 0.08 0.16 0.28 l 0 h 5 0.530.81 0.740.380.120.040.040.080.160.2924~ 0.540.820.740.380.130.040.040.070.160.2824~

0.70 0.78 0.56 0.27 0.10 0.03 0.03 0.09 0.17 0.43 26 min

Comparison o f t h e v a r i o u s schemes

The other schemes. The " i m p l i c i t i n equation", " s e m i - i m p l i c i t i n p r o j e c t i o n " and " f u l l y i m p l i c i t " schemes appear t o be s i m i l a r i n t h i s case. W

(SEC. 5 )

5.

PmaboZic inequaZities of type 11

453

INTRODUCTION TO PARABOLIC INEQUALITIES OF TYPE I1

To s t a r t w i t h we s h a l l g i v e two examples a r i s i n g i n t h e t h e o r y of thermal servo-control.

5.1

Example I

We c o n s i d e r a c o n t i n u o u s medium occupying a domain f2 w i t h boundary r a n d whose edge t e m p e r a t u r e i s servo-controlled s o as n o t t o d i m i n i s h w i t h t i m e , t h i s b e i n g a c h i e v e d by i n j e c t i n g h e a t a c r o s s r ( t h e w a l l t h i c k n e s s being n e g l i g i b l e ) . The problem may be f o r m u l a t e d as f o l l o w s ( s e e Duvaut-Lions /l/):

-

h e a t e q u a t i o n i n f2, au - Au=f. at

(5.1)

~ € 0 ~,E ] O , T [ ;

given i n i t i a l temperature, u(x, 0) = u()(x)j

- boundary c o n d i t i o n s on

x 10, T[ ( s e r v o - c o n t r o l )

,

aU

-30, at

-a Up o , -a U- =auo at an

By u s i n g G r e e n ' s formula, t h i s problem may b e e x p r e s s e d i n t h e form: f i n d u ( x , t ) s a t i s f y i n g ( 5 . 2 ) w i t h u 2 0 on r, such t h a t , a . e . t E [o, T] , we have

(5.2)

(u' - AU

- f,u - u') 2 0 ,

V v 2 0 on

r.

I n o r d e r f o r t h e problem t o be meaningful we s h a l l s e e k u such that,

(5.3)

u, 10 E L2(0,T ; H

'(a))

and such t h a t ( 5 . 2 ) i s s a t i s f i e d V V E H'(f2) w i t h v 3 0 on

r.

rn

454

Time-dependent variational inequalities

5.2

(CHAP. 6 )

Abstract formulation and existence theorem

We again use (1.1)-(1.3) and (2.21)-( 2.26) ( l ) , (2.28), ( 2 . 2 9 ) , and we assume, as is essential here ( 2 ) , that A is symmetric and independent of time ( 3 ) .

(5.4)

We put X = { u 1 U E W ( O , T ) , U ( ~ ) E Ka.e. , ~ E [ O , TI},

X, = { u I us W,(O, T ) , U ( ~ ) E Ka.e. , ~ E [ O ,T I } , X$ = { u I u E L2(0, T ; V),U' E Xo }

.

The general problem whose approximation and solution we shall study is: find (5.5)

U E

X,'

such that

+ Au - J u - d ) d t 2 0 ,

VUE X .

We recall (see Duvaut-Lions /l/, BrCzis /l/) the following existence result: Theorem 5 .l. Let f and u, be such t h a t : (5.6)

f,f'ELz(O,T;H),

(5.7)

3u,

E

u,EV,

K which s a t i s f i e s (u,

+ Au,

There then e x i s t s a unique function (5.8)

U , U ' E L ~ ( O ,VT);,

(5.9)

u(0) = u, ,

(5.10)

u'(t) E K ,

(5.11)

(u'+Au-f,u-u')>O,

(l)

(2)

c3)

a.e

.

t

E

[O,

- f (0), u - u,) u

20,

Vu E K

.

such that

TI , V U E K ,a.e. ~ E [ O , T ] ;

For simplification we shall not consider functionals j which come under the classification of parabolic inequalities of type 11. However, the proofs can be extended without difficulty to the case in which j # O . See also Remark 5.4. We require at least that the "principal part" of A be symmetric. This point is purely to simplify the proofs a little.

Parabolic i n e q u a l i t i e s of type 11

(SEC. 5 )

4 55

moreover (5.12)

u " ~ L ~ (T0;,H ) ,

(5.13)

~'(0)= u0 .

It can be shown without difficulty that the ineqRemark 5.1. ualities (5.5) and (5.11)are equivalent.

Remark 5.2. (5.14)

Assumption (5.7) is always satisfied if

K is bounded in V

o r indeed if (5.15)

- [ A u ~- f ( O ) ]

Remark 5 . 3 .

E K.

The following inequality can be established

This result enables us to conclude, when u O e x i s t s (see Remark 5.2), that u' is bounded in K for the norms L2(0,T ; v) and L"(0, T ; V) It is then easy to see that if K is not bounded, i t i s possible t o define a convex s e t

.

R sufficiently large, such that problem (5.11) defined on K,in We can then fact has the same solution as the initial problem. consider, without loss of generality, (but still under assumption (5.7)) that (5.16)

K is bounded in V .

This remark is useful for the study of the approximation.

Remark 5.4. We can likewise solve the following inequality, which is more general than (5.5): (5.17)

JOT

(u'

+ Au - 5 u - d ) df +

vu E

[j ( u ) - j(u')] df

x,u E Yo

where j is a functional of the type ( 2 . 2 7 )

.

>0,

4 56

Time-dependent variational i n e q u a l i t i e s

6.

( CHAP.

6)

APPROXIMATION OF PARABOLIC INEQUALITIES OF TYPE I1

6.1

Fundamental assumptions for the approximation

In studying the approximation we shall retain everything said in Section 3.1 with, in addition, the following assumptions: (6.1)

ah(uh, u h ) is sym??ZetriC,

I I

Kh is bounded in Vh, independently of h , i.e.

11 uh

d

c,

VUhEKh

and Vh,

h +0.

( F o r this latter assumption, see Remark 5.3).

We shall transpose the results given in Section 3.1.3 to N a n d to

xg+. In view of ( 5.6) we have f c V0(O, T;H ) and we take

as in Section 3.1.6 with f / defined by (3.27'). 6.2

Approximation scheme for parabolic inequalities of type I1

Taking account of the above assumptions, we shall now study the approximation of the inequality

We note that inequality (6.5) can be written:

(SEC. 6)

Approximation of inequaZities of type I1

457

This possesses a unique solution 6; ; thus one by one all the

up1,i = 0, 1, ..., (u;"

= u;

+ k&),

exist and are uniquely defined. Moreover we note (since Ah is symmetric) that the u;", for i=O,1, ... are uniquely defined from the respective solutions of the N minimisation problems (i = 0, ..., N - 1)

+

minimise f. I 6; ;1 f. ek(Ah $ ) h - (A,, ui - f;,6;)h subject to the constraints

where ui, (i 3 I), is given (the solution of the previous u i = uh.o We put u F 1 = ui' k6u; . problem), and

.

+

We use the following terminology:

- Explicit scheme if O = 0, - Crank-Nicholson scheme if - Implicit scheme if 0 = 1. Remark 6.1.

e=f.,

In the same spirit, the solution of the inequal-

ity (u'

+ AU -

u - u')

+ j ( ~-) j ( ~ 2' ) 0 ,

V DE

X,u E X,* ,

(where j ( u ) is defined as in ( 2.27)) leads to the schemes (omitting the suffix h )

+ A[u' + flu'+' ui+l

+j(u) - j (

-

- ui)]- fi,u -

"')2

-

,,'+I

0 , VDEK,- k

- U'

'

E

K,

o r , since A is symmetric, to the solution of the problems:

m h k i s e $ 1 6; I;+$ek(Ah &, 6;) - (Ah u; - f;, 4 ) h + j(&, subject to the same constraints as in (6.8).

4 58

Time-dependent variational ineqwzlities

( CHAP.

6)

Remark 6 . 2 . In the same way as for parabolic inequalities of type I, we can here also define decoupled schemes:

where P,is

6.3

the projection on & in the sense of the norm Convergence of the approximate inequalities

We shall now study scheme (6.5) which we rewrite (omitting the subscript h for clarity) as follows

I ( ~ ' + A U ' + ~ - ~ ~ , I I 2- ~0' ),

(6.9)

with

I uo = uo 6' =

V o E K , 6 ' E K , i = O , 1,

..., N ,

given,

- ui

ui+l

k

'

In addition we shall use the following notation: 6i+e =

6.3.1

ui

+ 1 +B

- u'

+@

k

=

6' + e [ g i + l - 6 1 ,

Stability

We shall now establish: Theorem 6.1. When (h, k ) + 0

(6.10)

i n a general fashion i f f Q 8 Q 1, under the s t a b i l i t y condition C k S W Q 1 - /3

if

0Q

we have, f o r a l l n = 0,1, ..., N, (6.11)

II u; Ilk Q c ,

e c f , (/3

fixed E]O,ID,

(SEC. 6 )

A p p r o ~ mtai o n of inequalities o f type 11

459

and

Proof. (6.14)

Writing

(6.9) for i +

+ Au""'

(6"'

-

v -

f i + ' 9

With v = 6"' in ( 6 . 9 ) ,

I 6i+l

- 6i 12

+

a'+' ) z o ,

v = 6'

- ui+B

a(ui+l+@

1 we have VVEK,~'+~EK.

in (6.14) and adding, we obtain gi+l

- 6') < ( f i + 1 - fi,

-

a'),

or, dividing by k ,

and hence, multiplying by 2, k I yi

l2

+ 2 ~ ( 6 ' ,gi+l - 6') + 2 ea(6'+l

- 6')

Q

k I gi

12,

or, since a is symmetric, (6.15)

k I y' 1'

+ ~(6"') - a($) + (2 0 - 1) ~ ( 6 " ' - 6') < k I g' 1'.

We then distinguish the following two cases:

Since the term in ( 2 0 -

1) is positive we can suppress it in i = 0 to n < N - 1 we obtain

(6.15), and by summing from k IyiIz i=O

+ ~(6"") d

k Ig'I2 i=O

+ a(6").

As a consequence of assumption (6.2)we have a(6O)G C. In view of (5.6) and (3.27') we then obtain (6.13) immediately. Using the coercivity of a(,) we deduce (6.12). The upper bound (6.11)is an immediate consequence of (6.12)and of the fact that ~(6') d C by noting that 4+1=

u:

+k

c 60. n

i=o

460

Time-dependent variationai! i n e q u a l i t i e s 2nd case : 0 Q

( CHAP. 6 )

e < 4.

We u s e t h e upper bound - 6') = h2 a(y') d Ck2 S(h)2 I y'

U(6'+'

12

,

whence

(2 e - 1) a(ai+l - 6') 2 (2 e - 1) ckz s(hl2I 7' l2 , and (6.15) i m p l i e s t h a t (6.16)

+ (2 6 - 1) CkS(h)'] I y' I' + ~(6'")

k[1

Hence i f we assume, i n t h e c a s e

1

- CkS(h)' 2 B > 0, /? f i x e d

- a(6') Q k 1 gi 1' .

2 8 - 1 < 0, t h a t €10, 1[,

it may be deduced from (6.16) t h a t

Bk I y' 1'

+

U(6'+1)

- 46') Q k I g' 12 ,

and by summation w e o b t a i n (6.11), (6.12), (6.13), as i n t h e 1st case. Thus Theorem 6 . 1 i s proved. rn

Remark 6 . 3 . I n t h e c a s e where V h , k + O , t h e i n e q u a l i t y ( 6 . 9 ) i s i n f a c t , f o r i = 0 , an equation (convex s e t n o t " s a t u r a t e d " ) ; we have 60 + AUO+O - f O = 0 , and hence

11 '6 11

=

(I

+ eks0) - f o I( Q II A U O

from which we deduce, w i t h

II + ek 11 ~6~ 11

+ 11 f o 11,

11 Auo 11 d c, 11 Ado 11 d c t h a t

(1 - ekc) II 60 II < c , that, for k sufficiently s m a l l , ' 6 i s bounded f o r t h e norm V,, independentzy o f h and k, and assumption ( 6 . 2 ) i s i n f a c t rn unnecessary. SO

h'eak convergence

6.3.2.

From Theorem 6.1, when h , k + O ( w i t h (6.10)i f 0 < 0 < 4 ) e x i s t s a subsequence, a g a i n denoted by uhc, and elements (6.17)

U E

L ~ OT ,; v),U E L ~ O T, ; v),w E ~ ' ( 0 ,T ; HI, CE

such t h a t (6.18)

ph uh,k + ou weakly" i n L"(0, T ; F) qh u , , ~+ u weakly" i n L"(0, T ; H)

v,

there

(SEC.

6)

Approximation of inequaZities of type I1

* *

(6.19)

ph Su,., + au weakly 9h 6uh.k + v weakly

(6.20)

9h S2uhsk+ W weakly i n

(6.21)

{

in in

461

L"(0, T ; F ) L"(0, T ; H )

L2(o,T ; H )

ph u: -+ at weakly i n F , 9h4 + weakly i n H .

I n t h e same s p i r i t as i n Lemma 3 . 1 it may t h e n be shown ( s e e V i aud /1/) t h a t

(6.22)

u = u',

= u(T), u ' E K ,

w = u",

s o t h a t u s a t i s f i e s (5.8), (5.9), (5.10), (5.12). t h a t u s a t i s f i e s t h e i n e q u a l i t y (5.11).

c uid .

We s h a l l show

N

Let

V E Xand vhsk=

be a sequence approximating u as i n

is0

Lenma 3.2. We t a k e u = u i i n (6.9), sum from i = O t o N - 1 and m u l t i p l y by k ; we o b t a i n ( o n c e more o m i t t i n g t h e s u f f i x h for t h e time being) N- 1

N- 1

(6.23)

1 [k(S', u') + kU(U'+@, .=

ui)

- k ( f ' , ui-

0

Si)]

2

1 k I 6'

12

+ x,

i=O

with N-1

(6.24)

x = 1 kU(Ui+@,6') i=O

Now N- I

with

Y = (2 0 - 1 ) .

c

U(Ui+'

- u')

i=O

We s h a i l d i s t i n g u i s h t h e f o l l o w i n g two c a s e s ( r e i n t r o d u c i n g the suffix h ) : 1st

(6.25)

case : f s 8 s 1.

lim inf X 2 h.t-0

f a(u(r)) - f u(uo).

462

Time-dependent variational inequalities

( CHAP.

6)

Using (6.23) and (6.31) together with the various consistency assumptions on . f , a ( , ) , f and the results (6.18), (6.19), we obtain :

IoT

(u'

+ Au - 5 v - u') dt 2 0

which is true V v e X From this we deduce that u is a solution of problem (5.5) o r (6.3).

2nd case : 0

< 0 c f.

We then have Y < 0, but the above result remains valid if we have Y - r 0. Now

Hence (6.27)

N

I YI < kC

k

< kC,

i=0

so that we indeed have Y - r O . To conclude, we note that as a consequence of the uniqueness of U , the results (6.24),. ,(6.27) are true for the entire sequence. We car1 now state:

..

Theorem 6.2. with

When h, k

< 1-p, p

CkS(h)'

we have ph uk*k

(6.28)

I

-r

P k 6uh,k

+

q h 'h.k + -r

q k 6uh.k q h 6'Uh.k ph

4

--*

u'

-r

U"

+0

cmd i n the case when 0

<0cf

fixed ~ ] 0 , 1 [ ,

weakly

*

in

L"(0, T ; F )

weakly

*

in

L"(0, T ; H )

weakly

*

i n L'(0, T ; H )

au(T) weakly i n F

Remark 6.4. We note that we have thus proved the existence of a solution of problem ( 5 . 5 ) , (6.3). 6.3.3.

Strong convergence

We shall now prove (cf. Viaud Theorem 6.3.

When h , k

+0

111) the

following result:

and i n the case where 0 < 0 < f

(SEC. 6 )

Appro&mation o f i n e q u a l i t i e s of type 11

463

with (6.29)

CkS(h)' d 1 - /7,

/7 f i x e d €10, 1 [ ,

we have :

Proof of Theorem 6 . 3 . We assume h e r e t h a t we can "extend" a(,) t o F ; i n o t h e r words, t h a t w e can d e f i n e a(,) on F x F such that ii(au, uu) = a(u, u ) , vu, 0 € v , and such t h a t a(,) r e t a i n s on F t h e same c o e r c i v i t y c o n s t a n t a and c o n t i n u i t y c o n s t a n t C as a ( , ) , and s t a y s symmetric ( s e e Remark 6.5 below) ; i n a d d i t i o n , w e t a k e ah(uh,oh) = ii(phuh,ph oh). We t h e n p u t (for s i m p l i f i c a t i o n o m i t t i n g s u b s c r i p t s i n d i c a t i n g c e r t a i n dependencies on h and k )

1st term : Xl(r). ( )

Z(U) = Z(U,u), u E F.

Time-dependen t v a r i a t i o n a 1 i n e q u a l i t i e s

464

( CHAP. 6 )

From u, u' E L"(0, T ; V), 4 0 ) = uo and t h e f a c t t h a t a(u, 0) is symmetric , w e deduce t h a t

and hence

X,(t) =

1:

v [ J u'

12

1 + a(u, u')] ds + zu(uo)

2nd term : X2(t). N-

We p u t

iik(f)

=

1

1 u((i + 1) k)Xi.

I t can t h e n r e a d i l y b e v e r i f i e d

i= 0

qh 0h.k

(6.35)

+

P h 0h.k +

q h 6Vh.k

u'

s t r o n g l y i n L2(0,T ; H )

uu' s t r o n g l y i n L2(0,T ; F ) U"

s t r o n g l y i n L2(0, T ; V ).

(SEC. 6)

Approximation o f i n e q u a l i t i e s of type I1

+

However f o r t ~ [ n k , ( n 1)k[ that

465

it can be shown w i t h o u t d i f f i c u l t y

By u s i n g , i n t h e c a s e 0 < 4 , t h e f o l l o w i n g upper bound:

we o b t a i n

Noting t h a t t h e l a s t i n t e g r a l on t h e r i g h t - h a n d s i d e t e n d s t o z e r o , and u s i n g t h e convergence assumptions ( 6 . 2 8 ) , (6.35),we o b t a i n , on p r o c e e d i n g t o t h e l i m i t : rl

C o l l e c t i n g t h e r e s u l t s obtained f o r

-21 a(u0) . X,,X,,X, , we have

X ( r ) - + O , when h , k - + O ( w i t h ( 6 . 2 9 ) ) .

Hence we deduce t h a t

ah huh,,

-+

u' s t r o n g l y i n L2(0,T ; H )

Ph uh(f) + o u ( / ) s t r o n g l y i n F , a. e .

fE

[0, TI

.

However t h e f a c t t h a t ph uh,k-+ ou weakly * i n L"(0, T ; F) and s t r o n g l y a . e . i n F i m p l i e s , by Lebesgue's theorem, t h a t Ph Uh,t

-+

ou s t r o n g l y i n Lz(O, T ; F ) .

Moreover, f o r

tE

[nk, (n + 1) k [ , w i t h

Time-dependent v a r i a t i o n a l i n e q u a l i t i e s

466

(CHAP. 6 )

N- 1

pn uh,k qh

+

6Uh.k

uu

+ u'

s t r o n g l y i n Lm(O,T ;F) strongly i n

Lm(O,T ; H )

which proves t h e theorem. I n g e n e r a l , w i t h V such t h a t H&2) c V c H'(B), Remark 6.5. we t a k e F = L'(Q) x LZ(62)N,where Q i s an open bounded r e g i o n o f RN, and with u(u,u) = j " U U d x

q -a

+

i=]

-dx, au

n a x i axi

we p u t

where u = (uo, uI, ..., uN)E F, ij = (u0, uI, ..., uN) E F and f o r

btl

=

(

u,-,...,*)

ax,

u E V,

we define

(SEC. 7 )

Numerical solution of i n e q u a l i t i e s of t y p e I1

467

With : Ph

:

uh

vh

(uh

1"

; V h l U h In,

...

7

vh#h

In)

and

11 uh

Ilk

=

11 P h uh

[IF

it can t h e n be v e r i f i e d w i t h o u t d i f f i c u l t y t h a t z(.) s a t i s f i e s t h e c o n d i t i o n s r e q u i r e d at t h e s t a r t o f t h e p r o o f of Theorem 6.3.

7.

NUMERICAL SOLUTION OF PARABOLIC INEQUALITIES OF TYPE I1

I n t h i s s e c t i o n we s h a l l i n d i c a t e some r e s u l t s c o n c e r n i n g t h e n u m e r i c a l s o l u t i o n o f t h e two examples g i v e n i n S e c t i o n 5. F i r s t , we r e c a l l t h a t t h e d i s c r e t i s a t i o n of t h e s e two examples l e a d s t o i n e q u a l i t i e s o f t h e form:

T h i s may also be s t a t e d i n t h e form: d e t e r m i n e , f o r i = 0 , 1,.. . , t h e s o l u t i o n w;" o f t h e i n e q u a l i t y

and d e f i n e

(7.3)

. u r l = ui

Putting

(7.5)

+ kw;".

A = Ahui - A

it can be s e e n t h a t ( 7 . 2 ) may be w r i t t e n

minimise J,,(wh) = %(I,, Wh E

+ k6Ah)Wh, wh)h - (2,wh),

Kh

t h e sol. u t i o n of t h i s l a s t problem g i v i n g w r ' . For t h e two examples which f o l l o w , problems ( 7 . 5 ) w i l l be s o l v e d by t h e method of r e l a x a t i o n w i t h c o n s t r a i n t s ( s e e Chapter Note t h a t t h e o p t i m a l v a l u e of t h e r e l a x a t i o n 2, Section 1.2). p a r a m e t e r i s d i r e c t l y r e l a t e d t o t h e t e r m (Ih + 6kAJ , and moreo v e r , depends on t h e zones o f s a t u r a t i o n o f t h e convex s e t Kh ( i . e . where t h e c o n s t r a i n t s a r e a c t i v e ) , as w e l l as, o b v i o u s l y ,

468

Time-dependent v a r i a t i o n a l i n e q u a l i t i e s

(CHAP.

6)

on h . In the following numerical applications we have chosen to take values of w giving results which are not too expensive in terms of computation time. The parameter was taken as indicated (as a function of 1 = Ok ) in Figure 7.1. To assess the merits of the various schemes and the stability and convergence assumptions, we shall rely on problems for which exact a n a l y t i c s o l u t i o n s are available.

2tw 1.9 1.8. 1.7. 1.6.

1.5. 1.4. 1.3.

.

12. 1.1.

*

1 .o

I

I

I

I

c

Values of the relaxation parameter w as a function of I = Ok for the two examples considered.

Fig.7.1.

For the two examples I and I1 we shall take: 62 = 10, 1[ x 10, I[, T = 0.1 , f ( x , 1) and u(x, 1) continuous.

Having specified the discretisation of 62 for each example, we shall take fi

h

-

1 f[M,ik]o f , = 1 u(M, 0) O? MERh

uO.h

9

M E &

where is the characteristic function of the panel z ( M ) . The investigation will be carried out for h = 1/20,

k between lo-' and ~ x I O - ~ , O=O (explicit scheme)

= f (Crank-Nicholson scheme)

= 1 (implicit scheme). rn The computations were performed on a CII 10070.

rn

( S E C . 7 ) Numerical s o l u t i o n o f i n e q u a l i t i e s o f type I1

7.1.

S o l u t i o n of Example I

The p a r t i c u l a r s r e l a t i n g t o t h i s example are:

{ K=

V = H ' ( Q ) , H = L*(Q),

(7.6)

On Q = ] O , l [

{ u E H ' ( Q ) ( u ( x ) ~ Oa.e. x E T ) x ]0,1[ we c o n s i d e r t h e g r i d

Rh = { M

= (ml

h, m2 h), m , , m 2 = 0, 1 , ..., 20 = l/h }

and we d e f i n e

I t cam be shown i n t h e p r e s e n t c a s e t h a t

S(h) = 2 $/h

.

470

Time-dependent variational inequalities

( CHAP.

6)

( i = 0,1, ...

As i n d i c a t e d p r e v i o u s l y , at each t i m e " i t e r a t i o n "

. .., N = T / k ) we

c a l c u l a t e w;+'-and hence deduce u r l -by s o l v i n g a problem o f t h e t y p e ( 7 . 5 ) by t h e r e l a x a t i o n method, u s i n g a r e l a x a t i o n parameter a s i n d i c a t e d i n F i g u r e 7.1. I n t h e d e t e r m i n a t i o n of , t h e r e l a x a t i o n algorithm i s t e r m i n a t e d when t h e d i f f e r e n c e i n t h e L" norm between two s u c c e s s i v e i t e r a t e s wX+ I(''and w ; ' ~ ~ ' ' )i s l e s s t h a n , i.e. when

wrl

(7.11)

1 I

wry^) - W : + l ( n - l ) (I ~s) 10-4.

MEnh

I n o r d e r t o assess q u a n t i t a t i v e l y t h e m e r i t s of t h e v a r i o u s schemes, w e s h a l l a l s o i n v e s t i g a t e t h e r e l a t i v e , e r r o r E(u) i n t h e Lz norm, r e l a t i v e t o t h e v a r i o u s e x a c t s o l u t i o n s , as a f u n c t i o n of t h e t i m e :

(7.12)

E(u) =

1

(u E *h

I u ( M ; ik) ~z)l'* .

I u;(M) - u ( M ; ik) MIEnh

( F o r t h e e x a c t s o l u t i o n c o n s i d e r e d h e r e , t h e denominator remains > 0). We c o n s i d e r t h e f o l l o w i n g e x a c t s o l u t i o n on Q = lo,T [ x 61 = ]0,0.1[ x lo, 1[ x 10, 1[

+

1) (x - 1)' yz[(sin 10 t - y)+I3 u(x, y , t ) = 1 + lOO(2x + 100(2y 1) ( y - 1)' xz[(sin 10 t - x)+13 - 100 XZ(1 - x)y(1 - y)Z [ ( y - sin 10 t ) + ] 3 - 100y~(1- y ) x(l - x)'[(x - sin 10 t)+]' , with, f o r a quantity g :

(7.13)

+

=g

if 9 2 0 ,

= O

if

+

gs0.

t h e problem

(where

K i s g i v e n by ( 7 . 6 ) and 3'2

i s d e f i n e d as i n S e c t i o n 5 . 2 )

(SEC. 7) Numerical solution of inequalities of type I1

471

has the exact solution (7.13)(’ ) . It may be noted, moreover, that an arbitrary constant can be added to u without modifying f (see (7.14)). Since it would therefore be possible to give results with as small a relative error as desired, we shall note that

max u(x, y , 1)

N

3.2

.

(x.y.0 E Q

1

I

I I

a

I

I

I

I

1

0 I I

II II

I

Fig. 7.3.

u on f.

I I I

a

I I

I I

Fig. 7.4.

du/dn on f

Figures 7.3 and 7.4 define the values of u and au/an on f. Figures 7.5, 7.6, 7.7 show the evolution of the error E(u) for various values of k and for the implicit, Crank-Nicholson and explicit schemes. With h = 1/20, the implicit and CrankNicholson schemes give a minimum error for k = and these schemes are almost equivalent for this value. We observe that and that in the explicit scheme overflow occurs for k > 7 ~ 1 0 - ~ gives results comparable to those obtained the value k = using the other schemes with k=10-3. The computation times (on a CII 10070), were as follows: -

-

(l)

Implicit and semi-implicit schemes with h = 1/20, k = : time N 2.5 min. Explicit scheme with h = 1/20, k = 2 ~ 1 0 :- ~time

N

6 min.

As a consequence of choosing f in accordance with (7.14), we note that the solution (7.13) also corresponds to problem (7.15) with K = H ’ ( O ) in place of (7.6). Thus the inequality (7.15)is here equivalent to an equation.

472

Time-dependent variational i n e q u a l i t i e s

E(4 4

0.01

0.05

Fig. 7.5.

0.0 1

0.0 1

t

0.1 t

Semi-implicit scheme.

0.05

Fig. 7.7.

c

Implicit scheme.

0.05

Fig. 7.6.

0.1

Explicit scheme.

0.1

t

I

(CHAP.

6)

(SEC. 7 )

7.2.

Numerical solution of inequalities of type I1

473

S o l u t i o n o f Example I1

The p a r t i c u l a r s r e l a t i n g t o t h i s example are

(7.16) Taking

V = Hd(Q), H = LZ(Q), K = { u ~ H i ( Q ) ( u ( x ) > 0a.e. Q = 10, I[ x 10, I[

XEQ}.

w e d e f i n e , w i t h Rh as i n S e c t i o n 7.1,

I

0

Fig. 7.8.

Sr, f o r h = 115.

I n t h i s c a s e t h e problem can a g a i n be w r i t t e n i n t h e form ( 7 . 5 ) and i s s o l v e d by r e l a x a t i o n , w i t h w t a k e n as i n F i g u r e 7 . 1 and u s i n g t h e t e r m i n a t i o n c r i t e r i o n g i v e n by ( 7 . 1 1 ) . We s h a l l assess t h e schemes i n terms o f E(u) g i v e n by (7.12), and w i t h

x

1 I u ' ( M ; (i + e) k ) 1' MERh

.

>"'

Here it can be shown once a g a i n t h a t S(h) = 2 f i / h . We s h a l l now i n v e s t i g a t e t h e approximate s o l u t i o n o f - t h e inequality

(where K i s g i v e n by

(7.16)and

i s d e f i n e d as i n S e c t i o n 5 . 2 ) .

474

Time-dependen t v a r i a t w n a l i n e q u a l i t i e s

(CHAP. 6 )

We take :

- Au - lOo(10 h t

(7.18)

f(x, Y, I ) =

(7 * 19)

u(x, Y. 0) = u d x , y ) ,

U'

- 9 - y')'

,

with the exact solution u(x, y ,

(7.20)

r)

=

10 sin x sin y(l - x) (1 - y ) -

-10x~l-x)(1-y)[(x~+y~-IOsinr)+]'.

Over the domain Q; =

we have u'

( X , Y ) E Q 1 10 sin t 2 x2

=0

+ y2 1,

(see Figure 7 . 9 ) .

I

Fig. 7.9.

sz

u' = 0 on Q;.

The results are shown in Figures 7.10, 7.11, 7.12, 7.13, the computation times being as follows :

- Implicit

-

and semi-implicit schemes with h = 1/20,

time * 1.2 min., Explicit scheme with h = 1/20,

k

= lob3 :

k =2

x

lo-* : time

= 3.5

For the explicit scheme, overflow occurred for k 2 7 x I O - * (see Figure 7.13). In this case the implicit scheme gives the best results.

min.

H

7.3. Conclusions In conclusion, f o r the two examples considered it was observed that f o r similar accuracy the implicit schemes were two o r three times faster than the explicit schemes, and that the latter Other examples diverge when the timestep.is not s m a l l enough. which corroborate those presented here f o r inequalities of type I1 are given in Viaud 111. H

(SEC. 7) Numerical solution of inequalities of type I1

Fig. 7.10.

Fig. 7.11.

Implicit scheme.

Semi-implicit scheme.

0.005

I I I

0

0.01

0.05

Fig. 7.12.

Explicit scheme.

0.1 t

475

Time-dependent variational i n e q u a l i t i e s

476

0

0.01

0.05

Fig. 7.13.

8.

0.1

( CHAP.

6)

i

E(u‘) f o r t h e e x p l i c i t scheme.

INTRODUCTION TO TIME-DEPENDENT INEQUALITIES OF THE SECOND ORDER I N t

8.1.

Example I

The t h e o r y o f v i s c o - e l a s t i c m a t e r i a l s w i t h edge f r i c t i o n l e a d s t o problems ( s e e Duvaut-Lions /l/) for which t h e f o l l o w i n g i s a model example: f i n d a f u n c t i o n u = u(x, t ) such t h a t a2u

au

at2

at

-- A - - A u = f

in ax]O,T[,

with t h e i n i t i a l d a t a dx, 0) = uo(x) 3

at4

5 (x, 0) = Ul(X),

and t h e boundary c o n d i t i o n s : u = 0 on one p a r t

(u”, u

(8.1)

T o x 10, r[

r

- u’) + al(u’, u - u’) + ao(u, u - u’) + j ( u ) - j(u‘) 2

B(f,u-u’)

VUEY,

40) = uo , u‘(0) = u1 , where

o f t h e boundary

U’ E V

x 10, T [ ,

(SEC. 8 )

InequaZities of order

2

in t

477

V = { u ( u E H ' ( R ) u, = O o n r , ) , ao(u, u ) = a,(u, u ) =

8.2.

1"

grad u grad u dx ,

Example I1

A number of p h y s i c a l s i t u a t i o n s ( s e e Duvaut-Lions t h e s e a r c h f o r a f u n c t i o n u such t h a t

/l/) l e a d t o

d'-Au-f>O, ~ ' 3 0 , - Au - f) = 0 i n D x 10, T [ ,

u'(u"

w i t h t h e i n i t i a l d a t a as above and w i t h t h e boundary c o n d i t i o n s g i v e n , f o r example, by u = 0 on

r

x 10, T [ .

T h i s problem can a l s o b e f o r m u l a t e d as a v a r i a t i o n a l i n e q u a l i t y o f t h e second o r d e r i n 1 , namely

(8.2)

1

(u",11 - u') + U(U, u - u') 3 (f,u - u') V U E K , u'(t) E K , ~ ( 0 )= u0 , ~ ' ( 0 = ) u1 ,

where

K = { u ( u ~ H d ( D )u ,> O

ah, u) = 8.3.

a.e. i n R } ,

1"

grad u grad u dx .

Abstract formulation

Using t h e same f u n c t i o n a l s e t t i n g as i n S e c t i o n 5 . 2 ( V c H c V') we i n t r o d u c e , w i t h K a c l o s e d convex s e t o f V, t h e convex s e t s

X = { U I U E E ( O , Tu )( ,i ) ~ K , a . e .

tE[O,T]},

x* = { u I UEL*(0,T ; V ) , U ' E x } , 30' = { 0 I u E x * , u(0) = uo, u'(0) = UI } , where uo,uI a r e g i v e n , and we s h a l l c o n s i d e r i n e q u a l i t i e s o f t h e type

(8.3)

L(u, U) =

foT (u"

AU - f, u - u') d t 3 0 , V U E X , u E Xg ,

w i t h u(0) = uo, u'(0) = u1 where A and J have t h e same p r o p e r t i e s as i n

Time-dependent v a r i a t i o n a 1 inequa t i t i e s

478

( CHAP. 6 )

S e c t i o n 5.2; i n p a r t i c u l a r A i s assumed t o be symmetric ( ’ ) (and i s c l e a r l y coercive and c o n t i n u o u s ) . The i n e q u a l i t y ( 8 . 3 ) i s more o f t e n w r i t t e n i n t h e form (u”-Au-~,v-u’)>O

(8.4)

VVEK, U E X ~ .

The e x i s t e n c e of u , t h e s o l u t i o n of ( 8 . 3 ) , i s given ( s e e Lions

111) by: Theorem 8.1.

With

(8.5)

f,S’E L2(0, T ;H),

(8.6).

A u E~ H ,

(8.7)

uIEK,

there e x i s t s a unique f u n c t i o n u s a t i s f y i n g ( 8 . 3 ) ( o r ( 8 . 4 ) ) s u c h that (8.8)

u,u’EL~(O T ,;V),

(8.9)

u ” ~ L ~ (T0;H , ).

Remark 8.1.

By i n t r o d u c i n g

X, = { v I v E L2(0, T ; V), V’ v ’ ( t ) ~ K , a.e.

E L2(0,T ;V), 40) =

u0,

~E[O,T]}

we can i n v e s t i g a t e t h e e x i s t e n c e o f weak s o l u t i o n s of t h e inequality JOT

(v”

- AU -

v’

- u’) dt 3 o ,

vv

E

ju,+ , u E

3 ~ .;

For t h i s problem w e r e f e r t h e r e a d e r t o BrCzis 121, BrCzisLions 111.

Remark 8.2. A formalism somewhat more g e n e r a l t h a n ( 8 . 3 ) i s t o consider t h e i n e q u a l i t y (8.10)

1:

(u”

+ .Au - f, v - u’) dt +

[&)

- &’)]

dr 2 0 ,

VVEX, U C X 2

(’)

It i s s u f f i c i e n t t h a t t h e p r i n c i p a l p a r t o f A be symmetric. We can a l s o consider problems i n which A i s dependent on 1 . The c a s e o f ( 8 . 1 ) may be t r e a t e d by t h e same methods.

I n e q u a l i t i e s of order 2 i n t

(SEC. 8 )

where j i s a f u n c t i o n a l o f t h e t y p e ( 2 . 2 7 ) .

479

8

Remark 8 . 3 . R e s u l t ( 8 . 8 ) a l l o w s us t o c o n s i d e r t h a t i f K i s n o t bounded t h e n it i s p o s s i b l e t o d e f i n e a convex s e t KR ( d e f i n e d as i n Remark 5.3) which i s bounded i n v, and which i s s u c h t h a t t h e problem ( 8 . 3 ) defined on KR ( r a t h e r t h a n K ) has t h e We can t h u s c o n s i d e r t h a t same s o l u t i o n as t h e i n i t i a l problem. K i s bounded i n V .

(8.11)

I n t h e same way as i n t h e p a r a b o l i c c a s e o f t y p e 11, t h i s remark i s o f u s e i n i n v e s t i g a t i n g t h e approximation. 8

9.

APPROXIMATION OF INEQUALITIES OF THE SECOND ORDER I N t

I n t h i s s e c t i o n w e embark on t h e i n v e s t i g a t i o n of t h e approxi m a t i o n o f h y p e r b o l i c i n e q u a l i t i e s o f t h e form ( 8 . 4 ) .

9.1.

Assumptions.

We s h a l l work w i t h i n t h e s e t t i n g d e f i n e d i n 6.1. s h a l l d e n o t e t h e approximate f u n c t i o n s by uhJ w i t h

A s u s u a l we

N

We s h a l l u s e t h e n o t a t i o n

For t h e approximation of uo, ul, X", u(u, u) = (Au, u), / ( I ) we make t h e following assumptions:

Approximation of the i n i t i a l values. With U ~ VE, U ,E K, we assume t h a t t h e r e e x i s t (9.1)

U0.h

(9.2)

Ul,h E K h

E

v h

Such t h a t pj, U0.h

+ 0110

Such t h a t

-B

P h U1,k

UUl

It h

c,

S t r o n g l y i n F and

(1

S t r o n g l y i n F and

I( U 1 . h [ ( h 6 c.

UOsh

6

480

Time-dependent variational inequalities

duh.k(t)

(9.3)

+

P h 0h.k

+

E

[o, r ]

u weakly

9

*

t E [o, TI , s t r o n g l y i n L2(0,T ; H ) s t r o n g l y i n L2(0, T ; F). + uu

E Kh u

9

q h 1)h.L +

p h I)h,k

(9.5)

9

i n Lm(O,T ; H ) uul weakly * i n L"(0, T ; F) p h U h , k ( T ) + UU2 and p h U h r . k ( 0 ) + UUp Weakly i n v q h d o h , , + U 3 weakly * i n L"(0, T ; H ) ( o r w i t h q h 6vh.k + u j ) P h d u h , k + UU4 weakly * i n L"(0, T ; F) ( O r w i t h q h 6Vh.k + U U 4 ) q h 6 U h , k ( T - k) + Us and q h b U h , k ( o ) + 0: Weakly i n H weakly * i n L"(0, T ; F) q h 7Vh.k + q h uh.k

uh.k(l)

(9.4)

Kh

(CHAP. 6 )

Kh i s bounded i n

vh

.

For t h i s l a s t assumption, c f . Remark

9.4.

Approximation of u(u, u). We s h a l l assume ( c f . (9.6)

IAhu0.h

Ih

(8.6)) that

G C , c o n s t a n t independent o f h .

Otherwise t h e assumptions a r e t h e same as i n S e c t i o n 6.2. p a r t i c u l a r , q , ( l ( k , u,,) i s symmetric.

In

Approximation of f. The same remarks a p p l y as i n S e c t i o n 6.2. here t h a t A f ' E

Lm(O,T ; H),

We s h a l l assume

(SEC. 9 ) Approximation of i n e q u a l i t i e s of order 2 i n t

481

and therefore f E V0([0, TI ; H ) ,

and we take N

=

fh.k(r)

i=O

such that (9.7)

9.2.

I

qhfh,k

-+

P h %,k

fi xi(r)

11 f h , k Ilk G f' in L.D(o,T ; H ) and 11 6 f h . k 1: < c .

f in

-t

L"(o,

H,

and

1

Approximation schemes

Taking account of the approximation assumptions and with

we shall

(9.8)

In th following we shall use the notation (omitting the suffix h )

find u2, u3, ..., # such that

Scheme ( 9 . 8 ) can thus be written: + Aui+@- f',u (9.9)

d'+l

(i21), VVEK,

(9.10)

uo =

~

0

U ,' = u0

-

,,i+l

- ,,i-l

2k

-

2k

ui-l

)2 0 ,

EK,

+ ku,

From the theory of elliptic inequalities, the inequality ( 9 . 9 ) ,

(CHAP. 6 )

Time-dependent variational inequazities

482

which by p u t t i n g =

- 8-l

+ k7 (Ad - BkAS'-'

can a l s o be w r i t t e n ((I 8kz A ) d'

+

- f'],

+ f',u - d') 2 0 ,

p o s s e s s e s a unique s o l u t i o n , d! T h i s e n a b l e s us one by one t o d e f i n e u n i q u e l y a l l t h e u'+l= u i - l

+ kd',

(i 2 l ) ,

w i t h t h e d a t a (9.10). S i n c e A i s symmetric, w e n o t e t h a t t h e d+' (i 3 1) can b e o b t a i n e d from t h e s o l u t i o n s d' o f t h e N 1 m i n i m i s a t i o n problems (i = 1, ..., N - 1) :

-

minimise

f I d' I'

+ f Bk'(Ad'. d') + (7,d') ,

subject t o the constraints

d'EK, where 8-', u'-', d = u'-' + k8-I a r e , f o r i 2 2 , g i v e n by t h e solutiorTt h e p r e v i o u s problem and, f o r i = 1, '6 = u1, uo = uo, u1 = uo k6'.

+

We u s e t h e f o l l o w i n g t e r m i n o l o g y :

- e x p l i c i t scheme i f 8 = 0 , - semi-impticit scheme i f 0 < 8 < 1, - implicit scheme i f 8 = 1. rn Remark 9.1.

I n t h e same s p i r i t , t h e s o l u t i o n of t h e inequa-

lity

(ff where

j(u)

+ AU - f, u - u') +Au)-j(u') 2 0 ,

VUEK,

~~2-2,

i s d e f i n e d as i n (2.27), l e a d s t o t h e scheme

-

(""1

$f+

u'-1

ui+l

+Au)-j(

+ Ad+' - f', u - d-1 2k

ui+l

2-k u i - l ) u'+l

VUEK,

)20,

+

- ui-l 2k

EK,

or, i f A i s symmetric, t o t h e s o l u t i o n o f t h e problems

minimise f 1 d' 1'

+ f 8k2(Ad',d') + (f"',d 9 + A d ' ) ,

subject t o t h e constraints

Remark 9.2.

d'EK.

rn

It i s a p p a r e n t from t h e above remark t h a t t h e

(SEC. 9 )

Approximation of i n e q u a l i t i e s of order 2 i n t

483

methods of Chapter 2 , which were o r i g i n a l l y d e s i g n e d f o r s o l v i n g e l l i p t i c variational i n e q u a l i t i e s , now a l s o p l a y a fundamental p a r t i n t h e s o l u t i o n o f time-dependent v a r i a t i o n a l i n e q u a l i t i e s .

Remark 9.3.

A v e r y l a r g e number of o t h e r schemes is a l s o Some o f t h e s e a r e mentioned below ( o m i t t i n g t h e

possible. suffix h ) :

Scheme 1 ui+l

- 2 ~ +' u i - l

(9.11)

,,i+l

VVEK,

+ Au' - f i , v

,,i+l

-

- ,,i-l 2k

).O,

- ,,i-l 2k

E K ,

with

(9.12)

I

,I

= 8-

ui-l

+ 80 ui + 8+

ui+l

8-, O0, 8+ E [O, 11

where -.

,

and

8-

+ Oo + 8+ = 1 .

Schemes 11

I

and

d

as i n ( 9 . 1 2 ) .

Schemes I11 ( d e c o u p l e d schemes)

I

ui+l =

U'

+ kP,,[

ui+l/2

-

"1

,

where ,PKh i s t h e p r o j e c t i o n on t h e (,)k norm.

& i n t h e s e n s e of

The a p p r o x i m a t i o n o f schemes I and I1 i s s t u d i e d i n TrPmo i : r e s

lbl. 9.3.

Convergence o f t h e approximate i n e q u a l i t i e s

We s h a l l now i n v e s t i g a t e t h e s t a b i l i t y and, for h, k + 0 , t h e convergence o f scheme ( 9 . 9 ) . Using t h e n o t a t i o n of S e c t i o n 9.2,

484

Time-dependent variational i n e q u a l i t i e s

( CHAP.

6)

(9.9) is expressed as follows: (yi, u - d') + u(ui+', u - d') 2 (fi, u - d') , (i 2 I ) , (9.14) scheme

VUEK, d ' E K .

Before starting the investigation we recall Gronwall's Lemma: I f pi, i = 0, I,

Lemma 9.1.

..., n, are numbers which s a t i s f y

0 Q p 1 6 Cl , constant

Z0,

n

0 Q pn+

< Cl + C2 1 ki p i ,

n = 1, ..., N ,

i=1

with C, constant 2 0 and k i 2 0 , i = 1 ,..., N , N

1 ki = T , p o s i t i v e

constant

i= 1

then

(

)<

p . Q C , exp C2 C ki i:,

C, exp(C2 T ) ,

n = 2 , 3 ,..., N .

for

rn

stability

9.3.1.

We shall now prove:

Vhen ( h , k ) 4 0

Theorem 9.1.

- in

an arbitrary manner i f 0 = 1 , i f 0 Q 8 < 1, with the s t a b i l i t y condition, CkzS(h)' Q 1 - /?, fi f i x e d E 10, 1[,

- and,

(9.15)

we have, f o r a l l n

(9.18)

I

= 0, 1,

- 2 4 + u;-' k2

...

lhQ

c.

First, we establish: Lemma 9 . 2 .

(9.19)

I y' ;1

Under assunptions ( 9 . 6 ) and ( 9 . 5 ) we have Q C,

where C is a constant independent of h and k .

(SEC. 9 ) Approximation of i n e q u a l i t i e s o f order 2 i n t

Proof o f Lemma 9.2. (y'

k 2

7'

=

VU E K ,

So, noting t h a t

and t h a t u 1 + 0 k ~ ' = u o + ( 1 + ~ ) k 6 ' + ~yk' ,Z

d1-6'=-y1

(

We note t h a t

+ A(u' + OkS') - f ',u - d') 2 0 ,

and hence by taking u

48 5

+ A(u' + (1 + 0) k6' + Okz 7')

- fl, 2 7'

k

,

Q 0,

and by m u l t i p l y i n g by 2/k, and s u p p r e s s i n g t h e term a ( y l , y l ) , which i s p o s i t i v e , we have

I y i iZ Q

1 a(uo, yl) I + (1 + e) 1

SO, 6'

-

so) 1,

and ( 9 . 1 9 ) can t h e n be deduced immediately u s i n g ( 9 . 6 ) and (9.5).

Proof of Theo~ern9.1. (9.20)

(y"',

11

- di+' ) +

a(ui+l+e,

Taking u = d i + ' i n ( 9 . 1 4 ) , . obtain ($+I

+ 1 , we - di+1) >, (f"', 0 - d ' + ' ) .

W r i t i n g ( 9 . 1 4 ) for i

u=d'

i n ( 9 . 2 0 ) , and a d d i n g , we

- y i , d i + l - di) + a ( u f + l + @ - d+e,di+l - di) < ( f i + l - p,di+1 - di)

Q

NOW

d'+'

- d'

g+l +

=

6i

si + ai-1

-- -= 2

(9.21)

(7'+'

- yi,

y'+'

61+1 -

2

and hence, a f t e r m u l t i p l i c a t i o n by

$-I

2

1Q n Q

+ 7') + ~(6'+', a'+' - a'-') < &g',

N - 1,

-

2/k,

b u t , n o t i n g t h a t a ( , ) i s symmetric, we have

and, f o r

have

y'+'

+ y'),

486

Time-dependent VariationaZ i n e q u a z i t i e s

( CHAP.

Now

+

1 4ao,a]) I Q f f a(al), lu(an,P+l - 691 Q f a ( 6 " ) + f k Z a ( y " + ' ) , and hence, suppressing the term

!a(6"+'), which is positive 2

Moreover

So that, summing ( 9 . 2 1 ) from 1 21 y " + I I2

i = 1 to

1 + p(69 Q c+c ckI I

i=1

+ 21 4ao)+ 1

p('(b1)

y'12

n Q

N - 1 , we obtain

1-8 +2 k2 a(y"") +

+ 1 y1 I2 .

From Lemma 9.2 (I yl 1 Q c ) , assumption ( 9 . 5 ) ( ~ ( 6 ' Q ) c) and assumption ( 9 . 2 ) (~(60)Q C) , we obtain , after multiplication by 2,

(9.22)

[i

- (1

-

e) c~ S ( ~ ) ZI] y n + l

12

+ 469 Q c + c i k I yi i=I

12.

6)

(SEC. 9 ) Approximation o f i n e q u a l i t i e s oy order 2 i n t

487

By t h e n imposing ( 9 . 1 5 ) i f 0 Q 0 < 1, we o b t a i n ( 9 . 1 8 ) by G r o n w a l l ' s lemma. The upper bounds ( 9 . 1 7 ) and (9.16) t h e n f o l l o w immediately ( s e e ( 9 . 2 2 ) ) . 8

Remark 9.4. I n t h e c a s e i n which Vh, k + 0, t h e i n e q u a l i t y ( 9 . 1 4 ) i s i n f a c t , for i = 1 , an e q u a t i o n ( t h e convex s e t i s n o t " s a t u r a t e d " ) ; we have

6' - 60 k

+ Au'"

11 6' 11

11 6'

-f o = 0,

and hence =

< II 6'

+ kAu' + Okz A6' II

+k

- k f o 11 Q

II Au' 1I + Okz II A6' II + k II f o II ,

from which it c a n be deduced t h a t when k i s s u f f i c i e n t l y s m a l l , 6' i s bounded for t h e vh norm i n d e p e n d e n t l y of h and k and assumption ( 9 . 5 ) i s in f a c t unnecessary. 8

Weuk convergence

9.3.2.

From aksumptions ( 9 . 1 ) , ( 9 . 2 ) , ( 9 . 3 ) and under t h e c o n d i t i o n s o f Theorem 9.1, t h e sequence uh,k

=

u;

xi

s a t i s f i e s , f o r an e x t r a c t e d subsequence a l s o denoted by uh,k,

(9.23)

Now l e t v b e a r b i t r a r y w i t h v E Lz(O, T ; V), v ( f ) E K, a. e . I E [o, r ] and l e t v h , k b e a sequence which converges t o v i n t h e s e n s e of

(9.4). R e p l a c i n g v by

+- :

(.:'I

v; i n

Upe,V ; ) -(fl, h

(9.24)

to

( 9 . 1 4 ) , we o b t a i n

l4- 1 +Ah

Vi-

up'-u;-')*2 2k

We s h a l l show t h a t by m u l t i p l y i n g by k and summing from i = 1 N - 1 we o b t a i n

(9..25)

Ior

(ii"

+ A;,

v ) df

- JOr ( J v - 2)dt 2 JoT(Z+ Aii, 2 )dl ,

Time-dependen t variational inequalities

488

(CHAP. 6 )

t h u s e s t a b l i s h i n g t h a t ?i i s a s o l u t i o n o f t h e problem. Since t h e l e f t - h a n d s i d e o f ( 9 . 2 5 i s o b t a i n e d i m m e d i a t e l y , it remains t o examine t h e r i g h t - h a n d s de o f ( 9 . 2 4 ) . M u l t i p l y i n g t h e r i g h t - h a n d s i d e of ( 9 . 2 4 ) by 2 k, we o b t a i n

2 k0i.k = (6' - ai-', 6' = 16'1:

+

- 1 8 - 1 ;1

#-I),

+ ( A , uF', u F 1 - .;-I)

+ (A, z p ,

=

- .;-I).

However ( o m i t t i n g t h e s u f f i x h )

(9.26)

- u(u'-') + - e) [a(ui - ui-11 - a(ui+l - ui)3 + ea(ui+l - U I - ~ ) .

2 ~ ( d " ,ui+'

+ (1

- d-l)

= a(ui+I)

Hence, summing from i = 1 t o N we o b t a i n

-

1, and o m i t t i n g t h e s u f f i x h,

Now w e have t h e f o l l o w i n g r e s u l t s

liminf16N-1)22 I;'(T)('

( f r o m (9.2311, ( f r o m ( 9 . 2 ) and ( 9 . 2 3 ) ) , liminfa(fl) 2 a(Ti(T)) ( f r o m t h e c o n s i s t e n c y of a(,) and lim infa(uN-I) 2 * ( T ) ) } (9.23) ) 16°1+IU112=

);'(o))'

a(#') + a(uo) = u(U(0))

}

4 u 0 ) + 4 u o ) = a(Si(0))

'

( f r o m t h e convergence o f (9.2311,

and hence

lim inf 2 Dh,k 2 2 D = I Z(T)1' - I Z(0) 1'

- u(Z(0)) + lim inf r , with

+ u(ii(T)) -

a(,) and

(SEC. 9 ) Approximation o f i n e q u a l i t i e s df order 2 i n t

489

Using G r e e n ' s f o r m u l a (Lemma 1.4)(l) and d i v i d i n g by 2 w e have,

D =

loT

(Ti",

+

2)dr

a@, 2)dr

+ lim inf I2

By summing and p r o c e e d i n g t o t h e l i m i t i n ( 9 . 2 4 ) we t h e n obtain

(9.28)

loT + (is"

Ai

- J u - Z') d t 2 lirn inf -2r '

We n o t e a t t h e o u t s e t t h a t i f 0 = 1 t h e n liminfr 2 0 and Z i s In the other indeed a s o l u t i o n of t h e i n i t i a l i n e q u a l i t y ( 8 . 3 ) . c a s e s (0 < 1) we have

riow a@') Q C

(1 6' 1:

'c k2a(d') < c

N- 1

i= 1 '

(from

(by hypothesis)

Q C ,

N- 1

k2

1hi

+26i-

1 112

c k2

N- 1


i= 1

i= 1

(9.17)),

and hence N- I

IrlQ(1 -0)k2C+6kC

1 kQ(1 -0)k2C+6kC,

i= 1

and we have lim inf I r I = 0 and hence c l e a r l y ,

(9.29)

lirn inf r = 0 ,

which p r o v e s ( s e e ( 9 . 2 8 ) w i t h ( 9 . 2 9 ) ) t h a t i i s i n d e e d a . s o l u t i o n S i n c e t h e s o l u t i o n ii i s u n i q u e , it of t h e i n i t i a l i n e q u a l i t y . i s n o t n e c e s s a r y t o e x t r a c t a subsequence, and t h e e n t i r e sequence u. We can now s t a t e : converges towards Theorem 9 . 2 .

-

when

(h, k) + 0

i n an arbitrary manner i f 0 = 1, and, i f O Q 0 < I , w i t h the s t a b i l i t y condition ck2s(h12Q 1 - j?, j? f i x e d

( l ) S i n c e A(t)

w e have

E

lo, 11,

i s symmetric, f o r a l l u such t h a t

u , u ' E L ~ ( O , V) T;

Time-dependent v a r i a t i o n a l i n e q u a l i t i e s

490

( CHAP.

.6)

we have, w i t h u a s o l u t i o n of t h e i n i t i a l i n e q u a l i t y ( 8 . 3 ) ,

* *

weakly weakly ph 6uhsk-B uu' weakly q h y u k k -B u" weakly qk uh,k + u

P h uh,k + uu

Remark 9.5.

L"(0, T ; H ) L"(0, Tr F) L"(0, T ; F) L"(0, T ; H ) .

The above proves t h e e x i s t e n c e of

s o l u t i o n o f 8.3.

9.3.3.

* *

in in in in

U E X, ~ the

8

Strong convergence

In t h i s last section w e s h a l l e s t a b l i s h :

When (h, k) + 0

Theorem 9.3.

-

-

i n an a r b i t r a r y manner i f 0 = 1, and, i f 0 Q 0 < 1, w i t h t h e c o n d i t i o n

CkzS ( I Z )Q~ 1 - /? , /? f i x e d

E

10, 1[ ,

we have, w i t h u a s o l u t i o n o f the i n i t i a l i n e q u a l i t y ( 8 . 3 ) , s t r o n g l y i n V , t E [O, u'(t) s t r o n g l y i n H , t E [0,

P h u h , k ( t ) + .m(t)

q,, &h,k(t)

-B

TI , .

8

Before commencing t h e proof o f t h i s theorem, it should be p o i n t e d out t h a t w e t a k e u(,), a(,) a s i n 6.3.3 and t h a t we s h a l l omit u s i n g t h e o p e r a t o r s qh,phru i n c a s e s where t h e r e i s no p o s s i b l e ambiguity. We s h a l l now e v a l u a t e t h e q u a n t i t y

X(t) =

I 11'

- qh

6Uh.k

+ ci(u(f)- ph u l t , k ( t ) ) .

1'

By expanding, we o b t a i n

W ) = X , ( t ) - 2 XZW '

+ Xdr) ,

with

Using t h e assumptions and t h e lemmas concerning i n t e g r a t i o n by-parts, t h e f i r s t term becomes

X,(t) = 2

1:

(U", U') dS

+2

1:

U ( U , U')

dS + U(U,)

L e t us now examine t h e second term.

Let

+ 1 u'(0) 1'. nk

8

be such t h a t

(SEC. 9 ) Approximation of inequalities of order 2 in t

IE

[nk, (n + 1) k[ and p u t N- 1

N

1ukx:

We r e c a l l t h a t uh.k=

and t h a t a(,) i s symmetric and does

i=O

We r e a d i l y o b t a i n ( o m i t t i n g PA) t h a t

n o t depend on t i m e . a(u(t),

p h uh,k(t))

1:

qu’, d ds + c(g,$1 .

..., N,

However f o r i = 1,

s“

=

we h a v e :

q u ’ , &’) & = qui, ui-1)

- qui-1,

.;-I),

(i- 1)k

and h e n c e , by a d d i n g Z(ui, u$ t o e a c h s i d e

qu‘, ui)

=

I&

qu‘, u;- 1) ds + qu’,u: - ui-

) +$u’-l,u:-l),

( I - 1)k

and f i n a l l y a(u(f),

ph u h , k ( I ) ) = -k

a(‘(0),

1;

4uf,p h

uh,k)

ds +

fa

C?(Tk

u, p k

auk,,)

ds +

.

uh,k(o>)

S t i l l for t h e second t e r m , l e t us now e v a l u a t e (u‘,6uk,&. have (u’, 6 u h . k ) =

I

(u”, an) ds

+ (u’(nk),6”).

But

I“

(u”, 6’- ) ds = (u’(ik),6’-

1)

- (u’((i - 1) k), 6‘- I ) ,

(i- l)k

and, by a d d i n g (u’(ik),8 - l ) t o e a c h s i d e

(u’(ik),a9 =

I&

(u”, ai-1) cis

+

(i- Ilk

+ (u’((i- 1) k), and hence (U‘, & . k )

=

I:(.”,

1”

(u’(ik),y9 ds

+

( i - l)k

8-1)

,

q h 6Uk.k) ds +

I.

mh

(Tk

u‘, q h p h , k )

dS +

+ (u’(o),q h 6”) . W e can t h e n p r o c e e d t o t h e l i m i t i n

X, and o b t a i n

We

Time-dependent v a r i a t i o n a l i n e q u a z i t i e s

492

6)

( CHAP.

.

lim X2(r) = A',([)

L e t u s now examine t h e t h i r d term. We f i r s t w r i t e t h e inequFor t h i s purpose w e s h a l l use t h e a l i t y i n i n t e g r a t e d form. notation

We can t h e n w r i t e

(9.14) i n i n t e g r a t e d form for r e [ & ,T - k]

:

(9

N- 1

where we t a k e

uh.k

=

"1ui

such t h a t

i= I

ph oh,& + nu'

s t r o n g l y i n L2(0, T ;F )

However, f o r r E [nk, (n

1; lk

+ 1) B[, n 2 1, we

( n + 1)k

(9.31)

=

(i+l)f

-

= i= 1

ui E 4.

have

1,

(n+l)k

-

and hence, u s i n g ( 9 . 3 1 ) , ( 9 . 3 2 ) and ( 9 . 2 6 ) , w i t h c(k) + 0, when k

+ 0,

(SEC. 9 )

Approximation of inequalities of order 2 in t

We then have, with ( 9 . 3 3 ) into (9.30):

t u(f+k(t))

= (1

493

- 4) a ( u h . k ( r ) ) , and substituting

Proceeding to the limit, we obtain lim X&) = 1 u'(0) 1'

+ a(u(0)) + 2

Assembling the results we have lim X(r) = lim X,(r)

1:

(u"

+ Au, u') ds

- 2 lim X 2 ( t ) + lim X3(r) = 0 ,

which establishes the theorem. 10. NUMERICAL SOLUTION OF INEQUALITIES OF THE SECOND ORDER IN t.

In this section we propose to solve, using simplified data, Examples I and I1 of Section 8. The numerical investigation will enable us to appreciate the relative merits of the implicit and explicit schemes, and to assess the approximations made, in terms of h and k.

10.1.

Solution of Example I

Considering Example I, we shall seek to solve the inequality

494

Time-dependent variational inequalities

(u".

(10.1)

0

- u')

+ a,(u',

0

- u3

> (f,u - u') ,

-j(u')

u(0) = u,

,

+ a&,

vu E

v,

u

- u') + j ( u ) v,

( CHAP.

6)

-

u' E

u'(0) = ub ,

We s h a l l compare t h e s o l u t i o n u of (10.1)w i t h t h e s o l u t i o n of t h e " q u a s i - s t a t i c " problem ( s e e Duvaut-Lions Ill),

(10.2)

n,(u", u

Vue

+ ao(t7, u - E') + i(u)

- ii')

v,

v.

i7E

ii(0)

-Air) 2 ( J 0 - 3), =zoo,ir(0) = 3,.

For t h e s e two problems we t a k e a, = a1 = a

w i t h a(u, u) =

1[,

= 10, 11 x 10,

r = To u rl

I

grad u grad u dx

,

( s e e F i g u r e 10.1),

f = 40sin2rrx, g = 0.2 .

rorlr

The d i s c r e t i s a t i o n of B i s c a r r i e d o u t i n t h e u s u a l manner and t h e d i s c r e t i s e d V , a(u, u) and f a r e d e f i n e d i n t h e normal way. We t a k e

h = 1/20,

k

= 10-2.

rl

Fig. 10.1. For t h e i n i t i a l d a t a w e t a k e

(10.3)

u,=O,

=o,

ub-0,

4 = 0 .

S o l u t i o n o f i n e q u a l i t i e s of order 2 i n t

(SEC. 1 0 )

(10.4)

{2

495

ub = 0.2,

0,

= uo = 0 ,

iib

=

0.2.

We approximate the two inequalities by semi-implicit schemes. Putting

6" =

u"+1

lP

- u"

k

'

+ u"+l = u " + yk F ,

the discretisation of (10.1)can then be written

+

a0

VO€

(vu", + -t

)+

6", u - 6"

j(u)

- j(6") 2 (1;u - 6") ,

and the discretisation of (10.2)can be written

01(6", u - 6") + a,

+ j ( u ) - j(6") 2 ( A v - P ) , u E V .

In order to test the convergence u ( t ) + C ( l ) , when t + m , we put

The results shown in Table 10.1 confirm the converaence results. 2

0.5 case (1 0.3) c a s e (1 0.4)

2.1x10-2 2.2x10-2

4~10-~ 9~10-~

4 1.5x10-' 4~10-~

8

1~10-~ 1.5~lO-~

For the two inequalities (solved by over-relaxation at each time-step) the solution required 5 min to reach 1 = 10 on a CII 10070. Results completely analogous to those of Table 10.1 were obtained by replacing V b y K = { U E H ' ( P ) ( U = O on

ro,uaO on

in (10.1)and (10.2) and taking g = 0. 10.2

Solution of Example I1

This involves solving the inequality (10.5)

(u" - AU - J u - u') 2 0 , VU E K , u E Y o

f,},

496

Time-dependent variational i n e q u a l i t i e s

( CHAP. 6 )

with

V = Hd(Q), Q an open bounded domain of R z , K = { u I o ~ V , u > O , a . e . on a } , .Xo = { u I u, U' E Lz(O, T ; V),U" E Lz(O, T ; V') , u(0) = uo given , u'(0) = u, given u'b0,a.e.

on Q = ] O , T [ x 6 2 ) .

We t a k e 62 = lo,1[ x ]0,1[ w i t h t h e exact solution i s

u =

j

lOx(1

100

5

uo, u, d e f i n e d in such a way t h a t

31 < t < -23,

- x)y(1 - y )

- s i n F t ) x(1- x)A1- y ) .

?2 < t < l = T .

The s o l u t i o n ( s e e F i g u r e 1 0 . 2 ) "ascends" from t = 0 t o f, s t a b i l i s e s between f and 5 , and "re-ascends" f o r t = 4 t o 1. I n Order t o r e p r e s e n t t h i s problem i n t h e form o f an a c t u a l i n e q u a l i t y ( r a t h e r t h a n an e q u a t i o n ) , we t a k e

f=

{

- AU , - Au - (t - f ) ( s - t ) , U" - AU ,

O < l < f ,

U"

f < t < 5, f < t < l .

Representation of t h e exact solution of the f i r s t - o r d e r constrained hyperbolic inequality. Cross-sections of t h e solution on [O, 11 x { 4 (v = 4) for v a r i o u s i n s t a n t s : t E [0, 11. 1 9

I

I

I

4

Values of t h e

0

Cross-sections o f t h e s o l u t i o n for y = i

Fig. 10.2.

1

S o l u t i o n of i n e q u a l i t i e s of order 2 i n t

(SEC. 10)

V , with

The s t a n d a r d d i s c r e t i s a t i o n w a s used f o r

h

497 =

1/20.

Description of t h e schemes

10.2.1

The i n e q u a l i t y was approximated by t h e scheme d e s c r i b e d below. I n each c a s e u: = u(0) = 0 ,

6;

=

Si(M) = u'(0, M ) and

u: = u:

+ k6;,

(UP'- u ; ) / k .

For t h e schemes ( o m i t t i n g t h e s u f f i x h ) w e have, w i t h

i 2 0,

E x p l i c i t scheme Tii

- 6'-1 k

6' = max

- Aui - f'=

[ai, 01,

0,

u i + l = ui

+ k6'.

"Implicit i n equation" scheme

8' - 6'-1 k

- Aiji - f i

= 0,

with

2

+ kd',

=

6' = max [Ti', 01, u i + l = ui + k 6 ' . "Semi-implicit i n i n e q u a l i t y " scheme

with

ui+l

= ui

+ k6'.

"Impl i c i t i n i n e q u a l i t y " scheme

w i t h ui+' = ui

10.2.2

+ k6'.

R e s u l t s of the various schemes

The f o l l o w i n g t a b l e ( T a b l e 1 0 . 2 ) g i v e s t h e r e s u l t s f o r t h e schemes d e s c r i b e d i n 1 0 . 2 . 1 . For schemes o t h e r t h a n t h e e x p l i c i t scheme, t h e s o l u t i o n method used a t each t i m e s t e p i s t h e Gauss-Seidel method, o r t h e GaussS e i d e l method w i t h p r o j e c t i o n , w i t h t h e v a l u e s of o corresponding to = 2[1

+

(1

- 2(1

2

h f )h f + (rrf/2) hf/kf

)"']

-

.

The t e r m i n a t i o n c r i t e r i o n f o r t h e Gauss-Seidel method i s

4 98

Time-dependent variational inequalities

( CHAP.

i s t h e solution calculated a t t h e i t h i t e r a t i o n of t h e The q u a l i t y o f t h e approximations i s measured i n terms

where method. of

ECART, =

lu;(M)

- u ( M ; i k )I .

(“1

MEnh

Table

10.2.

Hyperbolic i n e q u a l i t y of Example 11.

The problems were s o l v e d on a C I I 10070.

10.2.3.

Conclusions of Section 10

For t h e problem ( 1 0 . 5 ) c o n s i d e r e d h e r e , w e have:

E x p l i c i t scheme :

-

always t h e danger of divergence, w i t h overflow,

-

s t a b i l i t y a p p e a r s t o correspond t o

k/h Q ZxlO-’.

The other schemes:

(*)

6)

-

virtually identical,

-

an a d m i s s i b l e v a l u e o f k/h i s 2~10-’,

See Translator‘s Note on page 432.

(SEC. 11)

-

Nwnerical computation of Bingham f Z u i d flows

499

For t h e same p r e c i s i o n , t h e s e schemes a r e t h r e e t i m e s f a s t e r H t h a n t h e e x p l i c i t scheme.

11.

NUMERICAL COMPUTATION OF THE FLOW OF BINGHAM FLUIDS

11.1

N o t a t i o n and s t a t e m e n t of t h e problem

We s h a l l g i v e h e r e o n l y t h o s e r e s u l t s which a r e e s s e n t i a l t o t h e u n d e r s t a n d i n g of t h e n u m e r i c a l r e s u l t s , and we r e f e r t h e r e a d e r t o Duvaut-Lions /1/ f o r a more complete d i s c u s s i o n of t h e 61 w i l l d e n o t e a n open bounded domain o f R w i t h boundproblem; a r y r. Unless s t a t e d o t h e r w i s e , we s h a l l r e s t r i c t o u r a t t e n t i o n If u = { u l , ..., } h e r e a f t e r t o t h e two-dimensional c a s e (n = 2 ) . i s a f u n c t i o n w i t h v a l u e s i n R”, we p u t

(11.1)

(11.2) “

P

(11.3)

(11.4)

(1 1.5) (11.6)

V = { u I o ~ ( H i ( 6 1 ) ) ”and div u = 0 )

(11.7)

H = { u ( o ~ ( L ~ ( 6 1 ) and )” divu=O

and

u.nI,=O};

H i s t h e c o m p l e t i o n of V i n (L’(C2))” and we have: (11.8)

V c H c V‘;

V i s equipped w i t h t h e t o p o l o g y o f (Hd(61))” and we s h a l l d e n o t e by ( I u ( I t h e norm o f u i n V. Similarly of u i n H . For U E V, it i s c l e a r t h a t :

(11.9)

u(u,u)~aIIuIlz;

a>O.

We c o n s i d e r t h e f o l l o w i n g problem:

(uI

w i l l d e n o t e t h e norm

Time-dependent variational i n e q u a l i t i e s

500

in

Problem 1. 10, T [ :

6)

Find u(t) which s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s

(11.10)

u E L’(0, T ; V) n L“(0, T ; H), u’ E L’(0, T ; V‘)

(1 1.11)

(u‘(t), u - u(t))

(11.12)

( CHAP.

+ vu(u(r), u - u(t)) + b(u(r), u(t), U) + + g j ( u ) - gj(u(t)) 2 ( f ( t ) ,u - u(r)), Vu E V, a . e .

40) = U , E H ,

in I

.

~ E L ’ ( O , TV’). ;

The v a r i a t i o n a l i n e q u a l i t y (ll.ll), t o g e t h e r w i t h (11.10) and ( 1 1 . 1 2 ) c o n s t i t u t e s a model d e s c r i b i n g t h e flow o f a r i g i d v i s c o p l a s t i c Bingham f l u i d , w i t h v i s c o s i t y v and p l a s t i c i t y t h r e s h o l d g ( s e e Duvaut-Lions, loc. c i t ) . Note t h a t f o r g = 0, we r e c o v e r t h e c l a s s i c a l v a r i a t i o n a l f o r m u l a t i o n o f t h e Navier-Stokes equati o n s f o r Newtonian f l u i d s . For g > 0 , w e o b s e r v e zones i n which t h e behaviour i s t h a t o f a f l u i d w i t h v i s c o s i t y v . The v a r i a t i o n a l f o r m u l a t i o n a u t o m a t i c a l l y t a k e s account o f t h e f r e e boundary s e p a r a t i n g t h e two zones; t h e p r e c i s e determination of t h i s boundary w i l l be one o f t h e p r i n c i p a l d i f f i c u l t i e s involved i n t h e numerical t r e a t m e n t o f t h e problem. We r e c a l l h e r e simply t h a t i n a Bingham f l u i d t h e s t r e s s t e n s o r uij and t h e rate-of-deformation t e n s o r D , a r e r e l a t e d by t h e constitutive l a w : Qij

= - Paij

+ g(DJ(41)”’) + 2 @ i j -

This e x p r e s s i o n i s o n l y meaningful i f D,, # O . If D , , = O , t h e s t r e s s t e n s o r i s i n d e t e r m i n a t e and we have a r i g i d zone; i f g i n c r e a s e s , t h e r i g i d zones grow and are l i a b l e The t o block t h e flow completely f o r g s u f f i c i e n t l y l a r g e . s c a l a r -p r e p r e s e n t s t h e s p h e r i c a l p a r t of t h e s t r e s s t e n s o r . We may t a k e p t o be t h e p r e s s u r e .

Remark 11.1. I n p r a c t i c e , it i s probable t h a t i n t h e c a s e o f Bingham f l u i d s ( g > O ) it i s p e r m i s s i b l e t o n e g l e c t t h e nonl i n e a r terms i n t r o d u c e d i n t o (11.11)by t h e t r i l i n e a r form I f we a c c e p t t h i s s i m p l i f i c a t i o n , most o f t h e results qu, u, 4 * which we s h a l l s t a t e f o r t h e two-dimensional c a s e a r e a l s o v a l i d t o any dimension. The f o l l o w i n g r e s u l t s a r e proved i n Duvaut-Lions 111. Theorem 11.1 In two-dimensional space (n = 2)problem 1 possesses a unique solution. Theorem 1 1 . 2 For n = 2, i f u(t) denotes the solution of problem I, there e x i s t functions mij (1 < i , j < 2) such t h a t : (11.13)

m i j E L Q ( Q ) , mij = mr(Q = L! x 10, T o

Numerical computation of Bingham f l u i d f l o w s

(SEC. 11)

501

2

C

(11.14)

r n i j r n i j < 1 a.e.

in Q .

i,j=1

2

1

(11.15)

n1ij D i j ( U )

=

(

DiAu> DiAu))"*

a. e . i n Q .

iJ=1

i.j=I

(u', o)+vu(u, u)+b(u, u, o)+&g

(11.16)

VUEV, a.e.

u)

in t .

Conversely i f (11.13)-(11.16)are s a t i s f i e d , w i t h : u E L2(0,T ; V) n Lm(O,T ; H ) , u' E L2(0,T ; V')

(1 1.17)

then

u

, u(0)

= uo

is the solution of Problem I.

11.2

Numerical schemes

We s h a l l c o n s i d e r h e r e a n e x t r e m e l y s i m p l e scheme which has i n p r a c t i c e proved e f f e c t i v e f o r a p p r o x i m a t i n g Problem I. The problems posed by i t s p r a c t i c a l a p p l i c a t i o n w i l l b e examined i n S e c t i o n 11.3. A s r e g a r d s t h e s p a t i a l d i s c r e t i s a t i o n , we s h a l l assume f o r s i m p l i c i t y t h a t we a r e c o n s i d e r i n g a n i n t e r i o r approxwe s h a l l mention i n p a s s i n g t h e minor modi m a t i o n Vhof V (I); i f i c a t i o n s which must b e made i n t h e c a s e where a n e x t e r i o r apprused. It would be p o s s i b l e t o t r a n s p o s e t h e r e s oximation u l t s t o any o t h e r t y p e o f d i s c r e t i s a t i o n v e r y e a s i l y , w i t h o u t a f f e c t i n g t h e p r i n c i p l e o f t h e scheme. For any i n t e g e r n, 0 Q n d N 1, we p u t

rhis -

.J

r(n+l)t

(11.18)

=

f(r)drE

V'

(0 Q 8 d 1).

Ilk

Given a t r i a n g u l a t i o n ThOf 0, we assume ui g i v e n i n v h , chosen We s h a l l assume so t h a t u: t e n d s t o uo i n H when h t e n d s t o z e r o . We s h a l l d e f i n e by r e c u r r e n c e a sequence t h a t IIuill i s bounded. U;: o f e l e m e n t s o f V,with which we s h a l l t h e n a s s o c i a t e ' ' a p p r o x h a t i n g functions". We w r i t e :

To s i m p l i f y t h e n o t a t i o n , we d e n o t e by U; a sequence which a c t u a l l y depends on h and k. (l)

Some i n d i c a t i o n o f t h e c o n s t r u c t i o n o f t h e s p a c e s V,, p, i s g i v e n a t t h e end o f t h i s s e c t i o n .

Time-dependent variational i n e q u a l i t i e s

502

( CHAP. 6 )

For 8 > 0, we have (11.21)

and hence (11.22)

U;"-u;: k

- 4+'-24$ Bk

However problem (11.24) is not coercive on vh, which brings in difficulties regarding the existence (and computation) of 4". Using the methods of Chapter 4, we may verify:

H

Proposition 2.1. I n two-dimensionaz space ( n = 2), (11.20), (and thus also (11.23)),with 6 > 0 and k " s u f f i c i e n t l y smalt", define G+' cmd G+' uniquely. By methods analogous to those above, it can be shown (details ' of the proofs are given in M. Fortin /I./) that the U; and 4 defined by (11.20) satisfy, for e 2 f :

Numerical computation of Bingham f l u i d f l o w s

(SEC. 11)

503

C = Const. independent of k and h provided that:

kSz(h) d C,, an arbitrary constant.

(11.28)

For e have :

>$,

condition (11.28) is superfluous and in addition we

N- 1

(11.29)

k

1 (1 U;:" - U;: 112

d C1.

n=O

The above estimates are s u f f i c i e n t , in the case of inequalities f o r l i n e a r operators, to demonstrate the convergence, but they are insufficient in the case of inequalities (and equations) for nonl i n e a r operators. For these cases we have to obtain a supplementary estimate allowing us to proceed to the limit b y compactness. To this end we use a method due to L. Tartar and R. T6mam /1/ (see also the work of Ternam / 2 / and the bibliography therein). We write (11.20)in the form ( l ) (11.30)

1

- , ( u ; + ~- 4, G+@ -

Uh(t))

Q va(U;:+@,

- b(#'@? #+@, 4+@ - vh(t))

+ (f;+', 4+@ - Uk(f)) Vt E

[nk, (n

+ 1) k [ .

9

4+@ - Uh(f)) -

+ a(uh(t)) - d(4'B) t

voh E

L2(0, T ; vh)

9

Consider the left-hand side of (11.30). Defining uh(t)as the linear function over each interval [nk,(n l ) k ] which is equal to U;: at nk , we have:

+

( ')

Note that b(u, u, w ) = - b(u, w, u) -

"5

1 i=1

by parts) and Nu, u, u) = 0, with u E

(div u) uj w j d x

( integration

R

v

and Vo~(Ht(l2))".

Time-dependent variational i n e q u a l i t i e s

504

( CHAP. 6 )

Moreover

Moreover we have :

Thus, by (11.30), we have:

where hh denotes t h e s t e p f u n c t i o n e q u a l t o However we have:

and i n a d d i t i o n

4"

i n [nk,(n

+ I)k].

Numerical computation o f Bingham fluid flows

(SEC. 11)

505

Similarly:

I(

uh(t)dr)

u,O,

(11.38)

I 1- + s

U

dt

I u: 1’

Thus

f

IorAuh(f))

dt .

We s h a l l now m a j o r i s e t h e r i g h t - h a n d s i d e . t h e l i n e a r terms

F i r s t , consider

S i m il a r ly ,

(11.41)

’

lor

(f;+e, iih(t) - uh(t)) dt d C

loT1 :1 f(t)

dt

+

const.

For t h e n o n l i n e a r terms w e have

and hence by u s i n g t h e e s t i m a t e s o f Chapter h v a u t - L i o n s /1/:

6, S e c t i o n 3 - o f

506

Time-dependent variationaZ i n e q u a l i t i e s

( CHAP.

6)

Finally j(v,(t)) dt

G C

+g

11 u,,(t) 11’

dt

G

const.

Consequently, we have shown that the approximate function satisfies :

uh(t)

(1 1 .45)

Ior-uI

uh(t

on condition that E > 0, we have: (11.46)

u,(t)

+

U)

1’

- uJ,(t) d!

< CO

kSz(h) < C , (arbitrary constant) ; thus f o r aZZ

resides in a bounded set of H“’-‘(O T; , H).

8

The above estimates, combined with the compactness of the injection from V + H , allow us to proceed to the limit. We have: Theorem 11.3. We assume that h and k + 0, i n an arbitrary manner i f 8 > f , and d t h condition (11.28)i f 8 = f Then uJ,-+u weakly“ i n Lw(O,T;H)and weakZy i n L’(0,T;V). 8

.

Remark 11.2. We refer to M. Fortin /1/ for details of the proof of Theorem 11.3 and for the strong convergence results.

8

Construction of i n t e r i o r and exterior approximations of V. There are essentially three classes of possibilities for approximating V by spaces V, : (i) interior approximation v h t V ; (ii) interior approximation of (HA(f2))” (n = 2 or 3), the condition divu = 0 being satisfied only approximatezy ; (iii) approximation by spaces of functions which are not in (H’(62))”(non-conforming approximations). 8

An exhaustive study of this topic will not be undertaken here.

Nwnerical computation of Bingham fZuid flows

(SEC. 11)

507

We s h a l l merely g i v e some g e n e r a l i n d i c a t i o n s and r e f e r t h e r e a d e r t o t h e b i b l i o g r a p h y f o r p r o o f s , a n d i n p a r t i c u l a r t o t h e work o f Crouzeix and R a v i a r t /1/ We suppose t h a t B i s a p o l y h e d r a l domain o f l e t Fhbe a t r i a n g u l a t i o n i n t o non-degenerate s i m p l e x e s K ; i f h(K) ( r e s p . p ( K ) ) denotes t h e diameter of K ( r e s p . t h e diameter o f t h e sphere inscr i b e d i n s i d e K ) we assume

.

w";

+

K E F , ~ w i t h v e r t i c e s ui,,, i = 1, 2, ..., (n I ) , we d e n o t e by t h e b a r y c e n t r i c c o o r d i n a t e s of a p o i n t x o f W w i t h r e s p e c t t o t h e p o i n t s ui.,. If I.i,,(x)

.

Example 1 ( S i t u a t i o n ( i i )) I n two-dimensions ( t o s i m p l i f y t h e d e s c r i p t i o n ) we d e n o t e by W h t h e s p a c e o f f u n c t i o n s vh c o n t i n u o u s in such t h a t :

,1

1)

v h

2)

VhIr

E s p a c e o f polynomials o f d e g r e e =

< 2(P2),

V K EYh,

0.

A f u n c t i o n U h E wh i s c h a r a c t e r i s e d by i t s v a l u e s a t t h e v e r t i c e s ui,, and a t t h e m i d p o i n t s (lij., o f [ui,,uj,,]. We t h e n d e f i n e V h a s a s u b s p a c e o f ( H @ ) ) 2 , but not as a subspace of V , by : v h E ( W h ) ' and (11.47)

lKdiVUhdX

=

0 VKEFh.

If u h ( r e s p . u ) d e n o t e s t h e s o l u t i o n o f t h e approximate l i n e a r ( r e s p . c o n t i n u o u s ) s t e a d y - s t a t e problem, it can be shown ( s e e Crouzeix-Raviart /l/) t h a t

where h d e n o t e s t h e maximum d i a m e t e r o f

KEY,,.

Example 2 ( S i t u a t i o n ( i i ) ) . The e s t i m a t e (11.48) i s n o t what might be e x p e c t e d for t h e o r d e r o f a n a p p r o x i m a t i o n u s i n g second C l e a r l y t h i s a r i s e s from t h e f a c t t h a t , degree p o l y n o m i a l s . f o r m a l l y , t h e a p p r o x i m a t i o n (11.47)o f t h e c o n d i t i o n d i v u = 0 'I i s o n l y f i r s t order a c c u r a t e . We can improve upon t h i s w i t h o u t i n t r o d u c i n g general t h i r d degree p o l y n o m i a l s . If we w r i t e lii n p l a c e o f Ai,,(x), we i n t r o d u c e (11.49)

P , = s p a c e g e n e r a t e d by t h e polynomials I.,

I.2,

I.2

I.,,

I.3

I.,

and

1, i.2 2,

.

I.:, I.;, I.:,

508

Time-dependent variational inequalities

( CHAP.

6)

We then introduce

an element u, of Whis characterised by its value at the vertices q K , at the midpoints aigKand at the barycentres of K, K E y h ‘ k . We then introduce (11.51)

v, =

{

uh

I,

I u, s(wh$

q div uli cix = 0 vq E P, (polynomials

of degree Q 1 ) and V K E F , It can be shown (see Crouzeix-Raviart 111) that (1 1.52)

I( t l h - I( I I ( H L ( ~ ) ) ~

Q

ch2.

8

The method of Example 1 was introduced by Fortin /l/¶ in which the extension of the above example into three dimensions and the investigation of higher-order approximations can also be found. 8

In two dimensions, a classical Example 3 (Situation (i)). means of constructing “test functions” which satisfy the condition If divu = 0 is to work with the stream function and to make use of the fact that since 62 is a simply-connected bounded open domain, the mapping (11.53)

~

-+u =

establishes an isomorphism from Hi(62) onto v. We then construct an approximation of H&?) formed from polynomials of degree 5 on K, whose image is taken using (11.53). See Fortin 111, Thomasset /1/ and the account by R. T6mam 121. 8

Example 4 (Situation ( iii)) for simplicity that n = 2 ) : (11.54)

.

We introduce ( still assuming

W, = { uh I u, I K g P , VK, q,continuous at the midpoints

of on

[ai,K,aj,K],and

r}.

aij,r

zero at the points ui,,K which lie

We next introduce

Then a(u, u) i s not defined on V,

but we introduce

Numerical computation o f Bingham f l u i d f l o w s

(SEC. 11)

which e n a b l e s us t o d e f i n e t h e s o l u t i o n

(1 1 .57)

ah(l(k,

oh)

=

(J oh)

vu E

v h , Uh

E vh

&,Of

t h e S t o k e s problem

.

We t h e n o b t a i n t h e same e s t i m a t e as ( 1 1 . 4 8 ) ; R a v i a r t 111.

11.3

509

s e e Crouzeix-

Numerical r e s u l t s

Having chosen t h e s p a c e v h , t h e problem r e d u c e s t o t h e numerical s o l u t i o n of ( 1 1 . 2 3 ) ( l ) . If we i n t r o d u c e t h e n o n l i n e a r o p e r a t o r A ( i n f i n i t e d i m e n s i o n s ) by:

Using t h e d i s c r e t e a n a l o g u e o f Theorem 1 1 . 2 , it can b e s e e n t h a t (11.59) i s equivalent t o t h e f o l l o w i n g ( L a g r a n g i a n ) problem i s a saddle point): (which e x p r e s s e s t h e f a c t t h a t {4+',d+'}

Uzawa's algorithm g i v e s t h e f o l l o w i n g :

t o simplify t h e notat-

i o n we p u t :

(11.63)

#+' = w ,

d+e= m .

Then, s t a r t i n g from a r b i t r a r y m'we follows :

(l)

(*)

d e f i n e : w l , m ' , wz,m2. ...

Y

as

H e r e a f t e r we p u t :Du= { Di,(u)}. Where, i n t h e c a s e of non-conforming e l e m e n t s , t h e i n t e g r a l s o v e r 62 are r e p l a c e d by t h e sum o f t h e i n t e g r a l s o v e r t h e simplexes o f t h e t r i a n g u l a t i o n .

Time-dependent variational i n e q u a l i t i e s

( A d + ' , v,) (11.64)

m'+'/'

= m'

+ ke f i g ( & ,

+

Dv,) = (4 kefl",

+ pek & g D d + '

V&

'do, E

( CHAP.

6)

v,

,

ma+' = P K ( d " / 2 ) , PK p r o j e c t o r on K i n (L2(Q))4. l i

If i n

(1 1.65)

(11.64) we choose t h e parameter p

p>O,

such t h a t

< v/kegz

t h e n it can b e shown ( F o r t i n 111) t h a t t h e above a l g o r i t h m converges. F o r t i n , loc. cit., a l s o c o n t a i n s an i n v e s t i g a t i o n o f t h e ArrowHurwicz a l g o r i t h m t o g e t h e r w i t h v a r i o u s f u r t h e r remarks on t h e application of these algorithms. We a l s o r e f e r t o BCgis /1/ f o r an i n v e s t i g a t i o n o f t h e v a r i o u s parameters f o r a c c e l e r a t i n g t h e convergence. N a t u r a l l y , one o f t h e e s s e n t i a l f e a t u r e s o f t h e numerical experiments c a r r i e d o u t i s t h e d e t e r m i n a t i o n o f t h e r i g i d zones ( l ) . I n t h i s c o n t e x t w e n o t e t h a t i f E m ; < 1 then c o n d i t i o n ( 1 1 . 1 5 ) t o g e t h e r w i t h ( 1 1 . 1 4 ) can o n l y b e s a t i s f i e d i f D,,u = 0 . ThUS

(11.66)

if

c m $ x ) < 1 , x l i e s i n t h e r i g i d zone.

.

This i n f o r m a t i o n i s used i n c o n j u n c t i o n w i t h l~"(x)l This a l l o w s an a c c u r a t e p r a c t i c a l d e t e r m i n a t i o n o f t h e r i g i d zones.

Some numerical r e s u l t s . We p r e s e n t o n l y some o f t h e numerical r e s u l t s o b t a i n e d , notably by M. F o r t i n /1/ and BCgis 131. . The work o f F o r t i n u s e s t h e f i n i t e element method employed i n Example 1. We s h a l l p r e s e n t some r e s u l t s due t o BCgis loc. C i t . , who u s e s f i n i t e d i f f e r e n c e methods, and t h u s e x t e r i o r approximation ( t h e above c o n s i d e r a t i o n s may b e extended t o t h i s c a s e , b u t w i t h additional technical d i f f i c u l t i e s ) . F i g u r e 11.1 shows t h e , d o m a i n o,f r i g i d i t y ( h a t c h e d ) a t t h e i n s t a n t 0.005 f o r zero right-hand s i d e and non-zero bowzdary condi t i o n s ( c a s e s which come w i t h i n t h e g e n e r a l t h e o r y ) . F i g u r e 1 1 . 2 shows t h e domain o f r i g i d i t y f o r a v a l u e o f g g r e a t e r t h a n t h a t i n F i g u r e 11.1, a l l o t h e r q u a n t i t i e s b e i n g equal. (l)

About which t h e r e a r e v e r y few r e s u l t s o t h e r t h a n "numerical evidence"

.

( S E C . 11) Numerical computation

Fig. 11.1.

of Bingham f l u i d f l o w s

T i m e 0.005 : g = 2.5.

Fig. 11.2.

511

T i m e 0.005 : g = 5 .

T i m e 0.005 : g = 10.

Fig. 11.3. PARAMETERS VISCOSITY EXTERNAL FORCE F l ( X l , X 2 , T )

= 0.0

1.0 F2(Xl,X2,T)

= 0.0

BOUNDARY C O N D I T I O N S NORMAL COMPONENT O F V E L O C I T Y TANGENTIAL COMPONENT O F V E L O C I T Y

0.0 0.0 A T X l = 0, 1 AND X 2 = 1

I N I T I A L CONDITION I N I T I A L VELOCITY

0.0

- STREAM F U N C T I O N CONTOURS

mu

RIGID

ZONES

-

Time-dependent variational i n e q u a l i t i e s

Fig. 11.4.

T i m e 0.005.

Fig. 11.4'.

Fig. 11.5.

(CHAP.

6)

T i m e 0.010,

T i m e 0.040.

P W T E R S P L A S T I C I T Y THRESHOLD V I scos I TY EXTERNAL F O R C E

2.5 1.0 Fl(Xl,X2,T)=

300 x (X2 x (M

F2(XL,X2,T) =-300 . . BOUNDARY C O N D I T I O N S

NORMAL COMPONENT OF 'VELOCITY 0.0 T A N G E N T I A L COMPONENT OF V E L O C I T Y 0.0 I N I T I A L CONDITION I N I T I A L VELOCITY

- STREAM

0.0 F U N C T I O N CONTOURS R I G I D ZONES

- .5) - .5)

( S E C . 11) Numerical computation of Bingham f l u i d f l o w s

Fig. 11.6.

T i m e 0.005.

Fig. 11.7.

Fig. 11.8.

513

T i m e 0.010.

T i m e 0.040.

PARAMETERS P L A S T I C I T Y THRESHOLD VISCOSITY EXTERNAL FORCE

5.0 1.0 Fl(Xl,X2,T) = 300x (X2 F 2 ( X l , X 2 , T ) = -300 x (Xl

BOUNDARY C O N D I T I O N S NORMAL COMPONENT O F V E L O C I T Y 0.0 TANGENTIAL COMPONENT O F V E L O C I T Y 0.0 I N I T I A L CONDITION I N I T I A L VELOCITY

-

0.0 STREAM F U N C T I O N CONTOURS

I R I G I D ZONES

-

- .5) - .5)

Time-dependent variational inequalities

514

Fig. 11.9.

F&.11.10.

T i m e 0.005.

Fig. 11.11.

(CHAP.

6)

T i m e 0,010.

T i m e 0.040.

PARAMETERS P L A S T I C I T Y THRESHOLD VISCOSITY EXTERNAL VELOCITY

10.0 1.0

F1( Xl,X2,T) = 300 x ( X 2 F 2 ( X l , X 2 , T ) = -300 x ( X I

BOUNDARY CONDITIONS NORMAL COMPONENT OF VELOCITY 0.0 TANGENTIAL COMPONENT OF VELOCITY 0.0 I N I T I A L CONDITION I N I T I A L VELOCITY

-

0.0

STREAM FUNCTION CONTOURS llpll

R I G I D ZONES

-

-

.5)

- .5)

(SEC. 11)

5 15

Numerical computation of Bingham fluid flows

PARAMETERS PLASTICITY THRESHOLD VISCOSITY EXTERNAL FORCE

G 1.0

F1( x1,X 2 , T ) = 300 x ( X 2 - . 5 ) F 2 ( X L , X 2 , T ) = - 3 0 0 x ( X l - .5)

BOUNDARY CONDITIONS NORMAL COMPONENT OF VELOCITY 0.0 TANGENTIAL COMPONENT OF VELOCITY 0.0 INITIAL CONDITION I N I T I A L VELOCITY

0.0

110 0 0

112.0 . ...__._ G=_ 0 _ . ._

-.-G=

81.0

56.0

28.0

0.

0;

0.025

0 n 050

0.075

0 100

0 0 125

TIME VARIABLE, T

Fig. 11.12.

G- 2 G= 4 L= 6 8 G= 10 G= 12

.s

14

G=

16

516

( CHAP. 6 )

Time-dependent variational i n e q u a l i t i e s

PARAMETERS PLASTICITY THRESHOLD V I S cos ITY EXTERNAL FORCE

G 1.0 F l ( X l , X 2 , T ) = 300 x ( X 2 - . 5 ) F2(Xl,X2,T)= - 3 O O X ( X l - .5)

BOUNDARY CONDITIONS NORMAL COMPONENT OF VELOCITY 0.0 TANGENTIAL COMPONENT OF VELOCITY 0.0 I N I T I A L CONDITION I N I T I A L VELOCITY

0.0

6.0

4m8

________

~

3.6

.

-.

G= G. G= G= G. G= G= G= G=

0=(0.1

2.4

1.2

0. 0.

0.025

0.050

0 100

0.075

0

00125

TIME VARIABLE, T

Fig. 11.13.

0.

2

L

6

e

1C 12 11

1f

) d o ,1)

(SEC. 12)

Discussion

517

We observe that, in keeping with physical reasoning, the domain o f r i g i d i t y increases with g, which is again confirmed by Figure 11.3 in which g is even larger. Figure 11.4 presents a case where f#O the boundary conditions being zero. We note that in Figure 1 1 . 5 , which refers to a time greater than that in Figure 11.4, the zone of rigidity decreases as t increases. The foZzo&ng conjecture is supported by the figures and appears reasonable from a physical viewpoint: l e t u be the solution of &obZem I with f ( t ) = f independent o f t and with u,, = 0 ; then the zone of r i g i d i t y decreases ( l ) when t increases, t o "converge" towards the zone of r i g i d i t y o f the steady problem. Figures 11.6, 11.7, and 11.8 show a case analogous to those of Figures 11.4 and 11.5, with a larger g, and hence more extensive zones of rigidity. The same is true of Figures 11.9, 11.10, 11.11.

Figures 11.12and 11.13 represent graphs of the functions r-r 414&t)) and t + J(u,(t)) for various values of g. We note that a(u,(t)) and J(u,(t)) tend towards an equilibrium value, are increasing functions of time, and for fixed 1 , are increasing functions of g. It would be very interesting to obtain a proof oT these facts, which appear very plausible from a physical point of view. Furthe; numerical results are given in D. BCgis, zoc. c i t . 12.

DISCUSSION

The adaptation of the results of Raviart /I/ to the investigation of the-dependent inequalities was outlined in Lions 151, and then extended and subjected to a detailed study in Tr&nolisres / b / and Viaud /l/; we also cite Tr6moliPres / b / for a more detailed investigation than that presented here for parabolic inequalities of the first type and for inequalities of the second order in t , and to Viaud /1/ regarding parabolic inequalities; this latter work also contains an investigation of the various "delay" inequalities (for the case of delay partial differential equations see M. Artola 111, and for a study of the numerical approximation, Reverdy 111). We also refer the reader to Bourgat 111. A numerical investigation of.the inequalities governing the flow of a Bingham fluid (see Duvaut-Lions 111, Cioranescu-Margolis 111, 121) has been undertaken by M. Fortin /1/ (we refer to the works of this author for a more complete investigation than that Whilst the presented in Section 11 of the present chapter). Bingham inequalities themselves contain the Navier-Stokes equations as a special case, it would not be possible for our investigation

(l)

In the sense of the inclusion of the sets.

518

Time-dependent v a r i a t i o n a l i n e q u a l i t i e s

(CHAP.

6)

a l s o t o include as s p e c i a l c a s e s a l l t h e (innumerable!) works rela t i n g t o t h e numerical approximation of t h e Navier-Stokes equati o n s ; i n t h i s connection we r e f e r t o t h e work o f T6mam / 2 / and t h e bibliography contained t h e r e i n . Numerous problems remain t o be solved i n t h e numerical study of t h e flow of Bingham f l u i d s . For example, t h e proof of t h e convergence o f t h e approximate soldoes u t i o n s u, t o t h e s o l u t i o n u has been o u t l i n e d i n t h e t e x t ;

the " f l u i d - r i g i d " boundary f o r u, s i m i l a r l y converge touards the I t should be noted, moreover, t h a t t h i s f r e e boundary for u? type of (open) problem a r i s e s f o r a l l t h e problems o f t h i s chapter. An a d a p t a t i o n of t h e method due t o Baiocchi / 2 / t o t h e p a r a b o l i c case of t h e f i r s t type, f o r convex s e t s K of t h e type { u l v < $ on

6 2 ) , should be p o s s i b l e b u t has not been c a r r i e d o u t . Also y e t t o be i n v e s t i g a t e d a r e boundary l a y e r phenomena ( f o r t h e s u b j e c t o f s i n g u l a r p e r t u r b a t i o n s i n i n e q u a l i t i e s , which has not been considered h e r e , we r e f e r t o Lions / $ / ) . We have not considered h e r e t h e methods of s p l i t t i n g and decomp o s i t i o n which a r e introduced i n Lions-TCmam /1/ ( s e e i n p a r t i c u l a r Bensoussan-Lions-Ti5mam /l/) and which c o n s t i t u t e extensions o f t h e c l a s s i c a l methods o f f r a c t i o n a l s t e p s t o i n e q u a l i t i e s (see Yanenko /l/, Marchouk /1/ and t h e b i b l i o g r a p h i e s t h e r e i n ) . The numerical approximation o f various o t h e r problems of t i m e dependent i n e q u a l i t i e s i s p o s s i b l e using t h e methods described i n t h i s chapter. Examples a r e given i n Begis /l/. For coupled systems, such a s t h o s e a r i s i n g i n t h e r m o - e l a s t i c i t y or magnetohydrodynamics ( s e e Duvaut-Lions / 3 / ) which involve "energy" methods, t h e results o f t h i s c h a p t e r can b e extended. This i s not t h e case f o r t h e problem of Bingham f l u i d s whose v i s c o s i t y depends on t h e temperature; s e e Duvaut-Lions / 2 / ; f o r t h i s problem, t h e convergence of a s u i t a b l e approximation i s open t o question ( t h e method used i n Duvaut-Lions t o demonstrate t h e e x i s t e n c e of a s o l u t i o n i s i n f a c t not a c o n s t r u c t i v e method). Other problems of coupled-time-dependent i n e q u a l i t i e s are given i n Lions / 6 / , i n p a r t i c u l a r for problems of endo- o r exothermic s a t u r a t i o n ; many questions remain unresolved with regard t o t h i s work; a p a r t i a l l y h e u r i s t i c numerical a n a l y s i s of t h e s e problems i s c a r r i e d o u t i n Marrocco 121. Other problems of time-dependent i n e q u a l i t i e s a r e encountered i n problems o f S t e f a n t y p e ( s e e G. Duvaut /l/) and s i m i l a r l y with regard t o f i r s t - o r d e r hyperbolic o p e r a t o r s ( "Maxwell operators" ; c f . t h e problems o f i n s u l a t i o n breakdown i n antennas i n DuvautLions /l/, Chapter 7 ) ; t h i s l a t t e r t o p i c r a i s e s a general problem which w e have not considered h e r e ; it would b e i n t e r e s t i n g , i n p a r t i c u l a r , t o s e e t o what e x t e n t it i s p o s s i b l e t o extend t h e r e s u l t s o f Lesaint /1/ on f i r s t - o r d e r hyperbolic equations t o " f i r s t - o r d e r hyperbolic i n e q u a l i t i e s " . S i m i l a r l y we have t h e problem of t h e optimaz control of systems governed by time-dependent i n e q u a l i t i e s . We r e f e r t o Kernevez /l/, Yvon /l/, Mignot /l/, Saguez /l/. Questions of e r r o r estimation remain l a r g e l y open; it would

Discussion

(SEC. 12)

519

be v e r y i n t e r e s t i n g t o s e e t o what e x t e n t t h e e s t i m a t e s o f Douglas and Dupont /l/, a n d R a v i a r t / 2 / can b e extended t o i n e q u a l i t i e s . Time-dependent i n e q u a l i t i e s w i t h growth c o n d i t i o n s a t i n f i n i t y a r e encountered i n problems o f o p t i m a l c o n t r o l and games t h e o r y ( s e e Bensoussan-Lions /l/, A. Friedman /l/, /2/); t h e i r numerical i n v e s t i g a t i o n h a s n o t been a t t e m p t e d h e r e . For t h e c a s e o f v a r i a t i o n a l i n e q u a l i t i e s i n which t h e convex s e t a n d / o r t h e n o n d i f f e r e n t i a b l e f u n c t i o n a l depend on t i m e , t h e r e i s no s p e c i a l numerical d i f f i c u l t y ; on t h e o t h e r hand t h e r e a r e s e r i o u s t h e o r e t i c a l d i f f i c u l t i e s when t h e convex s e t s depend " i r r e g u l a r l y " on t i m e ; t h i s i s t h e c a s e o f t h e o b s t a c l e problem

dU - AU - f k 0,u at

-

JI 2 0 ,

u = 0 on t h e boundary, a n d f o r where

JI

= $(x, I )

f =

0

is a n " i r r e g u l a r " f u n c t i o n o f x and

f .

I n t h i s c a s e Mignot and P u e l /1/ have proved t h e existence of a (weak) m i n i m sozution, t h i s minimum s o l u t i o n b e i n g t h e l i m i t o f penalised-or semi-discretised solutions.

This Page Intentionally Left Blank

REFERENCES ABADIE, J. (Ed. ) /1/ Nonlinear p r o g r a m i n g , North Holland (1967). /2/ I n t e g e r and nonlinear p r o g r a m i n g , North Holland (1970)

.

ABADIE, J., CARPENTIER, J. /1/ Generalisation of the Wolfe reduced gradient method to the case of nonlinear constraints. In Gptimisation, R. Fletcher, Ed., Acad. Press, London and N.Y. (1.969)~ pp. 37-49. ABLOW, C.M., BRIGHAM, G . /1/ An analogue solution of programming problems. Gper. Research, Vol. 3 (1955), pp. 388-394. ALTMAN, M. /1/ Generalised gradient methods of minimising a funcpp. 313-318. tional. B u l l . Acad. P o l . Sci. , Val. 14 (1966)~ ANNIN, B.D. /1/ Existence and uniqueness of the solution of the elastic-plastic torsion of a cylindrical bar of oval crosspp. 1038-1047. section. J . of A p p l . Math. Mech. 29 (1965)~ ANTOSIEWICZ, H.A., RHEINBOLDT, W.C. /1/ Conjugate direction methods and the method of steepest descent, in A Survey of Numerical A n a l y s i s , J.Todd, Ed., McGraw Hill, N.Y. (19621,pp. 501-512. ARGYRIS, J.H. /1/ Elasto-plastic matrix displacement analysis of 3-D continua. J . Royal Aero. SOC. , 69 (1965), pp. 633-635. /2/ Elasto-plastic analysis o f three-dimensional media. Acta Technica Academia S c i . Hungaricae, vol. 54 (1966), pp. 219-238. /3/ Continua and discontinua. Proc. First Conference Matrix Methods i n S t r u c t . Mech. , Wright-Patterson Air Force Base, Ohio, Oct. 1965. ARGYRIS, J.H., BALMER, H., DOLTSINIS, J., WILLIAM, K.J. /1/ Finite element analysis of thermo-mechanical problems, Third Conference Matrix Methods S t r u c t . Mech. , Wright-Patterson Air Force Base, Ohio, Oct. 1971. ARGYRIS, J.H. , SCHARPF, D.W. /1/ Methods of elasto-plastic analysis. Proc. of t h e ISD-ISSC Sympbsium on F i n i t e Element Techniques. Institut fib Statik und Dynamik der Luft und Raumfahrt-Konstruktione, Stiittgart (1969). ARGYRIS, J.H., SCHARPF, D.W., SPOONER, J.B. /1/ The elasto-plastic calculation of general structures and continua. Proc. 3rd Conference on Dimensioning, Budapest (19691,pp. 345-384. /2/ Die elasto-plastische Berechnung von allgemeinen Tragwerken und Kontinua. Ingenieur Archiv. , Vol. 37 (May 1969), pp. 326-352.

References

522

ARROW, K.J., HURWICZ, L., UZAWA, H. /1/ Studies in linear and nonlinear programming. Stanford University Press, Stanford, California (1958). ARROW, K.J., HURWICZ, L. /1/ Gradient method for concave programming, in Studies in linear and nonlinear programing, Arrow, K.J., Hurwicz, L., Uzawa, H., Ed. Stanford University Press (1958)y pp 117-132. ARTOLA, M. /1/ Sur les perturbations des kquations d'kvolution. Applications 2 des problhmes de retard. Annales E.N.S. , 2 (1969)3 PP. 137-253. ATTOUCH, H., DAMLAMTAN, A. /1/ Probl&nes d'kvolution dans les Hilbert et Applications. J. Math. Pwles Appli. , 54 (1975),pp.

53-74. AUBIN, J.P., /1/ Approximation of variational inequalities, in Functional Analysis and Optimisation. Caianiello, Ed. , Acad. Press (19661,pp. 7-14. /2/ Approximation of elliptic boundary value problems. Wiley (1972). AUSLENDER, A. /1/ Une mkthode gkngrale pour la dkcomposition et la minimisation de fonctions non diffkrentiables. C. R. Acad. Sci., Paris, vol. 271 (1970),pp. 1078-1081. /2/ Mkthodes numkriques pour la dkcomposition et la minimisation de fonctions non diffgrentiables. Nwner. Math. , 18 (1972), pp. 213-223. / 3 / Rksolution numkrique d'inkgalitks variationnelles. C.R. Acad. Sci. , vol. 276 (1973), pp. 1063-1.066. AUSLENDER, A., GOURGAND, M., GUILLET, A. /1/ Rksolution numgrique d'inkgalitks variationnelles C.R. Acad. Sci. , Paris , vol. 279 (1974), PP. 341-344. BAIOCCHI, C. /1/ Su un problema di frontiera libera connesso a questioni di idraulica. Annali Mat. Pura ed Appli., 92 (1972)3 pp. 107-127. a /2/ Estimations d'erreurs dans L pour les inkquations 2 obstacle. In Mathematical Aspects of Finite Element Methods, I. Galligani, E. Magenes, Eds., Lecture Notes in Math., 606, Springer-Verlag, Berlin, (1977), pp. 27-34.

.

BAIOCCHI, C., COMINCIOLI, V., MAGENES, E., POZZI, G.A. /1/ Free boundary problems in the theory of fluid f l o w through porous media. Existence and imiqueness theorems. Annuli di Mat. Pura ed Appli. , XCVII (1973), pp. 1-82.

, NEUSTADT, L.W. (Ed. ) /1/ Computing methods in optimisation problems. Acad. Press, New York (1964). BAYADA, G. /1/ Indquations variationneZ2es elliptiques avec conditions aux limites pdriodiques. Application a la rdsolution de 1 'dquation de Reynold. Third-Cycle Thesis , Universitk de Lyon (1972).

BALAKRISHNAN, A.V.

References

523

BECKMAN, F.S. /1/ The solution of linear equations by the conjugate gradient method. In Mathematical Methods f o r Digital Computers, Ralston, A. and Wilf, H . S . , Ed., Wiley (19601, pp. 62-72. BEGIS, D. /I/Analyse nwngrique de l'dcoulement d'un f l u i d e de Bingham. Third-Cycle Thesis, Universitk de Paris VI (1972). /2/

Rdsolution nwndrique de probl2mes de convection e t LABORIA Report No. 15 (1973).

de t r a n s f e r t de chaleur.

/3/ Etude nwndrique du comporternent d'un f l u i d e de Bingham. LABORIA Report No. 42 (1973). BEGIS, D., GLOWINSKI, R. /1/ Dual numerical techniques for some variational problems involving biharmonic operators, in Techniques o f Optirnisation. A.V. Balakrishnan, Ed., Acad. Press, New York (1972), pp. 159-174. BENSOUSSAN, A., LIONS, J.L. /1/ Probl'emes de temps dlarr@t optimal et inkquations variaticnnelles paraboliques. Journal o f Applicable Analysis, Vol. 3 (1973), pp. 267-294. /2/ Book in preparation on "Quasivariational i n e q u a l i t i e s " . BENSOUSSAN, A., GOURSAT, M., LIONS, J.L. /1/ ContrBle impulsionnel et ingquations quasi variationnelles stationnaires. C.R. Acad. S c i . , Paris (9 May 1973), vol. 276, Skrie A, pp. 1279-1284. BENSOUSSAN, A., LIONS, J.L., TEMAM, R. /1/ Sur les mkthodes de d&omposition, de dgcentralisation et de coordinations et applications. Cahier de Z ' I R I A , No. 11 (1972), pp. 5-189. BERGE , C.

/1/ Espaces t o p o ~ o g i q u e s ,Dunod, Paris (1959).

BIDAUT , M. J. /1/ Thgorhe d 'existence e t d 'existence "en gdndraz" pour des s y s t h e s re'gis par des e'quations a m ddrivdes p a r t i e l z e s non lindaires. Thesis, Universitk de Paris VI (1973). /1/ A convergent gradient procedure in pre-Hilbert spaces. P a c i f i c J . of Mathematics, Vol. 18, No. 1 (1.966).

BLLIM, E.K.

BORSENBERGER, A. /1/ Internal IRIA report (1973). BOSSAVIT, A. /1/ Une mgthode de &composition de l'op6rateur biharmonique. Note H.I. 58512, Electricitk de France (1971). BOURGAT, J.F. /1/ Analyse nwndrique du problBme de l a torsion e'Zasto-pZastique. Third-Cycle Thesis, Universit.6 de Paris VI (1971). /2/ Numerical study of a dual iterative method for solving a finite element approximation of the biharmonic equation. Computer Meth. A p p l . Mech. Eng. , 9 (1976)~pp. 203-218.

References

524

BOURGAT, J.F., DUVAUT, G. /1/ Numerical a n a l y s i s of flow with or without wake p a s t a symmetric two-dimensional p r o f i l e with or without incidence. I n t . J . Nmer. Meth. Eng. , 11 (1977), pp. 975-993.

/1/ Analyse numgrique du r e f r o i BOURGAT, J.F., T R ~ O L I E R E S ,R . LABORIA Report No. 17 dissement d'un c y l i n d r e r a d i o - a c t i f .

(1973). BOX, M . J . ,

DAVIES, O . , SWA", W.H. /1/ Nonlinear o p t i m i s a t i o n I . C . I. Monograph , Oliver and Boyd, Edinburgh

techniques. (1969).

BRAM, J . , SAATY, T.C.

/I/ Nonlinear Mathematics.

McGraw H i l l

(1964 1. /1/ Rayleigh-Ritz-Galerkin methods BRAMBLE, J . H . , SCHATZ, A.M. f o r D i r i c h l e t ' s problem using subspaces without boundary con d i t i o n s . Corn. Pure and Applied Math. , 23 (1970), pp. 115-

175

-

BREZIS, H. /1/ Equations e t inkquations non l i n 6 a i r e s dans l e s espaces v e c t o r i e l s en d u a l i t 6 . Annales de l ' I n s t y t u t Fouri e r , 18 (19681, pp. 115-175. /2/ Problzmes u n i l a t g r a u x .

liqubes , 5 1 (1972) , pp. 1-168.

J . de Math. Pures e t App-

/3/ Opdrateurs maximam monotones e t semi-groupes de contraction dans Zes espaces de H i l b e r t . North-Holland/Else v i e r (1973).

/ 4 / Un problzme d'6volution avec c o n t r a i n t e s u n i l a t 6 r a l e s dkpendant du temps. C.R. Acad. S c i . , P a r i s , v o l . 274 (1972), PP. 310-312. stique.

/ 5 / M u l t i p l i c a t e u r de Lagrange en t o r s i o n klasto-plaArchive Rat. Mech. Anal. , 49 (1972), pp. 32-40.

/6/ Monotonicity methods i n H i l b e r t spaces and some a p p l i c a t i o n s t o n o n l i n e a r p a r t i a l d i f f e r e n t i a l equations , i n Contributions t o Nonlinear Functional Analysis. E. Zarantone l l o , Ed. , Acad. P r e s s (1971) , pp. 101-156. BREZIS, H . , DWAUT, G. /1/ Ecoulement avec s i l l a g e autour d'un p r o f i l symktrique sans incidence. C.R. Acad. S c i . , vol. 276 (1973) 3 PP. 875-8789

/1/ Sur c e r t a i n s problsmes u n i l a t k r a u x BREZIS, H . , LIONS, J . L . hyperboliques. C.R. Acad. S c i . , P a r i s , v o l . 264 (1967) , pp. 928-931. /1/ Equivalence de deux in6quations variaBREZIS, H . , SIBONY, M. t i o n n e l l e s e t a p p l i c a t i o n s . Archive Rat. Mech. Anal. , 4 1 (1971) 3 pp. 254-265.

References

525

/2/ Mkthodes d'approximation et d'itkration pour les ope'rateurs monotones. Archive Rat. Mech. Anal., 28 (1968), pp. 59-82. BREZIS, H., STAMPACCHIA, G. /1/ Une nouvelle mgthode pour l'ktude d'kcoulements stationnaires. C.R. Acad. SCi. , Paris, vol. 276 (19731, pp. 129-132. /2/ Sur la re'gularitk de la solution d'ine'quations elliptiques. Bull. SOC. Mathgmatique de France, 96 (1968), pp. 153-180. BROWDER, F.E. /1/ The fixed-point theory of multi-valued problems. Math. Annalen, 1977 (19681, pp. 283-301. BUI-TRONG-LIEU, HUARD, P. /1/ La mkthode des centres dans un espace topologique. N m . Math., 8 (1966), pp. 56-67. CANON, M., CULLUM, C., POLAK, E. /1/ Constrained minimisation problems in finite dimensional spaces. SIAM J . on Control, V O ~ .4, NO. 3 (19661, pp. 528-547. CAPURSO, M. /1/ Principi di minim0 per la soluzione incrementale dei problemi elasto-plastici. Rendi Conti Acad. Naz. L i n c e i , (1969), XLVI; I, pp. 417-425; 11, pp. 552-560. /2/ On the incremental solution of elasto-plastic continua in the range of large displacements. Meccanica, Vol. V, 1 (1970), pp. 1-9. /3/ A general method f o r the incremental solution of elasto-plastic problems. Meccanica, IV (1969),4, pp. 267280. CAPURSO, M., MAIER, G. /1/ Incremental elasto-plastic analysis and quadratic optimisation. Meccanica, V (1970), pp. 1-10. CAROLL, C.W. /1/ An operator research approach t o t h e economic o p t i m i s a t i o n o f a k r a f t pulping p r o c e s s , Ph.D. Dissertation, Institute of Paper Chemistry, Appleton, Wis. (1959). CAUCHY, A. /1/ Me'thodes gkne'rales pour la rksolution des systgmes d'e'quations simultankes. C.R. Acad. S c i . , Paris, Vol. 25 (1847), PP. 536-538. CEA, J. /1/ Private communication. /2/ @ t i m i s a t i o n .

ThQorie e t algorithmes.

Dunod, Paris

(1971). /3/ Approximation variationnelle des problsmes aux limites. Annales de l ' l n s t i t u t F o u r i e r , 14 (19641, pp. 345-

444. CEA, J., GLOWINSKI, R. /1/ Mkthodes numkriques pour 1'6coulement laminaire d'un fluide rigide visco-plastique incompressible. I n t . J . of COT. Math., B, Vol. 3 (1972), pp. 225-255.

References

526

/ 2 / Sur des m6thodes d'optimisation par relaxation. Revue Franqaise d 'Automatique , Informatique , Recherche Op&rationnelle (Dec. 1973), R-3, pp. 5-32.

CEA, J., GLOWINSKI, R., NEDELEC, J.C. /1/ Application des mkthodes d'optimisation, de diffkrences et d'616ments finis ?i l'andlyse numgrique de la torsion klasto-plastique d'une barre cylindrique , in Approximation e t me'thodes ite'ratives de re'solution d'in6quations variationnelles e t de problbmes non lin6aires. Cahier de l'IRIA, No. 1 2 (1974), pp. 7-138. / 2 / Minimisation de fonctionnelles non diffkrentiables , in Conference on Applications of Numerical Analysis. Morris Ed. , Lecture Notes in Math. , B 228, Springer-Verlag (1971), pp. 19-38.

CImLET, P.G., GLOWINSKI, R. /1/ Sur la r6solution num6rique du probl&ne de Dirichlet pour l'opkrateur biharmonique. C. R. Acad. S c i . , Paris, S6rie A, vol. 279 (1974), pp. 239-241.

/2/ Dual iterative techniques for solving a finite element approximation of the biharmonic equation. Comp. Meth. A p p l . Mech. Eng. , 5 (1975), pp. 277-

295. CIARLET, P.G., RAVIART, P.A. /1/ General Lagrange and Hermite interpolation in R"with applications to finite element methods. Archive Rat. Mech. Anal. 46 (1972), pp. 177-179. / 2 / Interpolation theory over curved element with applications to finite element methods. C O T . Meth. i n A p p l i . Mech. and Eng. 1 (1972), pp. 217-249.

/3/ A mixed finite element method for the biharmonic equation, in Mathematical aspects of f i n i t e elements i n partial d i f f e r e n t i a l equations. C. de Boor, Ed., Acad. Press, New York (1974), pp. 125-145. CIARLET, P.G., WAGSCHAL, C. /1/ Multipoint Taylor formulas and applications to the finite element method. Num. Math. , 17 (1971), pp. 84-100. CIORANESCU, D., MARGOLIS, B. /1/ Ecoulement d'un fluide de Bingham dans un domaine non-cylindrique avec singularit6 , J . Functional Analysis, 13 (1973), 2 , pp. 125-131. COMINCIOLI, V. /1/ Metodi di rilassamento per la minimizzazione in uno spazio prodotto. L.A.N. , C. N. R . , 20 (1971) , Pavia. COMINCIOLI, V., GUERRI, L., VOLPI, G. /1/ Analisi numerica di un problema di frontiera libera connesso col mot0 di un fluido attraverso un mezzo poroso. Publication No. 17 of the Numerical Analysis Laboratory of t h e University of Pavia

(1971)

.

References

527

COURANT, R . /1/ V a r i a t i o n a l method f o r t h e s o l u t i o n o f problems o f e q u i l i b r i u m and v i b r a t i o n s . B u l l . Amer. Math. S o c i e t y , Vol. 49 (1943) pp. 1-23. CROCKETT, J.B., CHERNOFF, H. /1/ G r a d i e n t methods o f maximisation. P a c i f i c J . of Math., Vol. 5 (19551, pp. 33-50. CROUZEIX, M. /1/ Etude d'une mgthode de l i n g a r i s a t i o n . Rgsoluti o n num6rique d e s k q u a t i o n s de S t o k e s s t a t i o n n a i r e s . Applica t i o n s aux g q u a t i o n s de Navier-Stokes s t a t i o n n a i r e s , i n

Approximations e t mdthodes i t e ' r a t i v e s de rSsoZution d'indquations v a r i a t i o n n e l l e s e t de problBmes non l i n d a i r e s . Cahi e r de l ' I R I A , No. 1 2 , pp. 139-244 (1974). CROUZEIX, M . , RAVIART, P.A. /1/ Conforming and nonconforming f i n i t e element methods f o r s o l v i n g t h e s t a t i o n a r y S t o k e s I. Revue Frmqaise d' Automatique, d 'Informatique equations. e t de Recherche Opdrationnelle, R-3 ( 1 9 7 3 ) , pp. 33-76. CRYER, C.W. /1/ The s o l u t i o n o f a q u a d r a t i c programming problem using systematic over-relaxation. SIAM J . on Control, Vol. 9 , NO. 3 (19711,pp. 385-392.

/2/ The method o f C h r i s t o p h e r s o n for s o l v i n g f r e e boundary problems. Math. C O W . , 25 (1971),pp. 435-443. CURRY, B. /1/ The method o f s t e e p e s t d e s c e n t f o r n o n l i n e a r minim i s a t i o n problems. Quart. of Applied Math., Vol. 2 (1944), pp. 258-261.

DANIEL, J.W.

/1/ The c o n j u g a t e g r a d i e n t method for l i n e a r and n o n l i n e a r o p e r a t o r e q u a t i o n s . SIAM. J . Nwn. A n a l y s i s , Vol. 4 (19671, pp. 10-26. / 2 / The approximate minimisation o f f u n c t i o n a l s . P r e n t i c e Hall (1970 )

.

/1/ V a r i a b l e m e t r i c method f o r m i n i m i s a t i o n . DAVIDON, W.D. A. G. G. Res. and Deu. Report. ANL-5990 (1959).

/1/ M i n i m i s a t i o n o f f u n c t i o n s on convex bounded DEMYANOV, V.F. s e t s . K i b e m e t i k a , Vol. 1, No. 6 (1965), pp. 65-74. DOUGLAS, J., DUPONT, T. /1/ G a l e r k i n methods f o r p a r a b o l i c equa t i o n s . SIAM J . Nwn. Analysis (1970), pp. 575-626.

/1/ S o l u t i o n s nwne'riques des dquations algdbriques. DURAND, E. Volumes 1 and 2 , Masson, P a r i s ( 1 9 6 0 ) . DUVAUT, G . /1/ R Q s o l u t i o n d'un problsme de S t e f a n . C.R. P a r i s , S 6 r i e A , Vol. 276 (19731, pp. 1461-1463. DWAUT, G . ,

A c . SCi.

/1/ Les ine'quations en mdcanique e t en LIONS, J . L . Dunod, P a r i s ( 1 9 7 2 ) .

physique.

/ 2 / T r a n s f e r t de c h a l e u r dans un f l u i d e de Bingham dont l a v i s c o s i t g dgpend de l a t e m p g r a t u r e . J . o f Funct. A n a l y s i s , 1 0 ( 1 9 7 2 ) .

References

528

/3/ Inkquations en thermo-klasticitk et magngto-hydrodynamique. Archive Rat. Mech. Analysis , 46 (1972) , pp. 241-279* /4/ Problsmes unilatkraux dans la thkorie de la flexion forte des plaques. (I) Le cas stationnaire, Journal de mdcanique , Vol. B y No. 1 (March 1974). /5/ Problkmes unilat6raux dans la thkorie de la flexion forte des plaques. (11) Le cas d'$volution, Journal de mdccmique, Vol. B y No. 1 (June 1974). EKELAND, I. /1/ Sur le contr6le optimal de systkmes gouvernks par des 6quations elliptiques. J . of E'unct. Analysis, 9 , No. 1 (19721, pp. 1-62. EKELAND, I., TEMAM, R. /1/ Analyse convexe e t probZBmes variationnels. Dunod, Gauthier-Villars , Paris (1974). ELKIN, R.M. /1/ Convergence theorems for Gauss-Seidel and other minimisation algorithms. Technical Report 68-59, University of Maryland, Comp. Sci. Center, College Park, Maryland (1968). EHRLICH, L.W. /1/ Solving the biharmonic equation as coupled finite difference equations. SIAM J . Nwn. Anal. , 8 (1971), pp. 278-287. FALK, R.S. /1/ Approximate solutions o f some variational inequali t i e s with order o f convergence estimates. Ph. D. Thesis, Cornell University (1971). /2/ Error estimates for the approximation.of a class of variational inequalities. Math. of &mp. , Vol. 28, 128

-

(1974), PP 963-971-

FENCHEL, W. /1/ On conjugate convex functions. Canadian J . of Math. , 1 (1949), PP. 73-77. FIACCO, A., McCORMICK, G. /1/ Nonlinear programing: unconstrained minimisation techniques. Wiley, New York (1968). FICHERA, G. /1/ Problemi elastostatici con vincoli unilateral<; il problema di Signorini con ambigue condizioni a1 contorno. Mem. Accad. N a z . d e i L i n c e i , 8 ( 7 ) (1964), pp. 91-140. FLETCHER, R.

/1/ Optimisation.

Academic Press, London (1969).

FLETCHER, R., POWELL, M. /1/A rapidly convergent method for minimisation. C o p . J . , vol. 6 (1963), pp. 163-168. FLETCHER, R., REEVES, C.M. /1/ Functional minimisation by conjugate gradients. Comp. Journal, Vol. 7 (1964), pp. 149-

153. FORTIN, M.

/1/ Calcul nwndrique des dcoulements des f l u i d e s de Bingham e t des f l u i d e s nmtoniens incompressibles par l a mdthode des dlbments f i n i s . Thesis, Universit6 de Paris VI

(1972).

I

References FORTIN, M., GLOWINSKI, R., MARROCCO, A.

529 /1/ To appear.

FRAIEJS DE VEUBEKE, B. /1/ Displacement and equilibrium models in finite element method. In S t r e s s A n a l y s i s , Chap. 9, O.C. Zienckiewicz and G.S. Molistek, Ed., Wiley (1965). FRANK, M., WOLFE, P. /1/ An algorithm for quadratic programming. Naval Res. Log. Quart., 3 (19561,pp. 95-110. FREMOND, M. /1/ Solide reposant s u r un solstratifi6. B u l l . du Laboratoire des Ponts e t Chaussdes, April 1972, pp. 85-95. /2/ Etude des s t r u c t u r e s v i s c o - d l a s t i q u e s s t r a t i f i des sownises d des charges harmoniques e t de s o l i d e s d l a s t iques reposant sur des s t r u c t u r e s . Thesis, Universitk de Paris VI (1971).

FRIEDMAN, A. /1/ Stochastic games and variational inequalities. Archive Rat. Mech. and Anal., Vol. 51, No. 5 (1973), pp. 321-346. /2/ Regularity theorems for variational inequalities in unbounded domains and applications to stopping-time problems. Archive Rat. Mech. and A n a l . , Vol. 52, 2 (1973), pp. 134-160. FUSCIARDI, A., MOSCO, U., SCARPINI, F., SCHIAFFINO, A. /1/ A dual method for the numerical solution of some variational inequalities. J . Math. Anal. AppZ. , 40 (1972), pp. 471-493. GABAY, D., MERCIER, B. /1/ A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comp. and Math. w i t h A p p l i c . , Vol. 2 (1976), pp.

17-40. GLOWINSKI, R. /1/ Mgthodes numkriques pour l'gcoulement d'un fluide rigide visco-plastique incompressible, in Proc. of t h e 2nd I n t . Conference on Numerical Methods i n Fluid Dynamics, M. Holt, Ed., Lecture Notes in Physics, Vol. 8, Springer Verlag (1971), pp. 385-394. /2/ Sur l'kcoulement d'un fluide de Bingham dans une conduite cylindrique. J . de MGcanique, Vol. 13, No. 4, Dee. ,1973,pp. 601-621. /3/ Mgthodes itkratives duales pour la minimisation de fonctionnelles convexes, in Constructive a s p e c t s of f u n c t i o n a l A n a l y s i s , G. Geymonat, Ed. , Cremonese, Rome (19731, pp. 263-290. atica,

/4/ La mdthode de r e l a x a t i o n . d i Roma (1971).

Rendiconti di Matem-

14 Universitg

/5/ S u r la minimisation par surrelaxation avec projection, de fonctionnelles quadratiques dans les espaces de Hilbert, C. R. Acad. Sci. , Paris, Vol. 276 (19731, pp. 1421-1423.

References

530

/6/ Approximations externes, par klgments finis de Lagrange d'ordre un et deux, du problhe de Dirichlet pour l'opkrateur biharmonique. Mkthodes itgratives de rksolution des problkmes approchks, in Topics i n Numerical Analysis, J.J.H. Miller, E d . , Acad. Press, London (1973), pp. 123-171. /7/ Numerical methods f o r some nonlinear e l l i p t i c boundary value problems. Notes du Sgminaire de Mathkmatiques supkrieures de 1'Universitk de Montrgal, ktk 1975 (to appear) /8/ Internal IRIA report (1975) (see also Sec. 5.2.5 of Glowinski, R., Pironneau, 0 . /lA/). /9/ Introduction to the approximation of elliptic variational inequalities. Report 76006, Laboratoire d'Ana1yse NwnQrique, LA 189, Univ. P a r i s VI (1976). GLOWINSKI, R., LANCHON, H. /1/ Torsion glasto-plastique d'une barre cylindrique de section multi-connexe. J . de M&?anique, 12 (1973), pp. 151-171. GLOWINSKI, finis d'une Acad.

R., MARROCCO, A. /1/ Sur l'approximation par glgments d'ordre un et la r6solution par pgnalisation-dualit6 classe de problgmes de Dirichlet non linkaires. C.R. S c i . , Paris, Val. 178, S6rie A (June 1974), pp. 1649-

1652. /2/ sur l'approximation par 6lkments finis d'ordre un et la rgsolution par pknalisation-dualit6 d'une classe de probl&nes de Dirichlet non lingaires. Revue Francaise d'dutomatiqug d'Informatique e t de Recherche Opdrationnelle, R-2 (1975), pp. 41-76. GODUNOV, S . K . , PROKOPOV, G.L. /1/ Variational approach to the solution of large linear systems arising from strongly elliptic problems. h b l i c a t i o n of t h e A p p l . Math. I n s t . o f the Academy o f Sciences of t h e U.S.S.R. , Moscow (1968) (in Russian). GOLDFARB, D. /1/ Extension of Davidon's variable metric method to maximisation under linear inequality and equality constraints. SIAM J . A p p l . Math. , Vol. 17, No. 4 (1969), pp.

739-764. GOLDFARB, D. , LAPIDUS, L. /1/ Conjugate gradient method for non. and linear programming problems with linear constraints. I E.C. Fundamental, Vol. 7 (1968), pp. 142-151. GOLDSTEIN, A.A. /1/ Cauchy's method of minimisation. N w n . Math., 4 (1962), pp. 146-150. /2/ Convex programming in Hilbert spaces. B u z z . Amer. Math. S o c i e t y , Vol. 70 (1964), pp. 709-710. GOURSAT, M.

/1/ Internal IRIA report

(1975).

References

531

/ 2 / Analyse numt?rique de p r o b l h e s d ' $ l a s t o p l a s t icite' e t v i s c o - p l a s t i c i t i & . Third-Cycle Thesis, Universitg de Paris VI (1971). GRAVES, R.L., WOLFE, P. (Ed.) /1/ Recent Advances i n Mathematical Programing. McGraw Hill, New York (1963).

GREENSTADT, J.L. /1/ On the relative efficiencies of gradient methods. Math. o f Computation, Vol. 21 (1967)~ pp. 360-367. HADLEY, G.

/1/ Nonlinear and dynamic programing.

Addison-Wiley

(19641. HAUGAZEAU, Y. /1/ Sur les ingquations variationnelles. C.R. Acade. S c i . , Paris (1967). /2/ Thkse d'Etat, Paris (1968). HERAKOVITCH, C.T. and HODGE, P.G. /1/ Elastic-plastic torsion of hollow bars by quadratic programming. I n t . Journal Mech. S c i e n c e s , Pergamon Press (1969), 11, pp. 53-63. HESTENES, M.R. /1/ The conjugate gradient method for solving linear systems. Proc. Symposium i n Applied Math., Vol. 6, Numerical Analysis, pp. 83-102, McGraw Hill, New York (1956). /2/ Multiplier and gradient methods, J.O.T.A. , 4, NO. 5 (19691, pp. 303-320. HESTENES, M.R., STIEFEL, E. /1/ Methods of conjugate gradient for solving linear systems. J . Res. Nut. Bureau o f Standards, Vol. 49 (1952), pp. 409-436. HODGE, P.G., HERAKOVITCH, C.T., STOUT, R.E. /1/ On numerical comparisons in elastic-plastic torsion. Transactions of the A.S.M.E. (19621, pp. 454-459. HORMANDER, L. /1/ S u r la fonction d'appui des ensembles convexes dans m espace localement convexe. A r k i v fiir Mat. , 3 (1954), pp. 181-186. HUARD, P . /1/ Resolution of mathematical programming with nonlinear constraints by the method of centers. Nonlinear prog r a m i n g . Abadie, J., Ed., North Holland (1967). HUET, D. /1/ Perturbations singulizres d'ingquations variationnelles, C.R. h a d . S c i . , Paris, 2 7 6 ~(1968)~ ~ pp. 932-934. JOLY, J.C.

/1/ Thesis, Facult6 des Sciences de Grenoble (1970).

KALKER, J. J. /1/ Variational Principles of Contact Elastostatics, J . Inst. Math. A p p l . , 20 (1977),pp. 199-219. /2/ A minimum principle for the law of dry friction with application to elastic cylinders in rolling contact. J . A p p l . Mech. , 38, E (1971), pp. 875-880 and 881-887.

532

References

KERNEVEZ, J.P. /1/ Systhmes bienzymatiques structurgs et systhmes gouverngs par des ingquations variationnelles, Report NO. 4, D p t . Bio-Math. , Universitk Technologique de Compisgne (1974). KOIfTER, W.I. /1/ General theorems f o r elastic plastic solids. Progress i n soZid Mechanics, Vol. 2, North-Holland, P.C. (1960)~ pp. 165-221. KOWALIK, J. , OSBORNE, M.R. /I/Methods f o r unconstrained optimi s a t i o n problems, Elsevier, New York (1968). KUHN, H.M., TUCKER, A.W. /1/ Nonlinear p r o g r m i n g , Proc. Second Berkeley Symposium on Math. S t a t . and B o b . , University of California Press (1951), pp. 481-492. KUNZI, M.P., KRELLE, W., OETTLI, W. Blaisdell (1966).

/1/ Nonlinear programming.

KUNZI, H.P., TZSCHACH, H.G., ZEHNDER, C.A. /1/ NwnericaZ Methods of MathematicaZ Optimisation. Acad. Press (1968). /1/ Sur un thkorkme minimax. v01. 259 (1964), pp. 3925-3928.

KY-FAN

Acad. S c i .

C.R.

, Paris,

LANCHON, H. /1/ Solution du problkme de torsion klasto-plastique d'une barre cylindrique de section quelconque. C.R. Acad. S c i . , Paris, vol. 269 (1969), pp. 791-794.

/2/ Torsion 5Zasto-plastique d ' u n arbre cyZindrique de s e c t i o n simplement ou multiplement connexe. Thesis, Universitk de Paris VI, 1972. LASDON, L.S. /1/ Optimisation theory f o r Large systems. Book Co. , London, 1970.

McMillan

LAURENT, P.J. /1/ Approximation e t o p t i m i s a t i o n . Hermann, Paris, 1972. LAVAINE, F. /1/ Mgthodes de dkcomposition rapidement convergentes pour des ingquations variationnelles Third-Cycle Thesis , Universitk de Paris-Sud, 1975.

.

LEMONNIER, P.

/1/ Internal IRIA report

(1971).

LERAY, J., LIONS, J.L. /1/ Quelques rksultats de Visik sur les problsmes elliptiques non lingaires par les mkthodes de Minty-Browder. BUZZ. Soc. Math. de France, 93 (1965), pp.

97-107. LEROY, D.

/1/ Internal IRIA report

(1971).

LESAINT, P. /1/Finite element methods for symmetric hyperbolic equations. Nwn. Math. , 21 (1973), pp. 244-255. LEVITIN, E.S., POLYAK, B.T. /1/ Constrained minimisation methods. U.S.S.R. Cornput. Math. Math-Physics, Vol. 6 (19661, pp. 1-50. /2/ Convergence of minimising sequences in conditional extremum problems. S o v i e t Math. Dokl. , Vol. 7 (19661,pp. 764-767.

References

533

L E W , H., STAMPACCHIA, G. /1/ On the regularity of the solution of variational inequality. Corn. Pure AppZ. Math., 22 (1969), pp. 153-188. /2/ On the smoothness of superharmonics which solve a minimum problem. Journal d 'amZyse mathdmatique , 23 (1970), pp. 227-236. LIONS, J.L. /1/ Quelques me'thodes de rdsolution des probZBmes a m Zimites non lindaires. Dunod, Gauthier-Villars, Paris , 1969. /2/ Contrdle optima2 des systBmes gouverngs par des dquations aux dgrivges p a r t i e l l e s . Dunod, Gauthier-Villars, Paris, 1968. /3/ Cours d 'analyse numdrique. Facultk des Sciences, Paris, 1961.

/ b / Sur un nouveau type de problzme non linkaire

.

Sgminaire pour opkrateurs hyperboliques du deuxisme ordre J . Leray, CollGge de France, 1965-1966, 11, pp. 17-33. /5/ S u r l'approximation de la solution d'ine'quations d'kvolution. C.R. Acad. S c i . , Paris, vol. 263 (1966), pp.

55-57. /6/ S u r l'analyse numkrique de probldmes d'inkquations couplkes. Rone, I s t i t u t o d i Alta Matematica (1972).

/ 7 / Approximation nmkrique des inkquations d'6volution, in Constructive Aspects of FunctionaL Analysis, C.I.M.E., 1971, G. Geymonat, Ed., Cremonese (1973), pp. 295361. /8/ Mzthodes de perturbations s i n p l i B r e s dans l e s probZBmes aux Zimites e t l e contrble optimal. Lecture Notes in Math., Springer, B 323, 1973. LIONS, J.L., MAGENES, E. /I/ ProbZBmes aux Zimites non homogBnes. Vol. 1, Dunod, Paris (1968). LIONS, J.L. , STAMPACCHIA, G. /1/ Variational inequalities. Pure AppZ. Math., XX (1967)~ pp. 493-519.

Corn.

LIONS, J.L., TF.MAM, R. /1/ Une mkthode d'kclatement des ope'rateurs e t des contraintes en calcul des variations. C.R. Acad. S c i . , Paris, v o l . 268 (1966), pp. 563-565. LOOTSMA, F.A. /1/ Boundary properties of penalty functions for constrained minimisation. Thesis, Eindhoven, 1970. LUENBERGER, D.G. /1/ O p t h i s a t i o n bg vector space methods. Wiley, 1969. /2/ The conjugate residual method for constrained minimisation problems. SIAM J . Nwn. AnaZ. , Vol. 7 , No. 3

(1970).

5 34

References

/1/ Quadratic programming and theory of e l a s t i c MAIER, G. p e r f e c t l y p l a s t i c s t r u c t u r e s . Meccanica, I V (1968) , pp. 1-9. /2/ Some theorems f o r p l a s t i c s t r a i n rates and p l a s t i c s t r a i n s . Journal de Me'canique, 8 (1969), pp. 5-19. MAGENES, E., STAMPACCHIA, G. /1/ I problem? a1 contorno p e r l e equazioni d i f f e r e n z i a l i d i t i p 0 e l l i t t i c o . Ann. Scuola Normale S u p e r i o r e , P i s a , 1 2 (19581, pp. 247-358.

/I/ Cours de me'crmique des m i l i e u x continus. Vol. 1: Mdcanique des f l u i d e s , Vol. 2: Mscanique des s o l i d e s . Gauthier-Villars , P a r i s (1966).

MANDEL, J.

MANDELBROJT, S. /1/ Sur l e s f o n c t i o n s convexes. P a r i s , vol. 209 (19391, pp. 977-978.

C.R.

Acad. Sci. ,

/1/Nonlinear p r o g r m i n ~ , McGraw H i l l , New

MANGASARIAN, O.L. York, 1969.

MARCHOUK, G . I . /1/ Mgthodes de mathgmatiques numgriques. b i r s k , 1972.

Novosi-

/1/ A p p l i c a t i o n de l a mdthode de p6naZisation a l a re'solution d'un problbme d ' e ' l a s t o - p l a s t i c i t s . Third-Cycle Thesis, P a r i s , 1970.

MARROCCO, A.

/2/ R6solution numkrique de problsmes de s a t u r a t i o n endo- ou exo-thermique. I R I A Report INF. 7203/72003, Jan. 1972. MARZULLI /1/ Risoluzione a l l e d i f f e r e n z e f i n i t e di equazioni a l l e d e r i v a t e p a r z i a l l i d i t i p o e l l i t t i c o con condizioni su un contorno l i b e r o . Calcolo, SuppZ. 1, 5 (1968), pp. 1-22. MERCIER, B. /1/ Approximation p a r 616ments f i n i s e t r k s o l u t i o n p a r p 6 n a l i sat ion-dualit 6 d' un problsme d' 6las t o - p l a s t i c i t k C.R. Acad. S c i . , P a r i s , vol. 280, S&ie A (1975) , pp. 287290.

.

/2/

I n t e r n a l I R I A r e p o r t (1971).

MEZLYAKOV /1/ On a r e l a x a t i o n method f o r s o l v i n g systems o f l i n e a r i n e q u a l i t i e s . J . Of Math. and Phys., Vol. 2 , No. 3 (1962).

MIELLOU, J . C . /1/ M6thodes de Jacob?, Gauss-Seidel, sous/surr e l a x a t i o n p a r b l o c s appliqukes 2 une c l a s s e de probltmes non l i n g a i r e s . C.R: Acad. S c i . , P a r i s , Vol. 273, Se'rie A (1971) 3 PP. 1257-1259. MIGNOT, F. , PUEL, J.P. /1/ In6quations d'6volution paraboliques avec convexe d6pendant du temps: applj.cation am I.Q.V.

Archiv. Rat. Mech. Anal. (1976). MINTY, G.

/1/ Monotone ( n o n l i n e a r ) o p e r a t o r s i n H i l b e r t space. , 29 (19621, pp. 341-346.

Duke Math. J .

References

535

MORFAU, J.J. /1/Fonctionnelles convexes. Sdminaire J . Leray, Collkge de France, 1966. /2/ Fonctions de r6sistance et fonctions de dissipation. Se'minaire d 'analyse convexe , Montpellier (1971), No. 6. /3/ SIX 1'6volution d'un systsme 6lasto-viscoplastique. C.R. Acad. S c i . , Paris, vol. 273 (1971), pp. 118-121. /4/ Rafle par un convexe variable. Se'minaire d'analyse convexe, Montpellier (1971). MORICE, P. /1/ Calcul parallsle et d6composition dans la r6solution d'kquations aux derides partielles de type elliptique. IRIA Report INF. 7214/72017 (1972). MOSCO, U. /1/ Dual variational inequalities. J. Math. Anal. and AppZ. , 39 (1972), pp. 202-206. /2/ Convergence of convex sets and of solutions of variational inequalities. Advances i n Math. , 3 (4) (1969), PP. 510-585. MOSCO, U., STRANG, G. /l/ One sided approximations and variational inequalities. B u l l . A.M.S. , 80 (1974), pp. 308-312. MOSSOLOV, P.P., MIASNIKOV, V.P. /1/ Variational methods in the theory of fluidity of a viscous-plastic medium. P.M.M. , 29

(1965), pp. 468-492.

MOTZKIN, SCHOENBERG /1/ The relaxation method f o r linear inequalities. Canadian J . o f Math. , Vol. 6 , 1954. NAYAK, G., ZIENKIEWICZ, O.C. /1/ Note on the "alpha"-constant stiffness method for the analysis of nonlinear problems. I n t . J . o f Ivwn. Meth. Eng. , 4 (19721, pp. 579-582. NECAS, J. /l/ Les me'thodes d i r e c t e s dans l a t h g o r i e des gquations aux de'rive'es p a r t i e l l e s . Masson, Paris, 1967. ORTEGA, J., RHEINBOLDT, W.C. /1/ Monotone iterations for nonlinear equations with applications to Gauss-Seidel methods. SIAM J . Nwn. Anal. , Vol. 4 ( ~ 9 6 7 pp. ) ~ 171-190. /2/ On a class of approximate iterative processes. Archive Rat. Mech. Anal. , Vol. 3 ( 1 9 6 ~ ) ~ pp. 352-356. /3/ I t e r a t i v e s o l u t i o n of nonlinear equations i n several v a r i a b l e s . Acad. Press, 1970. ORTEGA, J., ROCKOFF, M. /l/ Nonlinear difference equation and Gauss-Seidel type iterative methods. SIAM J. Nwn. Anal. , V O ~ .3, NO. 3 (19661, PP. 497-513. PEARSON, J.D. /1/ On variable metric methods of minimisation. COTut. J . , V O ~ .12 (1969), pp. 171-178.

5 36

References

PERALBA, J.C. /1/ Un problsme d'gvolution r e l a t i f 2 un opkrateur s o u s - d i f f e r g r e n t i e l dkpendant du temps. Sbminaire d 'Analyse convexe, Montpellier, 1972.

/1/ On t h e g e n e r a l i s e d over-relaxation method PETRISCHYN, W.V. f o r o p e r a t o r equations. Proc. Amer. Math. Society, 1 4 (1963) , PP. 917-921. /2/ Operator equations i n H i l b e r t space. , Vol. 29, 3 (1970), pp. 558-568.

J.

Math. A n a l . A p p l .

POW(, E. /1/ Computational methods i n optimisation. Press, 1971. POLYAK, B.T. ionals.

Academic

/1/ Gradient methods f o r t h e minimisation of funct, 3, No. 4 (19631, pp. 643-653.

Zh. fych. mat.

/2/ A g e n e r a l method f o r s o l v i n g extremum problems.

Soviet Math. Dokl., Vol. 8, No. 3 (1967), pp. 593-597. / 3 / Convergence of minimising sequences i n conditi o n a l extremum problems. Doklady, Vol. 168, No. 5 (19661, pp.

764-766. / 4 / Existence theorems and convergence of minimising

sequences i n extremum problems with r e s t r i c t i o n s . , Vol. 7 (1966) , pp. 72-75.

Soviet

Math. Dokl.

POWELL, M.J.D. /1/ A method f o r nonlinear o p t i m i s a t i o n i n minimisation problems, i n Optimisation, R. F l e t c h e r , Ed., Acad. P r e s s , New York, 1969. PRAGER, W. CO.

/1/ Introduction t o mechanics of continua, Ginn and

(1961). /2/ Problbmes de p l a s t i c i t 6 thborique, Dunod, 1958;

/ 3 / P l a s t i c design and thermal stresses. B r i t i s h Welding Journal (19561, pp. 355-359. /4/ On i d e a l locking materials. Transaction of the Society of Rheology, Vol. 1 (19571, pp. 169-175. /5/ U n i l a t e r a l c o n s t r a i n t s i n mechanics of continuum. A t t i del Simp. Lagrangian Accad. Sc. Torino, pp. 1-11.

/6/ L i n e a r i s a t i o n i n v i s c o - p l a s t i c i t y , Osterreiches Ingenieur-Archive, XV (19611, pp. 152-157. /7/ Composite s t r e s s - s t r a i n r e l a t i o n s f o r e l a s t o p l a s t i c s o l i d s . Proc. IUTAM Symp. Vienna (19661, pp. 315325.

/8/ On e l a s t i c , p e r f e c t l y locking materials.

PPOC.

I n t . Conf. Applied Mech. (19641, pp. 538-544. /l/ Theory of p e r f e c t l y p l a s t i c solids. PRAGER, W., HODGE, P.G. Wiley, 1961.

5 37

References /1/ Sur l ’ a p p r o x i m a t i o n de c e r t a i n e s kquations RAVIART, P.A. d ’ 6 v o l u t i o n l i n g a i r e s e t non l i n 6 a i r e s . J . Math. Pures e t A p p l . , 46 (1967), pp. 11-183.

/2/ The u s e o f numerical i n t e g r a t i o n i n f i n i t e element methods f o r s o l v i n g p a r a b o l i c e q u a t i o n s , i n Topics i n numerical a n a l y s i s , J . J . H . Miller, Ed., Acad. P r e s s , London ( 1 9 7 3 ) , pp. 233-264.

REVERDY, J. /1/ Sur une mkthode de Runge-Kutta pour d e s kquations & r e t a r d . Compt. Rend. A w ~ . S c i . P a r i s , 2 8 6 ~(1978) ~ , pp. 417420.

ROCKAFELLAR, A. T. /1/ Convex f u n c t i o n s , monotone o p e r a t o r s and v a r i a t i o n a l i n e q u a l i t i e s . I n NATO Course, Venice (1968) , O d e r i s i Ed. (1969), pp. 231-247. / 2 / D u a l i t y and s t a b i l i t y i n extremum problems i n v o l v i n g convex f u n c t i o n s . P a c i f i c J . Math. (1962) , pp. 167-187.

, 21

/3/ Convex a n a l y s i s , P r i n c e t o n U n i v e r s i t y

Press , P r i n c e t o n , N. J. ( 1 9 7 0 ) .

/4/ The m u l t i p l i e r method of Hestenes and Powell a p p l i e d t o convex programming. J . O p t . Th. and AppZ. , 1 2 (1973). / 5 / Augmented Lagrange m u l t i p l i e r f u n c t i o n s and d u a l i t y i n non-convex programming. SIAM J . on Control, Val. 1 2 , No. 2 (1974), pp. 268-285. /1/ The g r a d i e n t p r o j e c t i o n method f o r n o n l i n e a r ROSEN, J . B . programming. P a r t I , l i n e a r c o n s t r a i n t s . SIAM J . , Vol. 8, No. 1 (19601, pp. 181-217. /2/ The g r a d i e n t p r o j e c t i o n method f o r n o n l i n e a r programming. P a r t 11, n o n l i n e a r c o n s t r a i n t s . SIAM J . , Vol. 9 , No. 4 (19611, pp. 514-532. /3/ Approximate s o l u t i o n and e r r o r bounds f o r quasil i n e a r boundary v a l u e problems. SIAM J . N w n . Anal. , 7 (1970), pp. 80-103.

/1/ The t o r s i o n o f s o l i d and hollow prisms i n t h e SHAW, F.S. e l a s t i c and p l a s t i c range by r e l a x a t i o n methods. Report

ACA-11

(1944).

SCHECHTER, S.

/1/ I t e r a t i o n s method f o r n o n l i n e a r problems.

Trans. Amer. Math. S o c i e t y , 104 (1962), pp. 601-612. / 2 / Minimisation o f convex f u n c t i o n s by r e l a x a t i o n , Ch. 7 of I n t e g e r and nonlinear p r o g r a m i n g , Abadie Ed. , North-Holland (1970) , pp. 177-189.

/ 3 / R e l a x a t i o n method f o r convex problems. , Vol. 5 (19681, pp. 601-612.

J . of Num. Anal.

SIAM

538

References /I/ The'orie des distributions.

SCHWARTZ, L. (1967).

Hermann, P a r i s

/1/ Mgthodes ite'ratives pour l e s dquations e t indqua-

S I B O N Y , M.

tions tone.

~zurt:dgrivges

p a r t i e l l e s non line'aires de type mono-

C a l c o l o 2 (1970), pp. 65-183.

S I G N O R I N I , A.

/1/ Sopra alcune question2 d i E l a s t o s t a t i c a .

Atti

d e l l a Societh I t a l i a n a per il Progress0 d e l l a Scienza, 1933. /1/Existence de cols pour les f o n c t i o n s quasi convexes e t semi-continues. C.R. Acad. Sci. , P a r i s , V o l . 244 (1954), pp. 2120-2123.

S I O N , M.

/ 2 / On g e n e r a l minimax theorems.

Pacific J . Math.

,8

(1958) 9 pp. 171-176. S M I T H , J. /1/ The coupled equation approach t o t h e numerical s o l u t i o n of t h e biharmonic equation by f i n i t e d i f f e r e n c e s , I. SIAM J . Nwn. Anal. , 5 (1968), pp. 323-339. /2/ The coupled equation approach t o t h e numerical s o l u t i o n of t h e biharmonic equation by f i n i t e d i f f e r e n c e s , 11. SIAM J . N w n . Anal. , 7 (1970), pp. 104-111.

/3/ On t h e approximate s o l u t i o n of t h e f i r s t boundary value problem f o r V4u = f. SIAM J . Nwn. Anal. , 10 (1973), PP. 967-982. /1/ Applications de l'anatyse fonctionnelle a m e'quations de l a physique mathdmatique. Leningrad, 1950.

SOBOLW, S.L.

SOUTHWELL /I/ Relaxation methods i n theoretical physics, V o l . 1, Clarendon P r e s s , Oxford, 1946. SPANG, H.A. /1/ A review of minimising techniques f o r n o n l i n e a r functions. SIAM Revia), V o l . 4 , No. 4 (1962), pp. 343-365. STIEFEL, E.

/1/ Recent developments i n r e l a x a t i o n techniques. , Vol. 1 (1954), pp.

Proceedings I n t . Congress A p p l . Math. 384-391.

STOUT, R.B., HODGE, P.G. /1/ E l a s t i c - p l a s t i c t o r s i o n o f hollow c y l i n d e r s . I n t . J . Mech. Sci. , 1 2 (1970), pp. 91-108. STRANG, G . ,

method. TARTAR, L . ,

/1/ An analysis of the f i n i t e element F I X , G. P r e n t i c e H a l l , New Y o r k , 1973. TEMAM, R.

/1/ P r i v a t e communication.

BANICHOUK, N.V. /1/ ProbZ2mes variationnels en m6canique e t en contrbte, Moscow, 1973.

TCHERNOUSLO, F.L.,

TEMAM, R. /1/ S o l u t i o n s g6n6raliskes de c e r t a i n e s gquations du type hy-persurfaces minima. Archive Rat, Mech. Anal. , 44 (1371) , pp. 121-156. 121 Navier-Stokes equations.

1977.

North-Holland,

Amsterdam,

539

References

/ 3 / S u r l a s t a b i l i t k e t l a convergence de l a m6thode d e s p a s f r a c t i o n n a i r e s . AnnaZi d i Mat. Pum e AppZ., 4 , LXXIV (19681, pp. 191-380.

/1/ I n t e r n a l I R I A r e p o r t , 1972.

THOMASSET, T.

/ 2 / Etude d'une me'thode d'e'ZBrnents f i n i s de degre' cinq. AppZication aux probZ2mes de pZaques e t d 'e'coulement de f i u i d e s . Third-Cycle T h e s i s , U n i v e r s i t 6 de P a r i s X I (1974) T I N G , T.W. /1/ E l a s t i c - p l a s t i c t o r s i o n problem 111. Archive Rat. Mech. AnaZ, , 34 ( 1 9 6 9 ) , pp. 228-243.

.

/ 2 / E l a s t i c - p l a s t i c t o r s i o n o f convex c y l i n d r i c a l b a r s . J.

Math. Mech, 19 (1969), pp. 531-551.

/ 3 / E l a s t i c - p l a s t i c t o r s i o n of simply-connected c y l i n d r i c a l b a r s . Indiana Univers. Math. J . , Vol. 2 0 , No. 11 (19711, pp. 1047-1076.

/1/ La me'thode des centres ~3 troncature variable.

T R ~ O L I E R E S ,R .

Third-Cycle T h e s i s , P a r i s

(1968).

/2/ Optimisation p a r l a technique des s k r i e s divergentes.

InternaZ I R I A r e p o r t , (1969). / 3 / O p t i m i s a t i o n non l i n 6 a i r e avec c o n t r a i n t e s .

IRL4 report (1970 )

.

/4/ Ine'quations VariationneZZes: e x i s t e n c e , T h e s i s , U n i v e r s i t 6 de P a r i s V I approximations, r&soZu$ion. (1972).

UZAWA, H.

/1/ I t e r a t i v e methods f o r concave programming, i n

Studies i n Zinear and nonZinear programring, Arrow, K.J., Hurwicz, L., pp. 154-165.

Uzawa, H . ,

Ed.,

S t a n f o r d University P r e s s (1958),

VAINBERG, M.M. /1/ Le probldme de l a m i n i m i s a t i o n d e s f o n c t i o n n e l l e s non l i n g a i r e s , i n Problems i n nonlinear anaZysds. P r o c e e d i n g s of t h e C . I . M . E . , G. P r o d ? , Ed., Cr6mon&e (1971), PP. 465-574.

VARGA, R.S. /1/ Matrix i t e r a t i v e anaZySis. wood C l i f f s , New J e r s e y , 1962.

P r e n t i c e H a l l , Engle-

VARAYIA, P.P. /1/N o n l i n e a r programming i n Banach s p a c e s . J . AppZied Math., Vol. 1 5 , No. 2 , 1967. / 2 / Notes on optirnisation.

SIAM

Van N o s t r a n d , Rheinbold,

1972.

/1/ Approximation nume'rique des prob ZBrnes d 'asservissement e t d'asservissernent retard&. T h e s i s , U n i v e r s i t 6 de P a r i s

VIAUD, D.

V I , 1973.

5 40

References /1/ Methods of n o n l i n e a r programming, i n Recent advances Graves and P. Wolfe, Ed., McGraw H i l l , New York (19631, pp. 67-86.

WOLFE, P.

i n mathematical programming, R.L.

YAMADA, Y . /1/ Recent developments i n matrix displacement method f o r e l a s t i c problems i n Japan, i n Recent Advances i n Matriz Methods of Structural Analysis and Design, R.H. Gallagher, Y. Yamada, J . T . Oden, E d . , U n i v e r s i t y of Alabama P r e s s (1971) , pp. 283-316.

YANENKO, N.N.

/1/ Me'thodes de pas fractionnaires.

A. Colin,

1968. /1/ Etude de quelques problBmes de contrble pour des systzmes distribue's. Thesis, Universitk de P a r i s V I (1973).

YVON, J.P.

/2/ Contrdle optimal de systsmes gouvernks p a r des inkquations v a r i a t i o n n e l l e s . LABORIA Report No. 53,(1974). ZANGWILL, W . I . /1/ Nonlinear programming: P r e n t i c e H a l l , 1969.

a unified approach.

/1/ The f i n i t e element method i n structural and continuum mechanics. McGraw H i l l (1967). /2/ The f i n i t e element method i n Engineering Science. McGraw H i l l (1978).

ZIWKIEWICZ, 0.C.

ZIENKIEWICZ, O.C., CORMEAU, J . C . by f i n i t e element process. pp. 773-889, Warsaw.

/1/ V i s c o - p l a s t i c i t y s o l u t i o n

Archive of Mechanics, 24 (1972),

ZIENKIEWICZ, O.C. VALLIAPAN, S. /1/ Analysis of r e a l s t r u c t u r e s f o r creep p l a s t i c i t y and o t h e r complex c o n s t i t u t i v e l a w s , i n Structure, Solid Mechanics and Engineering Design. McTe'eni, Ed. , Wiley, I n t e r s c i e n c e (1971), pp. 27-48.

/1/ E l a s t o - p l a s t i c Z I E N K I E W I C Z , O.C., VALLIAPAN, S . , K I N G , J.P. s o l u t i o n s of engineering problems , i n i t i a l stress f i n i t e element approach. I n t . J. f o r Nwn. Methods i n Engineering, 1 (1969), pp. 75-100. ZLAMAL /I./ On t h e f i n i t e element method. pp. 394-409.

Nwn. Math.

, 12

(1968),

/2/ On some f i n i t e element procedures f o r s o l v i n g second o r d e r boundary value problems. Num. Math. , 1 4 (1969), pp. 42-48. ZOUTENDIJK, G. /1/ Nonlinear programming: a numerical survey. SIAM J'. on Control , Vol. 4 , No. 1 (1966) , pp. 194-210.

Appendix 1 FURTHER DISCUSSION

OF STEADY-STATE INEQUALITIES 1.

SYNOPSIS

In the following, we shall now consider in a rather more systematic manner the investigation of the existence, uniqueness and approximation properties of steady-state inequalities of the type

Find U E K such that (1.1A) a(u,v-u) t L(v-u) Vv

E

K

and

Find U E V such that (1.2A) a(u,v-u)+j(v)-j(u)

tL(v-u)

Vv E V .

In Sections 3 and 4 we shall consider the approximation, using conforming and nonconforming finite elements, of the obstacle problem f (v-u)dx

VV

E

K,

( 1 .3A)

with (1.4A)

1

K = { V E H (Q), v r J ,

a.e. on R , v l r

= g}

.

In Section 5 we give a brief outline of the work of BogomolniiEskin-Zuchowiskii relating to a stamp problem formulated as a variational inequality of type (l.lA), with the form a(*,*) relating to a pseudo-differential operator. In Section 6 we give an example of nonlinear elliptic equations, the investigation of which can be reduced to that of variational inequalities of type

Steady-state inequalities

542

( U P . 1)

F i n a l l y , i n Section 7, we g i v e a b r i e f i n t r o d u c t i o n t o (1.2A). t h e s u b j e c t of steady quasi-variational inequalities (Q.V. I. )

.

2.

EXISTENCE, UNIQUENESS AND APPROXIMATION RESULTS FOR PROBLEMS (1.1A) and (1.2A).

The r e s u l t s presented i n t h i s s e c t i o n were a l r e a d y known a t t h e time when t h e French e d i t i o n w a s being prepared, but t h e y had not t h e n been made e x p l i c i t . 2.1

The f u n c t i o n a l s e t t i n q

We s h a l l assume throughout t h i s s e c t i o n t h a t

-

V is

norm

-

R

r e a l H i l b e r t space with i n n e r product

(a,.)

and

11-11.

a : V x V + R i s a continuous coercive b i l i n e a r form (sometimes r e f e r r e d t o a s V - e l l i p t i c ) ;

there thus

e x i s t s an a > 0 such t h a t

It i s not assumed a p r i o r < t h a t a ( - , * ) i s symmetric.

- L : V + R i s a continuous b i l i n e a r form. - K i s a convex closed nonempty s e t i n V. - j : V-tK ( = R U { + m ) U [ - O O ) ) i s a functional

..

which i s

convex, lower semi-cont inuous (1.s c ) , and proper ( t h i s l a s t property implying t h a t j (v) > -00 that

2.2

V v E V,

and

j 3 +- 1.

Existence and uniqueness results f o r ( 1 . 1 A )

and (1.2A)

It w a s shown i n Chapter 1, Section 3.1, t h a t under t h e assumptions regarding V, K , a and L defined i n Section 2.1, we have e x i s t ence and uniqueness of a s o l u t i o n of problem ( 1 . l A ) . We r e c a l l t h a t t h e method used (due t o Lions-Stampacchia /l/) reduces ( 1 . 1 A ) t o a fixed point problem. We s h a l l show below, by a similar process, t h a t (1.2A) admits one and only one s o l u t i o n under t h e We t h u s have assumptions of S e c t i o n 2.1.

Existence, uniqueness and approximqtion

(SEC. 2 )

543

Theoren 2.1: Under the assumptions of S e c t i o n 2 . 1 on L and j , problem (1.2A) admits one and only one solution.

v,

a,

Proof: 1) Uniqueness Let u and u b e two s o l u t i o n s . 1 2

(2. IA)

a(ul ,v-uI> + (v)-j j (ul

2

(2.2A)

a(u2,v-u2>+j(v>-j(u2>

t

We t h e n have

L(v-ul )

vv

E

V,

~ ( v - u ~ V) V E V .

We n o t e t h a t j ( u . ) , i = 1 , 2 i s f i n i t e s i n c e , j ( * ) b e i n g proper, 1 we have j ( u . > - m and moreover t h a t (2.1A), (2.2A) imply, f o r 1 i=1,2 , t h a t

(2.3A)

j(u.)
vv

E

V.

S i n c e j 9 +co t h e r e e x i s t s vo E V such t h a t j(vo) < + m ; p u t t i n g v=v i n (2.3A) we hence deduce t h a t j(u.) < + m Under 1 t h e s e c o n d i ? i o n s we c a n piit v=u i n (2.1A) , v=ul i n (2.2A) and 2 add, g i v i n g

.

2)

Existence

We p r o c e e d as i n Lions-Stampacchia, l o c . c i t . , u s i n g a f i x e d p o i n t method. Let U E V (u a r b i t r a r y f o r t h e moment); we t h e n d e f i n e w as b e i n g t h e s o l u t i o n ( i f one e x i s t s ) o f t h e problem

(vp (u) 1

I

Find W E V such that

(w,v-w)+pj (v)-pj (w)

2

(u,v-w)+pL(v-w)-pa(u,v-w)

vv

E

V,

w i t h p > 0. S i n c e t h e v e c t o r u i s g i v e n , ( r p ( u ) ) i s a problem o f We s h a l l assme f o r t h e moment t h a t ( r p ( u ) ) admits t y p e (1.2A). a s o l u t i o n (which i s n e c e s s a r i l y u n i q u e ) . We d e n o t e by w = S,(u) t h e mapping u + w. correspond t o u1, u2 r e s p e c t i v e l y , we have

If w

1

and w

2

Steady-state i n e q u a l i t i e s

544

(UP. 1)

which, by a d d i t i o n , g i v e s

We have ( s e e Chapter 1, S e c t i o n 3.1) t h e e x i s t e n c e of a unique @Ek(V,V) such t h a t

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

Vu,v~v.

We then deduce from ( 2 . 3 A ) t h a t

II 2

llw2-w1

( ( I - r n (u2-u1),w21J1)

5

II 1-41 lIu2-u1II IIw2-w1II,

or

Taking account o f t h e V - e l l i p t i c i t y of a( , * )

(2.5A)

o < p < -

, it

is clear that

2a

ll@ll

We have t h u s shown t h a t if p s a t i s f i e s ( 2 . 5 A ) , t h e n Sp i s s t r i c t l y and uniformly a contraction mapping; t h e fixed-point problem

(2.6A)

u = Sp(U)

then admits one and only ,one s o l u t i o n which s a t i s f i e s , s i n c e u i s a s o l u t i o n of' ( a (u)) P

a(u,v-u)+j (v)-j ( u ) ~L(v-u)

I

V V E v,

UEV.

We have t h u s proved Theorem 2 . 1 up t o t h e proof of

( SEC .2)

54 5

Existence, uniqueness and approximation

Lemma 2.1:

The problem

(71

P

( u ) ) h i t s one and only one solution.

S i n c e t h e b i l i n e a r form on t h e l e f t - h a n d s i d e o f t h e Proof: (vp ( u ) ) i s e q u a l t o , * ) it i s symmetric. It t h e n inequality i n f o l l o w s ( s e e Chapter 1, S e c t i o n 2.1, C6a 121, Lions 121, EkelandT6mam 111) t h a t ( a , ( u ) ) i s e q u i v a l e n t t o t h e m i n i m i s a t i o n problem (

Find

0

such that

W E V

(2.7A)

J (u;w) < J (u;v) V V E V P

P

The f u n c t i o n a l v + J ( u ; v ) i s s t r i c t l y convex and 1 . s . c . Moreover i t s a t i s f i e s P

(2 . 8 A )

lim IIvII++

J (u;v) = P

.

+m,

The p r o p e r t y (2.8A) r e s u l t s from t h e f a c t t h a t s i n c e j(*) ( a n d t h e r e f o r e p i ( - ) ) i s convex, p r o p e r and l . s . c . , t h e r e e x i s t ( s e e F,keland-T6mam f l f ,f o r example) A : V -+ W , l i n e a r and c o n t i n u o u s , and p E IR , s u c h t h a t

I n view o f (2.9A) it can b e e a s i l y shown t h a t ( 2 . 8 ~ )h o l d s . From t h e above p r o p e r t i e s o f J P (u;-)w e have ( f r o m a s t a n d a r d r e s u l t o f convex a n a l y s i s f o r which we a g a i n c i t e t h e t h r e e above r e f e r e n c e s ) t h e existence and uniqueness o f a s o l u t i o n o f (2.7A), and t h e r e f o r e o f (71 ( u ) ) . P

5;

Some a p p l i c a t i o n s o f Theorem 2 . 1 were g i v e n i n Chapters f u r t h e r examples a r e g i v e n i n S e c t i o n 6.

2.3

On t h e i n t e r i o r a p p r o x i m a t i o n of ( 1 . 1 A )

4

and

and (1.2A)

I n t h i s s e c t i o n we s h a l l r e c o n s i d e r S e c t i o n s 4.3 and 4.4 of Chapter 1, weakening c e r t a i n a s s u m p t i o n s and expanding c e r t e i n remarks ( i n p a r t i c u l a r Remark 4.1).

546

Steady-state inequ.aZitiss

(APP. 1)

The assumptions on V, a , L, K and j are t h o s e of S e c t i o n 2.1. 2.3.3.

Approximation of ( 1 . 1 A )

With h + 0 we approximate V by t h e f a m i l y ( V ) where, Vh , V i s a c l o s e d subspace o f V ( i n p r a c t i c e dim + m ) . We t h e n h approximate K by t h e f a m i l y ( K h ) h where t h e Kh a r e , Vh , nonempty, c l o s e d convex s u b s e t s o f V ; we do not assume a p r i o r i t h a t We do assume kowever t h a t satisfies t h e f o l l %.'K * owing two p r o p e r t i e s :

f9

(s)h

If (Vh) h is such that vhe % v h , then the cluster points of (vh ) h f o r the weak topoZogy of V beZong t o K. There e x i s t

x c K,

= K and rh :

x + K,,

such that

(P2)

lim IIrhv-vll h+O

Remark 2.1: we n o t e t h a t i f

=

o

v

V E X

ul$

It r e s u l t s from (P,) t h a t c K ; moreover %cK Vh t h e n (P,) i s automaticaZZy s a t i s f i e d .

We t h u s approximate ( 1 . 1 A ) by

it i s c l e a r t h a t t h e approximate problem (2.10A) admits one and only one s o l u t i o n .

Remark 2.2: approximate a(

I n t h e m a j o r i t y o f c a s e s , it i s n e c e s s a r y t o and L( by , ) and $( ) ( g e n e r a l l y by a numerical integration procedure i n normal a p p l i c a t i o n s ) . The problems posed by t h e approximation o f a(*,=) and L ( - ) are t h o s e encountered i n t h e c a s e o f approximating l i n e a r v a r i a t i o n a l e q u a t i o n s ; w e t h e r e f o r e r e f e r t o C i a r l e t /1A/ and Strang-Fix /1/ f o r a d e t a i l e d a n a l y s i s of t h e s e problems ( i n t h e c o n t e x t o f f i n i t e element approximat i o n s ) rn

ah(

,)

.

With r e g a r d t o t h e convergence o f

(x) h

when h

-+

0, we have

Theorem 2.2: Let u be the solution of (1.U) and uh that of (2.10A); under the above assumptions on V, K , a and L, and i f (Pl), (P,) are s a t i s f i e d , we have

Existence, uniqueness and approximation

(SEC. 2 )

(2.11A)

IIy,-ull

lim h+O

Proof:

=

547

0.

T h i s i s a v a r i a n t o f t h a t o f Theorem 4.2 o f Chapter

1, S e c t i o n 4.4. 1)

A p r i o r i estimates for

"h

We deduce from (.2.10A) t h a t a(uh,\)

a(\,vh>-L(vh-uh>

Vvh

E

I$,

which i m p l i e s t h a t

where IIL\I*=

sup V E V 4 01

I L(v) I

and

II v II

d l i s t h e unique o p e r a t o r from

Let V ~ E X ; we deduce from (P,) ( w i t h C . d e n o t i n g v a r i o u s 1 c o n s t a n t s which depend on v b u t n o t on h) 0

11 'hvoll

co

which i m p l i e s

from which we f i n a l l y deduce t h a t , ( \ ) 2)

Convergence o f (

h

i s bounded i n V.

%)h

We can e x t r a c t from ( I + ) ~ a s u b s e q u e n c e - such t h a t

(2.15A)

lim h+O

yn

(P,) t h e n i m p l i e s (2.16A)

u*

E

K.

= u*

weakZy i n V;

-

a l s o denoted by

548

Steady -s t a t e inequa Z i t i e s

(APP. 1)

In view of (2.15A) and (P,) we bbtain by proceeding to the limit in a(%,%) la(yl,rhv)-L(r hVvveX 9

*

(2.17A)

lim inf a(%,%) h + O Moreover we have

VVEX.

la(u ,v)-L(v-u*)

a11%-U*112 ga(x-u*,Uh-u*)

=

* *

a(%,\)+a(u

,u )-a(x,u*)-a(u

*

,%) ,

giving, in the limit,

(2.18A)

lim inf a(%,%) h + O

2

* *

a(u ,u )

which is a well-known property of weak 1.s.c.. It then follows from (2.17A), (2.18A) that

(2.19A)

* *

*

a(u ,u ) la(u ,v)-L(v-u

*

) Vvex.

Since = K, (2.19A) in fact holds for all from (2.16~)we have:

I*

a(u* ,v-u*)

2 ~(v-u*)

vv

E

V E

K , Hence,

K,

u EK.

We have thus shown that u* is a solution of (1.lA); in view of the uniqueness we have u* = u and the e n t i r e family ( \ ) h converges weakly to u.

3) Strong converaence of

(~h)~.

We have

In view of (P,) and the fact that lim \ = u weakly in V, we h-tO obtain by proceeding to the limit in (2.20A)

= a(u,v-u)-L(v-u)

Vv

E X

.

Existence, uniqueness and approximation

(SEC. 2)

Since

X=

K we also have (by continuity)

a l i m sup 11yI-ull

(2.21A)

549

s a(u,v-u)-L(v-u)

vv

E

K.

Taking v=u in (2.21A) we obtain

-

which implies l i m (Iuh-uII = 0 , i.e. strong convergence. h+O

Some applications of Theorem 2.2 were given in Chapters 3 and

4.

Approximation of (1.2A)

2.3.2

Preliminary remark:

In this section we shall assume that

j(*) is continuous on

(2.22A)

v.

In Seition 6 we shall examine a class of inequalities of the type (1.2A) for which (2.22A) does not generally hold, but which can be investigated using methods very si.milar to those described below. m Assuming again that h + 0, we approximate V by (V ) where, h in Vh , Vh is a closed subspace of V (finite practical applications). We assume that (Vh)h satisfies

dimensions!?

1

There exists V c v that

, with T =v , and

r

h

:V

+

Vh such

We then assume that the functional j(*) is approximated by a family such that

(2.23A)

I

j,

[ j,

:

vh

+

E,

is convex, Z.S.C.

, and

uciformly proper in h;

the last property implies the existence of X E V ’ that

and p

E

IR such

Steady-state i n e q u a l i t i e s

550

(2.24A)

jh(vh) 2 A(vh)+v

V V ~ E h’ V

Vh

(APP. 1)

.

We s h a l l s i m i l a r l y assume t h a t t h e f o l l o w i n g a r e s a t i s f i e d

If vh

+

v weakly i n

l i m i n f jh(rhv)

2

v thm

j(v)

,

h + O

l i m j ( r v) = j ( v )

(P3)

h+O

h

h

VvcV

.

Remark 2.3: I n all t h e a p p l i c a t i o n s of which we a r e aware, it has always been p o s s i b l e , i f j(-) i s continuous on V, t o c o n s t r u c t j ( * ) , continuous on V and s a t i s f y i n g (2.24A), (P2), h h (P3). Remark 2.4: (P,)

, (P3)

If j ( v ) = j(vh) Vvh6Vh, a r e automakc&.ly s a t i s f i e d .

Vh

, then

(2.24A),

rn

We t h u s approximate (1.2A) by

Find \ c V h

such t h a t

(2.258) a(\’vh-%)

+jh ( vh ) - j h (I+> 2 L(vh-u,,)

Vvh

E

Vh.

The approximate problem (2.25A) admits one and o n l y one s o l u t i o n . Remark 2.2 of S e c t i o n 2.3.1 a l s o h o l d s f o r problem

(2.25A). With r e g a r d t o t h e convergence o f ( L + )

h

when h -+ 0, we have

Let u be the solution of (1.2A), yn t h a t of Theorem 2.3: (2.25A); under the above .assumptions on V, a, L and j , and i f (Pl), (2.23A), (2.24A), (P2), (P3) are s a t i s f i e d , we have (2.26A)

(2.27A)

l i m II%-ull h+O l i m jh(\) h+O

= 0

=

,

j(u).

(SEC. 2 ) Proof:

Existence, uniqueness and approximation

551

T h i s i s a v a r i a n t o f t h a t o f Theorem 2.2 of S e c t i o n

2.3.1.

Let vo c p j t h e C . which a p p e a r below d e n o t e v a r i o u s q u a n t i t i e s which depenh onVo , b u t n o t on h ; we have, t a k i n g account o f (P,) , ( P~),

If we t a k e v = vo i n (2.29A), we deduce from (2.30A)

'lluhll which i m p l i e s t h a t 2)

2

=3

%

IIu,,II

+

cq

Y

i s bounded i n V.

Weak convergence o f ( \ )

S i n c e t h e sequence ( \ )

h

h'

i s bounded i n V, we can e x t r a c t from such t h a t

it a subsequence, a l s o d e n o t e d by ( \ ) h , (2.31A)

l i m u,, =

,*

weakZy i n V .

WO Moreover we deduce from (2.25A) t h a t

(2.32A) Using (2.3l.A)

QvE'V:

, (P,) , (P,) , (P3)

we have i n t h e l i m i t i n (2.32A),

(APP. 1)

Steady-s t a t e inequalities

552

Using the density of ’Ir in V and the continuity of j ( deduce from (2.33A)

0

, we

)

*

a(u*,v-u*)+j(v>-j(u*) ~ ~ ( v - uQ

~ E v ,

(2.34A) U * E V ,

which implies u* = u and the convergence of the e n t i r e sequence ( ) towards u. Uhh

In view of (Pl), (P2), (P ) and from the weak convergence result given above, we haveY3in the limit, in (2.35A), Vv E

Using the density of ’v in V and the continuity of j deduce from ( 2.36A) that

(

0

)

, we

Obstacle problems

(SEC. 3 )

Taking

FU

- .I

5 53

i n (2.37A) we o b t a i n

j(u) = l i m inf jh(%) = l i m ( aII%-uII

2

+jh(%))

which c l e a r l y i m p l i e s (2.26A), (2.27A). Some a p p l i c a t i o n s o f t h e above Theorem a r e given i n Chapters

4 and 5.

3.

THE OBSTACLE PROBLEM. APPROXIMATIONS

( I ) GENERAL REMARKS.

CONFORMING

Synopsis

3.1

I n Chapter 1, S e c t i o n 3.6, Example 3.5, w e considered t h e s p e c i a l c a s e o f t h e o b s t a c l e problem (1.3A) corresponding t o JI = 0 , g = 0. A v a r i a n t o f t h i s s p e c i a l c a s e which a r i s e s i n l u b r i c a t i o n t h e o r y w a s a l s o c o n s i d e r e d i n Chapter 2, S e c t i o n 5 . I n t h e p r e s e n t s e c t i o n , which follows Glowinski / l A / , w e s h a l l b p a r t i c u l a r l y concerned w i t h t h e approximation of (1.3A) by means of conforming f i n i t e element methods, and we s h a l l g e n e r a l i s e t h e r e s u l t of Chapter 1, S e c t i o n 4.7, r e l a t i n g t o t h e s p e c i a l c a s e J, = 0, g = 0. The approximation o f (1.3A), u s i n g nonconforming f i n i t e elements o f mixed type, w i l l be c o n s i d e r e d i n S e c t i o n 4. It i s a p p r o p r i a t e t o p o i n t o u t t h a t v a r i a t i o n a l problems of type (1.3A), ( 1 . 4 A ) ( t h e obstacle problem), although p a r t i c u l a r l y simple, provide a good mathematical model i n s e v e r a l important a p p l i c a t i o n s ( s e e S e c t i o n s 3.2 and 3.3 below). Moreover, amongst v a r i a t i o n a l i n e q u a l i t y problems t h e y are t h o s e f o r which t h e most d e t a i l e d t h e o r e t i c a l and numeridal r e s u l t s have been obtained. Formulation o f t h e problem.

3.2

N

Physical i n t e r p r e t a t i o n

Let R be a bounded domain i n IR , w i t h r e g u l a r boundary N we c o n s i d e r t h e f o l l o w i n g With x = {xi]i=l, V = {a}N ax. i=ly

r=aQ.

1

obstacle problt?m, which has already been formulated i n ( 1 . 3 A ) , (1.4A)

,

Steady-s t a t e i n e q u a l i t i e s

5 54

( U P . 1)

Find u c K such that (3.1A)

Vu*V(v-u)dx

f(v-u)dx

2

Vv

E

K,

n

where, i n (3.1A),

( 3 .2A)

K =

fELL(Q)

{VEH

1

and K i s d e f i n e d by

(511, v z q

a.e.

on 0 ,

vlr = g)

where J, and g a r e g i v e n f u n c t i o n s . We assume 51 c n2 ; a s t a n d a r d i n t e r p r e t a t i o n o f ( 3 . 1 A ) , (3.2A) i s t h e n t h a t i t s s o l u t i o n , u, r e p r e s e n t s t h e v e r t i c a l displacements, assumed small, o f a horizontal e l a s t i c membrane R u n d e r t h e e f f e c t o f a d i s t r i b u t i o n of v e r t i c a l forces of i n t e n s i t y f ( f i s a s u r f a c e d e n s i t y o f vertical forces). T h i s membrane i s f i x e d a l o n g i t s boundary r ( u = g), and i s constrained t o l i e above an obstacle w i t h h e i g h t g i v e n by 6 (u2JI) ; F i g u r e 3.1 below d e s c r i b e s t h i s phenomenon geometrically :

PhysicaZ interpretation:

U=$

\

u=g \

\

/

\

/

\ \

I

\

Fig. 3.1 3.3

O t h e r phenomena r e l a t e d t o t h e o b s t a c l e problem

V a r i a t i o n a l i n e q u a l i t i e s o f t h e same t y p e as ( 3 . 1 A ) , (3.2A) b u t p o s s i b l y w i t h d i f f e r e n t boundary c o n d i t i o n s a n d / o r a nonsymmetric b i l i n e a r form) a r i s e i n t h e f o l l o w i n g a p p l i c a t i o n s :

Obstacle problems

(SEC. 3 )

-

555

I

- Lubrication phenomena; see for example Cryer 111, 121, Marzulli /l/ and Chapter 2, Section 5 of the present book for solution by finite difference methods and some further references. For the modelling itself see Capriz /lA/. - Seepage o f l i q u i d s i n porous media; see in particular Baiocchi /l/, /2/, /lA/, /2A/, Comincioli /lA/, Baiocchi-BrezziComincioli /lA/, Baiocchi - Comincioli - Magsnes-Pozzi /l/, /lA/, Cryer-Fetter /lA/, Baiocchi-Cape-lo /lA/, and also their associated bibliographies. - mo-dimensiona2 potentiaZ fZoiJ of p e r f e c t f l u i d s ; see BrgzisStampacchia /l/, /lA/, B r 6 z i s /lA/, Ciavaldini-Pogu-Tournemine /lA/, Roux /lA/, and also the references therein.

- Wake probZerns; see Brczis-hvaut 111, Bourgat-Duvaut Ill. The above list is far from exf-austive; numerous other applications also exist in biomathematics, mathematical economics and in semi-conductor physics (see Hunt-Nassif /lA/, etc.)

.

3.4

Interpretation of (3.1A), (3.2A) as a free boundary problem.

Let u be the solution of (3.1A), (3.2A); we then define

no

= {xlx E n

y =

u(x) = $(x)}

an+ n aao

and finally

u+ = u(

uo =

n+

UI

no

.

A classical formulation of (3.1A), (3.2A) is then: Find u and y ( t h e f r e e boundary) such that (3.3A)

-Au = f on R

(3.4A)

u =

(3.5A)

u = g on ,'I

I/J

on R

0

4-

,

556

Steady-state i n e q u a l i t i e s

(APP. 1)

The physical interpretation of (3.3A) to (3.6A) is as follows:

(3.3A) means that the membrane is s t r i c t l y above the obstacle and (3.4A) means that on $lothe has a purely e l a s t i c behaviour; membrane is on contact with the obstacle; (3.6A) is a transmission condition at the free boundary. In fact (3.3A) - (3.6A) are not sufficient to characterise u and y and it is necessary to introduce supplementary conditions concerning the behaviour of u on y o r alternatively the global regularity of u; for example, if JI is sufficiently regular, we could require the "contiquity" of Vu on 1 y (more specifically we could impose Vu E H ($2) X H ($2)).

Remark 3.1: In Chapters 1 - 6 we have already investigated other variational inequality probfems , interpreted as free-boundary problems. 3.5

Existence, uniqueness and regularity of the solution of (3.U) .(3.2A).

With regard to the existence and uniqueness of a solution of (3.1A), (3.2A) , we can easily prove Theorem 3.1:

Suppose that

r

i s regular, and that

$ ~ H l ( $ 2 ) , gEH1'2(r) with $ I r ' g a . e . on r ; then (3.1A), (3.2A) admits one and only one solution.

Remark 3.2: We already have existence and uniqueness for functions f and $ which are much less regular than those introduced in Theorem 3.1. W With regard to the regularity of u we recall the results below, which are due to Br6zis-Stampacchia /2/ If

r i s s u f f i c i e n t l y regular and

we have f g="gr

3.6

E

LP(Q) n ( H I ($2))'

with

ZE?'~($~),

,JIE

i f for p

E

11 *+ = C

4 *P(Q)

then u ~ b ? ~ ~ ( Q ) .

Finite-element approximations of problem (3.1A), (3.2A). (I) Piecewise-linear approximations

In this section we shall consider the approximation of (3.1A),

(3.2~)by a first-order conforming finite-element method (i.e. using continuous , piecewise-affine approximations). Approximation by second-order conforming finite elements will be considered in Section 3.7.

Obstacle problems

(SEC. 3 )

-

557

1

I n t h e f o l l o w i n g we d e s c r i b e t h e results o f Brezzi- Hager- Raviart ( ' ) / l A / , t h e r e b y g e n e r a l i s i n g t h o s e o b t a i n e d i n C h a p t e r 1, S e c t i o n 4.7 f o r t h e s p e c i a l c a s e JI = 0, g = 0. We assume f o r s i m p l i c i t y t h a t n i s a bounded polygonal domain i n IR2. We a l s o assume t h a t

(3.8A)

ncO(E)

$EH'(Q)

, gEH1'2(r)

nco(r>.

The n o t a t i o n i s t h a t of Chapter 1, S e c t i o n s 4 . 1 and 4.7, e x c e p t t h a t h w i l l now r e p r e s e n t t h e l e n g t h of t h e longest s i d e of Lhe t r i a n g u l a t i o n Suppose t h e n t h a t % i s a t r i a n g u l a t i o n C h a p t e r 1, of Q which s a t i s f i e s c o n d i t i o n s (4.7) - (4.9) S e c t i o n 4.1, a n d ( 4 . 5 1 ) o f Chapter 1, S e c t i o n 4.7, i . e .

%

u

(3.9A)

T =

09

n.

TECh We t h e n i n t r o d u c e

Ch = (P

,P

E

0

ch

= {P E Ch

T

a v e r t e x of

,PB

r}

E

ch},

,

=ChnQ

and we approximate H l ( s 2 ) and K by

(3.10A)

Vh = { v ~ E 0C(Q)

( 3 . 1 1A)

5=

{vh E Vh

where i n (3.11A)

, and

v h l T e P1 v T e C h }

, vh(P)

t $(P)

VP

E

Ch, vh(P) = g ( P ) V P E Chn

h e r e a f t e r , f o r k t 0 we have:

Pk i s t h e space o f polynomials i n two v a r i a b l e s o f degree Ik. We t h e n approximate (3.1A)

If

$1

5

g

w e have

P r o p o s i t i o n 3.1:

5#

, (3.2A)

0

by (3.11A) and

and it i s t h e n c l e a r t h a t :

The d i s c r e t e o b s t a c l e problem a d n i t s one and

only one s o l u t i o n . ( ' ) H e r e a f t e r a b b r e v i a t e d t o B.H.R.

r

5 58

Steady-s t a t e i n e q u a l i t i e s

(APP. 1)

Concerning the convergence of the approximate solutions when h + 0 , see Glowinski /1A/ for the case in which the solution u is not very regular. In the following we shall estimate the approxwhen u,JI E H2(n>, a E H2(R>; we imation error Ilu, - u I I Hl(S2)

follow very closely the investigation of B.H.R. / 1 A , Section (in which a(u,v) = J R ~ u * dx ~ v + jnuv dx). For this we use:

Assme that u

Lemma 3.1:

-

(3.13A)

-Au

(3.14A)

(-Au-f)(u-@)=

Proof:

f 20

E

4/,

H2 (52) ; then

a.e. on R a.e. on a .

0

See Br6zis /2/.

Lemma 3.2: Let u and yn be t h e r e s p e c t i v e s o l u t i o n s of (3.1A), (3.2A) and ( 3.11~) ( 3.12~). We then have

VvhE

( w i t h a(v,w)

VV,WE H

dx

=

1

Kh

(SZ))

..

(3.15A)

a(yl-u,yl-u)

Proof:

We follow B.H.R.

a(yl-u,y,-u)

/1A, Theorem 2.1/.

We have Vvhc I$,

h ) = a(y,-u,vh-u)+

= a(%-u,v,-u)+a(%-u,%-v

(3.16A)

+

a(u,vh-%)

-

f(vh-yl) +

)dx -a(%,vh-%).

s2

Since u satisfies (3.12A) we have h

I,

f (vh-%)dx

- a(ylyvh-%)

I 0 Vvh

E

\,

which combined with (3.16A) implies (3.15A). We thus have 2 2 H (Q), g=glr w i t h E H (SZ),and i f the angles of %hare bounded below by eo > 0 , independent o f h, then Theorem 3.2:

where u and

If f E L2(52') ,$

% are

E

t h e r e s p e c t i v e s o l u t i o n s o f (3.1~) (3.2~)and

(3.11A1, (3.1 A).

Proof:

Once again we follow B.H.R.

/1A, Theorem 4.1/ (see

Obstacle problems

(SEC. 3) a l s o Falk

(3.18A)

/lA/).

1

559

We h a v e , from G r e e n ' s i d e n t i t y

Auv dx +

a(u,v) =

It t h e n f o l l o w s from vh-%

(3.19A)

-

E

1

H,,(Q)

Vv

jrg

v dr

E

h K h

and

1

H (Q).

(3.18A) t h a t

Let n h b e t h e V h - i n t e r p o l a t i o n o p e r a t o r on C h ' i . e . o p e r a t o r d e f i n e d by 1

V E

(-Au-f)(v h-% ) d x V V h E % '

a(u,vh-uh)-

rh : H

V

(n) nco(C)

+

the

vh ,

1

V V E H (Q)n Co(C) , V P E Ch w e have

71 v ( P ) = v ( P ) . h 2 We h a v e R c IR , and h e n c e H Jfi) c C o ( z ) w i t h c o n t i n u o u s t h e c o n d i t i o n s U E H ( Q ) , u > $ on R, u = g on r , injection; t h e n imply T ~ U E

2

\.

We t a k e vh = ?rhu i n

(3.20A)

a(%-u,%-u)

I

(3.15A), (3.19A) and h e n c e o b t a i n a(\-u,nhu-u)

+

1,

(-Au-f) (nhu-y,)dx

.

We n o t e t h a t

( 3 . 2 1 A) Let w =

'rrhu-\

= ( T ~ u - u+ ) (u-$) + ( $ T ~ $ )+ (Th$-%)

-

2 -nu - f ; w e h a v e w E L (Q) and w e deduce from (3.21A)

From Lemma 3.1 we h a v e w 2 0 a.e. and more- w(u-$) = 0 a.e.; over s i n c e E It t h e n f o l l o w s from we have . ' r h $ - ~ s O m R . ( 3.22A) t h a t

u,,

From t h e c o n d i t i o n on t h e angles of w e h a v e , s i n c e u , $ E H 2 (Q)

s t a t e d i n t h e Theorem,

560

Steady-state inequalities

(APP. 1)

where, i n (3.2bA); (3.25A), C denotes various q u a n t i t i e s which a r e independent of h,u,$. We w r i t e

1.1

=

I ,Q

(1,

then r e s u l t s from (3.20A)

(3.26A)

I uh-uI

I V V ~ ~ ~ X )and ~ ' ~ IIvII

= IIvII

,

(3.238)

- (3.25A)

j

H

1,Q

it

(Q)

that

= O W .

w e note t h a t , s i n c e i-2 i s I n o r d e r t o e s t i m a t e IIy,-uII 1 ,n bounded, we have

(3.278)

I I v I I ~ ,< ~ C

1 l ~ l ~V v, c~H o ( Q ) ,

C independent o f v . 1

It t h e n follows from (3.26A) and from Y , ' ~ ~ u E H ~ ( Qt)h a t

Comments: Concerning t h e approximation of t h e o b s t a c l e problem by f i r s t - o r d e r f i n i t e elements, t h e O(h) e s t i m a t e w a s obtained by Falk /l/, and then by Mosco-Strang /1/ ( s e e a l s o Chapter 1, Section 4.7). I n Falk /2/, / 1 A / t h e v a l i d i t y of t h i s e s t i m a t e w a s shown under q u i t e g e n e r a l assumptions on i-2 ( t h e r e s u l t s o f The problem Falk / 1 A / were considered f u r t h e r i n C i a r l e t / l A / ) . .of o b t a i n i n g L2 e s t i m a t e s of optimal o r d e r ( i . e . O ( h 2 ) ) of \-u v i a a g e n e r a l i s a t i o n of the,Aubin-Nitsche method has not y e t been completely resolved; f o r incomplete r e s u l t s i n t h i s d i r e c t i o n we r e f e r t o N a t t e r e r /1A/ and Mosco / l A / . To conclude our discussion of piecewise-affine

approximations,

it should be pointed out t h a t using adequate assumptions Baiocchi /2/ ( r e s p . Nitsche /YL/)have been a b l e t o o b t a i n f o r t h e o b s t a c l e problem

11 %-uI I

=

O(h 2 -€ ) ,

E

>O

a r b i t r a r i l y s m a l l (resp.

Obstacle problems

(SEC. 3 )

-

561

I

3.7 Finite-element approximations of problem (3.1A), (3.2A) (11) Piecewise-quadratic approximations With C

0

h

and C

h

as in Section 3.6, we define

E, P

C ' = {P E

h triangle

T E

0

z;

= {pEz;

the midpoint of a side of a

'Ch} , , P $ r), , C;I = C h U C i 0

1;: =

ChUCi

0

.

0

We then approximate Hl(0) and K by (3.29A)

Vh = { v e C 0 ( E ) , v h l T € P 2 v T e C h }

(3.30A)1

\\

(3.30A)2

E;,

2

,

0

= EvheVh,

vh(P)2+(P)

\PEE{

, vh(P)

VPEC;lnT)

= g(P)

0

VPc C i

= { v h e V h , vh(P) > $ ( P I

, vh(P)=g(P)

We remark that in the condition vh(P) > $ ( P ) required at the midpoints of the sides.

V P E C;:nl')

is only

Finally we approximate the obstacle problem (3.1A), (3.2A) by

Find

4f E

such t h a t

(3.3 1 A)

11,

i i Vu,,*V(v,-u,,)dX

where i = 1, 2. If

$Ir

s g we have

<

2

1,

f(vh-\)dx i

VV,

E

E;,i ,

# 0 and thus:

Proposition 3.2: The approximate problems (3.31A);, adnit one and only one solution.

i = 1,2,

As regards the convergence of the.approximate solutions, we refer to Glowinski /1A/ where lim II<-uII = 0 is proved, h-+O H1(52) provided we a6sme that the condition on the angles of %h stated in Theorem 3.2 is satisfied. The investigation of the approximation error is much more complicated than in the case of piecewise-affine approximations and we therefore refer to B.H.R. /1A/ where, using suitable assumptions on f, u, $, g and the free boundary, it is shown that Vi=1,2 we have

.

562

Steady-state i n e q u a l i t i e s

i f t h e above c o n d i t i o n on t h e a n g l e s o f

4.

h

( U P . 1)

is satisfied.

THE OBSTACLE PROBLEM. (11) NON-CONFORMING APPROXIMATIONS USING MIXED FINITE ELEMENTS

4.1

Synopsis

I n t h i s s e c t i o n which f o l l o w s B.H.R. /2A/ ( s e e a l s o Hlavachek / l A / ) , we c o n s i d e r t h e a p p r o x i m a t i o n o f t h e o b s t a c l e problem (3.1A), (3.2A) by a mixed f i n i t e element method, v i a a dual formu l a t i o n of ( 3 . 1 A ) , (3.2A). 4.2

A d u a l f o r m u l a t i o n o f t h e o b s t a c l e problem (3.1A),

(3.2A).

It i s c l e a r t h a t (3.1A), (3.2A) i s e q u i v a l e n t t o

(4.1A)

Min { { v ,q ) E X

i

lqI2dx

R

-

f v dx

1,

52

where

I

v r ~ la.e. on n).

We a s s o c i a t e w i t h ( 4 . 1 A ) ,

(4.3A)

L(v,q,LJ) =

$-

(4.2A) t h e Lagrangian

lq12dx

-

i,f.

dx +

U*(Vv-q)dx

,

R 52 and we c o n s i d e r t h e dual problem ( s e e C h a p t e r s 1 and 2 f o r t h e concepts of d u a l i t y ) o f ( 4 . 1 A ) ,

(4.2A) r e l a t i v e t o

d , namely

where 'c.

(4.5A)

)I =

{{v,q)

EH

1

(R)X(L

2

(QBN,v=g on

r,

v ~ J Ia.e.

on 52).

It can b e shown q u i t e e a s i l y t h a t : Proposition

4.1:

The dual problem ( 4 . 4 A ) i s e q u i v a l e n t t o

where (4.7A)

C = {q ~ H ( d i v , R ) ,V-q+f 1 0 a.e.

on R)

(SEC.

4)

563

Obstacle problems- 11

Conversely ( 4 . 6 ~ )(4.7A) ~ admits a unique solution p such that p = vu where u i s the solution of the obstacle problem ( 3 . 1 A ) ,

(3.a). Remark 4.1:

I n t h e above, n d e n o t e s a u n i t v e c t o r normal t o We r e c a l l t h a t i f qE H ( d i v , a ) , p o i n t i n g outwards from R. S t r i c t l y , t h e t e r m & g q'n d r t h e n q * n e x i s t s i n H-2(I'). s h o u l d b e r e p l a c e d by < g , q * n > , w h e r e < - ,-: d e n o t e s t h e b i l i n e a r form of t h e d u a l i t y between H n ( r ) and H-P(I').

r,

Remark 4 . 2 :

Let

EL 2 (a),

A =

(4.9A)

1120 a . e .

on R}

.

We t h e n have t h e f'ollowing e q u i v a l e n c e

q-cJii q

( 4 . 1 OA)

E

H(div,R),

(V*q+f)p d x S O

V ~ E A .

52

4.3

A n a p p r o x i m a t i o n o f t h e d u a l problem mixed f i n i t e e l e m e n t s

( 4 . 6 ~ ) , (4.7A)

by

2We once a g a i n assume t h a t R i s a bounded polygonal domain i n i s a triangulation o f R. We t h e n approximate H(div,R a n) ,d % L ( R ) , A , C r e s p e c t i v e l y , by

IR

( 4 . 1 IA)

t

%=

iqh

V-qhlT

E

E

qh(T

H(div,R)

Pk VT

E

Ch, qh*nl

( n i s a u n i t v e c t o r normal t o

2

(4.12A)

Lh = {ph

(4.13A)

Ah

(4.14A)

ch = {qh E

=

E L

A n L~ = {ph

1,

and

'k+lxpk+1

E

Pk VS

s i d e of a T

s),

p h I T E P k V T E &h}

('),

%,

E

EL^, uh

20

a.e.

9

on 52)

,

(v* qh+f)Vh d x S O V p h ~ A h )

We t h e n approximate t h e d u a l problem

-

(4.6A), (4.7A) by

E-}

.h

Steady-state inequalities

564

Find p h E ch such that Vqh (4.15A)

1

ph' (qh-Ph)dx

E

(APP. 1)

Ch we have

g(qh-ph)'"

+

dr

52

I t is clear that (4.15A) adnits one and only one solution.

Let A be the Kuhn-Tucker vector (see Rockafellar 131) relating to (4.14fi); problem ( 4 . 1 5 A ) is thus equivalent to the variational sys tem

..

It results from B.H.R. /2A/ that (4.16A) admits one and only one solution, with ph the solution of ( 4 . 1 5 A ) . A s regards the convergence of ph' h h when h results ire proved in B.H.R. /2A/.

Theorem 4.1: bounded below by

-+

0, the following

We assume t h a t when h -+ 0 the angles of independent of h. Then

(i)

2 2 I f k = 0 and if f E L (521, u , $ E H (Q) we have

(ii)

I f k = 1, i f f u

E

5 are

go > 0 ,

WS'"(Q)

E

L"(Q), u,$

Vae]I

E

,+4 a d

d'O0(52 and ) if s <2+

1 P'

then

with

E

> O arbitrarily small.

rn

If k = 0 and if qh E \, then qhlT = (qIT'q2T) Remark 4.3: must necessarily satisfy (see J.M. Thomas, private communication): (4.19A)

I

qlT(X)

q1T

92T(X)

q2T+bTX2

+

bTXl

9

Stamp problem

(SEC. 5 )

565

(where x = {xl, x 2 1 ) .

Remark 4.4: The numerical s o l u t i o n o f ( 4 . 1 5 A ) , ( 4 . 1 6 A ) would appear t o be more complicated t h a n t h a t o f t h e approximate problems of S e c t i o n 3 corresponding t o conforming finite-element approrimations of t h e o b s t a c l e problem ( 3 . 1 A ) , (3.2A).

5.

ON A STAMP PROBLEM WHICH LEADS TO A VARIATIONAL INEQUALITY FOR A PSEUDO-DIFFERENTIAL OPERATOR

This s e c t i o n f o l l o w s . t h e work o f G . I . Eskin / 1 A / A. Bogomolnii, G. Eskin and S. Zuchowizkii / l A / .

5.1

and o f

Statement o f t h e problem

Consider a stamp w i t h an e x t e r n a l s u r f a c e r e p r e s e n t e d by

= where x

E

IR2

= $(XI

$ being a regular function with

@ (0)

(0) = 0

=

3x2

,

x # 0. The stamp i s a p p l i e d t o t h e e l a s t i c h a l f space z < 0. Let 6 ( > 0 ) be t h e displacement imposed on t h e stamp i n t h e d i r e c t i o n p a r a l l e l t o t h e z-axis. Let G be t h e c o n t a c t domain, which i s not known a p r i o r i . w(x) denotes t h e displacement normal t o t h e p l a n e z = 0 and u ( x ) t h e normal component of t h e stress t e n s o r . A s i n c l a s s i c a l t h e o r y , w and u a r e r e l a t e d by

k >O

(5.2A)

1x1

=

d

x.

The c o n d i t i o n s on z = 0 d i f f e r according t o whether one i s i n s i d e G or o u t s i d e G : i n G we have

(5.3A)

w ( x ) = $ ( ~ ) - 6 and

a(x) S O

,

and o u t s i d e G we have: (5.4A)

w<$(x)-~

and

a(x)

= 0

.

If we i n t r o d u c e t h e n o t a t i o n ( i n accordance w i t h t h e usual notation f o r t h i s subject)

566

Steady-state i n e q u a l i t i e s

(5.5A) then

( U P . 1)

u = - 0

/

u20,

-

*u

(5.6A)

(6-@)-> 0 ,

1 u [ k -*u

- (&-+)I =

I "I

0 i n IR

which i s an "obstacLe-type" problem

2

,

-

except t h a t t h e secondorder e l l i p t i c o p e r a t o r which normally appears has here been replaced by the convolution operator

(5.7A)

1

Au=k-*U.

I "I

We n o t e t h a t problem ( 5 . 6 A ) can be formulated f o r a bounded open domain ( a t l e a s t with a stamp of "reasonable shape"). I n f a c t , i f u 2 0, t h e n Au 20 and t h u s Au > 6- 4 when 6-$(x) < 0 ; ) u = 0 when 6-$(X) < 0 , t h u s , using t h e l a s t of conditions ( 5 . 6 ~ , It i s t h u s s u f f i c i e n t t o consider t h e problem analogous t o (5.6A) i n an open domain fi containing the region "6-@(x) < O " i n i t s i n t e r i o r , which i s p o s s i b l e with Q bounded i f t h e stamp z = $(x) i s "reasonable" ! We t h u s put

(5.8A)

6-6 = f

.

We f i n a l l y have t h e problem:

domain i n n2 (5 .9A)

, with

u20 ,

Au-f 2 0

Find u i n Q , a bounded open

, u(Au-f)

=

0 i n $2

where f i s given, with f < 0 i n the neighbourhood of the €owz&ry of n, and where i n (5.7A) u i s understood t o be extended by zero outside Q. 5.2

Functional formulation

We denote by 3 t h e F o u r i e r transform;

-2'ixs$ We n o t e t h a t (5.1

1

OA) =

151

formally, we have:

(x) dx

.

Stamp problem

567

so t h a t

( 5 . 1 IA)

~(Au)= k

-Q

.

I51 Consequently, i f

+

and

JI €,&(a) , we have

It i s t h e n n a t u r a l t o i n t r o d u c e :

1

E = completion of

H(n)

f o r t h e norm

t h i s space i s one of a family o f spaces o f ' f r a c t i o n a l Sobolev' t y p e , i n v e s t i g a t e d elsewhere. More g e n e r a l l y , f o r s > 0 , w e i n t r o d u c e ( s e e L. Hormander and J . L . Lions /lA/):

2(Q) = completion of (5.1 4A)

(1

1512'

&R)

f o r t h e norm

1G(5)12d5)1/2.

IR2

W e must t a k e n o t e o f t h e f a c t t h a t it i s not always p o s s i b l e t o i d e n t i f y t h i s space w i t h a subspace o f d i s t r i b u t i o n s on Q. However -1 / 2

(IR2 ) i s i d e n t i f i e d with a subspace o f

&

(5.15A)

( a n d , consequently,

i'/2(R)

8' (IR 2 ).

i s i d e n t i f i e d , f o r any Q , bounded o r

n o t , w i t h a subspace of &'(n)). To prove

j' 2 ( = 2 )

(5.15A), we c o n s i d e r a Cauchy sequence 4 n f o r ;

if

Q

4

dx =

I,

E

&lR2

),

w e have:

1C11/2 $n(lCl-1'2

$)dC

;

IR2 2 2 -1/2 s i n c e 2 )5 ) '"IRi s a Cauchy sequence-in L (lR ) and 15) 2 @ll i n L (IR ) we t h u s see t h a t @, JI dx converges. We n o t e t h a t

IlR2

.

JI i s

Steady-state inequalities

568

(5. I6A)

I

i 1 I 2 ( n )(a

E = t h e dual o f t h e space

j-I

i s g e n e r a l l y denoted by

(APP. 1)

12

space which

(n) 1

and from ( 5 . 1 1 A ) we see t h a t

A i s an isomorphism from j - ' l 2 ( s 2 )

(5. I7A)

onto B -+I '2(s2)

If ( f a $ ) denotes t h e s c a l a r product between .6-1'2(s2) - 1 12

B

(a),

(5.1 8A)

.

and

w e have:

(A$¶$) =kll$ll;

*

W e introduce

(5.19A)

K

= { v ) v ~ E v

~ iOn Sl

(9]

which d e f i n e s a nonempty c l o s e d convex s e t i n E. f i n d u E K such t h a t i s t h u s formulated:

(5.20A)

(Au,v-u)

where f i s given i n

2

(f ,v-u)

Vv

6

Problem (5.9)

K

8^'I2(n)

From (5.18A) t h i s problem comes w i t h i n t h e g e n e r a l t h e o r y of I t admits a unique solution. Variational Inequalities. 5.3

F i n i t e element approximat i o n

I n concept, any f i n i t e element method ( w i t h a s u i t a b l e approxThe d i f f i c u l t y l i e s imation o f K ) i s a p p l i c a b l e and convergent. i n t h e f e a s i b i l i t y o f calculating the c o e f f i c i e n t s . To t h i s end t h e following method i s proposed i n Bogomolnii, Eskin and Zuchowizkii /lA/.

We consider a p a r t i t i o n i n g o f RL i n t o squares of s i d e h a with edges p a r a l l e l t o t h e coordinate axes. We put

(5.21A)

$,(XI

= (l-lxJ)(l-

1x21>

i n t h e square lxll < 1 lx21 < 1 , and +o = 0 o u t s i d e t h e square. We t h e n consider t h e basis functions:

~~

(')

I n t h e sense o f d i s t r i b u t i o n s on Sl

(SEC.

6)

Nonlinear DirichZet problems

569

We assume t h a t Q i s approximated by t h e union o f s q u a r e s and t h e i r t r a n s l a t i o n s , and we approximate K by combina t i o n s of t h e 4 corresponding t o c o e f f i c i e n t s 20. This amounts t o calculating P

/xi1 < h

(5.23A)

(A($p.($q) = a p y q .

I n fact a depends o n l y on p-q: P39 (5.248) aPYq = aP = p-q where, a f t e r a simple c a l c u l a t i o n

(5.25A)

a

P

=

kh3 jj 2I.r

sin

eiP5

IR2

a

=

kh3

P

1 la1 S M

sin

-

1( 7 151

It i s p o s s i b l e t o c a l c u l a t e a (5.2618)

5, -

P

-

2 2 u s i n g an asymptotic formula

(-l)lal($a

+ O(-

D2a(L)

(PI

'

I P 12M+2

)

where we c o n s i d e r p as a continuous v a r i a b l e and t h e n t a k e p E i n t h e r e s u l t , and where t h e 4 are t h e Taylor c o e f f i c i e n t s a of t h e f u n c t i o n

2

We r e f e r t o Bogomolnii e t a l , l o c . c i t . , r e s u l t s obtained using t h i s technique.

6.

f o r numerical

SOLUTION OF NONLINEAR DIRICHLET PROBLEMS BY REDUCTION TO VARIATIONAL INEQUALITIES

6.1

Synopsis

I n t h i s s e c t i o n w e show t h a t t h e methodology of v a r i a t i o n a l i n e q u a l i t i e s can a l s o be u s e f u l i n t h e s o l u t i o n o f equations, We s h a l l i l l u s t r a t e a f a c t which h a s been used by h e 1 / l A / . t h i s u s i n g a f a m i l y of nonlinear Dirichlet probZems , c o n c e n t r a t i n g i n p a r t i c u l a r on t h e i r approximation by means o f a finite element method. We s h a l l omit most o f t h e p r o o f s , r e f e r r i n g for t h e s e t o Chan-Glowinski /lA/, Glowinski /2A/.

Steady-state inequalities

570

6.2

r

(APP. 1)

The continuous problem

L e t fi b e a bounded domain i n lRN (N t I ) = an; w i t h V = H A ( f i ) we c o n s i d e r :

w i t h a r e g u l a r boundary

-

L : V + IR, l i n e a r and continuous, i . e . L(v) = < f , v > V V E V , where f E V ' = H-I(S2) ( V ' i s t h e d u a l o f V and < * , * > i s t h e b i l i n e a r form o f t h e d u a l i t y between V' and V ) .

-

a : V X V +IR b i l i n e a r , c o n t i n u o u s , c o e r c i v e ; e x i s t e n c e of an a > 0 such t h a t

where

(1,

IIvIIv =

we t h u s have t h e

1VvI2 dx)1'2.

We do not assume a p r i o r i t h a t a ( * , * )i s symmetric.

-Q

: lR+IR,

$(t')

r$(t)

Q non-decreasing ( i . e .

QEC'(IR), Vt,t'EIR,

t ' rt),

Q(0) = 0.

We t h e n c o n s i d e r t h e nonZinear variational equation

(6.2A)

I

Find u

E

v such that Q (u)

E

L

1

(a) n V ' and

a ( u , v ) + <$(u) ,v> = Vv E V .

Let A E&V,V') such t h a t a ( v , u ) = V V , W E V ; it i s c l e a r t h a t ( 6 . 2 ~ )i s equivalent t o t h e nonlinear Dirichlet problem

Au + Q(u) = f , (6.3A) U E V ,

Example: (6.4A)

Let

$(u) E L

1 '

(a).

a. ~ L - ( f l ) such t h a t

ao(x) 2 a > O a.e.

on fi

and l e t B be a constant vector i n IRN

.

We d e f i n e a ( * , * )bilinear and continuous on H 1 ( n ) x H A ( Q ) by

a(u,v) = We have

o;j

0

Vu*Vv dx +

I,

B-Vu v dx

.

Nonlinear Dirichlet problems

(SEC. 6)

571

which, combined with (6.4A) , implies a(v,v) t

\

2 lVvl dx

1 Vv e H O ( Q )

,

R i.e. the coercivity of a(*,*).

6.3

6.3.1

Existence and uniqueness results for (6.2~).(6.3A).

Introduction. A variationa Z inequa Z i t y associated with (6.2A), (6.3A). 1

If N = l we have H (Q)c C0@) with continuous injection and the proof of the exist%nce and uniqueness of a solution of (6.2~1, (6.3A) is then almost immediate. If N 2 2 , as we shall assume in the following the essential difficulty is precisely the fact that HL(Q) # Co(&). The analysis which follows can also be applied Remark 6.1: to problems in which V = H1(Q), or in which V i s a closed subspace of H1(Q). Let Q : IR-+IR be defined by t

O(t> =

;

$(T)dT 0

@

is clearly C1, convex and

20

.

We then define j ( - ) by

It is easily shown (see Chan-Glowinski /lA/, Glowinski /2A/) that : Proposition 6.1: The functional j ( - ) i s convex, proper and on L~(Q). B Assume that a( ) is symmetric; it would then be natural to associate with (6.2A), ( 6 . 3 A ) the following calculus-ofvariations problem : 2.s.c.

9 ,

Find U E V such that (6.6A) J(u) I J(v) V v

where

E

V

572

steady-state inequalities

(6.7A)

J(v)

=

(APP. 1)

T1 a(v,v)+j(v)-L(v>.

I f a ( - , * ) i s non-symmetric t h e above approach i s not f e a s i b l e , at l e a s t d i r e c t l y , b u t , on t h e o t h e r hand, w e can a s s o c i a t e with (6.2A), (6.3A) t h e E l l i p t i c Variational Inequality ( o f t y p e ( 1 . 2 A ) ) :

[ Find u ~ such v that (6.8A)

t

a(u,v-u)+j (v)-j (u) ~L(v-u) V v

E

V

which i s i n f a c t a g e n e r a l i s a t i o n o f (6.6A), (6.7A). A s regards t h e e x i s t e n c e and uniqueness o f a s o l u t i o n o f

(6.8A), i n view of P r o p o s i t i o n 6.1 w e can apply Theorem 2.1 of Section 2.2, which g i v e s :

Under the above asswnptions on a, L, 4, problem Theorem 6.1: (6.8A) admits one and only one soZution.

6.3.2

Equivalence between (6.2A), (6.3A) and (6.8A).

It i s p o s s i b l e t o prove: Theorem 6.2: The solution o f (6.8A) i s necessarily a solution of (6.2A), (6.3A). Conversely , any soZution of (6.2A), (6.3A) i s a solution o f (6.8A). The above theorem implies t h a t ( 6 . 2 ~ ) , ( 6 . M ) admits one and only one s o l u t i o n . Another consequence of t h e above equivalence i s t h a t , from t h e p o i n t of view o f t h e numerical solution, we can e i t h e r work d i r e c t l y on (6.2A), (6.3A), or a l t e r n a t i v e l y consider problem (6.8A); i n f a c t , from t h e p o i n t o f view o f t h e approxima t i o n , it i s i n our opinion simpler t o work with (6.8A).

6.4

The f i n i t e element approximation of ( 6 . 2 ~ ) , ( 6 . 3 ) and

(6.8A). 6.4.1

Synopsis

W e s h a l l now approximate problem (6.2A), (6.3A) by working d i r e c t l y on t h e e q u i v a l e n t formulation (6.8A); however, w e should p o i n t out t h a t t h e c o n s i d e r a t i o n s o f S e c t i o n 2.3.2 cannot be a p p l i e d d i r e c t l y s i n c e assumption (2.228) ( i . e . j(-) continuous on V) i s not s a t i s f i e d .

(SEC.

6)

6.4.2

Nonlinear Dirichlet problems

Triangulation of R. prob Zem.

573

Definition o f the approximate

.

2 We assume t h a t R i s a polygonal ( ’ ) bounded domain i n IR o f t r i a n g u l a t i o n s of R which s a t i s f y We i n t r o d u c e a family t h e assumptions o f S e c t i o n 3.6 and w e approximate V = H1(.Q) by

(ch)h

0

1

Vh = {vh Ivh €C0(E),vh I ’ = 0 , V h I T E P I VT E Th}. It i s t h e n n a t u r a l t o approximate ( 6 . 2 ~ )(6.3A) ~ and (6.8A), r e s p e c t i v e l y , by:

Find

u,, e V h

(6.9A)

a(u,,,v,)

+

such t h a t

I

@(u,,)v, dx = L(vh)

VV,

cVh

R

and

It i s c l e a r t h a t t h e s e two finite-dimensional

problems a r e

equiualent. From a p r a c t i c a l p o i n t o f view it i s not g e n e r a l l y p o s s i b l e t o use (6.9~)and (6.10A) as t h i s r e q u i r e s t h e c a l c u l a t i o n of i n t e g r a l s which a r e i m p o s s i b l e t o e v a l u a t e e x a c t l y . It i s t h u s n e c e s s a r y t o modify (G.gA), (6.10~)v i a t h e use o f numerical i n t e g r a t i o n procedures. I n t h e f o l l o w i n g we s h a l l b e c o n t e n t t o approximate j ( * ) . (Remark 2.2 o f S e c t i o n 2.3.1 a l s o a p p l i e s i n t h i s c o n t e x t ) . Using t h e n o t a t i o n o f F i g u r e 6.1 below, w e approximate j ( * ) by j h ( * )d e f i n e d by

I n f a c t we could a l s o c o n s i d e r j ( v ) as b e i n g t h e exact integral of a piecewise-constant f u hn c t hi o n ; more p r e c i s e l y , i f we assume t h a t t h e s e t C of nodes o f %h i s o r d e r e d from 1 t o N = Card (1 ) , t h e n w i t k each Mi E C h w e a s s o c i a t e t h e s e t $2. h

6.2) o b t a i n e d by j o i n i n g t h e centre; Of t h e t r i a n g l e s w i t h common v e r t e x M. t o t h e midpoints o f t h e edges 1 which m e e t a t Mi. ( h o r n hatched i n F i g u r e

(1)

This assumption i s n o t e s s e n t i a l .

Steady-state inequa Z i t i e s

574

( U P . 1)

M3T

F i g . 6.1

F i g . 6.2

(If. Mi

E

r,

t h e m o d i f i c a t i o n r e q u i r e d i n Figure 6.2 i s t r i v i a l ) .

W e t h e n d e f i n e a space

where

x.1

4,of

piecewise-constant

i s t h e c h a r a c t e r i s t i c f u n c t i o n of 52 i , i . e .

xi(x)

= 1 if

xiS2.

1 '

.

xi(x) = 0 if x B Q i 1 We next d e f i n e qh : Co(E)n Ho(S2) + Lh by Nh (6.13A)

f u n c t i o n s by:

qhv =

1

i=l

v(Mi)Xi

.

(SEC. 6 )

Nonlinear Dirich l e t prob Zems

57 5

It t h e n f o l l o w s from ( 6 . 1 1 ~ )- (6.13A) t h a t

and hence t h a t

an a

Find u,,eVh

such t h a t

(6.17A)

a(U,,,Vh-Uh)+jh(Vh)-jh(U,,)

'L(vh-\)

Vvh

'h

'

It c a n t h e n e a s i l y b e shown t h a t ( 6 . 1 6 ~ )and (6.17A) m e equivalent and a h i t one and only one s o l u t i o n .

Four Zemas

6.4.3.

(5)

With t h e a i m o f p r o v i n g t h e convergence o f t o u when h h + 0, we s h a l l u s e t h e f o u r lemmas below, which a r e proved i n Chan-Glowinski / 1 A / and Glowinski /2A/: Lemma 1 . 6 : (6.18A)

1

Let v e H0 ( R ) ; for n e N we d e f i n e

T v = inf(n,sup(-n,v)).

n

we then have

lim n++ 03

T v = v

n

I

s t r o n g l y i n H (R) 0

T ~ V by

576

Steady-state inequalities

( U P . 1)

( 6 . 2 ~ ) ,(6.3A)) i s The solution u of ( 6 . 8 A ) (d

Lemma 6.2:

characterised by 1

1

U E H ~ ( Q > , @(u) E L

(a),

(6.19A) ~ ~ ( v - u ~v )

a(u,v-u)+j(v)-j(u)

I

EH~(Q)

nLoo(n).

Lemma 6.3: The space 8(Q) i s dense i n Hb(Q) n LOD@) , t h i s l a t t e r space being provided with the strong topology of V and the weak* topology o f ~ ~ ( f i ) .

Lemma 6.4:

For a l l p such that 1 < p i *

we have

Lemmas 6.1, 6.2, 6.3 are d i r e c t consequences of c l a s s i c a l r e s u l t s on t h e truncation of f u n c t i o n s of H1(Q) (and H A ( f i ) ) , for which w e refer t o Stampacchia /lA/.

6.4.4

Convergence of the approximate solutions

Regarding t h e convergence of t h e approximate s o l u t i o n s when h + 0, w e have

We asswne that when h + O the angles o f c h a r e Theorem 6.3: we then have bounded below by a constant g o > 0 . (6.21A) (6.228)

lim h+O

III+,-UII~

=

0,

l i m jh(I+,) = j ( u ) h+O

where u i s the solution of (6.2A), ( 6 . 3 ) and (6.8A) and that of ( 6 . 1 6 ~ )and (6.17A).

yn i s

Proof: We proceed as i n Chan-Glowinski, l o c . c i t . , Glowinski /2A/; . t h e praof i s along t h e same g e n e r a l l i n e s a s t h a t of Theorem 2.3, S e c t i o n 2.3.2. 1)

A prior; estimates for

I+,

W e t a k e vh = 0 i n (6.17A); f a c t t h a t j,(O)

(6.23A)

I.,

= 0 1

IILII.

9

we t h e n deduce from ( 6 . 1 ~ ) and t h e

(SEC. 6 )

where

IILll*=

2)

577

Nonlinear Dirichlet problems

.

sup

v€v-Io)

Weak convergence o f ( % ) h a

I t r e s u l t s from (6.23A) and from t h e compactness of t h e i n j e c t i o n from V = H1(Q) i n t o L2(Q) t h a t w e can e x t r a c t from 0 ( L + , ) ~ a subsequence - also denoted by ( u ~ -) such ~ that = u*

(6.258) lim

weakly i n V

WO

(6.26A) lim h+O

%

= u*

strongly i n L2(Q)

(6.278)

%

= u*

a.e. on Q .

lim W O

I n view of (6.23A) s i m i l a r l y have

- (6.26A)

and Lemma

6.4

(with p = 2 ) w e

(6.28A) lim qhu,, = u* strongly i n L2(Q).

NO We t h u s have, up t o a new e x t r a c t e d subsequence

(6.296) lim qhu,, = u* a.e. on Q

N O (6.30A) lim Q(qh%) = $(u*)

a.e. on a .

h+O Let V E B(Q) j it t h e n r e s u l t s from e.g. Ciarlet /1A/ t h a t under t h e assumption on t h e angles of b given i n Theorem 6.3, h w e have

where C denotes v a r i o u s q u a n t i t i e s which are independent of v and h , and where r i s t h e interpolation operator on't: defined by

h

h

(APP. 1)

Steady-state inequalities

578 vv

n01 (a) n co(Si) we have

E

r,v(P)

6.4 (with

lim kt0 Taking vh = rhv i n

E V ~and

VP E C ~ .

= v(P)

14oreover , Lemma

rhv

p = +

m)

implies

(6.336)

\ - L(rhv-yl) From

(6.358)

(6.17A) w e

t/vE B(Q)

(6.25A), (6.26~)and

then o b t a i n

.

Proposition

6.1, w e

deduce

* *

a(u ,u )+j(u*) 2 l i m i n f (a(u,,,y,)+j(q,u,,)). h + O

I n view o f (6.24A), (6.30A) we can apply Fatou's Lema t o {$(qh\))h from which w e o b t a i n :

(6.36A) (p(u*) E L 1 ( 5 2 ) . Moreover, we have

lim h+o

@(qhrhv)dx =

n

a(u* ,u*)+j (u*)

It t h e n follows from u*

@ ( v ) d x= j ( v )

(6.35A), i m p l i e s ,

which, with

(6.378):

1,

E

v,

5

V V E B(n)

i n the l i m i t in

(6.36A), (6.37A) t h a t

@ (U*) E L

1

(6.34A) vv

a(u* , v ) + j (v)-L(v-u*)

,

E

&n).

u* s a t i s f i e s

(521,

(6.38A) a(u* ,v-u*)+j (v1-j (u*) L ~ ( v - u * ) tlv

E

an).

w e deduce from Lemma 6.3 t h e e x i s t e n c e of Let v E V n Lm(s2); a sequence {vnIn such t h a t vn E J ( Q ) V n and

vn = v strongly i n V,

(6.39A)

lim n-

(6.40A)

l i m vn = v weakty* i n L m ( Q ) . n-t+oo

(SEC. 6)

From

579

Yonlinear Dirichlet problems

(6.38A) we u* E V ,

deduce

O(U*)

EL1(,)

( 6 . 4 1 A)

~ ~ ( v ~ - uv* n. )

a(u*,vn-u*)+j(vn)-j(u*)

(6.39A)

Moreover (6.42A)

lim

a(u*,v -u*) n

n* (6.43A)

implies

lim

L(vn-u*)

=

a(u*,v-u)

= L(v-u

,

*) ,

n* and, up t o a n e x t r a c t e d subsequence (6.44A)

lim n++co

vn = v a.e. on R ,

which i m p l i e s (6.45A)

a.e. on R.

lim O(vn) = O(V) n*

(6.40) we have

I n view of

and hence Vn : (6.46A)

0 5Q(vn)

I n view of

a.e. on R.

5 const.

(6.45A), (6.46A)

we can a p p l y t h e Lebesgue dominated

convergence theorem, which g i v e s : lim n++W

In t h e l i m i t i n

u* (6.47A)

E

*

I

@(v )dx =

( 6 . 4 ~ )we

t h u s have

j(v ) = lim n n++m

v,

O(U*)

EL

1

52

@(v)dx

= j(v)

.

(a),

a(u ,v-u*)+j (v)-j(u*> z ~(v-u*) v v

E

v n Lm ( 0 )

.

5 80

Steady s t a t e i n e q u a l i t i e s

( U P . 1)

From Lemma 6.2, (6.47A) implies u*= u where u is the solution of (6.2A), (6.3A) and (6.8A). In view of the uniqueness of u it is the entire sequence ( ~ h which ) ~ converges weakly to u.

3)

Strong convergence o f

(%Ih.

From (6.17A) and from the coercivity of a(=,-), it results that v v E &a) :

Using the various convergence results from the second part of the proof, we have, in the limit in (6.48A) < lim inf jh(%) I lim inf(a

11 s - u I I v2 +jh(%))

I

slim sup(a I(\-uIIV 2 + j h U ( h) ) s

Using, as above, the density of B(n) in V n Lm($), it can be shown that ( 6 . 4 ~ )also holds for all the v E V n L (Q); with T defined by (6.18A) we then have: n 2 j ( u ) slim inf j h ( \ ) slim inf(a IIu,,-ulIv + j h (% ) ) I

(6.50A)

I

I

lim sup (a II%-uIIv 2 + j,(%))

I

a(u,-r,v-u) +j( T ~ v -L(T~v-u) )

5

v v E V, v n .

In view of the properties of T (see Lemma 6.1) we have, in the n limit in ( 6 . 5 0 ~ )when n +. + m :

(SEC. 7 )

Quasi-variational inequa Z i t i e s

581

Taking v = u i n ( 6 . 5 1 ~ )w e t h u s deduce

which proves t h e Theorem. Descriptions of various i t e r a t i v e methods which can be used t o ) ( 6 . 1 7 ~ ) can be s o l v e t h e equivalent approximate problems ( 6 . 1 6 ~ , found i n Chan-Glowinski, l o c . c i t . , which a l s o contains t h e r e s u l t s of various numerical t e s t s .

7.

INTRODUCTION TO NUMERICAL ALGORITHMS FOR QUASI-VARIATIONAL INEQUALITIES ( )

7.1

Quasi-variational i n e q u a l i t i e s

Consider, i n an open bounded ( 2 ) domain R , an o p e r a t o r

with t h e assumption of coercivity on H1( 0 ) ( 3 , :

where

(7.3A)

a(u,v) =

I '52 aij

av J

dx

+

I/

1

ai

52

zi aU v ax + /52aouv ax.

We a l s o introduce a nonZinear operator J,

+

WJ,) m

OD

from L y ( Q ) + L + ( Q ) where L + ( R ) denotes t h e s e t of f u n c t i o n s ( a . e . 2 0 ) i n L m ( R ) ; we assume t h a t t h i s o p e r a t o r M has t h e (l)

Abbreviated t o Q.V.I.

(2)

I n ' o r d e r t o c l a r i f y t h e concepts.

(3)

This assumption can be weakened and w e can consider a - H1 (n) H i l b e r t space V with HL(R) 5 V c

.

582

Steady-state inequalities

(APP. 1)

following p r o p e r t i e s :

(7.4A)

I

M i s positive increasing i .e. 0 “4J,

‘ J I , # 0 s M ( J I l ) <M(J12)

,

Under t h e s e c o n d i t i o n s , t h e r e e x i s t s one and only one function, u , which is a solution of the Q.V.I.

a(u,v-u) 2 ( f ,v-u)

v v with 0 6 v sM(u)

,

(7.6A) O
.

In ( 7 . 6 ~, ) f i s given i n LoO,(n).

Q.V.I‘s have been introduced i n Bensoussan-Lions /lA/,/2A/,/3A/, /IrA/in t h e steady-state and time-dependent ( ) c a s e s , with the operator M given by (7.7A)

M(JI)(x) = k + i n f

Si’O

$(x+S)

, Vx

EQ

where i n (7.7A) $J(X + < ) i s taken f o r t h e 6 such t h a t 6. 2 0 and x + < E Q ; t h e motivation f o r t h i s work r e s i d e s in i h l s i v e stochastic control ( 2 ) ( 3 ) . L. Tartar /1A/ has made t h e observation t h a t , for a (non-constr u c t i v e ) existence proof, t h e assumption that M is increasing i s s u f f i c i e n t , and !I%. Laetsch /1A/ has proved t h e uniqueness o f t h e s o l u t i o n when t h e supplementary condition (7.5A) i s met.

Remark 7.1: If i n (7.6A) M(u) were known, we would t h e n have a v a r i a t i o n a l i n e q u a l i t y and, i n f a c t , an obstacle problem; however, s i n c e M(u) i s not known, t h e problem i s implicit and we have adopted t h e terminology Q.V.I.

(l)

Not considered here.

(2)

A systematic i n v e s t i g a t i o n of t h i s i s described i n A.

(3)

Impulsive c o n t r o l l e a d s t o numerous o p e r a t o r s M which are d i f f e r e n t from (7.7A) b u t which are s t i l l s i m i l a r i n nature.

Bensoussan , J.L.

Lions, Contrdle Impulsionnel Stochastique , Dunod-Gauthiers-Villars, t o appear.

QuasivariationaZ inequa2ities

(SEC. 7 )

Synops i s I n t h e f o l l o w i n g we s h a l l b e concerned s o l e l y w i t h a construct-

i v e method f o r a p p r o x i m a t i n g t h e s o l u t i o n . 7.2

I t e r a t i v e scheme

The i t e r a t i v e scheme t o b e i n t r o d u c e d i s c o m p l e t e l y s t a n d a r d ; 0 s t a r t i n g w i t h u 2 0 ( s u i t a b l y chosen; we shall return t o t h i s p o i n t l a t e r ) we s u c c e s s i v e l y d e f i n e u1 , u2,. by t h e soi!ution of

..

the variationa2 inequa2ities (obstacZe probZem) n n a(u ,v-u )

2

(f ,v-un)

b'v w i t h 0

5

n- 1 v 5M(u ),

(7.8A)

n 05u

5

M(un-').

T h i s i s t h e scheme i n t r o d u c e d i n Bensoussan, Goursat and Lions Various n u m e r i c a l e x p e r i m e n t s , m o t i v a t e d by p r a c t i c a l a p p l i c a t i o n s , n o t a b l y stock control ( s e e Goursat and Maarek /lA/), show a very rapid convergence o f un t o t h e s o l u t i o n . In t h e case where M i s g i v e n by (7.7A) J . L . Menaldi /lA/ h a s proved convergence as l / n by ' p r o b a b i l i s t i c methods. However, t h e r e is i n fact a very simp2e genera2 r e s u l t giving exponential convergence, aue t o B. Hanouzet and J . L . J o l y /1A/,which w e s h a l l now p r e s e n t ( s e e a l s o Cortey-Dumont /lA/).

111.

We s t a r t by i n t r o d u c i n g t h e f u n c t i o n

(7.9A)

s o l u t i o n of

V V E H1 ( a ) .

= (f,v)

a(;,v)

i,t h e

We have :

(7.10A)

-u t O

-

O

D

UEL

and

(a).

We u s e t h e i t e r a t i v e scheme

(7.11A)

OSU

0

-

su

(7.8A) t a k i n g uo such t h a t

.

We i n t r o d u c e A w i t h

(7.12A)

O<X< 1

,

XuIM(0)

a.e.

on R.

Remark 7.2: From (7.5A) and (7.lOA) there e x i s t s a A s a t i s f y i n g (7.1.2A). I n o r d e r t o o b t a i n a more a c c u r a t e r e s u l t ( s e e ( 7 . 1 4 A ) below) it i s advantageous t o choose X as large as possibte.

584

Steady-s t a t e inequalities

( U P . 1)

We assume that the operator M s a t i s f i e s (7.4A),

Theorem 7.1:

(7.5A) and i s concave i n the sense

*

wJ1'$2

M(O-8)J11+8J12) 2 (1-8)M($1)+8M($2),

if oseS1. We use the i t e r a t i v e scheme (7.8A) starting from uo with (T.llA), and choose h satisfying (7.12A). Then

11 un-uII

(7.14A)

I

~(1-h)"

,c

=

constant..

Lm(w

-

Remark 7.3: - If we can choose h = 1 s a t i s f y i n g (T.l2A), i . e . n have - - , then u-is a solution ,Of t h e Q.V.I. s i n cI ne wt he i tsh ecase a(u,v-u) = (f,v-u) V v and O
u 5 M(0)

Theorem i s t r i v i a l .

Proof. 1.)

We introduce t h e following n o t a t i o n : S ( $ ) denotes the i .e.

solution of the V. I . with obstacle M($),

(7.15A)

I

a(S($) ,v-S($)) 2 ( f ,v-S($)) V v S($> SM($)

with 0 I v SM($)

*

Note t h a t i s t h e s o l u t i o n of t h e V . I . t h a t t h e v d u e of t h e s o l u t i o n of t h e V . I . if the obstacle increases ; t h u s

(7.16A)

S($) I S ( - )

-

= u

with obstacle

+oo and with o b s t a c l e increases

,v$.

2 ) Since A(hu) = Mu = )if S f , we s e e t h a t hy can be considered as t h e s o l u t i o n of t h e o b s t a c l e problem when t h e o b s t a c l e i s Au, so t h a t Au 5 M(0) implies

(7.17A)

IS(0).

-

3) 0

S

For a l l v , w with 0 5 v , w 5 u , we can choose a,B w i t h a,@ S 1 such t h a t

(7.18~) -aw SV-w 5Bv ;

(SEC. 7)

Quasi-variational inequalities

585

we shall now show that (7.18~) implies (7.19A)

-(I-h)a

I(l-x)B

S(W) IS(V)-S(W)

S(V)

.

In fact, the second inequality (7.18A) can be written

However, the concavity of M implies the concavity of S, i.e.

so that

I

(7.2OA) implies S(o)

S(W) >(l-B)S(v)+B

>(by

(7.16A))

?(by (7.17A))(i-B)S(v)+gXZ

2

(I-B)S(v)+BXS(v)

which leads to the second inequality (7.19A). The first inequality (7.18A)can be vritten v 2(1-a)w and henceS(v) 2 ( 1 -a) S(w) +as(0) 2 ( I -a) S(w) +aAS(w) giving the first inequality (7.19A).

4)

+ a0

We now use (7.18A) 67.19A) by recurrence; we note that taking w = u , we choose ci and B in such a way that

S ( u ) = u;

0

0

-au
from this we deduce that -( I-A)aS(uO) I S(u)-S(uO)

< (I-A) B S(u)

i.e. -(I-X)au

1

su-u

1

I (I-A)Bu.

We now apply (7.18A) (7.19A) with v = u, w = ul; a has become (1 - X ) a and B has become (1 -'X)B; from this we deduce 1 1 2 2 S(u) -(I-A) a S(u ) sS(u)-S(u ) < ( I - A ) and so on by recurrence, thus (7.228)

-(l-A)"a

U"

Iu-u" < ( I - A ) n B u

giving, in particu Z a r , ( 7.14A).

586

Steady-state inequalities 0

Remark 7.4:

( U P . 1)

-

We s t a r t e d w i t h 0 I u 5 u. It i s s u f f i c i e n t t o s t a r t w i t h uo 5 upper s o l u t i o n , which i s chosen t o be a suffic-

i e n t Zy large constant. From a numerical p o i n t of view, we s h a l l t h u s use Remark 7.5: ( 7 . 8 A ) , each V . I . b e i n g s o l v e d by t h e methods d e s c r i b e d elsewhere i n t h e book.

A f e w iterations are sufficient.

Remark 7.6: It h a s been observed by C. Baiocchi / 3 A / , (and t h i s o b s e r v a t i o n h a s given r i s e t o a l a r g e number of i n v e s t i g a t i o n s ) a l l o w t h e s o l u t i o n o f free-boundary problems which that Q.V.I.'s a r i s e i n t h e seepage o f w a t e r i n non-rectangular dams. In t h i s c a s e , assumption ( 7 . 5 A ) is not s a t i s f i e d ; n e v e r t h e l e s s , from t h e numerical p o i n t of view, w e observe a ' v e r y r a p i d ' convergence o f scheme ( 7 . 8 A ) , and t h i s would seem t o i n d i c a t e t h a t Theorem 7.1 could be extended s t i l l f u r t h e r .

Appendix 2 FURTHER DISCUSSION

OF OPTIMISATION ALGORITHMS 1.

SYNOPSIS

In this appendix we shall present a fairly succinct discussion, In supplementary to Chapter 2, on optimisation algorithms. Section 2 we describe a constrained block-overrelaxation algorithm and present a convergence result. In Section 3 we return to the duality algorithms of Chapter 2, Section 4, in order to point out some new results on the convergence of the multipliers. Section 4 is devoted to complementarity methods and Section 5 to conjugate gradient methods for the minimisation of quadratic functionuls over the products of intervals. Section 6 describes alternating direction algorithms which can be applied to the solution of certain nonlinear variational problems. In Section 7 we show the relationship which exists between the augmented Lagrangian methods mentioned in Chapter 2, Section 6, Chapter 3, Section 10 and Chapter 5, Section 9 and the alternating direction methods; for further details on augmented Lagrangian methods and their application to the solution of linear and nonlinear variational problems the reader is referred to the monograph currently under preparation by Fortin-Glowinski /1A/ (and also the corresponding bibliography); see also Marrocco / U / . 2.

THE 'METHOD OF BLOCK OVERRELAXATION WITH PROJECTION

This section supplements Chapter 2, Section 1.4 on the minimisation of finite-dimensional quadratic functionals using overrelaxation with projection. We shall describe a block variant of algorithm (1.41)y (1.42), which is also valid in infinite-dimensional Hilbert spaces, and give a convergence result. We follow the presentation of Cea-Glowinski /2/ (see also Glowinski / 4 I y 151, Comincioli / l / yCottle-Golub-Sacher /lA/).

588

@timisation aZgorithms 2.1

(APP. 2 )

Statement of t h e problem

L e t V be a f i n i t e - o r infinite-dimensional r e a l v e c t o r space. N

We assume t h a t V i s of t h e form V = II V . where t h e Vi a r e Hilbert 1 spaces; i f v e V w e then have i=l

...vN) with vi

v = {v, ,

E

.

Vi v i = l , . . N .

II*Ili

We denote by ( ( * , * ) ) and t h e inner product and t h e a s s o c i a t e d with V;f V i s t h e n a H i l b e r t space f o r t h e i n n e r product and norm defined by

1101111

Let K be a nonempty closed convex subset of V, of t h e form N

(2.2A)

Il Ki

K =

i=l where i = 1, s e t s of Vi.

..... N ,

and t h e Ki a r e (nonempty) closed convex sub-

We next consider J : V

-+

IR defined by

E V and a: V x V -t lR i s a bilinear form which i s continuous, 2 symnetric and coercioe ( i . e . 3ct> 0 such t h a t a ( v , v ) ? ctllvll VVCV).

where f

It then follows from t h e above p r o p e r t i e s t h a t t h e minimisation problem

(2.4A)

I

Find J(u)

U E

S

K such that

J(v) V v e K

admits one and only one s o l u t i o n c h a r a c t e r i s e d by

(2.5A)

I

a(u,v-u) 2 ( ( f ,v-u))

VVE K.

Block o v e r r e k c a t i o n with projection

(SEC. 2)

589

Description of t h e algorithm

2.2

We n o t e t h a t

(2.6A)

J(v) = J(v I,...~N) =

T1

N

1

- 1

aij(vi,vj)

I < i ,jsN

((fi,vi))

i= 1

where f i e Vi and where t h e a i j a r e continuous b i l i n e a r forms on Vi

x

V

j'

t h e aii being symmetric and coercive (with

V v i e V i ) and, more g e n e r a l l y

aii(vi,v.)2a llv.112 1 i i

By using Riesz's theorem we deduce from (2.7A) t h e e x i s t e n c e of o p e r a t o r s A.. E k ( V . ,Vi) such t h a t

J

IJ

(2.8A)

aij(vi,vj) = ((vi,A. .v.))~ J

13

* , Aij -- *ji

9

where t h e Aii (which are s e l f - a d j o i n t ) are isomorphisms from V. onto Vi. As t h e forms aii are sylrnnetric and coercive, t h e y d e h n e a H i l b e r t s t r u c t u r e on V whose norm defined by

111 111

i'

i s equivalent t o t h e norm

II.II

we s h a l l denote by P. t h e

;

orthogonu2 projection operator from V. onto Ki according ko t h e

111 111. .

B

NOW l e t u s consider positive scalars 'to solve (2.4A) we use t h e following g e n e r a l i s a t i o n o f algorithm (1.41), (1.42) of Chapter 2:

norm w

,..:.wN;

(2.1 OA)

u

0

=

...+I

~uy),

given a r b i t r a r i l y i n K

then for n 2 0 successivezy determine ur'l, (2.11A)

'+:u

=

P.(u?-w.Ay!( 1

1

1 1 1

1

Aijuj m+l j
+

.., N, by

i = 1,.

1

j2 i

A

u?-fi)).

ij J

Remark 2.1: A v a r i a t i o n a l d e s c r i p t i o n of algorithm (2.10A) (2.11A) i s given i n Comincioli /l/. 2.3

Convergence o f algorithm (2.10A), (2.11A)

The following theorem i s proved i n Cea-Glowinski

...N,

Theorem 2.1: If 0 < wi < 2, v i = l , IunIn defined by (2.10A), (2.11A)

/2/:

we have f o r the sequence

(APP. 2)

optimisa tion algorithms

590

lim IIun-uII

=

o ,

n-

where u is the solution of (2.4A). 2.4

Various remarks

Remark 2.2: N

I n t h e case i n which dim V < +

,..

-, with

II Vi, and i n which V i - 1 .N w e have W i = w > 1 ( r e s p . i=l w = 1, w < l), algorithm (2.10A), (2.11A) coincides with t h e standard block overrelaxation ( r e s p . relaxation, underrelaxation) K = V =

algorithm a s s o c i a t e d with t h e s o l u t i o n of t h e l i n e a r system defined by Au = f ( s e e Varga 111, Young /U/) with A E L(V,V) a(u,v) = ((Au,~)) V U,VE V.

Remark 2.3: I n t h e case i n which V = lRN , we can i d e n t i f y t h e above o p e r a t o r A with an NxN symmetric p o s i t i v e - d e f i n i t e matrix. K =

If

N II 1 Ki with i=

Ki

=

{vila.
, then

algorithm (2.10A),

(2.11A) reduces t o algorithm (1.41), (1.42) o f Chapter 2, S e c t i o n

1.4. Remark 2.4: An a p p l i c a t i o n of algorithm (2.10A), (2.11A) w a s described i n Chapter 3, S e c t i o n 8.3. Further a p p l i c a t i o n s may be found i n Marrocco /lA/, Fortin-Glowinski /1A/ and GlowinskiMarrocco /lA/, i n connection with t h e use of augmented Lagrangian methods. 3.

DUALITY METHODS

-A

FURTHER DISCUSSION

W e s h a l l now supplement Theorems 4.1, 4.2 and Remark 4.7 o f Chapter 2, Section 4 ( I ) by d e s c r i b i n g t h e behaviour of t h e sequences {pnIn defined by algorithms (4.12), (4.13) and (4.31),

(4.32). To t h i s end we use t h e following lemma (see Opial /lA/):

O p i d ' s Lemma: Let L be a Hilbert space, X a subset of L and I p n l n a sequence in L such that for all p e X the sequence

is convergent. If every szbsequence { p"k )k which converges weakly has its limit in X then { p In converges to an element of X. {

11 pn-p)) L)n

We can t h e n supplement Theorem

(l)

4.1

The n o t a t i o n o f which we r e t a i n .

o f Chapter 2, Section

4 by

591

Duality methods

(SEC. 3 )

Assume that the conditions f o r the application Theorem 3.1: of Theorem 4.1 of Chapter 2, Section 4 are s a t i s f i e d ; i f the condition ( 3 . IA)

o
i s s a t i s f i e d we then have f o r the sequence { P " } ~defined by ( 4 . 1 2 ) , (4.13) ( 3 .2A)

lim 1lpn+l-pn~lL = 0, *+w

(3.3A)

lim p n++m

= p

weakly i n L

where p i s such that {u,P) i s a saddle point of d on MxA

Proof: (3.4A)

With t h e Lagrangian k ( *,*)

.

defined by

;c(v,q) = J0(V)+(S,$(VHL

( s e e Chapter 2 , Section 4.3, r e l a t i o n ( 4 . 9 ) ) we consider t h e differences

L(U"+l,pn+l)4?(un,pn) = Jo(u n+ 1 )-JO(un)+(pn+l

(3.5A)

We have

d(un+' ,pn+l)-&un,pn). =

,@(un+')),-(Pn,$(Un)),

2

(u") 1,

L (J'(un) ,un+l-un)+(p n+ 1 ,$ (un+I ) ) ,-(p% 0

= (JA(un) ,un+ 1 -un>+(pn,$(un+'

)-@(Un)),

=

+

n+ 1

+

(P

Since t h e v e c t o r un is a s o l u t i o n of t h e minimisation problem (3.6A)

Min d(v,pn) veM

we have (3.7A)

(J'(un) ,v-un)+(pn,$(V)-b(Un)),~

0 VV€M

0

and hence

(JA( u")

,un+' -u")

+ (pn ,$ (un+l ) -$ (u") )

which i n conjunction with (3.5A) i m p l i e s

,0 2

@timisation a Zgori thms

592

L(un+l

yp

n+l

)-l(U"YP")

2

(UP. 2 )

n+ 1 (P

or a l t e r n a t i v e l y

e(un+I ypn+l)-du n yp n ) z (3.8A) (Pn+I -pn ,Q

) -Q (u") +, (pn+I -pn

Y

0 (u")

)

,.

From t h e p r o p e r t i e s of t h e p r o j e c t i o n s , w e have

(q-pn+l ,pn+pQ(un)-pn+l), and hence i n p a r t i c u l a r which we deduce

I0

VqE A

,

(pn-pn+l ,pn+pQ(un)-pn+'), n+l

(3.9A)

S 0

from

n

Taking ( 3 . 9 A ) i n t o account , ( 3 . 8 A ) implies

(3.1 OA)

I

e(un+l

YP

n+l

)-du",p")

2

n+ 1 It now remains t o estimate (p -pnyQ(un+l)-Q(~n))L

p"+2 = P*(p n+ 1

+PQ(U"+"

; we have

Y

and hence, s i n c e t h e p r o j e c t i o n i s contractive

n+l

n

(3.11A)

or a l t e r n a t i v e l y ( 4 being Lipschitz continuous with constant C,)

Duality methods

(SEC. 3 )

593

By addition we then deduce from ( 3 . 1 0 A ) ,

(3.12A)

and hence by summation and 'dnt 0 L(U"+l

rpn+l)-yUO,p 0)

2

(3.13A) 1

n

n

"

-

2

We shall now show that under condition ( 3 . 1 A ) the sequence is convergent: in fact from ( 3 . 7 A ) we have

j =O (J:, (u") (J;(un+')

,un+' - u ~ ) ~(pn,$ + ( un+I ) -$ (u") ,un-u n+ 1 ),+(p

)

2

0

n+ 1

and hence, by addition (pn-pn+l ,$ (Un+l ) -$ (un) ) t ( J:, ( bn+l -J; (un) ,un+l -un)

(3.14A) n+l

2

4lu

which in conjunction with ( 3 . 1 1 A ) implies

and hence (by summation)

U'

n 2

IIv

Optimisation algorithms

594

(APP. 2 )

The proof o f Theorem 4.1 i n Chapter 2 , Section 4 . 3 shows t h a t under condition (3.1A) t h e sequence (pnIn i s bounded i n L; ( 3 . 1 5 8 n then implies t h a t t h e sequence u j + l - u j ;In i s convergent. 3'0 In-order t o prove t h e convergence of t h e sequence

1 11

11

f

, and hence t h a t lim IIpn+ 1 -pnlIL= 0 , it Ilpj+l-pjll2) L n n++w j=O remains t o show t h a t t h e term on t h e left-hand s i d e of t h e inequn s i n c e t h e sequence ( u In ( r e s p . a l i t y (3.13A) i s bounded; {p In) i s convergent ( r e s p . bounded) t h i s follows simply from t h e We c o n t i n u i t y of L ( * , * ) and t h e l i n e a r i t y o f q -t ( q , + ( v ) ) , . have t h u s proved ( 3 . 2 A ) . {

We s h a l l now use OpiaZ's Lennna i n o r d e r t o prove (3.3A) ; l e t M be t h e s o l u t i o n of t h e primal problem

UE

Inf SUP L(v,q) veM q e A and l e t

X

= (p E

A

, ( u , p ) is a saddle point

of d on M x A ) .

The proof of Theorem 4 . 1 i n Chapter 2 , Section 4 shows t h a t i f condition (3.1A) i s s a t i s f i e d we have convergence of t h e sequence { Ilpn-pllL)n Vp E X I n o r d e r t o be a b l e t o apply O p i a l ' s

.

Lemma, it t h e r e f o r e remains t o prove t h a t any weakly convergent subsequence {pnklk e x t r a c t e d from {pn) has i t s l i m i t i n X . We n know t h a t uX, p"k, p % + l s a t i s f y

I

(3.16A)

,"kEM,

(3.17A) 1p"k~A

.

\ Under condition (3.1A) we have lim unk = u strongly i n V , k++m

with U E M s i n c e M i s closed i n V;

lim k++m

pk

= p*

weakly i n L,

s i m i l a r l y we have

lim (p%+ip%)=o k++m

strongly i n L

(SEC. 4 )

Complementarity methods

(from (3.2A)) with p*

E

595

A s i n c e A i s convex and closed i n L.

In view o f t h e s e p r o p e r t i e s , w e have i n t h e l i m i t i n ( 3 . 1 6 ~ ,) (3.17A)

which shSjws t h a t {u,p*) i s a saddle-point of !s on M x A , and We can t h u s apply Opial's Lemma t o prove ( 3 . 3 A ) hence p E X thereby proving Theorem 3.1.

.

,

The above r e s u l t s a l s o hold for t h e sequence Rezark 3.1: {un,p In defined by algorithm ( 4 . 3 1 ) , (4.32) of Chapter 2 , Section 4.4.

4.

INTRODUCTION TO COMPLEMENTARITY METHODS

4.1

General remarks.

Synopsis.

This f o u r t h s e c t i o n c o n s t i t u t e s an i n t r o d u c t i o n t o complementarity methods, which have been widely used i n recent y e a r s f o r t h e numerical s o l u t i o n of v a r i a t i o n a l i n e q u a l i t y problems of t h e

obstacle problem t y p e ( s e e Appendix 1, Section 3 ) . The model problem f o r linear complementarity is as follows:

Given a matrix A of type N that Au-b 2 0, (4.1A)

x

N and

N , seek u eIRN such

b ER

u 2 0,

(Au-b,u) = 0 ; i%(4.1A) , ( , ) denotes t h e ordinary Euclidean i n n e r product. of IR , and i n g e n e r d

Dptimisation algorithms

596

v = {v, wtv

,. ..vN)

*

20

( U P . 2)

.

@ vi 2 0 Vi=I ,. .N,

w-vto. N

Hereafter we s h a l l use t h e n o t a t i o n Rf:= { V E R , v t 0)

.

It is c l e a r t h a t ( 4 . 1 A ) i s equivalent t o t h e variational inequ-

a l i t y problem

(4.2A) (Au,v-u)

2

N VV ER+

(b,v-u)

.

If A i s symmetric, ( 4 . 2 A ) is a necessary condition (and s u f f i c i e n t if A i s p o s i t i v e semi-definite) o f o p t i m a l i t y f o r t h e minimisation problem

(4.3A)

with

(4.4A)

1

J(u)

5

J(v)

N V v EX+

UER:

1

J(v)

(Av,v)-(b,v).

With t h e above problems we s h a l l a s s o c i a t e

a)

The least element problem

I

Find u E

(4.5A)

A

usv

where

(4.6A) b)

such t h a t

-

VVEK

N K = {vER+, Av-b

t 0)

The family o f l i n e a r p r o g r d n g problems

(4.8A)

(SEC. 4 )

Complementarity methods

5 97

where

(4.9A)

Sufficient conditions on A and p f o r all t h e above problems t o be equivalent a r e given i n Cryer-Dempster /1A/ ( s e e a l s o t h e i f t h i s equivalence can be e s t a b l i s h e d it bibliography t h e r e i n ) ; i s then p o s s i b l e t o s o l v e t h e v a r i a t i o n a l i n e q u a l i t y (4.2A) by pivoting methods i n s p i r e d by t h e simplex method of l i n e a r programming. We s h a l l show i n Section 4.2 (which i s i n s p i r e d by Cryer-Dempster, Zoc. cit.) t h a t t h e model problem f o r variationaz inequazities of

obstacle t y p e

where

(4.11A)

K

= {VC

1

H0 (521, v 2

o a.e.1

i s equivalent t o a l i n e a r complementarity problem (formally of t h e same type as (4.U))which i s i t s e l f equivalent t o an (infinitedimensional) linear programming problem. A d i s c u s s i o n of complementarity methods, t o g e t h e r with a number of r e f e r e n c e s , w i l l be given i n Section 4.3. 4.2

The o b s t a c l e problem from t h e p o i n t of view of compleme n t a r i t y methods

We follow t h e account of Cryer-Dempster /lA/. 4.2.1

Formulation of the problem.

standard results.

Let C2 be a bounded open s e t i n IRN with r e g u l a r boundary w e consider t h e variational inequality problem UE

K,

(4.12A)

vu*V(v-u)dx

2



VV E

K

r;

598

(APP. 2 )

Optimisation algorithms

where , i n ( 4 . 1 2 A ) , K i s given by ( 4 . 1 1 A ) between H-l(Q) and HA(Q) and f E H-* ( a ) .

,

,* >

denotes t h e d u a l i t y

W e know t h a t (4.12A) (which w a s i n v e s t i g a t e d e a r l i e r i n Chapter 1, Sections 3.6 and 4.7 and i n Appendix 1, Sections 3 , 4 ) admits one and only one s o l u t i o n c h a r a c t e r i s e d by

(4.13A)

1

J(u)

V V E K,

J(v)

5

UEK

(where J(v) =

I,

2 lvvl dx

-

< f ,v>) and a l s o , K being a convex

cone with vertex 0

( 4 . 1 4 A ) can also be w r i t t e n

(4.15A)

I(

V V E K,

<-Au-f,v> 2 0 <-Au-f ,u>

=

0

,

UEK.

2

L 2 ( a ) , which implies t h a t U E H 2 -Au-f E L (a), w e deduce from ( 4 . 1 5 A )

If f

E

(4.16A)

]

1

(a) fl H,(n) , and

hence t h a t

-Au-f 2 0 a.e. on 52 (-Au-f)u

= 0 a . e . on 52

UEK.

Conversely, it follows from (4.15A) t h a t t h e s o l u t i o n u of

( 4 . 1 6 A ) i s a s o l u t i o n o f (4.12A). It i s c l e a r t h a t t h e infinite-dimensional problems (4.15A)

,

( 4 . 1 6 A ) a r e formally of t h e same type as ( 4 . 1 A ) and should t h u s a l s o be considered as linear complementarity problems ( i n

H p )1 The above p r o p e r t i e s extend t o v a r i a t i o n a l inequRemark 4.1: a l i t i e s which are more complicated t h a n (4.12A) and which possibly r e l a t e t o nonlinear e l l i p t i c o p e r a t o r s .

(SEC. 4 )

4.2.2

Complementarity methods

599

Other properties associated with ( 4.12A)

I n view of S e c t i o n 4 . 1 it i s n a t u r a l t o a s s o c i a t e t h e f o l l o w i n g problems w i t h t h e e q u i v a l e n t problems (4.12A) , (4.13A) , (4.15A) :

a)

The l e a s t element problem

Find

UE

fi

such that

-

(4.17A)

u l v a.e. On

a,

V V ~ K

where

k

= {vc K, -Av-f t 0)

(4.18A)

Vv-Vw dx b)

2



V W E K)

The f a m i l y o f l i n e a r problems

Find u E

such that

(4.19A) A



l



VVE K

Here a g a i n we t h e r e f o r e have a family where p E I4-l (n) i s given. of l i n e a r programming problems w i t h parameter p. The dual problem t o (4.19A) i s given by

Find X

E

;;* such

that

(4.20A)



t



VpE

ii*

By u s i n g a s u i t a b l e p o s i t i v i t y assumption on p , we s h a l l now show t h a t t h e above problems are e q u i v a l e n t . This w i l l r e s u l t from t h e f o l l o w i n g lemma, due t o Stampacchia / l A / , /2A/;

Optimisation algorithms

600

Lemma

4.1:

(APP. 2)

then

If v1 ,v2 E

l'l-v21 2

VI+V2

w = inf ( v l ' v2 ) ( = - - 2

)

also belongs t o K.

and also from

Lemma 4.2: There i s equivalence between problems (4.12A), ( b . r n ( 4 . 1 5 ~ and ) the variational inequality problem

(4.228)

IiiK

Vu*V(v-u)d%2

VVE

t

.

The proof o f Lemma 4.2 i s immediate. We are now i n a p o s i t i o n t o prove

There i s equivalence between problems ( 4.12A) , Theorem 4 . 1 : (4.13A), (4.15A) and the least element problem ( 4 . 1 7 A ) . If, i n

addition , we have

then the linear p r o g r m i n g problem (4.19A) i s equivalent t o (4.12A), (4.13A), ( 4 . 1 5 A ) , ( 4 . 1 7 A ) . Proof: 1) The equivalence between (4.12A), (4.13A) , (4.'15A) has a l r e a d y been proved elsewhere. W e s h a l l now prove t h e equivalence between (4.15A) and_(4.17A). L e t u be the solution of ( 4 . 1 5 A ) ; w e t h e n have U E K , and hence (see Lemma 4 . 1 ) : A

w = inf (u,v) which i m p l i e s , s i n c e

(4.24A)

E

K

VVE K ,

u-wzo (i.e. u-wEK),

<-Aw-f ,u-w>

2

0.

We a l s o have, s i n c e u-is t h e s o l u t i o n o f and s i n c e w e K c K

(4.1%)) (4.25A)

<-Au-f,w-u>

2

( 4 . 5 ) (and hence of

0.

By adding (4.24A) and (4.25A) w e t h u s have

(SEC.

601

CompZementarit y methods

4)

0 2 <-A (w-U) ,W-U> =

1, I

V (W-u)

I 2 dx

and hence I

VVE K ,u

=

w = inf(v,u)

which implies u < v Vv E K i.e. u is a solution of (4.17A). Conversely if u is a solution of (4.17A) we have _-

since

-Au-f t 0

U E

K

and n

v-u~OVVEK

,

and hence

-

Vu*V(v-u)dx

{ !ti,

*

=

<-AU-f,V-U>

2

0 VVEK

which shows, from Lemma 4.2, that u is a solution of (4.12A) ,

(4.15A). V-u t 0 V v

2)- If u is a solution of (4.17A) we have if p satisfies (4.23) we thus have E K ; n

2 0 V V E K which shows that ( 4.17A) implies (4.19A)

.

Conversely, if u is a solution of (4.19A) we have n

VVE K ,w

CI

=

inf(u,v)

E

K

and hence (4.26A)

2 0 .

.

However ( 4.23A) implies, since u-w 2 0 ( i e. u-w (4.278)



2

0.

By adding (4.26A) , (4.27A) we obtain

1

= 0 u-w

2

,

0

and hence w = u, from (4.23A).

E

K)

,

602

(APP. 2)

Optimisation algorithms A

CI

W e t h u s hsve U E K, u = in€ (u,v) V v e K , and hence vv E K , which proves t h a t u i s a s o l u t i o n o f ( 4 . 1 7 A ) . v2 u theorem i s t h u s completely proved.

The

Remark 4.2: There i s no d i f f i c u l t y i n f i n d i n g p E H-'(Cl) such > 0 V V E K-(0). I n f a c t it i s s u f f i c i e n t t o define

that P by

1 dx V v e Ho(Q)

(4.28A)

where C i s a positive constant. 4.3

Discussion and r e f e r e n c e s

4.3.1

Discussion

From Section 4.2 we see t h a t it i s p o s s i b l e t o reduce t h e t a s k of s o l v i n g t h e o b s t a c l e problem (4.12A) t o t h a t of s o l v i n g an equivalent l i n e a r programming problem. From a p r a c t i c a l p o i n t of view a d i s c r e t i s a t i o n i s performed using finite differences or finite elements and t h e d i s c r e t e analogues of problems (4.12A) , ( 4 . 1 3 A ) , ( 4 . 1 5 A ) , (4.17A) , ( 4 . 1 9 A ) are considered; i n particular t h e d i s c r e t i s e d form of problem (4.lgA) can b e solved by a pivoting method of t h e simplex t y p e . I n t h i s case t h e equivalence Theorem 4.1 results from t h e f a c t t h a t t h e o p e r a t o r -A a s s o c i a t e d with D i r i c h l e t boundary c o n d i t i o n s obeys t h e maximum principze ( t h i s a l s o holds f o r more complicated second-order o p e r a t o r s and f o r o t h e r forms of boundary c o n d i t i o n ; s e e Stampacchia / l A / , /2A/; i n o r d e r t h a t t h e equivalence p r o p e r t i e s of Theorem 4 . 1 can be c a r r i e d over t o t h e d i s c r e t e case it i s necessary t h a t t h e operator approximating -A obey a discrete maximum principle (see CiarletRaviart / l A / ) . I n t h e case of t h e o b s t a c l e problem (4.12A) with Q c B2, d i s c r e t i s e d by a triangular finite-element method, t h e d i s c r e t e maximum p r i n c i p l e w i l l be s a t i s f i e d i f

(i) (i5)

w e use a f i r s t - o r d e r finite-element approximation ( i . e . g l o b a l l y continuous and piecewise a f f i n e ) a l l t h e angles of t h e t r i a n g u l a t i o n are

71 2

.

If some of t h e angles are obtuse and/or f i n i t e elements o f o r d e r 2 2 are used, t h e d i s c r e t e m a x i m u m p r i n c i p l e no longer applies.

We should a l s o p o i n t out t h a t a c e r t a i n number of o p e r a t o r s which are o f fundamental importance i n p r a c t i c a l a p p l i c a t i o n s , such as t h e b i h m o n i c operator $ +. A2$ with t h e boundary conditions

$1,

= gl

,

%Ir

= g2

,

and s i m i l a r l y t h e linear

(SEC. 5 )

Minimisation of quadratic functionals

603

e l a s t i c i t y operator do not s a t i s f y a maximum p r i n c i p l e . I n view of t h e above remarks, it would appear t h a t t h e s e compl e m e n t a r i t y methods by d e f i n i t i o n have a f a i r l y l i m i t e d range of a p p l i c a t i o n , a t least as f a r as t h e s o l u t i o n of v a r i a t i o n a l inequa l i t i e s i s concerned. A f u r t h e r d i s c u s s i o n o f t h e above t o p i c s can be found i n Cryer-Dempster / l A / . 4.3.2

References

Complementarity methods have been i n v e s t i g a t e d by numerous a u t h o r s , p a r t i c u l a r l y with r e g a r d t o t h e s o l u t i o n of v a r i a t i o n a l i n e q u a l i t i e s of t h e o b s t a c l e problem type. To attempt t o g i v e a l i s t of a l l t h e r e f e r e n c e s r e l a t i n g t o t h i s c l a s s of methods would be o u t of t h e question, and w e t h e r e f o r e r e s t r i c t ourselves t o mentioning only t h e r e f e r e n c e s below, t h e b i b l i o g r a p h i e s of which a r e a l s o worth consulting. Considering only works which a r e o r i e n t e d towards t h e s o l u t i o n o f free-boundary problems of t h e o b s t a c l e problem type, w e may mention among o t h e r s : Cottle-Sacher /1A/ , Cottle-Golub-Sacher / l A / , C o t t l e / 1 A / , /2A/ , Mosco-Scarpini / l A / , S c a r p i n i /1A/ and of course Cryer-Dempster,

loc. c i t . MINIMISATION OF QUADRATIC FUNCTIONALS OVER THE PRODUCTS OF INTERVALS. USING CONJUGATE GRADIENT METHODS

5.

5.1

Synopsis

In t h i s f i f t h s e c t i o n we s h a l l supplement t h e i n v e s t i g a t i o n s of Chapter 2 , Section 2.3 ( s e e a l s o t h e discussion i n Chapter 2 , S e c t i o n 6 ) with regard t o t h e c q j u g a t e g r a d i e n t method. We consider t h e model problem i n I R

Find u c K such that (5. IA) J(u)

5

J(v)

V

V E

K,

where (5.2A)

N K = ll i=l

Ki,

Ki = [ a

bilcR

i7

( ai ,bi f i n i t e or otherwise) ,

604

Optimisation algorithms

(UP. 2 )

where A is an N x N symmetric positive-definite matrix and b eRN ; as usual (in this b ok) ( - , * ) denotes the canonical Euclidean inner product of (and more generally of mp ) . In Section 5.2 we shall describe an algorithm of conjugate gradient type, which allows an efficient solution of (5.1A) - (5.3A), and does so i n a f i n i t e number of iterations (i.e. as i n the constraintfree case of Chapter 2 , Section 2.3).

IRa

5.2

Description of the method.

Convergence results

In this section we have followed the account of D.P. O'Leary /lA/ to which we refer for further information and numerical examples. 5.2.1

x

Let u = {Al,.

General remarks

.p

1 be the solution of (5.1A) - (5.3A), and let e de ined by

.N

(5.4A)

X-Au-b;

it can easily be shown that u i s characterised by

[

V i=l,...N, X.2 O i f

(5.5A)

1

we have

ui = ai'

X.< I

Xi = 0 i f a i < u i < b i

0

.

i f u.1

=

bi

,

Remark 5.1: The above characterisation contains X as a KuhnTucker multiplier (see e.g. Rockafellar /3, Section 2 8 / ) for the problem (5.1A) - (5.3A).

...

Remark 5.2: If V i = 1 , N , we have a. = 0, bi = + w then 1 the above characterisation reduces to the linear complementarity problem

(5.6A)

I

Find u e R N such that (Au-b,u) = 0 , N Au-b eR+

, u E R+N

the solution of which was the subject of Section Appendix.

4 of

this

We shall describe Polyak's /1A/ algorithm in detail in Section 5.2.2, but we first shall give a broad outline of this algorithm

(SEC. 5 )

Mixhisation of quadratic functionaZs

605

and attempt t o e x t r i c a t e i t s g e n e r a l p r i n c i p l e s . The algorithm n n u E K , and we generates a s e uence {u } nrO such t h a t V n, a d j u s t A"(= Au -b) i n such a way as t o s a t i s f y ( 5 . 5 A ) .

4

n

Given U O E K , for n 2 0 w e t h e n c o n s t r u c t u E K, and An s a t i s f y i n g ( 5 . 5 A ) ''as well as p o s s i b l e " , using a method which i s With regard t o t h e o u t e r i t e r a t i o n loop, with i t s e l f iterative. n u known w e f i r s t d e f i n e

I"

c {1

,...N)

n n as being t h e s e t o f i n d i c e s i f o r which {ui,Ai)

(5.5A),

i s c o n s i s t e n t with

2.e. n

(5.7A)

I

=

{;I.

n

1

.

with In and Jn (={1,. .N)

n uI = {uili b,: ,:b , A: In = 11,.

..

X i > O l u l i l u ~=

= a.1 and

-

n I ) we a s s o c i a t e t h e v e c t o r s

Jn ( w e s i m i l a r l y d e f i n e and U" = {uiIi In J We t h e n reorder t h e i n d i c e s i n such a way t h a t 1;). I n view Card ( I n ) } and Jn = {l+Card (I") . .N}.

,. .

of t h e above p a r t i t i o n i n g of {l , . . . N )

N

omposition of RN (which implies R we can w r i t e t h e r e l a t i o n

(5.8A)

bi and X i < O )

and t h e corresponding dec-

-- RCard(In.)gRCard(p))

Aun-b = A"

i n t h e form

n where, i n ( 5 . 9 A ) , AII definite matrices.

n

and AJJ

a r e square, symmetric, p o s i t i v e -

The b a s i c idea behind Polyak's method l i e s in modifying n u (u; remaining f i x e d ) so as t o attempt t o make h vanish; J J l e a d s t o t h e l i n e a r system ( 5 . 1 OA)

AYJvJ = b l

this

- AnJIunI .

Since t h e m a t r i x An i s symmetric and positive definite w e can JJ apply t h e conjugate gradient method of Chapter 2 , Section 2 . 3 for t h e s o l u t i o n of (5.10A); w e can t h u s solve (5.10A) exactly

;

606

O p t h i s a t i o n algorithms

(AFT. 2 )

i n a f i n i t e number of iterations (of the inner i t e r a t i o n loop for the overall algorithm). I n p r a c t i c e , it i s u n l i k e l y t h a t t h e exact s o l u t i o n of (5.10A) w i l l s a t i s f y t h e bounds a. <- v. < b.

(5.1 1A)

1

1

1

v i e J" ; we t h e r e f o r e have t o be content with s o l v i n g (5.10A) "as w e l l as p o s s i b l e " , w h i l s t keeping t h e i n t e r v a l c o n s t r a i n t s I n order t o do t h i s w e modify t h e conjugate (5.11A) s a t i s f i e d . g r a d i e n t method; i f during t h e descent phase o f t h e conjugate g r a d i e n t method ( s e e (2.15), ( 2 . 1 6 ) of Chapter 2, Section 2 . 3 ) some of t h e bounds (5.11A) are v i o l a t e d , we reduce the descent step (denoted by A i n ( 2 . 1 5 ) of Chapter 2 , Section 2.3) by as l i t t l e as possible t o ensure t h a t t h e bounds (5.11A) s t a y t h i s results i n some o f t h e bounds being s a t i s f i e d satisfied; n with e q u a l i t y ; t h e corresponding i n d i c e s are t h e n added t o I and t h e i n n e r i t e r a t i o n loop i s resumed with a new decomposition o f A and of t h e v e c t o r s . When t h e modified conjugate g r a d i e n t algorithm has converged, we know t h a t t h e v a r i a b l e s , now denoted by u?+l, of which t h e index i has remained i n Jn, s a t i s f y (5.11A) and t h a t t h e corresponding s c a l a r s A . a r e zero. 1

Using t h e modified conjugate g r a d i e n t method, w e have t h u s constructed a v e c t o r un+' E K ; i f I n + l = I n t h e n un+l s a t i s f i e s (5.5A) and w e t h e r e f o r e have un+l = u , where u i s t h e s o l u t i o n of (5.1A); i f In+1# In t h e i n n e r i t e r a t i o n loop i s repeated with In replaced by I n + l .

5.2.2

D e t a i l e d d e s c r i p t i o n o f Polyak's algorithm

We follow D.P. O'Leary's /lA/d e s c r i p t i o n ; phases of t h e algorithm a r e

A.

I n i t i a l i s a t i o n o f t h e o v e r a l l algorithm

(5.12A)

uo E K, given arbitrarily ,

(5.13A)

n = 0,

(5.14A)

I = {l,2,

B.

Outer i t e r a t i o n loop

For n 2 0, s e t

...N}.

t h e various

(SEC. 5 )

Minimisation o f quadratic functionals

(5.15A)

n + n + l , un

(5.16A)

I = I,

un-I,

607

= Aun-b,

and then define In by ( 5 . 7 A ) .

If In = In-' then un = u the solution of ( 5 . 1 A ) ;

i f I"

#

In-l

set (5.17A)

I = 1"

and s t a r t the inner i n t e r a t i o n loop. Inner i t e r a t i o n loop

C.

c* 1

n D e c o ~ E o s i t i o nof_A_an_d_~f_t_he_v~ct_ors-u_.blEot_ation

.

Let J = { 1,. . N } - I ; decompose and reorder A and u" ,b r e l a t i v e t o the partitioning I+J = {l,. .N}, giving

..

with AJJ s y m e t r i c and positive-definite. {w'}, {r'] r e s p e c t i v e l y , the sequence of We denote by (z'}, approximate solutions generated by the modified conjugate gradient algorithm (inner algorithm), the sequence of descent directions and the sequence of residuals rq bi AJJzq AJII

-

I

.

-

I n i t i a l i s a t i o n of t h e i n n e r a l g o r i t h m

C.2

Set (5.18A)

q = o

and

n

(5.19A)

zo =

(5.20A)

wo = yo

UJ

, I

by

- AJJz - AJ IunI 0

(UP. 2 )

Optimisation algorithms

608

c. 3

C a l c u l a t i o n of zq+l and rq+l ___--------_--_---__-----_-_

First determine the optimal descent step i n the direction of wq, namely P': =

(5.21A)

(r4,wq)

(rq,rq) (Ajjw4 ,W41 '

(Ajjw4 ,W4 )

and then the maximal descent step f o r which the bounds ( 5 . 1 1 A ) are s t i l l s a t i s f i e d , namely a -z4

b .-zq

fin

(5.22A)

j

J

J

The descent step f i n a l l y used i s then

and hence z q + l , rq+l are given by (5.24A)

zq+I = zq

(5.25A)

'+'r

C.4

pgwq

'

- p q ~ ~ ~. w q

Termination c r i t e r i o n for t h e i n n e r i t e r a t i o n looe

-If rq+' = loop. -If

= rq

+

{jlz!"

J

0,

n

put uJ = z q + l and r e s t a r t t h e inner i t e r a t i o n

-

-

0 go t o c.5. J put u? = zq+l and I =.{iluy = a i o r bi}. a. or b.) J

-Otherwise, I = {l Nl r e s t d t t h e outer i t e r a t i o n loop. the inner i t e r a t i o n loop.

,...

If Otherwise r e s t a r t

Alternating direction methods

6)

(SEC.

609

giving the new descent direction ( AJJ-conjugate t o t h e previous directions ) (5.27A)

wq+l

E

$+I

+ yqwq

.

Then replace q by q + 1 and go t o C. 3.

8

Remark 5 . 3 : We r e f e r t o O'Leary, ZOC. c i t . , f o r a number o f v a r i a n t s of t h e above algorithm, and a l s o p r a c t i c a l d e t a i l s of i t s implementation on a computer. 5.2.3

Convergence resuZts

I n O'Leary, loc. c i t . , t h e following theorem i s proved:

Polyak's algorithm of Section 5.2.2 converges Theorem 5.1: t o the solution o f problem ( 5 . 1 ~ )i n a f i n i t e number of iterations. 5.3

Discussion

A v a r i a n t ( c a l l e d t h e S.N. method) of Polyak's algorithm of S e c t i o n 5.2 i s given i n Tr6molieres /4, Chap. 7/. More s p e c i f i c a l l y , regarding t h e a p p l i c a t i o n of Polyak's algorithm t o t h e s o l u t i o n of variational inequality problems ( a f t e r s u i t a b l e d i s c r e t i s a t i o n using f i n i t e d i f f e r e n c e s o r f i n i t e elements) w e r e f e r t o O'Leary, ~ O C . c i t . , and O'Leary-Yang /1A/ where numerical a p p l i c a t i o n s t o t h e elasto-plastic torsion problem o f Chapter 3 can be found, i n t h e equivalent formulation ( 2 . 4 ) o f Chapter 3, Section 2.2 ( s e e a l s o Chapter 3, Section 3 ) .

6.

ALTERNATING D I R E C T I O N METHODS. APPLICATION TO THE SOLUTION OF NONLINEAR VARIATIONAL PROBLEMS

I n t h i s s e c t i o n we follow t h e d e s c r i p t i o n of P.L. Lions B. Mercier / l A / ; t h e n o t a t i o n i s t h a t of H. Br6zis 131, t o which we a l s o refer f o r t h e v a r i o u s d e f i n i t i o n s and p r o p e r t i e s connected with t h e concept of monotonicity used i n t h i s s e c t i o n .

@timisation aZgorithms

610

6.1

Formulation o f t h e problem.

(APP. 2)

Synopsis

L e t H be a r e a l H i l b e r t space of which t h e i n n e r product and We a s s o c i a t e d norm are denoted r e s p e c t i v e l y by ( , and consider t h e equation 0

(6.1A)

OE

)

I I.

C(u)

where C i s a monotone o p e r a t o r on H, which may p o s s i b l y be multivalued ( I ) ( i n which c a s e , i f V E H, C ( v ) i s a subset of H, p o s s i b l y reduced t o a s i n g l e p o i n t o r empty). O f p a r t i c u l a r i n t e r e s t i s t h e s o l u t i o n of

(6.1~)when

,

C = A+B (6.2A)

are maximal monotone.

A,B,C

L e t X > 0; t o s o l v e (6.1~) u s i n g t h e decomposition ( 6 . 2 ~ , ) we consider (as i n P.L. Lions-B. Mercier, loc. tit.) t h e following two algorithms : F i r s t algorithm:

(6.3A)

0

u given

n

and for n z O , u

known

Second algorithm:

(6.5A)

and f o r . (6.6A)

0

u given

n

nZO, u

known

un+'

.-

(I+AB)

-' (I+AA)-' [

(I-XB) +ABl un.

Remark 6.1: W e assume (for s i m p l i c i t y ) t h a t A,B,C are s i n g l e valued ( 2 ) o p e r a t o r s and we put

Translator's notes:

(l)

The terminology m l t i v o q u e i s sometimes used.

(2)

The terminology univoque i s sometimes used.

(SEC. 6 )

(6.7A)

611

Alternating direction methods

.

I2= (I+AA)-I (I-AB) un

U

Equation ( 6 . 7 ~ )i s equivalent t o ( I + X A ) U ~ + " ~ + X B ( U ~ ) allows us t o r e w r i t e t h e f i r s t algorithm i n t h e form

(6.8A)

u

0

n u , which

given

and f o r n z 0 , un known AA(un+l/2 )

(6.9A)

Un+1/2

(6.1 OA)

un+' +AB(u"+')

+

=

un

n+1/2 'U

-

XB(un)

,

AA( un+l I2 1 ;

l i k e w i s e t h e second algorithm i s equivalent t o

(6.11A)

u

0

given

n and f o r n z 0 , u known

(6.12A)

un+' /2+u(un+' 12)

(6. I3A)

n+ 1 u +AB(un+')

E

un-hB(un)

= un- AA(u

n+l/2

, ).

( r e s p . (6.11A)-(6.13A))algorithm has t h e appearance of an a l t e r n a t i n g d i r e c t i o n method of t h e Peaceman-Rachford /lA/ type ( r e s p . Douglas-Rachford /1A/ t y p e ) I n view of (6.8A)-(6.10A)

(6.31) , ( 6 . 4 A ) ( r e s p . (6.51)

, (6..6A))

.

I n Section 6.2 w e s h a l l g i v e some information on t h e converg( r e f e r r i n g t o P.L. Lions-B. ence of (6.3A), ( 6 . 4 A ) and ( 6 . 5 A ) , ( 6 . 6 A ) Mercier, Zoc. c i t . , and Gabay /lA/ f o r t h e p r o o f s ) . I n Section 6 . 3 w e s h a l l b r i e f l y d e s c r i b e t h e a p p l i c a t i o n o f (6.M) , (6.4A) and ( 6 . 5 ~ ,) ( 6 . 6 A ) t o t h e s o l u t i o n of t h e o b s t a c l e problem of Appendix 1, Section 3.

For f u r t h e r d e t a i l s regarding algorithms (6.3A), ( 6 . 4 A ) and ( 6 . 5 ~, ) (6.6A) , and a l s o for t h e i r implementation, w e refer t h e r e a d e r to t h e two r e f e r e n c e s c i t e d above, i n which various generali s a t i o n s a r e a l s o given. 6.2

Convergence of algorithms ( 6.3A)

, ( 6.4A)

and ( 6.5A)

, ( 6.6A)

W e follow P.L. Lions-B. Mercier /1A, Section 1/ t o which we r e f e r f o r t h e proofs o f t h e following r e s u l t s .

Optimisation algorithms

612

6.2.1

Asswnptions and additional notation.

(APP. 2)

Initialisation

We r e c a l l t h a t A,B,C are maximal (Assumption (6.2A)). We denote by D(A) t h e domain of A ( i . e . D(A) { v e H, A(v) c H, A(v) # 8 ) 1, and by R(A) t h e image of A ( i . e . R(A) = { v E H ? 3 w E H such t h a t v ~ A ( w ) } ) * We r e c a l l t h a t A i s monotone i f

-

(Y-z,u-v)

2

0

, VU E H,

y E A(u),

V V E H, z

E

A(v) ;

moreover t h e statement t h a t A i s maximal monotone i s equivalent t G saying t h a t t h e resolvent S? = (I+AA)-l ( a s i n g l e v a l u e d A o p e r a t o r ) i s a contraction defined on the whole of H. I n t h i s p r e s e n t s e c t i o n we s h a l l make t h e assumption t h a t (6.1~)admits a t l e a s t one s o l u t i o n , i . e .

(6.14A)

, 1

There e x i s t s

such t h a t

UE

H, a

E

A(u) , b E B(u) ,

a+b = 0 .

If A and B are multi-valued, it i s a p p r o p r i a t e t o d e s c r i b e algorithms (6.3A), (6.4A) and (6.5A) , (6.6A) i n r a t h e r g r e a t e r d e t a i l , i n p a r t i c u l a r with regard t o t h e i r i n i t i a l i s a t i o n . To i n i t i a l i s e (633A), (6.4~)and (6.5A), (6.6A) w e t a k e U ' E D ( B ) , then b E B(u ) , and we put

(6.15A)

v 0 = uo + Abo

A

uo = JB v

n We next d e f i n e t h e sequence { v }

n20

0

.

by

F i r s t algorithm ( ( 6 . 3 ~ (6.4A)) ~ (6.16A)

n+1 h A n v = (2JA-I)(2JB-I)v ;

Second algorithm ( (6.5Aj , ( 6.6A) ) (6.17A)

I n both cases we o b t a i n t h e sequence

(6.4A) or (6.5A), (6.6A)), s t a r t i n g from

{un} (from (6.3A), n} nrO n20' by

{V

(SEC. 6)

un = J

(6.18A)

6.2.2

613

Alternating direction vethods

p

. .

Convergence o f algorithm (6.3A), ( 6.4A)

We p u t = u+Xb

,w

= u+ha,

n n w n = 2 un- v n , b n = + -U , a We t h e n have ( s e e P.L. Lions-B. Proposition

n

n n+l w-w = r.

Mercier /1A, Section 11):

6.1: Assume (6.14A) t o be s a t i s f i e d ; we then

have (6.19A)

n n n n n the sequences u ,v ,w ,a ,b are bounded

( 6.20A)

lim re-

(bn-b,un-u)

(6.2 1A)

lim n+-

(an-a,

n+1

v

= 0

+wn

,

- u)

=

0.

The above p r o p o s i t i o n r e s u l t s i n t h e convergence p r o p e r t i e s given i n : Corollary 6.1:

I f B i s singleualued and s a t i s f i e s

For any sequence l x n j , x

(6.228)

lim

(Bx -Bx,x -x) n

D(B) Vn, such that

x weakZy and = 0, we have x = x

{Bx 1 i s bounded, xn *+Co

E

n

+

then we have (6.23A)

lim

n u = u

weakZy i n H ,

rrt+o3

where u i s the soZution of (6.1~)(unique, from (6.228)). Remark 6.2: Property (6.22A) i s s a t i s f i e d i n t h e following cases ( s e e P.L. Lions-B. Mercier, zoc. c i t . , f o r t h e ' p r o o f s ) : (i)

B i s coercive ( a l s o termed H - e l l i p t i c ) ,

i.e.

3ff

>O

such t h a t

Optimisation aZgorithms

614

(UP. 2)

(6.24A) I n t h i s case we a c t u a l l y have a s t r o n g e r c o n d i t i o n t h a t 66.23A), s i n c e ( 6 . 2 0 ~ ) md (6.24A) imply t h e strong convergence of {u 1 t o U.

(ii)

B-l

is coercive, i . e . ? B > O such t h a t

( 6.25A)

I n t h i s c a s e , i; a d d i t i o n t o

(6.23A), w e a l s o have t h e strong

convergence o f {Bu 1 t o Buy from (6.20~)and (6.258). ( i i i ) B i s s t r i c t l y monotone and weakly closed, i . e .

- v x , y ~ D ( B ) such t h a t (B(y)-B(x),y-x) = 0 we have y=x, - i f {xn is such t h a t xn ED(B) Vn, l i m xn = 'x weakly i n H and i f t h e r e e x i s t s y

n E Bxn with

i n H, then

Remark 6.3:

X E

D(B) and y = Bx.

If B i s l i n e a r , o r i f J

weakly closed.

Remark 6.4: t h a n B satisfies

If i n Corollary

n-

lim

y

= y

weakly

w n

B i s compact, t h e n B i s

6.1 w e assume t h a t A r a t h e r V"+ 1 +w"

(6.22~)~ then { 2 In converges weakly t o (6.1~). In f a c t everything s a i d i n

t h e unique s o l u t i o n of Remark

6.2 with regard t o {u"},,

6.2.3

a l s o holds f o r {

++I

+wn In 2

.

Convergence of aZgorithm ( 6.5A), (6.6A)

The n o t a t i o n i s t h a t of Sections 6.2.1 and 6.2.2; with regard t o t h e convergence of (6.5A), (6.6A), t h e following theorem i s proved i n P.L. Lions - B. Mercier /1A, Section 1.3/:

the I f (6.14A) i s s a t i s f i e d , then as n -t + Theorem 6.1: sequence {v") generated by (6.15A), (6.17A) converges weakZy t o v E H such t h a t u = J A v i s a soZutio; of (6.1~).Moreover f o r B by un = the sequence {u") defined , we have the following convergence properties:

J3

6)

(SEC.

-

I f B i s Zinear, u

- If

n

A and B are odd

A(x)

615

Alternating d i r e c t i o n .methods

= -A(-x)

converges weakly t o a soZution of (6.1~). (i.e.

VXED(A),

B(x)

V X E D ( B ) ) then {u")

= -B(-x)

converges strongZy t o a solution of (6.1~).

- If A +

B i s m d m a l monotone the weak c l u s t e r p o i n t s of

{ u n ) are soZutions of

(6.1~).

The f o l l o w i n g p r o p o s i t i o n s a r e a l s o proved i n P.L. M e r c i e r , loc. c i t .

Lions-B.

6.2: I f JBh i s weakZy closed and i f (6.14A) i s s a t i s f i e d , then u" converges weakly t o a solution o f (6.1~). Proposition

h P r o p o s i t i o n 6.3: I f JA i s compact and i f (6.14A)i s s a t i s f i e d , then un converges strongly t o a solution of (6.1~).

P r o p o s i t i o n 6.4: Assume t h a t B i s coercive and Lipschitz continuous, i. e . 3 a and M > 0 such t h a t Vx ,x E H we have 1

2

Under these conditions there e x i s t s a constant (6.28A)

Iv"-vl


,

Iun-uI
h

C

such t h a t

,

where u = J B v i s the unique solution o f (6.1A)and where 2Aa 112. k = (1( 1 +AM$ Remark 6 . 5 : The e s t i m a t e s (6.28~) a l s o h o l d for a l g o r i t h m

(6.3A), (6.4A)i f B s a t i s f i e s t h e n have Zinear convergence.

,

(6.26A), (6.278).

Both a l g o r i t h m s

,

Remark 6.6: We remark t h a t a l g o r i t h m (6.5A), (6.6A) h a s been i n v e s t i g a t e d by J. L i e u t a u d /1A/ who has e s t a b l i s h e d i t s convergence u n d e r l e s s r e s t r i c t i v e assumptions on A and B (which a r e assumed singZe-vaZued , however)

.

Remark 6.7: I n S e c t i o n 7 we s h a l l show t h e r e l a t i o n s h i p s which e x i s t between t h e above a l t e r n a t i n g d i r e c t i o n methods and augmented Lagrangtan methods.

616

(APP. 2 )

Optimisation algorithms

Application t o t h e s o l u t i o n of t h e o b s t a c l e problem

6.3

We follow P.L. Lions-B. Mercier /1A, S e c t i o n 31: l e t s2 be2& bounded open domain i n I R 2 , with r e g u l a r boundary, and f E L (a). The v a r i a t i o n a l i n e q u a l i t y

Find

u E H i ( Q ) n K such that

( 6 . 2 9A) Vu*V(v-u)dx

2

f (v-u)dx

VV

E

1 Ho(Q) n K,

where

2

K = {v E L (a), v(x)

(6.30A)

2

0

a. e. on Q),

The admits one and only one s o l u t i o n and we have u E H 2 ( n ) . s o l u t i o n u o f (6.29A) i s a l s o t h e s o l u t i o n o f t h e (multivoque ( * ) ) equation i n L ~ ( Q :)

( 6 . 3 1 A)

o ~ - A U

+ f +

aIKw ,

where 31 i s t h e sub-differential ( s e e BrCzis 131, Ekeland-Temam /1/ f o r !his concept) o f t h e indicator functional of t h e convex s e t K defined by (6.30A). Since t h e convex s e t K i s closed and it t h e n follows t h a t nonempty, I i s convex, proper and 1 . s . c ; K 31 i s maximal monotone. W e put H = L2(s2), A = aIK and we d e k n e B by

B(v) = -Av + f with D(B) = H

2

(a) n H o1( Q ) .

The domain $2 being bounded, w e have coercivity o f B s i n c e h l YV2

E

D(B)

where yo > 0 i s t h e s m a l l e s t eieenvalue of -A on H2 ($2) n H 1 (a). 0

h Moreover w e have JA = P where P is t h e p r o j e c t i o n o p e r a t o r K K onto K i n t h e L 2 ( Q ) norm, namely

(PKv)(x) = sup(O,v(x)), a.e. on il ~~

~

.

~

(*) Translator 's Note : The terminology multivoque equation r e f e r s t o an equation r e l a t e d t o a multivalued o p e r a t o r .

(SEC. 7 )

A.D. and augmented Lagrangian methods

617

K i s t h u s i n f a c t a truncation o p e r a t o r .

P

We can t h u s apply a l g o r i t h m s (6.3A), ( 6 . 4 A ) and ( 6 . 5 A ) , (6.6A) f o r t h e s o l u t i o n of (6.29A) , (6.31A) ; a t each i t e r a t i o n we have t o c a r r y o u t a t r u n c a t i o n o p e r a t i o n , which i s numerically t r i v i a l , and t o s o l v e a homogeneous D i r i c h l e t problem r e l a t i n g t o t h e o p e r a t o r I - A A , which i s a s t a n d a r d numerical p r o c e s s and nowadays poses few problems ( a t l e a s t i f s1 c R 2 ) . We r e f e r t o P.L. Lions-B. Mercier, loc. cit., f o r a comparison o f t h e performance o f t h e above two a l t e r n a t i n g d i r e c t i o n a l g o r i t h m s a p p l i e d t o t h e s o l u t i o n o f (6.29A) , ( 6 . 3 ~ )( a f t e r a s u i t a b l e d i s c r e t i s a t i o n ) ; it would appear t h a t for t h i s example ) t w i c e as f a s t as (6.5A), (6.6A). t h e a l g o r i t h m (6.3A), ( 6 . 4 ~ ~i s I n c o n t r a s t , however, examples a r e given i n Chan-Glowinski /1A/ r e l a t i n g t o t h e n o n l i n e a r D i r i c h l e t problems o f Appendix 1, S e c t i o n 6 , f o r which a l g o r i t h m (6.5A), (6.6A) appears much more e f f i c i e n t t h a n (6.3A), ( 6 . 4 A ) .

7.

RELATIONSHIP BETWEEN THE ALTERNATING DIRECTION METHODS AND AUGMENTED LAGRANGIAN METHODS

I n t h i - s s e c t i o n w e s h a l l supplement Remark 6.7 of S e c t i o n 6.2; w e s h a l l show t h a t t h e alternating direction methods o f S e c t i o n 6 a r e i n f a c t s p e c i a l c a s e s o f a more g e n e r a l f a m i l y of a l g o r i t h m s r e l a t e d t o t h e duality algorithms o f Chapter 2 , S e c t i o n 4 and t o t h e augmented Lagrangian methods mentioned i n Chapter 2, S e c t i o n 6, Chapter 3, S e c t i o n 10 and Chapter 5, S e c t i o n 9. For f u r t h e r d e t a i l s w e r e f e r t h e r e a d e r t o Bourgat-Way-Glowinski / 1 A , S e c t i o n 6.31 and Gabay / l A / .

7.1

A model problem i n H i l b e r t space

The n o t a t i o n i s t h a t o f S e c t i o n 6 ; f o r s i m p l i c i t y w e c o n s i d e r a p a r t i c u l a r model problem o f t h e t y p e ( 6 . 1 ~ ) . We t h e r e f o r e l e t H be a H i l b e r t space on IR w i t h i n n e r product and norm denoted by ( , ) and , r e s p e c t i v e l y . W e c o n s i d e r t h e problem

I I

(7.1A)

C(u) = 0 ,

where C : H + H i s a s i n g l e - v a l u e d o p e r a t o r . We assume ( a g a i n f o r s i m p l i c i t y ) t h a t C i s t h e d e r i v a t i v e ( i n t h e Frechet or Gateaux s e n s e ) o f a f u n c t i o n a l J ( i . e . C = J'), strictly conz)ex, such t h a t

lim

II VII

J(v) =

+a.

-++OD

It f o l l o w s from t h e p r o p e r t i e s o f J t h a t C i s s t r i c t l y monotone and t h a t t h e minimisation problem

aptimisation aZgorithms

618

Find

UE

( U P . 2)

H such t h a t

(7.2A) J(u)

I

Vv

J(v)

E

H,

admits one and only one s o l u t i o n c h a r a c t e r i s e d by consequently admits a unique s o l u t i o n .

(7.1A), which

We assume t h a t

(7.3A)

J = JA+JB,

where J ,J a r e a l s o convex, d i f f e r e n t i a b l e f u n c t i o n a l s ; A B

we

write :

A,B a r e monotone o p e r a t o r s and we have A 7.2

Decomposition-coordination augmented Lagranaian

+

B = C.

of (T.lA), (7.2A) using t h e

The problems (7.lA) , ( 7 . 2 A ) a r e c l e a r l y equivalent t o t h e minimisation problem

(7.5A)

where

(7.7A)

3=

v,q)

E

Hx H, v-q = 0).

We s h a l l handle t h e l i n e a r e q u a l i t y c o n s t r a i n t v-q = 0 by t o t h i s end w e introduce penalty and Lagrange muZtipZier; (with r > 0 ) t h e augmented Lagrangian dr : Hx Hx H + R defined by

(7.8A)

L,(v,q,p)

= j(v,q)

+

$11

v-ql\

+

bSv-4)

(SEC. 7 )

A.D.

619

and augmented Lagrangian methods

The f o l l o w i n g p r o p o s i t i o n may r e a d i l y be proved ( s e e e . g . Fortin-Glowinski /2A/, Glowinski /2A/) P r o p o s i t i o n 7.1:

Under the above assumptions on C , J , J A Y the augmented Lagrangian admits a unique saddle point B { u , p , A l on H x H x H and we have p = u , h = B ( u ) = -A(u), where u is the solution of ( 7 . 1 A ) , (7.2A).

%.

J

7.3

Solution of ( 7 . 1 A ) ,

(7.2A) v i a d u a l i t y a l g o r i t h m s f o r

dr

I n view o f P r o p o s i t i o n 7 . 1 it i s n a t u r a l t o e n v i s a g e s o l v i n g (7.2A) v i a t h e d e t e r m i n a t i o n o f t h e s a d d l e p o i n t s of d we shals' u s i n g t h e d u a l i t y a l g o r i t h m s o f Chapter 2 , S e c t i o n 4 ; r e s t r i c t o u r a t t e n t i o n t o Uzawa's a l g o r i t h m ( 4 . 1 2 ) , (4.13) of Chapter 2 , S e c t i o n 4 . 3 ; we t h u s have:

(7.1A),

(7.9A) then f o r An by

AOE

H given n

n 2 0, assuming that A n is known, determine u ,p

Find {un,pn)

(7.1 OA)

H x H such that V (v,q)

E

and

HxH

we have n n n dr(u ,p

(7.11A)

E

n

An+1

=

,A 1 Sdr(v,q,x") ,

An +p(u"-p")

,p>0

;

Rendering (7.10A) e x p l i c i t , we have

run+A(un)-rp"

=

- A",

rpn+B(Pn)-run

=

A".

(7.1 OA) '

The e s s e n t i a l d i f f i c u l t y i n t h e a p p l i c a t i o n o f (7.9A)-(7.11A) i s t h u s t h e s o l u t i o n o f t h e system o f two coupled e q u a t i o n s (7.lOA)'. A s p o i n t e d o u t p r e v i o u s l y i n Chapter 3, S e c t i o n 10 and Chapter 5 , S e c t i o n 9 i n a s i m i l a r c o n t e x t , it i s p o s s i b l e t o e n v i s a g e u s i n g block v a r i a n t s o f t h e relaxation methods of Chapter 2 , S e c t i o n 1 ( t h e convergence o f which h a s been e s t a b l i s h e d by Cea-Glowinski / 2 / u n d e r q u i t e g e n e r a l a s s u m p t i o n s ) t o i n t h e c a s e i n which o n l y a single r e l a x a t i o n s o l v e (7.10A)'; i t e r a t i o n i s performed, n a t u r a l l y u s i n g t h e r e s u l t s o f t h e p r e v i o u s i t e r a t i o n as i n i t i a l c o n d i t i o n s , we a r e l e d t o t h e f o l l o w i n g

Optimisation algorithms

620

( U P . 2)

algorithm (introduced by Glowinski-Marrocco 1 2 1 ) : (7.12A)

p 0 , x 1 given i n H,

then f o r n 2 1 , assuming that pn-l and An are known, determine un,pn and An" by

-

(7.13A)

n n- 1 ru + ~ ( u " )= r p

(7.14A)

rpn

(7.15A)

An+' = An + p ( un-p n) .

A*,

+ ~ ( p =~ run ) + A~,

A variant (due to Gabay /lA/, to which we also refer for the convergence results) is given by

(7.16A)

p 0 , ~ ' given i n H,

tenfor n 2 1 , assuming t h a t pn-l and An are known, detennine u ,p and by (7.17A)

run+A(un)

= rpn- 1 -An,

rpn+B(ph)

= run+An + l / 2

(7.18A) (7.19A)

9

= A

(7.20A)

n+l/2

+

p(u"-p")

.

Under "very reasonable" monotonicity and continuity assumptions for A and B y it is shown in Fortin-Glowinski / 2 A / , Glowinski /2A/ 0 that for any p ,A' E H and if (7.21A)

we have

p

E

10,

I + 6 rC 2

621

A . D. and augmented Lagrangian methods

(7.22A)

lim Ilp"-ull

\

lim

X"

=

= 0,

X weakZy in H.

TI++-

If JB is linear (or a f f i n e ) , it is proved in Gabay-Mercier /1/ that (7.22A) also holds if p E 10,ZrC

.

In fact the convergence proofs (based on energy Remark 7.1: methods) given in the above references also hold if A,B satisfy suitable monotonicity and continuity properties. without necessarily being the derivatives of convex functionals (and possibly if they are multivalued)

.

7.4

An "alternatina direction" interpretation of algorithms (7.12A)-(7.15A) and (7.16A)-(7.20A)

The case of aZgorithm (7.12A)-(7.15A): We assume that that

p =

r; it then follows from (7.14A), (7.15A)

1 = B(pn)

(and hence

)\"

= B(pn'l)

)

,

which combined with (7.13A) implies

(7.23A)

run +

A(U~)

n-1

= rp

-~(pn-1 1.

We have, from (7.14A), mn + An = rpn+B(pn), and hence, by using this in (7.13A)

(7.24A)

rpn+B(pn)

rp

n- 1 -A(un).

Putting pn-1/2 = un and r = ( 7.24A)

(7.25A)

Pn+1'2+

1 h,

XA(p n+1'2)

we finally deduce from (7.23A),

= pn-AB(pn)

,

622

(7.26A)

@timisation

pn+'+hB(pn+l)

a lgorithms

(APP. 2 )

pn-XA(p n+1/2

=

1.

R e l a t i o n s (7.25A), (7.26A) show t h a t f o r p = r , (7.12A)-(7.15A) i s i n f a c t e q u i v a l e n t t o t h e Douglas-Rachford Alternating Direction

algorithm o f S e c t i o n 6.1, Remark 6.1. The case of algorithm ( 7 . 1 6 A ) - ( 7 . 2 0 ~ ) : Here a g a i n we assume t h a t p = r; from (7.17A)-(7.2OA) t h a t 1 12 = -A(un),

An+'

it can t h e n r e a d i l y be deduced =

B(pn)

(hence

hn = B(pn-l 1) ,

and u s i n g t h i s i n (7.17A), (7.19A) g i v e s

(7.27A)

run+A(un)

= rpn'l-B(pn'l)

(7.28A)

rpn+B(pn)

= run

-

, A(un).

1 P u t t i n g p n - l I 2 = un and r = - we f i n a l l y deduce from (7.27A), A (7 . 2 8 ~

(7.29A)

rPn+1/2+A(p n+l/2

= rpn-B(pn),

(7.30A) It follows from (7.29A), (7.30A) t h a t , i f p = r , (7.16A)(7.2OA) i s i n f a c t e q u i v a l e n t t o t h e Peaceman-Rachford Alternating Direction algorithm o f S e c t i o n 6.1, Remark 6.1.

Remark 7.1: To t h e b e s t of o u r knowledge, t h e r e l a t i o n s h i p s which e x i s t between t h e alternating direction and augmented La_mangian methods were f i r s t demonstrated i n Chan-Glowinski /U/, /2A/, i n connection w i t h t h e numerical s o l u t i o n o f t h e n o n l i n e a r D i r i c h l e t problems o f Appendix 1, S e c t i o n 6.

Appendix 3 FURTHER DISCUSSION OF THE NUMERICAL ANALYSIS

OF THE ELASTO-PLASTIC TORSION PROBLEM

SYNOPSIS

1.

The aim of this appendix is to supplement Chapter 3 relating to the numerical analysis of the elasto-plastic torsion problem; we shall, in fact, restrict our attention to several supplementary results concerning the finite-element approximation of this problem and its i t e r a t i v e solution. More specifically, we shall return in Section 2 to the approximation using f i n i t e elements of order one (i.e. piecewise a f f i n e ) , previously investigated in Chapter 3, Sections 4 and 6; it will be shown that under reasonable assumptions we "almost" have = O(h1I2) (in

I I+,-uI

HL (52)

fact we have

IIuh-uII

1 Ho

= O(h) if R clR )

(a)

.

In Section 3,

which follows Falk-Mercier /lA/, it will be shown that by using an equivalent formulation it is possible to obtain an approximation error of optimal order, even if R c lR2 Finally, in Section 4, we shall give some additional information on the iterative solution of the above problem.

.

Let us conclude this first section by recalling (see Chapter

3, Section 1) that the problem under consideration is defined by Find u c K such that ( 1 .1A)

a(u,v-u)

2

L(v-u)

Vv

E

K

where 1

(1.2A)

K =

(1.3A)

a(v,w) =

( 1 .4A)

L(v) =

I V E H

0

(R),

1,

(VV~ 51

Vv-Vw dx

a.e. 1

, 1

VV,WEH (R),

1 VVEH~(R)

,f

EH-'(R)

1

= (Ho(R))'

.

624

Elas to-p l a s t i c torsion

(APP. 3 )

.

THE FINITE-ELEMENT APPROXIMATION OF PROBLEM ( 1 . 1 A ) (I) ERROR ESTIMATES FOR PIECEWISE-LINEAR APPROXIMATIONS

2.

This s e c t i o n , which follows Glowinski /2A, Chapter 2 , Section 3/ w i l l supplement t h e results of Chapter 3 , S e c t i o n 6.2. 2.1

One-dimensional case

2 Suppose t h a t n = lO,l[ and t h a t i n ( 1 . 4 A ) we have f E L (52). Problem ( 1 . 1 A ) can t h e n be w r i t t e n

( 2 . IA)

1;,

Find U E K = { V E H 01 ( O , l ) ,

du (-dv - &)dx du dx d x

2

f (v-u)dx

L e t N be an i n t e g e r > 0 and l e t h = N and f o r i = OJ,

...

e 1. =

[ X ~ - ~ , X ~ ] i, = 1 , 2

dv 1x 1

51

a.e.1 such t h a t

VV E K .

1 f; we consider xi = i h

,...N .

We t h e n approximate Hl(0,l) and K r e s p e c t i v e l y by 0

(2.2A)

Voh =

{ V h E C 0 c0,11,

vh(o)=vh(l)=o,

vhlei E P ~ , i = 1 , 2 ,

...N}

and

( w i t h , as u s u a l , P I = space o f polynomials of degree

51 ) .

The approximate problem i s t h e n defined by

1

Find \

6%

such that

Problem (2.4A.) c l e a r l y admits a unique solution. t h e approximation e r r o r IIuh-uII 1 we have H O W , 1)

Regarding

Finite-e Zement approximation

(SEC. 2) Theorem 2.1:

If

(2.5A)

II y,-uII

Proof:

Since y , r \c

f

I

625

\

Let u and

( 2 . U ) and ( 2 . h ) .

-

CE

be the r e s p e c t i v e soZutions of L2(0,1) we then have

= O(h).

K it r e s u l t s from (2.1A) t h a t

(2.6A) We deduce, by adding ( 2 . 4 A ) and (2.613) , t h a t

VvhE

$

and hence Vvhe

2 since f E L ( 0 , l ) implies t h a t Section 2 . 1 ) we have

UE

2 H (0,I)n K ( s e e Chapter 3,

1

(2.8A)

lodx !du

dx

(vh-u) dx =

We a l s o have ( t h e proof being l e f t as an exercise f o r t h e reader) :

which combined with ( 2 . 7 A ) , ( 2 . 8 ~ )implies vvhE K h

Let v r K ; we define t h e linear interpolate rhv by

Ih

r vrVoh

(2.1 OA)

(rhv) (xi)

=

v(xi)

, i=O,l,.

..N.

E Zasto-p lastic torsion

626

We have d (rhv)le. dx 1

-

X.

v(xi)-v(xi-l 1

1

h

h

(APP. 3 )

x

dv -dx, dx

i-1

and hence

d dx( rhv)

e.

1 1 since

1

dv I- dxl < 1 -

a.e. on J O , ~ C ;

we t h u s have

(2.11A)

rhVE%I

V V E K,

Replacing v by r u i n ( 2 . 9 A ) we t h e n have h h

The reguzarity property

u

E

H'(0,l)

and

imply

where C denotes v a r i o u s c o n s t a n t s independent of u and h. The e s t i m a t e (2.5A) then follows t r i v i a l l y from (2.12A)-(2.14A). 2.2

Two-dimensional case

Suppose t h a t R i s a bounded convex polygonal domain i n l R 2 with boundary r , and t h a t f E LP(n) with p > 2 (a reasonable assumption s i n c e f o r a p p l i c a t i o n s i n mechanics we have f = c o n s t . ) . we t h u s approxWe use t h e n o t a t i o n of Appendix 1, Section 3.6; imate HA(S1) and K r e s p e c t i v e l y by

(2.15A)

Voh = {vh E Co(E) ,vhIr = 0, vhlT E P I

VT

E

5)

Finite-element approximation

(SEC. 2 )

-

I

627

and

(2.16A)

Kh = K n V oh

'

g i v i n g the approximate problem

Find

\ such

I+,€

that

(2.17A) Vyl*V(vh-~)dx

2

.

VV~E

f(v,-yl)dx

Problem (2.l7A) admits a unique s o l u t i o n , and w i t h r e g a r d t o t h e convergence of t o u we have

5

Theorem 2 . 2 :

b e t a , by0,>Oas h and f we have

Suppose t h a t the angles of r h a r e bounded Then under t h e above assumptions on $2 + 0.

where u and u a r e t h e r e s p e c t i v e s o l u t i o n s of (1.1A) and (2.17A). h

Proof: (2.1 9A)

It r e s u l t s from Chapter 3, S e c t i o n 2 . 1 t h a t

u

E

d'P(Q).

By proceeding as i n t h e proof o f Theorem 2 . 1 , and u s i n g t h e fact that \ c K we o b t a i n

+a(u,vh-u)-

(2.20A)

*1 ' 71 IIVh-"Il Ho(Q>

1

(-Au-f)(vh-u)dx

Q

Vvh e

Kh'

Using t h e Halder i n e q u a l i t y we deduce from (2.20A)

\Yvhe\,

with -1+ - ' 1 P P

= 1 ( i . e . p'

=L). 1'P

L e t q be such t h a t 1 5 q + a ; t h e n suppose t h a t t h e assumptions o f Theorem 2.2 and t h a t p > 2. If

t,,satisfies

628

Ekzsto-plastic torsion

(APP. 3)

d Y p ( T ) c WISq(T) (with continuous i n j e c t i o n ) , t h e n it results from Ciarlet /1A/ and from SoboZev 's embedding theorem ( W2,P(T) c W1 ,OD(T) c Co(T) , with continuous i n j e c t i o n ) t h a t and Vv 6 WzsP(T)we have VT

E%

( 2 22A)

IIV('rrTv-v) I I

1 1 ChT1+2(--- p) IIvll$,p(T) L ~ ( T x) L ~ ( T )

I n (2.22A), I T V i s t h e linear interpolate of v at t h e t h r e e T v e r t i c e s of T, h i s t h e l e n g t h o f t h e l o n g e s t s i d e of T , and C L e t v ~ W ~ r P ( n )and l e t i s a constant inxependent of T and v. IT : H :(n) n C o ( n ) + Voh be defined by

h

s i n c e p > 2 implies ?''(a) C C o ( n ) , with continuous i n j e c t i o n , we can d e f i n e Thv( ) but i n contrast t o the one-dimensional case

we have i n general IThV

' Kh

i f

v

E

f ~ ? ' ~ ( an)K.

Since $"(a) cW""(~) with continuous i n j e c t i o n i f p > 2 , it follows from (2.22A) t h a t almost everywhere on SZ we have

Iv(Thv-v) (x)

I Irh1-2/p

vv

E

2 'qn)

from which we deduce t h a t

( (2.23A)

a h o s t everywhere on sz we have

1

t h e constant r i n (2.23A) i s independent of v and h. define rh ' Hi(Q) n?'P(Q) + Voh by

We t h e n

ITV

h

(2.24A)

(

rhv

l+rhl-2/p

We can, i n f a c t , d e f i n e lThv Vv

Y

E

K

since K

C W

1 '"(a)

C

Co(rr>.

Finite-element approximation

(SEC. 2)

-

629

I

it t h e n follows from (2.23A), (2.24A) t h a t (2.258)

rhv

Vv

%I

E

$'p(52)

n K.

Since u ~ $ ' ~ ( 5 2 )nK, it follows from (2.25A) t h a t w e can t a k e = r u i n (2.21A), g i v i n g "h h

which i m p l i e s

+ rh'-2'plI ull

2 ,p (52)

llu

II

LP ' (52) Since p > 2 we have Lp(n) C L P ' ( G ) , with continuous i n j e c t i o n ; it t h e n r e s u l t s from Strang-Fix /1/ and Ciarlet /1A/ t h a t under t h e assumptions on s t a t e d i n t h e Theorem w e have

p;n

(2*298)

I!nhu-ull 1

Ho (52)

' Chllull w29w

'

w i t h , i n (2.29A), (2.30A), C independent of h and u. (2.18A) t h e n r e s u l t s t r i v i a l l y from (2.26A)-(2.30A).

Estimate

Remadi 2.1: It follows from Theorem 2.2 t h a t i f f = constant (which i s t h e c a s e f o r a p p l i c a t i o n s i n mechanics) and i f 52 i s a convex polygonal domain, t h e n w e have an approximation e r r o r which i s l f p r a c t i c a l l y ' l of o(&) s i n c e U E w ~ , P @ )~p (<+I.

Etas to-plus t i c torsion

630 3.

(APP. 3 )

FINITE-ELEMENT APPROXIMATION OF PROBLEM (1.lA). (11) OPTIMAL ORDER ESTIMATES THROUGH THE USE OF AN EQUIVALENT FORMULATION I n t h i s s e c t i o n , which f o l l o w s t h e account o f Falk-Mercier

/lA/, we show t h a t t h e u s e o f an e q u i v a l e n t f o r m u l a t i o n e n a b l e s u s t o o b t a i n f i n i t e - e l e m e n t approximations which l e a d t o e r r o r e s t i m a t e s o f optimal order.

3.1

Equivalent f o r m u l a t i o n o f problem (1.1A)

The f o l l o w i n g e q u i v a l e n c e lemma forms t h e fundamental r e s u l t of t h i s t h i r d s e c t i o n : Lemma 3.1: Suppose t h a t fl i s a simpZy-connected bounded domain i n l R 2 ; there i s then equivaZence between problem (1.1A) and the variational probZem ( l )

Find p E A n n such that

I where

+

= {+ly+2}

i s an ( a r b i t r a r y ) solution of

and where (3.3A)

A = { q E L 2 (Q) X L2 ( a ) , lql < I

(3.4A)

n

2

= 19 E L (Q) X L

a.e. on a )

q*vw dx = 0

VWEH~(Q)}.

The soZutions u o f (1.1A) and p of (3.1A) are connected by the retation

(SEC.

Finite-element u p p o x h a t i o n

3)

-

11

631

P r o o f : ( i ) L e t u be t h e s o l u t i o n of (1.1A) with f s a t i s f y i n g ; w e use t h e n o t a t i o n

(3.2A) and with v E HA(,) (3.6A) Clearly we have (3.7A)

VEK=-+~EI\,

1

fv dx =

(3.8A)

v

$*q d x

v ~ H1 ~ ( a ) ,

s2

p'q d x

(3.9A) Moreover w e have,

by t h e d e n s i t y of vq =

I-av , ax2

(3.11A)

av

ax,

vw

E

v v ~ H1~ ( S 2 ) .

€I1 (a),

B ( n ) i n HA(Q), (3.10A) a l s o holds

1 ,

1 v6Ho(S2),

so t h a t

av , -

1 v € H O ( n ) 3 q = {ax2

By u s i n g (3.6A)-(3.9A), (1.1A) t h a t p = {.k

(3.11A) we can aU

e a s i l y deduce from } i s t h e s o l u t i o n o f (3.1A).

- ax,

ax2

( i i ) I t now remains t o prove t h e converse; l e t p be t h e s o l u t i o n of (3.1A) and l e t q E A n H The condition q E H i s i n f a c t equivalent (from Green's formula) t o

.

(3.12A)

Vaq = 0 a.e.

(3.13A)

q - n = 0 on

in

51,

r,

where n = (nl,n2) i s t h e u n i t normal p o i n t i n g outwards from Q ; t h e v e c t o r T = (n2,-nll i s t h u s t a n g e n t i a l t o .'l Conditions (3.12A), (3.13A) imply t h e e x i s t e n c e o f v E H

t h e simple c o n n e c t i v i t y of Q defined t o w i t h i n an

(a),

Eksto-pZastic torsion

632

(APP. 3)

a d d i t i v e constant , such t h a t (3.14A) (3.15A)

av = 0 on r ++ v a.r

= constant

on

r. 1

We can t h u s t a k e v = 0 on r , which combined with v e H (a) implies v E HI (a) ; i n view o f t h e above we t h e n c l e a r l y have 0

(3.16A)

qehnH W v e K .

Since t h e above r e l a t i o n s a l s o hold f o r p , we deduce from (3.2A), (3.14A) t h e e x i s t e n c e o f a u which satisf ies

(3.1A),

ueK

Vu*V(v-u)dx 2

I,

f(v-u)dx

i . e . u i s t h e (unique) s o l u t i o n of ( 1 . l A ) .

v EK,

fl

Remark 3.1: It results from Duvaut-Lions /1, Chapter 5 / t h a t (3.1A) i s i n f a c t t h e n a t u r a l mechanical formulation f o r t h e problem of t h e e l a s t o p l a s t i c t o r s i o n o f a c y l i n d r i c a l b a r under t h e p h y s i c a l assumptions o f Chapter 3, Section 1.2. Remark 3.2: If f = c o n s t . = C (which corresponds t o t h e p h y s i c a l s i t u a t i o n ) , it follows immediately t h a t Q = {Ql,Q2} defined by

satisfies condition (3.2A), VA (uniquely) determine uo E H;(Q)

-

E

IR

.

I f f is not constant, w e

from

Auo = f i n a , u putting

=

0 on

au0 ax,

9

r;

$2

au0 , it -ax,

i s c l e a r t h a t t h e function Q

t h u s defined sat isfies (3.2A).

Remark 3.3: The regularity results r e l a t i n g t o problem (1.1A) (proved i n Br6zis-Stampacchia /2/ and quoted i n Chapter 3, S e c t i o n 2.1) and r e l a t i o n (3.5A) imply t h a t if r i s s u f f i c i e n t l y regular,

Finite-element approx6nation

(SEC. 3 )

-

I1

633

t h e n we have f o r t h e s o l u t i o n p of problem (3.1A)

Remark 3.4: This remark concerns t h e e x i s t e n c e of a Lagrange m u l t i p l i e r , a s s o c i a t e d with t h e c o n d i t i o n p E H , for problem ( 3 . 1 A ) : it follows from ( 3 . 1 A ) , ( 3.4A) t h a t a n a t u r a l Lagrangian a s s o c i a t e d with problem (3.1A) i s & : (L2(n))2 X H 1 ( Q ) -C IR defined

by

which l e a d s t o t h e saddle-point problem

We t h e n have Proposition 3.1:

There i s equivalence between (3.20A) and

moreover i f { p y x } i s a solution of (3.201, (3.21A), then p i s the solution o f problem ( 3 . 1 ~ ) . Proof:

( i ) (3.20A)

(3.21A) :

From t h e left-hand i n e q u a l i t y

i n (3.20A) w e deduce t h a t

and hence

(3.22A)

I,

p*V(w-X)dxz 0

I,

p-Vw dx = 0

Vw E H

1

(a),

Vw E H I ( ~ )

~ E . H;

t h e right-hand i n e q u a l i t y i n (3.20A) implies (as is s t a n d a r d )

EZas t o p h s t i c torsion

634

Now

lim

(UP. 3)

4 p + t ( q - p ) ,XI- 2(P,X)

t+o

t

t#O

which with (3.23A) implies t h e i n e q u a l i t y i n (3.21A) ; i n view of (3.22A) we have t h u s proved (3.20A) (3.21A).

( i i ) (3.218) on t h e one hand

p€H+t(p,w) and hence

(3.20A) : This i s immediate since

=$I

a

2 Iql d x -

I

0 - q dx

n

On t h e o t h e r hand we have, by t h e COnVc?Xitg of

q

~ E H ~ ( Q ) ,

+

&(q,X),

which combined with ( 3.2l.A) implies

The double i n e q u a l i t y i n (3.20A) r e s u l t s from (3.24A)

, (3.25A).

(iii) It i s moreover, immediate t h a t (3.2l.A) implies t h a t p s a t i s f i e s (3.1A); it i s s u f f i c i e n t t o r e s t r i c t q t o h n H i n (3.21A) and t o note t h a t

Finite-element approximation

-

II

635

x

The e x i s t e n c e of a m u l t i p l i e r s a t i s f y in g (3.20A),(3.21A) i s a t r i c k y problem: however, it i s shown i n Duvaut-Lions /l/, u s i n g t h e results of Br6zis / 5 / , t h a t such a m u l t i p l i e r x E H1(Q) e x i s t s i n t h e case i n which f = c o n s t . ( t h e c a s e i n mechanics); it should be noted ( s e e Duvaut-Lions, l o c . cit.) t h a t , i n mechanics, i s i n t e r p r e t e d as a displacement. We note t h a t t h e m u l t i p l i e r d i s c u s s e d above i s defined only t o w i t h i n an a r b i t r a r y a d d i t i v e constant.

x x

3.2

Approximation of problem (3.1A) by a finite-element method

We s h a l l assume i n t h e following t h a t Q i s a bounded convex polygonal domain i n I R ~ .

3.2.1

Definition o f the approximate problem

Using n o t a t i o n similar t o t h a t o f Appendix 1, S e c t i o n 3.6, we approximate H1(Q), L 2 ( Q ) x L 2 ( Q ) , H a n d A r e s p e c t i v e l y by:

(3.26A)

Vh = {wh

(3.278)

Lh

E

Co(n),

2 { q h E L (1 '

whlT

E

xL2(n),

P1

VTE

eh},

qh T E P ~ X PV ~T E ~ ~ } 9 ( ~ )

We t h e n approximate problem ( 3.1A) by

( ' ) Po: space of constant real-valued f u n c t i o n s .

Elas to-plas t i c torsion

636

( U P . 3)

It i s easy t o prove t h e following

Proposition 3.2 : 3.2.2

Problem ( 3.30A) admits a unique solution.

Investigation of the convergence

We s h a l l assume i n t h e following t h a t f E L 2 ( n ) ; s i n c e $2 i s convex and bounded, t h i s implies ( s e e Chapter 3, Section 2.1) t h a t t h e s o l u t i o n u o f (1.l.A) s a t i s f i e s u E H2(S2), and hence f o r t h e s o l u t i o n p of (3.1A)

s i m i l a r l y i f Q i s obtained from f using t h e r e l a t i o n s of Remark 3.2, w e have

with

i n (3.31A), (3.33A), C denotes c o n s t a n t s . We a l s o assume t h e e x i s t e n c e of a m u l t i p l i e r t h a t {p,x) satisfies (3.20A), (3.21A).

x

E

H l ( S 2 ) such

I n o r d e r t o i n v e s t i g a t e t h e convergence, w e s h a l l need t h e following r e s u l t s : 2

2

Let -mh : L (Q) X L (Q) + 4, be the orthogonal projection operator onto Lh ( i n the norm L 2 ( n ) x L2(n)); we then hme the following propertzes : Proposition 3.3:

Finite-element approximation

(SEC. 3 )

('lrhq-q)-zh dx = 0

(3.36A)

vq

E

-

637

I1

L2(52) xL2(52)

,

Vzh

E

I,,,

Proof: P r o p e r t i e s (3.34A)-(3.36A) are almost immediate; w e s h a l l now prove (3.37A) : We have Vh c H1(Q), g i v i n g

q

E

Since Vwh H then

E

1 5 2

$

it follows from (3.36A) and (3.38A) t h a t i f

q*vwhdx = 0 'lrhq'Vw dx = h 1 5 2

Vwh EVh*'IThq

El#,

.H

Moreover, by u s i n g t h e r e s u l t s of Ciarlet / l A / , f o r example, we may prove t h e following p r o p o s i t i o n r e g a r d i n g 71 h:

We have

P r o p o s i t i o n 3.4:

(3*39A)

11*hq-qll

2

\I q E

H1(52) x H 1(52)

2 Ch 114Il

L (52) x L2 (a)

H1(a)X H' (52)

'

with C independent of h and q . Regarding t h e convergence of ph t o p , we have Theorem 3 . 1 ( Falk-Mercier / l A / ) : Lhzder the above asswrrptions on n , f ,x, ( t h e assumptions o n eh being those o f Theorem 2 . 2 ) we have the (optima2 order) error estimate

where , i n (3.40A) ,p i s t h e so2ution o f problem ( 3 . 1 A ) and ph that of the approximate prob2em ( 3.30A)

.

Proof: problem

We have, from t h e d e f i n i t i o n of t h e approximate

EZas to-plas tic torsion

638

and a l s o

By adding (3.41A) and ( 3 . 4 3 A ) , we o b t a i n a f t e r some s t r a i g h t forward manipulation, Vqh E Ah n %, Vwh E Vh:

I n t h e following w e s h a l l use t h e n o t a t i o n

t h e n deduce from (3.44A) ( v i a Schwarz's inequality and t h e r e l a t i o n 2IxIIyl

x2+y2 ~ x , y c n ) ,

VqhE\n%

,

vWhEvh

:

Finite-element approximation

(SEC. 3 )

We t h e n approximate p and rh

: Co($ i.e.

-f

x

by

TI

h

-

I1

639

p and rh x where

Vh i s t h e ( l i n e a r ) V -interpoZation o p e r a t o r on

r v EV h h

h

wVECO(Q,

(rhv)(P)=v(P) V P

v e r t e x of

T

E

%

;

it i s p e r f e c t l y a c c e p t a b l e t o d e f i n e rhx, s i n c e X E H 2 ( R ) cCo(n) i n t h e two-dimensional c a s e under c o n s i d e r a t i o n . We t h e n have

s i m i l a r l y we have ( s e e P r o p o s i t i o n 3.3)

(3.488)

ThP E % n %

(3.49A)

jRPh'(ThP-P)

9

dx = 0

Moreover, s i n c e p E H1(Q) x H1(Q) and x E H2(Q) ( s e e (3.39A) and Strang-Fix 111, C i a r l e t /1A/ f o r t h e following estimates) , we have

where i n (3.50A) , (3.51A) , C denotes c o n s t a n t s which a r e independe n t of h , p , and x. I n view of (3.47A), (3.488) w e can, i n (3.46A), t a k e q = vhp and w = rhx; w e t h e n deduce from (3.46A) (3.49A) , ( 3 . h A ) h

640

Elastoplastic t o r s i a

(APP. 3)

From Proposition 3.3, r e l a t i o n (3.36A) , w e have

I and hence

we then deduce from (3.31A)-(3.33Aly (3.39A) and from (3.50A), (3.52A), (3.53A)

which proves t h e estimate (3.40A) and renders it more p r e c i s e .

4.

FURTHER DISCUSSION OF ITERATIVE METHODS FOR SOLVING THE ELASTO-PLASTIC TORSION PROBLEM

4.1

General d i s c u s s i o n .

Synopsis

I n Chapter 3, v a r i o u s methods based on t h e formulation ( 1 . 1 A ) were described f o r t h e i t e r a t i v e s o l u t i o n of t h e e l a s t o - p l a s t i c t o r s i o n problem; without going i n t o g r e a t e r d e t a i l ( f o r which we refer t o Marrocco / l A / , Gabay-Mercier 111, Fortin-Glowinski / l A / , /2A/ , Glowinski /2A/ and Glowinski-Marrocco / l A / ) we can state t h a t t h e most e f f i c i e n t c a l c u l a t i o n methods c u r r e n t l y a v a i l a b l e are based on algorithm (10.5)-(10.7) o f Chapter 3, S e c t i o n 1 0 (of the augmented Lagrangian t y p e ) ; t h i s algorithm ( ) can be g e n e r a l i s e d i n a p a r t i c u l a r l y simple manner t o c a s e s i n which t h e cross-section of t h e b a r 2s mZtipZy-connected.

We s h a l l conclude t h i s f o u r t h s e c t i o n (and t h i s appendix) by i n v e s t i g a t i n g , i n Sections 4.2 and 4.3, t h e i t e r a t i v e sozution of (l)

Together with i t s v a r i a n t s f o r which c i t e t h e above r e f e r e n c e s .

(SEC. 4 )

641

I t e r a t i v e methods

(1.U) v i a the equivalent formulation (3.1A) , u s i n g d u a l i t y - t y p e algorithms based on t h e formulation (3.20A); i n Section 4.2 we d e s c r i b e a Uzawa-type algorithm and i n S e c t i o n 4.3 a v a r i a n t of conjugate-gradient t y p e .

4.2

S o l u t i o n of problem (3.1A) by a d u a l i t y method

4.2.1

Genera2 discussion.

Description of the a2gorithm

Given t h e p a r t i c u l a r s t r u c t u r e of problem (3.1A) , we s h a l l r e s t r i c t our a t t e n t i o n t o a d u a l i t y algorithm of t h e same t y p e as t h o s e considered i n Chapter 2 , S e c t i o n 4.3 ( s e e a l s o Appendix Using t h e d u a l i t y results i n Remark 3.4 of 2, S e c t i o n 3 ) . S e c t i o n 3.1 of t h i s appendix, w e s h a l l reduce t h e s o l u t i o n of (3.1A) (and t h e r e f o r e t h a t of ( 1 . 1 A ) ) t o t h e s o l u t i o n of t h e saddle-point problem (3.2OA) , which as we r e c a l l i s

with

which i s j u s t i f i e d s i n c e p i s i n t h i s case t h e solution of (3.1A). I n view of Chapter 2 , S e c t i o n 4.3, t h i s l e a d s t o t h e following (Uzawa-type) algorithm:

xo E HI

(4.3A)

then for n using

2

0,

(Q) given

(xo = o

xn E HI (Q) known,

f o r example),

determine p"

E

A and

xn+l E

H' (Q)

(APP. 3)

EZas to-p Zas t i c torsion

642

The p r a c t i c a l implementation o f (4.3A)-(4.5A) p r e s e n t s no major d i f f i c u l t i e s ; i n e f f e c t (4.4A) i s equivalent t o (4.6A)

p" = pA(@vxn)

S

2 2 2 PA : (L (f2))2 -P (L (62)) i s t h e orthogonaz projection operator onto h ( i n t h e norm L 2 ( Q ) x L2(Q) ) ; we t h e n have

where

(4.7A)

p"

NVX"

=:

SUP(1

Regarding

I

I@Vfl)

(4.6~)~ if we

define

(a,.

)Hl(Q) by ( ' 1

n+l n from x and pn ( v i a (4.5A)) reduces t h e n t h e determination of x t o t h e s o l u t i o n of a Newnann problem f o r t h e o p e r a t o r - A + I ; this problem i s well-posed s i n c e t h e l i n e a r mapping

w

-t

-



H1(62)

I,

P Vwep"dx

i s obviously continuous on H ' ( Q ) . It can t h u s be seen t h a t t h e implementation o f (4.3A)-(4.5A) ( i n p r a c t i c e , i t s finite-dimensi o n a l v a r i a n t s ) i s s t r a i g h t f o r w a r d provided t h a t e f f i c i e n t programs f o r t h e numerical s o l u t i o n of l i n e a r e l l i p t i c problems of o r d e r two a r e a v a i l a b l e .

(l)

o t h e r choices are p o s s i b l e

(2)

and hence

llwll

4' w

EH

1

(62).

4)

I t e r a t i v e methods

4.2.2

Convergence of algorithm ( 4.3A

(SEC.

Regarding t h e convergence o f algorithm Theorem

643

(4.3A)-(4.5A)

4.1: Suppose that the saddle-point problem (4.1A), is defined

(4.2A) h i t s a solution { p y x ) and that by (4.8A); then i f (4.9A)

w e have

0

we have f o r the sequence

defined by algorithm (4.3A)-

{p n

(4.5A) (4.1OA)

l i m Ilpn-pI( = 0 n++m L~(QxL~(s~)

vxo

CHI(,)

where, i n (4.10A), p i s the solution of (3.1A). This f o l l o w s t h e same g e n e r a l l i n e s as t h a t of Proof: Theorem 4:1 i n Chapter 2, S e c t i o n 4.3; d e s p i t e t h i s we s h a l l g i v e t h e proof i n full i n o r d e r t o demonstrate t h e convergence c o n d i t i o n

(4.9A). L e t { p , x ) be a s o l u t i o n o f (4.1A),.(4.2A); t h e n ( s e e Proposi t i o n 3.1) p must n e c e s s a r i l y be a s o l u t i o n o f (3.1A). I n view o f t h e f a c t t h a t p E H, w e have

I,

vw ~ H ' ( i 2 )

p * V w dx = 0

and hence

which, by d i f f e r e n c i n g w i t h

p"

=

Pn-P

9

2n

(4.5A) and w r i t i n g I

xn-x

,

leads t o

Moreover, by convexity and by t a k i n g account of right-hand i n e q u a l i t y i n (4.1A), we have

(4.4A) and t h e

644

EZas t o p Zastic torsion

( U P . 3)

we deduce, by adding (4.12A) and (4.13A)

A t t h i s s t a g e it i s convenient t o i n t r o d u c e defined ( u n i q u e l y ) by

f" E

H

1

(a),

7 E H1(Q)

¶

(4.1 5A)


=

H'(Q)

I

dx

n

W e t h e n have, from ( b . l l A ) ,

in+'

t

in

-

V w CHI(,).

(4.15A)

,

giving

II in+' I I

€2(a)

or alternatively

By p u t t i n g

w =

in

-n

i n (4.15A) w e deduce from ( 4 . 1 4 A ) , ( 4 . 1 6 A )

by now p u t t i n g w = y i n ( 4 . 1 5 A ) we thereby deduce, using Schwarz ' s i n e q u a l i t y and (4.8A)

(SEC.

4)

645

Iterative methods

and hence

It t h e n f o l l o w s from (4.17A)

,

(4.18A) t h a t

It i s c l e a r i n view o f ( 4 . 1 9 A ) '-n (4.10A) s i n c e p - pn-p.

, that

condition ( 4 . 9 A ) implies

We deduce from ( 4 . 5 A ) , (4.10A) and from t h e f a c t Remark 4 . 1 : t h a t p E H t h a t under t h e assumptions for which Theorem 4.1 above a p p l i e s , we have

I n f a c t w e know a g r e a t d e a l more about t h e behaviour of t h e n sequence {x }n20, s i n c e , by a p p l y i n g Theorem 3.1 o f Appendix 2 , S e c t i o n 3, it can be shown t h a t under t h e above assumptions we have

(4.21A)

lim

xn = x*

weakly in H~(G)

n+*

where x* ( E H I ( $ ? ) ) i s such t h a t { p y x ? i s a s o l u t i o n of t h e saddlep o i n t problem ( 4 . M ) , (4.2A). We n o t e ( s e e Lions-Magenes

/l/) t h a t (4.2lA) i m p l i e s

and hence, i n p a r t i c u l a r

Remark 4.2: The above can e a s i l y be "extended" t o t h e approximate problem (3.30A) v i a t h e s o l u t i o n of t h e a s s o c i a t e d saddlep o i n t problem

EZas to-pks tic torsion

646

(APP. 3)

i n p a r t i c u l a r , t h e convergence c o n d i t i o n ( 4 . 9 A ) s t i l l holds f o r finite-dimensional v a r i a n t s of algorithm ( 4 . ?,A)-( 4.5A) (provided, i s s t i l l defined by ( 4 . 8 A ) ) . of course, t h a t ( ,-) H1(Q)

W e n o t e t h a t (4.23A) always admits a saddle p o i n t ; i n f a c t (3.30A) admits a (unique) s o l u t i o n and t h e r e l a t i o n s d e f i n i n g Hh are linear and are finite i n number ( t h i s number being equal t o t h i s i s s u f f i c i e n t t o ensure (see Rockafellar 131) t h e dim V h ) ; e x i s t e n c e of a s o l u t i o n t o t h e saddle-point problem (4.23A) i n t h e finite-dimensional case.

4.3

On conjugate-gradient-type

v a r i a n t s o f algorithm ( 4 . 3 A ) -

(4.5A) 4.3.1

General discussion

It i s o f t e n advantageous ( l ) t o attempt t o accelerate the convergence of gradient-type algorithms by u s i n g a v a r i a n t of conjugate-gradient t y p e i n t h e sense of Chapter 2, S e c t i o n 2.3 (see a l s o Appendix 2, Section 5 ) ; t h i s i s particularly true for algorithms of t h e Uzawa t y p e , s i n c e t h e s e can o f t e n be i n t e r p r e t e d

as g r a d i e n t algorithms a p p l i e d t o a problem which i s t h e duaZ o f t h e o r i g i n a l problem ( t h e primal problem). The conjugate-gradient algorithm (2.23)-( 2.25) o f Chapter 2 , Section 2.3 w a s a p p l i e d t o t h e s o l u t i o n o f t h e problem

1J(u) < J ( v )

N VveIR

,

(4.24A) lU€RN, with

A t t h e expense of a c e r t a i n i n c r e a s e i n t h e complexity of t h e programming.

(SEC.

4)

647

Iterative methods

we a l s o mentioned a variant which can be applied to non-quadratic cases (in fact this variant is known as the Fletcher-Reeves method, see Pol& 111). The above algorithms can in fact be generalised to much more complicated situations, both finite- and infinitedimensional (see Daniel /I/,/2/, Pol& /1/ and the additional references given in Section 4.3.3); in particular, as we shall see in Section 4.3.2, we can define conjugate-gradient-type variants of algorithm (4.3A)-( 4.5A) (and of its finite-dimensional variants associated with the solution of the approximate problem (3.30A)). 4.3.2

Dual functional. Description of conjugate-gradienttype variants of algorithm (4.3A)-( 4.5A)

2 Let 4 : (L x H 1 (a) + IR be the Lagrangian defined by (4.2A): it is convenient to consider the functional : H1(Q) + IR defined by (4.268)

R(w) =

- Min

d(q,w).

qEA

If we denote by p(w) the solution of the problem

and hence

(4.29A)

+vw

dx

- -21

We deduce from (4.29A) that' (4.30A)

n(w) =

M(w) dx

n

with the integrand M(*) in (4.30A) defined by

+vw

dx

.

648

EZasto-plastic torsion

Starting from (4.26A)-(4.28A) easy to prove:

(APP. 3)

and (4.30A), (4.3lA), it is quite 1

: H (n) IR defined by The functional Lipschitz continuous and Gateaux differentiable; i t s Gateaux dzfferential R' i s also Lipschitz continuous, and i s defined by +

(4.32A)


=

I

p(w)=Vn dx

n

V W , ~E H ' ( ~ )

;

where, i n (4.3s) , p(w) i s defined by (4.27A) , (4.28A) and uhere <*,*>denotesthe bilinear form of the duality between (H1(Q))' and H1(Q). B Remark 4.3:

It follows from (4.26A) that the problem

is the dual of problem (3.1A) associated with the Lagrangian d ; moreover, algorithm (4.3A)-( 4.5A) is simply a ( fixed-step) gradient-methodapplied to the solution of the dual problem (4.33A). Problem (4.33A) is an infinite-dimensionaZ minimisation problem, non-quadratic but unconstrained; the generalisation of the conjugate-gradient algorithms of Chapter 2, Section 2.3 to this problem leads to the following algorithms: Stage

0:

(4.34A)

Initialisation

xo E HI (Q), given

then determine go by

(4.350

(SEC. 4)

I t e r a t i v e methods.

649

and put (4.36A)

z o = gO .

Next , f o r n 2 0 and assuming that xn,g n ,z n are known determine Xn+l , gn+l , zn+l using Stage 1:

Descent

Solve the onevariable minimisation problem F i n d An

E

IR such that

(4.37A)

m(Xn-Xnzn)

Stage 2 :

I

?7?(x"-Az")

v

A

E

IR

Construction of the new descent direction

Define gn+l by g"+I

E

(n) ,

HI

(4.39A) (gn+I ,w)

H (Q)

and then yn+'

by

( o r alternatively

and f i n a l l y

= <@'

(xn+I),w>

1

Vw E H (Q) ,

EZas t o p Zas t i c torsion

650

(4.41A)

z

n+l

n+l

= g

+

(APP. 3)

yn+lzn

The next s t e p , of course, i s (4.428)

n = n + l , go t o (4.37A).

The above algorithms call for a number of comments: The conjugate-gradient algorithm (4.34A)-(4.39A) , Remark 4.4: (4.40A) , (4.4I.A), (4.428) (resp. (4.34A)-(4.39A), (4.40A)2, of Fletcher-Reeves ( resp. Polak-RibiBre) type (see (4.42A)'is Pol& /1/ for the finite-dimensionaZ convergence of these algorithms). If 8 is quadratic it can be shown that the tUo algorithms coincide; however it appears that methods of the Polak-Ribisre type in general perform better than those of the Fletcher-Reeves type (for an explanation of this phenomenon, see Powell /lA/). rn The two non-trivial stages in the implementation Remark 4.5: of the two algorithms above are:

(i)

The solution of the onevariabze minimisation problem (4.37A); this problem can be solved (without calculating derivatives) using methods of dichotomy or Fibonacci type, etc... (see Pol& 111, Wilde-Beightler /lA/, Brent /lA/); it is also possible to use Newton or secant type methods. We should point out that it is not computationally expensive to I evaluate v ( w ) , vw E H (a), from relations (4.28A), (4.29A).

n+l requires the solution of (ii) The calculation of gn+l from x the Neumann problem (4.39A); we note that in view of (4.32A) we have, in (4.39A), (4.43A)


~ 7 =

I

pn+'*Vw dx

n

VwcH

1

(a),

where in (4.43A) we have put (see (4.28~))

In view of the above remarks it is clear that, Remark 4.6: as for algorithm (4.3A)-(4.5A) , an efficient method for solving Neumann problems will be a fundamental tool f o r the implementation of the two conjugate-gradient algorithms above; it will in fact

I t e r a t i v e methods

(SEC. 4 )

651

be finite-dimensional variants of these two algorithms which will be applied to problem ( 3 . 3 0 A ) , and under these conditions it will be advisable to factorise once and for all the (symmetric, positive definite) matrix associated with .the bilinear form Vh x Vh -+ IR defined by

to this end we shall employ either a Gauss or a ChoZesky factorisation. The above also holds, of course, for finite-dimensional W variants of algorithm ( 4 . 3 A ) -( 4.5k)

.

The convergence of the two algorithms ( 4 . 3 4 A ) Remark 4.7: ( 4 . 4 2 8 ) is a tricky problem, and to the best of our knowledge is still an open question. 4.3.3

Discussion.

Conclusion.

The algorithm ( 4 . 3 A ) - ( 4 . 5 A ) (and its finite-dimensional variants) falls within the class of gradient methods with a m i z i a r y operator, previously encountered in Chapter 1, Section 3 . 1 and Chapter 5, Section 6.4. A s regards the two algorithms of Section 4 . 3 . 2 above, these are

conjugate gradient methods with auxiliary operator (in the specialist literature these are usually referred to as conjugate gradient algorithms with preconditioning) ; such algorithms have been used in particular for the numerical solution of problems involving linear or non-linear partial differential equations, some of which ( see Bristeau-Glowinski-Periaux-Perrier-PironneauPoirier /1A/ , Bristeau-Glowinski-Periaux-Perrier- Pironneau /1A/ , Periaux /lA/and Le Tallec / l A / ) are much more complicated than those considered in this current section. In addition to the above references, further details may be found in Bartels-Daniel / 1 A / , Axelsson /U/, Douglas-Dupont /1A/ , Concus-Golub-0' Leary / 1 A / , GlowinskirPironneau / l A / , and their respective bibliographies. It would certainly appear that the algorithms discussed in this section, combined with the finite-element approximation methods (due to Falk-Mercier / l A / ) of Section 3, provide methods which are efficient and relatively easy to program for solving, via formulation ( 3 . 1 A ) , the problem of the elasto-plastic torsion of a cylindrical bar with simply-connected cross-section under the physical assumptions of Chapter 3, Section 1.2.

This Page Intentionally Left Blank

Appendix 4 FURTHER DISCUSSION OF BOUNDARY UNILATERAL I’ROBLEIG AND ELLIPTIC VARIATIONAL INEQUALITIES OF ORDER

4.

APPLICATION TO FLUID MECHANICS. 1.

SYNOPSIS

This appendix supplements Chapter 4 on boundary u n i l a t e r a l problems and on e l l i p t i c v a r i a t i o n a l i n e q u a l i t i e s of order 4. I n S e c t i o n 2 w e c o n s i d e r t h e approximation, u s i n g both conforming and non-conforming f i n i t e elements, of t h e u n i l a t e r a l problem

(1.2A)

K = { V E H1 ( a ) , v t g a.e. on

r}.

I n S e c t i o n 3 w e r e t u r n t o t h e method o f mixed f i n i t e elements for f o u r t h - o r d e r problems, which formed t h e s u b j e c t o f S e c t i o n 4.5 of Chapter 4; we supplement t h e approximation r e s u l t s and d e s c r i b e an a p p l i c a t i o n o f t h i s method t o t h e s o l u t i o n o f t h e problem o f the (fourth order) e l l i p t i c variational inequality

with

(1.4A)

2

K = {VEH~(Q), a

I AvIB

a.e. i n Q}.

I n S e c t i o n 4 we show t h a t t h e t e c h n i q u e s used f o r s o l v i n g fourth-order e l l i p t i c v a r i a t i o n a l i n e q u a l i t i e s f a c i l i t a t e t h e numerical t r e a t m e n t o f c e r t a i n problems i n f l u i d mechanics; as an example we have t a k e n t h e transonic p o t e n t i a l flow of a p e r f e c t

654

Unilateral problems and e Z l i p t i c inequalities

compressible f l u i d ;

(NP.

4)

f u r t h e r examples w i l l be given i n Appendix 6.

F i n a l l y i n Section 5 we give a supplementary bibliography r e l a t i n g t o t h e s u b j e c t s contained i n t h i s a?pendix, or t o c l o s e l y r e l a t e d subjects.

CONFORMING AND NON-CONFORMING APPROXIVATIONS OF THE BOUNDARY UNILATERAL PROBLEM ( l . l A ) , (1.2A).

2.

The numerical a n a l y s i s of problem ( l . l A ) ,

using

(1.2A)'l)

confomring f i n i t e eZements of order one and two w a s t h e s u b j e c t of Chapter 4, Section 3, i n which v a r i o u s i t e r a t i v e methods f o r solving t h e continuous problem and t h e corresponding approximate I n t h e following, w e supplement problems were a l s o i n v e s t i g a t e d . t h e above r e s u l t s by those of Brezzi-Hager-Raviart( 2, /1A/ , /2A/ regarding t h e finite-element approximation o f (1.lA), (1.2A) and i n p a r t i c u l a r t h e problem of o b t a i n i n g optimal order e r r o r e s t i mates f o r t h e approximation. I n Section 2 . 1 ( r e s p . 2 . 2 ) we cons i d e r t h e approximation of (1.1A), (1.2A) by first-order conforming ( r e s p . mixed-type non-conforming) f i n i t e elements. 2.1.

Approximat ion of t h e u n i l a t e r a l problem ( 1 . 1 A ) by f i r s t - o r d e r conforming f i n i t e elements.

,

(1.2A)

We suppose i n what follows t h a t R i s a bounded polygonal domain i n I R 2 and t h a t f E L P ( a ) ( w i t h p > l), g E C o ( r ) . 2.1.1.

The approximate problem.

We use t h e n o t a t i o n of Appendix 1, Section 3.6, namely

(2.1A)

ch

(2.2A)

Vh = { v , E C

= {P

E n ,

Chi,

P a vertex o f

T

E

(a),

v

Tech)

0 -

vhlTeP,

.

We then approximate t h e convex s e t K ( d e f i n e d i n (1.2A)) by

(2.3A)

(') (')

%=

vh,

vh(p) > g ( p )

With g = 0 i n (1.2A). Abbreviated h e r e a f t e r t o B.H.R.

VP

E

\ n TI

,

(SEC. 2 )

655

Conforming and non-conforming approximations

It is then c le a r t h a t P r o p o s i t i o n 2 . 1 : The approximate problem ( 2 . 3 A ) , (2.4A) admits one and only one solution. 2.1.2.

Convergence o f the approximate solutions. mates of optimal order.

Error e s t i -

A s r e g a r d s t h e convergence of t h e a2proximate s o l u t i o n s when h -+ 0 , we r e f e r t o Chapter 4, S e c t i o n 3.5 f o r t h e c a s e i n which t h e s o l u t i o n u i s not very r e g u l a r ( a n d g = 0 ) . I n t h e following we s h a l l e s t i m a t e t h e approximation e r r o r 1 using a y (Q) c e r t a i n number o f ( r e a s o n a b l e ) r e @ l a r i t y assumptions on u , g and t h e f r e e boundary ( i . e . t h e boundary between = {P E U(P) = g ( p ) l a d = {P E u ( ~> ) g(P)) )

II,-uII

r+

ro

.

r,

r,

I n o r d e r t o o b t a i n t h e s e e r r o r e s t i m a t e s we s h a l l make use of

Suppose that u

Lemma 2 . 1 :

(2.5A)

-h+u =

(2.6A)

-aU> O an -

aU

(2.7A)

Proof:

2 H ($2) ; then

f a.e. i n R,

a.e. on

(u-g)

E

r, r.

= 0 a.e. on

See Chapter 1, S e c t i o n 1.1.

Lemma 2 . 2 : Let u and (1.2A) and ( 2 . 3 A ) , (2.4A).

a(v,w) =

be the respective solutions of ( l . l A ) , We then have ( w i t h [Vv-Vw + vw] dx V v,w E H1 (Q) ) , v vh E%:

i,

a ( Y1-u 9 s - u )

(2.8A)

Proof:

'a (\-u

9

vh'u> +a(

3

"h-uh)

-

i,

f (Vh-\)

See t h a t of Lemma 3 . 2 of Appendix 1, S e c t i o n 3.6.

We t h e n have Theorem 2 . 1 : Suppose t h a t the solution u of ( l . l A ) , (1.2A) s a t i s f i e s u E W ~ , P ( Q )with p > 2, t h a t g E WI s o o ( r ) and t h a t the s e t I f the o f p o i n t s on t h e f r e e boundary ( i . e . 7 nF+) i s f i n i t e . angles of r h a r e bounded below by eo >OO, zndependent of h , we then have

(2.9A)

dx

.

656

VnilateraZ problems and eZZiptic inequazities

where

yn is

(MP.

4)

the soZution of the approzimate probZem (2.3A), (2.4A).

Proof: We follow B.H.R. / 1 A , ~emma~6.1/. From Green's formula and s i n c e u E W2,P(Q), we have , V v E H (n) ,

and hence

v

vh

EY,

(2.1 OA)

L e t IT^ be t h e V - i n t e r p o l a t i o n o p e r a t o r on ch (see Agpendix 1, h Section 3 . 6 ) ; w e have n cIR2 , and hence Wz,P(Q) c C l ( @ , with continuous i n j e c t i o n , i f p > 2; t h e conditions u E W2sP (n) and u Z g on r then imply t h a t 'rrhu E Kh W e t a k e vh = IT u i n ( 2 . 8 ~ ,) (2.10A) giving h

.

1

a(%-u,yl-u)

+

I

2

a(yl-u,rhu-u)

(-Au+u-f) (vhu-yl)dx +

n

If

(rhu-%)dI'

,

which i m - l i e s , i n view of (2.5A) .(2.1 1A)

a(yl-u, yl-u) 2 a(yl-u,rhu-u)

+

(rhu-%) dr

.

It can e a s i l y be deduced from (2.11A) t h a t

It r e s u l t s from C i a r l e t / l A / , p > 2 implies

under t h e assumptions on

f o r example, t h a t

f% s t a t e d

u

with EgSp(fl)

i n t h e theorem.

I n view of (2.12A), (2.13A) it w i l l s u f f i c e t o e s t a b l i s h

(SEC. 2) Conforming and non-conforming approximations

t o prove t h e e s t i m a t e

657

(2.9A).

be i t s Vh-interpolate Let i e C 0 ( n ) be a l i f t i n g of g and l e t IT h we denote by g t h e t r a c e of IT on r . This function % on Ch; i s i n f a c t t h e i n t e r p o p a t e ( ' ) o f g f r o b t h e values taken on r n ch We then have

.

(2.15A)

gh'%

On

which combined with

It then follows from

r,

(2.6~) implies

(2.7A), (2.16~) that

Let rOh ( r e s p . r ) be t h e s u b s e t of I' which i s t h e ucion of th e t h e s i d e s of t h e t r i a n g l e s of c h c o n t a i n e d i n To ( r e s p . r+). W denote by Sh t h e g e n e r i c element of t h e s e t of s i es o f t h e 9U t r i a n g l e s of %h l y i n g on r ; i f S h C r+h., t h e n - = O ( s e e an (2.7A)); i f s h c roh then u=g on Sh, whlch i m p h e s ~h = nhu on Sh. From t h e s e r e l a t i o n s w e deduce

i

(2.18A)

'h

an c(rhu--&)-(u-g)1 aU

dr = 0

V s h c roh

" r+h

'

I n view of t h e assumption on t h e f r e e boundary s t a t e d i n t h e theorem, we have

(2.19A)

I

rOhu

I'+his less than or equal t o the number of p o i n t s on the free boundary. the number of sides sh $

ro

u r + h t h e r e e x i s t s P E sh such t h a t u ( P ) = g(P). If sh # Since g and uyr belong t o W 1 * 0 3 ( r ) , we have (from Taylor's theorem and c e r t a i n standard i n t e r p o l a t i o n r e s u l t s f o r which w e r e f e r t o C i a r l e t /lA/) ( 1 ) piecewise l i n e a r on

r

65 8

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

Since u

edYp(,) with

p > 2 implies u

E

C'

(a)we

(APP.

4)

have

(2.228) It then follows from (2.20A)-(2.22A) and from

dr = O(h) t h a t

(2.23A)

The r e q u i r e d r e l a t i o n (2.14A) then r e s u l t s t r i v i a l l y from ( 2 . 1 7 A ) , (2.18A), ( 2 . 1 9 A ) , (2.23A).

Remark 2.1: I t i s shown i n B.H.R. /1A/ t h a t t h e e s t i m a t e (2.9A) a l s o holds i f R i s a convex domain ( p o s s i b l y not polygonal) and t h e t r i a n g u l a t i o n 0: and t h e approximate convex s e t Kh a r e h s u i t a b l y chosen. 2.2.

Approximation o f t h e boundary u n i l a t e r a l problem ( l . l A ) , (1.2A) using non-conforming f i n i t e elements of mixed t y p e .

I n t h i s s e c t i o n , which follows B.H.R. /2A/, we consider t h e approximation of t h e boundary u n i l a t e r a l problem ( 1 . 1 A ) , (1.2A) using a mixed finite-element method v i a a dual formulation of ( l . l A ) , (1.2A); we adopt t h e n o t a t i o n of Appendix 1, Section 4 and we assume t h a t g , ~ 1 / 2 ( r ) . 2.2.1.

A duaZ formulation o f the bcundary unilateral problem (l.lA),

(1.2A).

I t i s c l e a r t h a t problem ( l . l A ) ,

( 1 . 2 A ) i s equivalent t o

(SEC. 2 )

Conforming and non-conforming approximations

659

With ( 2 . 2 4 A ) , (2.25A) w e a s s o c i a t e t h e Lagrangian

(2.268) and w e c o n s i d e r t h e dual problem o f ( l . l A ) ¶ (1.2A) r e l a t i v e t o k, n me l y

where

It i s q u i t e e a s y t o p r o v e : P r o p o s i t i o n 2.2:

The dual problem (2.27A) i s equivalent t o

where

(2.308)

c

(2.31A)

H(div,n)

= {q

E

q'n 2 0 on

H(div,R),

= cq

2

E

(L

,

(n))N, v * q E L2 @ ) I .

Conversely ( 2 . 2 9 ~ )(2.30A) ~ a h i t s a unique s o l u t i o n p such t h a t p = V u where u i s the s o l u t i o n of the boundary u n i l a t e r a l problem ( l . l A ) ¶ ( 1 . 2 A ) . Remark 2 . 2 : Remark 4 . 1 o f Appendix 1, S e c t i o n 4.2 a l s o h o l d s f o r (2.29A) (2.30A). Remark 2 . 3 :

Let

We t h e n have t h e f o l l o w i n g e q u i v a l e n c e

q

E

H(div,R),

(2.338)

0 Vp E A

,

660

h i Z a t e r a l problems and e l l i p t i c i n e q u a l i t i e s

4)

(WP.

where <*,*> denotes t h e b i l i n e a r form of t h e d u a l i t y between H 1 i 2 ( I-) and H-1/2( r). The following saddle-point result can then e a s i l y be proved: Theorem 2 . 2 : Suppose t h a t the solution u of ( 1 . 1 A ) , ( 1 . 2 A ) belongs t o H 2 ( i 2 ) ; then l e t (2.34A)

I

=

k(q,p)

2 (lq12+(V*ql )dx + +

52

I,

f V*q dx

be the Lagrangian associated with ( 2 . 2 9 A ) , ( 2 . 3 0 A ) ; d &its unique saddle point (p,A} on H ( d i v , n ) x A such t h a t (2.35A)

p = Vu i n 51

(2.368)

X

= g-u on

Moreover Ip , A 1 (2.37A) (2.38A)

(2.39A) 2.2.2.

a

r.

i s characterised by

(p,X) EH(div,n)

XA

,

I,

[p*q+V*p V*q]dx + = - f V-q d x v q < 0 tf p E A

E

H(div,Q)

,

.

An approximation t o the dual problem (2.29A),(2.30A) using mixed f i n i t e elements.

20nce again we assume t h a t Cl i s a bounded polygonal domain i n and t h a t ?& i s a t r i a n g u l a t i o n of R. We t h e n approximate H ( d i v , n ) , H 1 I 2 ( r ) , A , C r e s p e c t i v e l y by

%=

{qh

H(div,Q) 9 l,q

T

'k+l "k+l

and

(2.40A)

V'qhlT

E

VTEPP,,

pk

qh*nl

e p k V S a s i d e of T E

ch}

(n i s a u n i t v e c t o r , normal t o S ) (2.41A)

I,,

= {%EL

2

(r), p h I S ~ p k Vscr, s

a side o f T E C ~ } ,

( SEC

-

2

(2.43A)

Conforming and non-conforming approximations

ch = (qh

'%,

qh*n ph dr s o vph €Ah)

661

-

We t h e n approximate t h e d u a l problem ( 2 . 2 9 A ) , (2.30A) by

Find ph < C h such that vqh eCh we have [ph. (qh'ph)

+v'ph

Qf g(qh7h)'n Jr

dr

v* (qh'ph)

-

1,

lax

fV* (qh-Ph)dx

9

which i s t h e d i s c r e t e analogue o f ( 2 . 2 9 A ) .

I t i s clear that (2.44A) admits one and only one solution. Let h be t h e Kuhn-Tucker v e c t o r a s s o c i a t e d w i t h (2.43A); t h e h approximate d u a l problem (2.44A) i s t h e n e q u i v a l e n t t o t h e v a r i a t i o n a l system

Find {Ph,+,)

+

( 2.45A)

1r

ph'"(ph

E H h

\r

XAh such t h a t

(Xh-g)qh*n dr =

-X h ) dr

0

1, -1,

vphEAh

[Ph '9 h + V*ph V.qh]dx f V*qh dx v q h

h

when h

%'

,

which i s t h e d i s c r e t e a n a l o g u e of (2.37A)-(2.39A), s o l u t i o n of (2.44A). Regarding t h e convergence of p /2~/!' r e s u l t s a r e p r o v e d i n B.H.R.

E

-+

ph b e i n g t h e

0, t h e following

Theorem 2 . 3 : Suppose that when h + 0 the angles o f are Then i f k = 0 ,h i f bounded below by Oo> 0, independent of h . f € H1(n) n L"(Q) and i f the s e t u E H'(Q) n w 1 '"(Q) , g E W1 '"(r), of points on the f r e e boundary ( i . e . To nT+ ) i s f i n i t e , we have

B.1I.R. /2A/ i n c l u d e s an i n v e s t i g a t i o n of t h e c a s e i n which fi F u r t h e r m o r e , Remarks 4 . 3 and 4 . 4 of Appendix i s not polygonal. 1, S e c t i o n 4.3 a l s o h o l d f o r ( 2 . 4 4 A ) , (2.45A).

Unilateral problems and e l l i p t i c inequalities

662

3.

(APP.

4)

FURTHER DISCUSSION ON THE APPROXIMATION OF FOURTH-ORDER VARIATIONAL PROBLEM USING MIXED FINITE-ELEMENT METHODS

3.1

Synopsis

The a i m of t h i s s e c t i o n i s t o supplement Section 4 o f Chapter 4 concerning t h e approximation of c e r t a i n fourth-order v a r i a t i o n a l problems. Section 3.2 gives some information on t h e convergence in of t h e mixed finite-element method of Chapter 4, Section 4.5; Section 3.3 w e b r i e f l y d e s c r i b e c e r t a i n r e s u l t s of GlowinskiPironneau /1A/ concerning t h e s o l u t i o n of t h e approximate biharmonic problem and which g e n e r a l i s e t h o s e o f Chapter 4 , S e c t i o n

4.6. F i n a l l y , i n Section 3.4, w e apply t h e above methods t o t h e s o l u t i o n of t h e fourth-order v a r i a t i o n a l i n e q u a l i t y problem The n o t a t i o n i s t h a t of Chapter 4. defined by (1.3A), (1.4A). 3.2

F u r t h e r d i s c u s s i o n of t h e converpence of t h e mixed finite-element method of Chapter 4, S e c t i o n 4.5.

Let R be a bounded domain i n I R 2 with r e g u l a r boundary; homogeneous D i r i c h l e t problem f o r A2 i s then defined by

1

U'A

the

= f i n R,

(3.1A)

au

o

on

r.

I n Chapter 4, S e c t i o n 4 . 5 w e described an approximation u s i n g mixed f i n i t e elements of order k, f o r which w e s t a t e d t h e convergence result

due t o Ciarlet-Raviart 131; t h e estimate (3.2A) supposes t h a t U E Hk+2(n) and makes various r e g u l a r i t y assumptions on t h e family ch)h , f o r which we refer t o Chapter 4 , Section 4.5. I n f a c t it results from t h e works of Scholz / l A / , /2A/, Rannacher / l A / , /2A/ and Gb-ault-Raviart /1A/ t h a t under t h e same and with s u i t a b l e (and reasonable) regulassumptions on a r i t y assumptions on u, we have 'd k ? 1 :

q),

(3.3A)

II \-uI I

€2(Q)

= O(hk-€)

(with

E

0 '

i f k22

Mixed finite-element methods

(SEC. 3 )

II Uh-4

(3.4A)

=

2 L (52)

with

0(hk+"')

663

if k 22

E: = O

,

Indeed, it i s even proved i n Scholz /3A/ t h a t f o r k l l we have ( l )

52

v'occ

(3.6A)

11 Ph-(-Au) 1 I

=

O(hk*).

We remark t h a t t h e e s t i m a t e s (3.3A), ( 3 . 4 A ) ,

(3.6A) a r e of

optimal o r quasi-optimal order. 3.3

Discussion supplementing Chapter 4 , Section 4.6 on t h e s o l u t i o n o f t h e approximate biharmonic problem (4.72)

3.3.1.

Review of t h e approximate bihamonic problem

I n Chapter spaces

4, S e c t i o n 4.5.2

we d e f i n e d t h e following d i s c r e t e

0 -

(3.7A)

Vh = {vh(vh E C (521, vhl

(3.8A)

voh

=

{vhlvh

E

vh,

vhlr =

E

VT

Pk

01

=

vhn

cCh), 1

H~(Q),

and a l s o M~

;

complement of Voh i n Vh,

(3.9A)

~ ~+ € vhlT =5 o V T E ' C ~ ,a T n r Ml-, i s defined uniquely by (3.9A).

We next consider

(1)

RoCC52

#Eocn

.

=

8

;

664

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

(UP.

4)

Problem (3.11A) admits one and only one s o l u t i o n . 3.3.2.

Solution of (3.11A) by reduction t o a variational problem i n 4 .

We have vh 3 voh @ M h ; l e t { Uhyph) be t h e Solution and l e t hh be t h e component of Ph i n % y i . e .

Of

The following theorem then r e s u l t s from Ciarlet-Glowinski

(3.11A)

12.1:

Theorem 3.1: Let {uh,ph) be t h e solution o f (3.11A) and l e e Ah be t h e component of ph i n {uhyph,hh) i s then t h e unique element o f Voh xVh x Mh such t h a t

s;

(3.14A)

1,

Vph*Vvhdx =

(3.15A)

\n

V\*Vvhdx

(3.1 6A)

n

a

vyl*vphdx

f vh dx

In

lzh%dx j$h+,

vvh

E

Voh, ph-Ah

vvh EVoh dx

\d \

3

Voh,

E

% EVoh

9

'%

z

L e t Ah€ M h a n d approximate D i r i c h l e

(3.17A)

I,

, $h

be t h e r e s p e c t i v e s o l u t i o n s of t h e problems

V%*Vvhdx

0

v vh E Voh ,

E

Vh, %-Ah

E

Voh

Mixed f i n i t e - e lernent met hods

(SEC. 3 )

V$h*Vvhdx = ln%vhdx

(3.18A)

We then define t h e b i l i n e a r form

665

vvh

y,

:

%xs

$h +

IR by

The following lemma i s proved i n Glowinski-Pironneau /lA/:

The b i l i n e a r form

Lemma 3.1:

definite.

%(*,*) i s symmetric and p o s i t i v e -

I

Now l e t uo and Poh be t h e s o l u t i o n s of t h e two agproximate D i r i c h l e t proklems

(3.20A)

(3.21A)

1

n

i,

VPoh'~hdx

I

VUoh'whdx

jKohvhdx

n

vh dx

vvh EVoh 9 poh EVoh 9

v vh EVoh,

Uoh EVoh ;

t h e fundamental r e s u l t concerning t h e s o l u t i o n of (3.11AXvia Mh, i s then Theorem 3.2: Let {uh,ph} be t h e solution of (3.11A) and i!et be t h e component o f ph zn %; Ah i s then the unique solution of t h e linear variational problem Find Ah \E such t h a t Ah

(3.22A)

%(\,%)

I:ohov%dx

-

]npoh'hdx

which i s equivalent t o a linear system with a symmetric positived e f i n i t e matrix. It can t h e n e a s i l y be shown t h a t algorithm (4.77)-(4.80) of Chapter 4, Section 4.6.1 i s a c t u a l l y a fixed-step gradient algori t h m a p p l i e d t o t h e s o l u t i o n of problem (3.22A); Glowinski-Pironneau /lA/ g i v e s ( l ) some algorithms f o r s o l v i n g (3.11A), v i a (3.22A), which a r e more e f f i c i e n t than t h e above algorithm, and i n p a r t i c u l a r some algorithms of t h e conjugate-gradient t y p e , which a r e no more complicated t o implement, each i t e r a t i o n r e q u i r i n g e s s e n t i a l l y t h e s o l u t i o n of two approximate D i r i c h l e t

( I ) See a l s o Vidrascu /1A/ for f u r t h e r d e t a i l s and numerous numerical experiments.

666

h i l a t e r a l problems mad e l l i p t i c i n e q u a l i t i e s

(APP.

4)

problems f o r - A .

3.4

Appiication t o t h e numerical s o l u t i o n of problem ( 1 . 3 A ) (1.h).

I n t h i s s e c t i o n we s h a l l consider t h e numerical s o l u t i o n of t h e problem of t h e ( f o u r t h o r d e r ) E l l i p t i c V a r i a t i o n a l I n e q u a l i t y (1.3A), ( 1 . 4 A ) . 3.4.1.

F o m l a t i o n of the continuous problem. results.

Regularity

L e t S2 be a bounded domain i n I R 2 with r e g u l a r boundary f E H - ~ ( Q ) = (HZ(L-2))' and l e t a,B E IR be such t h a t Q 5 6 .

r,

let

We t h e n consider t h e v a r i a t i o n a l problem:

Find u E K such that (3.23A) Au A(v-u) dx 2

~ V E K

where

(3.24A)

2 K = (vEH~(R)

,a

5

AvsB a.e. on SZl

and where <=,*> denotes t h e s t a n d a r d b i l i n e a r form of t h e d u a l i t y between H-2(S2) and Hg(S2).

Remark 3.1: (3.258)

We have ( c f . Green's formula)

Av dx =

1, $

dr = 0

tfv

E

H,(Q). 2

Hence it follows t h a t K = @ i f Q 0 o r 5 < 0; it a l s o follows from (3.25A) t h a t K = ( 0 ) i f Q = 0 o r 5 = 0. We s h a l l t h e r e f o r e assume h e r e a f t e r t h a t

(3.26A)

a
B.

Remark 3.2: There i s equivalence between (3.238) and t h e minimi s at ion problem

The mapping

v

+

d e f i n e s a norm i n Hg(S2) equivalent

t o t h a t induced by H2(S2); moreover i f K # @ t h e n K i s n e c e s s a r i l y From t h e s e p r o p e r t i e s it i s easy t o deduce: closed i n Hg(S2).

Mixed f i n i t e - e l e m e n t methods

(SEC. 3)

P r o p o s i t i o n 3.1: If K # and only one s o l u t i o n .

# ¶

667

then problem (3.23A) admits one

Regarding t h e r e g u l a r i t y o f t h e s o l u t i o n , it i s shown i n B r C z i s Stampacchia /2A/ ( s e e a l s o T o r e l l i /lA/) t h a t

We r e f e r t o Caffarelli-Friedman-Torelli /1A/ f o r a d e t a i l e d i n v e s t i g a t i o n o f t h e f r e e boundary a s s o c i a t e d w i t h problem (3.23A) n aQ+where (3.27A), i . e . t h e s e t

ano

Ro

= {x E Q ,

Av(x) = a o r Av(x) =

R+ = { x E R , a
61,

.

A d e n s i t y lemma.

3.4.2.

I n o r d e r t o i n v e s t i g a t e t h e convergence o f t h e f i n i t e - e l e m e n t approximations o f S e c t i o n 3.4.4 we s h a l l r e q u i r e

Lemma 3.2: (l)

domain;

Suppose t h a t a < 0 < fi and t h a t R i s a s t a r s h a p e d let = K n B(Q); we then have

x

X = K.

(3.29A) Proof:

We can always assume t h a t R i s a s t a r s h a p e d domain Let v r K and E >o. W e H2(IR2) by

with respect t o the origin ( 0 , O I . define

E

v(x) i f x e n (3.30A)

,

G(x) = O i f x

P Q.

We have (3.31A)

a SAG

We n e x t d e f i n e

(3.32A)

a.e. i n m 2 .

GE E H2(IR2)

GE(X) = G((l+E)x)

We n o t e t h a t (')

SB

by YX€lR

2

.

GE has compact support i n Q.

All convex domains a r e s t a r - s h a s e d .

668

Unilateral problems and e l l i p t i c inequalities

We next define v,

(3.33A)

=

VE

E H

2 (n)

0

1 -

(App.

4)

bY

..,

We then have

(3.348) (3.358)

v,

2 H0(W

E

9

= v strongly i n H ~ ( R )

l i m v, €+O

(3.368)

Av,

5

5

B a.e. i n n.

From (3.34A)-(3.36A) w e deduce t h a t t h e s e t

X,

has compact support i n

v

= {v E K ,

QI

i s dense i n K. Let v and l e t {pnIn be a r e p l a r i s i n g sequence ( s e e Remark 3.3 of Chapter 3 , S e c t i o n 3.3 f o r t h e d e f i n i t i o n of such sequences); we t h e n define a e H 2 ( l R 2 ) by (3.30A) and {G } n n' Gnc &(m2) t/n, by

(3.37A)

an

=

a*&.

We t h e n have

(3.38A)

lim

..,

v

c

=

2

2

strongly i n H (IR ) .

n++co

Moreover w e have

(3.39A)

A"", = G * P n

or

(3.40A)

AGn(x) =

I

,

A%y)P,(x-y)dy

V

XE

P2

We deduce from (3.4QA) and from u S we have

A; 56 a.e.

2 IR

.

that

VICE

2

Mixed finite-element methods

(SEC. 3 )

669

For n s u f f i c i e n t l y large w e have (3.42A)

SUPP(Gn)

=a

;

i f we d e f i n e v by vn = GQ1? , it follows from (3.38A), ( 3 . 4 1 A ) , (3.42%) t h a t " f o r n s u f f z c z e n t l y large we have

(3.44A)

lim v n++m

(3.45A)

OL

strongly i n H ~ , ( R )

= v

Av < B n

on R.

X1

I n view of t h e density of imply = K.

7

3.4.3.

i n K , r e l a t i o n s (3.43A)-(3.458)

Solution of (3.23A), (3.27A) by a d u a l i t y algorithm.

By analogy w i t h t h e v a r i o u s o b s t a c l e - t y p e problems encountered i n t h i s book it i s n a t u r a l t o a s s o c i a t e w i t h problem (3.23A), (3.27A) t h e Lagrangian : H 2 ( n ) X ( L 2 ( n ) ) 2 -+ IR d e f i n e d by

d(v,p) =

n

(3.46A) +

where

2 IAvl dx

jn

-

(Av-BIdx

IJ

+

j

p2(a-Av)dx

52

LI = ( p l , p 2 } .

We n e x t d e f i n e ( 3 .4 7 A )

2 L+@)

2

= {q E L

(a),

and

(3.48A)

.

2 2 A = L+(n) x L+@)

We t h e n have

q(x) 2 0 a.e.

i n R)

,

Unilateral problems and e l l i p t i c inequalities

670

(WP. 4 )

Theorem 3.3: Let {u,h} ~ H t ( 5 2 ) X A be a saddle point of d o n 2 H,(Q) x A ; we then have

(3.49A)

u i s the solution o f (3.23A) , (3.27A)

and

(3.518)

Proof:

I,

( a - h ) (p2-X2)dx

and

(3.548)

0

From t h e d e f i n i t i o n of a s a d d l e p o i n t , we have

R e l a t i o n s ( 3.50A) and imply

(3.53A)

5

a

I

5

n

Au

5

,

( 3 . 5 1 A ) a r e immediate consequences of ( 3.52A)

B a.e. on 52,

X1(AU-B)dx = 0

,

I

We put

it follows from (3.54A) t h a t

We have (see (3.52A), (3.56A))

X2(a-bu)dx = 0

n

(3.57A)

If v

671

Mixed f i n i t e - e Zement methods

(SEC. 3)

X I (Av-B)dx +

E

K we have ( s i n c e A

E

I,

X2(a-Av)dx

2 V V E Ho($2).

A)

and hence, i n view of (3.57A),

J(u) < J(v)

vv E K

a

which proves (3.49A).

I n view of Theorem 3.3, X = {Xl,X2} E A appears Remark 3.3: as a Kuhn-Tucker v e c t o r a s s o c i a t e d with problem (3.23A), (3.27A). Moreover it follows from (3.54A) and

(3.586)

A,(Au-B)

= X2(a-Au)

=

X

E t~h a t w e have

0 a.e. on Q.

Remark 3.4: The e x i s t e n c e of a m u l t i p l i e r X E A i s a t r i c k y t h i s would not be t,he case i L w e were t o seek a m u l t i problem; p l i e r a s s o c i a t e d with K i n (L ($2)): x (L (52)):

.

We r e f e r t o Remark 9.2 of Chapter 3, Section 9.1.4 f o r an analogous d i s c u s s i o n concerning t h e e l a s t o - p l a s t i c t o r s i o n problem. N a t u r a l l y t h e above d i f f i c u l t i e s vanish f o r t h e finite-dimensional v a r i a n t s of problem (3.23A) , (3.27A). I n view of Theorem 3.3 we can s o l v e problem (3.23A), (3.27A) u s i n g t h e d u a l i t y algorithms of Chapter 2 , S e c t i o n 4; i f we apply algorithm ( 4 . 1 2 ) , (4.13) of Chapter 2, Section 4.3 (with p n = p ) we a r e l e d t o

(3.59A)

Xo

=

{AoI ’ A’)2

then for n 20, An

E

A

E

A, given a r b i t r a r i l y ,

known, determine un and An”

from

672

(mp. 4 )

Unilateral problems and e l l i p t i c inequalities

(3.60A)

(3.61A)

We n o t e t h a t (3.60A) i s e q u i v a l e n t t o t h e s o l u t i o n i n H 2 ( Q ) of 0 t h e biharrnonic problem

(3.62A)

A

(3.63A)

u " = -a=~

~

=U f ~+ A(X~-X;)i n SZ,

an

"

0 on

r.

Concerning t h e convergence of (3.59A)-(3.61A) we have Theorem 3.4: Suppose that L admits a saddle point on 2 QXOE A and i f H,(Q)xA ; then

(3.644)

o
,

we have strongly i n H2~ ( Q )

(3.65A)

l i m u" = u n++=

(3.66A)

l i m . An = A* weakly i n L 2 ( Q ) n++ 00

x

L2(Q)

where u i s the solution of probZem (3.23A) (3.2'j'A) and where A * ( E A) i s such that (u,X*I i s a saddle point of 2 on H ~ ( Qx )A. 0

proof: L e t (u,XI be a saddle p o i n t of r e l a t i o n s ( 3.50A), ( 3.51A) imply

d

on

2

H0 (Q) x A ;

Mixed f i n i t e - e l e m e n t methods

(SEC. 3)

XI = CAI

673

p(AU--B)l+ ,

+

(3.67A)

X2

=

CX,

+ p(~r-Au)l+ ;

relation (3.52A) implies

(3.68A)

Au AV dx = < f ,v> +

(A2-Al)Av dx

2 v v E H,(Q) .

u"

1"

= An- A , = un-u ; since the mapping sup ( 0 , q ) is a contraction from L ~ ( Q )+ J~:(Q), then by subtracting (3.678) from (3.61A) (with 141 = IlqlI ) ve have L2 (Q)

We+put

q + q

=

IX1 -n+l I 2

5

I

[ A-n+l 2 2 s

I

]TI2+ 2p n 13l2- 2p I T

A;"dx A?dx

I ,

+ p 2 IAu -n 2 +

p 2 IAu - nI2

,

n and hence by addition

By subtracting (3.68A) from (3.60A) and putting v = deduce

]A;"/

=

I

n

(X;-XY)AGn

dx

in,we

,

which in combination with (3.69A) implies

If p ~ 1 0 , 1 [ the inequality (3.70A) clearly implies (3.65A); as regards (3.66A), this follows from Theorem 3.1 of Appendix 2, Section 3.

674

UniZateraZ problems and eZZiptic inequazities

3.4.4

(UP.

Approximation of t h e variationaZ probZem ( 1 . 3 A ) using a mixed finite-eZement method

4)

, (1.4A)

2

W e assume i n t h e following t h a t f E L ( a ) ; w i t h t h e notation of Chapter 4 , Section 4.3 ( l ) it can e a s i l y be shown t h a t t h e r e is equivalence between (1.3A) , ( 1 . 4 A ) and

where

7

(3.72A)

j(v,q) =

(3.73A)

x = {{v,q)

1n

(q12dx

E W,

-B

-

f v dx

n

.

q <-a a.e. on 0)( 2 )

Hereafter w e assume t h a t Q i s a polygonal domain i n IRL , and w e confine our a t t e n t i o n t o triangular f i n i t e elements o f order 2 ; i n view of t h e formulation (3.71A)-(3.73A) it i s n a t u r a l t o approxi m a t e problem ( 1 . 3 A ) , ( 1 . 4 A ) u s i n g t h e mixed finite-element method of Chapter 4 , Section 4.5, which i n t h e n o t a t i o n o f Chapter 4 , Section 4.5.2 and of Section 3.3.1 of t h i s appendix i s

where

with

(I) (*)

See ( 4 . 5 2 ) , i n p a r t i c u l a r , f o r t h e d e f i n i t i o n of W. We r e c a l l t h a t p = - Au.

(3.768)

a side of

675

Mixed finite-element methods

(SEC. 3)

Ch = {PIP e n , P a v e r t e x of T

TE'€'~I

.

E

ehor

the midpoint o f

It can e a s i l y b e shown t h a t i f a < 0 < 6 , the approximate

probZem admits one and only one solution.

3.4.5

Convergence o f the approximate solutions

We s h a l l assume i n t h e f o l l o w i n g t h a t Q i s a bounded, polygonal, I n o r d e r t o prove t h e convergence o f t h e approximate s o l u t i o n s { u h l p h ) t o t h e s o l u t i o n (u,-Au) o f (3.71A) when h + 0 , we s h a l l r e q u i r e t h e f o l l o w i n g lemmas:

convex domain i n IR2

.

Let ({vh,qh))hbe such t h a t

Lemma 3.3:

v h,

( 3 . 7 7A)

1vh

(3.78A)

l i m {vh,qhI = { v , q ) weakZy i n Ho(R)

9

qh) Xh

1

x

L2(Q),

h+O

eo

Then i f , when h + 0, the angles o f > 0 independent of h , we have

( 3 . 7 9A)

Proof: (3.80A)

{v,ql

E X

vhremain bounded below by

.

Since {vh,qh)

i,

V v h * V vhdx

E

=

Why

we have

lQqhvhdx

vvhEVh

*

L e t l~ E H 1 ( Q ) and l e t ph be t h e p r o j e c t i o n o f p on Vh i n t h e H 1 ( Q ) norm; we t h e n have

Under t h e assumptions on

lim h+O

IIY~-vII

h =

H1 (Q)

s t a t e d i n t h e lemma, we have

0.

676

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

(mp. 4)

From t h i s w e deduce i n t h e l i m i t i n (3.80A) t h a t

which s i n c e

(3.8 1A)

( v , q ) EH:(!J) (v,q)

E

XL2(Q)

implies t h a t

w.

We s h a l l now show t h a t define

{v,q)

E

%

.

Let $

E

Co(n) ,$ 2 0 ; we

where, i n (3.82A)

GT i s the centroid of the triangle T, and

xT

i s the c h m a c t e r i s t i c function of the triangle T.

We have (uniform continuity of 4 on

1)

0 .

Moreover we have

where, i n (3.848), t h e mi are t h e midpoints o f t h e s i d e s o f t h e t r i a n g l e T; from t h e d e k n i t i o n o f 8, ( s e e (3.75A) ) we have i n (3.8W

B+qh(miT) 2 0

tf i = 1 , 2 , 3 ,

VT

6

rh,b'{vh,qh)

EX^ ,

giving, since 4 t 0 ,

(3.85A)

(6+qh)sh$ dx

2

0

d ' $ E Co(n) ,$ 2

2 0.

We have ( 3 . 8 3 ~ )and l i m qh = q weakly i n L (Q), and hence h-to

677

Mixed finite-eZement methods

(SEC. 3 )

i n t h e l i m i t of ( 3 . 8 5 ~ )w e o b t a i n :

R

v 4 E Co(@,

(B+q) 4 dx 2 0

4 50 ,

which i s e q u i v a l e n t t o

-6

I q

a.e. on Q ;

s i m i l a r l y it can be shown t h a t q <-a a . e . on Q , g i v i n g {v,q)

E

X ( i n view of ( 3 . 8 1 ~ ) ) .

x = K n J(Q) d e f i n e x* =

Let

We

u

(3.86A)

-

X*

= K.

AX ; it i s c l e a r t h a t

O
*

If we d e f i n e X

x*

; it follows from Lemma 3.2 t h a t

=

X by

IIv,q)

E

w,

v

E X

*

1

then it follows from (3.86A) and t h e d e f i n i t i o n of W t h a t 4

K =

(3.87A)

L e t {v,q)

E

x.

%*; w e d e f i n e

{rhv, rhq)

E

Wh

by

We then have Lemma 3.4: Suppose t h a t the assumptions on of Lema 3.3 are s a t i s f i e d , along with condition (4i73) of Chapter 4, Section 4.5.4. Then we have V {v,q) E XI (3.89A)

{r v,r q) h h

E

l$, f o r h s u f f i c i e n t l y small,

1

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

678

(3.90A)

l i m {r v,rhql = {v,ql h h+O

strongly i n H 1 ( 0 ) 0

(APP. 4 )

.

XL2(S2)

Proof: Problem (3.88A) i s t h e s p e c i a l case o f problem (3.11A), (3.12A) which corresponds t o f = A2v, i . e . t h e approximate problem a s s o c i a t e d with t h e biharmonic problem (3.1A) when f = A2v; t h e s o l u t i o n of (3.1A) i s t r i v i a l l y equal t o v when f = A2v. We have {v,q) E giving

x*

(3.91 A) (3.928)

V E

&Q,

-B < q

= -Av

< -a

on a ;

it t h e n follows from Rannacher /2A/y and from (3.91A) , t h a t under s t a t e d i n t h e lemma we have ( w i t h C indept h e assumptions on endent o f v and h ) h

which proves (3.90A);

moreover we deduce from (3.93A) t h a t

and hence, i n view of (3.92A)

(3.94A)

-B < r q <-a h

for h s u f f i c i e n t l y small.

Property ( 3 . 8 9 ~ )t h e n r e s u l t s from (3.758)

, ( 3 . 8 8 ~ )and

(3.94A).

I n view of t h e two preceding lemmas, w e have

Theorem 3.5:

Lema 3.4 (3.95A)

With the same assuntptions on G)h as i n

we have l i m {\,ph)

= {u,-Au) strongly i n

1

Ho(n) xL2(52)

h+O

where i n (3.95A) { u h y p 1 i s the s o l u t i o n o f the approximate problem (3.74A) and u !!hat of problem (1.3A), ( 1 . 4 A ) . Proof: The proof follows t h e same g e n e r a l l i n e s a s t h a t o f Theorem 2.2 of Appendix 1, Section 2.3.

1)

679

Mixed finite-element methods

(SEC. 3 )

A p r i o r i estimates for

{\,ph}.

I n what f o l l o w s , C d e n o t e s v a r i o u s c o n s t a n t s ; since 0 < B w e have { O , O } c H h , which, on p u t t i n g {vh,qh) = { O , O } i n (3.74A) , g i v e s

CY

<

(3.96A)

1

1

lphI2dx

-

R

1

fy, dx10.

R

From (3.96A) w e deduce t h a t

From t h e d e f i n i t i o n o f W

h ( s e e (3.10A)) we have

(3.98A)

i,

phuhdx

\j ph

E

Vh.

which i n combination w i t h (3.97A) i m p l i e s

2)

bleak convergence

I n view o f ( 3 . 9 9 6 ) , (3.100A) w e can e x t r a c t from at } )t h ~ subsequence - a l s o denoted b y ' ( ~ ~ , p - ~such

(3.101A)

lim

u,, =

u*

weakZy i n H 1 o(Q)

h+O (3.102A)

l i m ph = p* h+O

S i n c e {%,ph}

Exh

vh

2

strongly i n L ( Q )

,

it f o l l o w s from Lemma 3 . 3

that

UniZateraZ probZems and eZtiptic inequalities

680

(3.103A)

{u*,p*)

E:

X

(UP. 4)

.

L e t {v,q} E: X*; it follows from Lemma 3.4, r e l a t i o n (3.89A), t h a t f o r h s u f f i c i e n t l y s m a l l w e have (rhv,rhq) e x h ; we then deduce from (3.72A) , (3.74A) t h a t f o r h s u f f i c i e n t l y s m a l l w e have v{v,q} E: X *

(3.104A)

I

lphI2dx

n

-

\nf\dxsT 1

I

(rhq12dx - Infrhv dx.

n

It t h e n r e s u l t s from (3.90A)2and from t h e (weak) lower semic o n t i n u i t y o f z + 211 2 : L (n) + IR t h a t

11

4

I n view o f H; ( 3.105A)

=

H

L

(a)

and t h e continuity of j ( , - ) w e deduce from

which combined w i t h (3.103A) shows t h a t {u*,p*} i s t h e s o l u t i o n o f ( 3 . 7 1 A ) , i . e . u* = u, p* = p = -Au where u i s t h e s o l u t i o n o f (1.3A) , (1.4A). The uniqueness of u implies t h a t t h e whole family {\,ph) + {u,p) and not simply an e x t r a c t e d subsequence. 3)

Strong convergence Since I\,ph1

E

Wh and {u,p}

It w a s shown e a r l i e r t h a t

E

W (with p = -Au), we have

Mixed finite-element methods

(SEC. 3)

(3.109A)

681

2

l i m p = p weukly i n L ( Q ); h h+O 1

from the compactness of the canonical injection from H ( Q ) into 0

L 2 ( Q ) we a l s o have

(3.1 IOA)

.

2 l i r n I+, = u strongly i n L ( Q ) h+O

We then deduce from (3.107A)-( 3.110A) that

I

lirn h+O

Q

lV%I2

dx = l i r n h+O

I

phI+,dx =

52

I,py dx =

which, in combination with the weak convergence of H;(Q) , implies

n

% to

lVul

2

dx

u in

It now remains to demonstrate the strong convergence of p to it follows from (3.74A) atd from (3.89A) that For h sufficiently small we have, \J {v,q) E K

p in L 2 ( n ) ;

It results from (3.90A) that, proceeding to the limit in (3.112A) we have, v{v,q} E X "

$ (3.1 13A) 5

I

2

1

lpl d x 5 T l i r n i n f n h + O

3

I

R

I

2

1

lphl d x 5 T l i r n sup n h + O

( q I 2 d x - 1,f (v-u)dx

2

52

lphl

dx

.

By density and continuity, (3.113A) a l s o holds we can thus put {v,q} = (u,p) in (3.l.13A) giving limIIPhlI 2 = llPll 2 h+O L L which combined with ( 3.109A) proves that

l i m 11 ph-p h+O

V{v,q)

11

E

X

= o L (52)

;

682

hilateral problems and elliptic inequalities

(A ~ P . 4)

and concludes t h e proof o f t h e theorem. In view of t h e proof o f Lemma 3.3, i f i n (3.75A) Remark 3.5: we merely imposed t h e c o n d i t i o n

-B ‘q (P) h

‘-a

a t t h e midpoints of t h e s i d e s of t h e T r e s u l t s of Theorem 3.5 would s t i l l apply.

E

ch

, the

convergence

I

Remark 3.6: An i n v e s t i g a t i o n o f o t h e r methods f o r approximati n g problem - ( 1 . 3 A ) , ( 1 . 4 A ) by f i n i t e elements can be found i n Glowinski-Marini-Vidrascu / 1 A / .

3.4.6

8

NwnericaZ solution of the approximate problem

I n o r d e r t o s o l v e t h e approximate problem (3.74A) we can u s e , a s w e do below, a f i n i t e - d i m e n s i o n a l v a r i a n t of algorithm (3.59A)(3.61A) of Section 3.4.3; f i r s t we define

where, i n ( 3 . 1 1 4 A ) , (note t h a t q E V +

h

(3.115A)

Ah =

h

C h i s a s defined i n S e c t i o n 3.4.4 by (3.76A) qh 2 0 on ?i s i n c e k = 2 ) . W e next d e f i n e

+ + vh X V h

9

and t h e n a new i n n e r product on Vh

, namely

-

( ,*),,

by

vyh,Zh E V h 9 where 5 i s t h e area of t h e domain c o n s i s t i n g o f t h e union of t h o s e t r i a n g l e s of Yh which have P as a common v e r t e x or as t h e midpoint of a common s i d e . Using t h e i n n e r product ( ,* l h , w e d e f i n e t h e symmetric positive-definite o p e r a t o r Sh: Vh -+ Vh, by ( ’ )

(l)

S i s i n f a c t an approximation t o t h e i d e n t i t y o p e r a t o r . h

683

Mixed finite-element methods

(SEC. 3 )

The d i s c r e t e analogue of a l g o r i t h m (3.59A)-( 3.61~)i s t h e n =

(3.1 16A)

then with An

h

E

A

h

{ ~ y ~ , ~ y ~given, } E\

n n k n o m , determine u,, E IToh, ph E

vh and

A:+'

E

\

from

( 3 . I 1 7A)

( 3 . 1 8A)

(3.1 9A)

Regarding t h e convergence o f a l g o r i t h m (3.116A)-(3.119A) can b e shown t h a t t h e r e e x i s t s a 6h > 0 such t h a t VA; E for a l l p s a t i s f y i n g

, it and

we have

where { y ~ , p ~is} t h e s o l u t i o n of ( 3 . 7 4 A ) . t h a t l i m bh = 1. h+O

It can a l s o be shown

684

UniZateraZ problems and elziptic inequalities

(Wp.

4)

Remark 3.7: The problem (3.117A) is simply an approximate mixed formulation of the biharmonic problem (3.60~); a formulation ivalent to (3.117A) is given by

For solving the approximate biharmonic problem (3.117A), (3.122A) each iteration. we refer to the discussion in Section 3.3 of this appendix and-to Chapter 4, Section 4.6; see also GlowinskiPironneau /lA/, Vidrascu /lA/, and Glowinski-Marini-Viarascu /lA/. 3.4.7

Application to the numerical solution of

a model problem

We shall now consider the particular case of the problem (1.3A), f = const., a = - 1, f3 = + 1. (1.h) in which ~ = ] O . , l [ X ] O , l [ , The triangulation used to define the approximation with finite elements of order tw is shown in Figure 1.10 of Chapter 4, Section 1.7, and consists of 128 triangles and 289 nodes; we thus have In implementing algorithm (3.116A)dim V = 289, dim Voh = 0225. (3.11$A), we have used Ah = { O , O ) ; furthermore, the optimum value for p , for various values of f, is close to 0.8. For f = 100, Figures 3.1-3.4 show the contours of pn corresponding to n = 0, 20, 40 and 240; the contours of % %e shown in Figure 3.5 (convergence is in fact virtually complete after 50 iterations).

5

Figures 3.6-3.10 show the contours of pn corresponding to n = 0, 60 and 180, for f = 1000; the conkours of % are indicated in Figure 3.11.

20, 40,

In the above figures the regions where ph ( = -Au) attains its limiting values are clearly visible; ph is equal to + 1 in the central region and - 1 in the four regions in contact with the edges. However, it should be noted that the contours of ph depend quite appreciably on especially near the free boundaries. h'

(SEC. 3 )

Mixed f i n i t e - e Zement methods

F i g . 3.1 (f = 100,

contours o f Po) h

Fig. 3.2 ( f = 100, contours o f p 1' h

686

Unilateral problems and elliptic inequalities

Fig. 3.3 (f = 100, contours of p 40) h

Fig. 3.4 (f = 100, contours

Of p

h

240

(APP.

4)

Mixed f i n i t e - e l e m e n t methods

(SEC. 3 )

\ F i g . 3.5 ( f = 100, contours o f u ) h

F i g . 3.6 ( f = 1000, contours of p:)

688

Unilateral probZems and eZZiptic inequazities

Fig. 3 . 7 (f = 1000, contours of p 20) h

F i g . 3.8 (f = 1000, contours of p ‘ 0 )

h

( U P . 4)

(SEC. 3)

Mixed finite-element methods

F i g . 3.9 (f = 1000, contours of p 6 0 ) h

F i g . 3.10 ( f = 1000, contours of p 180 h

Unilateral problems and e Z l i p t i c inequazities

690

(APP.

4)

/ F i g . 3.11 ( f = 1000, contours of

4.

y~)

NUMERICAL SIMULATION OF THE TRANSONIC POTENTIAL FLOW OF IDEAL COMPRESSIBLE FLUIDS USING VARIATIONAL INEQUALITY METHODS

4.1

Synopsis

A s we have a l r e a d y remarked i n Appendix 1, Section 6, t h e methodology of v a r i a t i o n a l i n e q u a l i t i e s can provide an e f f e c t i v e t o o l f o r t h e s o l u t i o n o f problems i n i t i a l l y formulated i n a more A f i r s t example has a l r e a d y been given i n t r a d i t i o n a l manner. Appendix 1, Section 6; i n t h e p r e s e n t s e c t i o n we s h a l l consider another example which is much more important from t h e a p p l i c a t i o n s point of view, but much more d i f f i c u l t mathematically, namely t h e numerical simulation of t h e .transonic potential f l o w of perfect compressible f l u i d s . Within t h e l i m i t e d scope o f t h i s appendix, it would be out of t h e question t o attempt t o provide a d e t a i l e d w e s h a l l t h e r e f o r e be content t o give account of t h i s s u b j e c t ; only an o u t l i n e of t h e material considered i n more depth i n BristeauGlowinski-Periaux-Perrier-Pironneau-Poirier ( ) /lA/, /2A/ (which a l s o contains a s u b s t a n t i a l bibliography on t h e s u b j e c t .

'

(1)

Abbreviated h e r e a f t e r t o B.G.4P.

4)

(SEC.

4.2 4.2.1

Transonic p o t e n t i a l f l o w

691

Mathematical formulation

The equation of c o n t i n u i t y

If R is the flow domain and r its boundary, it is shown in Landau-Lifchitz /1A/ that the flow satisfies (4.1A)

+

V*pu = 0 i n 1

with (4.2A) (4.3A)

+

u=V$,

where, in (4.1A)-(4.3A) a) b) c) d)

4.2.2

$ is the p is the

velocity potential d e n s i t y of the fluid

y is the ratio of specific heats ( y = 1.4 in air) &'is the c r i t i c a l speed of sound.

Boundary conditions

In the case of an a e r o f o i l such as B (see Figure 4.1), if we assume that the flow is uniform at infinity and tangential to the aerofoil surface r we have

B'

F i g . 4.1

692

UnilateraZ problems and elliptic inequalities

(4.4A)

(4.5A)

+

u =

+ U,

(APP. 4)

at infinity

san= O o n r B'

I n p r a c t i c e we bound R by p l a c e of (4.4A) w e use

roDas

shown i n Figure 4 . 1 and i n

It t u r n s out t h a t f o r t h e above c o n d i t i o n s t h e p o t e n t i a l 4 i s to determined only t o w i t h i n an a r b i t r a r y a d d i t i v e c o n s t a n t ; remedy t h i s s i t u a t i o n we can p r e s c r i b e a v a l u e of 4 a t some p o i n t i n R ; f o r example w e can use

(4.6A)

4.2.3

4

= 0 at the trailing edge

T.E. of B.

Lifting aerofoila and the Kutta-Joukowsky condition

The Kutta-Joukowsky c o n d i t i o n i s not p a r t i c u l a r t o t r a n s o n i c flows, as it a l s o a p p l i e s i n t h e flow of incompressible p e r f e c t f l u i d s and i n t h e subsonic flow o f compressible p e r f e c t f l u i d s . We r e f e r t o B.G. 4P /1A/ and Periaux /1A/ f o r t h e numerical treatment of t h e Kutta-Joukowsky c o n d i t i o n i n t h e c a s e of two-' and three-dimensional flows.

4.2.4

Shock-jwnp relations

The flow must s a t i s f y t h e Rankine-Hugoniot r e l a t i o n s a c r o s s a shock, namely

(4.7A)

(4.8A)

++

++

(pu*n)+ = (pu*n>-

+

(where n i s normaz t o t h e shock l i n e o r surface)

the tangential component of the ve tocity is continuous.

A s u i t a b l e v a r i a t i o n a l formulation of ( 4 .lA)-(4.3A) w i l l t a k e

(4.7A)-(4.8A)

i n t o account a u t o m a t i c a l l y .

(SEC. 4)

4.2.5

Transonic potential f l o w

693

Entropy condition

This can be formulated as follows (see Landau-Lifchitz /1A/ for more details):

There cannot be an increase i n the flow v e l o c i t y across (4.9A)

a shock as there would then be an associated decrease i n the entropy, which i s physically impossible.

The numerical implementation of (4.9A) will be discussed in Seetion 4.5.

Remark 4.1: Since the boundary between the subsonic region and the supersonic region is unknown, the problem to be solved i s in fact a free-boundary problem; the numerical treatment of this problem will take this into account. 4.3

Least-squares formulation of the continuous problem

In this section we shall not consider the practical implementation of (4.9A); we shall only discuss the variational formulation of (4.1A)-(4.5A), (4.7A), (4.8~~) and of an associated nonlinear least-squares formulation. 4.3.1

A variational formulation of the equation of continuity

We consider for simplicity the situation shown in Figure 4.2, representing a symmetric flow, subsonic a t i n f i n i t y , around a symmetric aerofoil; since the flow is symmetric, the KuttaJoukowsky condition is automaticalZy satisfied and the aerofo'il is non-li fting

.

F i g . 4.2.

694

Unilateral problems and e l l i p t i c i n e q u a l i t i e s

(UP.

4)

For p r a c t i c a l i t y ( a l t h o u g h o t h e r approaches a r e p o s s i b l e ) w e imbed t h e a e r o f o i l i n a " l a r g e " domain; u s i n g t h e n o t a t i o n o f S e c t i o n 4 . 2 , t h e c o n t i n u i t y e q u a t i o n and t h e boundary c o n d i t i o n s are

(4.1 On)

V*p($)V$

= 0 in R

with

and

(4.1 3A)

g=O on

rB,

+ +

g = prnuo3*nrn on

ra, .

We c l e a r l y have

(4.14A)

!:

p-=

g on

r

and

An e q u i v a l e n t v a r i a t i o n a l f o r m u l a t i o n o f (4.10A), (4.14A) i s

(4.15A)

The f u n c t i o n constant.

0

c a n b e d e t e r m i n e d o n l y t o w i t h i n an a r b i t r a r y

i s a n a t u r a l choice f o r 4 s i n c e Remark 4.2: The s p a c e dym(Q) physical f l o w s r e q u i r e (among o t h e r p r o p e r t i e s ) a p o s i t i v e density p; t h e r e f o r e from (4.11A) I$ h a s t o s a t i s f y

(SEC. 4)

Transonic p o t e n t i a l flow

695

(4.16A)

4.3.2

A least-squares formulation of (4.15A)

For a genuine transonic f z o w , problem (4.15A) i s n o t equivalent to a standard problem in the Calculus of Variations (as is the case for purely subsonic flows); to remedy this situation and - in some sense - "convexify" the problem under consideration, we introduce a nonlinear least-squares formulation of the transonic flow problem (4.15A), as follows:

Let X be a s e t of f e a s i b l e transonic flow s o l u t i o n s ; least-squares problem i s then (4.17A)

the

Min J(<>,

SEX

w i t-h (4.18A)

where, i n (4.18A) y(s) ( = y) i s a soZution o f the s t a t e equation Find y

E

HI (Q)

such t h a t

( 4 . 1 9A) p(c)V<*Vv dx

-

gv d r

v

E

.

H' (Q)

If the solutions of the transonic flow problem exist, then these are solutions of the least-squares problem and give the value zero to the objective function J.

4.4

Finite-element approximation and least-squares-conjugategradient solution of the approximate problems

We shall consider here only two-dimensional problems, although the following methods can also be (and have been) applied to threedimensional problems.

696

Unilateral problems and elliptic inequalities

4.4.1

(APP. 4)

Finite-element approximation of the nonlinear variat-.. ional equation ( 4 . 1 5 A )

We a g a i n c o n s i d e r t h e n o n - l i f t i n g s i t u a t i o n o f S e c t i o n 4 . 3 . 1 . The flow r e g i o n i s imbedded i n a l a r g e bounded domain R , which w e With a sta dard triapproximate by a p o l y g o n a l domain R we approximate H l h i R ) ( a n d i n f a c t W''p(R), angulation of R h' v p r l ) by

%

(4.20A)

€(, = {vhlvh

E

Co(Eh), vhlT E PI Y T E G,)

where, i n (4.20A), P i s t h e s p a c e o f polynomials i n two v a r i a b l e s 1 of d e g r e e I 1. We p r e s c r i b e t h e v a l u e z e r o f o r t h e p o t e n t i a l a t t h e t r a i l i n g edge; t h i s l e a d s t o

vh

(4.2 1A)

=

1 (

~

~

6

, %vh

(T.E.) = 0)

;

we c l e a r l y have 1 dim I#, = 1 + dim Vh = number of vertices of

(4.228)

rh.

We t h e n approximate t h e v a r i a t i o n a l e q u a t i o n ( 4 . 1 5 A ) by

Find @ 6 V h such that h (4.23A)

1

n

P(@h)V@h*Vvh dx

jnphvhdr

"hEvh

where, i n (4.23A)' gh i s a n a p p r o x i m a t i o n o f t h e f u n c t i o n g i n (4.13A). The above d i s c r e t e v a r i a t i o n a l f o r m u l a t i o n i m p l i e s t h a t = g i s a u t o m a t i c a l l y s a t i s f i e d "approximately".

1-an r

Nh

{wi)i=l be a v e c t o r b a s i s o f Vh; t h e n (4.23A) i s e q u i v a l e n t t o t h e nonlinear finite-dimensional system Let

=

(SEC.

4)

Transonic p o t e n t i a l flow

@ j = @(P.) v j = l , J set of the vertices of

where

...Nh

,

697

assuming that {F’j)?l

is the

different from T.E.

<

With the above choice of and V , there is no problem with Vvh, the numerical integration, since in ?4.23A), (4.24A), V$, Vwi (and therefore p ( +h) ) are piecewise constant.

4.4.2

Least-squares formulation of the discrete problem (4.23A).

Combining the results of Sections 4.3.2 and 4.4.1, we introduce the following least-squares formulation of the approximate problem (4.23A)

where, in (4.25A), and where

3 is the

set of feasible discrete solutions,

with yh ( = yh ( 5h ) ) the solution of the discrete variational state equation

Find y h e V h such t h a t (4 . 2 7A)

4.3.3

1,r

Vyh*Vvhdx =

-

p(~h)V~h*Vvhdx

ghvhdr

VV,

E

Vh.

Conjugate-gradient solution of the least-squares problem (4.25A)-(4.27A)

We follow B.G. 4P /lA/, /2A/ and Periaux /lA/; a preconditioned conjugate-gradient algorithm for’solving (4.25A)-(4.27A) - with % = Vh - is Step 0 :

(4.288)

Initialisation

@: 0

Evh

then compute gh from

given,

U n i l a t e r a l prob lems and e l Z i p t i c i n e q u a l i t i e s

698

(APP.

0

(4.29A) VgE*Vvh dx

=

<J'($h)0 ,vh>

~

V Vh~

,E

and s e t 0

(4.3 OA)

0

Z h = & .

Step 1: .Descent

Compute An = Arg min J ($"-Xz:)

(4.3 1 A)

XElR

h

h

,

(4.328) Step 2: Construction of the new descent direction n+l

Define gh

by

(4.338) '';gV then

h h *Vvhdx = <J'($"+')

,vh>

V V ~ EVh ,

4)

Transonic p o t e n t i a l flow

(SEC. 4)

699

n = n + l , go t o (4.31A). The two n o n - t r i v i a l s t e p s o f a l g o r i t h m s ( 4 . 2 8 ~ ) - 4.35A) ( are :

( i ) t h e s o l u t i o n o f t h e single-variable m i n i m i s a t i o n problem (4.31A); p a r t ( i ) o f Remark 4 . 5 i n Appendix 3, S e c t i o n 4.3.2 s t i l l h o l d s f o r ( 4 . 3 l A ) . I t should be noted t h a t each e v a l u a t i o n of J ( 5 ) , f o r a given argument ,C, r e q u i r e s t h e s o l u t i o k o? t h e l i n e a r approximate Neumann problem (4.27A) t o o b t a i n t h e corresponding y h . n+l n+l gh from , which r e q u i r e s t h e s o l u t i o n o f two l i n e a r approx'mate Neumann problems n+f and ( 4 . 3 3 A ) ) . ( namely ( 4.27A) with 5, - +h

( i t ) the calculation

Of

Calculation o f J ' ( S n

n

I n view o f t h e importance o f and g : h h s t e p n ( i i ) , we s h a l l now d e s c r i k e i n d e t a i l t h e c a l c u l a t i o n o f n J A ( E h ) and gh ( s u p p o s i n g f o r s i m p l i c i t y t h a t p 0 = 1 ) . We have by d i f f e r e n t i a t i o n

where 6yh i s , from ( 4 . 2 7 A ) , t h e s o l u t i o n o f

6yh€Vh (4.37A)

1

and V v € V h we have h

v6yh*Vvh dx =

s2

S i n c e p(ch) w e have

=

p,(l-K(Vch(

\

p(ch)V6ch*Vvhdx + R with K =

It f o l l o w s from ( 4 . 3 6 ~-() 4 . 3 8 ~ )t h a t

6p($)V~,*Vvhdx

1 y-l y+, - and

c*

a = I/(y-l),

.

Unilateral problems and e Z l i p t i c inequalities

700

(APE'.

4)

I I n f a c t <J'( ) T) > h k ' h functional

can be i d e n t i f i e d w i t h t h e l i n e a r

It i s q u i t e e a s y t o o b t a i n gn" h

from

n+l

, using

(4.33A), (4.40A).

Remark 4.3: An e f f i c i e n t d i s c r e t e Poisson s o l v e r w i l l form a fundamental p a r t of t h e method i f t h e above a l g o r i t h m (4.28A)(4.35A) i s used.

4.4.4

Numerical solution o f a t e s t problem

W e s h a l l now apply t h e above methods t o t h e numerical s i m u l a t i o n o f t h e flow around a d i s c , t h e f l o w b e i n g uniform and subsonic a t infinity. Owing t o t h e symmetry o f t h e flow, t h e Kutta-Joukowsky condition i s s a t i s f i e d automatically. If i s sufficiently small t h e flow i s p u r e l y subsonic and a v e r y good s o l u t i o n i s o b t a i n e d i n a v e r y small number o f i t e r a t i o n s of a l g o r i t h m (4.28A)(4.35A) (=: 5 i t e r a t i o n s ) . For g r e a t e r v a l u e s o f ~~~w~~a super-

llzwll

s o n i c pocket a p p e a r s , and i f t h e computed ( l ) Mach number d i s t r i b u t i o n on t h e s u r f a c e o f t h e body i s p l o t t e d , we o b t a i n t h e d i s t r i b u t i o n i n F i g u r e 4.3 showing a r a r e f a c t i o n (or expansion) shock ; t h e l a t t e r cannot e x i s t p h y s i c a l l y . The c o r r e c t Mach number d i s t r i b u t i o n i s shown i n Figure 4.4.

('1

The convergence i s s t i n very f a s t ( = 10 i t e r a t i o n s )

(SEC.

4)

Transonic potential fZow

Expans i o n shock

Physical

F i g . 4.3

Phys ica 1 shock

M <

< 1

F i g . 4.4

702

Unilateral problems and e l l i p t i c inequalities

(QP.

4)

From t h e above t e s t s it i s apparent t h a t w e have t o i n c o r p o r a t e i n our numerical procedure some device ( i n f a c t some d i s s i p a t i o n ) aimed a t suppressing t h e s e non-physical shocks which v i o l a t e t h e

entropy condition (4.9A). The numerical implementation of Section 4.5.

4.5

(4.9A) i s

discussed below i n

Numerical implementation o f t h e entropy c o n d i t i o n

4.5.1

General discussion.

Synopsis.

Several methods based on penalisation and/or a r t i f i c i a l v i s c o s i t y have been d i s c u s s e d and numerically t e s t e d i n B.G. 4P /lA/; w e introduce i n S e c t i o n 4.5.2 an i n t e r i o r penalty method which i s e f f e c t i v e i n suppressing r a r e f a c t i o n shocks and g i v e s a good approximation o f t h e s o l u t i o n i n t h e neighbourhood of p h y s i c a l shocks. If we were t o go back t o t h e example of S e c t i o n 4.4.4 and t r y t o r e p r e s e n t t h e corresponding l e a s t - s q u a r e s f u n c t i o n a l 5, + J (5 ) , it would+most probably look l i k e t h e graph i n Figure This f e e l i n g about 4.5 ( i v whe assume u,II s u f f i c i e n t l y - l a r g e ) . Figure 4.5 i s based on t h e f a c t t h a t 5 i s a very powerful a t t r a c t o r h f o r almost any i t e r a t i v e method, and i n p a r t i c u l a r f o r algorithm (4.28A)-(4.35A); i n c o n t r a s t t h e p h y s i c a l s o l u t i o n seems t o l a c k t h e s e a t t r a c t o r p r o p e r t i e s (except p o s s i b l y i n a very s m a l l neighbourhood of t h i s s o l u t i o n ) . Based on t h e s e o b s e r v a t i o n s , one p o s s i b l e i d e a c o n s i s t s of r e p l a c i n g t h e continuous curve i n Figure 4.5 by t h e discontinuous one, corresponding t o a f u n c t i o n a l -t J*(F 1, t a k i n g very l a r g e v a l u e s f o r - t h e non-physical s o l u t i o n h 5, and t a k i n g i t s minimal value c l o s e t o 5,.

11

4.5.2 4.5.2.1

An i n t e r i o r penalty method with truncation The p h y s i c a l motivation .......................

It i s known from Landau-Lifchitz

/1A/ t h a t

i n t h e case of a

weak shock w e have, following the streamlines,

(4.4 1 A)

@I

= O(Cvl3),

where i n ( 4 . 4 ~,) cg1 ( r e s p . [v] ) denotes t h e jwnp i n entropy ( r e s p . t h e jump i n V e l o c i t y ) a c r o s s t h e shock, with t h e s i g n o f opposite t o t h a t of [v] Now f o r a p h y s i c a l shock we must have

u]

.

(SEC. 4)

(4.42A)

Combining (4.43A)

Transonic p o t e n t i a l f l o w

703

C8120 ;

(4.41A), (4.42A) we shall use CVI

3

50

as entropy condition

.

T

‘h

‘h F i g . 4.5

‘h

704

Unilateral problems and elliptic inequalities

(APP.

4)

This choice is based on the fact that our finite element method gives us direct access to velocity variables.

Consider two adjacent triangles, like those shown in Fig. 4.6 (in fact the following method can be (and ha5 been) applied to three-dimensional problems); we denote by n the unit vector of the normal to their common side AB, directed from T into T 1 2' Since we are using linear elements, we have

(4.448) 0

Moreover since we are ysing C -conforming elements the component along AB of the velocity v = V + is continuous, unlike the normal h h component which is cleasly discontinuous; from these considerations the discontinuities of v are supported by the local normal components of the velocity. #,om these observations and in view of (4.44A) we introduce the functional % defined as ,follows

F i g . 4.6

(SEC.

4)

Transonic p o t e n t i a l f l o w

705

where, in (4.45A), the summation is carried out over the common sides of all the pairs of adjacent triangles in the triangulation 0 . being a coefficient which takes account of the l o c p size of the triangulation, and where the symbolic notation t is defined as follows:

ch

(4.46A)

t+ = max (0;t)

v t €IR .

Remark 4.4: We observe that (4.45A) is in fact very similar to the integral over Q of the sixth power of a truncated secondorder derivative; for homogeneity this suggests we take (4.478)

0. = 1

ci di

where, in (4.47A), d. is the length of the common side and where 1 a = -5 for one-dimensional problems, a = -4 for two-dimensional 8 problems and a = -3 for three-dimensional 'problems. Combining the above functional

%

and the approximate problem

(4.25A)-(4.27A) of Section 4.4.2, we obtain the following discrete problem, which incorporates a dissipative device aimed at SUPPressing non-physical shocks

where, in (4.48A), and where

3 is the

set of feasible discrete solutions

with r > 0 and y (=y (€, ) ) the solution of the discrete variath h ional state equakion

To solve (4.48A)-(4.50A) we can use a variant of the conjugate gradient algorithm (4.28A)-(4.35A) (with Jh replaced by Jrh).

706

u n i l a t e r a l problems and e l l i p t i c i n e q u a l i t i e s

(APP.

4)

The numerical results produced by the above methods are fairly good, the sonic transition from the subsonic region to the supersonic region being very well approximated, (non-physical "expansion shocks" having been suppressed); furthermore the computed compression shocks are located accurately and are very sharp; we refer to Section 4.6 and also to B.G. 4P /2A/ and Periaux /lA/, in which numerical results produced by the above methods are presented and discussed.

A practical and very important problem concerns the proper choice of r, since for a given aerofoil the optimal value of r , of seems to be a complicated and sensitive function of the angle of attack, etc.; we shall discuss in Section 4.5.2.3 a technique which we have recently discovered, which seems to be very effective in removing (or at least reducing) this sensitivity to the choice of r.

l~mll

Remark 4.5: We may supplement Remark 4.4 as follows: since F,(Eh) appears like the integral over R of the sixth power of a discrete truncated second-order derivative of Eh, we can, by differentiation, associate with E a discrete fourth-order nonlinear operator. Since % is a convex &nctional its differential is a monotone operator. From these properties the addition of r J may be interpreted as a non-linear fourth-order h

process.

Remark 4.6: We may justify the title of Section 4.5.2 by noting that the functional E which we have introduced to regularise our problem is in fact a ]Tmore complicated) variant of the i n t e r i o r penalty f u n c t i o n a l s discussed in Douglas-Dupont /2A/ and Wheeler

/1A/. Remark 4.7: We have performed computations using exponents other than 2 and 3 in (4.45A); these two exponents in fact appear to be the optimal combination, based on the quality of the computed solutions.

4.5.2.3

A---____--___ nonlinear12---_ weighted interior Eenaltx method ___-_------------- ---___-

From the comments made in Section 4.5.2.2 regarding the sensitivity to the choice of r in the definition of J - which is rh clearly related to the strong nonlinearity of our problem - it is very tempting to control of the regularisation process associated with 3+ ) 2 . To achieve this goal, we have introduced in the above functional I$, (defined in (4.45A)) a nonlinear weighting directly related to the local value of the

Transonic potentiaZ ' f z o w

(SEC. 4 ) density; by

more precisely, instead of using E

1 3 is

where, in ( 4 . 5 1 A ) ,

707 we use

% defined

a symbolic notation defined either by

Pi 1

(4.52A)1

1

1

1

T=T(K+z) Pi

or by

with

2 1 (4.538)

Pij

=(I-

where in (4.53A)

T. Ti$ being the adjacent triangles of si e o

th which have the i

the triangulation in common.

Remark 4 . 8 : Our computer experiments indicate that a "good" value for n in (4.51A) is n = 2. rn

7 5,

Replacing by we obtain a variant of the approximate problem ( 4 . 4 8 ~- ( 4 . 5 A ) which can be solved by the same type of iterative methods; computer experiments with Eh have shown that for a NACA 0012 aerofoil the same value of r was optimal (or nearly optimal) for the following conditions

708

Unilateral problems and elliptic inequalities

Angle o f a t t a c k (degrees)

(UP.

4)

Mm

0.6 0.78 0.8

6 1 0 0

0.85 Table 4.1

The c o r r e s p o n d i n g r e s u l t s a r e d e s c r i b e d i n S e c t i o n

4 5

-2 4 *

4.6.

An_jnt_er_i~r_Ee_nal_t_~-~e_t_h_o~_usjn_g_d~_nsit_~-i~~

The i n t e r i o r p e n a l t y method d i s c u s s e d i n S e c t i o n 4 . 5 . 2 . 3 two e f f e c t s :

( i ) It p e n a l i s e s e x p a n s i o n s h o c k s v i a

[vl

combines

+

( i i ) By v i r t u e o f t h e n o n l i n e a r w e i g h t i n g t h a t we have i n t r o d u c e d , t h e r e g u l a r i s a t i o n e f f e c t i s a m p l i f i e d i n r e g i o n s where t h e Mach number i s h i g h , s i n c e i n t h e s e r e g i o n s p i s " s m a l l " . It i s t h e r e f o r e n a t u r a l t o l o o k f o r a v a r i a n t of ( 4 . 5 1 A ) which combines t h e s e two e f f e c t s more c l o s e l y ; w e s t a r t by n o t i n g t h a t f o r a p h y s i c a l shock we must have ( s t i l l f o l l o w i n g t h e s t r e a m l i n e s )

T h i s jump c o n d i t i o n (4.54A) l e a d s t o t h e f o l l o w i n g v a r i a n t of t h e entropy f u n c t i o n a l s a n d % o f S e c t i o n s 4 . 5 . 2 . 2 and 4 . 5 . 2 . 3 ( t h e n o t a t i o n o f which h a s been r e t a i n e d )

%

(4.556) The n o t a t i o n i n (4.55A) i s s e l f - e x p l a n a t o r y , which i s d e f i n e d by

e x c e p t f o r w.

1

(SEC. 4)

Transonic p o t e n t i a l fZow

709

the indices j = 1,2 corggsponding to the two adjacent triangles side of the triangulation in common. which have the i of

ch

It is then-quite easy to formulate an approximate problem in is replaced by R ; moreover the same type of ods can be applie$ to this new approximate problem. Numerical experiments have yet to be carried out to check the validity of this new approach and also to determine an optimal choice for the two exponents 6 and n in (4.55A). 4.5.2.5

Further comments

The interior penalty methods with truncation that we have discussed in the above sections have been directly inspired by some of the methods used for the numerical treatment of variational inequalities (cf. Chapter 2, Section 3 and also Glowinski /2A/, In the present case the problem to be solved definitely /?,A/). lacks those monotonicity properties which are so useful in the theory and approximation of variational inequalities; however the least-squares formulation and a l s o the formulation of an entropy c b n d i t i o n as a set of inequality relations suggest very clearly a-methodology founded on techniques which have proved successful for simpler types of inequality problems.

4.6

Numerical experiments

In this section we shall present some of the numerical results obtained using the above methods,. The results of Section 4.6.1 relate to a NACA 0012 aerofoil, and those of Section 4.6.2 to the flow around a two-component aerofoil.

4.6.1

Simulation of flows around a NACA 0012 a e r o f o i l

As a first example we consider the flow around a NACA 0012 aerofoil at various angles of attack and various freestream Mach numbers. The corresponding pressure d i s t r i b u t i o n s are shown in Figures 4.7-4.11, in which the.isomach l i n e s i n t h e supersonic region are also shown. We observe that the physical shocks are quite well-defined and also that the computed transition from the subsonic region to the supersonic region is smooth and shock-free, implying that the entropy condition has been satisfied. The above numerical results lie very close to those obtained by various authors using finite difference methods (see particularly Jameson

/MI.

710

Unilateral problems and e l l i p t i c inequalities

(APP. 4 )

NACA 0012 Aerofoil M = 0.6 a = ' 6

CF

OI

A A

'

--

* - - - - - . . . , *

-L -L

Fig. 4.7

a

L I

A

iSEC. 4)

Transonic potential flow

711

NACA 0012 Aerofoil M, = 0.78

a = l

I J

F i g . 4.8

0

712

Uni Zateral problems and e 2 l i p t i c inequalities

( APP. 4)

CI NACA 0012 Aerofoil M m = 0.8 a =

Fig. 4.9

'0

(SEC.

4) Transonic potentiaZ flow

-i

rl

713

0 7

714

UnilateraZ probZems and eZZiptic inequazities

CI

I #

#

#

.

#

#

#

#

A

NACA ,0012 Aerofoil M, = 0.85 a =

00

F i g . 4.11

#

&

(APP.

4)

(SEC.

4)

715

Transonic potential fZow

Bi NACA 0012 M, = 0.6 Q = ' 6

CI

F i g . 4.12

hailateral problems and e l l i p t i c inequalities

716

4.6.2

(AFT. 4)

Flow around a two-component aerofoil

The two-component aerofoil investigated is shown on Figure 4.12; each component is a NACA 0012 aerofoil (the upper one being component No. 1). The pressure distribution and the isomach lines are shown on Figure 4.12. We observe that the region between the two aerofoils acts as a nozzle; we also observe two supersonic regions, one between the two aerofoils and one adjacent to the upper surface of body No. 1. 4.6.3

Concluding remark

It appears from the above numerical results that the method used leads to sharp shocks and to a smooth transition from the subsonic to the supersonic region.

5.

SUPPLEMENTARY BIBLIOGRAPHY

In Section 2 of this appendix we have already given various references relating to the approximation of second-order boundaryunilateral problems using the method of conforming or non-conforming finite elements; it is also appropriate to mention, amongst other references on this and related subjects: Hlavacek /2A/, /?A/, Haslinger /lA/, Hlavacek-Lovisek /lA/, Mosco /lA/, ScarpiniVivaldi /lA/, Kawohl /lA/, Johnson / 1 A / and also the monograph ( ) of Kikuchi-Oden /1A/ on the numerical analysis of contact problems. Regarding the numerical analysis of fourth-order variational inequalities, we cite Mercier /lA/, / 2 A / , Brezzi-Johnson-Mercier /lA/, Haslinger /2A/. The numerical simulation of transonic flows has given rise to a large number of works, and we shall therefore merely cite the references on this subject contained in B.G. 4P /lA/, /2A/, Jameson /lA/, Periaw /1A/ and Hunt /lA/.

( I ) The substantial bibliography of this work will also reward

investigation.

Appendix 5 FURTHER DISCUSSION OF T H E NUMERICAL ANALYSIS OF THE STEADY FLOW OF A BINGHAM FLUID IN A CYLINDRICAL DUCT 1.

SYNOPSIS

The aim of this Appendix is to supplement Chapter 5 on the Numerical Analysis of the ppoblem of the steady flow of a Bingham We shall give some supplementary fluid in a cylindrical duct. results on the finite-element approximation of this problem and on its iterative solution.

In Section 2 we shall return to approximation by finite elements of order one (i.e. piecewise affine), previously investigated in we shall show that under reasonable Chapter 5, Sections 3 and 5; assumptions we "almost" have 11 \-ull 1 = O(h) ( l ) . In Section 3, Ho (0) which follows Falk-Mercier flA/, it will be shown that the use of an equivalent formulation allows an approximation to be obtained which is of optimal order, even if SlclR? Finally, in Section 4, we give supplementary information on the iterative in particular we describe consolution of the above problems; jugate-gradient methods similar to those in Appendix 3, Section

.

4.3. To conclude this first section we recall (see Chapter 5, Section 1) that the problem under consideration is defined by

( 1 .IA)

where

I

Find u E H

1 0

(n)

such t h a t

a(u,v-u)+gj (v)-gj (u) 2 L(V-u) Vv

1

(1.2A)

j(v) =

(1.3A)

a(v,w) = PI vv*vw dx

(1.4A)

L(v) =

n

lVvl dx

E

I H,(Q

1

Vv~H~(fi), vv,w

n

1

v v eHo(n)

,f

E

1

Ho(n)

,

1 E H - ' ( ~ ) = (H,(n))'

,

(APP. 5 )

Numerical analysis of Binghum fluid flow

718

and g and p a r e p o s i t i v e constants.

2.

FINITE-ELEMENT APPROXIMATION OF PROBLEM ( 1 . 1 A ) . ESTIMATES FOR PIECEWISE-LINEAR APPROXIMATIONS

( I ) ERROR

I n t h i s s e c t i o n , which i n p a r t f o l l o w s Glowinski / 1 A l 3 /2A3 Chapter 2, S e c t i o n 6.71 and /3A/, w e s h a l l supplement t h e r e s u l t s o f Chapter 5, S e c t i o n 5 . 1 .

2.1

One-dimensional c a s e

We t a k e n = q O , l [ and assume t h a t i n ( 1 . 4 A ) Problem ( 1 . 1 A ) can t h e n b e w r i t t e n

Find

we have f

-du 1 dx Let N be an i n t e g e r >O and l e t h = N f o r i=O,l, N and

...

,xi]

L

2

such t h a t

u e H oI ( O , I )

ei =

E

,

i=l,2

dx

w e c o n s i d e r x. = i h 1

,... N.

1 We t h e n approximate H ( 0 , l ) by 0

(2.2A)

Voh =

{Vh

€C0CO,11, vh(o)=vh(l)=o

I

,

vh ei E P ~ ,i=1,2

,...N).

(l)

The approximate problem i s t h e n d e f i n e d by

P = s p a c e o f polynomials i n one v a r i a b l e o f d e g r e e I 1. 1

(a).

Finite-element approximation - I

(SEC. 2 )

Problem (2.3A) admits a unique s o l u t i o n . approximation error, w e have

719

Concerning t h e

Theorem 2.1: Let u and u be the respective solutions of If f E k 2 ( 0 , 1 ) then we have (2.1A) and ( 2 . 3 A ) .

Proof: We r e c a l l t h a t V c HL(0.I) t h e V o h - i n t e r p o l a t e of u, i .eoh1et

cC0[O,I 1 ; l e t

h

be

r UEV h oh (2.5A) rh'(xi)

= u(xi)

Vxi,

i-I ,2

,... N.

Proceeding as i n Chapter 5 , S e c t i o n 5 . 1 it can r e a d i l y be shown that

w e deduce from (2.6A) t h a t

a(yl-u,yl-u) la(%-u,rhu-u)+g(j OF

(rhu)-j (u))+a(u,r,u-u)

i n a more e x p l i c i t form ( a f t e r u s i n

Ho(O,l) and from t h e f a c t t h a t

Schwarz's i n e q u a l i t y i n

21x11y$Sx2+y2

vx,y E I R ) :

2 The c o n d i t i o n f E L (0,l) i m p l i e s ( s e e Chapter 5, S e c t i o n 2 . 1 and B r 6 z i s 161) t h a t

with i n f a c t

Nwne&caZ m Z y s i s of Bingham f Z u i d fZow

720

(APP. 5 )

(2.9A) t h e proof o f t h i s i s l e f t as an e x e r c i s e f o r t h e r e a d e r .

By i n t e g r a t i n g by p a r t s , we deduce from (2.7A)-(2.9A)

(2.1 OA)

I

I n view of (2.8A) we have

The e s t i m a t e (2.4A) w i l l t h u s f o l l o w from (2.10A)-(2.12A) i f we can succeed i n proving t h a t j ( rhu ) S j ( u ) Vh ; w e have

However, on

J X ~ - ~ , X ~we [

have

We deduce from (2.14A) t h a t , on

]xi-,,xi[

,

(SEC. 3 )

Finite-element approximation

-

I1

and hence by i n t e g r a t i n g t h e i n e q u a l i t y (2.15A)

which i s e x a c t l y t h e i n e q u a l i t y we were t r y i n g t o p r o v e .

2.2

Two-dimensional c a s e

2

If RclR , t h e n a somewhat more c o m p l i c a t e d argument (which among o t h e r t h i n g s makes u s e o f t h e c h a r a c t e r i s a t i o n ( 3 . 4 8 ) - ( 3 . 5 1 ) of C h a p t e r 1, S e c t i o n 3.4 and ( 7 . l ) , ( 7 . 2 ) o f Chapter 5, S e c t i o n 7.1) shows t h a t i f s u i t a b l e r e g u l a r i t y assumptions ( ’ ) . f o r u and f o r t h e f r e e boundary ( 2 ) a r e s a t i s f i e d , t h e n f o r t h e f i r s t - o r d e r f i n i t e - e l e m e n t a p p r o x i m a t i o n of C h a p t e r 5, S e c t i o n 3 , and u s i n g t h e same a s s u m p t i o n s on as i n Theorem 5 . 1 o f Chapter 5 , h S e c t i o n 5.1, we have

or “ a l m o s t ” O(h); we r e f e r t o Glowinski / l A / , d e r i v a t i o n of t h e e s t i m a t e ( 2 . 1 6 ~ ) .

3.

/2A/,

/3A/ f o r t h e

FINITE-ELEMENT APPROXIMATIONS OF PROBLEM (1.1A). (11) OPTIMAL-ORDER ERROR ESTIMATES THROUGH THE USE OF AN EQUIVALENT FORMULATION

In t h i s s e c t i o n we f o l l o w t h e a c c o u n t of Falk-Mercier / l A / ; w e show t h a t t h e u s e of an e q u i v a l e n t f o r m u l a t i o n e n a b l e s us t o o b t a i n f i n i t e - e l e m e n t a p p r o x i m a t i o n s which l e a d t o e r r o r e s t i m a t e s of optimal order. I n view of t h e a n a l o g i e s which e x i s t w i t h S e c t i o n 3 s a t i s f i e d i n p a r t i c u l a r i f s1 i s a d i s c and i f f = c o n s t .

( (

2,

i . e . t h e i n t e r f a c e between

R+ = (Xe52, lVu(x)I > O )

.

no

(X

en,

IVu(x)

I

= 0) and

Numerical analysis of Bingham f l u i d fZow

722

(APP. 5 )

of Appendix 3, many of t h e r e s u l t s w i l l be s t a t e d without p r o o f .

3.1

Equivalent f o r m u l a t i o n of problem ( 1 . 1 A )

The f o l l o w i n g e q u i v a l e n c e lemma forms t h e fundamental r e s u l t i t s proof i s a v a r i a t i o n o f t h a t of of t h i s t h i r d section; Lemma 3.1 of Appendix 3, S e c t i o n 3.1, t h e n o t a t i o n of which i s retained. Lemma 3.1: Suppose t h a t n i s a simply-connected bounded domain i n IR2 ; t h e r e i s then equivalence between problem ( 1 . 1 A ) and t h e v a r i a t i o n a l problem

where I$ = {1$1,~21i s an ( a r b i t r a r y ) soZution of

The s o l u t i o n s u o f (1.U) and p of ( 3 . 1 A ) are connected by t h e relation

Remark 3.1:

W e consider t h e vector

U = {O,O,u)

i n IR3 and

, "

!

m

a

a

--}

a

{ y * a x 29 ax3

; we t h e n have w

-,

6

VXU

- , , "

5

{-,au

ax2

I n view o f (3.4A) we can t h e r e f o r e r e g a r d p as b e i n g a vectorvalued v o r t i c i t y f u n c t i o n a s s o c i a t e d w i t h t h e v e l o c i t y f i e l d d e f i n e d by u.

Finite-e lement approximation - I1

(SEC. 3 )

723

Remark 3.2: Remark 3.2 o f Appendix 3, S e c t i o n 3.1 r e l a t i n g t o W t h e c o n s t r u c t i o n of C$ from f i s a l s o a p p l i c a b l e h e r e . The r e g u l a r i t y r e s u l t s r e l a t i n g t o problem ( 1 . 1 A ) Remark 3.3: (proved i n B r & z i s /6/ and r e f e r r e d t o i n Chapter 5, S e c t i o n 2.11, t o g e t h e r w i t h r e l a t i o n (3.4A), imply t h a t i f r i s s u f f i c i e n t l y r e g u l a r , t h e n f o r t h e s o l u t i o n p o f problem (3.1A) w e have

Remark 3.4:

This remark concerns t h e e x i s t e n c e o f Lagrange

m u l t i p l i e r s a s s o c i a t e d w i t h t h e c o n d i t i o n p E H f o r problem (3.1A); it f o l l o w s from (3.1A), (3. A) t h a t a n a t u r a l Lagrangian a s s o c i a t e d w i t h problem (3.1A) i s & : L2(n))2'XH1(s2) + IR , d e f i n e d by

which l e a d s t o t h e saddle-point problem

We t h e n have t h e f o l l o w i n g p r o p o s i t i o n , t h e proof of which i s an immediate v a r i a n t of t h a t of P r o p o s i t i o n 3.1 i n Appendix 3, S e c t i o n 3.1:

There i s equivaZence between (3.7A) and

P r o p o s i t i o n 3.1:

/

{pyx)

E

HXH'

(a) ,

moreover if {p,x1 i s a soZution of ( 3 . 7 A ) , (3.8A), then p is t h e s o l u t i o n o f problem (3.1A).

724

Numerical a n a l y s i s o f gingham f l u i d f l m

(UP. 5 )

I n c o n t r a s t t o problems (3.20A), (3.21A) of Appendix 3, S e c t i o n 3.1, which a r e formally similar, t h e e x i s t e n c e of x such t h a t (?,XI i s a s o l u t i o n of problems (3.7A) , (3.8A) poses no major d i f f i c u l t y . Hence w e have P r o p o s i t i o n 3.2: Using t h notation o f Proposition 3 . 1 above, we have the existence o f x E Hf ( Q ) , such that ( p y x } i s a solution of problems (3.7A) , ( 3 . 8 ~ ) .

Proof:

I f we assume t h e e x i s t e n c e of

of the dual problem

x,

then

x

i s a solution

the converse i s also true. We denote by p(w) t h e (unique) s o l u t i o n of t h e problem

it can e a s i l y be shown t h a t ( ' )

If w e d e f i n e

371 : H 1 ($2)

+

IR by

t h e n we may deduce from (3.6A), (3.9A)-(3.12A) t h a t t h e dual problem (3.9A) admits t h e formulation

(l)

w e use t h e n o t a t i o n

t

+

= sup(O,t),

v tcIR.

Finite-e lement approximation

(SEC. 3 )

-

I1

725

i s convex ( b u t not s t r i c t l y It f o l l o w s from (3.14A) t h a t convex) and continuous on H1(Q) ; moreover we have

(3.15A)

lim

Ilwll.~'

Mw)

= + m ,

H (52) 1 where, i n (3.15A), we have used t h e n o t a t i o n ;'(a) = H (n)/IR. imply t h e e x i s t e n c e o f s o l u t i o n s of t h e The above p r o p e r t i e s o f d u a l problem ( 3 . 9 A ) , t o w i t h i n an a d d i t i v e c o n s t a n t .

3.2

Approximation o f problem (3.1A) by a f i n i t e - e l e m e n t method

We assume h e r e a f t e r t h a t R i s a bounded, convex, polygonal domain i n IR2.

3.2.1

Definition of the approximate problem

Using t h e n o t a t i o n of Appendix 3 , S e c t i o n 3.1, we approximate (3.1A) by

I t can r e a d i l y be proved t h a t P r o p o s i t i o n 3.3:

3.2.2

Problem (3.16A) a h i t s a unique sozution.

Investigation of the convergence 2

We s h a l l assume h e r e a f t e r t h a t f E L ( a ) ; with Q convex and bounded, t h i s i m p l i e s ( s e e B r & z i s /6/)t h a t t h e s o l u t i o n u of ( 1 . 1 A ) s a t i s f i e s u E H2(R) w i t h

(3.17A)

IIuII

726

Nwnerical analysis of Bingham f l u i d fZow

(APP. 5 )

where y depends o n l y on Q. W e deduce from ( 3.&A), 0 t h e s o l u t i o n p of ( 3 . l A ) s a t i s f i e s

( 3.17A) t h a t

S i m i l a r l y i f $I i s o b t a i n e d from f u s i n g t h e r e l a t i o n s o f Remark 3 . 2 i n Appendix 3, S e c t i o n 3.1, t h e n

with

(3.20A)

1144

(n))2

11 11 L2 (a

We a l s o assume t h e e x i s t e n c e o f a m u l t i p l i e r t h a t { p y x ] s a t i s f i e s (3.7A) , (3.8A).

x

E

H

2

( a ) such

H e r e a f t e r we s h a l l use TI^ t o denote t h e o r t h o g o n a l p r o j e c t i o n o p e r a t o r onto L , i n t h e norm ( (L2(C2))2; we r e f e r t o P r o p o s i t i o n s 3.3 and 3.4 of fappendix 3, S e c t i o n 3.2.2 f o r t h e v a r i o u s p r o p e r t i e s o f TI , which a r e u s e f u l i n i n v e s t i g a t i n g t h e convergence o f t h e solukion p o f (3.16A) t o t h e s o l u t i o n p o f (3.1A). More s p e c i f i c a l l y , wikh r e g a r d t o t h i s convergence w e have Theorem 3 .I ( Falk-Mercier / l A / ) : Under the above assumptions on n , f ,x and with the same assumptions on r b as i n Theorem 3 . 1 of Appendix 3, Section 3 . 2 . 2 , we have the optzmal-order error estimate

where, i n (3.21A), p i s t h e s o l u t i o n o f problem (3.1A) and ph t h a t o f t h e approximate problem (3.16A).

Proof: Proceeding as i n t h e proof of Theorem 3 . 1 of Appendix 3, S e c t i o n 3.2.2, w e o b t a i n t h e e s t i m a t e

(SEC.

4)

727

Iterative methods

The theorem w i l l be proved i f

We have

,

but

where, i n (3.25A), m ( T ) = measure of T .

We t h u s deduce

The i n e q u a l i t y (3.23A) t h e n r e s u l t s t r i v i a l l y from (3.24A), (3.26~).

4.

SUPPLEMENTARY INFORMATION ON ITERATIVE METHODS FOR SOLVING THE PROBLEM OF THE STEADY FLOW OF A BINGHAM FLUID I N A CYLINDRICAL DUCT

4.1

General d i s c u s s i o n .

Synopsis

I n Chapter 5 w e d e s c r i b e d v a r i o u s i t e r a t i v e methods f o r s o l v i n g i n t h i s c a s e , t h e most problem ( 1 . 1 A ) i n i t s d i r e c t f o r m u l a t i o n ; e f f i c i e n t c a l c u l a t i o n methods a g a i n seem t o be t h o s e b a s e d on d g o r i t h m ( 9 . 6 ) - ( 9 . 8 ) of Chapter 5, S e c t i o n 9 ( o f t h e augmented f o r f u r t h e r d e t a i l s we r e f e r t o Marrocco /lA/, Lagrangian t y p e ) ;

hk.unerica1 analysis of B ~ h g h a mf l u i d fZm

728

(APP. 5 )

Gabay-Mercier /1/, Fortin-Glowinski /lA/, /2A/, Glowinski /2A/, Glowinski-Marrocco /lA/. Furthermore, it is extremely likely that we could improve the speed of convergence of algorithm (6.93)-(6.96) of Chapter 5 , Section 6.4 (which relates to the regularised problems (6.3)) by using a variant of the conjugate-gradient type (and thus with preconditioning in the sense of Appendix 3, Section 4.3.3) ; the algorithm used would then be similar to the algorithms (4.34A)(4.42A) of Appendix 3, Section 4.3.2. In Sections 4.2, 4.3 below, we shall conclude this fourth section with an investigation of the i t e r a t i v e solution of (1.lA) via the equivalent formulation (3.lA), using duality-type algorithms based on the formulation (3.7A); in Section 4.2 we describe a Uzawa-type algorithm, and variants of the conjugate-gradient type are described in Section 4.3.

4.2

Solution of problem (3.1A) by a duality method

The material discussed in this section is similar to that in Appendix 3, Section 4.2. 4.2.1

Description of the algorithm

In view of the duality results of Remark 3.4 of Section 3.1 of this appendix we shall reduce the solution of (3.1A) (and hence of (1.1A)) to that of the saddle-point problem (3.7A), namely Find { p , ~ ) E ( L 2 ( n ) I 2

XH'(Q)

such that

(4.1A) &,w)

Ap,x)

L(q,x)

v Cq,w)

E

( L 2 ( W 2 X H 1 (Q)

with

which is justified since p is then the soZution o f (3.1A). By analogy with Appendix 3,.Section 4.2.1, we are led to the following algorithm:

(SEC. 4)

I t e r a t i v e methods

then f o r n 20,

xn+' cH1(61)

xn E H1 (52)

729

known, determine p"

E

(L2(52))

2

and

by

w i t h , i n (4.5A),

P >

0.

The practical implementation of (4.3A)-(4.5A) does not pose any major difficulties; in fact (4.4A) is equivalent (see (3.11A) of Section 3.1) to (4.6A)

pn = p(x")

=

1 u (~f$+vx*~-g)+ 0.Vx" I4+vxnI

,

In connection with (4.5A), we refer to the comments in Appendix 3, Section 4.2.1. 4.2.2

Convergence of aZgorithm (4.3A)-( 4.5A)

By proceeding as in Appendix 3, Section 4.2.2, the following theorem on the convergence of algorithm (4.3A)-( 4.5A) can be proved.

meorem 4.1:

then if

Suppose t h a t (

,

0 )

(Q

i s defined by

730

Numerical a n a l y s i s of Bingham f l u i d flo w

n

(UP. 5 )

defined by algorithm (4.3A)-(4.5A)

then f o r the sequence {p we have

.

where, i n (4.8A), p i s t h e s o l u t i o n of (3.1~) Remarks 4.1, 4.2 and 4.3 of Appendix 3, Section 4.2.2 and 4.3.2 also hold for algorithm (4.3A)-( 4.5A)

.

4.3

On conjugate-gradient type variants of algorithm (4.3A)(4.5A).

As previously mentioned in Appendix 3, Section 4.3.1 in relation to a similar problem, we might expect to accelerate the convergence of algorithm (4.3A)-(4.5A) by using a variant of conjugate-gradient type. In fact this possibility is justified by 1

The f u n c t i o n a l r/l : H (52) + R defined by (3.14A) i s convex, L i p s c h i t z continuous and G a t e a u x - d i f f e r e n t i a b l e ; i t s Gateaux d i f f e r e n t ' i a l 711' i s a l s o L i p s c h i t z continuous and i s defined by Proposition 4.1:

(4.9A)



=

p(w)*Vq

n

1

dx Vw,q E H (52)

w i t h ( s e e (3.11A))

i n (4.9A) denotes t h e b i l i n e a r form of t h e d u a l i t y between (H'(i2))' and H'(i2). <*,a>

By proceedjpg as in Appendix 3, Section 4.3.2, we are led to the following conjugate-gradient algorithms:

(SEC.

4)

Stage 0 :

I t e r a t i v e methods

731

Initialisation

xo E H1 ( a ) ,

(4.11A)

then determine g

0

given,

using

and put zo = go.

(4.13A)

Next, for n n+l

x

n+l

,g

n+l

,z

S t a g e 1:

2 0

and assuming t h a t

xn ,g n ,z n

are known, determine

using

Descent

So Zve tha one-variable minimisation problem

Find XncR such t h a t (4.14A) ~((X"-X"z") 5

S t a g e 2:

Define g

nC'(x"-Az")

vx

E

IR

C o n s t r u c t i o n of t h e new d i r e G t i o n o f d e s c e n t n+l

by

gn+'

E

H'

(a) ,

(4.1 6A) (gn+l ,w)

=

H1(a)


,w>

vw

E

H1( a ) ,

732

Numerical analysis of Bingham f l u i d flow

(up. 5)

and then yn+' by

(or alternative Zy by

and finally

(4.18A)

n+* 'g n+l+Yn+l2

.

Naturally, the next step is

(4.19A)

n = n + l , go to ( 4 . 1 4 A ) .

Remarks 4 . 4 , 4.5 and 4.6 o f Appendix 3, S e c t i o n 4 . 3 . 2 a l s o a p p l y t o t h e above a l g o r i t h m ( b . l l A ) - ( b . l g A ) ; i n particular n + l gn" i s determined from x v i a ( 4 . 1 6 A ) and t h e f o l l o w i n g relations

where

Appendix 6 FURTHER DISCUSSION OF THE NUMERICAL ANALYSIS OF TIME-DEPENDENT VARIATIONAL INEQUALITIES

1.

SYNOPSIS

This appendix is intended to provide a fairly brief supplement to Chapter 6 on the numerical analysis of time-dependent inequalIn Section 2 we give a number of references concerning ities. the approximation of time-dependent inequalities, mainly in the modelling of problems related to the mechanics of continuous Sections 3 and 4 supplement the discussion in Chapter 6, media. Section 11 concerning the time-dependent flow of incompressible Section 3 considers the case of the flow in a Bingham f l u i d s . cylindrical duct, and thus forms an extension to Chapter 5 and In Section 4 we return to the numerical Appendix 5 of this book. investigation of Chapter 6, Section 11, concerning the flow i n a two-dimensional c a v i t y ; by introducing a stream function we are led to a time-dependent variational inequality problem of parabolic type, which is of fourth order with respect t o the space variables, and which we shall discretise (spatially) by a method of mixed f i n i t e elements similar to that discussed in Chapter 4, Section 4.5 and in Appendix 4, Section 3. 2.

SUPPLEMENTARY BIBLIOGRAPHY

The numerical analysis of problems which arise in general from the mechanics of continuous media and which can be reduced to time-dependent inequalities has given rise to a number of works, among which we cite the following: a) Johnson /2A/, Berger /lA/, Berger-Falk /la/ concerning the approximation in space and time of the model time-dependent obstacle problem, which in the notation of Chapter 6, Sections 1 and 2 can be written

I (2.1A)

Find u(t) E K a.e. on IO,T[, ,v-U)

+

I,

Vu*V(v-u)dx

2 (f

such that a.e. on lO,T[

,v-u)

t/v

E

K,

(UP. 6)

Time-dependent variational i n e q u a l i t i e s

734

where

(2.2A) where

K

=

1

( V E H0 (Q), vrJ, a.e. on

J, E H ' ( S ~ ) nCo(n)

with JI 5 0 on

n)

an.

We assume t h a t 52 i s a bounded polygonal domain i n IR2 and t h a t a s t a n d a r d family of t r i a n g u l a t i o n s of R ; we t h e n , i n t k e standard way, approximate H1(R) and K by means o f a f i r s t 0 order finite-element method, g i v i n g

(eh) i s

voh

(2.3A)

where C

=

IVh E c0(El,vh

=

o

i s t h e s e t of v e r t i c e s of

oh

on

5

an, vh IT

E

VT E eh>,

p1

interior t o R.

This l e a d s t o t h e approximation of ( 2 . 1 A ) by - amongst o t h e r schemes - t h e following implicit scheme (with A t > 0 )

/

with

\ where u

oh

<+I E

<

6%

k n m , determine

,

n=0,1,2

xn+'by

<

,... ,-

Voh i s an approximation of u

= Uoh

0.

It i s shown i n Berger-Falk /1A/ t h a t under s u i t a b l e (and reasonable) assumptions on uO,f,$,u and u oh- u0 we have

q)h,

with, i n ( 2 . 6 ~ ) , and

E

> O a r b i t r a r i l y s m a l l (we have

E

= 0 i f R c IR)

(SEC. 2)

where

wn

SuppZementary bibliography

735

is the characteristic function of the interval l0,TC n l ( n -

1 T)At,

(n+

1 T)AtC

.

It is also shown in Berger-Falk, loc. cit., that the above estimate (2.6~)also holds for more explicit variants of scheme (2.5A) , provided a stability condition of the type At I Ch2 is satisfied. b) P.L. Lions-B. Mercier /lA/, which deals with the investigation of the convergence of implicit schemes of altemzating direction type (A.D.I.), applicable to the numerical integration of a large number of parabolic time-dependent inequa Zities.

c) Baiocchi-Pozzi /1A/ which investigates the finite-difference approximation, in space and time, of a free-boundary problem with one space variable, after first reducing it to a parabolic variational inequality. This problem has its origin in bio-mathematics This paper derives (diffusion-absorption of owgen in tissues). estimates of the approximation error, together with results on the convergence of the free boundary in the approximate problem towards that in the continuous problem. Comincioli-Torelli /1A/ which investigates the finite-element approximation of parabolic time-dependent inequalities in the modelling of the unsteady flow due to the seepage of an incompressible fluid in a porous medium. d)

e) Johnson /3A/, /4A/, Laborde /lA/, Mercier /1A/ and SamuelssonFroier /1A/ which investigate the finite-element approximation of time-dependent inequalities modelling nonlinear phenomena in the mechanics of continuous media (elasto-plasticity, visco-plasticity, hardening, etc.. )

..

f) L6tstedt /lA/-/kA/ which investigates the numerical solution of variational inequalities which are of second order in t, in the modelling of dynamic friction and contact problems in the Mechanics of Structures. a We also refer to Glowinski /2A, Chapter 31 which gives a description of various time-discretisation schemes for parabolic inequalities ("quasi" explicit, implicit, Crank-Nicolson, multistep schemes, etc ). We conclude this section by mentioning the use of techniques for the numerical integration of timedependent inequalities for solving nonlinear vibration problems

...

736

Time-dependent variational inequalities

(APP.

6)

i n incompressible or inextensible media ( l ) ; w e r e f e r t o Bourgath a y - G l o w i n s k i /1A/ f o r an a p p l i c a t i o n i n t h e study of large-displacement v i b r a t i o n s i n inextensible elastic pipelines.

3.

ON THE STEADY FLOW OF A BINGHAM FLUID I N A CYLINDRICAL DUCT. ASYMPTOTIC PROPERTIES OF THE CONTINUOUS AND DISCRETE PROBLEMS.

Following Glowinski /2A, Chapter 3, S e c t i o n 6 / ( s e e a l s o B r i s t e a u /1A/ and Bristeau-Glowinski /U/) w e consider t h e timedependent problem a s s o c i a t e d with t h e e l l i p t i c v a r i a t i o n a l inequa l i t y of Chapter 5 and Appendix 5 , and study i t s asymptotic properties along with t h o s e of a r e l a t e d approximate problem.

3.1

Formulatioe of t h e problem. results.

Existence and uniqueness

L e t Q be a bounded domain i n lR2 w i t h a r e g u l a r boundary r ( Q i s t h e cross section of t h e d u c t ) . Using t h e n o t a t i o n o f Chapters 5 , 6 and Appendix 5 , w e consider

. V = Ho(Q), 1 . a(v,w) = .A

H = L2 (Q), V' = H-'(Q),

Vv-Vw dx

I,

v v , w 6 H o1( Q ) ,

time interval [O,T], 0 < T <+-

. f E L2 (O,T;V'),

,

uo E H ,

(see Chapter 5 , S e c t i o n 1 . 2 f o r t h e mechanical i n t e r p r e t a t i o n o f P,t3).

( I)

The c o n d i t i o n s f o r i n c o m p r e s s i b i l i t y or i n e x t e n s i b i l i t y are expressed as nonlinear equality-type constraints a t each p o i n t of t h e deformable medium under c o n s i d e r a t i o n .

Bingham f l u i d flow i n a cylindrical duct

(SEC. 3)

737

We t h e n have t h e following theorem:

For every uo E H and every parabolic variational inequalzty Theorem 3.1:

au ,v-u) (at

+ ua(u,v-u) + g j (v)

a. e. i n t

(3.1A)

U(X,O) = u

E

-

2 f E L (0,T;V')

g j (u) 2 (f,v-u)

, the Vv

E

lO,TC, p

has a unique solution u such t h a t I

u

(3.2A)

E L

2

aU

=EL

(0,T;V) nCo(CO,T1;H),

2

(0,T;V').

For a proof of t h i s theorem s e e Duvaut-Lions

3.2

/1, Chapter 6 / .

On t h e asymptotic behaviour of t h e continuous s o l u t i o n

We assume t h a t f i s independent of t and t h a t f consider t h e following steady-state problem Pa(u,v-u)

(3.3A) u

+ gj(v)

- gj(u)

2

I

f(v-u)dx

E

L2(n).

We

\dvEV,

R

EV

t h e numerical s o l u t i o n o f which w a s d i s c u s s e d i n Chapter 5 and Appendix 5 . It i s proved i n Duvaut-Lions /1, Chapter 6 / t h a t

where

v

(UP. 6 )

Time-dependent variational inequalities

738

We then have t h e following theorem: Theorem 3.2: Suppose that f E L ~ ( Q )with i f u i s the solution of (3.1A) w e have

IIf 1 L2 (52)

1

where Xo i s the smallest eigenvalue o f -A i n Ho(S2) 2

1.1

Proof:

We use Since f HA(Q)-norm.

E

< Bg; then

(Ao >o).

II*II

f o r t h e L (Q)-norm and for the Loo(IR+;L2(S2)) it follows from Theorem 3.1

t h a t t h e s o l u t i o n of (3.1A) i s defined on t h e whole of IR+.

I

We now observe t h a t i f gf3 > If , t h e n zero i s t h e unique it t h e n follows from Theorem 3 . 1 t h a t i f s o l u t i o n of (3.3A); u ( t o ) = 0 f o r some to 2 0, t h e n

(3.7A)

Vt

u(t) = 0

2 to

a

Taking v = 0 and v = 2 u i n (3.1A), we o b t a i n

(3.8A)

aU

2 dv Since V E L (O,T;V), - E L

result) t h a t t d dv -lv12.= 2 ( dt

(3.W

fu dx IS2

( z , u ) + va(u,u) + g j ( u ) =

+

a.e. i n t .

2

(0,T;V') imply ( t h i s being a general dt ( v ( t ) I 2 is absozuteZy continuous w i t h

, ~v) , we

T1Z ~dU I

2

o b t a i n from ( 3 . 8 A ) t h a t

+ pa(u,u)

+

gj(u) =

Since a ( v , v ) 2 h O ( v l 2 V V E V ,

and

fu d x s 1 5 2

I f I IuI

a.e. i n t.

j(v) 2 B ( v l V V E V (from

(3.5A)) it follows from (3.9A) t h a t

(3*1OA)

1 d

~ I u 2I I - I X ~ ~ U I 2 +

Suppose t h a t u ( t )

f

+

< g B - l f l > l u l 2 0 a.e. i n t c IR+.

0 V t 1 0 . then, since t

-+

lu(t)I2 is

QbsohteZy continuous with [ u ( t ) l 7 0, t h i s i m p l i e s t h a t t +- lu(t)I is a l s o absoZuteZy continuous. Therefore from (3.10A) and from I d

d

we o b t a i n :

(SEC. 3 )

(3.11A)

Bingham f Z u i d fZow i n a cyZindricaZ duct

d dt

l u ( t ) l + p X o l u ( t ) l + (gB-

739

lfl) S O a . e . i n t

E

33,.

It follows from (3.11A) t h a t

d dt

(3.12A)

lu(t)l+

Define y by

I

I u w gB-

2 -pho

f~

a.e. in t

; t h e n y > 0.

y =

E

IR,.

It follows t h e n by

i n t e g r a t i n g (3.12A) t h a t (3.13A)

lu(t>l

+

Y

5

( I u o l + y ) e -PAo t

Vt€m+

(3.13A) i s c l e a r l y absurd f o r t s u f f i c i e n t l y l a r g e . have u ( t ) = 0 i f

Y 2

;

I n f a c t we

(1 uol +v)e -vXot ,

i.e.

which completes t h e proof of t h e theorem. 2 Remark 3.1: Let f E L ( R ) , p o s s i b l y w i t h If\ 2 gf3, and deno t e by u , t h e s o l u t i o n of (3.3A); it can be t h e n e a s i l y proved that

where u ( t ) is t h e s o l u t i o n of (3.1A).

3.3

Approximation of (3.1A). discrete solution.

Asymptotic p r o p e r t i e s of t h e

W e s t i l l suppose t h a t f E L'(S-2); t o approximate (3.1A) w e proceed as follows: assuming t h a t R i s a polygonal domain, we approximate V = HA(R) by

vh

= Cvh

~cO(i;i),

vh =

o

on

r-, V h l T

E P ~

V T €51,

Time-dependent variationa 2 inequa 1iti e s

740

(APP. 6 )

i . e . by means of a first-order finite-element method. approximate (3.1A) by t h e following implicit scheme

I

with

<

n=0,1,2, We assume t h a t

u

oh

known, determine <+I

We then

by

... ; u,, = uoh .

E

0

Vh \dh

, and

that

(3.1 6A) r

We a l s o assume t h a t f i s approximated by

J,

fhvhdx

{fhIh such

that

can be computed e a s i l y , and t h a t

Using t h e methods of Chapter

6 , it i s p o s s i b l e t o prove t h e

convergence ( f o r some topology) of t h e discrete solution t o t h e continuous one, as h -+ 0, A t +. 0 , i f w e assume t h a t t h e usual angle condition h o l d s as h +. 0. Concerning t h e asymptotic p r o p e r t i e s o f t h e d i s c r e t e s o l u t i o n , we have t h e following d i s c r e t e v a r i a n t of Theorem 3.2: Theorem 3.3: Suppose that I f ] < Bg; i f (3.16A), (3.17A) hold and i f h i s s u f f i c i e n t l y small, we have

( 3 . I8A)

<

=

o

for n s u f f i c i e n t l y large,

(an estimate being given i n (3.26A) below). Proof:

n+l Taking vh = 0 and vh = 2% i n (3.15A)

, we

obtain

(SEC. 3)

Bingham fluid f l o w in a cylindrical duct

741

using Schwarz's inequality in L2(Q), it follows from (3.19A) that V n 2 0:

Since l i m Ifh - fl = 0 we have h+O

-

gB

(3.21A)

Ifh] > O for h sufficiently small.

It then follows from (3.20A), (3.21A) that m

(3.22A)

=

o

n implies tj, = 0 for n l m if h is

sufficiently smatl. Suppose that

(3.23A)

<

# 0 Vn 2 0 ;

1 <+I I -I
I

+UXol<+l

We define yh by yh = gf3- Ifh[j

(3.248)

then (3.20A) implies

+ gB

- Ifh!

then

yh > O for h sufficiently small,

l i m yh = y = gB- l f l . h+O It follows from (3.23A) that

which implies that

Vn10.

Time-dependen t variations l inequalities

742

(APP. 6 )

Since y > 0 f o r h s u f f i c i e n t l y s m a l l , (3.25A) i s imposzible More p r e c i s e l y , we w i l l have y n = o f o r n s u f fhi c i e n t l y Large. if

which implies

If h i s s u f f i c i e n t l y small, then

<

= 0

r

(3.26A)

Log( 1 +vXoAt) Relation (3.26A) makes t h e statement of Theorem 3.3 more n p r e c i s e . Moreover, i n terms of time, (3.26A) i m p l i e s t h a t yn i s equal t o zero i f

(3.27A)

nAt

We note t h a t

lim h+O 3

t+o

<

Hence t a k i n g t h e l i m i t i n (3.27A) we o b t a i n a f u r t h e r proof converges t o u i n some topology) of t h e estimate ( 3 . 6 A ) (since given i n t h e statement of Theorem 3.2.

<

Remark 3.2: Let be t h e s o l u t i o n of t h e steady-state d i s c r e t e problem a s s o c i a t e d with f h (and ( 3 . 1 5 A ) ) , p o s s i b l y with W e t h e n have Ifh/ 2 Bg. (3.28A)

The above estimate (3.28A) i s t h e d i s c r e t e analogue o f t h e estimate i n Remark 3.1.

(SEC. 4 )

3.4

Bingham fluid flow in a 2-D

cavity

743

F u r t h e r remarks

We can g e n e r a l i s e Theorem 3.1 t o t h e case of a Remark 3.3: Bingham flow i n a two-dimensional bounded c a v i t y .

Numerical examinations of t h e above asymptotic Remark 3.4: p r o p e r t i e s are performed i n Glowinski /2/ , B r i s t e a u /1A/ , BristeauGlowinski /1A/ and Begis 131, and show them t o be c o n s i s t e n t with see a l s o Begis /1A/ i n which the theoretical predictions; numerical simulations of two-dimensional Bingham flows i n a c a v i t y are performed u s i n g t h e method described i n Section 4 below.

4.

NUMERICAL SIMULATION OF THE FLOW OF A BINGHAM FLUID I N A TWODIMENSIONAL CAVITY, USING A STREAM FUNCTION METHOD

4.1

Synopsis

The o b j e c t of t h i s s e c t i o n , which i n p a r t follows Begis / l A / , i s t o supplement Chapter 6, S e c t i o n 11, on t h e numerical a n a l y s i s of t h e flow of Bingham f l u i d s . A f t e r f i r s t reviewing t h e i n i t i a l formulation of t h e problem ( i . e . t h a t of Chapter 6, Section 11) i n S e c t i o n 4.2, w e show i n Section 4 . 3 t h a t t h e i n t r o d u c t i o n of a stream function enables t h e problem t o be reduced t o a paraboZic variational inequality, which i s of fourth order with respect to

the space variables. I n Sections 4.4 and 4.5 we examine t h e approximation of steadystate and time-dependent problems using mixed finite-eZement methods ( f o r t h e spatial approximation) and finite differences ( f o r t h e time approximation). Next, it i s shown i n Section 4.6 t h a t t h e methods of decomposition-coordination using an augmented Lagrangian, mentioned i n Appendix 2, S e c t i o n 7 (see a l s o Chapter 3, S e c t i o n 10 and Chapter 5 , S e c t i o n 9 1 , allow t h e above approximate problems t o be solved i n a p a r t i c u l a r l y simple manner. F i n a l l y , i n Section 4.7 w e p r e s e n t some numerical results obtained u s i n g t h e above methods, which enable us t o demonstate c e r t a i n p r o p e r t i e s of t h e Bingham f l u i d flows under c o n s i d e r a t i o n . 4.2

Review of t h e v e l o c i t y formulation of t h e Bingham flow problem

L e t R be a bounded domain i n 1R2 , with r e g u l a r boundary r . 1 i s a f u n c t i o n with v a l u e s i n B2 , w e put ( u s i n g e s s e n t i a d ; $he same n o t a t i o n as i n Chapter 6, Section 11.1)

If v = {v v

(UP. 6 )

Time-dependent variational inequalities

744

(4.2A)

(4.3A)

a(v,w)

= 2

f

i,j=l

(4.6A)

,

2 2 H = { v l v (a) ~ ~ X Z (Q), V * V = 0, van =

xH'/2(r)

If i n a d d i t i o n z € H i / 2 ( r )

(4.7A)

D. .(v)D. .(w)dx 1J 1J

z-n dr = 0

o

on

r).

with ( l )

,

w e a s s o c i a t e with z t h e a f f i n e space (which i s nonempty, see Cattabriga / l A / ) :

Hereafter w e s h a l l n e g l e c t t h e nonlinear terms a s s o c i a t e d with t h e t r i l i n e a r form b ( . , * , = ) defined by (11.5) i n Chapter 6, SecWe a r e t h e n l e d (see Duvaut-Lions /1, Chapter 6 1 ) t i o n 11.l. t o modelling t h e flow of a Bingham f l u i d i n R , s a t i s f y i n g u = z on r , by

( ' ) n:

u n i t normal v e c t o r p o i n t i n g outwards from R .

(SEC.

Bingham f l u i d f l o w i n a 2-D cavity

4)

\

2

~ ( 0 )= u 0 c H, f E L (0,T;V') 0

We recall that i n

745

.

(4.9A)

viscosity of the f l u i d , g is t h e p l a s t i c i t y threshold, ( i . e . t h e yield s t r e s s ) , f i s t h e density of the extermal ,forces. v i s the

Duvaut-Lions /1 , Chapter 6 / proves ( f o r z = 0 ) t h e existence and uniqueness o f a s o l u t i o n of problem (4.9A); t h e case z # 0, with z s a t i s f y i n g (4.7A), may be t r e a t e d i n similar fashion.

Remark 4.1: I n t h e above w e have assumed t h a t z ( = ul,) does n o t depend on t: i n f a c t t h e r e i s no e x t r a d i f f i c u l t y i n handl i n g such a c a s e numerically, u s i n g t h e following methods.

4.3

A stream-function formulation of t h e Bingham flow problem

I n t h i s s e c t i o n w e make t h e following two simplifying assumptions

(i)

i s simply connected,

(ii) z = uIr = 0; however, Remark 4.1 above s t i l l holds f o r s i t u a t i o n s i n which ( i ) and/or (ii)a r e not s a t i s f i e d . I n view of our a t t e n t i o n t h e condition ( t o within an

(4.1 OA)

t h e f a c t t h a t i n t h i s appendix we are r e s t r i c t i n g t o two-dimensional flows, it i s n a t u r a l t o e l i m i n a t e V-u = 0 by i n t r o d u c i n g a sti-eam function, defined a d d i t i v e c o n s t a n t ) by

u1

=

a9

ax2

a9

I - -

u2

*

Time-dependent variationaZ i n e q u a l i t i e s

746

The c o n d i t i o n u = 0 on

(4.11A)

J, =

(4.12A)

-=

(4.I 3A)

Vo;

r,

r;

0 on

r

w e s h a l l t a k e JI = 0 on above). E

implies

constant on

an

Let v

r

(which f i x e s t h e constant mentioned

with v w e t h e n a s s o c i a t e @

-

a$

v 1 - ax2

(APP. 6)

’

v2 =

E

2 Ho(Q) by

a$ - ax, ’

S u b s t i t u t i n g t h e f u n c t i o n s (4.10A), (4.1%) reduces (4.9A) t o t h e following paraboZic variational inequality (of f o u r t h o r d e r with r e s p e c t t o t h e space v a r i a b l e s )

(4.15A)

where

(4.16A)

(l)

I f , i n ( 4 . 9 A ) , (f,v) =

f * v dx

then

f =

af2 - af, . ax,

ax2

Bingham f l u i d f l o w i n a 27D c a v i t y

(SEC. 4)

i

(4.17A)

."

Remark 4.2:

747

- q)2] 2 1 I 2d x

) 2 + ($

C(2

R

axlax2

ax

ax

2

1

In fact we have

(4.18A)

in the following we shall use (4.16A) and (4.18A) simultaneously.

4.4

Approximation of the steady-state problem

4.4.1

Synopsis.

FormuZation of t h e ste a d y -sta te probZem

With a view to investigating the approximation of problem (4.15A) by a mixed finite-element method - we shall first investigate the.approximation of the corresponding steady-state problem, i .e. the fourth-order elliptic variational inequality ( )

( 4 . 1 9A)

where a and j are defined by (4.16A), (4.17A); we remark that (4.19A) is equivalent to the minimisation problem

(4.20A)

where, in (4.20A)

(')

Hereafter we shall omit the symbol

-.

748

(APP. 6 )

Time-dependent variational inequalities

-1 H e r e a f t e r w e s h a l l assume t h a t f E H (Q); however, t h e r e i s no d i f f i c u l t y i n considering t h e case i n which

(f,$) =

I

V$EHt(n)

f A$dx

n

; fcL

2

(a).

2 Since t h e form a ( = , * )i s Ho(R)-eZZiptic ( i . e . coer i v e ) from (4.18A) , and s i n c e j ( ) i s continuous and convex on Ho(R), problem ( 4 . 1 9 A ) , (4.20A) admits a unique solution.

5

Approzimation o f (4.19A),(4.20A) b y a method o f mixed f i n i t e elements

4.2.2

I n t h e following w e s h a l l assume t h a t R i s a convex polygonal denote a standard family o f t r i a n g d domain i n IR2 ; l e t We s h a l l approximate (4.19A), (4.20A) by a method a t i o n s of R . of mixed f i n i t e etements, a v a r i a n t ( i n s p i r e d by Miyoshi / l A / ) of t h e mixed method considered i n Chapter 4 , S e c t i o n 4 and Appendix 4, Section 3, of which we r e t a i n most of t h e n o t a t i o n . We t h u s have 0 -

( a ) , vhlT

(4.22A)

Vh = {vh E C

(4.23A)

Voh = Vh n Ho(!J)

L e t ,$

2 Ho(R);

v T reh>,

.

1

E

E P ~

f o r 1 S i, j 5 2 we define n

(4.246) We t h e n have

3%

qij = qji =

v 1.1

E

ax;axj

H' ($2)

(4.2 5A)

Conversely i f ,$ E H1 o ( R ) and q = {qij]lSi5j52 s a t i s f y (4.25A), 2 t h e n 4 E Ho(R) and q and ,$ are r e l a t e d by (4.24A). This l e a d s t o t h e i n t r o d u c t i o n of t h e space Wh c Voh x (Vh) defined by

3

(SEC. 4 )

Bingham f l u i d flow i n a 27D cavity

749

L e t a,B E l O , l [ be such t h a t a + 8 = 1; i n view of ( 4 . 1 6 A ) , (4.18A) we "approximate" t h e f u n c t i o n a l J of (4.20A), (4.2lA) by

(4.27A)

where

(4.28A)

We t h e n approximate t h e problem (4.19A),

(4.20A) by

(4.29A)

4.4.3

Solvability of the approximate problem (4.298)

Regarding t h e s o l v a b i l i t y of t h e approximate problem (4.29A) we have

The approximate problem (4.29A) adnits one and Theorem 4.1: onZy one solution. Boof: The f u n c t i o n a l j i s convex and continuous; moreh over i f we p u t p = 0, i n t h e r e l a t i o n s which d e f i n e Wh (see h (4.26A)), we have ( 9 )

7 50

Time-dependent variationaZ inequaZities

(AFT. 6 )

where, i n (4.30A), C i s independent of {+h,qh) and h , and where

I n view of t h e above p r o p e r t i e s it i s s u f f i c i e n t , i n o r d e r t o prove t h e e x i s t e n c e of a s o l u t i o n , t o show t h a t

with Co > 0 ;

it may t h e n r e a d i l y b e v e r i f i e d t h a t

(4.32A)

which i m p l i e s (4.3I.A) ( w i t h Co = 2 Min ( a , B ) ) . The i t e r a t i v e s o l u t i o n of (4.29A) w i l l form t h e s u b j e c t of S e c t i o n 4.6 below.

4.4.4

Convergence of the approximate soZutions

W e s h a l l c o n s i d e r only t h e c a s e k = 2 ( s e e Remark 4 . 3 below r e g a r d i n g t h e convergence of t h e approximate for k # 2); s o l u t i o n s as h + 0, we have

Suppose t h a t when h -+ 0 the angles of c h remain Theorem 4.2: bounded below, uniformZy i n , h , by 8 > 0 ; suppose also that condition (4.73) of Chapter 4 , Secteon 4.5.4 i s s a t i s f i e d . We then have

(SEC.

gingham fZuid flozl, i n a 2-D cavity

4)

7 51

where { $ , p 1 i s the soZution of the approximate probZem (4.29A), J, t h a t tke continuous problem (4.19A) , (4.201, and where a2* P ‘pij}l
04

j This i s a v a r i a n t of t h e proof of Theorem 2.3 of Proof: Appendix 1, Section 2.3.2 and t h a t of Theorem 3.5 of Appendix Section 3.4.5, so t h a t w e have:

4,

A p r i o r i e s t i m a t e s f o r {J,,,p,)

I)

I n t h e following, C denotes v a r i o u s c o n s t a n t s . Since i n view of E Wh we can t a k e (0 ,q 1 =(O,O) i n (4.29A);

{O,O)

h

jh(qh)

2

0 Vq,

E

(Vh)3

h

and ( 4 . 3 1 A ) w e t h u s deduce

(4.348)

I n combination with (4.30A) , r e l a t i o n (4.34A) implies

( 4 . 3 6A)

2)

lphl < C

Vh.

Weak convergence

I n view of (4.35A), (4.36A) w e can e x t r a c t a subsequence from ( C ( ~ ~ , p-~ all s) o~ denoted by ({$h,phl)h - such t h a t 1

(4.376)

$h

+

JI*

weakZy i n Ho(Q)

(4.38A)

ph

+

p*

weakZy i n

(L2(Q>)3.

It can be shown, u s i n g a v a r i a n t of t h e proof of Lemma 3.3 of Appendix 4, Section 3.4.5, t h a t

752

(APP. 6 )

Time-dependent variationat inequazities

we proceed as i n t h e proof It remains t o prove t h a t $* = JI; l e t us suppose t h e n of Lemma 3 . 4 of Appendix 4 , S e c t i o n 3 . 4 . 5 : t h a t $ E B(n) j we define {q$,rhql E wh by s o l v i n g ( u s i n g t h e mixed method of Chapter 4 , S e c t i o n 4 ) t h e approximate biharmonic problem

1 {r,$

1h

(4.41A)

,uh}

wfh

E

/nluhl 2 dx

/

-

and v{v,,w,l A2$ rh$ dx s-y 1

n

E

I Iw,~

we have 2dx

/

-

n

A2+ vhdx

n

where

* wh

{{vh,Wh} EVoh

xvh,

1,

problem (4.41A) i s w e l l posed.

(UniqUeZy) determine rhq

on

E

(V,)

Vvh*Vvhdx

dx

vv,

E

;

The f u n c t i o n rh$ being known, we

3

such t h a t {rh$,rhq)

E

Wh

, from

It then follows from Rannacher /2A/ t h a t under t h e assumptions (Pr;l)h s t a t e d i n t h e theorem, we have

*

(4.43A)

rh$

(4.44A)

(rhq)i j

Taking {$,,qh}

strongZy i n Ho(52). 1

$

-

*

axiax

strongty i n L 2 (n)

,

v 1 < i l j s-2.

j

{rh$,rhq)

i n (4.29A) w e deduce from (4.3711)-

(4.40A) and from ( 4 . 4 3 A ) , ( 4 . 4 4 A ) , by usiiig t h e weak tower semicontinuity of convex continuous f u n c t i o n a l s , t h a t

2 From t e density o f B(n) i n Ho(Q) and t h e continuity of J(*) on Ho(Q), (4.45A) a l s o h o l d s E H ~ ( Q ) which proves t h a t

B

v$

0

(SEC.

4)

Bingham f l u i d flotl i n a 2-D cavity

$* = $, p i j = P i j

=-.

a2+ axiax j

753

hence w e have convergence of t h e

e n t i r e family (I$h,ph))h. 3)

Strong convergence

Without going i n t o d e t a i l s , it can be shown, by using a v a r i a n t of t h e proof of Theorem 2.3 of Appendix 1, Section 2.3.2 and of Theorem 3.5 of Appendix 4, S e c t i o n 3.4.5, t h a t (4.46A)

l i m $, = $ strongly i n

Ho(n) 1

h+O

strongly i n L 2 (Q)

(4.478)

h+O

1

J

(4.48A)

(4.49A)

Remark 4.3: I n t h e above we have assumed t h a t k = 2; i n f a c t , similar convergence r e s u l t s would be obtained f o r approximations based on f i n i t e elements of o r d e r k 2 3 but given t h e limited regularity of t h e s o l u t i o n s (JI 4 H4(Q) n H0(Q), h i n general) t h e use of such high-order elements i s not j u s t i f i e d . Regarding t h e case k = 1, it follows from Miyoshi /1A/ t h a t t h e above convergence results also hold i f q ) h i s generated by 3 f a m i l i e s of p a r a l l e l s t r a i g h t l i n e s ; i n f a c t it i s reasonable t o c o n j e c t u r e t h a t t h e e s t i m a t e s of Scholz /2A/, Girault-Raviart /1A/ ( e s t a b l i s h e d f o r t h e mixed finite-element method of Chapter 4, S e c t i o n 4 ) can be g e n e r a l i s e d t o t h e Hermann-Miyoshi scheme ( s e e Miyoshi / l A / ) ; i f t h i s were t h e c a s e , t h e n Theorem 4.2 would hold f o r k = 1. (I)

4.4.5

Approximation using numerical integration

On t h e p r a c t i c a l l e v e l , it i s necessary t o use a numerical integration process t o approximate t h e f u n c t i o n a l J ( - , a ) i n (4.27A), h (4.29A); we s h a l l examine only t h e case k = 1. L e t Ch be t h e set of v e r t i c e s of PP we approximate on Vh t h e i n n e r product h ’

-

(l)

Very r e c e n t r e s u l t s o f L. Reinhart show t h a t t h i s c o n j e c t u r e is c o r r e c t .

754

Time-dependent variationa2 inequaZities

induced by L2(Q), i.e. {Ahyph?

(4.50A)

(Ah’Uh)h

=

7

c

-+

I,

Ahvhdx

(APP. 6)

, by

rn(P)Xh(P)Uh(P)

ch where m(P) is the sum of the areas of the triangles which have In view of (4.50A) we shall actually use P as a common vertex. in (4.29A) the functional Jh( , ) defined by Jh (@h 9 qh) aV

=

2

BV

(ql lh+q22h’ql Ih+q22h)h

+

2 c(2q12h’2q12h)h

(4.51A)

where f is an approximation of f . h Similarly, in place of using Wh defined by (4.268), if k = 1 we shall use Wh defined by

Using the relations in (4.52A), it is easy to express the ( P ) , v P E Ch, explicitly in terms of the values taken by ohJ$; we refer to Begis /1A/ for further details. q.

.

4.5 4.5.1

JIh

Approximation of the time-dependent problem (4.15A)

Semi-discretisation in time

Let k = At(> 0) be a time step; we then approximate (4.15A) by the following implicit scheme (where JIn = $(nk) and in which the -.have been omitted)

(SEC.

Bingham f l u i d f l o w i n a 2-D cavity

4)

1

with Jln knowz, determine qn+'

755

from

Using the above semi-discrete scheme we have thus reduced the solution of (4.15A) to that of a sequence of elliptic variational inequalities, equivalent to the sequence of minimisation problems (for n 2 0 )

(4.548)

where

-

v$"*v6 dx JR The solution of (4.55A) by mixed finite elements will form the subject of Section 4.5.2. -(f(n+l)k),6)

4.5.2 $

0

FUZZ d i s c r e t i s a t i o n of (4.15A)

The notation is that of Section 4.4.2; we approximate JI0 by Jlh0 E Voh, and then the semi-discrete scheme (4.53A) by:

=

n+l n+l with the function JlE E V known, obtain {Jlh , ph I by s o h ing the following minimisatton problem for n = 0, 1..

..

Find {9;+l,ph n+ll e w h such t h a t (4.568)

where (with the notation of Section

4.4.2)

Time-dependent variational inequalities

756

(APP. 6 )

(4.57A)

It can e a s i l y be shown t h a t problem (4.56A) admits one and moreover t h e comments i n S e c t i o n 4.4.5 on only one s o l u t i o n ; t h e numerical i n t e g r a t i o n a l s o hold f o r problem ( 4 . 5 6 ~ . ) Using t h e techniques developed i n Chapter 6, and under s u i t a b l e r e g u l a r i t y assumptions on JI ,f and $, it i s p o s s i b l e t o prove t h e 0 convergence of {$El t o $ as h, k -t 0, provided t h a t t h e assumpts t a t e d i n Theorem 4.2 are s t i l l s a t i s f i e d . ions on

q)h

4.6 4.6.1

S o l u t i o n of ( 4 . 1 9 A ) , methods

(4.54A) by augmented Lagrangian

Synopsis

I n t h i s s e c t i o n we s h a l l show t h a t it i s p o s s i b l e t o s o l v e t h e steady-state problem (4.19A), o r t h e sequence o f problems (4.54A) (obtained by t h e semi-discretisation in time of t h e time-dependent problem (4.15A)), by aueplented Lagrangian methods which are v a r i a n t s We s h a l l of algorithm ( 9 . 6 ) - ( 9 . 8 ) of Chapter 5, S e c t i o n 9. r e s t r i c t our a t t e n t i o n t o t h e case which i s continuous with respect t o t h e space v a r i a b l e s , b u t t h e extension t o approximate problems i n space and t i m e does not p r e s e n t any p a r t i c u l a r d i f f i c u l t i e s ( a p a r t from t h e manipulation of a somewhat cumbersome formalism). 4.6.2

The model problem. Lagrangian

Introduction of an augmented

Problems (4.19A) and (4.54A) l e a d us t o consider t h e minimisa t i o n problem

Bingham f l u i d fZm i n a 2-D cavity

(SEC. 4 )

7 57

1 and y 2 0 ( y = 0 i n t h e s t e a d y - s t a t e case, y = k i f (4.58A) The e s s e n t i a l d i f f i c u l t y i n t h e solo r i g i n a t e s from ( 4 . 5 4 A ) ) . u t i o n of (4.58A) arises from t h e non-differentiubze f u n c t i o n a l

hence, t a k i n g (with q = {ql,q2))

it i s c l e a r t h a t (4.58A) i s equivalent t o

where (with I q ( x ) ( )-=

j($,q)

=

)

)W12dx

n

+

I A o I ~ ~ x +.gJ

2 n

\q\dx

52

-

(f,$)-

It i s t h e n n a t u r a l t o a s s o c i a t e with (4.60A) t h e augmented

7 58

Time-dependent variational i n e q u a l i t i e s

dr

H f ( a ) x (L2(Q))2 x problem ( 4 . 5 8 A ) and on

4.6.3

(L2(f2)I2,

(PP. 6)

t h e n J, i s a s o l u t i o n o f

An augmented-Lagrangian algorithm

I n view of S e c t i o n 4.6.2 it i s n a t u r a l t o s o l v e (4.58A) by seeking saddle p o i n t s of dr on H i ( a ) X ( L 2 ( n ) ) 2 X ( L 2 ( a ) ) 2 ; by analogy with Chapter 2, S e c t i o n 4 , we are l e d t o t h e following (Uzawa t y p e ) algorithm: (4.61 A)

10E ( L (a) ~

then, for n 20, A"

E

given

2

(L (52))

2

known, determine $n,pn,Xn+l using

(4.628)

(4.63A)

( 4 .64A)

1

I

=

P+' = 2

;A + p(2 A2n

+

--

a2@

p(2

ax2

a2P- n -2 p 2 ) . ax 1

It t h e n follows from Fortin-Glowinski /1A, Chapter 3 1 , Glowinski /2A, Chapter 5 / and Theorem 3.1 o f Appendix 2, Section 3 t h a t r e g a r d i n g t h e convergence o f (4.61A)-(4.64A) we have Theorem 4.3:

Suppose t h a t dr admits a saddle point {$,p,A)

on Iit(Q)x ( L ~ ( Q ) ) x~ ( L 2 ( Q I 2 ; (4.6%)

0 < P <2r

(4.66A)

l i m JI" n++-

=

l i m p"

rn

(4.67A)

n++"

then i f

2

strongly i n H,(Q) p

strongly i n (L 2 (Q))2

(SEC.

4)

(4.68A)

Bingham f l u i d f l o w i n a 2-.D cavity

lim n++m

An

=

X weakly i n

759

( L2 ( a ) )2 ,

i s such that {$,p,A} i s a saddle point of 2 Ho(Q) x (L2(Q))2 x ( L 2 ( Q ) ) 2 .

where

dr on

It i s c l e a r t h a t t h e e s s e n t i a l d i f f i c u l t y i n t h e s o l u t i o n of , lies i n view of t h e i n s o l v i n g problem (4.62A) a t each i t e r a t i o n ; s p e c i a l s t r u c t u r e of w?, , we can s o l v e t h i s problem by a blockoverrelaxation method o f t h e type examined i n Cea-Glowinski 1 2 1 ; we r e f e r t o Fortin-Glowinski /1A, Chapter 3 1 , Glowinski /2A, Chapter 5 / and Bourgat-Dumay-Glowinski /1A/ f o r f u r t h e r d e t a i l s on t h e s o l u t i o n of problems of t h e t y p e ( 4 . 6 2 ~ )and on t h e optimal choice of t h e parameter r; t h e optimal choice f o r p appears (on t h e b a s i s of a l a r g e number of numerical experiments) t o l i e very close t o p = r.

( 4 . 5 8 ~ ,) v i a t h e intermediary of algorithm ( 4 . 6 1 A ) - ( 4.64A)

4.6.4

On variants of algorithm ( 4 . 6 1 A ) - ( 4 . 6 4 A )

I f , when s o l v i n g problem (4.62A) by block r e l a x a t i o n , we perform on-ly a s i n g l e i n n e r i t e r a t i o n , we are l e d t o t h e following v a r i a n t of algorithm ( 4 . 6 ~ ) - ( 4 . 6 4 A ): (4.69A)

I$-'

then, for

n 20,

pn,$n,An+l

from

,XO)

EH:(Q)

,A"}

x ( L * ( Q ) ) ~ given

known, successively determine

(4.70A)

(4.71A)

(4.72A)

Determine An+'

using ( 4 . 6 3 )

, ( 4.64A)

.

Time-dependent variational inequalities

760

( APP

. 6)

The convergence r e s u l t s of Theorem 4 . 3 a l s o hold (see FortinGlowinski, Glowinski, l o c . c i t . ) i f i n p l a c e of (4.65A) we have

(4.738)

O < p 1 +Js < 2 r.

It w i l l be seen t h a t t h e s o l u t i o n of problems (4.70A) , (4.7l.A) does not pose any d i f f i c u l t i e s ; i n f a c t problem (4.71A) i s equivalent t o s o l v i n g t h e following problem ( o f l i n e a r biharmonic type ( l ) ) :

which adnits one and only one s o h t i o n . Regarding ( 4 . 7 0 A ) , it can e a s i l y be shown t h a t t h i s problem admits a unique s o l u t i o n , given e x p l i c i t l y by

(4.75A)

p"

I

where Xn = IXy,X;)

X" rr +I x"l

,

i s defined by

X nI

a2J;"-'

,

(4.76A)

n X1 =

(4.77A)

n a2p-l a2p'l X;=X 2 + r ( T - 2 ) * ax2

and where 1q1

- d x

+ 2r

ax, ax2

if' q = (q1*q2).

Once again it would appear t h a t t h e optimal choice f o r p i s close t o p = r ; t h e optimal choice f o r r , however, i s a more

('1

Relating t o t h e o p e r a t o r

-yA +

(V+T)A

2

Bingham fZuid flow in a 2-D cavity

(SEC. 4 )

761

complicated problem, s i n c e it i s apparent t h a t t h e optimal value f o r r i s a f u n c t i o n of - amongst o t h e r t h i n g s - v and g. To conclude t h i s s e c t i o n we should mention t h e v a r i a n t (due t o Gabay / l A / ) of a l g o r i i p (4.69A)-(4.72A) i n which, with (4.69A) , i s determined using (4.70A) unchanged, An

xn+If2

(4.78A)

=

A;

&E! - p;),

+ p ( 2 ax ax

1

1

2

and t h e n $n i s determined u s i n g

(4.80A)

n+1/2 in and f i n a l l An+1 i s obtained from (4.631, (4.64A) with h p l a c e of A We remark t h a t i n t h e above algorithm t h e v a r i a b l e s t h i s i s not t h e case i n algorithm L + and q play a symmetric r o l e ; (4.69A)-(4.72A) in which t h e o r d e r chosen i s governed by e l l i p t i c i t y c o n s i d e r a t i o n s ( f o r t h e d e t a i l s of which we r e f e r t o FortinGlowinski /1A, Chapter 3 / ) .

K

.

S e c t i o n 4.7 below p r e s e n t s t h e results of numerical experiments i n which t h e above algorithms ( a c t u a l l y t h e i r finite-dimensional v a r i a n t s ) have been a p p l i e d t o t h e s o l u t i o n of problems ( 4 . 1 5 A ) and (4.19A).

4.7

Numerical experiments

I n t h i s s e c t i o n w e s h a l l d e s c r i b e c e r t a i n numerical results of Begis / l A / , obtained by t h e methods described i n t h e previous t h e above r e f e r e n c e also g i v e s various comments and sections; o t h e r numerical experiments.

4.7.1

Formulation of the test probZem

Let = ]O,lCxlO,l[ flows i n 0 , f o r which

; we consider a family of Bingham f l u i d

(APP. 6 )

Time-dependent variational inequalities

762

(4.81A)

v=l ,

(4.828)

f=O

, = z = {z, * z 2 ] with z2 =

(4.838)

Zl(O,X2) = Zl(l,X2) = Z,(X1,1) =

0

0 Yx, E l 0 , l C

o

wx,

,

on E

r,

10,lC

,

z1(x1,0)= 1 vx, E 1 0 , l C .

These a r e t h u s problems of t h e sZiding wall t y p e ; that

z

E

r

z-n d r = 0, b u t a l s o t h a t

H S ( r ) x H S ( r ) with

4.7.2

s

1

< 2

-

€ 9

z f!H1’2(r)

xH1’2(I’)

we remark ( w e have

V € > O 1.

Numerical r e s u l t s

We r e f e r t o B6gis /1A/ f o r d e t a i l s of t h e a c t u a l implementation

of t h e approximation and i t e r a t i v e s o l u t i o n methods described i n t h e previous s e c t i o n s .

For various v a l u e s of g, Figures 4.1Steady-state cases: 4.5 show t h e r i g i d (hatched) and v i s c o - p l a s t i c ( b l a n k ) r e g i o n s , t o g e t h e r with t h e s t r e a m l i n e s . The growth of t h e r i g i d region with i n c r e a s i n g g can be observed. Time-dependent c a s e s : For various values of g , w e have examined t h e case i n which t h e medium f i l l i n g t h e c a v i t y n i s i n i t i a l l y a t rest (hence u(x,O) = 0 x $(x,O)= 0 V x ~ i l) * and i s

v

s e t i n motion by t h e s l i d i n g ( d e f i n e d by (4.83A)) of t h e lower w a l l of s2. When steady conditions have more o r less been a t t ained, we a r r e s t t h e motion of t h e lower w a l l i n o r d e r t o examine t h e r e t u r n t o t h e corresponding steady s t a t e , which i s u = 0 on

n

+$=o

onn.

For various v a l u e s of t h e behaviour of

Q

( i n c l u d i n g g = 0 ) , Figure 4.6 shows

(SEC. 4)

Bingham f l u i d f l o w i n a 2-D cavity

763

w e observe t h a t f o r g > 0 t h e r i g i d s t a t e i s attained a f t e r a t h i s i s i n agreement f i n i t e time which d e c r e a s e s as g i n c r e a s e s ; w i t h mechanical i n t u i t i o n .

F i g . 4.1 ( g = 0 . 5 )

Time-dependent variational inequalities

Fig. 4.2 (g = 1)

F i g . 4 .3 (g = 2.5)

(UP. 6)

(SEC.

4)

Bingham f l u i d f l o w i n a 2-D cavity

F i g . 4.4 (g = 5 )

Fig. 4 . 5 (g =

10)

765

766

Time-dependent variationaZ inequa Zities

II

0

t4

(UP. 6 )

c,

BIBLIOGRAPHY OF THE APPENDICES AXELSSON, 0. /1A/ A class of iterative methods for finite element equations. Comp. Meth. A p p l . Mech. E n g . , 9 (1976), 2, pp. 123-138. BAIOCCHI, C. /1A/ Sur un problsme 8. frontisre libre traduisant le filtrage de liquides travers des milieux poreux. Comptes Rendus Acad. Sc. Paris, 273A, (1971), pp. 1215-1217. /2A/ Free boundary problems in the theory of fluid flow through porous media. Proc. of the I n t . Congress of Math. (Vancouver 1974), 2, 1975, pp. 237-243. /3A/ Studio di un problema quazi-variazionale connesso a problemi di frontiera libera. Boll. U.M. I . 11, (1975), 4 (suppl. fasc. 3) pp. 589-613. BAIOCCHI, C., BREZZI, F., COMINCIOLI, V. /lA/ Free boundary problems in fluid flow through porous media. 2nd I n t . Symp. Finite Element Methods i n Flow Problems, Santa Margharita, 1976, pp. 407-420.

BAIOCCHI, c., CAPELO, A. /u/ Dissequazioni G'ariazionali e &uasiApplicazioni a problemi d i frontiera libera. variazionali Quaderni 4 e 7 dell' Unione Mathematica Italiana, Pitagor Editrice , Bologna 1978. BAIOCCHI, C., COMINCIOLI, V., VAGENES, E., POZZI, G.A. /1A/ Fluid flow through porous media; a new theoretical and numerical approach. Publicazioni 69,L.A.N. - C.N.R., Pavia (1974). BAIOCCHI, C., POZZI, G.A. /1A/ Error estimates and free-boundary convergence for a finite-difference discretisation of a parabolic variational inequality. Rev. Franc. Autom. Info. Rech. Gp. , Anal. Nwn. , 11, (1977), 4, pp. 315-340.

.

BARTELS, R., DANIEL, J . W . /lA/ A conjugate-gradient approach to nonlinear boundary-value problems in irregular regions; in

Conference on the Nwne,nicaZ Solution of Diffepential Equations , Dwzdee, 1973, G.A. Watson (ed.), Lecture Notes in Math., 363, Springer-Verlag, Berlin, 1974.

BEGIS, D. /1A/ Analyse Num6rique de 1'6coulement d'un fluide viscoplastique de Bingham par une m6thode de Lagrangien augment6. LABORIA Report 355, 1979. BENSOUSSAN, A., LIONS, J.L. /1A/ Nouvelle formulation de problsmes de contr6le impulsionnel et applications. Comp. Rend. Acad. Sc. P a r i s , 276A (19731,pp. 1189-1192. /2A/ Nouvelles mithodes en contr6le impulsionnel. A p p l . Math. & O p t . , 1 , (1975), 4, pp. 289-312. /3A/ Optimal impulse and continuous control: method of nonlinear quasi-variational inequalities. Trudy Mat. Inst. Steklov, 134 (1975),A.M.S. transl. pp. 7-25. /4A/ Contrale impulsionnel et ingquat-

768

Appendix ions quasi-variationnelles d'6volution.

S c i . P d s , 276A, (1973), pp. 1279-1284.

Compt. Rend. Acad.

BERGER, A.E. /1A/ The truncation method for the solution of a class of variational inequalities, Rev. Franc. A u t a . I n f . Rech. @. , Anal. N w ~ ., 10 (1976)3, pp. 29-42. BERGER, A.E., FALK, R.S. /1A/ An error estimate for the truncation method for the solution of parabolic obstacle variational inequalities. Math. of Comp. , 31 (1977), pp. 619-628. BOGOMOLNII, A., ESKIN, G., ZUCHOWISKII, S. /1A/ Numerical solution of the stamp problem. Comp. Meth. AppZ. Mech. Eng. , 15 (1978) 2, pp. 149-159. BOURGAT, J.F., DUMAY , J.M. , GLOWINSKI, R. /1A/ Large displacement calculations of flexible pipe lines by finite elements and nonlinear p r o g r d n g methods. SIAM J. Sc. and Stat. Computi n g , 1 , (1980), 1, pp. 34-81. BRENT, R. /1A/ Algorithms for ndnimisation without derivatives. Prentice Hall, Englewood Cliffs, N.J. , 1973. BRFZIS, H. /lA/ A new method in the study of subsonic flows. In Partial Differential Equations and Related Topics, J. Goldstein ( e d . ) , Lecture Notes in Math. , 446, Springer-Verlag, Berlin, 1975, pp. 50-60. BREZIS, H., STAMPACCHIA, G. /LA/ The hodograph method in fluid dynamics in the light of variational inequalities. Arch. Rat. Mech. Anal. 61, (19761, pp. 1-18. /2A/ Remarks on some fourth-order variational inequalities, Ann. Scu. Norm. Sup. Pisa, 4, (1977), 4, pp. 363-371. BREZZI, F., HAGER, W.W., RAVIART, P.A. /lA/ Error estimate for the finite-element solution of variational inequalities. (I) Primal theory; Nmerische Math. , 18, (1977), pp. 431-443. / 2 A / Error estimates for the finite element solution of variational inequalities. (11)Mixed Methods, Numerische Math. , 31 (1978), pp. 1-16. BREZZI, F., JOHNSON, C., MERCIER, B. /LA/ Analysis of a mixed finite-element method for elasto-plastic plates. Math. of Con@. , 31 (1977)140, pp. 809-817. BRISTEAU, M.O. /lA/ Application de l a msthode des dlgments f i n i s tr l a rSsolution d 'indquations VariationneZles d 'dvolution de type Bingham. Third-wcle Thesis, Universit6 Pierre et Marie Cufie, Paris, 1975. BRTSTEAU, M.O., GLOWINSKI, R. /LA/ Finite-element analysis of the unsteady flow of a viscous-plastic fluid in a cylindrical pipe; in Finite Element Methods i n F l o w Problems, J.T. Oden, O.C. Zienkiewicz, R.H. Gallagher, C. Taylor (eds.), Univ. of Alabama Press, Huntsville, Alabama, 1974, pp. 471-488.

Bibliography

.

.

769

.

BRISTEAU, M.0 , GLOWINSKI, R , PERIAUX, J , PERRIER , P . , PIRONNEAU, 0. /lA/ On the numerical solution of nonlinear problems in Fluid Dynamics by least-squares and finite-element methods. (I) Least-square formulations and conjugate-gradient solution of the continuous problems, Comp. Meth. AppZ. Mech. mg., 17/18 (1979), pp. 619-657. /2A/ Transonic flow simulations by finite elements and least-square methods; in Proc. of the Third C m f . on Finite Ezements i n Flow PPoblems, Calgary, Canada, June 1980. BRISTEAU, M.0., GLOWINSKI, R ., PEBIAUX, J . , PERRIER, P ., PIRONNEAU, O., POIRIER, G. /lA/ Application of Optimal Control and Finite Element Methods to the Calculation of Transonic Flows and Incompressible Viscous Flows; in Numerical Methods i n Applied Fluid Dynamics , B. Hunt (ed.) , Acad. Press, London, 1980 (and also LABORIA Report 294 (1978)). CAFFARELLI, L.A., FRIEDMAN, A., TORELLI, A. /lA/The free boundary for a fourth-order variational inequality. Pub. 227, Lub. d i Anal. N w n . del C.N.R. , Pavia, 1979. CAPRIZ, G. /lA/ Variational techniques for the analysis of a lubrication problem; in Mathematical Aspects of Finite Element Methods, I. Galligani, E. Magenes (Eds.), Lecture Notes in Math. , 606, Springer-Verlag, Berlin, 1977, pp 47-55. CHAN, T.F., GLOWINSKI, R. /1A/ Numerical methods for solving some mildly nonlinear elliptic partial differential equations. Stanford Report STAN-CS-674, Comp. Sciences Dept . , Stanford University, 1978. /2A/ Numerical methods for a class of mildly nonlinear elliptic equations. Atos do Decime Primeiro Coloquio BrasiZiero de Matematica, Vol. I, IMPA, Rio de Janeiro, 1978, pp. 279-318. CIARLET, P.G. /1A/ The f i n i t e element method for e l l i p t i c problems, North-Holland, Amsterdam, 1978.

.

CIARLET, P.G. RAVIART, P.A. /1A/ Maximum principle and uniform convergence for the finite element method, Comp. Meth. A p p l . Mech. Eng. , 2, (19731, pp. 17-31. CIAVALDINI, J.F., POGU, M., TOURNEMINE, G. /1A/ Une m6thode variationnelle non lin6aire pour l'6tude dans le plan physique d'6coulements compressibles subcritiques en atmosphCre infinie. Compt. Rend. Acad. Sc. Paris, 281 A, (1975), pp. 1105-1108. COMINCIOLI, V. /lA/ On some oblique derivative problems arising in the fluid flow in porous media. A theoretical and numerical approach. Applied Math. and @timisation (1975), pp. 313-336.

770

Appendix

COMINCIOLI, V. , TORELLI, A. /1A/ A new numerical approach t o a nonsteady f i l t r a t i o n problem, Cazcozo, 1 6 , (1979), 1, pp. 93-124.

CONCUS, P . , GOLUB, G.H., O'LEARY, D.P. /1A/ Numerical s o l u t i o n o f nonlinear p a r t i a l d i f f e r e n t i a l equations by a generalised conjugate-gradient method. Computing, 1 9 , (1977) , pp. 321-340. CORTEY-DUMONT, P. /1A/ Contribution 21 l ' 6 t u d e de l'approximation p a r l a mCthode des 6lCments f i n i s d ' i d q u a t i o n s quasi-variatCompt. Rend. Acad. Sc. Paris, ionnelles elliptiques 2S8A (1979) , p . 141-143.

.

COTTLE, R.W. / l A / Computational experience with l a r g e - s c a l e l i n e a r complementarity problems; i n Fixed Points: AZgor ithms and AppZications, S . Karamardian ( e d . ) , Acad. P r e s s , New-York , 1977 , pp. 281-313. /2A/ Numerical methods for complementarityproblems i n engineering and a p p l i e d science; i n Computing Methods i n AppZied Sciences and Engineering, 1977, 1, R . Glowinski, J . L . Lions ( e d s . ) , Lecture Notes i n Math., Vol. 704, Springer-Verlag, B e r l i n , 1979, pp. 37-52.

COTTLE, R.W., SACHER, R.S. / 1 A j On t h e s o l u t i o n o f l a r g e , s t r u c t u r e d l i n e a r complementarity problems: t h e t r i d i a g o n a l c a s e , AppZied Math. Optimiz. , 3 , (1977) 4, pp. 321-340. COTTLE, R.W., GOLUB, G.H., SACHER, R.S. /1A/ On t h e s o l u t i o n of l a r g e , s t r u c t u r e d l i n e a r complementarity problems: t h e block p a r t i t i o n e d case. Apptied Math. Optimiz. 4 , (1978) , 4, pp. 347-364.

CRYER, C.W., DEMPSTER, M.A.H. /1A/ Equivalence of l i n e a r complementarity problems and l i n e a r programs i n v e c t o r l a t t i c e H i l b e r t spaces, SIAM J . Control and Optimiz. , 18 (1980) 1, PP. 76-90. CXER, C.W., FETTER, H. / l A / The numerical s o l u t i o n of axi-symme t r i c free-boundary porous-flow well problems u s i n g v a r i a t i o n a l i n e q u a l i t i e s . M. R. C. TechnicaZ S m a r y Report 1761, U n i v e r s i t y of Wisconsin , Madison, 1977. DOUGLAS; J . , DUPONT, T. /1A/ Preconditioned conjugate g r a d i e n t i t e r a t i o n a p p l i e d t o Galerkin methods f o r mildly nonlinear D i r i c h l e t problem$; i n Sparse Matrix Computations , J . R . Bunch, D . J . Rose ( e d s . ) , Acad. P r e s s , New-York, 1976, pp.

333-348. /2A/ I n t e r i o r p e n a l t y procedures f o r e l l i p t i c and p a r a b o l i c Galerkin methods; i n Computing Methods i n AppZied Sciences , R. Glowinski , J .L. Lions ( e d s . ) , Lecture Notes i n Physics, Vol. 58, Springer-Verlag, B e r l i n , 1976, pp. 207-216.

Bibliography

771

DOUGLAS, J., RACHFORD, H.H. /lA/On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans. Amer. Math. Soc. , 82 (19561, pp. 421-439. ESKIN, G.I. /1A/ About a variational-difference method for solution of elliptic pseudo-differential equations. Uspeki Mat. Nauk. , 28, (19731, pp. 255-256. FALK, R.S. /1A/ Approximation of an elliptic boundary value problem with unilateral constraints. Rev. Franc. A u t a . Rech. Op. , Anat. N m . , R2, (1975), pp. 5-12. FALK, R.S., MERCIER, B. /U/Error estimates f o r elasto-plastic problems. Rev. Fr. Auto. Inf. Rech. Up., Anal. k. , 11,(1977), PP. 135-144. FORTIN, M., GLOWINSKI, R. /1A/ RDsoZution numgrique de probZ2mes a m limites p a r des m&thodes de Lagrangien augment&. Book in preparation. /2A/ Mhthodes de d6 compos ition-coordination par lagrangien augment&. Chapter 3 of R&solution num-

&Pique de p r o b l h e s a m limites p a r des mgthodes de lagrangi e n augment&, M. Fortin, R. Glowinski (eds.) (to appear). GABAY, D. /1A/ Rdsolution et dgcomposition des indquations variationnelles par des mgthodes de multiplicateurs; in ThOse d’Etat, Universit6 Pierre et Marie Curie, Paris, 1979. GIRAULT, V.,RAVIART, P.A. /lA/ An Analysis of a Mixed Finite Element Method for the Navier-Stokes Equations. Nwnerische Math. , 33, (19791, pp. 235-271. GLOWINSKI, R. /1A/ Introduction t o the approximation of variational i n e q u a l i t i e s . Report 76006, Laboratoire d‘ Analyse Numdrique , Universitg Pierre et Marie Curie, 1976. /2A/ Numerical Methods f o r Nonlinear Variational Problems. Lecture Notes, Tata Institute, Bombay, and SpringerVerlag, Berlin, 1980. /3A/ Finite elements and variational inequalities. Chapter 12 of The Mathematics of Finite Elements and A p p l i c a t i o n s , 111, MFELAP 1978, J.R. Whiteman (ed.), Acad. Press, London, 1979, pp. 135-171. /4A/ Sur l’approximation d’une ingquation variationnelle elliptique de type Bingham. Rev. Franc. Autom. I n f . Rech, Op. , Anal. N u m . , 10 (1976), pp. 13-30. GLOWINSKI, R., MARINI, D., VIDRASCU, M. /l/ Finite-element approximations and iterative solution of a fourth-order variational inequality (to appear). GLOWINSKI, R. , MARRACCO, A. /lA/ Numerical solution of two-dimensional magnetostatic problems by augmented Lagrangian methods. Cow. Meth. A p p l . Mech. Eng. , 1 2 (1977), pp. 33-46. /2A/ Chapter 5 of Rgsotution Nwngrique de Prob12mes a m Limites p a r des Mgthodes de Lagrangiens Augment&, M. Fortin, R. Glowinski (eds.) (to appear).

772

Appendix

GLOWINSKI, R., PIRONNEAU, 0. /1A/ Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review, 21 (1979)2, pp. 167-212. GOURSAT, M., MAAREK, G. /1A/ Nouvelle approche des probl8mes de gestion de stocks. Comparaison avec les mgthodes classiques. LABORIA Report 148, March 1976. HANOUZET, B., JOLY, J.L. /1A/ Convergence uniforme des it6r6s dh finissant la solution d 1 ingquations quasi-variationnelles et application B la regularitb, N m . Funct. Anal. and G p t i m i z . 1, (19791, 4, pp. 399-414. HASLINGER, J. /1A/ Finite-element analysis for unilateral problems with obstacles on the boundary. Aplikace Matematiky , 22, (1977) 3, pp. 180-188. /2A/ On numerical solution of a variational inequality of the 4th order by finite element method. Aplikace Matematiky, 23, (1978) 5, PP. 334-345.

HLAVACEK, I. /lA/ Dual finite-element analysis for unilateral boundary value problems. Aplikace Matematiky , 22, (1977), pp. 14-51. /2A/ Dual finite-element analysis for elliptic problems with obstacles on the boundary. Aplikace Matematiky , 22, (1977), pp. 244-255. /3A/ Dual finite-element analysis for semi-coercive unilateral boundary value problems. Aplikace Matematiky , 23, (1978), pp. 52-71. HLAVACEK, I., LOVISEK, J. /1A/ A finite-element analysis for the Signorhi problem in plane elastostatics. Aplikace Matematiky, 22, (19771,pp. 215-228. HOFWNDER, L. , LIONS, J.L. /1A/ S u r la compl6tion par rapport 8. une intggrale de Dirichlet. Math. Scand. , 4 (1956), pp.

259-270. HUNT, B. (ed.) /1A/ Numerical Methods i n Applied FZuid Dyxurrics. Acad. Press, London, 1980. HUNT, C., NASSIF, N.R. /lA/ On a variational inequality and its application in the theory of semi-conductors. SIAM J . N m . Anal. , 12, (1975), 6, pp. 938-950. JAMESON, A. /lA/ Transonic flow calculations; in Numerical Methods in .Fluid Dynamics, H..J Wirz , J.J. Smolderen (eds ) , McGraw-Hill , New-York , 1978, pp. 1-87.

.

.

JOHNSON, C. /lA/ A n elasto-plastic contact problem. Rev. Franc. Aut. I n f . Rech. Gp. , Anal. Num. , 12, (19781, 1, pp. 59-74. /2A/ A convergence estimate for an approximation of a parabolic variational inequality. SIAM J . Nwn. Anal. , 13 (19%) , 4, pp. 599-606.

Bib tiography

773

/3A/ A mixed finite-element method for plasticity with hardening. STAM J. N w n . Anal. , 14 (1977), 4, pp. 575-583. /4A/ A finite-element method for consolidation of clay. Comp. Meth. AppZ. Mech. Eng. , 16, (19781, 2, pp.

177-184. KAWOHL, B. /1A/ On a mixed Signorini problem. N w n . E'unct. Anal. and Optimiz. , 1, (1979), 6, pp. 633-645. KIKUCHI, N. , ODEN, J.T. /lA/ Contact problems i n Elasticity. TICOM Report 79-8, Univ. of Texas at Austin, July 1979.

LABORDE, P. /lA/ On visco-plasticity with hardening. N m . Fwrc. Anal. and Optimiz. , 1, (1979), 3, pp. 315-339. UETSCH, T.H. /1A/ A uniqueness theorem for elliptic Q.V.I., J. Functional Analysis, 18, (1975), pp. 286-288. LANDAU, L., LIFCHITZ, E. /lA/ Mcanique des Fluides. Mir, Moscow, 1953 LE TALLEC, P . /lA/ Simulation nwndrique d 'kcoulements visquem incompressibtes p a r des mdthodes d 'dldments f i n i s mixtes. Third-Cycle thesis, Universit6 Pierre et Marie Curie, 1978. LIEUTAUD, J. /1A/ Approximation d'opdrateurs p a r des msthodes de d&composition. These d'Etat , Universit6 Paris VI , 1968. LIONS, P.L., MERCIER, B. bA/ Splitting algorithms for the sum of two nonlinear operators. SIAM J . Nwn. Analysis , 16, (1979), 6, PP. 964-979. LOTSTEDT, P. /1A/ Analysis of some difficulties encountered in the simulation of mechanical systems with constraints. Dpt. of Num. Anal. and Comp. Sci. , Royal Inst. of Tech. , Stockholm, TRITA-NA-7914 , 1979. /2A/ On a penalty function method for the simulation of mechanical systems subject to constraints. Dpt. of N w n . Anal. and Comp. S c i . , Royal Inst. of Tech. , Stockholm, TRITA-NA-7919, 1979.

-

/3A/ A numerical method for the simulation of mechanical systems with unilateral constraints. Dpt. of Nwn. Anal. and Comp. Sci. RoyaZ Inst. of Tech. , Stockholm, TRITA-NA-7920, 1979. /4A/ Interactive simulation of the progressive collapse of a building, revisited. D p t . of N w n . Anal. and Comp. S c i . , Royal Inst. of Tech. , Stockholm, TRITA-NA-7921,

-

1979

MARROCCO, A. b.A/ Expdriences numeriques sur des problemes non lindaires rCsolus par Cl6ments finis et lagrangien augmentb. LABORIA Report 309, May 1978.

774

Appendix

MENALDI, J.L. /1/ Sur le contrale impulsionnel et 1'I.Q.V. associge. Compt. Rend. Acad. S&. Paris , 284A (1977), pp. 1499-1502. MERCIER, B. /lA/ SUP Za Thdorie e t Z'AnaZyse Nwndrique de ProbZBmes de PZasticitg. Thkse d'Etat, Universitd Pierre et Marie Curie, Paris, 1977. /2A/ Computation of the limit load and elasto-plastic bending of thin plates. I n t . J. N w n . Math. Eng. , 14, (1979), pp. 235-250. MIYOSHI, T. /lA/ A finite-element method for the solution of fourth-order partial differential equations. Kunamoto J . S c i . (Math.), 9, (19731, pp. 87-116. MOSCO, U. /lA/ Error estimates for some variational inequalities; in MathematicaZ Aspects of Finite Element Methods, I. Galligani, E. Magenes (eds.) , Lecture Notes in Math. , 606, Springer-Verlag, Berlin, 1977, pp. 224-236. MOSCO, U., SCARPINI, F. /lA/ Complementarity systems and approximation of variational inequalities. Rev. Franc. Autom. I n f . Rech. Op., Anal. Nwn. , R-1 (1975), pp. 83-104. NATTERER, F. /1A/ Optimale L2-Konvergenz finiten Elemente bei Variationsungleichungen. Bonn Math. Schr. , 89, (1976), pp. 1-12.

NITSCHE , J. / l ~ /Lm-convergence of finite-element approximation; in tdathematicaz Aspects o f Finite Element Approximations , I. Galligani, E. Magenes (eds.) , Lecture Notes in Math. , 606, Springer-Verlag, 1977, pp. 261-274. O'LEARY , D.P. /1A/ A generalised conjugate-gradient algorithm for solving a class of quadratic programming problems. Stanford Report, Comp. Science Dept., STAN-CA-77-638, Dec. 1977. O'LEARY, D.P.? YANG, W.H. /1A/ Elasto-plastic torsion by quadratic programming. Comp. Meth. AppZ. Mech. Eng. , 16, (1978), 3, pp. 361-368. OPIAL, 2 . /lA/ Weak convergence of the successive approximations for non-expansive mappings in Banach spaces. Buzz. A.M.S. , 73, (19671,PP. 591-597. PEACEMAN, D.H., RACHFORD, H.h. /lA/ The numerical solution of parabolic elliptic differential equations. J . SOC. Ind. AppZ. Math. , 3, (1955), pp. 28-41. PERIAUX, J. /lA/ RdsoZution de queZques problames m n Zindaires en aSrodynamique p a r des mdthodes d'dZhents f i n i s e t de moindres carrgs fonctionnezs. Third-C'ycle thesis , Univers i t 6 Pierre et Marie Curie, 1979. POLYAK, B.T. /1A/ The conjugate-gradient method in extremal problems. U.S.S.R. Comp. Math. and Math. Phys. , 9 (1969), pp. 94-112.

Bib liography

775

POWELL, M.J.D. /lA/ Restart procedure for the conjugate gradient method. Math. P r o g r m i n g , 12, (1977), pp. 148-162. PUEL, J.P. /lA/ Existence, comportement B l'infini et stabilit6 dans certains problemes quasilingaires elliptiques et paraboliques d'ordre 2. Annali Scuola Normale Superiore , Pisa , CZ. d i Scienze, Serie I V Y 3 , (1976), 1, pp. 89-119. RANNACHER, R. /1A/ Punktweise Konvergenz der Methode der finiten Elemente heim Platten problem. Manuscripta Math., 19, (1976), pp. 401-416. /2A/ On nonconforming and mixed finite element methods for plate bending problems. The linear case. Rev. Franc. Autom. I n f . Rech. Op. , Anal. Nwn. , 1 3 , (1979), 4, pp.

369-387. ROUX, J. /lA/ R6solktion num6rique d ' u n probleme d'6coulement subsonique de fluide compressible. Rev. Franc. Autom. I n f . Rech. Op., Anal. N w n . 11 (1977), pp. 197-208. SAGUEZ, Ch. /lA/ ContrBZe de SystBmes h FrontiBre Libre. T h h e d'Etat, Universitg Technologique de Compiegne, 1980. SAMLJELSSON, A., FRdIER, M. /1A/ Finite elements in plasticity. A variational inequality approach. Chapter 9 of The Mathematics of Finite Elements and Applications, 111,MAFELAP 1978, J.R. Whiteman (ed.), Acad. Press, London, 1979, pp. 105-115. SCARPINI, F. /lA/ Some nonlinear complementarity systems algorithms and applications to unilateral boundary value problems. Fasc. 6, I s t i t u t o Matematico G. Castelnuovo , Universita degli studi di Roma, 1977-78. SCARPINI, F., VIVALDI, M.A. /1A/ Error estimates for the approximation of some unilateral problems. Rev. Franc. Autom. I f . Rech. Op. , Anal. Nwn. , 11 (1977) 2, pp. 197-208. SCHOLZ, R. /lA/ Approximation von Sattelpunkten mit finiten

Elementen. Math. Schriften, 89, (1976), pp. 53-66. /2A/ A mixed method for 4th order problems using linear finite elements. Rev. Franc. Autom. I n f . Rech., Anal. N w n . , 12, (19781,1, pp. 85-90. /3A/ Interior error estimates for a mixed finite element method. N m . Funct. Analysis Optimiz. , 1, (1979), 4, pp. 415-429.

STAMPACCHIA, G. /lAf Equations Elliptiques oh Second Ordre tt Coefficients Discontinus. Presse de l'Universit6-de Montr6d , 1965. /2A/ Le problhe de Dirichlet pour les 6quations elliptiques du second ordre B coefficients discontinus. Ann. I n s t . Fourier, GrenobZe, 15, (1965), pp. 189-258.

776

Appendix

TARTAR, L. /lA/Ingquations Quasi Variationnelles Abstraites. Compt. Rend. Acad. Sc. Paris, 278A, (1974), pp. 1193-1196. TORELLI, A. /1A/ Some regularity results for a family of variational inequalities. Ann. Scu. Norm. Sup. Pisa, to appear. VIDRASCU, M. /1A/ Sur la rksolution nwnkrique du probZBrne de D i p ichZet pour Z 'opkrateur biharmonique. Third-cycle thesis, Universitk Pierre et Marie Curie, Paris, 1978. WHEELER, M.F. /U/ An elliptic collocation finite-element method with interior penalties. SIAM J. N w n . A m Z . , 15, (19781, 1, pp. 152-161. WILDE, D.J. , BEIGHTLER, C.S. /1A/ Foundations of Optidsation. Prentice Hall, Englewood Cliffs, N.J., 1967. YOUNG, D.M. /1A/ I t e r a t i v e solution of large linear systems. Academic Press, New York, 1971.