Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems

Augmented Lagrangian Methods: Applications to the numerical solution of boundary-value problems STUDIES IN MATHEMATIC...

Author: Michel Fortin | R. Glowinski | Trans-Inter-Scientia

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Augmented Lagrangian Methods: Applications to the numerical solution of boundary-value problems

STUDIES IN MATHEMATICS AND ITS APPLICATlONS VOLUME 15

Editors: J. L. LIONS,Puris G. PAPANICOLAOU,New York R. T. ROCKAFELLAR, Seattle H. FUJITA, Tokyo

NORTH-HOLLAYD - AMSTERDAM

0

NEW YORK

OXFORD

AUGMENTED LAGRANGIAN METHODS: APPLICATIONS TO THE NUMERICAL SOLUTION OF BOUNDARY-VALUE PROBLEMS MICHEL FORTIN Professor at the Universitk Laval, Quebec ROLAND GLOWINSKI Professor at the Universitk Pierre et Marie Curie, Paris Scientific Director at INRIA

1983

NORTH-HOLLAND

-

AMSTERDAM

NEW YORK

0

OXFORD

Elsevier Science Publishers B.V., 1983 All rights reserved. N o part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN 0 444 86680 9 Translation of: Methodes de Lagrangien Augment6 Applications a la resulotion numerique de problemes aux limites. 0 Bordas (Dunod), Paris, 1982 Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM T H E NETHERLANDS Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

English version edited, prepared and produced b y

TRANS-INTER-SCIENTIA

P . 0 .Box 16, Tonbridge, T N l l g D Y , England

1

Library of Congress Cataloging in Publication Data

Fortin, Micbel. Augmented Lagrangian methods. (Studies in mathematics and its applications ; v. 15) Translation of: Mithodes de Lagrangien augnent6. Bibliography: p. 1. Boundary value problems--Numerical solutions. 2. Differential equations, Partial--Numerical solutions. 3. Lagrangian functions. I. Glowinski, R. 11. Transinter-scientia (Firm) 111. Title. IY. Series. QA379.F6713 1983 515.3'5 83-6802 ISBN 0-444-86680-9

PRINTED IN T H E NETHERLANDS

INTRODUCTION

The e s s e n t i a l p u r p o s e of t h i s volume i s t o p r e s e n t t h e p r i n c i p l e s of t h e Augmented L a g r a n g i a n M e t h o d , t o g e t h e r w i t h numerous a p p l i c a t i o n s o f t h i s method t o t h e n u m e r i c a l s o l u t i o n of b o u n d a r y - v a l u e

p r o b l e m s for 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 r i n e q u a l i t i e s a r i s i n g i n M a t h e m a t i c a l P h y s i c s , i n t h e M e c h a n i c s o f C o n t i n u o u s Media and i n the Engineering Sciences.

-

Simultaneous developments i n c o m p u t e r s and i n N u m e r i c a l A n a l y s i s

i n p a r t i c u l a r t h e F i n i t e E l e m e n t Method - have g r a d u a l l y l e d r e s e a r c h workers, e n g i n e e r s , e t c . . .

m a t h e m a t i c a l m o d e l s of g r e a t e r

t o use

and g r e a t e r complexity f o r t h e r e p r e s e n t a t i o n of t h e phenomena which a r i s e i n t h e i r respective disciplines.

C e r t a i n c a l c u l a t i o n s which

h i t h e r t o had been c o n s i d e r e d i m p r a c t i c a b l e have become r o u t i n e m a t t e r s , and it h a s become p o s s i b l e t o abandon c e r t a i n s i m p l i f y i n g assumptions

-

i n p a r t i c u l a r , l i n e a r i t y - and t h e r e b y g a i n a more

r e a l i s t i c s i m u l a t i o n of t h e phenomena c o n s i d e r e d . T h i s development h a s l e d , f o r N u m e r i c a l A n a l y s t s i n p a r t i c u l a r , t o a s e a r c h f o r e f f i c i e n t methods f o r s o l v i n g t h e " l a r g e s y s t e m s " which a r i s e from t h e d i s c r e t i s a t i o n of n o n l i n e a r b o u n d a r y - v a l u e

prob lems, v a r i a t i o n a l i n e q u a l i t i e s , optimal c o n t r o l problems, etc..

.

Many o f t h e s e problems can b e e x p r e s s e d i n t h e form of a s e a r c h

f o r a minimum o f ' a f u n c t i o n a l - o r , a t any r a t e can be e a s i l y reduced t o t h i s .

I t i s t h e r e f o r e t e m p t i n g t o a p p l y t o t h e s o l u t i o n of

t h e s e problems methods t a k e n from M a t h e m a t i c a l Programming

(i.e.

t h e f i e l d of O p t i m i s a t i o n A l g o r i t h m s ) . I t s h o u l d be remarked t h a t , w h i l s t a c c e p t i n g t h a t t h e e x i s t i n g

t e c h n i q u e s of Mathematical Programming p r o v i d e a s a f e and r e l i a b l e

V

INTRODUCTION

vi

s t a r t i n g p o i n t , t h e i r 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 problems i n v o l v i n g 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 or i n e q u a l i t i e s r e q u i r e s c e r t a i n p r e c a u t i o n s t o be t a k e n ; t h e r e a r e i n r e a l i t y few methods c a p a b l e of e f f i c i e n t l y m i n i m i s i n g f u n c t i o n a l s which depend on s e v e r a l t h o u s a n d v a r i a b l e s , o v e r s e t s d e f i n e d by s i m i l a r numbers of linear or nonlinear constraints.

A s one might e x p e c t , i t i s nec-

e s s a r y t o a d a p t t h e g e n e r a l methods and t a k e a c c o u n t o f t h e p a r t i c u l a r s t r u c t u r e of t h e problems t o be s o l v e d .

T h i s i s what w e

s h a l l be a t t e m p t i n g t o do i n t h i s book, by showing how a s m a l l number of v e r y s i m p l e i d e a s c a n b e a p p l i e d t o t h e s o l u t i o n of problems which a p r i o r i a p p e a r c o m p l e t e l y d i f f e r e n t . The approach which w e s h a l l b e f o l l o w i n g may a t f i r s t seem s u r p r i s i n g ; i n f a c t , w e s h a l l v e r y o f t e n b e modifying t h e o r i g i n a l problem by i n t r o d u c i n g s u p p l e m e n t a r y c o n s t r a i n t s and v a r i a b l e s , which a t f i r s t s i g h t , may a p p e a r t o have t h e e f f e c t of i n c r e a s i n g

i t s complexity.

However, t h i s c o m p l i c a t i o n w i l l t h e n b e s e e n t o

be l a r g e l y compensated by t h e s t r u c t u r a l s i m p l i f i e a t i o n which

it introduces. The b a s i c p r i n c i p l e s which w i l l s e r v e t o g u i d e us t h r o u g h o u t t h i s work w i l l b e t h e f o l l o w i n g : (i)

There a r e a t t h e p r e s e n t t i m e e f f i c i e n t methods a v a i l a b l e f o r s o l v i n g l i n e a r s y s t e m s , even of v e r y l a r g e o r d e r ; t h i s i s p a r t i c u l a r l y t r u e f o r s y s t e m s i n which t h e m a t r i x

i s s y m m e t r i c and p o s i t i v e d e f i n i t e . ( i i ) The a v e r a g e c o s t of t h e above s o l u t i o n s i n c r e a s e s o n l y

m a r g i n a l l y i f w e s o l v e , u s i n g a d i r e c t method,

linear

s y s t e m s r e l a t i n g t o a common m a t r i x , i n t e r n a l l y w i t h i n an i t e r a t i v e p r o c e s s . ( i i i ) N o n l i n e a r problems which depend on a s m a l l number of

variables (say,

< < 10

t o b e more s p e c i f i c ) a r e e a s y t o

s o l v e ( o r a t any r a t e much e a s i e r t o s o l v e t h a n t h o s e which depend on a l a r g e number of v a r i a b l e s ) . Methods f o r s o l v i n g l a r g e n o n l i n e a r s y s t e m s a r e u n i v e r s a l l y b a s e d ( s e e ORTEGA-RHEINBOLDT i t e r a t i v e process.

C11) on a l i n e a r i s a t i o n embedded w i t h i n an

The u p d a t i n g of t h e l i n e a r i s e d problem and i t s

s o l u t i o n g e n e r a l l y c o n s t i t u t e t h e most e x p e n s i v e p h a s e o f t h e o v e r a l l solution process.

I n t h i s p e r s p e c t i v e , a s o l u t i o n method which

u s e s t h e same m a t r i x a l l t h e t i m e m a y b e e x t r e m e l y a t t r a c t i v e . The methods which w e p r o p o s e u t i l i s e a d e c o m p o s i t i o n - c o o r d i n a t i o n p r i n c i p l e ( i n t h e s e n s e o f BENSOUSSAN-LIONS-TEMAM

111) which means

v ii

INTRODUCTION

t h a t t h e n o n l i n e a r i t y i s t r e a t e d a t a l o c a l l e v e l , and i n which t h e c o o r d i n a t i o n i s e f f e c t e d through t h e s i m u l t a n e o u s use of Lagrange

m u l t i p l i e r s and a p e n a l i s a t i o n method: i n t h i s w e f o l l o w a methodology f i r s t i n t r o d u c e d around 1 9 7 0 by HESTENES i l l and POWELL 111. T h i s p r i n c i p l e of l o c a l i s a t i o n of t h e n o n l i n e a r i t i e s i s i n f a c t one of t h e g o v e r n i n g p r i n c i p l e s of t h i s volume.

Additionally, the

methods of d e c o m p o s i t i o n which w i l l b e found i n t h i s book a r e w e l l s u i t e d t o P a r a l l e Z C o m p u t a t i o n , which c e r t a i n l y a p p e a r s t o b e one of t h e d i r e c t i o n s of t h e f u t u r e i n Numerical A n a l y s i s , h a v i n g r e g a r d t o t h e a r c h i t e c t u r e s now b e i n g a d o p t e d f o r l a r g e modern s c i e n t i f i c computers.

W e t h e r e f o r e have r e a s o n t o hope t h a t t h e methods de-

v e l o p e d i n t h i s book w i l l f i n d wide a p p l i c a t i o n i n f u t u r e y e a r s . W e now come on t o t h e c o n t e n t s of t h i s volume: C h a p t e r I i n t r o d u c e s t h e Augmented L a g r a n g i a n method i n t h e c l a s s i c a l c o n t e x t of Q u a d r a t i c P r o g r a m m i n g w i t h l i n e a r c o n s t r a i n t s i n

f i n i t e dimensions.

W e h e r e s t u d y i n d e t a i l some s t a n d a r d a l g o r -

i t h m s - and o t h e r s which are r a t h e r less s o

-

and g i v e a number o f

r e s u l t s , some o f them new, r e l a t i n g t o t h e i r convergence. I n C h a p t e r I1 w e a p p l y t h e r e s u l t s of t h e p r e v i o u s c h a p t e r t o t h e s o l u t i o n of t h e S t o k e s and N a v i e r - S t o k e s

viscous f l u i d s .

e q u a t i o n s for i n c o m p r e s s i b l e

C e r t a i n o f t h e r e s u l t s of C h a p t e r I a r e v e r i f i e d

e x p e r i m e n t a l l y on t h e b a s i s of n u m e r i c a l t e s t s . I n C h a p t e r I11 w e i n t r o d u c e , w i t h i n a r a t h e r g e n e r a l H i l b e r t i a n framework, t h e p r i n c i p l e of d e c o m p o s i t i o n - c o o r d i n a t i o n

r e s t of t h e book w i l l be b a s e d . convergence,

on which t h e

Here w e s t u d y i n p a r t i c u l a r t h e

under q u i t e g e n e r a l a s s u m p t i o n s , o f t h e two b a s i c a l -

g o r i t h m s , A L G l and A L G 2 , p r i m a r i l y devoted.

t o which t h e r e m a i n d e r o f t h e book i s

T h i s c h a p t e r i s i l l u s t r a t e d by numerous

examples t a k e n from Mechanics and from P h y s i c s . I n C h a p t e r I V w e a p p l y t h e r e s u l t s of t h e p r e c e d i n g c h a p t e r t o t h e s o l u t i o n o f m i l d l y n o n l i n e a r problems o f t h e form Au + $(u) = f,

where

A

i s a l i n e a r e l l i p t i c o p e r a t o r of o r d e r two and $ i s a W e a l s o c o n s i d e r i n t h i s c o n t e x t t h e u s e of

numerical function.

hybrid f i n i t e elements. I n C h a p t e r V , w e a p p l y t h e r e s u l t s o f C h a p t e r I11 t o t h e s o l u t i o n of s e c o n d - o r d e r 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 e q u a t i o n s and i n e q u a l i t i e s , i n which t h e n o n l i n e a r i t y r e l a t e s t o t h e g r a d i e n t of t h e solution.

The methods which a r e d e s c r i b e d h e r e a p p l y i n p a r t i c u l a r

viii

INTRODUCTION

t o t h e s o l u t i o n o f boundary-value

problems f o r 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 such as -V*(V(x,Vu)Vu)

= f.

I t w i l l be e s t a b l i s h e d t h a t such problems can be s o l v e d w i t h

remarkable e f f i c i e n c y . I n C h a p t e r s V I and V I I w e c o n s i d e r a p p l i c a t i o n s o f t h e methods o f C h a p t e r I11 t o t h e s o l u t i o n o f , r e s p e c t i v e l y , p r o b l e m s i n

Elasto-Plasticity

and i n t h e s t e a d y and t i m e - d e p e n d e n t

f l o w of

v i s c o - p t a s t i c f Z u i d s of B i n g h a m t y p e i n t w o - d i m e n s i o n a 2 c a v i t i e s . The p r o b l e m c o n s i d e r e d i n C h a p t e r V I i s f o r m u l a t e d i n i t i a l l y a s a n

e l l i p t i c variational inequality r e l a t i n g t o t h e Linear E l a s t i c i t y operator, while t h a t considered i n Chapter V I I is reformulated i n

t e r m s of a v a r i a t i o n a l i n e q u a Z i t y of o r d e r 4 , w h i c h may b e e l l i p t i c o r p a r a b o l i c d e p e n d i n g on t h e c i r c u m s t a n c e s , v i a t h e i n t r o d u c t i o n

of a s t r e a m f u n c t i o n . W h i l s t t h e p r o b l e m s d e a l t w i t h i n C h a p t e r s I11 t o V I I a l l f a l l w i t h i n t h e s c o p e o f C o n v e x A n a l y s i s and Monotone Operator.s,

those

c o n s i d e r e d i n C h a p t e r V I I I q u i t e d e f i n i t e l y d e p a r t from t h i s framework; t h e s e are n o n l i n e a r p r o b l e m s a r i s i n g f r o m F i n i t e N o n l i n e a r E l a s t i c i t y , a n d r e d u c e t o t h e m i n i m i s a t i o n o f f u n c t i o n a l s (which may b e convex) o v e r non-convex

sets.

N o n e t h e l e s s , t h e de-

c o m p o s i t i o n - c o o r d i n a t i o n p r i n c i p l e s o f C h a p t e r I11 s t i l l l e a d t o e x t r e m e l y powerful a l g o r i t h m s f o r s o l v i n g t h e s e problems, even though s t r i c t l y s p e a k i n g w e are n o l o n g e r w i t h i n t h e r a n g e o f a p p l i c a t i o n o f t h e s e methods. I n C h a p t e r s I1 t o V I I I r e s u t s o f n u m e r i c a l e x p e r i m e n t s are g i v e n which e n a b l e t h e e f f i c i e n c y of t h e p r o p o s e d methods of s o l u t i o n t o be assessed. C h a p t e r I X , which c o n c l u d e s t h i s volume, i s much more a b s t r a c t i n nature;

i t t a k e s up, i n a g e n e r a l s e t t i n g , c e r t a i n o f t h e i d e a s

o r i g i n a l l y c o n s i d e r e d i n C h a p t e r s I11 a n d I V and shows, i n p a r t i c u l a r , t h e l i n k s w h i c h e x i s t b e t w e e n t h e a l g o r i t h m s A L G l and ALG2 o f C h a p t e r I11 a n d c e r t a i n c l a s s i c a l d t e r n a t i n g d i r e c t i o n methods. I t a l s o p r o v i d e s a t h e o r e t i c a l framework w h i c h i s w e l l s u i t e d t o t h e

s t u d y o f t h e c o n v e r g e n c e o f a number o f a l g o r i t h m s b a s e d on t h e u s e of L a g r a n g e m u l t i p l i e r s . Augmented L a g r a n q i a n methods h a v e been t h e subject o f numerous p u b l i c a t i o n s , and it i s v e r y d i f f i c u l t t o select from t h e s e a b i b l i o g r a p h y which i s a n y t h i n g l i k e c o m p l e t e .

W e have t h e r e f o r e

i n d i c a t e d i n t h i s book o n l y r e f e r e n c e s w i t h which w e are p e r s o n a l l y

ix

INTRODUCTION

f a m i l i a r and which have a d i r e c t r e l e v a n c e t o t h e q u e s t i o n s addressed herein; we thus advise t h e reader i n t e r e s t e d i n obtaining f u r t h e r i n f o r m a t i o n t o r e f e r t o t h e b i b l i o g r a p h i e s of t h e s e volumes, together with the following journals: J o u r n a l of O p t i m i z a t i o n T h e o r y a n d A p p Z i c a t i o n s , Mathernatica l Programming, S i a m J o u r n a l , of C o n t r o Z a n d O p t i m i z a t i o n . W e would l i k e t o t h a n k Messrs

BGgis, B o u r g a t , Chan, Gabay,

L e T a l l e c , Marrocco, M e r c i e r and Thomasset who p a r t i c i p a t e d i n t h e p r e p a r a t i o n of t h i s book.

W e would a l s o l i k e t o acknowledge o u r

p a r t i c u l a r indebtedness t o M r s .

F r a n f o i s e Weber o f I N R I A who

p a i n s t a k i n g l y t y p e d t h e e n t i r e French o r i g i n a l of t h i s work ( t h e t y p e d e q u a t i o n s of which have been r e t a i n e d i n t h e p r e s e n t E n g l i s h language e d i t i o n ) , t o t h e T r a n s l a t o r s , Messrs B . D.

Hunt and

S p i c e r , and t h e i r wives who h e l p e d produce t h i s work f o r

"Trans-Inter-Scientia",

and t o t h e N o r t h - H o l l a n d

P u b l i s h i n g Company

f o r agreeing t o publish t h i s t r a n s l a t i o n i n t h e s e r i e s S t u d i e s i n Mathematics and i t s A p p l i c a t i o n s . The f i r s t e d i t o r (M.

F o r t i n ) w i s h e s t o t h a n k CRSNG (Canada) and

t h e M i n i s t r y o f E d u c a t i o n o f Quebec f o r t h e f i n a n c i a l a i d which t h e y have g i v e n t o t h i s work, a n d , s i n c e a l a r g e p a r t o f t h e work was d r a f t e d d u r i n g a p e r i o d s p e n t by t h e s e c o n d e d i t o r Glowinski) a t t h e Mathematical Research Center (M.R.C.)

(R.

of t h e

U n i v e r s i t y of Wisconsin a t Madison ( f i n a n c e d under c o n t r a c t DAA 629-80-C-0041),

w e would l i k e t o t h a n k P r o f e s s o r John Nohel,

t h e d i r e c t o r o f t h e M.R.C.,

f o r t h e f a c i l i t i e s which w e r e made

a v a i l a b l e f o r c a r r y i n g o u t t h e f i n a l d r a f t i n g of t h e work. L a s t b u t n o t l e a s t , o u r t h a n k s go t o t h e Management of I N R I A f o r allowing various people a t t h a t i n s t i t u t i o n t o p a r t i c i p a t e i n the p r e p a r a t i o n of t h i s book; i n p a r t i c u l a r , i t was a t I N R I A t h a t t h e m a j o r i t y of t h e n u m e r i c a l e x p e r i m e n t s p r e s e n t e d h e r e i n were c a r r i e d out.

Madison, U.S.A.

Michel F o r t i n ,

1 9 t h August 1981

Roland Glowinski.

x The f o l l o w i n g i n d i v i d u a l s c o n t r i b u t e d to t h e p r o d u c t i o n of t h e o r i g i n a l French-language e d i t i o n of t h i s book p u b l i s h e d by E d i t i o n s Dunod:

D.

BEGIS

I N R I A , B.P.

1 0 5 , 78153 L e Chesnay Cedex, F r a n c e . 105, 78153 L e Chesnay Cedex, F r a n c e .

J.F.

BOURGAT

I N R I A , B. P .

T.F.

CHAN

Computer S c i e n c e Department, Yale U n i v e r s i t y , Box 2158, New Haven, C o n n e c t i c u t 06520, U.S.A.

M.

FORTIN

Departement d e Mathgmatiques, U n i v e r s i t C L a v a l , F a c u l t 6 d e s S c i e n c e s , Quebec G 1 K 7 P 4 , Canada.

D.

GABAY

C.N.R.S. , U n i v e r s i t g P i e r r e e t Marie C u r i e , L a b o r a t o i r e d ' A n a l y s e Numgrique LA 1 8 9 , 4 , P l a c e J u s s i e u , 75230 P a r i s Cedex 0 5 , and INRIA.

R.

GLOWINSKI

U n i v e r s i t e P i e r r e e t Marie C u r i e , L a b o r a t o i r e d ' A n a l y s e Numerique LA189, 4 P l a c e J u s s i e u , 75230 P a r i s Cedex 0 5 , and I N R I A .

P.

LE TALLEC

L a b o r a t o i r e C e n t r a l d e s P o n t s e t Chaussges, S e r v i c e Mathgmatiques, 58 Boulevard L e f e b v r e , 75732 P a r i s Cedex 15.

A.

MARROCCO

INRIA,

B.

MERCIER

C.E.A., S e r v i c e MA, C e n t r e d ' E t u d e s de L i m e i l , B . P . 2 7 , 9 4 1 9 0 V i l l e n e u v e S a i n t Georges, F r a n c e .

F.

THOMASSET

INRIA,

The p r e s e n t E n g l i s h - l a n g u a g e and D.C.

B.P.

B.P.

1 0 5 , 78153 L e Chesnay Cedex, F r a n c e .

1 0 5 , 78153 L e Chesnay Cedex, F r a n c e .

e d i t i o n was t r a n s l a t e d by Messrs. B.

S p i c e r , and w a s produced by Trans-Inter-Scientia.

0

0 0

Hunt

TABLE O F CONTENTS

Chapter 1

:

Augmented Lagrangian methods i n q u a d r a t i c

.

............. 1. P r i n c i p l e s o f t h e m e t h o d .................................. 2. A f i r s t a l g o r i t h m f o r s a d d l e - p o i n t c a l c u l a t i o n ............ 2.1 D e s c r i p t i o n o f t h e a l g o r i t h m ................. ....... 2.2 Convergence r e s u l t s .......................... ....... 2 . 3 I n t e r p r e t a t i o n o f a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) . Rate of convergence i f pn = p and c h o i c e of r ..... ....... 3. Variable s t e p - l e n g t h algorithms . Conjugate gradien m e t h o d ............................................. ....... 3 . 1 G e n e r a l n o t e s ........................................ * 3.2 A p p l i c a t i o n t o t h e m i n i m i s a t i o n of J r ............... programming

4.

M .F o r t i n .

.

R Glowinski

3 3

4 7

18 18 21

method o f

........................................... S y n o p s i s ..................... ....................... Study of a l g o r i t h m ( 4 . 1 ) - ( 4 . 4 ) ....................... Study of a l g o r i t h m ( 4 . 6 ) - ( 4 . 8 ) ....................... 4.3.1 General n o t e s ......... .......................

A r r o w .H u r w i c z

26

4.1

26

4.2 4.3

4.3.2

Reduction of

(4.6)-(4.8

form o f a second-order 4.3.3

.

1

On c e r t a i n v a r i a n t s o f t h e m e t h o d s o f S e c t i o n 2 : I n t r o d u c t i o n of a r e l a x a t i o n p a r a m e t e r ;

5

1

:

34 34

t o the discrete

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

d i f f e r e n t i a l system

Convergence o f a l g o r i t h m ( 4 . 6 ) - ( 4 . 8 )

M i s c e l l a n e o u s remarks and d i s c u s s i o n

Chapter 2

27

34 35

42

A p p l i c a t i o n t o t h e S t o k e s and Navier-Stokes

. M . F o r t i n , F . T h o m a s s e t . .............. 1. I n t r o d u c t i o n .............................................. 1.1 M o t i v a t i o n ........................................... equations

xi

47 47

47

TABLE OF CONTENTS

xii

............................ 1.3 S t o k e s problem and q u a d r a t i c programming ............ D i s c r e t i s a t i o n of t h e S t o k e s probZern ..................... A l g o r i t h m s and d i s c u s s i o n o f r e s u l t s ..................... 3.1 E x p l i c i t f o r m u l a t i o n of t h e a l g o r i t h m s .............. 3.2 R e s u l t s and d i s c u s s i o n .............................. 3.2.1 F i x e d - s t e p Uzawa a l g o r i t h m s .................. 1.2

. 3.

2

S t a t e m e n t of t h e problem

3.2.2

E f f e c t of t h e i n c o m p l e t e s o l u t i o n of

3.2.3

V a r i a b l e s t e p l e n g t h and c o n j u g a t e - g r a d i e n t

(3.3)

...

...................................... .......... 3.2.5 A l g o r i t h m s of Arrow - Hurwicz t y p e ........... 3.2.6 C o n c l u s i o n s .................................. 4. N a v i e r - S t o k e s e q u a t i o n s . s t e a d y - s t a t e n o n l i n e a r c a s e ..... 4.1 S t a t e m e n t of t h e problem ............................ 4.2 B a s i c a l g o r i t h m ..................................... 5. V a r i a n t s and a p p r o x i m a t i o n s o f t h e b a s i c a l g o r i t h m o f S e c t i o n 4 ............................................. 5 . 1 V a r i a n t s o f Uzawa t y p e .............................. 5.2 V a r i a n t s of A r r o w - Hurwicz t y p e .................... 5.3 Numerical r e s u l t s ................................... 6. N a v i e r - S t o k e s e q u a t i o n s . T i m e - d e p e n d e n t c a s e ............ 6.1 S t a t e m e n t of t h e problem ............................ 6.2 S o l u t i o n a l g o r i t h m .................................. G e n e r a l d i s c u s s i o n on C h a p t e r 11 ........................ 7. 3.2.4

Chapter 3

:

47 49 52 57 57 62 62 66

methods

70

Algorithm w i t h r e l a x a t i o n parameter

71

On d e c o m p o s i t i o n - c o o r d i n a t i o n

74 15 75 7a

a3 a4 a6 a7

aa aa 91 95

methods u s i n g an

. M .F o r t i n . R . G l o w i n s k i . . . I n t r o d u c t i o n ............................................. 1. 1.1 M o t i v a t i o n . Examples ............................... 1 . 2 P r i n c i p l e of t h e method ............................. 2. I n v e s t i g a t i o n o f p r o b l e m ( P l and o f t h e s a d d l e - p o i n t s of d and e , ........................................... 2.1 E x i s t e n c e and u n i q u e n e s s p r o p e r t i e s f o r problem (P) . augmented L a g r a n g i a n

d

2.2

P r o p e r t i e s of t h e s a d d l e - p o i n t s of

2.3

R e l a t i o n s w i t h p e r t u r b a t i o n t h e o r y i n Convex Analysis

73

and of

dr

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

..

97 97 97

100

103 103 104 106

xiii

TABLE OF CONTENTS

.

............................ .............................. 3.2 Second a l g o r i t h m (ALG2) ............................. 3.3 A p p l i c a t i o n t o t h e examples of S e c t i o n 1 ............ 4. C o n v e r g e n c e of A L G l ...................................... 4.1 G e n e r a l c a s e ........................................ 4.2 The f i n i t e - d i m e n s i o n a l case ......................... 4.3 On t h e c h o i c e of r and of { p n I n .................. 5. C o n v e r g e n c e o f A L G Z ...................................... 5.1 G e n e r a l c a s e ........................................ 5.2 F i n i t e - d i m e n s i o n a l c a s e ............................. 5 . 3 D i s c u s s i o n . Choice of p and of r ................ A p p l i c a t i o n t o some n o n l i n e a r p r o b l e m s ................... 6. 6.1 I n t r o d u c t i o n ........................................ 3

Description of the algorithms 3.1

F i r s t a l g o r i t h m (ALG1)

108 108 110 111

113 113

117 119 122 122 126 126 127 127

6.2

Case of t h e examples of S e c t i o n 1.1 (Flow of a Bingham f l u i d and e l a s t o p l a s t i c t o r s i o n )

128

6.3

A n o n l i n e a r D i r i c h l e t problem

128

6.4

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 m i l d l y n o n l i n e a r

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

systems and r e l a t i o n s h i p w i t h a l t e r n a t i n g

................................... n o n l i n e a r p r o g r a m m i n g p r o b l e m s ............

135

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

135

d i r e c t i o n methods

7

.

Application t o 7 . 1 An augmented Lagrangian i n t h e c a s e of i n e q u a l i t y constraints

7.2

M i n i m i s a t i o n of a f u n c t i o n a l o v e r an i n t e r s e c t i o n of

7.2.1

......................................... S t a t e m e n t of t h e problem .....................

7.2.2

S o l u t i o n of t h e problem by A L G l and ALGZ

convex s e t s

7.3

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 t h e Weber

.

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

S t a t e m e n t of t h e problem

7.3.2

I n t r o d u c t i o n of an augmented Lagrangian f o r

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

7.3.3

A p p l i c a t i o n of ALGZ t o t h e s o l u t i o n of

7.3.4

Numerical a p p l i c a t i o n s

G e n e r a l d i s c u s s i o n o n C h a p t e r 111

Chapter 4

..... problem ....

7.3.1

t h e s o l u t i o n of problem ( 7 . 3 2 )

8

131

:

(7.32).

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

137 137 139 141 141 142. 142 143 145

Numerical s o l u t i o n of m i l d l y n o n l i n e a r problems

. M.Fortin, ..........................

by augmented Lagrangian methods

.

R Glowinski.

. .

T F Chan

147

xiv

. 2. 1

TABLE OF CONTENTS

............................................. A c l a s s of m i l d l y n o n l i n e a r e l l i p t i c p r o b l e m s ............ 2.1 Formulation of the problem .......................... Introduction

2.2

Approximation of problem (2.5), (2.6) by finite element methods

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

3

.

3.1 3.2

.

148 148 150

Augmented L a g r a n g i a n and d e c o m p o s i t i o n o f t h e problem 1 2 . 5 ) , ( 2 . 6 )

4

147

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

Construction of the augmented Lagrangian . (I) Continuous case Construction of the augmented Lagrangian (11) The discrete case

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

153 153 154

A l g o r i t h m s f o r s o l u t i o n of t h e a p p r o x i m a t e p r o b l e m

...................................... 5. N u m e r i c a l e x p e r i m e n t s .................................... 5.1 Formulation of a model problem . General notes ...... Discussion

12.131.

5.2

Comments on the implementation and the convergence of ALGl and ALG2 Discussion on the choice of the parameters r and p

157 161 161

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

162

........................................... 6. Some r e m a r k s o n h y b r i d m e t h o d s ........................... 7. G e n e r a l d i s c u s s i o n on C h a p t e r I V .........................

165

5.3

Chapter 5

:

b

166 170

Application to the solution of strongly

. ............. 1. I n t r o d u c t i o n ............................................. 2. G e n e r a l framework of t h e p r o b l e m s i n C h a p t e r V ........... 3. A c l a s s of n o n Z i n e a r D i r i c h l e t p r o b l e m s .................. 3.1 Formulation of the problems . Augmented Lagrangians . 3.1.1 The continuous case .......................... 3.1.2 The approximate problems ..................... 3.2 The basic algorithm and its convergence properties .. 3.3 Numerical experiments ............................... nonlinear second-order boundary value problems

M.Fortin,

3.4 4.

A.Marrocco.

Continuity constraints on the gradient; penalty methods

A magneto-static

4.1

R.Glowinski,

Formulation

171 171 171 173 173 173 174 176 181

Interior

..................................... p r o b l e m ................................. of the problem ..........................

185 186 186

TABLE OF CONTENTS

4.2 4.3 5

.

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

Formulation using an augmented Lagrangian Numerical experiments

.

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

5.1

Formulation of the problem

5.2

Formulation via an augmented Lagrangian algorithms

Further 6.1

.

6.2

.......................................... a p p l i c a t i o n s .....................................

Flow of a viscoplastic Bingham fluid in a cylindrical duct 6.1.1 Formulation of the problem Solution by augmented Lagrangian methods 6.1.2 Numerical results 6.1.3 Elastoplastic torsion of a cylindrical bar Formulation of the problem 6.2.1 6.2.2 Solution of ( 6 . 1 3 ) , ( 6 . 1 4 ) by augmented Lagrangian methods 6.2.3 Numerical results Application to the solution of the minimal

.....

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

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

6.3

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

surfaces problem Formulation of the problem 6.3.2 Solution of problem ( 6 . 2 1 ) by augmented Lagrangian algorithms Numerical results 6.3.3

6.3.1

.

190

D i s c u s s i o n on Chapter V

Chapter 6

:

193 194

Solution

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

7

188

C a l c u l a t i o n of s u b s o n i c and t r a n s o n i c p o t e n t i a l f l o w s o f compressible ideal f l u i d s

6

xv

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

Application of algorithm ALG2 to a two dimensional elastoplasticity problem B.Mercier.

. ..................................... I n t r o d u c t i o n .................................................. 1. The c o n t i n u o u s p r o b l e m ................................... The p r o b l e m ( P i .......................................... 2. 3. A p p r o x i m a t i o n by f i n i t e e l e m e n t s ......................... A p p l i c a t i o n of a l g o r i t h m A L G Z ............................ 4. 5. C o n v e r g e n c e of a l g o r i t h m A L G Z ............................ 6. N u m e r i c a l a p p l i c a t i o n .................................... 6.1 Description of the mechanical problem ...............

196 199 199 199 200 201 203 203 204 206

207 207

208 208 210

217 211 217 219 220 222 224 225 225

xvi

TABLE OF CONTENTS

6.2 6.3 6.4 6.5 7

.

Discussion

Chapter 7

. 2. 1

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

Choice of constants Gradient method Conjugate gradient method Choice of the parameters for algorithm ALG2

:

General

Application to the numerical solution of the two dimensional flow of incompressible viscoplastic fluids. D . B e g i s . R . G l o w i n s k i

................... n o t e s . S y n o p s i s ..................................

227 227 228 229

231

233 233

F o r m u l a t i o n o f Bingham f l o w s u s i n g t h e v e l o c i t y and

.............................................. F o r m u l a t i o n o f B i n g h a m f l o w s u s i n g a s t r e a m f u n c t i o n ...... 3. 4. A p p r o x i m a t i o n o f t h e s t e a d y - s t a t e p r o b l e m ................. 4.1 Synopsis . Formulation of the steady-state problem ... the pressure

4.2

Approximation of ( 4 . 1 ) , ( 4 . 2 ) by a mixed finite element method Solvability of problem ( 4 . 1 1 ) Convergence of the approximate solutions Approximation using numerical integration

....................................... 4.3 ........................ 4.4 ............. 4.5 ............ A p p r o x i m a t i o n o f t h e e v o l u t i o n p r o b l e m 1 3 . 7 1 .............. 5. 5.1 Semi-discretisation with respect to time ............. 5.2 Complete discretisation of ( 3 . 7 ) ..................... 6. S o l u t i o n o f 1 4 . I). ( 5 . 2 ) b y a u g m e n t e d L a g r a n g i a n m e t h o d s ... 6.1 Synopsis ............................................. 6.2 The model problem . Introduction of an augmented Laqranqian ........................................... 6.3 Application of ALG1 to seeking a saddle point of dr . 6.4 On variants of algorithm ( 6 . 6 ) - ( 6 . 8 ) ................. 7. N u m e r i c a Z e x p e r i m e n t s ..................................... 7.1 Formulation of the test problem ...................... 7.2 Numerical results .................................... Chapter 8

:

233 235 237 237 237 239 239 240 241 241 242 243 243

243 245 246 248 248 249

Application to the solution of finite nonlinear

. J.F.Bourgat, R.Glowinski, ..................................... n o t e s . S y n o p s i s .................................. elasticity problems

P.Le

.

1

General

Tallec

257 257

TABLE OF CONTENTS

2

.

xvii

D e c o m p o s i t i o n of v a r i a t i o n a l p r o b l e m s .

Associated

2.1

A f a m i l y of v a r i a t i o n a l problems

258

2.2

A

............................................... .................... decomposition p r i n c i p l e ...........................

259

2.3

An augmented Lagrangian a s s o c i a t e d w i t h (n)

.........

260

algorithms

2.4 2.5

2.6 2.7

................... A second a l g o r i t h m f o r s o l v i n g ( P ) .................. Remarks on t h e c h o i c e of p and r ................ R e l a t i o n s w i t h a l t e r n a t i n g d i r e c t i o n methods . F u r t h e r d i s c u s s i o n .................................. A f i r s t a l g o r i t h m f o r s o l v i n g (P)

2.7.1

258

261

262 263

264

R e l a t i o n s between a l g o r i t h m s ( 2 . 1 7 ) - ( 2 . 2 0 ) , ( 2 . 2 1 ) - ( 2 . 2 5 ) and c e r t a i n a l t e r n a t i n g

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

d i r e c t i o n methods 2.7.2

i n t e r m s of t h e n u m e r i c a l

and ( 2 . 2 1 ) - ( 2 . 2 5 )

2.7.3 3

.

264

I n t e r p r e t a t i o n of algorithms ( 2 . 1 7 ) - ( 2 . 2 0 )

........... ........................... n o n l i n e a r e l a s t i c i t y . (I) L a r g e -

i n t e g r a t i o n of e v o l u t i o n e q u a t i o n s

265

Further discussion

265

A p p l i c a t i o n s i n finite

d i s p l a c e m e n t c a l c u l a t i o n o f t h e e q u i l i b r i u m p o s i t i o n s of ineztensibLe. 3.1

fLezible pipelines

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

266

3.1.1

266

3.1.2

.......................... G e n e r a l d i s c u s s i o n ........................... S i m p l i f y i n g a s s u m p t i o n s ......................

3.1.3

Modelling o f s t a t i c problems

267

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

3.2

R e s u l t s on t h e e x i s t e n c e of s o l u t i o n s f o r t h e

3.3

Numerical s o l u t i o n of t h e s t a t i c problem .

3.4


s t a t i c problem

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

( I ) General n o t e s

(11) Approximation

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

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

3.4.1

Approximation of t h e s p a c e H2(0,L) and

3.4.2

Approximation of

3.4.3

Approximation o f problem ( 3 . 2 )

3.4.4

Convergence of t h e approximate s o l u t i o n s

the functional J

3.5

266

F o r m u l a t i o n of t h e problem

............................ B .........................

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

267

268

269 270 270 271 272 273


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

(111) I t e r a t i v e methods of s o l u t i o n

274

3.5.1

G e n e r a l n o t e s and s y n o p s i s

274

3.5.2

S o l u t i o n o f problem ( 3 . 2 ) by an augmented Lagrangian method

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

2 76

TABLE OF CONTENTS

xviii 3.5.3 3.5.4 3.6

4

.

k r ........... 277 A second iterative method using d r .......... 278

A first iterative method using

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

280 280

Numerical experiments 3.6.1 Description of the test problem 3.6.2

Further information concerning the numerical solution

280

3.6.3 3.6.4

Presentation of the numerical results Further discussion

281 284

Applications i n f i n i t e

........................... ........ ........................... n o n l i n e a r e l a s t i c i t y . (Is) T'wo

dimensional c a l c u l a t i o n s i n v o l v i n g large displacements and l a r g e s t r a i n s f o r i n c o m p r e s s i b l e m a t e r i a l s of Mooney-Rivlin

4.1 4.2

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

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

285 285 285 285 286

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

281

Formulation by equilibrium equations.

287 288

type Synopsis Formulation of the problem 4.2.1 Notation Mechanical assumptions 4.2.2 Mathematical formulations 4.2.2.1 Formulation by minimisation of the energy functional

.

4.2.2.2 4.2.2.3 4.2.2.4

4.3

4.4 5

.

Formulation by augmented Lagrangian On some relations between formulations (4.6), (4.8) and (4.12) Solution of problem (4.12) 4.3.1 A first algorithm for solving (4.12) 4.3.2 A second algorithm for solving (4.12) Numerical tests

.

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

Some r e m a r k s o n t h e a p p l i c a t i o n of

.

288 289 289 289 292

t h e a l g o r i t h m s of

S e c t i o n 2 t o t h e s o l u t i o n of e i g e n v a l u e and e i g e n v e c t o r problems Chapter 9

:

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

293

Applications of the method of multipliers to D.Gabay. variational inequalities

299

. ............ 1. I n t r o d u c t i o n ............................................. 1.1 Monotone operators .................................. 1.2 The method of multipliers ........................... The p r o x i m a l p o i n t a l g o r i t h m ............................. 2. 3. 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 d u a l i t y ...................... 4.

The m e t h o d of m u l t i p l i e r s f o r v a r i a t i o n a l i n e q u a l i t i e s

...

299 299 301 305 310 313

TABLE OF CONTENTS

5

.

D e c o m p o s i t i o n b y m u l t i p l i e r s : (I) A l t e r n a t i n g d i r e c t i o n methods

5.1

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

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

......................... D e c o m p o s i t i o n b y m u l t i p l i e r s : (II) P r o j e c t i o n m e t h o d s .... G e n e r a l d i s c u s s i o n .......................................

References

319

The Peaceman-Rachford variant of the method of multipliers: algorithm ALG3

. 7.

318

The Douglas-Rachford variant of the method of multipliers: algorithm ALG2

5.2

6

xix

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

323 326 330 333

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CHAPTER I

AUGMENTED L A G R A N G I A N METHODS I N Q U A D R A T I C PROGRAMMING

M. F o r t i n , R .

1.

Glowinski

P R I N C I P L E S OF THE METHOD

I n t h i s c h a p t e r and w i t h a view t o s i m p l i f y i n g t h e p r e s e n t a t i o n ,

w e s h a l l l i m i t ourselves t o a p a r t i c u l a r l y simple finite-dimensional problem: Let

be a symmetric, p o s i t i v e d e f i n i t e

A

that

b

E

with

IRN;

A

J: IRN +IR

functional

J(v)

(1.1)

=

and

b

d e f i n e d by

1 (Av,v) 2

-

(b,v),

where i n (1.1), ( .

,. )

in

be a l i n e a r mapping from IRN

IRN.

Let

B

m a t r i x and suppose

N x N

we associate the quadratic

denotes t h e canonical Euclidian inner product

b e i n g i d e n t i f i a b l e w i t h an

M

matrix.

x N

i n t o ,'RI

t h i s thus

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

m i n i m i s a t i o n problem (1.2)

I

J ( u ) 5 J(v)

vv

E

Ker B = { v e R N , Bv=O},

u ~ K e rB.

I t is a classical r e s u l t t h a t

(1.2)

admits a u n i q u e s o t u t i o n .

F o l l o w i n g a w e l l known t e c h n i q u e , we i n t r o d u c e a L a g r a n g e

multiplier problem',

p

t

IRM

which t r a n s f o r m s

( 1 . 2 ) i n t o an u n c o n s t r a i n e d

namely

W e a l s o use

(.,.)

f o r t h e inner product i n

d a n g e r of a m b i g u i t y .

1

BM, t h e r e b e i n g no

2

AUGMENTED LAGRANGIAN METHODS

(1.3)

{J(v) + (p,Bv)}.

Min V€RN

The L a g r a n g e m u l t i p l i e r f o r example, problem.

1)

(CHAP.

p

a p p e a r s as a n e x t r a unknown which may,

be obtained through t h e s o l u t i o n of a saddLe-point : b x RM + R

More p r e c i s e l y , w e d e f i n e

and w e r e c a l l t h a t

{u,p} w i l l be a s a d d l e - p o i n t

of

by

2 on lRN x IRM,

if

and a l s o t h a t ( 1 . 5 ) i m p l i e s

It c a n be shown t h a t I?N x IRM, where

u

d

saddle-points of

2

a d m i t s a t l e a s t one s a d d l e - p o i n t

i s t h e s o l u t i o n of on

IRN x lRM.

c o n d i t i o n of u n i q u e n e s s f o r

(1.2)

{u,p}

on

a n d i s common t o a l l t h e

A n e c e s s a r y and s u f f i c i e n t

{u,p},

in fact for

p, is t h a t

B

be

Rank B = M.

s u r j e c t i v e , i.e.

The f o l l o w i n g ( c l a s s i c a l ) r e s u l t i s e s s e n t i a l t o t h e s u b s e q u e n t discussion : THEOREM 1.1:

e x i s t e n c e of

I

(1.7)

The r e l a t i o n s

on

mN

x

mM.

p

The s o l u t i o n E

lRM

u

of

(1.2) i s c h a r a c t e r i s e d by t h e

such t h a t

t Au + B p = b, Bu = 0.

( 1 . 7 ) aLso c h a r a c t e r i s e a l l t h e saddZe-points

of

P

A FIRST ALGORITHM

2)

( S E C.

3

Following HESTENES 1 1 1 , and POWELL 111 w e i n t r o d u c e t h e augmented Lagrangian ( 1 -8)

= J(v)

Zr(v,q)

1.1

where, i n (1.8),

dr defined, f o r +

(q,Bv)

+

$

r

lBvl

2

>

by

0,

=d(v,q)

d

1.1:

(1.9)

Bv = 0

, M IR

= J(v) +

Lr

.

is a saddlerlBvI2

is satisfied).

I t s h o u l d be n o t e d t h a t f o r

L,(v,o)

2

( t h i s i s due t o t h e f a c t t h a t

and v i c e - v e r s a

v a n i s h e s when t h e c o n s t r a i n t Remark

$Bvl

d e n o t e s t h e c a n o n i c a l E u c l i d i a n norm on

I t can e a s i l y be proved t h a t any s a d d l e - p o i n t of

p o i n t of

+

q = 0

w e have

5 IBvl 2 ,

t h i s being the c l a s s i c a l p e n a l i s e d f u n c t i o n a l r e l a t i v e t o the constraint

Bv = 0.

The advantage of t h e augmented Lagrangian i s t h a t , because of t h e presence of t h e t e r m

(q,Bv),

t h e e x a c t S o l u t i o n of problem ( 1 . 2 )

r

can be d e t e r m i n e d w i t h o u t making

tend t o i n f i n i t y , unlike

o r d i n a r y p e n a l i s a t i o n methods where t h i s h a s t h e e f f e c t of c a u s i n g a d e t e r i o r a t i o n i n t h e c o n d i t i o n i n g of t h e systems t o be s o l v e d . Furthermore t h e a d d i t i o n of t h e q u a d r a t i c t e r m Lagrangian

2

5IBvl

t o the

w i l l improve t h e convergence p r o p e r t i e s of t h e

d u a l i t y algorithms described l a t e r . Remark 1 . 2 :

The case where

s i n c e i n t h i s case

u = 0

B

i s i n j e c t i v e i s of no i n t e r e s t

i s t h e unique s o l u t i o n o f ( 1 . 2 ) . K e r B # {O}.

In t h e

f o l l o w i n g t e x t w e s h a l l t h u s assume

2.

A FIRST ALGORITHM FOR SADDLE-POINT CALCULATION

2.1

D e s c r i p t i o n of t h e a l g o r i t h m

I t f o l l o w s from S e c t i o n 1 t h a t t h e r e i s e q u i v a l e n c e between s o l v ing (1.2)

and f i n d i n g a s a d d l e - p o i n t of

d,

on

IRN x I R M ; from

4

(CHAP. 1)


ARROW-HURWICZ-UZAWA

r l l , GLOWINSKI-LIONS-TFEMOLIEFESz

S e c t i o n 4 1 , e t c , such a s a d d l e - p o i n t

[I, Chapter 11,

can be c a l c u l a t e d u s i n g t h e

f o l l o w i n g a l g o r i t h m , t h e v a r i a n t s of which w e s h a l l d e n o t e i n t h e f o l l o w i n g t e x t under t h e g e n e r a l name of Uzawa's a l g o r i t h m : p o RM, ~

(2.1)

with

pn

known,

specified arbitrarily; n+l un t h e n p by

calculate

J

(2.2)

(2.3)

p n + ] = pn + p,

W e note t h a t ( 2 . 2 )

2.2

$ 0 .

is equivalent t o

(A + r B t B ) u n

(2.4)

,~

Bun

+ Btpn

= b.

Convergence r e s u l t s

Regarding t h e convergence of t h e a l g o r i t h m , w e now prove t h e following: THEOREM 2 . 1 : For O r and t h a t f o r t h e same v a l u e of r a l g o r i t h m opt i s i t e r a t i v e l y f a s t e r w i t h p g i v e n by ( 2 . 4 8 ) t h a n w i t h

r; of c o u r s e ,

( 2 . 4 8 ) i n v o l v e s Am

g e n e r a l a r e n o t known a p r i o r i . c hoice of

p,

with (2.46)

r e p l a c e d by ( 2 . 5 0 )

W e now go t o f i n i s h o f f Remark 2 . 4 ,

t o d e f i n e some n o t a t i o n .

IlLlI

but f i r s t it is appropriate

W e s h a l l d e n o t e by

IvI

and f o r a l i n e a r o p e r a t o r

IRN

w e s h a l l d e n o t e by

q u a n t i t i e s which i n

i s again valid f o r t h i s

c h o i c e of r .

C o n d i t i o n n u m b e r of A r ;

E u c l i d i a n norm on

AM,

and

Remark 2 . 4

llLl1

the standard L

d e f i n e d on

1.1 ,

the norm a s s o c i a t e d w i t h

N IR

namely

=

V € S

where

F o r t h e c o n d i t i o n number of

Ar

when

r

+

+

m,

w e t h e n have t h e

following: PROPOSITION 2 . 3 :

T h e c o n d i t i o n n u m b e r &(Ar)

(2.51)

where

Proof:

W e have &(Ar)

= llA,ll

I t c a n e a s i l y b e shown t h a t

Il

*

wi(p),

b o t h r o o t s a r e r e a l and w e deduce from ( 4 . 1 7 ) t h a t

ci+

and

asymptotic i n the

(5,

t h e g r a p h s of equations a r e

ci-

w)

a r e a r c s of h y p e r b o l a s , r e s p e c t i v e l y p l a n e t o t h e s t r a i g h t l i n e s whose

(SEC.

4)

VARIANTS OF METHODS

29

\

2

AUGMENTED LAGRANGIAN YZTHODS

30

(CHAP.

W e have shown i n F i g u r e s 4 . 1 and 4 . 2 t h e b e h a v i o u r of max ( a s a f u n c t i o n of >

p

I

r+'i

w,

when, r e s p e c t i v e l y ,

the scalar 2(1

Max

wi

(4.20)

pXi I + ( p + r ) Xi

IEfI

I t can e a s i l y b e shown t h a t

z

1

(P)

f

i i n F i g u r e 4.1 h a s t h e v a l u e

uit

-

r+

wi

(p)

and

Xi,

(4.21)

et(p) =

If we p l o t ,

w

= wi,

f o r given and t h a t it

4 7 -_Il+(r-p)Xi I w.(P)

a s a f u n c t i o n of

p,

I +( r + p ) Xi

t h e convergence r a t e s

corresponding t o t h e d i f f e r e n t eigenvalues (4.21),

where

I+(r+PIX 2 i

t h e convergence r a t e i s o p t i m a l if

is then equal t o

and

+ rX.

I t can t h e n b e shown (see a l s o t h e above f i g u r e s ) t h a t , p

1 ti+ 1 , 1 Ei+ I )

.

)

1 if

1)

t h e g r a p h s i n F i g u r e 4.3.

Xi,

*

Oi

w e o b t a i n , from

(SEC. 4 )

31

VARIANTS OF METHODS

r , t h e convergence r a t e w i l l be w i l l be t h e c a s e i f p = P where

I t then f o l l o w s t h a t f o r a given O;(p)

optimal i f

; this

= e;(p)

opt

(4.22)

S u b s t i t u t i n g t h i s optimal value of

Xi

(or

=

X m ) , we o b t a i n f o r

p

i n t o (4.19),with

Ai

=

AM

t h e optimal value

w

(4.23)

which g i v e s u s t h e o p t i m a l convergence r a t e

(4.24)

t h e asymptotic r a t e and w = w opt' opt of convergence i s b e t t e r t h a n t h e o p t i m a l convergence r a t e of a l g Remark 4 . 1 :

For

p = p

o r i t h m ( 2 . 1 ) - ( 2 . 3 ) , which c o r r e s p o n d s t o t h e c a s e particular,

a

=

Xm/XM),

for I -a I+a

Remark 4.2: W*(P),

r

=

J = 1.

0, the rates are respectively,

i n the case

w = 1

and

'

-

In

(writing

i n t h e case

w

w

1 +&

For g i v e n

is given, i f

p

5

p,

popt,

(4.25)

g i v i n g t h e convergence r a t e

t h e b e s t c h o i c e of by

w,

d e n o t e d by

opt.

32


(CHAP.

(4.26)

and for

(4.27)

g i v i n g t h e convergence r a t e

I n a l l c a s e s , t h e maximal v a l u e of

w

( i . e . t h e v a l u e above which

t h e a l g o r i t h m d i v e r g e s ) i s g i v e n by

(4.29)

F i g u r e 4 . 4 shows t h e g r a p h s of

w* ( p )

Figure 4.4

and of

wMax

(p)

.

1)

( SE C.

VARIANTS O F METHODS

4)

W e note t h a t i f

1

2(r + -)

>

p

is precisely w e have

we have

44 for

corresponds t o t h e f a c t t h a t

33

wMax(p)

This

< 1.

w = 1 t h e maximal v a l u e o f 2.2).

2 (r

Further, i f

44 4 4 '

pop t

T h i s w i l l b e t h e case, i n p a r t i c u l a r , f o r

r

=

p

1 3r < -

0 , when

w

t h e parameter v a l u e s of

g u a r a n t e e s t h e c o n v e r g e n c e of

p,

(4.1)-(4.4)

as l o n g a s w e t a k e

Remark 4.3:

w

(4.1)-(4.4)

(4.1)-(4.4)

Algorithm

c a n b e w r i t t e n i n t h e form

(4.31)

pn+l = pn + p ( ~ B u " + " ~ + ( I w ) B u n ) .

= A;'

(b-Btpn),

-Bun+$

Observing t h a t , with t h e n o t a t i o n of S e c t i o n 3, the gradient

gn

of

J*

at

p"+l

= p"

(4.32)

Writing

1 n

= -(p

-= p

P

pw

-p

n-1

and

In

sufficiently small.

Un+l

BU"

for

f o r any p o s i t i v e

(4.30)

since

of

g r e a t e r t h a n t h e bounds d e t e r m i n e d i n S e c t i o n 2 .

p

f a c t , w e can o b t a i n convergence o f v a l u e of

4.

>

This p o i n t c l e a r l y i l l u s t r a t e s t h e f a c t t h a t t h e introductio;

is i n f a c t

p n , we deduce from ( 4 . 3 1 ) t h a t :

- pw(gn

(I+)

n

(P -P

n-l

))

).

A

, PW

= (Iw)

we then obtain

p n + l = pn + E ( 9 , + h ( P n - p " - 5 ) .

(4.33)

W e t h u s o b t a i n , f o r p a r a m e t e r s s u i t a b l y c h o s e n a n d d e p e n d i n g on

t h e c o n j u g a t e - g r a d i e n t method o f S e c t i o n 3.

n,

W e could also use i n

t h i s a l g o r i t h m v a r i a b l e p a r a m e t e r s c o r r e s p o n d i n g to, a s e m i -

i t e r a t i v e Chebychev method. Remark 4.4:

I t i s p o s s i b l e t o o n c e more re-write

(4.31), i n the

form

Writing (4.35)

-

pn+I = p"-l

(4.34)

3

= (2-w)

and

a

=

(2-w)(pw 2-w

f i , we 2-w

pn+l = p n- 1 -D(ag,-pn+pn-l)

Algorithm ( 4 . 1 ) - ( 4 . 4 )

n

-pn+pn-l)*

obtain

.

is thus equivalent t o t h e two-step Richardson

aZgorithm applied t o t h e minimisation of t h e dual f u n c t i o n a l

*

J

.

34


(CHAP.

1)

I t c a n b e shown t h a t , s u b j e c t t o making t h e a p p r o p r i a t e c h a n g e s of n o t a t i o n , t h e optimal parameters given i n (4.22)

- (4.23)

correspond

e x a c t l y t o t h e s t a n d a r d o p t i m a l parameters i n R i c h a r d s o n ' s method

(see GOLUB C11).

4.3.1

rn

General notes

I n t h i s s e c t i o n w e s h a l l f i n i s h o f f Remark 2 . 5 , algorithm (4.6)-(4.8) (2.1)-(2.3)

since i n f a c t

corresponds t o t h e v a r i a n t of algorithm

o b t a i n e d when, i n t h e c a l c u l a t i o n o f

a s i n g l e i t e r a t i o n of a gradient-type

sr),

un+l, w e use only

algorithm (with t h e

un. W e r e c a l l also (see i s a p a r t i c u l a r case o f ( 4 . 6 ) - ( 4 . 8 ) - Ar. I n t h e case w h e r e Sr = I , w e g e t b a c k c o r r e s p o n d i n g t o Sr t o t h e s t a n d a r d method, c a l l e d t h e A r r o w - H u r w i c z method, i n t r o d u c e d I n t h e case where [ d n / o n = 6 , Vn, i n ARROW-HURWICZ-UZAWA E l l . w e r e c a l l t h a t t h i s method c o n s i s t s of s e a r c h i n g f o r t h e s a d d l e p o i n t s o f t h e L a g r a n g i a n Zr v i a t h e a p p r o x i m a t e i n t e g r a t i o n o f t h e auxiliary operator

Section 4 . 2 )

that

s t a r t i n g from

(4.1)-(4.4)

differential system

(4.36)

I n t h e c a s e w h e r e w e d i s c r e t i s e u s i n g an E u l e r t y p e scheme, w e o b t a i n a scheme w h i c h i s s l i g h t l y d i f f e r e n t f r o m ( 4 . 6 ) - ( 4 . 8 ) , namely (4.37) (4.38)

pn+l = pn

The scheme ( 4 . 6 ) - ( 4 . 8 )

+ pnBun. c a n t h u s b e c o n s i d e r e d as a " s e m i - i m p l i c i t "

v a r i a n t of t h e E u l e r scheme ( 4 . 3 7 ) ,

4.3.2

Reduction of

(4.6)-(4.8)

order d i f f e r e n t i a l system

(4.38).

t o t h e d i s c r e t e form o f a second-

(SEC.

4)


W e s h a l l assume f o r s i m p l i c i t y t h a t

35

= w,

wn

pn

= p,

Vn.

We

t h e n d e d u c e from ( 4 . 7 1 , by s u b t r a c t i ' o n ,

t h e n from ( 4 . 8 )

hence

Again w r i t i n g

iin = un

-

u

Bu = 0 , w e deduce from

and n o t i n g t h a t

(4.39) t h a t

W e observe t h a t

( 4 . 3 9 ) i s a d i s c r e t i s e d form, w i t h

u s i n g an e x p l i c i t scheme, o f t h e s e c o n d - o r d e r 2

7 d u +

(4.41)

wS;]A

dt

*

r dt

A t = 1

and

d i f f e r e n t i a l system

+ pus - 1 B tB u = O wSilAr

W e n o t e t h e p r e s e n c e o f t h e damping term

and f u r t h e r m o r e

t h a t , o t h e r t h i n g s being equal, t h e n a t u r a l frequencies of t h e undamped s y s t e m grow w i t h

pw.

may e x p e c t t h a t t h e b e h a v i o u r o f

Looking a t ( 4 . 4 0 )

i?

=

-

un

and ( 4 . 4 1 ) , w e

u , as a f u n c t i o n of

w i l l d e p e n d i n c o m p l i c a t e d f a s h i o n on t h e p a r a m e t e r s

w

and

n,

p.

W e r e c a l l t h a t e v e n i n t h e case o f t h e s c a l a r e q u a t i o n

..x + 2 k* x + w2 x = 0 t h e behaviour of t h e s o l u t i o n , as

c r i t i c a l damping.

A p r i o r i , when

, t

-+

commute t h e s t u d y o f t h e b e h a v i o u r o f

s p e c t r a l methods,

+

m,

SLIAr iin

brings i n t h e notion of and

S;lBtB

do n o t

as a f u n c t i o n o f

n , by

(see S e c t i o n 4 . 2 ) would s e e m i m p r a c t i c a b l e f o r t h e

moment; t h e r e f o r e i n t h e f o l l o w i n g s e c t i o n , w e s h a l l u s e e n e r g y

i?.

methods t o s t u d y t h e convergence t o z e r o of

4.3.3

C o n v e r g e n c e of a l g o r i t h m ( 4 . 6 ) - ( 4 . 8 ) .

W e s h a l l now s e e k c o n d i t i o n s on convergence o f

p

and

w

which a s s u r e t h e

( 4 . 6 ) - ( 4 . 8 ) ; w e f i r s t d e f i n e some n o t a t i o n .

36


(CHAP.

1)

With t h e s t a n d a r d E u c l i d i a n i n n e r p r o d u c t s t i l l w r i t t e n as ( . , . ) ,

we associate w i t h t h e symmetric p o s i t i v e - d e f i n i t e

operator

Sr

t h e norm (4.42)

IVI

2

=

.

(SrV,V)

r W e likewise w r i t e

11 vII

(4.43)

2

(Arv,v).

=

W e t h e n h a v e , by t h e e q u i v a l e n c e o f t h e norms on

2 IlvIlr

(4.44)

lql

2

B

the existence

such t h a t

o f a c o n s t a n t cir

The o p e r a t o r

, ' R I

2 arlVIS

*

b e i n g c o n t i n u o u s , w e have f u r t h e r m o r e ( w i t h

(SlS))

=

IBVI

(4.45)

2 5

B,

IIVII

2 r

and IBvI

(4.46)

W e next define

i?,

pn

8"

and

2

2 '~rIvIs

p"

.

as i n ( 4 . 9 ) a n d w e e s t a b l i s h t h a t

s a t i s f y the equations

(;n+l

(4.47)

-n -u ,Srv) + w(ArGn,v)

(pn+l-pn,q)

(4.48)

-

p(B;"+l,q)

+ w(p",Bv) =

0

vqcRM

= 0 VvrR

En

by

,

.

I n t h e following t e x t , w e s h a l l f o r s i m p l i c i t y denote and

N

En

by

u

pn. n

( 4 . 4 7 ) a n d w e m u l t i p l y by

We t h e n s e t

v = u

Furthermore,

from ( 4 . 4 8 ) , w e h a v e

2pw(pn,Bun)

=

in

2p;

we obtain

2w(p n ,pn-pn-I ) = w I p n 1 2 ~ ( P n - 1 ( 2 + w ( P n - P n - 1 1 ~

and by s u b s t i t u t i n g i n ( 4 . 4 9 ) w e t h e r e b y d e d u c e

n

(SEC.

4)


37

I = (4.50) n+I

= PJU

-u

n 2

Is . r

W e now h a v e t o o b t a i n a n a p p r o p r i a t e u p p e r b o u n d f o r t h e r i g h t - h a n d

s i d e ; f r o m (4.47) we have (4.51)

pI~"+~-u~l= f

r

-

-

pw(Arun,un+'-un)

t h e n , by a v a r i a n t o f t h e Cauchy-Schwarz

pw(pn,Bun+'

-

Bun)

,

i n e q u a l i t y and from ( 4 . 4 4 ) ,

w e have (4.52) Ipw(Arun,un+'-un)I

5

Is

p ~ l n+l u -u n 2 r

+ k%-4 €

Q.

n 2 r Ilu II, *

VE

' 0.

W e f u r t h e r deduce from ( 4 . 4 8 ) t h a t

Substituting (4.52),

(4.53) i n t o (4.51) w e o b t a i n

(IPn+l-Pn12

+

n n-1 2 I ) IP -p

,v

Next, r e p l a c i n g t h e r i g h t - h an d s i d e o f

0
o

1 -E*

n = 1 , . ..,fi,

we o b t a i n

1)

(SEC. 4)

VARIANTS OF METHODS

39

The left-hand side contains the term R+1 2 R+1 R 2 (IP -IP -P 1

I

(4.63)

I

2(1-E)

for which we shall now obtain a lower bound. (4.64)

IP

I -IP

-P

I

=

-

IpA12+2(pR+1,pR) =

-

To do this we write IpR12+2(p

R +pBu R + l , pR ) ,

We then substitute this lower bound into (4.62), regrouping the terms in lpRI2 and utilising (4.46), i.e. ~~61Bu''~)~ We then have, with 6 stiil to be chosen,

If we then suppose E = positive, which implies

(4.67)

E*,

6 2

5

p26y,luR+lI; r

.

the coefficient of lpNI2 must be

-=-

1

2( 1- E* )

2+PBr 2+2pur

'

On the other hand, in order for the coefficient of ain positive, we must have

IuN+'I2

to rem-

C o n s e q u e n t l y , i f w e can c h o o s e

1)

(CHAP.


40

so t h a t

6

(4.69)

t h e n c o n d i t i o n (4.68) w i l l i n f a c t be a consequence o f t h e r e f o r e have t o seek

6

(4.60).

s a t i s f y i n g simultaneously (4.67)

We

and

(4.69), i.e.

(4.70)

We t h u s n e e d t o show t h a t t h e i n t e r v a l d e f i n e d by ( 4 . 7 0 ) i s n o t

.

empty, i e . t h a t

or, regrouping t h e t e r m s , t h a t (4.72)

Br(Y,-a,B,)PL+2(Yr-arBr)P

I t can e a s i l y be seen t h a t

-

a, s 0 .

(4.72) w i l l b e s a t i s f i e d f o r any

p

if

w e have (4.73)

y,

s a B

r r

.

Before confirming t h i s p o i n t , w e s h a l l conclude t h e proof of converg e n c e ; w e h a v e s e e n i n f a c t , a s s u m i n g ( 4 . 7 3 ) i s t r u e f o r t h e moment, t h a t s u b j e c t t o obeying t h e c o n d i t i o n (4.60) w e c a n choose t o have i n t h e l e f t - h a n d s i d e o f

6

( 4 . 6 6 ) o n l y p o s i t i v e terms.

then follows i n p a r t i c u l a r t h a t we have

(4.74)

which i m p l i e s t h e c o n v e r g e n c e of t h e s e r i e s w i t h g e n e r a l t e r m

IIunll:

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

so as It

41


(SEC. 4 )

(4.75)

Likewise,

i s bounded; w e c a n t h u s

it f o l l o w s from (4.66) t h a t

e x t r a c t from

lp’I2 a c o n v e r g e n t sub-sequence.

pn

prove, as i n S e c t i o n 2 , t h a t pn n and p a c t u a l l y d e n o t e tin and

We shall in fact

converges t o zero (recall t h a t

fjn

here).

8

I t now r e m a i n s f o r u s t o p r o v e t h e i n e q u a l i t y

t h i s w e must f i r s t d e t e r m i n e t h e c o n s t a n t s

un

(4.73).

Br

clr,

To do

yr.

and

It

c a n e a s i l y b e shown t h a t t h e s e a r e r e s p e c t i v e l y t h e l a r g e s t e i g e n v a l u e s o f Sr-1A r , Ar-1B t B and Si1BtB. The i n e q u a l i t y ( 4 . 7 3 ) c a n t h u s be w r i t t e n , i f

denotes t h e s p e c t r a l r a d i u s of t h e

p(M)

m a t r i x M, a s

T h i s i n e q u a l i t y i s o n l y o n e p a r t i c u l a r case o f t h e f o l l o w i n g g e n e r a l result: LEMMA 4 . 1 :

I f

A

and

B

a symmetric p o s i t i v e - d e f i n i t e P(AB)

(4.77)

Proof:

5

a r e two s y m m e t r i c m a t r i c e s and

~(Ac)P(c-’B).

W e i n f a c t have

Remark 4.5:

is

m a t r i x , we h a v e

P(AB)

=

P(C’/2ABC-1/2)

r a d i u s b e i n g i n v a r i a n t under a change o f b a s i s ;

Sr = A r ,

C

,

the spectral

furthermore

I n t h e case (which w a s s t u d i e d i n S e c t i o n 4 . 2 )

t h e c o n d i t i o n ( 4 . 5 9 ) c l e a r l y covers t h e c o n d i t i o n

where

(4.29);

t h u s i t i s p r o b a b l e t h a t t h e b e s t p o s s i b l e r e s u l t h a s been o b t a i n e d . 8

Remark 4.6:

It should be noted t h a t (4.61) allows

t a k e n a s l a r g e a s d e s i r e d , as l o n g a s w e t a k e

small.

w

p

t o be

sufficiently

T h i s r e s u l t may seem p a r a d o x i c a l b e c a u s e t h e a l g o r i t h m may

b e c o n s i d e r e d a s a v e r s i o n of Uzawa’s a l g o r i t h m w i t h a n i n c o m p l e t e solution.

B u t , i n t h e c o m p l e t e - s o l u t i o n case w e h a v e s e e n i n

Section 2 t h a t

p

must be t a k e n s u f f i c i e n t l y s m a l l .

8

42

(CHAP. 1)


Remark 4.7:

It i s c l e a r t h a t t h e s p e e d o f t h e c o n v e r g e n c e w i l l

depend e s s e n t i a l l y on t h e c h o i c e of t h e a u x i l i a r y o p e r a t o r The s i m p l e s t c h o i c e s a r e of c o u r s e

Sr

and Sr = A r .

= I

a l s o show t h a t i n t h e case where t h e s y s t e m

Sr. W e can

t n Arun = -B p +b

s o l v e d by a method o f s y m m e t r i c s u c c e s s i v e o v e r r e l a x a t i o n

is

(SSOR),

( f o r w a r d and b a c k ) i s e q u i v a l e n t t o

p e r f o r m i n g one d o u b l e p a s s

which c a n b e c o n s t r u c t e d e x p l i c i t l y and ( 4 . 6 ) f o r an o p e r a t o r Ss F o r more d e t a i l s which i s c l e a r l y s y m m e t r i c p o s i t i v e - d e f i n i t e . 0. AXELSSON C11 c a n b e c o n s u l t e d .

Remark 4 . 8 :

W

The c h o i c e o f t h e o p t i m a l p a r a m e t e r s

algorithm (4.6)-(4.8)

for the

i s , t o t h e b e s t o f o u r knowledge,

p r o b l e m ( e x c e p t i n t h e case Sr = A r ,

see S e c t i o n 4 . 2 ) .

an o p e n We refer

t h e r e a d e r t o C h a p t e r I1 i n which w e s h a l l g i v e some i n f o r m a t i o n (of e x p e r i m e n t a l o r i g i n ) on t h i s s u b j e c t , i n c o n n e c tio n w i t h 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 S t o k e s and t h e N a v i e r - S t o k e s

5.

equations.

MISCELLANEOUS REMARKS AND DISCUSSION

Remark 5 . 1 :

A l l t h e d i s c u s s i o n i n S e c t i o n s 1 , 2 , 3 and 4 i s

s t i l l valid i f we consider,

i n s t e a d o f p r o b l e m (1.2), t h e p r o b l e m

(5.1) with

K

(5.2)

= {VER N , Bv = c )

,

c~IrnB.

I t is actually s u f f i c i e n t t o replace (A+rBtB)u"

= b-Btpn,

(5.3) pn+l = p" + p Bun,

by (A+rBtB)u" = b-Btpn+rBtc, (5.4)

P"+l

Remark 5 . 2 :

Suppose

necessariZy symmetric,

= pn + p (Bun-c)

.

A E d(Rn,Rn) i s p o s i t i v e d e f i n i t e , not

and suppose t h a t

K

i s d e f i n e d by ( 5 . 2 )

It

(SEC. 5 )

43

MISCELLANEOUS REMARKS

can be shown t h a t t h e v a r i a t i o n a l problem6

1

(5.5)

(Au,v-u)

2 (b,v-u)

VVEK,

ueK

a d m i t s one and o n l y one s o l u t i o n 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 p

BMs u c h t h a t

E

t A u + B p = b ,

(5.6)

Bu = c .

I n view of

( 5 . 6 ) , w e can a p p l y t o t h e s o l u t i o n of

(5.5) t h e alg-

orithm (5.7)

with

po pn

E

chosen a r b i t r a r i l y ; n+l then p , by

JRM

known, c a t c u t a t e

un,

(5.8)

t n (A+rBtB)un = b-B p

(5.9)

P

n+l

p

n +

pn(Bun-c)

By p r o c e e d i n g a s f o r Theorem 2 . 1 ,

+ sBtc,

,

pn 2 0

.

i t can e a s i l y b e shown t h a t a l g -

o r i t h m ( 5 . 7 ) - ( 5 . 9 ) c o n v e r g e s , w h a t e v e r t h e v a l u e of

PO,

subject t o

the condition t h a t

BL

where

i s d e f i n e d by

B2

(5.11)

= Max

vfO

(A,,v,v)

and where i n (5.11), Au i s t h e symmetric component of A , i . e . Au =

71

By c o n t r a s t , a " f i n e l y d e t a i l e d " s t u d y .of conver-

(A+At).

gence r a t e s s e e m s much more d i f f i c u l t , s i n c e t h e s p e c t r a l methods of S e c t i o n 2 . 2

c a n n o t t h e n be used.

Likewise, t h e e x t e n s i o n t o

problem ( 5 . 5 ) of t h e v a r i a b l e s t e p - l e n g t h

and c o n j u g a t e - g r a d i e n t

methods of S e c t i o n 3 may pose d i f f i c u l t i e s ; t h i s a p p l i e s p a r t i c u l a r l y t o t h e c o n j u g a t e - g r a d i e n t method. The proof of t h e convergence o f a l g o r i t h m ( 4 . 6 ) - ( 4 . 8 ) , b a s e d on energy e q u a l i t i e s and i n e q u a l i t i e s , e x t e n d s w i t h o u t t o o many e x t r a d i f f i c u l t i e s t o t h e c a s e where i n ( 4 . 7 )

A

is positive-definite,

T h i s i s an e l e m e n t a r y case of a v a r i a t i o n a l i n e q u a l i t y ( s e e f o r example G.L.T.

C11,

C21).

44

(CHAP. 1)


non-symmetric. I n C h a p t e r I1 w e s h a l l u t i l i s e , f o r t h e s o l u t i o n o f t h e N a v i e r Stokes equations, algorithms of t h e type (4.6)-(4.8)

with

Remark 5.3:

A

(5.7)-(5.9)

and

non-symmetric.

I n c e r t a i n p r o b l e m s i t may b e a d v a n t a g e o u s , as

s u g g e s t e d by FLETCHER 1 1 1 , t o u s e i n s t e a d o f t h e p e n a l i s a t i o n t e r m rlBvI2

a term o f t h e f o r m

(RBv,Bv)

where t h e m a t r i x

R

symmetric, p o s i t i v e - d e f i n i t e , and " a p p r o p r i a t e l y chosen". c l e a r l y a p p a r e n t t h a t t h e i n t r o d u c t i o n of

R,

is It is

however, w i l l

c o m p l i c a t e t h e s t u d y of c o n v e r g e n c e and o f t h e c h o i c e o f o p t i m a l parameters for t h e algorithms of Sections 2 , 3 , Remark 5 . 4 :

and 4 .

H

The augmented L a g r a n g i a n methods i n t r o d u c e d by

HESTENES 111 and POWELL 111 h a v e g i v e n r i s e t o a l a r g e number Of w o r k s , which i t would b e q u i t e i m p o s s i b l e t o r e c o r d i n d i v i d u a l l y . W e t h u s r e f e r t h e r e a d e r t o t h e t i t l e s g i v e n below and t o t h e corresponding b i b l i o g r a p h i e s . W e f i n d i n ROCKAFELLAR [l],

1 2 1 , 1 3 1 , a s t u d y o f augmented

L a g r a n g i a n m e t h o d s , 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 convex and non-convex

f u n c t i o n a l s , w i t h convex i n e q u a l i t y c o n s t r a i n t s ( p o s s i b l y

nonlinear).

T h i s s t u d y i n t r o d u c e s t h e augmented L a g r a n g i a n method

w i t h i n t h e framework o f t h e t h e o r y o f d u a l i t y i n convex a n a l y s i s . W e f i n d a l s o i n BERTSEKAS C11,

C21 a s t u d y o f t h e c o n v e r g e n c e of

algorithms closely r e l a t e d t o those considered i n Section w i t h i n a r a t h e r more g e n e r a l framework.

2 , but

I n KORT-BERTSEKAS [ 1 1

t h e r e are a l s o t o be found o t h e r p e n a l i s a t i o n p r o c e d u r e s a p p l i e d t o t h e c o n s t r u c t i o n o f augmented L a g r a n g i a n s o f a t y p e d i f f e r e n t from t h o s e c o n s i d e r e d i n t h i s volume.

F i n a l l y , t h e r e a d e r who d e s i r e s

a g e n e r a l view o f s o l u t i o n methods f o r c o n s t r a i n e d o p t i m i s a t i o n p r o b l e m s , i n c l u d i n g augmented L a g r a n g i a n m e t h o d s , c a n p r o f i t a b l y r e f e r t o G I L L a n d MURRAY C11. Remark 5 . 5 :

I n FORTIN 1 1 1 , 1 2 1 , G . L . T .

H

i l l , 121 and i n C h a p t e r

I1 o f t h e p r e s e n t volume t h e r e c a n be f o u n d s t u d i e s and a p p l i c a t i o n s

o f a l g o r i t h m s of t h e Uzawa and Arrow-Hurwicz t y p e s , t o t h e s o l u t i o n o f p r o b l e m s w h i c h a r e much more complex t h a n t h o s e c o n s i d e r e d i n t h i s chapter.

8

(SEC.

Remark 5 . 6 :

W e s h a l l round o f f Remark 5 . 1 by c o n s i d e r i n g t h e

c a s e where i n ( 5 . 2 ) we n o l o n g e r h a v e have

45

MISCELLANEOUS REMARKS

5)

K = pI

s i n c e it h a s no s o l u t i o n . (5.12)

po

t h e n for

n

2

E

E

We t h e r e f o r e

ImB.

C o n s i d e r , however, t h e a l g o r i t h m

BMc h o s e n a r b i t r a r i l y ;

0 , w i t h pn

(A+rBtB),"

(5.13)

c

i n t h i s c a s e , and problem ( 5 . 1 ) becomes i l l - p o s e d

=

kzown, d e f i n e

b+rBtc

-

un

by

pnfl

and

Btpn ,

W e can show t h a t under t h e c o n d i t i o n

w e have (5.16)

where

u

*

lim u" = u*, n-

i s t h e s o l u t i o n of t h e problem, { V ~ V E R ~ B,~ ( B ~ - = )=

U * ~ K *=

01

(5.17)

J(u*)

5

J(v)

Vv

where w e once a g a i n have W e know t h a t

K

*

(f

0)

E

K*,

J(v)

(Av,v)-(b,v).

=

i s t h e s e t of

the solutions of the

normal e q u a t i o n

(5.18)

B t Bz = Btc.

We can l i k e w i s e show t h a t t h e convergence of

un

to

u*

is linear

( i . e . a t l e a s t a s r a p i d a s t h a t of a g e o m e t r i c sequence w i t h r a t i o o, it f o l l o w s {pnIn A s r e g a r d s t h e sequence less t h a n 1 )

.

from ( 5 . 1 4 ) ,

~

and from t h e f a c t t h a t

l i k e an a r i t h m e t i c p r o g r e s s i o n . r a p i d " t h a n t h e convergence of

c p ImB, t h a t t h i s d i v e r g e s

T h i s d i v e r g e n c e i s "much less un, which means t h a t i n p r a c t i c e

t h e r e w i l l be no r i s k o f " o v e r f l o w " . The convergence r e s u l t s t a t e d above h o l d s o n l y f o r positive;

necessary.

r

strictly

t h e u s e o f a s t r i c t l y augmented Lagrangian i s t h e r e f o r e T h i s a l s o shows t h e r o b u s t n e s s of t h e methods d e s c r i b e d

46


(CHAP.

i n t h i s c h a p t e r , i n t h e p r e s e n c e p a r t i c u l a r l y o f r o u n d i n g errors. In actual f a c t t h e condition

c

E

ImB

can no l o n g e r b e s a t i s f i e d

e x a c t l y b e c a u s e o f t h e s e e r r o r s ; n o n e t h e l e s s t h e above c o n v e r g e n c e r e s u l t s show t h a t t h e augmented L a g r a n g i a n method r e m a i n s u s a b l e and p r o v i d e s t h e

b e s t p o s s i b l e r e s u l t (in t h e l e a s t s q u a r e s

s e n s e i i n t h i s "noisy" environment.

1)

CHAPTER I 1

A P P L I C A T I O N TO THE STOKES AND N A V I E R - S T O K E S M.

1.

Fortin, F.

EQUATIONS

Thomasset

INTRODUCTION 1.1

Motivation

The objectives of this chapter are twofold. First, we shall show that the a u g m e n t e d L a g r a n g i a n method can be applied directly to the solution of certain problems in H y d r o d y n a m i c s . Secondly, we shall illustrate by numerical examples the results of Chapter I, comparing the properties of the different algorithms on specific examples. In particular we shall see, through concrete cases, the importance of the choice of the parameters; on this topic we shall give experimental results for an Arrow-Hurwicz type algorithm (see Chapter I, Section 4.3). These experimental results will be given in the case of the l i n e a r i s e d S t o k e s e q u a t i o n s ; these equations can be written as the optimality conditions of a quadratic programming problem in the sense of Chapter I. We shall then show that the algorithms used can be extended to the case of the s t e a d y - s t a t e n o n l i n e a r N a v i e r - S t o k e s e q u a t i o n s . Finally we shall indicate briefly how similar techniques can be applied in the time-dependent case.

1.2

Statement of the problem

W e consider an open domain

n

in IR2

or in IR3 , b o u n d e d , with

(for example, Lipschitz continuous). The regular boundary r coordinates in IRN (N = 2 or 3 ) will be denoted by x = {x,, x21 W e seek to determine in n the characteror x = Ix1, x2, x3j. istics of the flow of an i n c o m p r e s s i b l e v i s c o u s f l u i d . Thus let

47

4a

STOKES & NAVIER-STOKES EQUATIONS

u(x,t) =

(1.1)

IU, (x,t),u2(x,t),u3(x,t)

(CHAP.

2)

1

d e n o t e t h e v e Z o c i t y of t h e f l o w and l e t

p

be t h e h y d r o s t a t i c

pressure. W e s h a l l now t r y t o c a l c u l a t e s o l u t i o n s of t h e N a v i e r - S t o k e s e q u a t i o n s which i n s t a n d a r d nondimensional form a r e w r i t t e n :

:(x,o)

(1.5)

= u

-0

(x)

R

in

We c o n s i d e r h e r e t h e ( n o t v e r y r e a l i s t i c , p h y s i c a l l y , b u t s i m p l e r t o h a n d l e ) c a s e where t h e boundary c o n d i t i o n s a r e of homogeneous D i r i c h l e t t y p e , and where t h e f l u i d i s d r i v e n by a d i s t r i b u t e d external force

.

f = {f1,f2,f,}

The t r a n s i t i o n t o more r e a l -

i s t i c c a s e s p o s e s no problem w i t h r e g a r d t o t h e n u m e r i c a l t r e a t m e n t . I n t h e g r e a t e r p a r t of t h i s c h a p t e r w e s h a l l concern o u r s e l v e s w i t h

2

and w e s h a l l n o t t h e r e f o r e have ( = O_) t o s p e c i f y an i n i t i a l c o n d i t i o n of t h e t y p e ( 1 . 5 ) . the steady-state case

The R e y n o Z d s number

Re,

t h e r e c i p r o c a l of which a p p e a r s i n ( 1 . 2 )

i n f r o n t of t h e v i s c o s i t y t e r m

Ag,

p l a y s , a s w e know, a c r i t i c a l I t is written,

r o l e i n d e t e r m i n i n g t h e b e h a v i o u r of t h e s o l u t i o n s . i n g e n e r a l , i n t h e form Vd Re=-,

(1.6) where

V

is a reference velocity,

is t h e kinematic v i s c o s i t y .

d

i s a r e f e r e n c e l e n g t h and

The c o n s t a n t s

V

and

i n s u c h a way t h a t , f o r example, t h e d i a m e t e r of

mum v e l o c i t y of t h e f l o w a r e of o r d e r u n i t y .

R

d

v

a r e chosen

and t h e maxi-

L e t u s s a y immedi-

a t e l y t h a t t h e methods which w i l l be p r e s e n t e d h e r e a r e v a l i d f o r f l o w s w i t h s m a l l o r i n t e r m e d i a t e v a l u e s of

Re,

and t h a t t h e c a l c u -

l a t i o n of s o l u t i o n s w i t h l a r g e Reynolds numbers p r e s e n t s c o n s i d e r a b l e d i f f i c u l t i e s which i t would t a k e t o o l o n g t o d e s c r i b e h e r e ( s e e , f o r example, FORTIN 1 2 1 , FORTIN-THOMASSET 111 f o r f u r t h e r details). I n o r d e r t o convey q u i t e c l e a r l y t h e approach which w i l l a l l o w u s

(SEC. 1)

49

INTRODUCTION

to u t i l i s e here t h e r e s u l t s of Chapter I , w e s h a l l f i r s t consider i n d e t a i l t h e case o f t h e s t e a d y - s t a t e

l i n e a r i s e d Stokes equations

which w e w r i t e a s f o l l o w s :

which c a n b e d e d u c e d from ( 1 . 2 ) - ( 1 . 4 )

terms

~-pu

valid i f

a n d by t a k i n g

aU

-=

at

by n e g l e c t i n g t h e n o n l i n e a r

!.

This approximation w i l l be

(1.6), f o r a f l o w w i t h low

is very small, i.e.,from

Re

v e l o c i t y , o r f o r a very viscous f l u i d .

1.3

S t o k e s p r o b l e m and q u a d r a t i c programming

W e s h a l l now show t h a t e q u a t i o n s ( l . 7 ) - ( 1 . 9 ) are t h e o p t i m a l i t y

c o n d i t i o n s o f a q u a d r a t i c programming p r o b l e m , s i m i l a r t o t h o s e s t u d i e d i n Chapter I. zero-divergence

The e s s e n t i a l p o i n t w i l l b e t o c o n s i d e r t h e

c o n d i t i o n ( 1 . 8 ) a s a l i n e a r c o n s t r a i n t on t h e

g , t h e p r e s s u r e t h e n a p p e a r i n g as a L a g r a n g e m u l t i p l i e r .

solution

The problems whichwe c o n s i d e r w i l l b e f o r m u l a t e d i n H i l b e r t s p a c e s , o f i n f i n i t e d i m e n s i o n , which w e f i r s t d e f i n e . Suppose t h e n t h a t t i o n s on

Cl,

L'(R)

i s t h e s p a c e o f square-summable f u n c -

e q u i p p e d w i t h t h e u s u a l norm and i n n e r p r o d u c t , i . e .

W e define, i n standard fashion, (1.11)

1

H (R)

=

2 ~ v l v E(R) ~

av ,axi E L ~ ( R ) ,

,...,N)

i=~

,

t h i s S o b o t e v s p a c e b e i n g e q u i p p e d w i t h t h e norm,

I t c a n be shown (see, f o r e x a m p l e , LIONS-MAGENES 1 1 1 ) t h a t t h e t r a c e

a t t h e boundary, and w e p u t

vIr, of a f u n c t i o n

v

from

H1(R)

h a s a meaning

50

STOKES

1

(1.13)

= {vlvcH

Ho(R)

The s p a c e

The norms

H~

(CHAP. 2 )

NAVIER-STOKES EQUATIONS

(a) , v l r

= 0)

.

w i l l b e e q u i p p e d w i t h t h e norm (which i s o n l y a

HA(Q)

semi-norm on

1

Ei

(a))

( 1 . 1 2 ) and ( 1 . 1 4 )

a r e e q u i u a Z e n t on

1 Ho(Q), this result

b e i n g a d i r e c t consequence o f P o i n c a r e ’ s i n e q u a l i t y ,

av lx,lo, i=l ,..., N

(1.15)

Ivjo~c(Q)

,

1

for functions

v

open s u b s e t o f

which a r e z e r o a t t h e b o u n d a r y o f

il,

a bounded

IRN.

Suppose t h e n t h a t 1

V = {yIy E (Ho(n))N, _O-y = 0 i n

(1.16)

For

v =

I

{V

. .vNl c V,

fl}.

we w r i t e

W e now c o n s i d e r t h e f u n c t i o n a l , d e f i n e d f o r

~_E(L,’(Q))~

and

y

E

I N (H,(fl))

by

Having d e f i n e d t h e q u a d r a t i c f u n c t i o n a l of t h e problem

J(-J),we can seek a s o l u t i o n

(SEC. 1)

51

INTRODUCTION

J(2)

5

J(y)

Vy e V ,

(1.20)

The e x i s t e n c e of a u n i q u e s o l u t i o n i s an immediate consequence of t h e Lax-Milgram theorem ( s e e , f o r example, EKELAND-TEMAM C11, LIONS ill).

Problem ( 1 . 2 0 ) c l e a r l y c o n s i s t s o f m i n i m i s i n g a quad-

r a t i c f u n c t i o n a l under a l i n e a r c o n s t r a i n t

(uEV,

i.e.

_V - u- =

0) ;

it i s t h e r e f o r e n a t u r a l t o s e e k t o impose t h i s c o n s t r a i n t by means of a Lagrange m u l t i p l i e r , t h e r e b y t r a n s f o r m i n g ( 1 . 2 0 ) i n t o a s a d d l e 1 2 W e thus define, f o r Y E (Ho(0))N and q e L ( a ) ,

p o i n t problem.

t h e Lagrangian (1.21)

-

=t(v_,s)

J ( _ V ) - ( ~ . V . ~=)

and w e s e e k a p a i r ( H L ( R ) ) ~ ~ L ~ ( R ) ,i.e.

{;,PI

a(v,y)-(f,v)-(q,P.v),

d e f i n i n g a s a d d l e - p o i n t of

d

on

a s o l u t i o n of t h e problem

The e x i s t e n c e of a s a d d l e - p o i n t ,

( a c t u a l l y of t h e pressure

p),

i s a more s u b t l e problem h e r e t h a n i n t h e f i n i t e - d i m e n s i o n a l c a s e of Chapter I.

I n t h e c a s e of t h e S t o k e s e q u a t i o n s , t h e r e s u l t i s i n

f a c t a s t a n d a r d one (see TEMAM 1 1 1 , EKELAND-TEMAM

111) and i t can

Hahn-Banach theorem, s u b j e c t t o c e r t a i n r e g u l a r i t y c o n d i t i o n s on t h e boundary of R (see FORTIN C31, TARTAR [ll). be deduced d i r e c t l y from t h e

W e now g i v e an i n t e r p r e t a t i o n of problem ( 1 . 2 2 ) i n o r d e r t o

verify that the pair

(1.7), ( 1 . 8 ) . respectively

and

I;,

p l , a c t u a l l y i s a s o l u t i o n of problem

The o p t i m a l i t y c o n d i t i o n s f o r ( 1 . 2 2 ) a r e

STOKES & NAVIER-STOKES EQWATIONS

52

Condition (1.23)

y

is true i n particular for

= $E(

(CHAP. 2 )

B(R))

and

we t h u s have, i n t h e d i s t r i b u t i o n a l s e n s e ,

-

(1.25)

~ A u+ pY!

in

= f

R.

I

Condition ( 1 . 2 4 )

is c l e a r l y equivalent t o (1.8).

I n view o f t h e r e s u l t s o f C h a p t e r I , w e are t h u s l e d , f o r t h e s o l u t i o n of

( 1 . 2 2 ) , t o use i n p l a c e of

(1.21)

t h e augmented

Lagrangian

For t h i s problem i n i n f i n i t e dimensions, w e could consider d i r e c t l y algorithms s i m i l a r t o t h o s e of Chapter I.

In particular,

Uzawa's

a l g o r i t h m c o n v e r g e s i n t h i s case u n d e r t h e same c o n d i t i o n s a s i n Chapter I , S e c t i o n 2 . I t need h a r d l y be s a i d t h a t , i n p r a c t i c e , w e f i r s t o f a l l have t o t r y t o f i n d a d i s c r e t i s e d v e r s i o n of ( 1 . 2 6 ) ; t h i s t a k e s us t o a

finite-dimensional

problem which t h e n f a l l s e x a c t l y w i t h i n t h e

framework o f C h a p t e r I .

Remark 1.1:

The f a c t t h a t w e c a n p r o v e t h e c o n v e r g e n c e o f t h e

a l g o r i t h m i n i n f i n i t e dimensions allows us t o a n t i c i p a t e t h a t t h e convergence rate w i l l t o a c e r t a i n e x t e n t b e in d e p e n d e n t o f t h e d i s c r e t i s a t i o n employed. mesh i n a f i n i t e - e l e m e n t

In particular,

t h e f a c t of r e f i n i n g t h e

(or f i n i t e d i f f e r e n c e ) approximation o f

t h e problem shoul d n o t i n i t s e l f b r i n g ab o u t a d r a s t i c d im in u tio n of t h e speed of convergence of t h e algorithm. choice of of

A study of t h e

t h e o p t i m a l p a r a m e t e r s h a s b e e n c a r r i e d o u t ( i n t h e case

r = 0 ) f o r t h e i n f i n i t e - d i m e n s i o n a l p r o b l e m , by C R O U Z E I X 111.

DISCRETISATION OF THE STOKES PROBLEM

2.

The u s e of t h e methods o f C h a p t e r I , f o r t h e s o l u t i o n o f i n compressible viscous f l u i d flow problems,

f a l l s naturally within t h e

framework o f f i n i t e - e l e m e n t methods where t h e i n c o m p r e s s i b i l i t y c o n s t r a i n t i s t r e a t e d by p e n a l i s a t i o n .

T h e s e methods h a v e

r e c e n t l y e n j o y e d c o n s i d e r a b l e p o p u l a r i t y and t h e c o r r e s p o n d i n g

,&R)

=

{ @ I @E cw(Z), @

h a s compact s u p p o r t i n

Cl)

(SEC.

DISCRETISATION OF STOKES PROBLEM

2)

53

t h e o r e t i c a l developments have e n a b l e d c e r t a i n of t h e s e whose o p e r a t i o n i s r e l i a b l e and e f f i c i e n t t o be p i c k e d o u t .

It is d i f f i -

c u l t , w i t h i n t h e i n e v i t a b l y r e s t r i c t e d s c o p e of t h i s volume, t o q u o t e even a s m a l l p a r t of t h e works d e v o t e d t o p e n a l i s a t i o n methods, a p p l i e d t o t h e n u m e r i c a l t r e a t m e n t of t h e Navier-Stokes e q u a t i o n s ; w e s h a l l t h e r e f o r e make do w i t h r e f e r r i n g t o

111, T A Y L O R - Z I E N K I E W I C Z

BERCOVIER [ 1 1 , ODEN

[ 1 1 , MALKUS-HUGHES

[ 11

and t o t h e b i b l i o g r a p h i e s i n t h e s e works. I n o r d e r t o h i g h l i g h t c l e a r l y t h e u s e f u l n e s s of t h e augmented Lagrangian method w i t h i n t h e f i e l d of p e n a l i s a t i o n methods, i t i s worth r e c a l l i n g h e r e s e v e r a l r e s u l t s .

This l e a d s us f i r s t l y t o

i n t r o d u c e a v e l o c i t y - p r e s s u r e mixed v a r i a t i o n a l f o r m u l a t i o n d i s c r e t i s i n g (1.23), (1.24), t h e n t o i n t r o d u c e a s u p p l e m e n t a r y p e n a l i s a t i o n t e r m (see GIRAULT-RAVIART of t h e S t o k e s and Navier-Stokes

Ch

f o r v a r i o u s mixed f o r m u l a t i o n s

problems)

.

i s a t r i a n g u l a t i o n of

Suppose, t h e n , t h a t i a t e with

[ 11

an approximation

Wh

Of

1

(Ho(Q))N

Q

;

w e assoc-

g e n e r a t e d by

conforming o r nonconforming f i n i t e e l e m e n t s ( f o r s i m p l i c i t y w e

l i m i t ourselves t o t h e case approximation of

L2

(n).

N = 2) ; likewise,

Qh

w i l l be an

I t i s n o t n e c e s s a r y t o impose matching

c o n d i t i o n s f o r t h e e l e m e n t s of

I n t h e c a s e of n o n c o n f o r m i n g

Qh.

e l e m e n t s we have i n g e n e r a l , i f

u,, = { u l h , u Z h } ~ l $ ,

-

(in fact i s a m e a s u r e ) ; w e cannot t h e r e f o r e u t i l i s e d i r e c t l y aaUih x. 7 t h e b i l i n e a r form a ( . , . ) d e f i n e d i n (1.19), s o w e “ a p p r o x i m a t e ” a(.,.)

by

ah(.,.)

d e f i n e d by

we of c o u r s e have

W e n e x t d e f i n e on

divergence,

Wh

a linear operator

with values i n

Qhn

by

divh, of d i s c r e t e

54


(CHAP.

i n t h e f o l l o w i n g d i s c u s s i o n we s h a l l u s e t h e n o t a t i o n

and w e approximate t h e S t o k e s problem (1.7)-(1.9) by t h e problem

Find - J E V ~ such t h a t (2.4)

(u

% ( s > v h ) = (,f,vh)

problem ( 2 . 4 )

Find

ehEvh ;

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 -%6vh

such t h a t

Introducing t h e m u l t i p l i e r

ph

E

Qh

c o n d i t i o n of z e r o d i v e r g e n c e , namely t o t h e d i s c r e t e m i x e d problem

t o impose t h e a p p r o x i m a t e

~ h ' ~ h= 0

,

w e g e t down

2)

(SEC. 2 )

Putting

55

DISCRETISATION O F STOKES PROBLEM

qh = 0 , w e o b t a i n t h e p e n a l i s a t i o n problem

W e observe t h a t t h e p e n a l t y term c o n t a i n s t h e d i s c r e t e d i v e r -

gence

!'a.

and n o t t h e e x a c t d i v e r g e n c e

f o r t h i s i s s i m p l e : when

r

is

The r e a s o n

l a r g e , t h e s o l u t i o n of

(2.10)

a p p r o a c h e s t h a t of t h e mixed problem ( 2 . 6 ) , ( 2 . 7 ) and -r(oh*s) c o n v e r g e s t o ph; w e c a n t h e r e f o r e o n l y o b t a i n a c o r r e c t s o l u t i o n of t h e p e n a l i s e d problem i f t h e mixed problem i s w e l l posed.

i s w e l l known t h a t t h e a p p r o x i m a t i o n s

and

Wh

Qh

It

c a n n o t be chosen

independently; i n o r d e r t o o b t a i n a convergent approximation, t h e

Babuska-Brezzi

c o n d i t i o n must be s a t i s f i e d ( s e e BREZZI 1 1 1 ,

BABUSKA 1 1 1 , FORTIN 1 4 1 , GIRAULT-RAVIART

where t h e c o n s t a n t

k

1 1 1 ) which i s h e r e w r i t t e n :

i s i n d e p e n d e n t of

h.

W e s h a l l n o t d w e l l h e r e on t h e meaning of t h i s c o n d i t i o n , b u t One of t h e s e i s t h a t i n g e n e r a l w e

s o l e l y on i t s consequences. cannot d e f i n e

Qh

by Qh = I-CW,),

even f o r conforming e l e m e n t s .

It

t h e r e f o r e f o l l o w s t h a t , e x c e p t i n s p e c i a l c a s e s f o r which we r e f e r t o e.g.

MERCIER 111 and GIRAULT-RAVIART

111, w e cannot i n t h e d i s -

c r e t i s e d problem make t h e d i v e r g e n c e v a n i s h c o m p l e t e l y , hence t h e 2 Il*yhlo ; t h i s f a c t rapidly i m p o s s i b i l i t y of p e n a l i s i n g w i t h

-

became a p p a r e n t t o u s e r s o f p e n a l i s a t i o n methods, and one s o l u t i o n which h a s been a d o p t e d h a s been t o e v a l u a t e Iv.yhlo2 I,Iy.yh/2 dx

by a n i n e x a c t q u a d r a t u r e formula.

T h i s method o f p r o c e d u r e h a s

become known under t h e name of r e d u c e d i n t e g r a t i o n ; b u t it m u s t be u n d e r l i n e d (see HUGHES-MALKUS C 11, t h a t t h i s p r o c e d u r e i m p l i c i t l y defines an operator

divh

and a s p a c e

is i n f a c t evaluated exactly, used.

Qh

f o r which

1,

phqh dx

by t h e q u a d r a t u r e formula

Yph,qhc Q, I n summary, t h e p e n a l i s a t i o n i s i n d i s s o c i a b l e from a mixed

( v e l o c i t y - p r e s s u r e ) method, and m u s t b e c o n s i d e r e d as a s o l u t i o n t e c h n i q u e f o r t h i s l a t t e r method, and n o t a s an a p p r o x i m a t i o n technique i n i t s e l f .

I n t h i s sense t h e u s e of augmented Lagrangian

56

STOKES

Ei


(CHAP. 2 )

methods i s q u i t e n a t u r a l and t h e t e c h n i q u e s o f C h a p t e r I p r o v i d e some advance on t h e more u s u a l methods, s i n c e s e v e r a l i t e r a t i o n s a c t u a l l y e n a b l e t h e e r r o r due t o t h e p e n a l i s a t i o n t o b e e l i m i n a t e d . We do n o t t h e r e f o r e have t o choose v a l u e s of p u r e p e n a l i s a t i o n method.

r

as large as i n a

T h i s p o s s i b i l i t y a l l o w s an improvement , and t h i s i s p a r t i c u l a r l y i n t h e c o n d i t i o n i n g of t h e problems i n

uh

u s e f u l i f one i s u n a b l e t o u s e d o u b l e p r e c i s i o n , o r i f t h e problem in

u i s t o be s o l v e d by an i t e r a t i v e method. -h Regarding t h e n u m e r i c a l e x p e r i m e n t s which w e a r e a b o u t t o d i s -

c u s s , l e t i t b e s a i d immediately t h a t o u r o b j e c t i v e h e r e i s t o check t h e e f f i c i e n c y of t h e a l g o r i t h m s of C h a p t e r I , r a t h e r t h a n t o o b t a i n p r e c i s e s o l u t i o n s t o a s p e c i f i c hydrodynamic problem.

We

a r e t h e r e f o r e s a t i s f i e d w i t h q u i t e a c o a r s e approximation i n which, n o n e t h e l e s s ,

a l l t h e d i f f i c u l t i e s i n h e r e n t i n t h e problem

i n question a r e s t i l l present. We t h e r e f o r e u s e a (nonconforming) a p p r o x i m a t i o n of f i n e d on a t r i a n g u l a t i o n o f t h e open domain space

Wh

n.

1

Ho(R), de-

This d i s c r e t e

i s made up o f f u n c t i o n s whose r e s t r i c t i o n t o e a c h t r i -

a n g l e i s a polynomial of d e g r e e 1, and which a r e c o n t i n u o u s a t t h e This is therefore a

m i d p o i n t s of t h e s i d e s of t h e t r i a n g l e s .

s p a c e of nonconforming f i n i t e e l e m e n t s i n t h e u s u a l s e n s e (see C I A R L E T 1 1 1 , CROUZEIX-RAVIART

C11, STRANG-FIX [ l ] ) .

W e t h u s de-

fine (2.12)

Vh =

{Yh

=

I Yh€

Iv,h,v2hl

The d i s c r e t e o p e r a t o r

Yh

Wh

, phVh

= 01

.

represents discrete differentiation i n

t h e s e n s e o f nonconforming e l e m e n t s , i . e .

r e s t r i c t e d t o t h e in-

t e r i o r of e a c h t r i a n g l e . In t h i s case t h e d i s c r e t e divergence e a c h t r i a n g l e and t h e z e r o - d i v e r g e n c e

yh.vh

is constant over

condition is thus expressed

by a l i n e a r c o n s t r a i n t a s s o c i a t e d w i t h e a c h o f t h e t r i a n g l e s . T h i s b e i n g s o i t i s n a t u r a l t o choose f o r t h e s p a c e

of t h e Qh d i s c r e t e p r e s s u r e s , t h e s p a c e of f u n c t i o n s which a r e c o n s t a n t on

each t r i a n g l e of

5 .

I t i s shown i n CROUZEIX-RAVIART C11, t h a t a p a i r

t h i s being a saddle-point of t h e approximation t o o r d e r

h

%,

f_y,,phj, d e f i n e d i n ( 2 . 8 ) , i s an

(h being t h e longest s i d e belonging t o

(SEC. 3 )

ALGORITHMS AND RESULTS

57

t h e t r i a n g l e s i n t h e t r i a n g u l a t i o n ) of t h e s o l u t i o n

{u,p}

of t h e

I

S t o k e s problem.

Recall t h a t

p

and

ph

a r e defined t o within

an a d d i t i v e c o n s t a n t . R e f e r e n c e may b e made t o THOMASSET 111 f o r a complete d i s c u s s i o n of t h e i m p l e m e n t a t i o n o f t h i s a p p r o x i m a t i o n . W e have c o n s i d e r e d i n o u r e x p e r i m e n t s a model problem, namely

t h e ( t w o - d i m e n s i o n a l ) f l o w between two n o n - c o n c e n t r i c c y l i n d e r s , t h e i n n e r c y l i n d e r b e i n g f i x e d and t h e o u t e r c y l i n d e r r o t a t i n g w i t h a uniform a n g u l a r v e l o c i t y t h e r e g i o n of

(2.13)

I

lR2

I n d e t a i l , w e have t a k e n f o r

W.

whose b o u n d a r i e s a r e

C1:

c i r c l e of r a d i u s 5 and c e n t r e

C2:

c i r c l e o f r a d i u s 2 a n d c e n t r e (1,O).

(O,O),

The d i s c r e t i s a t i o n employed used 1 2 6 t r i a n g l e s , t h e number of i n t e r i o r midpoints being 1 7 2 .

For e a c h m i d p o i n t we have t o d e t e r -

mine t h e components

u2

u1

and

of t h e flow v e l o c i t y , and f o r

e a c h t r i a n g l e t h e v a l u e of t h e p r e s s u r e . Our d i s c r e t e problem i s t h e r e f o r e a q u a d r a t i c p r o g r a m m i n g prob-

l e m w i t h 3 4 4 v a r i a b l e s , r e l a t e d by 1 2 6 l i n e a r c o n s t r a i n t s ( o f which 1 2 5 a r e l i n e a r l y i n d e p e n d e n t ) . a t i o n of C h a p t e r I , t h e m a t r i c e s

A,

Looking back a t t h e n o t B and Bt

correspond

(save

possibly f o r a s i g n ) r e s p e c t i v e l y t o t h e Laplacian, t o t h e divergence and t o t h e d i s c r e t e g r a d i e n t . T o i l l u s t r a t e t h e c o n c e p t s , w e have d i s p l a y e d on F i g u r e 2 . 1 t h e

domain and t h e t r i a n g u l a t i o n u s e d , and on F i g u r e 2 . 2 l i n e s of t h e s o l u t i o n

3.

yh

t h e stream-

obtained.

ALGORITHMS AND DISCUSSION OF RESULTS

3.1

E x p l i c i t f o r m u l a t i o n of t h e a l g o r i t h m s

W e s h a l l now, f o r t h e c a s e of t h e S t o k e s problem, g i v e an exp l i c i t d e s c r i p t i o n of t h e algorithms of Chapter I; w e w i l l then be i n a p o s i t i o n , by i n s p e c t i n g t h e n u m e r i c a l r e s u l t s o b t a i n e d , t o compare t h e i r e f f i c i e n c y and t h e i r ease of i m p l e m e n t a t i o n .

We

have of c o u r s e used i n o u r e x p e r i m e n t s t h e augmented Lagrangian

58

STOKES

&


Figure 2.1 Triangulation of Ci

(CHAP. 2 )

(SEC.

3)


Figure 2 . 2

Streamlines

59

60


(CHAP. 2 )

The s i m p l e s t a l g o r i t h m f o r t h e s o l u t i o n of o u r problem i s Uzawa's a l g o r i t h m of C h a p t e r I , S e c t i o n 2 , which w e w r i t e h e r e a s :

specified arbitrarily;

p:

(3.2)

with

pt

known, c a l c u l a t e a s o l u t i o n

then

n+ 1 ph by n+ 1 Ph = Pi

(3.4)

-

uhn

of

PYh.-"h".

W e have a l s o used t h e v a r i a b l e s t e p l e n g t h methods of Chapter I , S e c t i o n 3 and t h e c o n j u g a t e - g r a d i e n t method.

We s h a l l now b r i e f l y

review t h e o p e r a t i o n of t h e s e a l g o r i t h m s by g o i n g t h r o u g h them i n t h e p a r t i c u l a r c a s e of t h e Lagrangian

(3.1).

W e present within a

s i n g l e a l g o r i t h m t h e v a r i a b l e - s t e p g r a d i e n t methods and t h e

conjugate-gradient

method which d i f f e r o n l y t h r o u g h t h e c h o i c e of Thus suppose w e have :

t h e descent d i r e c t i o n . pz

(3.5)

and

uo -h

On i t e r a t i o n (3.7)


i s satisfying

wn

=

n,

c a l c u l a t e t h e d e s c e n t d i r e c t i o n by

Oh.$

i n t h e methods of s t e e p e s t d e s c e n t and o f minimum r e s i d u a l .

I n the conjugate-gradient (3.8)

wo =

xh*$

i f

m e t h o d , we p r o c e e d a s f o l l o w s :

n=O,

61


(SEC. 3 )

Knowing now t h e d i r e c t i o n

wn,

solve i n

I n t h e minimum r e s i d u a Z m e t h o d , r e p l a c e

Wh

t h e problem

(3.12) b y

W e may r e c o l l e c t t h a t t h e p u r p o s e o f i n t r o d u c i n g t h e i n t e r m e d i a t e

zhn

. i s s o t h a t only a s i n g l e l i n e a r system needs t o be n n+ 1 s o l v e d a t e a c h i t e r a t i o n f o r t h e c a l c u l a t i o n of uh,pn,ph

vector

.

T h i s method of p r o c e d u r e s l i g h t l y i n c r e a s e s t h e memory r e q u i r e m e n t s , b u t t h i s i n c r e a s e would o n l y become a l i m i t i n g f a c t o r f o r v e r y l a r g e s y s t e m s and h a r d l y ever p o s e s a p r o b l e m w i t h modern c o m p u t e r s . F i n a l l y w e h a v e c o n s i d e r e d t h e a l g o r i t h m s of C h a p t e r I , S e c t i o n 4 . I n a g e n e r a l way t h e s e c a n be w r i t t e n , i n t h e case of t h e S t o k e s problem, i n t h e f o r m

The a u x i l i a r y o p e r a t o r

Srh

must b e p o s i t i v e d e f i n i t e .

course considered the canonical choice Section 4.2,

(Sr = A r )

W e h a v e of

of C h a p t e r I ,

i n which case ( 3 . 1 5 ) c a n b e w r i t t e n i n t h e f o r m


62

(CHAP.

h'

which i s v e r y s i m i l a r t o ( 3 . 3 ) .

2)

'

The u s e of t h i s a l g o r i t h m t h u s

r e q u i r e s t h e s o l u t i o n of a l i n e a r s y s t e m a t e a c h i t e r a t i o n . W e h a v e , h o w e v e r , a l s o used a n o t h e r c h o i c e of

matrix is not given e x p l i c i t l y .

Srh,

i n which t h i s

The i d e a c o n s i s t s of s o l v i n g

( 3 . 3 ) by a symmetric s u c c e s s i v e o v e r r e l a x a t i o n method (SSOR), c a r r y i n g o u t , a t e a c h i t e r a t i o n of t h e Uzawa a l g o r i t h m , o n l y one double pass

( f o r w a r d and b a c k ) i n t h e o v e r r e l a x a t i o n method.

It

can e a s i l y b e shown t h a t t h i s p r o c e d u r e i s e q u i v a l e n t t o ( 3 . 1 5 ) f o r an a u x i l i a r y o p e r a t o r

which can be c o n s t r u c t e d e x p l i c i t l y

Srh

i f n e c e s s a r y (see AXELSSON 1 1 1 , 1 2 1 ) .

Any o t h e r i t e r a t i v e method

c o u l d s i m i l a r l y be used f o r t h e s o l u t i o n o f

( 3 . 3 ) and an e x p r e s s i o n

of t h e form ( 3 . 1 5 ) c o u l d b e o b t a i n e d ( i m p l i c i t l y ) by t a k i n g j u s t a s i n g l e i t e r a t i o n a t e a c h s t a g e of t h e Uzawa method.

The a u x i l i a r y

o p e r a t o r t h u s i n t r o d u c e d i s n o t i n g e n e r a l symmetric; t h i s i s t h e c a s e , i n p a r t i c u l a r , f o r t h e u s u a l o v e r r e l a x a t i o n method ( S O R ) . I t would be p o s s i b l e t o e n v i s a g e a complete f a m i l y of i n t e r m e d i a t e

a l g o r i t h m s by c a r r y i n g o u t a t e a c h s t e p a f i x e d number o f i t e r a t i o n s f o r t h e s o l u t i o n of

(3.3).

However, a s w e s h a l l see from t h e

e x p e r i m e n t a l r e s u l t s , t h e o p t i m a l number of i t e r a t i o n s seems t o be one ( o r a t any r a t e s m a l l ) .

Another p r o c e d u r e might be t o s o l v e

( 3 . 3 ) w i t h an a c c u r a c y which is low i n t h e i n i t i a l s t e p s and which becomes h i g h e r and h i g h e r a s w e a p p r o a c h t h e s o l u t i o n .

Supp-

lementary d e t a i l s and p r o o f s of convergence f o r s u c h t e c h n i q u e s c a n be found i n BERTSEKAS 121 and KORT-BERTSEKAS C11.

W e have r e t -

a i n e d t h e l i m i t a t i o n on t h e number of i t e r a t i o n s b e c a u s e of i t s s i m p l i c i t y of i m p l e m e n t a t i o n and frcm t h e f a c t t h a t i t l e a d s t o a l g o r i t h m s of t h e Arrow-Hurwicz

type s t u d i e d i n Chapter I , S e c t i o n

4.3.

3.2 3.2.1

R e s u l t s and D i s c u s s i o n

Fixed-step

UZAWA A l g o r i t h m s

The i m p l e m e n t a t i o n of t h i s t y p e of a l g o r i t h m i s v e r y s i m p l e , and

(SEC.

3)


MAXIMUM VALUES OF

IE ~ .

63

AFTER 'In" ITERATIONS

1 n=15

+,-

II /

Y

Figure 3 . 1

9.6

10

P 11

STOKES

64

&


(CHAP.

Figure 3.2 optimal p

, theoretical and experimental, as a function of r

-

theoretical

+

experimental

2)


(SEC. 3 )

65

the e s s e n t i a l d i f f i c u l t y as regards algorithm ( 3 . 2 ) - ( 3 . 4 ) determine t h e optimal parameter.

is t o

To i l l u s t r a t e t h e i m p o r t a n c e of

t h i s c h o i c e , F i g u r e 3 . 1 shows, a s a f u n c t i o n o f

Yh-luh

o b t a i n e d i n r e s p e c t of t h e c o n s t r a i n t

=

t h e accuracy

p,

,

0

after

n

W e a c t u a l l y show t h e l o g a r i t h m ( t o b a s e 10) of t h e

iterations.

maximum, o v e r t h e t r i a n g l e s , of t h e a b s o l u t e v a l u e of t h e d i s c r e t e (which i s h e r e c o n s t a n t o v e r e a c h t r i a n g l e ) .

divergence

r

v a l u e of

used h e r e i s e q u a l t o 6 ,

be o b t a i n e d f o r any o t h e r v a l u e

The

though s i m i l a r c u r v e s would

r.

of

A s w e might e x p e c t from

t h e r e s u l t s of Chapter I , a very " s h a r p " o p t i m a l value i s obtained. The a p r i o r i d e t e r m i n a t i o n of t h i s o p t i m a l v a l u e r e q u i r e s knowl e d g e of t h e s m a l l e s t and l a r g e s t e i g e n v a Z u e s of

It

AilBtB.

would be p o s s i b l e t o d e t e r m i n e t h e s e v a l u e s by t h e power method. We have a c t u a l l y e s t i m a t e d t h e s e v a l u e s by c a r r y i n g o u t , w i t h

r

=

0 , one p a s s of t h e a l g o r i t h m f o r a s m a l l v a l u e o f

smaller than t h e optimal

l i m i t value.

p),

p

(hence

then f o r a value very close t o t h e

AM

Then knowing t h e r e l a t i o n l i n k i n g

and

Am

to

t h e convergence r a t e and h a v i n g been a b l e t o o b t a i n an e s t i m a t e of t h e l a t t e r , we can t h u s c a l c u l a t e t h e d e s i r e d e i g e n v a l u e s .

I n our

case t h e s e values a r e (3.19)

.

AM = 2 , Am = .07425

The v a l u e

AM

=

2 c o i n c i d e s w i t h a t h e o r e t i c a l bound, which can be

(l~h*~hlo2

o b t a i n e d t h r o u g h an e n e r g y i n e q u a l i t y

%

2 (Ivh((, 2)

i n t h e d i s c r e t e s p a c e s used. W e have t h u s been a b l e t o c o n f i r m e x p e r i m e n t a l l y t h e agreement

between t h e o b s e r v e d o p t i m a l given by f o r m u l a ( 2 . 4 8 )

p

v a l u e s and t h e t h e o r e t i c a l v a l u e s

of Chapter I.

Figure 3 . 2 presents t h e

r e s u l t s of t h i s comparison; t h i s shows a p e r f e c t agreement between t h e p r e d i c t e d v a l u e and t h e e x p e r i m e n t .

I n p r a c t i c e it i s o f t e n

p o s s i b l e , a s i s t h e c a s e h e r e , t o o b t a i n an a p r i o r i e s t i m a t e f o r t h e l a r g e s t eigenvalue.

I n c o n t r a s t t h e d e t e r m i n a t i o n of

Am

i s c o s t l y and, i n f a c t , i f one employs a t e c h n i q u e a n a l o g o u s t o t h a t which w e h a v e u s e d , r e q u i r e s t h e s o l u t i o n of t h e S t o k e s problem.

T h i s c a l c u l a t i o n o n l y becomes p r o f i t a b l e i f c a l c u l a t i o n s

w i t h t h e same d i s c r e t i s a t i o n ( a n d t h e r e f o r e w i t h t h e same m a t r i c e s ) have t o be c a r r i e d o u t many t i m e s . For

r

large take

An approximate r u l e might b e : p

slightly l a r g e r than

r.

66

STOKES

&

(CHAP. 2 )


W e n o t e t h a t from ( 3 . 1 9 ) , t h e c o n d i t i o n number o f t h e d u a l p r o b l e m

is here of t h e order of 2 1 . c o n d i t i o n e d problem.

W e t h u s have h e r e a r e l a t i v e l y w e l l

W e have s e e n f u r t h e r m o r e , i n Chapter I , t h a t

r

t h e c o n d i t i o n i n g improves as

i n c r e a s e s , a n improvement which

o u g h t n o r m a l l y t o show i t s e l f t h r o u g h a n a c c e l e r a t i o n o f t h e c o n v e r I n T a b l e 3 . 1 w e show t h e number o f i t e r -

gence of t h e a l g o r i t h m .

a t i o n s o f a l g o r i t h m ( 3 . 2 ) - ( 3 . 4 ) which w e r e n e c e s s a r y ( w i t h t h e optimal

p)

t o obtain

on e v e r y t r i a n g l e .

5

This

p o s i t i v e l y e s t a b l i s h e s t h a t t h e s i t u a t i o n improves r a p i d l y as

r

increases.

r

I

2

3

5

10

30

n

42

28

22

I9

10

6

Table 3 . 1

Consequently, i f

( 3 . 3 ) i s s o l v e d by a d i r e c t method,

Ar

being f o r

example f a c t o r i s e d o n c e and f o r a l l , it s e e m s c l e a r t h a t t h e o p t i m a l s t r a t e g y c o n s i s t s of t a k i n g

r

as l a r g e as p o s s i b l e , a s l o n g a s w e

m a i n t a i n good a c c u r a c y i n t h e f a c t o r i s a t i o n . s i z e o f o u r model p r o b l e m , v a l u e s of

r

For a problem o f t h e

o f t h e o r d e r o f lo4 s t i l l

a p p e a r q u i t e r e a s o n a b l e i f t h e c a l c u l a t i o n s are performed i n d o u b l e precision.

The number o f i t e r a t i o n s is t h e n o f t h e o r d e r of

3 , d e p e n d i n g on t h e a c c u r a c y d e s i r e d .

2

or

For f u r t h e r examples,

r e f e r e n c e may b e made t o t h e a r t i c l e by SEGAL 111.

For l a r g e r

problems, i n p a r t i c u l a r i n t h r e e dimensions, i t i s probable t h a t f o r large

r

3.2.2

t h e i l l - c o n d i t i o n i n g of

Ar

would b e a g r e a t e r c o n s t r a i n t ,

E f f e c t of the i n c o m p l e t e s o l u t i o n of ( 3 . 3 )

W e h a v e s p e c i f i c a l l y c o n s i d e r e d t h e case where p r o b l e m ( 3 . 3 ) i s s o l v e d by a n iterative method, i n t h i s case i n t h e s h a p e o f a n

o v e r r e l a x a t i o n method.

S i n c e it seems p o i n t l e s s , a p r i o r i , t o

c a r r y o u t t h e s o l u t i o n f u l l y and c o m p l e t e l y i n t h e i n i t i a l s t a g e s ,

it i s n a t u r a l t o l i m i t 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 t o a v a l u e which may b e q u i t e s m a l l . I n o u r model p r o b l e m , t h e d e t e r n m inat i on of u i n t h e i n i t i a l s t e p s r e q u i r e d a p p r o x i m a t e l y 50 -h iterations. F i g u r e 3 . 3 shows t h e number o f i t e r a t i o n s o f

(SEC.

3)


Figure 3 . 3

67

68

STOKES

&


Figure 3 . 4

(CHAP.

2)

(SEC.


3)

69

550

500

A 50

40 0

3%

300

2 50

20c

I5C

1oc

5c

I

i

5

10

i

I

I

I

I

1

I5

20

25

30

35

Figure 3.5

m

70

STOKES

&

n -7 I._Oh-uhl 210

( 3 . 2 ) - ( 3 . 4 ) which w e r e r e q u i r e d t o o b t a i n t r i a n g l e , a s a f u n c t i o n o f t h e number

m

i t e r a t i o n s a l l o w e d f o r t h e s o l u t i o n of

(3.3).

m

2

10

(CHAP. 2 )

NAVIEK-STOKES EQUATIONS

on e v e r y

of o v e r r e l a x a t i o n I t shows t h a t f o r

t h e method undergoes p r a c t i c a l l y no f u r t h e r change, t h e

( 3 . 3 ) i n t h e i n i t i a l s t e p s having very l i t t l e

poor s o l u t i o n of

e f f e c t on t h e o v e r a l l p r o c e s s .

For

m

0

p

(J

The v a l u e of

p

By c o n t r a s t , t h e s o l -

v

small.

t h e r e s u l t s w e have

The s p e e d o f c o n v e r g e n c e d i m i n i -

b u t i t seems h e r e t o b e a r e a l a d v a n t a g e t o t a k e

v v

s m a l l , c o n t r a r y t o what h a p p e n e d i n t h e l i n e a r case.

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

least i n a sub-space of t h e space o f "admissible" s o l u t i o n s .

88


6.

NAVIER-STOKES EQUATIONS.

TIME-DEPENDENT

(CHAP.

2)

CASE.

T h i s s e c t i o n does n o t p r e t e n d t o b e e x h a u s t i v e .

The aim i s

simply t o show t h a t t h e t e c h n i q u e s d e v e l o p e d i n t h e s t e a d y - . s t a t e c a s e remain u s a b l e i n t h e time-dependent

c a s e , and t o i l l u s t r a t e

t h i s a s s e r t i o n t h r o u g h a s i m p l e example.

The methods p r o p o s e d can

e a s i l y be a d a p t e d t o more complex s i t u a t i o n s and t o more e l a b o r a t e schemes

6.1

. S t a t e m e n t of t h e problem

W e c o n s i d e r h e r e t h e n o n l i n e a r Navier-Stokes

time-dependent c a s e .

a:

VAU + (u.0)~ - - . . + Vp

(6.1)

-a t-

(6.2)

v-u = 0, -

(6.3)

I

=

equations i n the u(x,t)

of

f,

I

I

u(x.0) =

I

-Ulr

(6.4)

W e t h u s seek a s o l u t i o n

:I(x)

given,

= 0. I

W e c o n s i d e r a d i s c r e t i s a t i o n w i t h r e s p e c t t o t i m e by a v e r y

s i m p l e scheme of i m p l i c i t t y p e

with

-fR+'

= f

(6.7)

u

(6.8)

u

0

I

(x,(n+l)k),

I

n+ 1

.'0

(6.6)

(k = A t ) :

= 0,

= u0'

n+ 1

=

_o.

I n p r a c t i c e we s h a l l o b v i o u s l y b e a t t e m p t i n g t o s o l v e a d i s c r e t i s e d form of t h i s problem.

W e c o u l d , f o r example, use t h e f i n i t e -

e l e m e n t method d e s c r i b e d i n S e c t i o n 2 of t h i s c h a p t e r . The i m p o r t a n t p o i n t t o c o n s i d e r i s t h a t ,

yn

b e i n g known, t h e

s o l u t i o n of ( 6 . 5 ) - - ( 6 . 8 ) i s a problem of t h e same t y p e a s t h o s e d e s c r i b e d i n S e c t i o n 5 ; t h e o p e r a t o r s i n v o l v e d a r e merely m o d i f i e d slightly.

t Y Pe

More s p e c i f i c a l l y , w e have t o s o l v e a problem of t h e

(SEC. 6 )

89

NAVIER-STOKES; TIME-DEPENDENT CASE

- kVAu + k(u.V)u + kVp

u

(6.9)

I

=

I

(6.10)

-V-u-

=

0

(6.11)

:I r

=

2.

I

B,

in

Remark 6 . 1 :

F,

I n s t e a d of

(6.5) we could consider a semi-implicit

scheme, r e p l a c i n g ( 6 . 5 ) b y

Remark 6 . 2 :

(6.5)-(6.8)

c a n b e w r i t t e n i n v a r i a t i o n a l form

.

With t h e n o t a t i o n o f S e c t i o n 4 , w e o b t a i n

(6.14)

~

'

E

v

given. b ( y , y , y ) , it can e a s i l y

By v i r t u e o f t h e a n t i - s y m m e t r y o f t h e form

b e shown t h a t s u c h i m p l i c i t schemes are u n c o n d i t i o n a l l y s t a b l e .

Remark 6 . 3 :

The e x i s t e n c e and u n i q u e n e s s o f

-

un+l

-

i n t h e semi-

u n+l

i m p l i c i t scheme d e f i n e d by ( 6 . 1 2 ) p o s e n o p r o b l e m s .

In fact,

i s t h e s o l u t i o n of a l i n e a r p r o b l e m r e l a t i n g t o t h e operator which i s V - e l l i p t i c .

I+k(-vA+_u"-F)

By u s i n g , a s i n S e c t i o n 4 , of a s o l u t i o n out d i f f i c u l t y . imation of

u -

a fixed-point

of p r o b l e m ( 6 . 9 ) - ( 6 . 1 1 )

theorem, t h e e x i s t e n c e

can b e demonstrated with-

W e s h a l l now show t h a t i f w e c o n s i d e r a n approx-

(6.9)-(6.11)

i n a space of f i n i t e dimensions

(for

example by means of f i n i t e e l e m e n t s a s i n S e c t i o n 2 ) , w e h a v e

k

uniqueness of t h e s o l u t i o n f o r

As i n S e c t i o n 4 , w e d e n o t e by norms o f

a space

(L2(B))N

and

V.

sufficiently small.

1.1,

and

II.II1

t h e respective

W e s h a l l i n f a c t b e working w i t h i n

V hc V , w h e r e i n t h e m a j o r i t y o f c a s e s t h e p a r a m e t e r

r e p r e s e n t s t h e s i z e o f t h e mesh u s e d f o r t h e a p p r o x i m a t i o n . then e x i s t s a mesh-dependent

constant

S(h)

such t h a t

h There

90

STOKES

&


(CHAP.

2)

W e c a n now p r o v e t h e f o l l o w i n g :

LEMMA 6.1: N

=

2,

i n finite d i m e n s i o n s ,

t h e solution of

and f o r t h e s p a t i a l d i m e n s i o n is unique if

t h e p r o b Z e m (6.9)-(6.11)

k

is sufficiently small. Suppose

Proof:

y1

y2

and

a r e t w o s o l u t i o n s o f t h e problem.

I n v a r i a t i o n a l form w e can t h u s w r i t e

subtracting (6.17)

from ( 6 . 1 6 )

and p u t t i n g

y

=

u1-y2,

we obtain,

a f t e r v a r i o u s m a n i p u l a t i o n s o n t h e n o n l i n e a r terms and t a k i n g adv a n t a g e of t h e a n t i - s y m m e t r y o f

b

(see S e c t i o n 4 )

U t i l i s i n g t h e i n e q u a l i t y o f C a g l i a r d o (see LIONS [ 2 1 ) , w e c a n , i n two d i m e n s i o n s , o b t a i n t h e f o l l o w i n g u p p e r bound f o r t h e n o n l i n e a r

term:

W e s h a l l t h e r e f o r e have

u1

=

y2

i f w e can choose

k

such t h a t w e

have (6.22)

(I-kCS(h)

1(u2((I ) D 0 .

S t i l l using ( 6 . 1 5 ) , w e can transform (6.22) i n t o (6.23)

2 kCS (h)

But s e t t i n g

I I I ~ / ~1 ~.

v- = u- 2

i n ( 6 . 1 7 ) , it c a n e a s i l y b e s e e n t h a t

i s bounded above by a c o n s t a n t d e p e n d i n g o n l y on p o s s i b l e t o v e r i f y ( 6 . 2 3 ) , and h e n c e t h e r e s u l t .

f.

ILII~~

It i s thus

rn

91

NAVIER-STOKES; TIME-DEPENDENT CASE

(SEC. 6 )

Solution algorithms

6.2

W e have a l r e a d y n o t e d t h e analogy between problem ( 6 . 9 )- ( 6 . 1 1 )

and t h e cases d i s c u s s e d i n S e c t i o n s 4 and 5 of t h i s c h a p t e r .

We

can t h e r e f o r e u s e , f o r i t s s o l u t i o n , a l g o r i t h m s c l o s e l y r e l a t e d t o t h o s e d e s c r i b e d f o r t h e s t e a d y - s t a t e Navier-Stokes

equations.

To

i l l u s t r a t e t h e i d e a s , we s h a l l discuss h e r e an algorithm of t h e UZAWA t y p e .

UZAWA a l g o r i t h m , Suppose w e h a v e

u

- - kVA! -"'u = 0 ,

(6.24)

u

(6.25)

?Ir

(6.26)

=

semi-implicit

-

E

V

given.

case W e wish t o s o l v e

+ k ( i . j ' ) F + kj'p = _F

,

0.

Remark 6 . 4 :

In practice,

n g = g ,

= En+kf_. R

I n o r d e r t o s o l v e ( 6 . 9 ) - ( 6 . 1 1 ) , w e can t h e r e f o r e use t h e following algorithm: (6.27)

then


po

with

p

s+l

(6.29)

us

known, c a l c u l a t e t h e s o l u t i o n

ps

of

by p

s+l

= p

s

- psy.g.

W e n o t e t h a t problem

( 6 . 2 8 ) i s l i n e a r (nonsymmetric)

.

The c o n v e r -

g e n c e o f t h i s a l g o r i t h m i s a d i r e c t c o n s e q u e n c e of t h e r e s u l t s of Chapter I. (6.30)

I n p a r t i c u l a r , w e have f o r

ps

the condition

0 < p, < 2(r+v).

I t i s a l s o p o s s i b l e , a s i n S e c t i o n 4 , t o u t i l i s e t h e method of m i n i -

mum r e s i d u a l s f o r t h e d e t e r m i n a t i o n of Remark 6 . 5 :

ps.

The u s e of s u c h a n i t e r a t i v e method i n a t i m e -

d e p e n d e n t scheme would c l e a r l y b e p r o h i b i t i v e i f t h e c o n v e r g e n c e a t

(CHAP. 2 )


92

each t i me-st ep w e r e n o t ex t r emel y r a p i d .

In actual fact the situ-

a t i o n h e r e i s much more f a v o u r a b l e t h a n i n t h e s t e a d y - s t a t e case. In practice

-

t h e i n i t i a l i s a t i o n can b e e f f e c t e d u s i n g n+l p ,

pn

which w i l l i n

g e n e r a l be close t o

-

t h e problem i s w e l l conditioned f o r

r

k

s u f f i c i e n t l y s m a l l and

sufficiently large.

I n p r a c t i c e , w e may e x p e c t t o c a r r y o u t o n l y a v e r y s m a l l number o f i t e r a t i o n s a t each t i m e s t e p ( 2 or 3 ) . I W e a r e now g o i n g t o c o n s i d e r t h e f u l l y i m p l i c i t case and show

t h a t t h e UZAWA a l g o r i t h m i s s t i l l a p p l i c a b l e .

U Z A W A a l g o r i t h m i n t h e i m p l i c i t scheme c a s e . W e h a v e a l r e a d y shown i n S e c t i o n 6 . 1 t h a t t h e i m p l i c i t scheme (6.5)-(6.8)

i s w e l l posed f o r

k

sufficiently small.

see t h a t , u n d e r t h e same c o n d i t i o n s , w e c a n c a l c u l a t e n o n l i n e a r UZAWA a l g o r i t h m . equivalent t o t h e s o l u t i o n of

Since t h e calculation of

W e . s h a l l now n+l u by a - n + l is

(6.9)- (6.11), we s h a l l consider t h e

s o l u t i o n o f t h i s l a t t e r problem.

W e s h a l l u t i l i s e here t h e anti-

symmetrised form o f t h e n o n l i n e a r t e r m i n t r o d u c e d e a r l i e r i n S e c t i o n The a l g o r i t h m c o n s i d e r e d i s t h e n w r i t t e n :

5.1.

-

-

s-1

u

then

pS+l

(6.33)

chosen a r b i t r a r i l y ;

uo, p1

(6.31)

with

and

ps


by

ps+] = ps

-

p(0.gS).

I n v a r i a t i o n a l form, w e c a n w r i t e ( 6 . 3 2 ) as f o l l o w s :

us

of

(SEC. 6 )

NAVIER-STOKES ; TIME-DEPENDENT CASE

Remark 6 . 5 :

93

I n t h e c o n t e x t o f t h e i m p l i c i t scheme ( 6 . 5 ) - ( 6 . 8 ) , I n u0 = un , p = p . I

a n a t u r a l c h o i c e would b e t o t a k e

Remark 6 . 6 :

-

-

Problem ( 6 . 3 2 ) i s l i n e a r

(us-'

b e i n g known) a n d

I

Use of t h e a n t i - s y m m e t r i s e d f o r m

non-symmetric.

-

s t a t e t h e e x i s t e n c e of a unique s o l u t i o n 1 ( H ~ ( R )) N - e l l i p t i c .

us,

b

a l l o w s us t o

t h e operator being

I n t h e following d i s c u s s i o n , we consider algorithm (6.31)- (6.33) i n f i n i t e d i m e n s i o n s , i . e . f o r t h e s o l u t i o n of a d i s c r e t i s e d version of

(6.5)- (6.8)

.

We h a v e a l r e a d y p r o v e d a u n i q u e n e s s r e s u l t We s h a l l now p r o v e t h e f o l l o w i n g

f o r t h i s case i n Lemma 6 . 1 . result:

I n f i n i t e dimensions,

PROPOSITION 6 . 1 :

dimension small,

N = 2,

-

u

t o the solution

-

pair

{ u,p*} s a t i s f i e s

of p r o b l e m

= us-u

and

ps

=

k

sufficiently

(6.9)-(6.11).

p*

The s e q u e n c e

of t h i s s e q u e n c e t h e

(6.9)-(6.11).

We w r i t e ( 6 . 9 ) i n v a r i a t i o n a l f o r m .

Proof: X

and f o r t h e s p a t i a Z

converges f o r

us

i s bounded and f o r any c l u s t e r p o i n t

ps

Es

t h e sequence

ps-p,

Putting

w e o b t a i n by s u b t r a c t i n g ( 6 . 9 ) f r o m ( 6 . 3 2 )

I

Taking t h e i n n e r product of

(6.33) w i t h

p

s+l

and u s i n g

( 6 . 1 0 ) we

obtain

We c a n w r i t e t h e l a s t t e r m of

(6.36) i n t h e form

F u r t h e r m o r e , by t h e Cauchy-Schwarz

We m u l t i p l y ( 6 . 3 5 ) b y

p

i n e q u a l i t y we have

and ( 6 . 3 6 ) by

u s i n g ( 6 . 3 7 ) a n d ( 6 . 3 8 ) , we see t h a t

k.

T a k i n g t h e sum and

STOKES

94

A s i n Lemma 6 . 1 ,

&


(CHAP. 2 )

w e o b t a i n a n u p p e r bound f o r t h e n o n l i n e a r term by

using Cagliardo's inequality

By t h e e q u i v a l e n c e of norms ( 6 . 1 5 ) a n d t h e Cauchy-Schwarz

inequality,

we obtain

S u b s t i t u t i n g i n t o (6.39) we o b t a i n

W e w i l l b e able t o a p p l y t h e a r g u m e n t o f Theorem 2 . 1 o f C h a p t e r I i f

w e have

i.e (6.44)

2

Thus w e a g a i n h a v e t h e uniqueness condition o f Lemma 6 . 1 .

The

s t a t e m e n t of t h e p r o p o s i t i o n c a n b e d e d u c e d f r o m ( 6 . 4 2 ) by t h e usual procedures. The p r i n c i p l e s b e h i n d t h i s p r o o f c o u l d i n f a c t b e e x t e n d e d t o p r o b l e m s of t h e same t y p e a s s o c i a t e d w i t h o t h e r i m p l i c i t schemes which are more a c c u r a t e t h a n

(6.5) - (6.8).

This algorithm is

v e r y s i m i l a r t o t h o s e o f S e c t i o n 5 . 1 and it i s o f c o u r s e p o s s i b l e t o

(SEC. 7 )

BLSCUSSION

95

introduce into it variants such as the use of a relaxation parameter. The fact that we have been able to prove convergence here is clearly linked with the existence of a uniqueness result.

7.

GENERAL DISCUSSION ON CHAPTER I1 The results obtained for the solution of the Stokes problem show

that the use of an augmented Lagrangian can be an efficient method of approach for this problem. Its attraction ought to become even more pronounced in three-dimensional problems where the size of the matrices involved makes the use of direct methods difficult. The extension to The analogy the nonlinear case is often found to be efficient. between the ARROW-HURWICZ type algorithm of Section 5.2 and the method of CHORIN 111 should be mentioned. A similar approach has been used in FORTIN-PEYRET-TEMAM 111 and in BEGIS E l l , this latter work relating to the calculation of non-Newtonian fluids. In the nonlinear case, the benefit of the penalty term lies in the fact that it improves the dissipation of the system without perturbing the solution. We shall see in the following chapters that it can sometimes actually bring about the convergence of algorithms which would otherwise be ill-posed.


C H A P T E R 111 ON DECOMPOSITION-COORDINATION METHODS USING AN AUGMENTED LAGRANGIAN M.

Fortin, R.

Glowinski

INTRODUCTION

1.

1.1

Motivation.

Examples.

A large number of problems in Mechanics, in Physics, in Economics, etc... ( s e e Chapter IV, V, VI, VII, VIII) can be stated in the form

(P) Min { F ( B v )

(1.1)

+ G(v))

,

V€V

where

* V, H are normed vector spaces (real for simplicity) of finite or infinite dimension,

* F, G are functionals which are convex, proper, and lower semi-continuous (1.s.c.) on, respectively, H and V. The formulation (1.1) is quite general, since as will be seen in the Chapters which follow (see also ROCKAFELLER C 4 1 , CKELAND-TEMAM I l l ) , such a formulation encompasses the minimisation of functionals which are possibly nondifferentiable over convex sets, the nondifferentiability or the constraint relating to illustrate this by two examples: Example 1:

Let

(R

v

and/or

Bv.

We

F l o w of a Bingham f l u i d i n a c y l i n d r i c a l d u e t .

R be a bounded open domain in IR2 with regular boundary is the c r o s s s e c t i o n of the duct). W e define V, H by

97

r

98

DECOMPOSITION-COORDINATION

H

(1.3)

Let

w

=

and

(L2 (n)) 2 . b e two p o s i t i v e constants ( r e f e r t o Chapter V,

g

i o n 5 f o r t h e p h y s i c a l meaning of considered) . (1.4)

(CHAP. 3 )

w

and

g

Sect-

a n d of t h e example

Now s u p p o s e w e h a v e t h e p r o b l e m

Min v EV

w h e r e , i n (1.4),

f

ations considered)

.

Problem ( 1 . 4 )

E

L2(n)

(actually f = constant i n the applic-

i s o b v i o u s l y a p a r t i c u l a r form o f problem

(P)

ob-

t a i n e d by p u t t i n g (1.5)

B = v

and by d e f i n i n g

F, G

by

(1.7)

I n (1.6) w e h a v e p u t

191 =

4

An a l t e r n a t i v e c h o i c e f o r F , G

The above f u n c t i o n s

nondifferentiubze

on

F, G H

i s g i v e n by

are c o n v e x , and c o n t i n u o u s , and

F

is

b e c a u s e of t h e p r e s e n c e of t h e t e r m f n l q l d x .

The c h o i c e s (1.6)’ (1.7) o r (1.8), ( 1 . 9 ) w i l l l e a d t o s l i g h t l y d i f f e r e n t algorithms for t h e solution of cribed i n Section 3.

(1.4), by t h e methods d e s -

I n c i d e n t a l l y t h i s p o s s i b i l i t y of c h o i c e i n

t h e decomposition allows u s t o p r e d i c t a c e r t a i n d e g r e e o f v e r s a t i l i t y i n t h e u s e of t h e methods s t u d i e d i n t h i s c h a p t e r .

1)

(SEC.

E l a s t o p l a s t i c t o r s i o n of a c y Z i n d r i c a Z b a r .

Example 2 :

Q

Let (Q

99

INTRODUCTION

b e a bounded open domain i n

2

I R , w i t h r e g u l a r boundary

i s t h e c r o s s s e c t i o n of t h e b a r ) .

W e define

by

V, H

r

(1.2),

( 1 . 3 ) and w e c o n s i d e r t h e p r o b l e m ( I . 10) v EK

where

f

L2(n)

E

( f = c o n s t a n t i n t h e f l u i d mechanics a p p l i c a t i o n s

c o n s i d e r e d ) , and (1.11)

K =

1

{ V ~ V ~ H ~ ( Q ,

W e r e f e r t o Chapter V,

(l.lO), (1.11). lem

.

S e c t i o n 5 , f o r t h e p h y s i c a l meaning of

P r o b l e m (1.10) i s a l s o a p a r t i c u l a r form o f p r o b -

o b t a i n e d by p u t t i n g

(P)

B = 8 ,

(1.12)

with

5 1 a.e.1

d e f i n e d by

F, G

(1.14)

12

I n (1.13),

d e n o t e s t h e i n d i c a t o r f u n c t i o n o f t h e convex s e t

A

(1.15)

K = IqIqEH,

Iq/ 2 1

a.e.1

.

W e t h u s h a v e , by d e f i n i t i o n o f t h e i n d i c a t o r f u n c t i o n ,

(1.16)

L(q) = + m

if

q#K.

i s nonernpty(0

S i n c e t h e convex s e t

c o n v e x , p r o p e r , and Z . S . C . the same properties.

uous o n

E

8)

Furthermore

F

= 1K

F, G

and cLosed

it t h u s f o l l o w s t h a t G

in F

H,

Ii;

is

satisfies

i s c l e a r l y convex and c o n t i n -

V.

An a l t e r n a t i v e c h o i c e f o r (1.17)

on H ;

i s g i v e n by

100


3)

(CHAP.

The v a r i o u s r e m a r k s made i n r e l a t i o n t o Example 1 a p p l y e q u a l l y t o Example 2 .

rn

1.2

P r i n c i p l e of t h e method

The e s s e n t i a l i d e a i n t h e w h o l e of t h e f o l l o w i n g d i s c u s s i o n i s b a s e d on t h e f a c t t h a t t h e r e i s t r i v i a l l y a n e q u i v a l e n c e b e t w e e n (PI and

(n)

(1.19)

Min {F(q)+G(v)} {v,q) E W

,

with (1.20)

W = {{v,q} E V

XH,

Bv-q = 0 )

.

W e have t h u s i n t r o d u c e d a s u p p l e m e n t a r y v a r i a b l e

through t h e l i n e a r e q u a t i t y r e l a t i o n

Bv = q

.

q

,

linked t o

v

To h a n d l e t h i s

c o n s t r a i n t w e s h a l l , as i n C h a p t e r I , u t i l i s e a L a g r a n g e m u l t i p l i e r and r e d u c e t h e p r o b l e m (II)

( t h u s a l s o (P)) t o a s a d d l e - p o i n t p r o b l e m .

I n t h e f o l l o w i n g d i s c u s s i o n w e s h a l l assume t h a t t h e s p a c e s and

H

a r e HiZbert'spaces;

d e n o t e by

t h e inner product i n

(.,.)

V

i s i d e n t i f i e d w i t h i t s d u a l and w e

H

H,

a n d by

1.1

the associated

norm ( i n c e r t a i n cases t h e r e s u l t s w i l l a p p l y t o t h e case where

i s a r e f l e x i v e Banach s p a c e . ) LI

E

H

(1.21)

W e then define, f o r

v

E

V,

q

H H,

E

t h e Lagrangian & ( v , q , V ) = F(q) + G(v)

(U9Bv-q)

3

r 2 0 , t h e augmented L a g r a n g i a n 2

and t h e n f o r (1.22)

+

k,(v,q,U)

=

d(v,q,v)

+

f

(Bv-q[

.

I n Section 2 w e s h a l l study t h e problem o f t h e e x i s t e n c e of s a d d l e -points f o r of

Z !

L

&,

and

and t h e r e l a t i o n s b e t w e e n t h e s a d d l e - p o i n t s

Lr.

and t h o s e o f

m

The a p p r o a c h f o l l o w e d up t o now may s e e m somewhat c o n t r i v e d and c o m p l i c a t e d , s i n c e w e have i n t r o d u c e d a supplementary v a r i a b l e and a supplementary c o n s t r a i n t

Bv - q = 0 .

q

By d o i n g t h i s w e h a v e

i n a c t u a l f a c t s i m p l i f i e d t h e n o n l i n e a r s t r u c t u r e o f p r o b l e m (PI by decoupling

'

F

and

B

.

To i l l u s t r a t e t h i s f a c t we s h a l l apply t h e

T h e r e w i l l t h u s be no d i f f i c u l t i e s i n f i n i t e d i m e n s i o n s .

(SEC.

101

INTRODUCTION

1)

above p r i n c i p l e t o t h e two examples of S e c t i o n 1.1; we remark t h a t t h e n u m e r i c a l u t i l i s a t i o n of t h i s t y p e of method seems t o have been f i r s t i n t r o d u c e d by GLOWINSKI-MARROCCO [I] f o r t h e n u m e r i c a l s o l u t i o n of

the

(1 < p
O , and v i c e F u r t h e r m o r e u is a solution of (P), and we h a v e p = Bu.

V x H x H;

versa.

Proof:

Let

{u,p,A}

b e a s a d d l e - p o i n t of

k

on

W e t h u s have (2.4)

~ ( u , P , P )5 k ( u , p , X ) 5 k ( v , q , h ) ,

From t h e f i r s t i n e q u a l i t y i n ( 2 . 4 )

V { v , q l c VXH,

VP

w e deduce

hence

(2.5)

Bu

=

p.

From t h e second i n e q u a l i t y i n ( 2 . 4 ) w e t h e n deduce

hence a f o r t i o r i F(Bu) + G(u)

5

F(Bv) + G(v) V v c V ,

6

H.

V

X

H

X

H.

(SEC.

2)

which p r o v e s t h a t I n view of

( 2 . 5 ) w e immediately have

UU,P,U)

=

V{v,q}

VxH

E

(P).

i s a s o l u t i o n of

u

Zr(u,p,!4

, V~.J E

= & ( u , p , A ) = ;e,(u,p,A)

0 t h e d u a l problem

i s a r e g u l a r i s a t i o n by an i n f - c o n v o l u t i o n of t h e d u a l problem f o r

r = 0.

T h i s p r o p e r t y i s e s p e c i a l l y u s e f u l f o r t h e c o n s t r u c t i o n of

a l g o r i t h m s u s i n g t h e g r a d i e n t o f t h e f u n c t i o n a l of t h e d u a l problem.

3.

DESCRIPTION OF THE ALGORITHMS 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 two i t e r a t i v e methods of s o l u t -

i o n of

which a r e i n f a c t methods f o r c a l c u l a t i n g t h e s a d d l e -

(P)

p o i n t s of

3.1

L r r t h i s a p p r o a c h b e i n g j u s t i f i e d by Theorem 2 . 1 . F i r s t algorithm (ALG1)

I n view of Theorem 2 . 1 it i s n a t u r a l , f o r c a l c u l a t i n g t h e s a d d l e p o i n t s of

on

Pr

V x H x H I t o u t i l i s e t h e algorithm o f

t h e Uzawa

t y p e g i v e n below ( s e e C h a p t e r I , S e c t i o n 2 . 1 ) : Xo

(3.1)

with

An

(3.3)

E

n specified arbitrarily;

known, d e t e r m i n e

{unlpn}, t h e n

xn+l

by

- n +Pn(Bun-pn).

In t h e following t e x t a l g o r i t h m ( 3 . 1 ) - ( 3 . 3 ) w i l l o f t e n be r e f e r r e d t o

as A L G 1 .

The convergence o f A L G l w i l l be s t u d i e d i n S e c t i o n 4 .

f o l l o w s from ( 3 . 2 ) t h a t t h e p a i r c o u p l e d system

{un,pn}

i s c h a r a c t e r i z e d by t h e

It

(SEC. 3 )

DESCRIPTION O F ALGORITHM

G(v)-G(un)+(Xn,B(v-un))+r(Bun-pn,B(v-un)) (3.4) un

E

109

2 0 V v E V,

v,

Remark 3.1: The above algorithm can also be written as a saddlepoint calculation algorithm for Lr (see (2.15)). In fact we have seen in Section 2.3 that we actually have (3.6)

L,(v,LJ)

=

inf =!,(v,q,'s). 9

We thereby deduce that if have

{un,pn]

is a solution of (3.2) we also

It is shown in FORTIN 111 that problem (3.7) is in general nonlinear with respect to v and that its d i r e c t solution is difficult. Through the introduction of q, then of P,, this problem (3.7) has been decomposed into the system which is the equivalent of the two inequalities (3.4), (3.5). Expressing the problem in the form of a system will lead to efficient procedures for solving (3.7) (and (3.2)).

rn

Remark 3.2: The reader may verify that algorithm ALGl can be interpreted as a s u b g r a d i e n t algorithm for the dual functional

It can actually be shown that, whatever the values of have h,(u) 0 , hr(X)

is

always d i f f e r e n t i a b l e .

Thus i n t h i s c a s e we i n f a c t have a s t a n d a r d

g r a d i e n t a l g o r i t h m when

r

is s t r i c t l y positive.

Second a l g o r i t h m (ALG2)

3.2

I n t h e i m p l e m e n t a t i o n of A L G l t h e e s s e n t i a l d i f f i c u l t y i s c l e a r l y t h e s o l u t i o n a t e a c h i t e r a t i o n of t h e s y s t e m ( 3 . 4 )

,

(3.5).

A nat-

u r a l s o l u t i o n p r o c e d u r e c o n s i s t s o f u s i n g t h e b l o c k r e l a x a t i o n method g i v e n below ( w h e r e pn,o

(3.8)

then f o r

k

i s fixed):

n

n-1 = P

I

1

L

)+r (au"' k-p n,k-I

G(V)-G(~"*~)+

,B(v-unYk)) 2 0

V

v

E

'J,

(3.9)

F(q)-F(pn'k)-(Xn,q-pnyk)+~(pn,k_Bun~k,~-pn~k)

5

0Vq

E

H

,

(3.10)

p n Y k E H.

The a l g o r i t h m ( 3 . 8 ) - ( 3 . 1 0 ) i s c o n v e r g e n t u n d e r q u i t e g e n e r a l assumpt i o n s on

F

G (see f o r example, CEA-GLOWINSKI C11).

and

Taking

i n t o a c c o u n t t h e r e s u l t s of C h a p t e r s I and I1 c o n c e r n i n g i n c o m p l e t e m i n i m i s a t i o n i n t h e UZAWA a l g o r i t h m (see C h a p t e r I , S e c t i o n 2.3, Remark 2.5 and C h a p t e r 11, S e c t i o n s 3.2 and 5 . 2 ) w e o b t a i n n a t u r a l v a r i a n t s of A L G l by r e s t r i c t i n g o u r s e l v e s when s o l v i n g ( 3 . 4 1 ,

(3.5)

by ( 3 . 8 ) - ( 3 , 1 0 ) t o a f i x e d number of b l o c k r e l a x a t i o n p a s s e s .

In

t h e l i m i t i n g c a s e of a s i n g l e p a s s w e o b t a i n t h e a l g o r i t h m (3.11)

with

{po,X1} {pn-l,Xnl

E

H x H

arbitrarily specified; n n An+l u , p , by

known, d e t e r m i n e s u c c e s s i v e l y

3)

(SEC.

A"+'

(3.14)

111

DESCRIPTION O F ALGORITHM

= A"

+ p (Bun-pn).

In t h e following t e x t algorithm (3.11)-(3.14) w i l l often be referred

t o a s ALG2.

The c o n v e r g e n c e o f ALG2 w i l l b e s t u d i e d i n S e c t i o n 5.

The a l g o r i t h m ALG2 seems t o h a v e b e e n f i r s t i n t r o -

Remark 3 . 3 :

duced by GLOWINSKI-MARROCCO 111 i n c o n n e c t i o n w i t h t h e n u m e r i c a l s o l u t i o n of problem (1.23).

The c o n v e r g e n c e o f ALG2 f o r

p,

= p = r

w a s d e m o n s t r a t e d i n MERCIER C21 i n r e l a t i o n t o t h e n o n l i n e a r e l a s t i -

c i t y p r o b l e m o f C h a p t e r V I , and t h e n e x t e n d e d by GABAY-MERCIER 111 t o t h e case where O < p n = p < 2 r

, under

q u i t e g e n e r a l assumptions on

(G being l i n e a r ) .

F

3.3

A p p l i c a t i o n t o t h e examples of S e c t i o n 1

I n o r d e r t o c l a r i f y t h e concepts introduced i n Sections 3.1 3.2,

and

w e s h a l l now s e t o u t A L G l and ALG2 e x p l i c i t l y f o r t h e two model

p r o b l e m s o f S e c t i o n 1.1.

Case o f Example 1 of S e c t i o n 1.1 With t h e L a g r a n g i a n

t o account (1.26)-(1.31)

( F l o w of a Bingham f l u i d )

ier d e f i n e d by ( 1 . 2 4 ) ,

(1.25),

and t a k i n g i n -

and ( 3 . 4 ) , ( 3 . 5 ) , A L G l t a k e s t h e f o l l o w i n g

form :

(3.15)

t h e n , An

lo

a r b i t r a r i l y chosen i n

b e i n g known, d e t e r m i n e

system

- rAu"+rV*p"-V*A" (3. 16) Unlr

= 0,

= f

on R ,

(L2 ( Q ) ) 2 ;

{un,pn}

by s o l v i n g t h e coupled

112


Remark 3 . 4 :

In practice

(CHAP.

3)

w i l l be applied t o an

(3.15)-(3.18)

approximation

(by 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 e l e m e n t s ) o f t h e pro-

blem ( 1 . 4 ) .

Looking a t ( 3 . 1 6 ) ,

( 3 . 1 7 ) , it a p p e a r s t h a t t h e imple-

w i l l present n o p r a c t i c a l d i f f i c u l t i e s p r o v i d e d t h a t an e f f i c i e n t p r o g r a m i s

m e n t a t i o n of t h e b l o c k r e l a x a t i o n method

(3.8)-(3.10)

-A.

a v a i l a b l e f o r s o l v i n g t h e D i r i c h l e t problem f o r

a

T o p a s s from A L G l t o ALG2 it s u f f i c e s t o r e p l a c e ( 3 . 1 5 ) by

0

{p , A

(3.19)

1

1

a r b i t r a r i Z y chosen i n

( L 2 ( n ) ) 2 x (L2(Q))2;

a n d ( 3 . 1 6 ) by

-

rAun+rV-pn-]-V-An

= f

on

Q,

(3.20)

with t h i s algorithm t h e determination of

un, pn

is sequential.

Case o f Example 2 o f S e c t i o n 1.1 ( E Z a s t o p Z a s t i c t o r s i o n ) With t h e L a g r a n g i a n account o f

(1.34),

Pr

defined by ( 1 . 3 2 ) ,

( 1 . 3 5 ) and o f

( 1 . 3 3 ) and t a k i n g

( 3 . 4 ) , ( 3 . 5 ) , A L G l t a k e s t h e follow-

i n g form:

Ao

(3.21)

then,

a r b i t r a r i Z y chosen i n


An

-

rAun+rV-pn-V.An

=

( L 2 (R)) 2 ;

{un'pn}

by

f on Q,

(3.22)

I u n lr (3.23)

pn =

= 0,

An+rVun

1

sup( I+r, in+rvunl ) (3.24)

An+'

'

= An+pn(Vun-pn).

This algorithm i s very c l o s e l y r e l a t e d t o (3.15)-(3.18) 3.4 i s e q u a l l y v a l i d f o r ( 3 . 2 1 ) - ( 3 . 2 4 ) .

and Remark

4)

(SEC.

113

CONVERGENCE O F A L G l

T o p a s s from A L G l t o A L G 2 , i t s u f f i c e s t o r e p l a c e ( 3 . 2 1 1 , by ( 3 . 1 9 )

,

(3.22)

(3.20).

Remark 3.5:

A l l o t h e r t h i n g s b e i n g e q u a l , t h e " c o s t " o f an

i t e r a t i o n i s h i g h e r f o r ALGl than f o r ALG2,

and i n a l a r g e number of

problems it i s p r e f e r a b l e t o u s e t h i s l a t t e r a l g o r i t h m .

less w e s h o u l d p o i n t o u t t h a t i n c e r t a i n v e r y s t i f f example ( 1 . 4 )

g

with

Nonethe-

problems, f o r

" l a r g e " o r a l t e r n a t i v e l y (1.23) w i t h

p

" c l o s e " t o 1 o r " l a r g e " , A L G l i s f a s t e r t h a n ALG2 b o t h i n t h e number o f i t e r a t i o n s and i n t h e computation t i m e .

A l l these points w i l l

be i l l u s t r a t e d t h r o u g h v a r i o u s examples i n t h e f o l l o w i n g c h a p t e r s .

4.


In section 4.1,

w e s h a l l s t u d y t h e convergence o f A L G l when

H a r e H i l b e r t s p a c e s of a r b i t r a r y d i m e n s i o n ;

and

V

then i n Section 4 . 2

w e s h a l l examine t h e e x t e n t t o which t h e a s s u m p t i o n s o f S e c t i o n 4 . 1 can be weakened when

4.1

V

and

a r e of f i n i t e d i m e n s i o n .

H

General c a s e

To s t u d y t h e convergence of ALGL, w e s h a l l be making s e v e r a l supplementary a s s u m p t i o n s .

B

(4.1)

W e s h a l l f i r s t assume t h a t w e have

i s i n j e c t i v e and

ImB

i s closed i n

n.

I n a d d i t i o n , w e s h a l l make t h e f o l l o w i n g assumption on

F

concerning

growth a t i n f i n i t y

Taking i n t o a c c o u n t t h e p r o p e r t i e s o f G I

w r i t e p = Bu. W e n e x t assume t h a t 1.s.c.

~~

u

a d m i t s a unique s o l u t i o n

(P)

~

F = F

on H , and where

0

Fo

+

F1,

*.

(4.1)

and ( 4 . 2 )

imply t h a t

I n t h e following t e x t we s h a l l

where

F1

i s convex, p r o p e r and

i s convex, d i f f e r e n t i a b l e on

H I and

~

P r o v i d e d w e assume t h a t j:X

-tit h e n

Dom(PoB) n D o m ( G ) # 0

D o m ( j ) = {xc X, j ( x ) cR) ) .

(we r e c a l l t h a t i f

DECOMPos ITION - COORDINAT ION

114

convex

uniforniZy

O U ~ Tt

(CHAP. 3 ;

in the following

H

h e b o u n d e d s u b s e t s of

sense:

M > 0, t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n 6,(0) = 0 , s u c h t h a t CO,2M1 + 1 R , s t r i c t l y i n c r e a s i n g , w i t h

For any

6M: for a l l

p,qcH, l p ( S M , ( q l S M ,

(F;(q)-F;(P),q-P)

(4.3)

Idhere, i n ( 4 . 3 ) , FA

V x H x

if

pn

$&-PI).

H.

L r admits a saddle-point

Under t h e above a s s u m p t i o n s o n

{u,p,X}

and

F

G,

and

O < C ~< p S a l < 2 r , o n

we h a v e f o r

ALGl

n

t h e f o l l o w i n g convergence r e s u l t s :

(4.5)

u

+

u

strongly i n

V,

(4.6)

pn

+

p

strongly i n

H,

(4.7)

x"+~-x"

(4.8)

An

tu,p,h*}

Proof:

+

o

strongly i n

i s bounded i n

Furthermore i f

A*

H,

H.

A"

i s a ( w e a k ) c l u s t e r p o i n t of

is a s a d d l e - p o i n t

of

v

zr(v,q,p) on

x

n

x

in

H,

n.

In the following text we shall write

We thus have to show that

(4.10)

B,

satisfies

(4.4)

ded;

Fo.

denotes the gradient of

Suppose t h a t

THEOREM 4 . 1 :

on

2

we h a v e :

{u,p,X}

;" 0, p" -t

+

being a saddle-point of

G(v)-G(u)+(X,B(v-u))+r(Bu-p,B(v-u))s

0, and that

;er, w e 0, V v

E

v,

have

-n

A

remains boun-

(SEC. 4 )

115


X

(4.12)

=

A + pn(Bu-p), (3.2 ) - ( 3.5)

Furthermore

.

Vn

imply n n -p ,B(v-u"))

,

(4.13)

G(v)-G(u")+(X",B(v-u"))+r(Bu

(4.14)

(FA(pn) .q-pn)+FI (q)-FI (pn)-(An,q-pn)+r(pn-Bu?q-pn)

(4. I S )

An+'

= hn+pn(Bu

W e t h u s set

I

r Bin 1 '-r

(4.16)

n

(4.18)

in (4.10),

(p",

Bin) + (h", Bin)

and ( 4 . 1 7 )

q = p

n

in (4.11),

S

, v q E H,

v

then

=

u

in

0,

.

and r e g r o u p i n g t h e t e r m s , w e o b t a i n

,pn-p)+r(B;"-pnI2+(Xn,Bin-Fn)

(F;(pn)-FA(p)

5

0

.

f r o m ( 4 . 1 5 ) and t a k i n g t h e s c a l a r s q u a r e i n

Also, s u b t r a c t i n g (4.12) H,

0

2

.

(F~(pn)-F~(p),pn-p)+r~~n~2-r(~n,B?)-(~n,~n)S O

Summing ( 4 . 1 6 )

V v EV,

i n ( 4 . 1 4 ) , s o t h a t , by a d d i t i o n , w e h a v e

q = p

(4.13) and

(4.17)

v = u

n n -p )

t0

we obtain

Ihn/2-1Xn+112 = -

(4.19)

-n -n -n 2 -n -n 2 2pn(Bu -P ,A )-pnlBu -P 1 ,

I t t h e n f o l l o w s f r o m ( 4 . 1 8 ) and

lhn/2-lXn+1 1'

(4.20)

5

(4.19)

2pn(FA(pn)-F;(p),pn-p)+pn(2r-pn)

I n v i e w of t h e a s s u m p t i o n ( 4 . 4 ) on

(4.21)

l i m I ~ u ~ - p=~ ol n++m

(4.22)

lim (F;(P~)-F;(P),P"-P) n++m

Xn

and w e h a v e

ional

F

(4.23) Since

( since

IBu -n -p-n I 2

.

w e t h u s have

pn

~u=p),

= 0,

bounded i n H .

W e now show t h a t

pn

i s bounded i n H ; i n f a c t , s i n c e t h e f u n c t -

is proper, there e x i s t s

Then p u t t i n g

that

q =

3

6

E

H,

such t h a t

--

0 , such t h a t

( p n l S M , Vn.

(4.25)

Furthermore, f o r (4.26)

M

s u f f i c i e n t l y l a r g e , w e a l s o have

/PI 5I.i.

I t t h e n f o l l o w s from ( 4 . 2 5 ) ,

of

(4.2),

that there exists

Fo

(4.27)

(4.26)

o v e r t h e bounded s u b s e t s of ( ~ ; ( p ~ ) - ~ ; ( p ,pn-p) )

2

&M(I~n-~I)

and from t h e u n i f o r m c o n v e x i t y H,

t h a t we have

v n.

W e t h u s have l i m 6 (Ip"-p\) n++m M

(4.28)

= 0.

I t t h e n f o l l o w s from t h e p r o p e r t i e s of

6'M

that

l i m Ipn-p( = 0, n++m

(4.29)

and from ( 4 . 2 1 )

that

l i m ~u~ = p ( = ~ u ) , s t r o n g z y i n n++m.

(4.30)

Since the operator

is i n j e c t i v e with

B

p l i e s ( s e e Remark 2 . 2 ) l i m un = u n++m

(4.31)

,

H.

ImB

closed i n H,

t h i s im-

that

s t r o n g l y in

The p r o p e r t i e s of t h e sequence

An,

V. g i v e n i n t h e s t a t e m e n t of t h e

Theorem, a r e t h e n immediate consequences of t h e convergence p r o p e r t i e s of

un

and of

pn

and a r e o b t a i n e d by p a s s i n g t o t h e l i m i t i n

(3.3)-(3.5).

Remark 4.1:

a

W e have proved t h e convergence o f t h e a l g o r i t h m by I

making u s e of t h e a s s u m p t i o n of c o e r c i v i t y ( 4 . 3 )

on F o ( . ) .

It is

e a s y t o show t h a t an analogous r e s u l t i s o b t a i n e d by assuming t h a t G(')

is differentiable, its derivative satisfying a condition

(SEC. 4 )

117

CONVERGENCE OF ALGl

s i m i l a r t o (4.3).

I n f a c t t h e c o n v e r g e n c e p r o o f s i m p l i f i e s and i t

i s n o l o n g e r n e c e s s a r y t o assume

B

t o be injective.

We shall

e n c o u n t e r i n C h a p t e r I V a case where t h i s v a r i a n t w i l l b e u s e f u l .

4.2

The f i n i t e - d i m e n s i o n a l

If

and

V

case

are o f f i n i t e d i m e n s i o n , c o n v e r g e n c e o f ALGl c a n b e

H

o b t a i n e d under weaker assumptions t h a n t h o s e g i v e n i n S e c t i o n 4.1.

h r t o admit

Firstly, for

a s a d d l e - p o i n t i t w i l l be s u f f i c i e n t

to a d m i t a s o l u t i o n (see Remark 2 . 3 ) . Furthermore, I m is still closed. A s regards F , i t follows from CEA-GLOWINSKI

for

(P)

[I, S e c t i o n 2 . 2 1 t h a t t h e u n i f o r m c o n v e x i t y p r o p e r t y o f

€3

given

Fo

i n Section 4 . 1 is s a t i s f i e d i f (4.32)

i s s t r i c t l y c o n v e z a n d of c l a s s C1.

Fo

Fo

In fact i f

s a t i s f i e s ( 4 . 3 2 ) and if we assume t h a t (P) a d m i t s a

s o l u t i o n w e can d i s p e n s e w i t h assumption ( 4 . 2 ) . t h e f a c t t h a t t h e s t r i c t convexity of

T h i s f o l l o w s from

Fo i m p l i e s t h e s t r i c t

I

monotonicity

of

LEMMA 4 . 1 :

F o , and from t h e f o l l o w i n g :

Suppose

H

i s of f i n i t e d i m e n s i o n ;

be a c o n t i n u o u s a n d s t r i c t l y monotone o p e r a t o r , and

(p"),

(4.33)

a s e q u e n c e of e l e m e n t s of l i m (A(P?-A(~), n++m

pn-p)

H

let

H

A:

-t

p a n e l e m e n t of

H H

such t h a t

= 0.

We t h e n have lim

(4.34)

p" = p .

n++w

Suppose t h a t ( 4 . 3 4 ) i s n o t t r u e ; i n t h i s case t h e r e

Proof: exists

6

> 0

and a s u b - s e q u e n c e e x t r a c t e d from

m

(p"),,

(p

Im

say,

such t h a t Ipm-pI 2 6

(4.35)

6

Let

S(p;z)

ine

zm

E

Vm

.

be t h e s p h e r e w i t h c e n t r e

S(p;

6

2)

p

and r a d i u s

6 2.

W e def-

by

W e r e c a l l t h a t A: H -t H i s s a i d t o b e s t r i c t l y monotone i f ( A ( q ) - A ( p ) , q-p) > 0 V p . 9 8, P # 9.


118

m z m = p + ; x € L

(4.36) z

m

thus belongs

;

I Pm-P I t o the

"open i n t e r v a l "

m

6

-

tm t h e q u a n t i t y

of t h e s p a c e

lp,p [

(see F i g u r e 4 . 1 ) . W e s h a l l d e n o t e by

(CHAP. 3 )

;

H

w e t h u s have

2 Pm-P 1 1

Figure 4 . 1

(4.37)

zm = p

+ t m ( pm - p ) ,

a n d , from ( 4 . 3 5 ) , (4.38)

A

Since the operator (4.39)

(A(pm)-A(p)

i s s t r i c t l y monotone w e h a v e

,pm-p) > ( A ( p + t ( p m - p ) ) - A ( p )

hence i n p a r t i c u l a r ( w e set (4.40)

m

>O

Vt

E

10,1[

,

i n (4.39))

( A ( P ~ ) - A ( P ) ,pm-p) > ( A ( Z ~ ) - ( A ( P ) ,pm-p) > 0.

I n view o f

(4.37),

Since the sphere sub-sequence, (4.42)

t = t

.P"-P)

lim

( 4 . 3 8 ) i t f o l l o w s from ( 4 . 4 0 )

36 )

S(p;

i s compact w e can e x t r a c t from

i t s e l f a l s o d e n o t e d by zm = z

,

z

that

E s ( ~ ;

6

y)

m

(z )m

a

m

( z )m, such t h a t

.

m++m

Since the operator

(4.42)

that

A

i s c o n t i n u o u s w e deduce from ( 4 . 3 3 ) ,

(4.41),

(S E C .

4)

119

CONVERGENCE OF AL G l

being s t r i c t l y monotone, (4.33) i m p l i e s t h a t z = p, 6 which i s a b s u r d s i n c e lp-zl = - > O . W e c a n n o t t h e r e f o r e have ( 4 . 3 5 ) ; 2 n hence p converges t o p. The o p e r a t o r

A

I n view o f t h e s e v a r i o u s r e m a r k s , o n e c a n e a s i l y p r o v e t h e f o l l owing v a r i a n t o f Theorem 4 . 1 :

Suppose t h a t V and H a r e o f f i n i t e d i m e n s i o n (P) a d m i t s a s o l u t i o n u. We make t h e f o l l o w i n g a s s u m p B, G , F

THEOREM 4.2:

and t h a t t i o n s on

- B i s injective,

-

G i s c o n v e x , p r o p e r and Z . S . C .

- We h a v e F = Fo o n H , and Fo T h e s o l u t i o n of

f

F1

with

F1

on V , c o n v e x , p r o p e r and l . s . c .

s t r i c t l y c o n v e x a n d of c l a s s

C1

on H .

i s t h e n u n i q u e , and under t h e c o n d i t i o n

(P)

we h a v e for A L G l t h e f o l l o w i n g c o n v e r g e n c e r e s u l t s lim

un = u,

n++m

lim pn = n++m

BU,

i s b o u n d e d i n H.

An

M o r e o v e r , if

i s a c l u s t e r p o i n t of

h

a saddle-point

Remark 4 . 2 :

of

Lr on

V x H x

An

i n H, then

{u,Bu,h}

is

n.

W e s h a l l m e e t i n C h a p t e r V, i n c o n n e c t i o n w i t h t h e

minimum s u r f a c e s p r o b l e m , a s i t u a t i o n i n which t h e a s s u m p t i o n ( 4 . 2 ) on

F

i s n o t s a t i s f i e d , e i t h e r f o r t h e continuous problem o r f o r

t h e a p p r o x i m a t e p r o b l e m , and i n which t h e c o n v e r g e n c e o f A LG l w i l l f o l l o w ( f o r t h e a p p r o x i m a t e p r o b l e m ) from Theorem 4 . 2 .

4.3

On t h e c h o i c e o f

r

a n d of

{pnIn.

I n t h e g e n e r a l case t h e d e t e r m i n a t i o n o f t h e o p t i m a l p a r a m e t e r s

i s a c o m p l i c a t e d m a t t e r , as a c o n s e q u e n c e o f t h e n o n l i n e a r i t y of problem (3.4)

,

(3.5) ; furthermore, t h e convergence p r o p e r t i e s of t h e

120

(CHAP


sequences

{unln

and

Ip

n

3)

l n , {An}n may be d i f f e r e n t , a s w e s h a 1

see below. We c o n s i d e r t h e c a s e where 1

lq(

2

(4.44)

F(q) =

(4.45)

G(v) = -((f,v)), B

still satisfying (4.1).

(P)

is then equivalent t o

the operator The problem t

B Bu

(4.46) and

Zr

,

=

f,

admits a s a s a d d l e - p o i n t

{u,Bu,Bu}; w e t h u s have

I t i s a p p a r e n t t h a t t h e u s e of A L G l i s of no p r a c t i c a l

A = p = Bu.

i n t e r e s t f o r s o l v i n g ( 4 . 4 6 ) s i n c e , a t e a c h i t e r a t i o n , i t w i l l be

t

n e c e s s a r y t o s o l v e l i n e a r problems r e l a t i n g t o B B .

Nonetheless,

t h i s t r i v i a l ' case o f f e r s a c e r t a i n t h e o r e t i c a l i n t e r e s t f o r the The l a t t e r i s w r i t t e n ,

s t u d y o f t h e convergence of t h e a l g o r i t h m . ( l i m i t i n g o u r s e l v e s t o t h e c a s e where (4.47)

then,

b e i n g known, c a Z c u l a t e

+

(4.48)

BtAn

(4.49)

p" = A"

(4.50)

A"+l

=

rBt(Bun-pn)

+

=

n n r ( ~ -p u 1,

A n + p (Bun-pn)

(4.51)

B'X~+~B~(B;"-;") =

o

(4.52)

pn

v nL 0,

(4.53)

In+I

i;"+r(~;"-p~) = in+p

. pn-p,

xn

= An-A,

vn t 0,

Bt,

w e deduce from ( 4 . 5 1 ) t h a t

~ ~ =6o "V n ? 0 .

We t h e n deduce from ( 4 . 5 1 ) and from ( 4 . 5 4 ) t h a t (4.55)

BtXn

=

-rBtB;"

w e have

v n t 0.

(B;"-;")

M u l t i p l y i n g ( 4 . 5 2 ) by (4.54)

by

un,pn,An+'

f,

n Again u s i n g t h e n o t a t i o n ii" = u -u,

=

= p),

a r b i t r a r i l y specified i n H;

A'

A"

p,

.

V n t 0.

(SEC. 4 )

1 21


M u l t i p l y i n g ( 4 . 5 3 ) by Bt

we obtain Btg;n+l

(4.56)

and t a k i n g a c c o u n t o f

(y)

=

r

W e t h e r e b y deduce t h a t i f

v

B~B;"

Btgin = (I-:)

= p,

( 4 . 5 4 ) and ( 4 . 5 5 ) ,

n S 0.

w e have4

un = u, v n 2 1 ;

(4.57)

w e t h u s have convergence o f value of

Ao

sequence

u

n

For

H.

E

p

i n two i t e r a t i o n s , whatever t h e

un

# r

t h e rate of convergence o f t h e

d e p e n d s on t h e r a t i o

f.

m

(4.55) w e have

B;~=

- 1 B(B~B)-'B'x~

I n view

An.

W e now c o n s i d e r t h e c o n v e r g e n c e o f t h e s e q u e n c e of

,

or a l t e r n a t i v e l y (4.58)

B;"

=

-

I -n -PA

r

i s t h e p r o j e c t i o n o p e r a t o r from H o n t o ImB.

where P = ( B t B ) - l B t By s u b s t i t u t i n g

I"+'

(4.59)

= (1

( 4 . 5 8 ) i n t o ( 4 . 5 3 ) , and by u s i n g ( 4 . 5 2 ) , w e o b t a i n

-P ) xn+p(l 1 +r I+r

By p r o j e c t i o n o f K e r Bt (4.60)

r

SO.

(4.59) o n t o ImB, then o n t o t h e orthogonal subspace

w e t h u s deduce pxn+' = ( 1

- P)PX",

I n t h e case where

iterations. the factor

!l+r'

l)pxn, t/n

p

vn>O,

= r

w e t h u s have convergence o f

The p r o j e c t i o n o f P (1- l+r), i.e.

I f we choose

Ao

An-A

o n t o K e r Bt

i n t h e case where

p

PAn

i n two

" d e c r e a s e s " by

= r , a f a c t o r of

i n ImB we thus have, f o r

=

p

r , converg-

i n two i t e r a t i o n s . If Ao i s chosen a r b i t r a r i l y i n A n t o A = Bu w i l l b e f a s t e r t h e l a r g e r t h e H t h e convergence of ence of

An

v a l u e o f r , when

p

= r.

A s regards t h e sequence p

n

it f o l l o w s

f r o m t h e preceding relations t h a t

' BtB

B i n j e c t i v e with ImB c l o s e d i n H implies t h a t i s o m o r p h i s m o f V o n t o V.

W e recall t h a t

i s an

122


t h e sequence

pn

(CHAP. 3)

t h u s behaves l i k e t h e sequence

m

(I-P)xn.

The p r e c e d i n g a n a l y s i s i n d i c a t e s t h a t i n c e r t a i n c a s e s w e may e x p e c t a f a s t e r convergence f o r

un

An

than f o r

or

pn.

Such

a phenomenon h a s i n f a c t b e e n e s t a b l i s h e d e x p e r i m e n t a l l y (see

[ l l ) i n t h e case o f t h e e l a s t o p l a s t i c t o r s i o n p r o b l e m (see S e c t i o n 1.1, Example 2 ) . I t a l s o a p p e a r s , i n t h e l i g h t of

GABAY-MERCIER

numerous n u m e r i c a l e x p e r i m e n t s , t h a t t h e c h o i c e o f

is

"

quasi-op t i m a l "

r

= p =

pn

.

I t i s a l s o e a s y t o show t h a t t h e c o n v e r g e n c e of A L G l w i l l b e

f a s t e r t h e l a r g e r t h e v a l u e of

r.

From a p r a c t i c a l p o i n t o f

v i e w , however, t h e s i t u a t i o n i s r a t h e r more complex: c o n d i t i o n i n g of t h e system ( 3 . 4 ) ,

in fact, the

( 3 . 5 ) g e t s worse as

r

increases,

s o t h a t t h e s p e e d o f c o n v e r g e n c e of t h e r e l a x a t i o n a l g o r i t h m ( 3 . 8 ) Moreover t h e c h o i c e o f t h e t e r m i n a t i o n t e s t f o r

(3.10) decreases.

the internal iterations

(3.8)-(3.10)

e r r o r s also play a p a r t .

and t h e e f f e c t o f r o u n d i n g

E x p e r i m e n t a l l y t h e combined e f f e c t o f

t h e s e f a c t o r s - namely, w i t h a n i n c r e a s e o f

r

an a c c e l e r a t i o n of

A L G l b u t a s l o w i n g down o f t h e i n t e r n a l r e l a x a t i o n a l g o r i t h m

-

leads

i n many cases t o a n a l g o r i t h m whose o v e r a l l s p e e d o f c o n v e r g e n c e ( i n t e r m s o f c o m p u t a t i o n t i m e ) d e p e n d s r e l a t i v e l y l i t t l e on t h e r ; t h i s f a c t w i l l b e i l l u s t r a t e d by t h e v a r i o u s e x a m p l e s

choice of

considered i n t h e following chapters.

5.

CONVERGENCE OF ALGZ

I n t h i s s e c t i o n w e s h a l l show t h a t u n d e r q u i t e g e n e r a l assumpti o n s on

F

and I+&

G

w e h a v e c o n v e r g e n c e o f ALGZ u n d e r t h e c o n d i t i o n

r ; w e d o n o t know w h e t h e r t h i s r e s u l t i s o p t i m a l , s i n c e i n c e r t a i n p a r t i c u l a r cases ( G l i n e a r , f o r e x a m p l e ) t h e u p p e r

o < pn

= p < (T)

bound of t h e i n t e r v a l o f c o n v e r g e n c e c a n b e r e p l a c e d by

2r

.

In

f a c t t h i s q u e s t i o n becomes somewhat a c a d e m i c i n c h a r a c t e r ( i n o u r o p i n i o n ) s i n c e i n t h e v a r i o u s a p p l i c a t i o n s of ALGZ which h a v e b e e n undertaken t h e optimal choice f o r

5.1

p

seems t o b e

p =

r

.

G e n e r a l case

W e a r e now g o i n g t o c o n s i d e r t h e c o n v e r g e n c e o f ALGZ u n d e r t h e same a s s u m p t i o n s as t h o s e u s e d i n S e c t i o n 4 . 1 f o r A L G 1 .

123

CONVERGENCE OF ALG2

(SEC. 5 )

W e have

V

x

H

(5.1)

Under t h e a s s u m p t i o n s on

x H.

and i f

4.1,

Lr admits a saddle-point

We a s s u m e t h a t

THEOREM 5.1:

on

{u,p,A}

used i n S e c t i o n

G

F,

B,

satisfies

pn

02p(F:,(pn)-F~(p),pn-p)+p

W e now t r y t o modify t h e f i n a l t e r m on t h e r i g h t - h a n d s i d e o f

(5.16). Starting from B;n = (B;n-B;n-I

)+(B;n-l-pn-l

)+pn-l

From ( 5 . 1 7 ) and f r o m

w e deduce t h a t

C o n s i d e r i n g ( 5 . 1 0 ) on i t e r a t i o n (5.19)

(FA(pn-'),q-p

Taking

q = pn- 1

n- I

)+Fl(q)-FI(p

n

-

1

i n s t e a d of

n , w e have

n-1 ,q-p ) 20.

n-l)-(Xn-l,q-pn-l)+r(pn-l-B~n-~

i n ( 5 . 1 0 ) and

q = p"

i n ( 5 . 1 9 ) w e o b t a i n by

addition

1

I t f o l l o w s from t h e m o n o t o n i c i t y of (5.21)

Fo,

a n d from ( 5 . 2 0 ) , t h a t

rlpn-pn-112-(~n-~n-l-n -n-l)-r(B;n-B;n-l 3P -P

-n -n-l .p -p ) 50.

(SEC. 5 )

W e have

125

CONVERGENCE OF ALG2

(from ( 3 . 1 4 ) )

In

(5.22)

=

In-1 p(B;n-

1 --n-1

+

I t t h e n f o l l o w s from (5.21)

1 2-p(B;n-I

1;"-pn-I

).

and ( 5 . 2 2 ) t h a t -n-1 -n -n-I p -p )-r( B?-Bin-'

-p

,;"-in-')

s0

,

i.e. r(B;n-B;n-l

(5.23)

-n -n-l) rrlpn-pn-lj2-p(B;n-l -n-1 -n -n-1 .P -P -P .P -P )

I t t h e n f o l l o w s from ( 5 . 1 8 ) ,

2pr(B;n

-n -n-I

,P -P

(5.24) +

zP(r-P)(Bu

)

(5.23) t h a t

-n 2 >Pr(lp I

-IP -n-1 I 2 )+prlp-n-P-n-l I 2

+

-n-I -n-1 -n -n-I -P ,p -P ).

F i n a l l y , combining ( 5 . 1 6 ) ,

(5.241, we o b t a i n

Then u s i n g t h e Cauchy-Schwarz i n e q u a l i t y it f o l l o w s f r o m ( 5 2 5 ) t h a t t/a>o

If

we have

p =

r

i t i s c l e a r t h a t b y u t i l i s i n g t h e same method as i n t h e

p r o o f o f Theorem 4 . 1 w e h a v e ( 5 . 2 ) - ( 5 . 5 ) . If

O < p < r,

taking

c1 = 1

and o b s e r v i n g t h a t

Ir-pl = r-p,

and

t a k i n g i n t o account (5.261, w e have

w h i c h i m p l i e s , as i n Theorem 4 . 1 ,

Ir -

PI

= p

r w e have - r ; it t h e n f o l l o w s f r o m ( 5 . 2 6 ) t h a t t h e c o n v e r g e n c e (5.2)-(5.5).

If

p

>

126


r e s u l t s (5.2)-(5.5)

w i l l apply i f we have

By e l i m i n a t i o n of 2

pM

i.e.

(since

-

rpH

PM

pM,

where

w e deduce from ( 5 . 2 7 ) t h a t

u

-

p

(CHAP. 3)

r2 =

o

’ 0)

Taking i n t o a c c o u n t t h e c o n v e r g e n c e r e s u l t s ( 5 . 2 ) - ( 5 . 5 ) , t h e i n t h e s t a t e m e n t of t h e

weak c l u s t e r p o i n t p r o p e r t y o f

Theorem c a n e a s i l y b e d e d u c e d , by p r o c e e d i n g t o t h e l i m i t i n (3.12)(3.14).

a

F i n i t e - d i m e n s i o n a l case

5.2

Using a v a r i a n t o f t h e p r o o f o f Theorem 5 . 1 ,

t o g e t h e r w i t h Lemma

w e can e a s i l y prove t h e f o l l o w i n g :

4.1,

THEOREM 5.2:

S u p p o s e t h a t t h e a s s u m p t i o n s on

a r e t h o s e i n t h e s t a t e m e n t of T h e o r e m

4.2.

V,

H,

F, B , G

Then i f

t h e c o n c l u s i o n s i n t h e s t a t e m e n t o f Theorem 4.2 a r e s t i l Z v a l i d .

5.3

C h o i c e of

Discussion.

p

and of

r.

W e b e g i n w i t h some r e m a r k s :

Remark 5 . 1 :

[ll

If

G

i s Z i n e a r i t h a s b e e n p r o v e d by GABAY-MERCIER

t h a t ALG2 c o n v e r g e s i f O O ) and the operator fined by a(u,v) = V u , v e V.

A E P(V,V')

is de-

The problem to be solved can then be written in the form of a variational inequality: (7.12)

1

, V v e K,

a(u,v-u) 2 UEK.

This formulation can be extended to the case where

a(u,v) is not

symmetric. In the latter case (7.12) is no longer equivalent to a minimisation problem. We shall now introduce for the solution of this problem a decomposition principle whose aim is to obtain a family of optimisation problems, coordinated via a Lagrange multiplier. We thus put (7.13)

w

= { { v , q } ~ v x vM , v-qi = 0

x

= {{v,q}E

, vi=l, . . . 3 1,

and (7.14)

W , qi

E

Ki, V i = l ,

...,Pl}.

It is clear that the original problem is equivalent to (7.15)

inf {v,ql E

x

fo(q)

,

where we have written,

Suppose I K ,

is the indicator function of the convex set

1

write

It is then natural to consider the augmented Lagrangian

Ki.

We

(SEC.

NONLINEAR PROGRAMMING PROBLEMS

7)

139

The exWe thus look for a saddle-point of L r on V x $1 x VM. istence of such a saddle-point poses no problem in the finite dimensional case. The formulation (7.18) will be simpler to work with is identified with its dual. This poses no problem in

Remark 7 . 3 :

if

V

It will suffice to equip IRN

f i n i t e dimensions.

Euclidian metric.

with the

If another metric is used, we shall indicate at

that time the modifications which need to be made to the algorithms. 7.2.2.

S o l u t i o n o f t h e p r o b l e m by ALGl a n d ALG2

In accordance with the general results, algorithm ALGl is here written as follows: (7.19)

AO

c

$1


n 2 0, a n d w i t h of the system for

(7.22)

A*+' i =

:A

+ p,(u

An

n n -pi),


{un,pn}

i=l, ...,M.

We note that (7.21) is a system of variational inequalities, each n of these inequalities involving only a s i n g l e constraint pi E Ki. In many cases each of the problems decoupled in this way will be easier to solve than the original problem. For example if we use an algorithm based upon a projection onto K, it is in general much easier to project onto each of the Ki independently than onto their intersection. The same remark also applies for algorithms requiring the construction of an admissible solution. A l s o , such a process is well adapted to p a r a l l e l c o m p u t a t i o n , the possibility


140

(CHAP. 3)

of which can be anticipated on future computers. We can clearly pass from algorithm ALGl to ALGZ by replacing (7.19) and (7.20) by: (7.23)


{ p o . x I ) c?WM

and

The calculation of un and pn has become s e q u e n t i a l and no longer s i m u l t a n e o u s , but the M components of pn can be calculated i n d e p e n d e n t l y of one another, and in particular can be calculated in p a r a l l e l . By way of an example, we consider ALGZ in the particular case In order to fully describe the where F is of the form (7.11). algorithm we introduce the operator S of isomorphism between V and V'. (We have S = I if V is identified with its dual). We clearly have (7.25)

(U,V),

= <su,v>

V'XV

.

Using this notation, the algorithm ALG2 can be written: (7.26)

(po,X1}

E

io010 a(pi ,qi-pi)+r(pi

(7.28)


?x#

vqiEKi,

,qi-pi)"

P ~ E K i =~ l , ,

2

< f , q .1- p n1> + <S(=U"+X:)

,qi-p2

...,M,

Such an algorithm can also be applied in the case where the bilinear form a(u,v) is V-elliptic, and non-symmetric. If, in the symmetric case, we equip V with the norm 1 1 ~ 1 1 =~ a ( v , v ) = < A v , v > , (7.28) becomes (I+r) <Ap;,qi-p4>

2

+

(SEC.

7)


Remark 7 . 4 :

141

The a l g o r i t h m ALG2 h a s been d e d u c e d f r o m ALGl v i a

a s o l u t i o n method b a s e d on a b l o c k r e l a x a t i o n .

The r e a d e r c a n

e a s i l y d e d u c e a n a l g o r i t h m i n which p r o b l e m s o f t h e t y p e ( 7 . 2 1 ) o r ( 7 . 2 8 ) a r e s o l v e d s e q u e n t i a l l y and no l o n g e r i n p a r a l l e l f a s h i o n . F i n a l l y , w e n o t e t h a t t h e augmented L a g r a n g i a n which w e h a v e u s e d

i s o n l y one p o s s i b l e e x a m p l e .

W e c o u l d f o r example i n ( 7 . 1 3 ) h a v e

d e f i n e d a1t e r n a t i v e l y

(7.31)

w

= { { v , q j E v x V M ; v=ql ; qi=qi-l

, i=l ,...,MI

which would o f c o u r s e h a v e l e d t o q u i t e d i f f e r e n t a l g o r i t h m s .

Remark 7 . 5 :

The methods d e s c r i b e d i n t h e p r e s e n t s e c t i o n 7 . 2

can b e v i e w e d a s f r a c t i o n a l - s t e p methods w i t h m u l t i p l i e r , which i n a c e r t a i n s e n s e g e n e r a l i s e t h e methods d e s c r i b e d i n BENSOUSSANLIONS-TEMAM [ l , C h a p t e r 2 1 , and

which,amongst

other things, allow

us t o a v o i d t h e u s e of t h e d i v e r g e n t series u t i l i s e d i n t h e above reference.

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 Weber p r o b l e m S t a t e m e n t of

t h e problem

C e r t a i n a u t h o r s (see COOPER-KATZ

E l l , f o r example) u s e t h e des-

i g n a t i o n Weber p r o b l e m f o r t h e f o l l o w i n g n o n d i f f e r e n t i a b L e m i n i m i s a t i o n problem:

(7.32) where

(7.33)

J(y)

M =

1

i=I

oi IIy-”ll

9

with, i n (7.33),

(7.34)

ai>0 Yi=l,

...M

; x. , R ~ Y i = 1 , -1

...M,

P r o b l e m ( 7 . 3 2 ) a d m i t s a t l e a s t o n e solution.

142


(CHAP.

3)

I n t r o d u c t i o n o f an augmented Lagrangian f o r t h e s o l u t i o n

7.3.2.

of p r o b l e m

(7.32)

Problem ( 7 . 3 2 ) i s c l e a r l y e q u i v a l e n t t o t h e p r o b l e m

M (7.35)

1

Min {q,_y}Ew

-

ai

11 pill},

i=l

where

I t f o l l o w s from ( 7 . 3 5 ) ,

(7.36),

and f r o m t h e p r e c e d i n g s e c t i o n s , t h a t

a n augmented L a g r a n g i a n n a t u r a l l y a s s o c i a t e d w i t h p r o b l e m ( 7 . 3 2 ) i s g i v e n by:

where on

(

. ,.)

denotes, i n (7.37), t h e ordinary Euclidian inner product II-II 1 and y = { y i IMi e l ( ~RNM) .

(i.e. t h a t associated with

IRN

Max a. .

Remark 7 . 6 :

I n t h e case where

r o p r i a t e t o use, i n s t e a d of

1

L>> I, it would b e appMin ai

i ( 7 . 3 7 ) , t h e augmented L a g r a n g i a n d e f i n e d

by

7.3.3.

A p p l i c a t i o n of ALG2 t o t h e s o l u t i o n o f

The a p p l i c a t i o n o f ALG2 t o t h e s o l u t i o n o f m i n a t i o n of t h e s a d d l e - p o i n t s , Lagrangian ( 7 . 3 7 ) ,

pn ,,x n

(7.40)

and

n

,_An}

=

-

IRNM

x

(7.32),

v i a t h e deter-

BNM,o f t h e augmented

r(xn-'-,xi)

known, de*ermine s u c c e s s i v e l y

I

in+' by -1 a?

x


1, a s s u m i n g

2

IRN

leads t o t h e following algorithm:

{_xo,_X1l cRNxRNM,

(7.39)

t h e n , for

in

(7.32)

+

P,i = l , . ..M,

(SEC. 7 )

(7.41)

143


( p:

=

,O

othemise

; i=1,

...M,

(7.42) (7.43) I t s h o u l d b e n o t e d t h a t w e are n o t s a t i s f y i n g h e r e t h e c o n d i t i o n s of a p p l i c a t i o n o f Theorems 5 . 1 a n d 5 . 2 o f S e c t i o n 5 . 4 . i n g r e g a r d t o t h e c h o i c e made f o r

2,

I n f a c t , hav-

(see ( 7 . 3 7 ) ) , w e h a v e

M

and

G E 0; F i s t h e r e f o r e n o n d i f f e r e n t i a b l e

7.3.4.

and n o t s t r i c t Z y convex.

Nurnerica 1 a p p l i c a t i o n s .

W e s h a l l now a p p l y a l g o r i t h m ( 7 . 3 9 ) - ( 7 . 4 3 )

t o t h e s o l u t i o n of a

p a r t i c u l a r Weber p r o b l e m ; t h e p r o b l e m i n q u e s t i o n ( c o n s i d e r e d i n COOPER-KATZ, ( i = 1,

... 10)

loc. c i t . ) i s d e f i n e d i n i n T a b l e 7.1: i

ai

IR2

by t h e

-1 X.

1

3

(89,731

2

8

(36.891

3

3

t39,91

4

7

I14.51

5

1

{46,121

6

3

(55.11

7

9

c53.641

a

6

(32,571

9

7

(68,421

10

5

{63,921

Table 7.1 W e have used a l g o r i t h m (7.39)-(7.43)

with

ai

and

xi

144

(CHAP. 3)

DECOMPOS I T I O N - C O O R D I N A T I O N

xo

(7.44)

0,

=

I

x1

= 0, I

P = r,

(7.45)

and t h e t e r m i n a t i o n t e s t

(where llyll, = Iy11+Iy21 i f 2: = {y,,y2}) . I n T a b l e 7.2 w e h a v e r , t h e number o f i t e r a t i o n s n e c indicated, f o r s e v e r a l values of e s s a r y f o r convergence (under t h e c o n d i t i o n s

( 7 . 4 4 ) - ( 7 . 4 6 ) ) and t h e

corresponding c a l c u l a t e d s o l u t i o n s .

I

r

I number

of i t e r a t i o n s

I

calculated s o h .

I

(51 . 6 7 0 , 6 2 . 1 5 9 } { 5 1 .669,62.1591

710

{51.666,62.154}

T a b l e 7.2

I t w i l l be noted t h a t

xo

calculated solutions.

=

{O,O} i s " r a t h e r " f a r away f r o m t h e

The r e s u l t s o b t a i n e d by means o f

(7.39)-

(7.43) c o i n c i d e , t o very high accuracy, w i t h t h o s e obtained i n COOPER-KATZ,

loc. c i t . , by a s t e e p e s t d e s c e n t method; i n a c t u a l f a c t

t h e c o n v e r g e n c e of a l g o r i t h m ( 7 . 3 9 ) - ( 7 . 4 3 ) s i n c e (see T a b l e 7 . 3 ) ,

i s v e r y f a s t ( f o r r = 0.1)

as e a r l y a s t h e f i f t h i t e r a t i o n , w e a l r e a d y

h a v e a v e r y good a p p r o x i m a t e s o l u t i o n t o t h e Weber p r o b l e m con-

si d e r e d . I f i n s t e a d o f i n i t i a l i s i n g a l g o r i t h m ( 7 . 3 9 ) - ( 7 . 4 3 ) by ( 7 . 4 4 ) w e use

A1 =

and ( a s i n COOPER-KATZ,

2

xo =

i=l

loc.

cit.)

a.x.

1-1

I

w e have convergence, f o r

(

i.e.

t h e barycentre of t h e

xi),

r = 0.1, i n 2 5 i t e r a t i o n s ( i n s t e a d o f 4 1 ) .

GENERAL DISCUSSION

(SEC. 8 )

-

-

R"

5

0 . 9 5 0 ~I 0-3

{ 5 l .773,62.166)

10

0.1 17x1 0-3

I51 .699,62.098]

15

0.682X10-3

{ 5 l .687,62.1441

20

0.185~10-~

I51 .679,62.154}

n

n

X

I0,OI

0

25

0.957X10-5

{ 5 l .674,62.156}

30

o.512x10-5

{ 5 l .672,62.158)

35

0 . 2 3 4 ~ 1o

-~

{51.671,62.159}

40

0.1 ogx1 o

-~

I 5 1 .670,62.159}

41

0.944X10-6

-

Table 7 . 3 8.

145

{ 5 l .670,62.159}

(r = 0.1)

GENERAL DISCUSSION ON CHAPTER I11

A s w e have mentioned s e v e r a l t i m e s i n t h e p r e c e d i n g t e x t , t h e

methods of t h i s c h a p t e r can be e x t e n d e d t o v a r i a t i o n a l - i n e q u a l i t y problems which a r e n o t e q u i v a l e n t t o o p t i m i s a t i o n problems.

They

can a l s o be u s e d , a s i n BEGIS [ 2 1 , f o r t h e s o l u t i o n of n o n l i n e a r problems o f o r d e r 4 , c o r r e s p o n d i n g t o a problem i n v o l v i n g t h e flow

of a Bingham f l u i d which i s more g e n e r a l t h a n t h e c a s e of Example 1 of S e c t i o n 1.1.

W e s h a l l r e t u r n t o t h e above t o p i c i n C h a p t e r V I I .

The d e c o m p o s i t i o n - c o o r d i n a t i o n method which w e have p r e s e n t e d can b e r e l a t e d t o t h e methods d e s c r i b e d i n BENSOUSSAN-LIONS-TEMAM

C11.

H i s t o r i c a l l y s p e a k i n g , i t would a p p e a r t h a t t h e u s e of an augmented L a g r a n g i a n f o r s o l v i n g n o n l i n e a r v a r i a t i o n a l problems' ALG2 i s due t o GLOWINSKI-MARROCCO 111, C21, [ 3 1 .

convergence of A L G 2 GABAY-MERCIER

( i n t h e c a s e where

G

v i a A L G l and

The f i r s t p r o o f of

i s l i n e a r ) i s due t o

[ 11.

I t s h o u l d a l s o be p o i n t e d o u t t h a t , depending on t h e t y p e of problem c o n s i d e r e d , n a t u r a l v a r i a t i o n s of t h e a l g o r i t h m s d e s c r i b e d may l e a d t o more r a p i d convergence. T o c o n c l u d e t h i s c h a p t e r , i t s h o u l d b e mentioned t h a t by making u s e of t h e r e s u l t s of OPIAL [ l l , w e i n f a c t o b t a i n i n Theorems 4 . 1 and 5 . 1 sequence

'

( r e s p e c t i v e l y 4 . 2 and 5 . 2 ) t h e convergence of t h e whole {An}

to

A*,

such t h a t

O f boundary v a l u e t y p e .

{u,p,A*}

i s a s a d d l e - p o i n t of

146

d ( a n d of d r ) on V

(CHAP. 3)


x

H x H.

W e refer t o G.L.T.

[2,

f o r a proof of t h i s r e s u l t i n a m o r e g e n e r a l c o n t e x t .

A p p e n d i x 21

CHAPTER I V

N U M E R I C A L S O L U T I O N OF M I L D L Y N O N L I N E A R P R O B L E M S BY AUGMENTED L A G R A N G I A N METHODS

M. F o r t i n , R . G l o w i n s k i , T . F .

1.

Chan

INTRODUCTION

T h i s c h a p t e r p a r t l y c a r r i e s on t h e work o f CHAN-GLOWINSKI E l l , 1 2 1 and e x t e n d s t h e a l g o r i t h m i c p a r t o f i t , i n p a r t i c u l a r t h e p a r t d e a l i n g w i t h a p p r o x i m a t i o n by f i n i t e e l e m e n t methods and w i t h t h e u s e of W e s h a l l a l s o see how, by a j u d i c i o u s c h o i c e

quadrature formulas.

of t h e f u n c t i o n a l s p a c e s and o f t h e d e c o m p o s i t i o n , w e c a n o b t a i n h y b r i d f i n i t e e l e m e n t methods and s o l v e t h e c o r r e s p o n d i n g a p p r o x i -

mate p r o b l e m s by augmented L a g r a n g i a n methods. W e s h a l l p r e s e n t some numerical r e s u l t s i l l u s t r a t i n g t h e potent-

i a l i t i e s o f t h e methods d e s c r i b e d b e l o w and w e s h a l l show t h e close l i n k s which e x i s t b e t w e e n t h e s e a l g o r i t h m s and t h e a l t e r n a t i n g d i r e c t i o n methods of Peaceman-Rachford

and D o u g l a s - R a c h f o r d .

I n t h e remainder of t h i s c h a p t e r w e s h a l l t h u s be considering t h e n u m e r i c a l s o l u t i o n of m i l d l y n o n l i n e a r p r o b l e m s o f t h e f o l l o w i n g t y p e ,

fl

on a domain

of

lRN

w i t h boundary

aR

=

r

w h e r e , i n (1.1), w e h a v e : (i)

A i s a second-order

e l l i p t i c operator, possibly

non-symmetric, (fi)

@

i s a n i n c r e a s i n g mapping ( i n t h e w i d e s e n s e ) i n t o IR, i s a f u n c t i o n d e f i n e d o v e r R.

c o n t i n u o u s f r o m IR (iii) f

A s w e s h a l l see l a t e r , t h e r e s u l t s o b t a i n e d c a n b e e x t e n d e d t o

147

148

MILDLY NONLINEAR PROBLEMS

mu 1t i v a l u e d e q u a t i o n s

4)

(CHAP.

(*Iof t h e type

f e Au + a j ( u ) ,

(1.2)

1

where

aj(u)

denotes t h e sub-differeniial

convex f u n c t i o n a l

with respect t o

u

of a

.

j (. )

W e s h a l l f i r s t b r i e f l y r e v i e w t h e r e s u l t s o f CHAN-GLOWINSKI 111,121 c o n c e r n i n g t h e e x i s t e n c e and t h e u n i q u e n e s s o f a s o l u t i o n o f p r o b -

l e m ( l . l ) ,t h e n w e s h a l l n e x t d e s c r i b e a p r o c e d u r e f o r a p p r o x i m a t i n g F i n a l l y w e s h a l l show how

t h i s p r o b l e m by a f i n i t e e l e m e n t method.

-

u s i n g t h e methods of C h a p t e r I11

-

w e c a n decompose p r o b l e m (1.1)

t h r o u g h t h e u s e o f a s u i t a b l e augmented L a g r a n g i a n s o as t o o b t a i n t h e c l a s s i c a l a l t e r n a t i n g d i r e c t i o n methods.

2.

A CLASS OF M I L D L Y NONLINEAR ELLIPTIC PROBLEMS

2.1.

F o r m u l a t i o n of t h e problem

W e c o n s i d e r a b o u n d e d domain

r

u l a r boundary

R

in

IRN, w i t h s u f f i c i e n t l y r e g -

(say, Lipschitz continuous i n t h e sense of

NECAS Ell), a l s o (see C h a p t e r s I1 and I11 f o r t h e n o t a t i o n ) 1

,

(i)

V = Ho(n)

(ii)

a c o n t i n u o u s linear form where

f E

v'

=

H-'(fi)

L

form o f t h e d u a l i t y between (iii)

(2.1)

a:

: V +R,

and w h e r e V'

i.e.

L(v)

=

,

is t h e b i l i n e a r

and V ,

V x V +IR, a c o n t i n u o u s b i l i n e a r form, which i s

V-elliptic

(i.e.

a(v,v) 2 a l v 1 :

Y V v~

3 a>O

such t h a t

,

where w e w r i t e

(*)

T r a n s l a t o r ' s Note:

The t e r m " m u l t i v a l u e d e q u a t i o n " i s u s e d t o

d e n o t e an e q u a t i o n a s s o c i a t e d w i t h a m u l t i v a l u e d o p e r a t o r ; s u c h a n e q u a t i o n i s sometimes known u n d e r t h e F r e n c h name m u l t i v o q u e equation. 1

S e e , f o r example, ROCKAFELLAR 1 4 1 , EKELAND-TEMAM 111 f o r t h e d e f i n i t i o n of s u b d i f f e r e n t i a l s .

(SEC. 2 )

149

M I L D L Y NONLINEAR ELLIPTIC PROBLEMS

(2.2) t h e u s u a l norm’ on

H:(n)

)

.

W e d o n o t assume a p r i o r i t h a t

a(.

,. )

i s symmetric.

a c o n t i n u o u s f u n c t i o n $ : IR +IR, i n c r e a s i n g i n t h e

(iv)

w i d e s e n s e and s a t i s f y i n g

$ ( O ) = 0;

we put

(2.3)

the function

@

i s t h e n c o n v e x , C1

i t c a n b e shown t h a t ( t h u s a f o r t i o r i on

Remark 2 . 1 :

and n o n - n e g a t i v e

w i t h @ ( O ) = 0;

j(.) i s c o n v e x , p r o p e r and 1 . s . c . 1

V = H

on

L1(n)

(a)).

The c o n t i n u i t y of

is e s s e n t i a l f o r obtaining

$(.)

c e r t a i n o f t h e r e s u l t s of CHAN-GLOWINSKI 1 1 1 , 1 2 1 ; f o r m a l l y , a t

l e a s t , i t i s i n n o way n e c e s s a r y f o r t h e i m p l e m e n t a t i o n o f t h e a l g o r i t h m s which w e d e s c r i b e i n t h e remainder o f t h i s c h a p t e r . Consider then 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 l i t y

Find U E V ,

$(u) E L I ( , ) n V ‘

such t h a t

problem

and

(2.5) a(u,v) + =

w e associate with (2.5)

(2.6)

I

t h e v a r i a t i o n a l i n e q u a l i t y problem

uaV’ a(u,v-u) + j(v)

-

j(u)t

Under t h e a b o v e a s s u m p t i o n s on problems

YvcV ;

f,

a(.,.),

YvtV.

@(a),

i t c a n b e shown t h a t

( 2 . 5 ) and ( 2 . 6 ) are e q u i v a l e n t and admit a u n i q u e s o l u t i o n ;

n o n e t h e l e s s problem ( 2 . 6 ) remains meaningful

A t l e a s t , when

i s bounded.

(see LIONS 1 1 1 ,

150

(CHAP. 4 )

M I L D L Y NONLINEAR PROBLEMS

G.L.T.

ill, C21) even when

j(.)

i s n o t d i f f e r e n t i a b l e ; t h i s is f o r

example t h e c a s e w i t h

.

j(v) = \,lv\dx

(2.7)

The i n e q u a l i t y ( 2 . 6 ) u a t i o n of t h e t y p e a ( . ,. )

c o r r e s p o n d s , i n g e n e r a l , t o a m u l t i v a l u e d eq-

(1.2).

I n t h e c a s e where t h e b i l i n e a r form

i s s y m m e t r i c , problem ( 2 . 6 ) i s e q u i v a l e n t t o t h e f o l l o w i n g

problem i n t h e C a l c u l u s of

(2.81,

with

I

Find

J(u) 2 J(v)

J(v) =

(2.8)2

uev

Variations:

such t h a t h eV

T1 a(v,v)

,

+ j(v)

-

;

under t h e p r e c e d i n g a s s u m p t i o n s on

f,

a(-,-),

$(.),

t h e minimisation

problem ( 2 . 8 ) p o s s e s s e s a u n i q u e s o l u t i o n ; t h i s comes from t h e f a c t that,

j(.) b e i n g convex, p r o p e r and 1 . s . c .

t h e problems

(2.6)

on

V , we can a p p l y t o

and ( 2 . 8 ) a number of g e n e r a l r e s u l t s c o n c e r n 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 and t h e m i n i m i s a t i o n of convex f u n c t i o n s ; these r e s u l t s a r e e s t a b l i s h e d i n e.g.

LIONS-STAIQACCHIA 1 1 1 ,

LIONS [ l l , GLOWINSKI 111, 121, EKELAND-TEMAM C11.

I t i s shown

i n CHAN-GLOWINSKI [ 1 1 and GLOWINSKI 111 t h a t t h e s u f f i c i e n t c o n d i t i o n s of a p p l i c a t i o n a r e f u l f i l l e d ; it i s f u r t h e r shown t h a t $(u) and t h a t t h e r e i s e q u i v a l e n c e between ( 2 . 5 ) , a(.,.)

is symmetric).

(2.6)

E

L’(Q)

n V;

(and (2.8) i f

We s h a l l n o t d w e l l any f u r t h e r on t h e s e

t h e o r e t i c a l q u e s t i o n s c o n c e r n i n g problem (1.1) and i t s s e v e r a l variational formulations; i n t h e following sections w e s h a l l discuss t h e a p p r o x i m a t i o n o f problem (l.l), and i t s i t e r a t i v e s o l u t i o n v i a

d e c o m p o s i t i o n m e t h o d s of t h e same t y p e a s t h o s e i n C h a p t e r 111.

2.2.

Approximation of problem ( 2 . 5 ) ,

(2.6)

by f i n i t e e l e m e n t

methods I n t h e f o l l o w i n g w e c o n s i d e r a c a s e where problem ( 2 . 5 ) , o r one of its equivalent formulations (2.6)

o r ( 2 . 8 ) , i s a p p r o x i m a t e d by a

method of conforming f i n i t e e l e m e n t s of t h e most f r e q u e n t l y used type.

The t e r m i n o l o g y and t h e n o t a t i o n used i n r e l a t i o n t o t h e

method of f i n i t e e l e m e n t s a r e t h e s a m e a s i n C h a p t e r 11; t h u s l e t

be a t r i a n g u l a t i o n of t h e two-dimensional

domain

f o r s i m p l i c i t y w e s h a l l assume t o b e p o Z y g o n a Z . s pa c e of conforming f i n i t e el emen t s o f d e g r e e

where, i n ( 2 . 9 ) , 5

k

151

MILDLY NONLINEAR ELLIPTIC PROBLEMS

(SEC. 2 )

R

which

W e then consider a

k

(2

l), namely

Pk (K) d e n o t e s t h e s p a c e o f p o l y n o m i a l s o f d e g r e e

on t h e e l e m e n t

K.

Next w e c o n s i d e r t h e a p p r o x i m a t e p r o b l e m

w e have h e r e used t h e f o r m u l a t i o n (2.61,

b u t w e could equally w e l l

h a v e u s e d t h e e q u i v a l e n t f o r m u l a t i o n ( 2 . 5 ) and ( 2 . 8 ) . it i s n o t p o s s i b l e t o u s e ( 2 . 1 0 )

from

3

In general

p r a c t i c a l p o i n t of view;

in

r e a l i t y it i s n o t p o s s i b l e t o o b t a i n a n e x a c t a n a l y t i c a l e v a l u a t i o n of t h e i n t e g r a l s d e f i n i n g

j ( v h ) , and i n o r d e r t o o b t a i n a n u m e r i c a l l y

t r a c t a b l e p r o b l e m it i s n e c e s s a r y t o u s e n u m e r i c a l i n t e g r a t i o n W e thus consider i n t h e formulas i n o r d e r t o approximate j ( v h ) . element xiK,

K

i = 1,

(2.11)

of t h e t r i a n g u l a t i o n

...s ,

eh,s

numerical i n t e g r a t i o n points o . s u c h t h a t w e have

each being assigned a o e i g h t

I,

1’

S

f ( x ) d s = Area (K)

1

Wi

f(xiK).

i=l

W e then p u t

(2.12)

and w e c o n s i d e r t h e a p p r o x i m a t e p r o b l e m

Find

%E

Wkh

such t h a t

(2.13)

I t i s clear t h a t t h e accuracy of our approximation w i l l be i n f l u -

e n c e d by t h e a c c u r a c y o f t h e n u m e r i c a l i n t e g r a t i o n f o r m u l a , t h e c h o i c e of which w i l l b e g u i d e d by t h e p r o p e r t i e s o f t h e f u n c t i o n

a .

F o r r e a s o n s w h i c h w i l l become a p p a r e n t i n t h e f o l l o w i n g s e c t i o n i t

152


(CHAP.

4)

w i l l b e e s p e c i a l l y d e s i r a b l e t o u s e q u a d r a t u r e f o r m u l a s which are capable of i n t e g r a t i n g e x a c t l y t h e i n n e r product

L 2 ( Q ) i n Wkh,

hence

T h i s r e q u i r e m e n t i s n o t m a n d a t o r y however; i n p a r t i c u l a r i t i s n o t s a t i s f i e d i n CHAN-GLOWINSKI 111 and GLOWINSKI 1 1 1 .

w i l l be e s s e n t i a l t h a t t h e set of p o i n t s

s o l v e n t on s

xiK

However, i t

should be

Pk-uni-

t h a t i s t o s a y (see C I A R L E T [ l j ) t h a t knowing t h e

K,

v a l u e s o f a p o l y n o m i a l of d e g r e e

s k, a t the quadrature points,

determines t h i s polynomial u n i q u e l y .

I n CHAN-GLOWINSKI 111, GLOWINSKI [I, C h a p t e r 41 a n

ExampZe 2 . 1 :

a p p r o x i m a t i o n o f t h e p r o b l e m by c o n f o r m i n g f i n i t e e l e m e n t s o f d e g r e e

one on t r i a n g l e s i s u s e d ; t h e i n t e g r a t i o n p o i n t s a r e t h e v e r t i c e s o f t h e t r i a n g l e s , a'ssigned w e i g h t s 1 / 3 , and t h e y c o r r e s p o n d t o t h e d e g r e e s of freedom o f t h e approximation.

This q u a d r a t u r e formula

i s o f o r d e r one and d o e s n o t s a t i s f y c o n d i t i o n

( 2 . 1 4 ) ; nonetheless

i t does l e a d t o co n v er g en t ap p r o x i mat i o n s .

Example 2 . 2 :

W e c o n s i d e r , s t i l l , a n a p p r o x i m a t i o n by f i n i t e

e l e m e n t s o f d e g r e e one and w e u s e as i n t e g r a t i o n p o i n t s t h e midp 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 , t h e s e b e i n g a s s i g n e d w e i g h t s 1/3.

T h i s q u a d r a t u r e formula i s e x a c t f o r p o l y n o m i a l s of degree

two a n d s a t i s f i e s c o n d i t i o n ( 2 . 1 4 ) .

I t w i l l be noted t h a t t h e

n u m e r i c a l i n t e g r a t i o n p o i n t s c o r r e s p o n d t o t h e d e g r e e s o f freedom o f an a p p r o x i m a t i o n by n o n c o n f o r m i n g f i n i t e e l e m e n t s o f d e g r e e one ( i n f a c t t h a t u t i l i s e d i n C h a p t e r I1 f o r t h e s o l u t i o n o f t h e S t o k e s and Navier-Stokes

Example 2 . 3 : degree two.

problems)

.

W e c o n s i d e r a n a p p r o x i m a t i o n by f i n i t e e l e m e n t s o f

I t i s known

(see f o r example LYNESS-JESPERSEN C11)

t h a t i t i s p o s s i b l e t o c o n s t r u c t a q u a d r a t u r e f o r m u l a on a t r i a n g l e , which i s e x a c t f o r p o l y n o m i a l s o f d e g r e e 4 , tegration points.

and w h i c h u s e s s i x i n -

W e can moreover u s e t h e s e s i x p o i n t s t o d e f i n e

u n i q u e l y a p o l y n o m i a l o f d e g r e e two.

(SEC. 3 )

DECOMPOSITION O F PROBLEM ( 2 . 5 ) , ( 2 . 6 )

Examples 2 . 2

and 2 . 3 show t h a t c o n d i t i o n ( 2 . 1 4 )

153

can be s a t i s f i e d

by t h e trianguZar f i n i t e e l e m e n t s m o s t u s u a l l y employed.

The case

of q u a d r i l a t e r a l el emen t s i s even s i m p l e r because t h e c o rre sp o n d in g q u a d r a t u r e f o r m u l a s ( d e d u c e d from t h e G a u s s i a n f o r m u l a s ) a r e w e l l known and e a s i l y o b t a i n e d , a t l e a s t o n t h e r e f e r e n c e r e c t a n g l e . I t i s shown i n CHAN-GLOWINSKI L 1 1 ,

GLOWINSKI [l, C h a p t e r 4 1 t h a t t h e

a p p r o x i m a t i o n o b t a i n e d by u s i n g t h e f u n c t i o n a l

j h (v,)

d e f i n e d by

t h e q u a d r a t u r e f o r m u l a i n Example 2 . 1 l e a d s t o an a p p r o x i m a t e s o l u t i o n which conver g es , as h of problem ( 2 . 5 ) ,

0 , t o the exact solution (i.e. t h a t

+

(2.6)).

The p r o o f g e n e r a l i s e s w i t h o u t d i f f i -

c u l t y and i t i s e v e n p o s s i b l e t o o b t a i n e s t i m a t e s of t h e a p p r o x i mation e r r o r ( t h e s e b r i n g i n t h e n o n l i n e a r i t y o f t h e p ro b le m ).

We

s h a l l n o t d w e l l any f u r t h e r o n t h e s e p o i n t s b e c a u s e o u r o b j e c t i v e i n t h e p r e s e n t work i s r a t h e r t o d e s c r i b e i t e r a t i v e methods o f s o l ution.

AUGMENTED LAGRANGIAN AND DECOMPOSITION O F THE PROBLEM

3.

(2.6)

(2.5),

W e s h a l l assume i n t h i s s e c t i o n , a l t h o u g h t h i s i s n o t i n f a c t

essential, that the

3.1

b i l i n e a r form

is symmetric.

a(.,.)

C o n s t r u c t i o n o f t h e augmented L a g r a n g i a n .

(I)

Continuous

case. I n a c c o r d a n c e w i t h t h e g e n e r a l p r i n c i p l e s i n t r o d u c e d i n Chap-

t e r 111, w e s h a l l f i r s t t r y t o decompose p r o b l e m ( 2 . 5 ) , i n t r o d u c i n g a supplementary a r t i f i c i a l v a r i a b l e .

(2.6)

by

The c o o r d i n a t i o n

i s t h e n a c h i e v e d by means o f a L a g r a n g e m u l t i p l i e r and a p e n a l i A s w e s h a l l see a t a l a t e r s t a g e t h e c h o i c e o f t h e

sation term.

decomposition i s n o t unique,

and t h e one which w e s h a l l c o n s i d e r i n

t h i s s e c t i o n i s t h e one which t o u s a p p e a r s t o b e t h e s i m p l e s t o f the various possible choices. R e f e r r i n g back t o t h e n o t a t i o n of C h a p t e r 111, w e p u t

V

(3.1)

=

1

2

Ho(n), H = L (a)

and w e t a k e a s t h e o p e r a t o r into

H;

we then d e f i n e

G(.)

B

t h e canonical i n j e c t i o n of and

F(.)

by

V

(CHAP.


154

G(v) =

(3.2)

1

a(v,v)

-

4)

and

respectively.

It is clear that i f

i s a s a d d l e - p o i n t on

{u,p,X}

t h e augmented Lagrangian d r

,

v

-

q = 0

in

Lagrange m u l t i p l i e r

X

L2 ( Q )

.

of

is the

u

( 2 . 8 ) ; w e have o b t a i n e d ( 3 . 4 ) by

s o l u t i o n of problem ( 2 . 5 ) , ( 2 . 6 ) ,

q

introducing the a r t i f i c i a l variable straint

V x H x H

w e t h e n have u = p , where

and by imposing t h e con-

Proving t h e e x i s t e n c e of a

p o s e s no d i f f i c u l t i e s i n t h i s p a r t i c u l a r

case.

3.2

C o n s t r u c t i o n of t h e augmented Lagrangian.

(11)

The

d i s c r e t e case T o approximate t h e augmented Lagrangian

( 3 . 4 ) v i a a f i n i t e ele-

ment method, it i s n e c e s s a r y t o d e f i n e an approximation o f

q

i n o r d e r t o approximate t h e f u n c t i o n s

and

L2(Q)

appearing i n ( 3 . 4 ) ;

moreover w e need t o keep i n mind t h a t o u r o b j e c t i v e i s t o o b t a i n a l g o r i t h m s i n which t h e t r e a t m e n t of t h e n o n l i n e a r p a r t i s p u r e l y local. I n regard t o t h e approximation of

L

2

(fl) a n a t u r a l c h o i c e i s t o

consider

(3.5)

Qkh =

I qhl

qhl K c P k ( K )

YKc

5)

s

t h a t i s , t o use t h e same f i n i t e e l e m e n t s a s i n t h e c o n s t r u c t i o n o f Wkh,

b u t s u p p r e s s i n g t h e matching c o n d i t i o n s a t t h e i n t e r f a c e s of

t h e s e elements.

(2.12)

W e t h e n approximate t h e f u n c t i o n a l

and we c o n s i d e r f o r

vhc Wkh,

augmented Lagrangian d e f i n e d by

qhc Qkh,

p h c Q,

I(.) using

the discrete

(SEC.

3)

,( 2 . 6 )

DECOMPOSITION O F PROBLEM ( 2 . 5 )

drh(vh,qh,uh) =

(3.6)

1

-

7 a(vh,vh)+jh(qh)

155

(uh,vh-qh)+

+

3vh-qhl 2

0.

Now c o n s i d e r t h e case w h e r e t h e q u a d r a t u r e f o r m u l a d e f i n i n g (via 2.12))

s a t i s f i e s the condition (2.14),

product i n

L2(n)

o f two f u n c t i o n s from

jh(.) allows t h e inner

i.e.

t o b e c a l c u l a t e d ex-

Qkh

a c t l y ; i n t h i s case w e t h e n h a v e

(3.7)

i s n o t s a t i s f i e d , t h e L a g r a n g i a n s d e f i n e d by

(2.14)

I f condition

( 3 . 6 ) and ( 3 . 7 ) a r e d i s t i n c t .

However i f

vh = q h , whenever t h e s e

two f u n c t i o n s c o i n c i d e a t t h e q u a d r a t u r e p o i n t s , it i s p e r m i s s i b l e

t o u s e ( 3 . 7 ) 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 t h e approximate problem ( 2 . 1 3 ) ; t h i s i s t h e case i n p a r t i c u l a r f o r t h e q u a d r a t u r e f o r m u l a i n Example 2 . 1 . Before d e s c r i b i n g t h e a l g o r i t h m s which w i l l e n a b l e t h e s a d d l e p o i n t s of t h e augmented L a g r a n g i a n

(3.7) t o be c a l c u l a t e d , we w i l l

f i r s t i n t r o d u c e a c e r t a i n amount o f n o t a t i o n ; i t w i l l a l s o b e usef u l t o i d e n t i f y t h e optimality conditions of t h i s saddle-point problem. The s p a c e imation of

Wkh

1

Ho(Cl)

d e f i n e d by

( 2 . 9 ) i s a s t a n d a r d s p a c e f o r approx-

by t h e method o f f i n i t e e l e m e n t s ; w e s h a l l assume

t h a t f u n c t i o n s from

Wkh

are c h a r a c t e r i s e d by

conforming

(resp. a t t h e v e r t i c e s

and t h e m i d p o i n t s of

t h e s i d e s ) of t h e t r i a n g l e s i n t h e t r i a n g u l a t i o n on

r)

t o c o m p l e t e l y d e f i n e a f u n c t i o n from

As regards

Wlh

5

(not situated

(resp.

WZh).

Q k h , t h e n a t u r a l c h o i c e of d e g r e e s o f f r e e d o m w i l l be t o

c o n s i d e r t h e v a l u e s t a k e n by t h e f u n c t i o n s f r o m

rature points; points

finite

(resp. two) we s h a l l use t h e values

elements of degree one taken a t t h e v e r t i c e s

scalars, t h e

Nh

d e g r e e s of f r e e d o m ; f o r example i n t h e case o f

w e d e n o t e by

( n o t s i t u a t e d on

r).

Ph

Qkh

a t the

quad-

t h e t o t a l number o f q u a d r a t u r e

I n o r d e r n o t t o over-complicate t h e

notation unnecessarily, we s h a l l henceforth w r i t e

N = Nh,

P = P

h;

i n a d d i t i o n w e s h a l l d e n o t e by y , g , y whose components c o r r e s p o n d t o t h e d e g r e e s o f f r e e d o m a s s o c i a t e d w i t h the vectors i n

lRN, IRp, lRp,

156


respectively, operator

S

vh

E

from

,,Q

uh

into

iRp

qh

Wkh,

IRN

-

we a s s o c i a t e w i t h

i.e.

E

v

E

W e then consider t h e l i n e a r

Q,,.

d e f i n e d by

t h e v a l u e s t a k e n by t h e f u n c t i o n

r.

a t t h e q u a d r a t u r e p o i n t s n o t l o c a t e d on W e also define t h e

linear operator

from

M

o c i a t e d w i t h t h e a p p r o x i m a t e i n n e r p r o d u c t on

p h ( x i K ) = q (x. IK) = 0

where, i n ( 3 . 9 ) , M = M t.

-

4)

(CHAP.

if

into

IRp L2

xiK

“h

lRp,

‘kh

ass-

( n ) , by

t

T;

w e have

lRN

into

-

I f c o n d i t i o n ( 2 . 1 4 ) i s s a t i s f i e d w e have

F i n a l l y w e d e n o t e by

A

t h e l i n e a r o p e r a t o r from

N IR

I

d e f i n e d by

(&,y)

(3.11)

= a(u,,,v,)

Y\,vh

E

Wkh.

RN

With r e g a r d t o t h e n o n l i n e a r i t y , w e d e n o t e by v e c t o r o b t a i n e d by a p p l y i n g of

@

(resp.

a)

@

(9) ( r e s p .

Q

(3) )

9. T a k i n g a c c o u n t of t h e a b o v e n o t a t i o n , t h e augmented L a g r a n g i a n

( 3 . 7 ) can be w r i t t e n i n t h e form

where

1 = {l,

-

...1 1

( E

the

to e a c h o f t h e components

1RP ) .

The o p t i m a l i t y c o n d i t i o n s of o u r p r o b l e m c a n t h e n b e w r i t t e n

(SEC. 4 )

1 57

ALGORITHM FOR PROBLEM ( 2 . 1 3 )

(3.13)3

Su = p.

I n cases where

0

is n o t d i f f e r e n t i a b l e ,

(3.13)2

would h a v e t o

b e r e p l a c e d by t h e v a r i a t i o n a l i n e q u a l i t y

\

(3.14)

lp c~

P

,and

~q

5

~

'we

have

i n p r a c t i c e t h i s i n e q u a l i t y h a s t o b e s o l v e d p o i n t w i s e a t e a c h of t h e q u a d r a t u r e p o i n t s , which i n g e n e r a l c r e a t e s n o d i f f i c u l t i e s

(-A

-

u

and

b e i n g known).

W e a r e now i n a p o s i t i o n t o d e s c r i b e t h e a l g o r i t h m s f o r s o l v i n g t h e approximate problem (2.13)

( o f t h e same t y p e as t h o s e c o n s i d -

e r e d i n C h a p t e r 111) a s s o c i a t e d w i t h t h e augmented L a g r a n g i a n ( 3 . 1 2 ) .

4.

ALGORITHMS FOR SOLUTION OF THE APPROXIMATE PROBLEM ( 2 . 1 3 ) . DISCUSSION B e a r i n g i n mind t h e e q u i v a l e n c e b e t w e e n t h e a p p r o x i m a t e p r o b l e m

( 2 . 1 3 ) and t h e s y s t e m ( 3 . 1 3 ) , w e s h a l l b e a p p l y i n g , f o r t h e s o l u t i o n In the following

o f t h e l a t t e r , t h e a l g o r i t h m s o f C h a p t e r 111.

y, F , h

discussion

again denote t h e v e c t o r s i n

i a t e d , respectively, with (3.12),

uh, p h , A h .

IRN, lRp, IRp

assoc-

Thus, h a v i n g r e g a r d t o

( 3 . 1 3 ) , w e have t h e following algorithms:

ALG1:


(4.1)

ho c R p ,

then for

nzO, X n e ~ '

-

-

-I

b e i n g known,

Remark 4 . 1 : have

+

rsksu"

-

,

pn

and

An+'

by

+ ~$1"

-

- -- -

rSkpn=F,

I n t h e case d e s c r i b e d i n CHAN-GLOWINSKI Lll, C21, w e

S = St = M = I . -

u",

-

(4.2) AU"

determine

- - .

158

(CHAP. 4 )

M I L D L Y N O N L I N E A R PROBLEMS

Remark 4 . 2 :

I t s h o u l d b e n o t e d t h a t by n u l t i p l y i n g t h e f i r s t

StM, _ - t h e n by a d d i n g t h e r e s u l t o b t a i n e d t o t h e

r e l a t i o n i n ( 4 . 2 ) by

second r e l a t i o n i n ( 4 . 2 ) , w e o b t a i n

- - 4 (p") -

Aun + S k

(4.4)

I -

=

-F.

The i t e r a t i v e reZaxation method d e s c r i b e d i n C h a p t e r I11 i s a p p l i c a b l e f o r t h e s o l u t i o n of system ( 4 . 2 ) ; hence f o r

G i v e n the v e c t o r

n

2

0

choose u n r 0 a r b i t r a r i l y ,

An,

unru = un-l), then-for

k

O , - u- n r k

2

w e have: (for e x a m p l e

b e i n g known, s o l v e

successive l y

(4.6) The r e s u l t s of CEA-GLOWINSKI 1 1 1 , GLOWINSKI 1 2 , C h a p t e r 51 a p p l y t o (4.5),

( 4 . 6 ) a n d , u s i n g t h e a s s u m p t i o n s a l r e a d y made, w e c a n

p r o v e t h e c o n v e r g e n c e of m e n t a t i o n of

(1)

(4.5),

(4.6)

(4.6), to

(4.5),

u n t i l t h e d i f f e r e n c e between t w o s u c c e s s -

Continue t o i t e r a t e

i v e i t e r a t e s i s s m a l l e r t h a n some t h r e s h o l d advance, b e f o r e p r o ceed i n g t o u p d at e small f o r system ( 4 . 2 )

s m a l l " number

chosen i n

-

E

is s u f f i c i e n t l y

t o be solved t o high accuracy.

L i m i t t h e number of r e l a x a t i o n i t e r a t i o n s "

E,

An v i a (4.3); t h i s

c o r r e s p o n d s e x a c t l y t o ALGl i n s o f a r as

(2)

I n t h e imple-

{un,pn}.

two s t r a t e g i e s can b e u s e d :

(4.5),

(4.6) t o a

then update An v i a (4.3) ; t h e limiting kmaxI o b v i o u s l y g i v e s t h e a l g o r i t h m ALG2 d e s c r i b e d 5

case

kmax = 1

below.

ALG2 :

uo,A1

(4.7)

for n

p

i

n n

2

-

5-1


1, u and lAn+l and by

b e i n g known, d e t e r m i n e s u c c e s s i v e l y

An I

(4.8)

r pn +

(4.9)

(A+rStMS)un = rStMpn .

--

$ ( p n ) = rSun-l

-_-_

I-

+

-

A",

--

- StMAn I

+ F- ,

4)

(SEC.

ALGORITHM FOR PROBLEM ( 2 . 1 3 )

-A n+l

(4.10)

159

X n + p,(Su"-p").

=

--

I

I

T h i s l a t t e r a l g o r i t h m i s w o r t h y o f f u r t h e r a t t e n t i o n b e c a u s e it cont a i n s a s p a r t i c u l a r cases a number o f t h e c l a s s i c a l a Z t e r n a t i n g

d i r e c t i o n methods.

and ( 4 . 1 0 ) w e o b t a i n

(4.9)

I n f a c t combining

the relation

- - --Au" +

--

= F

SkX""

(4.11)

I

(p,-r)Sk(Su"-p"). I

-

-

I

-

e n a b l e s us t o e l i m i n a t e

The r e l a t i o n ( 4 . 1 1 )

An

from ( 4 . 8 )

and

5

(4.9);

f i r s t w e c o n s i d e r t h e case

- -S k S u n - l )

- --

(4.12)

r(Skp"

-

r(StMSu"

(4.13)

+ Aun-'

--I

I--

-

Suppose w e p u t

I

+ skgpn)

-..

I--

pn = un-';

I n t h e g e n e r a l case

(pn

- -stMpn-') --

I I

--

= F.

AU"

I

a s i s t h e case i n

if S = M = I

i s used) w e o b t a i n from ( 4 . 1 2 ) ,

..

+

I

-

121 (where t h e q u a d r a t u r e f o r m u l a o f Example 2 . 1

method o f t h e D o u g l a s - R a c h f o r d

r(Skpn

...--

I

CHAN-GLOWINSKI C11,

(4.14)

+ Sk$(pn) = F,

I I

sk~u"-') I

= p = r; we then obtain

pn

+

(4.13) an a l t e r n a t i n g d i r e c t i o n t y p e (see DOUGLAS-RACHFORD 1 1 1 ) .

# r) ( 4 . 1 2 ) would b e r e p l a c e d by AU"-' -I

+

-s -k $(pn) I

= F, I

where w e h a v e p u t (4.15)

sun-^

=

!"-I

+

(r-pn)

yn-l

Pn

I n p r a c t i c e i t i s e a s i e r t o work w i t h ( 4 . 7 ) - ( 4 . 1 0 )

t h a n ( 4 . 1 2 1 , (4.13).

I t i s a l s o i n t e r e s t i n g t o observe t h a t w e can d e r i v e 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

t y p e (see PEACEMAN-

RACHFORD [ l l ) t h r o u g h a v a r i a n t of ALG2; i n d e e d , c o n s i d e r t h e algorithm ALG3:

u0, X1

-

(4.16)'

for n

? ' -

(4.17)

-

n- 1

u,

n 2 1,

n '

!

!

I

rpn


and An+l by -

+

hn

$(En) = rSun-I I -

b e i n g known, d e t e r m i n e s u c c e s s i v e l y

-

+ A",

(CHAP. 4 )


160

-A"+"2

(4.18)

p(su"-'-p"), --. ."

= 1" + I

W e t h u s c a r r y o u t a f i r s t update of

en, t h e n

solution for

-A n

(by ( 4 . 1 8 ) ) f o l l o w i n g t h e

a second (by ( 4 . 2 0 ) ) f o l l o w i n g t h e s o l u t i o n pn and t h e un

un; i t w i l l a l s o b e n o t e d t h a t i n ALG3 t h e

for

I

p l a y r o l e s which a r e s y m m e t r i c , w h i c h i s n o t t h e case i n ALG2. From t h e p o i n t of view o f t h e s e a r c h f o r a s a d d l e - p o i n t

and i n r e l -

t h i s a l g o r i t h m i s " l e s s i m p l i c i t " t h a n ALG2, and i n

a t i o n t o ALG1,

f a c t f o r " s t i f f " p r o b l e m s ALG3 i s u s u a l l y less r o b u s t t h a n A L G 2 . If

p =

we deduce from ( 4 . 1 7 ) - ( 4 . 2 0 )

r

I2 I

=

$(En)

zk A,"+'

and

=

F-A~". I

I I

W e then obtain (4.21)

(4.22)

r(Skp"-SkSu"-')

-

I I

r(StMSun

I

1-1

-

I -

-."

I

-

I

I

+ Au" = F ;

StMp") + S%$(p") I

I

--

+ Aun-l + S k $ ( p n ) = F,

I

_I

I

n w e i n d e e d o b t a i n t h e method o f a l t e r n a t i n g dirp = un-', e c t i o n s o f Peaceman-Rachford.

putting

I

-

I t i s i n t e r e s t i n g t o n o t e t h a t w e h a v e b e e n a b l e t o l o c a t e some

a l t e r n a t i n g d i r e c t i o n methods w i t h i n a m o r e g e n e r a l framework, namely t h e s e v e r a l p o s s i b l e v a r i a n t s o f t h e a l g o r i t h m ALG1. With r e g a r d t o t h e c o n v e r g e n c e of t h e a l g o r i t h m s , t h e r e s u l t s o f C h a p t e r I11 c a n be a p p l i e d w i t h o u t d i f f i c u l t y .

W e have i n f a c t

t o d i s t i n g u i s h two c a s e s , d e p e n d i n g on w h e t h e r o r n o t t h e quadr a t u r e f o r m u l a u s e d i n t e g r a t e s e x a c t l y t h e i n n e r p r o d u c t on r e s t r i c t e d t o t h e space

Qkh.

L2(n)

I f i t d o e s , t h e r e s u l t s o f Chap-

t e r I11 (Theorem 4 . 1 and Remark 4 . 1 ) a r e a p p l i c a b l e i n t h e i r e n t irety.

Otherwise they apply t o t h e system i n f i n i t e dimensions;

however t h e e q u i v a l e n c e o f t h e norms p l a y s n o o v e r r i d i n g r o l e i n t h e p r o o f of c o n v e r g e n c e and w e c a n e x p e c t a n o v e r a l l r a t e of c o n v e r g e n c e which i s a l m o s t i n d e p e n d e n t of t h e d i s c r e t i s a t i o n .

(SEC. 5 )

5.

161

NUMERICAL EXPERIMENTS


5.1

F o r m u l a t i o n o f a model problem.

General notes.

I n o r d e r t o i l l u s t r a t e t h e r e s u l t s of t h e preceding s e c t i o n s , we s h a l l now summarise t h e n u m e r i c a l e x p e r i m e n t s o f CHAN-GLOWINSKI C11 r e l a t i n g t o t h e c o n v e r g e n c e of t h e a l g o r i t h m s A L G l a n d ALG2 a p p l i e d t o t h e s o l u t i o n o f a p a r t i c u l a r p r o b l e m (1.1). W e h a v e t h e r e f o r e c o n s i d e r e d t h e model p r o b l e m

t h e function (5.2)

$I ( . )

being defined, f o r

@(t) = sgn(t)ltlR = tltl

L- 1

R > 0 , by

.

W e h a v e shown i n F i g u r e 5 . 1 t h e f o r m of t h e f u n c t i o n

values of Q.

I n t h e m a j o r i t y of o u r tests we took

@

for three

Q = 0.1,

which

l e a d s t o a p r o b l e m which i s q u i t e d i f f i c u l t n u m e r i c a l l y ; w e i n f a c t have

@'(O)

=

r e g i o n s where

+- , s o t h a t w e c a n e x p e c t some d i f f i c u l t i e s i n t h e u ( x 1 , x 2 ) = 0.

u(x) = sin 2nxl

s i n 2nx

W e took

( f o r x = {x1,x2})

2'

(5.3)

1

Figure 5.1

t

1 62

(CHAP. 4 )


and w e a t t e m p t e d t o s o l v e a d i s c r e t i s e d v e r s i o n o f p r o b l e m ( 5 - 1 )t

(5.2).

APP i c a t i o n o f t h e a l g o r i t h m s d e s c r i b e d i n S e c t i o n 4 l e a d s t o h a v i n g t o s o l v e o n e - d i m e n s i o n a l p r o b l e m s o f t h e form

rc

(5.4)

where

b

+

+ ( 5 ) = b,

i s given.

Note t h a t t h e s i n g u l a r i t y o f

+'

at

ob-

0

l i g e s us t o take c e r t a i n precautions during t h e numerical s o l u t i o n of

( 5 . 4 ) , even f o r such an e l e m e n t a r y problem. W e s h a l l f i r s t make a number o f g e n e r a l comments on t h e b e h a v i o u r

o f t h e v a r i o u s a l g o r i t h m s t e s t e d ; t h e n w e s h a l l go i n t o r a t h e r more d e t a i l on c e r t a i n p o i n t s which s e e m t o u s t o p o s s e s s some i m p o r t a n c e i n relation to t h e algorithms.

5.2

Comments on t h e i m p l e m e n t a t i o n and t h e c o n v e r g e n c e of A L G l and ALG2

AS

w e h a v e s e e n i n S e c t i o n 4 , a l g o r i t h m ALGZ i s a s p e c i a l case o f

ALGl i n which t h e number o f i n t e r n a l i t e r a t i o n s h a s b e e n l i m i t e d t o

one.

I n p r a c t i c e a g r e a t e r d e g r e e of g e n e r a l i t y would b e o f f e r e d

by i n c o r p o r a t i n g ALGl i n a p r o g r a m a l l o w i n g t h e number o f i n t e r n a l i t e r a t i o n s t o b e l i m i t e d by a t e r m i n a t i o n t e s t b a s e d e i t h e r on t h e d e c r e a s e o f s o m e r e s i d u a l , o r on a maximum number of i n t e r n a l iterations.

A s r e g a r d s t h e v a r i a n t ALG3, t h i s would i n v o l v e o n l y

a minor m o d i f i c a t i o n . Concerning t h e s p eed o f convergence,

t h e main d i f f i c u l t i e s en-

c o u n t e r e d r e l a t e d t o t h o s e r e g i o n s where t h e f u n c t i o n u l a r , t h a t i s , where

$'

is sing-

u(x) = 0 ; i t i s i n f a c t o b s e r v e d t h a t t h e

a s y m p t o t i c r a t e of c o n v e r g e n c e o f ALG2 i s v e r y s l o w a t t h e s e p o i n t s , and t h i s f a c t i s c l e a r l y i l l u s t r a t e d i n F i g u r e 5.2 which corresponds t o a n u m e r i c a l t e s t i n which w e h a v e i n i t i a l i s e d ALGZ f i r s t with

uo

- . A.1

= 0,

= 0 , and t h e n

uo

=

10

(=

{lo,.. . l o } ) ,

5'

=

0.

In

t h e f i r s t case t h e s o l u t i o n h a s a l r e a d y been a t t a i n e d a t t h e s i n g u l a r p o i n t s b e c a u s e it c a n e a s i l y b e s e e n t h a t t h e a l g o r i t h m w i l l l e a v e t h e v a l u e s unchanged a t t h e s e p o i n t s ;

i n t h e second c a s e t h e

s o l u t i o n h a s t o r e a c h t h e v a l u e z e r o a t t h e above p o i n t s .

uo = 0 w e h a v e a v e r y f a s t l i n e a r conve r g e n c e ; i n t h e s e c o n d case- (i.;. uo = 1,O) w e h a v e a s u b l i n e a r F i g u r e 5 . 2 shows t h a t f o r

-

(SEC. 5 )

163


convergence.

A c l o s e e x a m i n a t i o n of t h e

results reveals that the

s l o w n e s s of t h e convergence i s l o c a l i s e d a t t h e p o i n t s where u ( x ) = 0 , t h e o t h e r v a l u e s h a v i n g a l r e a d y been o b t a i n e d t o a r e a s o n I t i s t h e r e f o r e c l e a r t h a t t h e c h o i c e of good i n -

able accuracy.

i t i a l v a l u e s c a n l e a d t o a v e r y s i g n i f i c a n t improvement; however i t

i s s t i l l v e r y i m p o r t a n t t o make t h e a l g o r i t h m more r o b u s t because 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 i t w i l l b e p o s s i b l e t o f i n d i n i t i a l v a l u e s which w i l l e n a b l e t h e d i f f i c u l t y a s s o c i a t e d w i t h t h e s i n g c$' t o b e circumvented.

u l a r p o i n t s of

. .

1

2

3

4

.

5

6

7

8

iter.

9

*

10-I

lo-*

I

o - ~

Io

I

p=r=5

-~

o-~

-xo = 0p0=0 I

-

Figure 5.2 Convergence of ALG2 ( t h e v e r t i c a l a x i s r e p r e s e n t s t h e L2 e r r o r i n un) I

164

(CHAP. 4 )

XILDLY N O N L I N E A R PROBLEMS

I n p r a c t i c e two r e m e d i e s a r e p o s s i b l e :

(1)

Increase

r

d u r i n g t h e c o u r s e o f t h e c a l c u l a t i o n ; t h i s nec-

essitates t h e r e f a c t o r i s a t i o n of t h e matrix

=

A

+

-

rStMS, -I

which may be e x p e n s i v e . Carry o u t f u r t h e r i t e r a t i o n s w i t h i n t h e i n t e r n a l r e l a x a t i o n

(2)

loop ( 4 . 5 ) ,

(4.6);

t h i s i n f a c t means u s i n g A L G l w i t h a more

or less c o m p l e t e i n t e r n a l s o l u t i o n . 1

2

3

4

I

I

5

6 I

7

8

9

I

1 I

0 I

1

,

1

1 I

2 I

*

iter.

-T

1 o-6

.,

1

\

+'

\ Figure 5.3

\

\

uo = 1-0

i

=

ALG 1

-c

5)

(S EC .

165


F i g u r e 5 . 3 i l l u s t r a t e s t h e r e s u l t s o b t a i n e d by u s i n g t h e s e c o n d s t r a t e g y ; i n t h e corresponding numerical tests we stopped t h e i n t e r nal iterations (4.5),

( 4 . 6 ) when t h e d i f f e r e n c e b e t w e e n two s u c c -

e s s i v e i t e r a t e s was s m a l l e r i n norm t h a n convergence r e s u l t s obtained w i t h

E

=

10

F i g u r e 5 . 3 shows t h e

E.

-2

and

lo-*

; t h e horiz-

o n t a l a x i s shows t h e t o t a l number o f i n t e r n a l i t e r a t i o n s o f (4.5),

n e e d e d t o a t t a i n t h e a c c u r a c y shown on t h e v e r t i c a l

(4.6)

norm o f t h e e r r o r c o r r e s p o n d i n g t o

yn).

The c i r c l e d p o i n t s i n d i c a t e t h a t t h e r e h a s been a n u p d a t e o f

in,

axis ( t h i s being t h e

L2

(4.3), a t that iteration.

via

By way o f c o m p a r i s o n w e h a v e r e p l o t -

t e d on t h i s f i g u r e t h e r e s u l t s f o r ALG2 a l r e a d y p r e s e n t e d i n Fig-2 u r e 5.2. For E = 10 , algorithm ALGl degenerates r a p i d l y i n t o ALGZ, s i n c e f o r convergence of n

1

no;

(4.5),

(4.6)

i n a s i n g l e i t e r a t i o n as soon a s

it can be seen t h a t convergence i s a t t a i n e d r a p i d l y a t

p o i n t s f o r which For

of t h e o r d e r of t e n t o f o r t y o r s o w e have

n

E =

-4 10

uh(x)

#

0.

a l a r g e r number o f i n t e r n a l i t e r a t i o n s ( 4 . 5 1 ,

An

w a s carried out before updating c o n v e r g e n c e i s , however,

(4.6)

v i a ( 4 . 3 ) ; t h e mean r a t e o f

g r e a t l y improved i n c o m p a r i s o n t o

a n d i s i n f a c t close t o t h a t o b s e r v e d f o r ALG2 w i t h

yo

=

E

0;

=10

-2

,

thus

w e h a v e i n l a r g e p a r t e l i m i n a t e d t h e e f f e c t o f t h e i n i t i a l conditions.

I n view o f t h e above n u m e r i c a l t e s t s t h e a l g o r i t h m

5.1:

Remark

ALGl i s a p p a r e n t l y m o r e r o b u s t t h a n ALGZ w h i c h i s i n f a c t e q u i v a l e n t t o a n a l t e r n a t i n g d i r e c t i o n method.

W e can t h e r e f o r e c o n s i d e r

o b t a i n e d by i n t r o d u c i n g a n augmented L a g r a n g i a n , a s a method

ALG1,

w h i c h e n a b l e s t h e r o b u s t n e s s and t h e s p e e d o f conver'gence o f t h e s e a l t e r n a t i n g d i r e c t i o n methods t o be i n c r e a s e d .

D i s c u s s i o n on t h e c h o i c e of t h e p a r a m e t e r s

5.3

r

&

p.

W e s h a l l c o n c l u d e S e c t i o n 5 by d i s c u s s i n g t h e c h o i c e o f p a r a -

meters

r

and

p.

A s f a r as

p

i s concerned w e have s y s t e m a t i -

r a l l t h e t i m e ; t h i s v a l u e i s a l w a y s a v e r y good one even i f it is n o t a b s o l u t e l y o p t i mal . The n u m e r i c a l t e s t s c a l l y used

p

=

showed t h a t i n t h e case o f ALG2 a v a l u e o f (p

p

r a t h e r l a r g e r than

1.1 r ) a c c e l e r a t e d t h e c o n v e r g e n c e v e r y s l i g h t l y ; i n t h e abs-

e n c e o f a p r e c i s e method f o r d e t e r m i n i n g t h e o p t i m a l v a l u e of

p,

r

166


however, w e recommend t h e c h o i c e of

p =

r.

4)

(CHAP.

With r e g a r d t o t h e c h o i c e

r , F i g u r e 5.4 i n d i c a t e s t h e number o f i t e r a t i o n s n e e d e d f o r conr.

vergence of ALGl f o r d i f f e r e n t v a l u e s of

r

orded near choice of

r

=

An optimum w a s rec-

5 , b u t t h i s optimum i s n o t c l e a r l y d e f i n e d and t h e

i s n o t c r i t i c a l w i t h i n a r a t h e r wide i n t e r v a l ;

t h i s is

due t o an e f f e c t o f p a r t i a l c a n c e l l a t i o n b e t w e e n two phenomena w i t h What h a p p e n s i s t h a t i n c r e a s i n g

opposite actions.

-A n

t h e speed of convergence o f

increases

i n ALG1, b u t d e c r e a s e s t h a t o f t h e

(4.6);

internal iterations (4.5),

r

t h e combined e f f e c t i s h i g h l y

complex b u t t h e r e s u l t i s an a l g o r i t h m w h i c h i s n o t v e r y s e n s i t i v e t o t h e choice of

r.

40

30 20 10

.

OL

5

1

. 10

+ 20

r

Figure 5.4 E f f e c t of t h e c h o i c e of

r

o n t h e c o n v e r g e n c e o f ALGl

The r e a d e r c a n r e f e r t o CHAN-GLOWINSKI 1 1 1 f o r f u r t h e r d e t a i l s r e l a t i n g t o t h e c o n v e r g e n c e o f A L G l and ALG2 a p p l i e d t o t h e s o l u t i o n of problem ( 5 . 1 ) ,

( 5 . 2 ) , t o g e t h e r w i t h a number o f c o m p a r i s o n s

w i t h o t h e r i t e r a t i v e methods.

6.

SOME REMARKS ON H Y B R I D METHODS I n t h i s s e c t i o n w e c o n s i d e r a v a r i a n t o f t h e p r e c e d i n g methods

which may b e u s e f u l f o r c e r t a i n p r o b l e m s and w h i c h i s l i n k e d w i t h t h e hybrid-primal

f i n i t e e l e m e n t methods

( h y b r i d e s primaux i n t h e

o r i g i n a l F r e n c h t e r m i n o l o g y of THOMAS [l]).

The s t a r t i n g p o i n t

W i l l

o n c e a g a i n be t h e methodology o f C h a p t e r 111, t h e g e n e r a l i t y o f which w i l l a g a i n b e f u r t h e r i l l u s t r a t e d h e r e . I n h y b r i d f i n i t e e l e m e n t methods i t i s s t a n d a r d p r a c t i c e t o d e c o u p l e a boundary-value problem i n t o a f a m i l y o f local problems

16 7

H Y B R I D METHODS

6)

(SEC.

d e f i n e d o v e r each element of t h e t r i a n g u l a t i o n .

The r e c o u p l i n g

of t h e s e l o c a l problems i s g e n e r a l l y c a r r i e d o u t by means of a Lagrange m u l t i p l i e r a s s o c i a t e d w i t h t h e matching c o n s t r a i n t a t t h e i n t e r f a c e of t h e e l e m e n t s .

I n t h e l i g h t of what we have p r e s e n t e d

e a r l i e r , t h e a d v a n t a g e of such a method i s a p p a r e n t i n t h e c a s e of n o n l i n e a r p r o b l e m s i i n f a c t t h e s o l u t i o n a t t h e element l e v e l inv o l v e s a problem h a v i n g a s m a l l number of v a r i a b l e s , which i s much e a s i e r t h a n d e a l i n g w i t h a g l o b a l n o n l i n e a r problem.

However two

obstacles a r i s e , i n regard t o the algorithms, f o r t h e e f f i c i e n t e x p l o i t a t i o n of h y b r i d methods: The l o c a l problems a r e o f t e n i l l posed and c a n n o t be s o l v e d

(i)

i n d e p e n d e n t l y of one a n o t h e r . C o o r d i n a t i o n a l g o r i t h m s b a s e d on t h e convergence of t h e

(ii)

m u l t i p l i e r s a r e slow, and t h e i r e f f i c i e n c y d e t e r i o r a t e s r a p i d l y a s t h e number o f e l e m e n t s i n c r e a s e s . W e s h a l l now show t h a t t h e u s e of a s u i t a b l y d e f i n e d augmented

Lagrangian e n a b l e s t h e above d i f f i c u l t i e s t o b e overcome, and a l l o w s e f f i c i e n t a l g o r i t h m s t o be c o n s t r u c t e d . W e s h a l l t h u s c o n s i d e r a model problem of t h e form ( 2 . 5 ) ,

Find

HA(Q)

U E

such t h a t

$(u)

E

L1(Q) n H - I ( Q ) ,

namely:

satisfying

(6.1) a(u,v) + = < f ,v> Yv

1

on Ho(n) of the (6.2)

Thus, l e t

Ho(Q). 1

i s symmetric, (6.1) is equivalent to the minimisation

a(.,.)

When

E

1

J(v) =

Ch

by ( 2 . 8 ) , , i . e .

functional defined a(v,v) + j(v)

-

be a t r i a n g u l a t i o n of

. Q ; t h e p r i n c i p l e of h y b r i d

methods c o n s i s t s of d e f i n i n g t h e problem on a l a r g e r s p a c e , c o n s i d e r i n g membership of (6.3)

H =

nCh

HA(Q)

as a c o n s t r a i n t .

To do t h i s w e p u t

H * W ,

KE

t h i s s p a c e b e i n g equipped w i t h t h e p r o d u c t t o p o l o g y , i . e . associated with t h e inner product

that

(CHAP. 4 )

M I L D L Y N O N L I N E A R PROBLEMS

168

Suppose t h a t onto

H,

notation

f

L

E

2

(Q); t h e f u n c t i o n a l ( 6 . 2 ) t h e n e x t e n d s n a t u r a l l y i s a c l o s e d s u b s p a c e of H ( w e r e t a i n t h e

1

V = Ho(Q)

and

a(.,.)

I(.))

and

.

The s t a n d a r d a p p r o a c h would con-

s i s t o f i n t r o d u c i n g a L a g r a n g e m u l t i p l i e r on t h e i n t e r f a c e s o f t h e triangulation

Ch

i n o r d e r t o e n f o r c e matching.

This approach,

however, d o e s n o t l e n d i t s e l f w e l l t o t h e u s e o f a n augmented L a g r a n g i a n b e c a u s e t h e n a t u r a l p e n a l i s a t i o n t e r m u t i l i s e s t h e norm t h e m a n i p u l a t i o n o f which i s somewhat awkward.

Hf(aK),

on

It is

i n f a c t n e c e s s a r y t o use a l i f t o n t o t h e e l e m e n t t o o b t a i n a c a l c u l a b l e e x p r e s s i o n , and t h i s i s w h a t w e s h a l l d o i n d i r e c t l y i n t h e work which f o l l o w s . G 5 0.

W e s h a l l u t i l i s e t h e methodology o f C h a p t e r I11 w i t h Thus f o r

vcV, q c H ,

Remark 6 . 1 :

For

~ E H w , e d e f i n e t h e augmented L a g r a n g i a n

r

= 0,

t h e Lagrangian ( 6 . 5 ) i s t h e s t a n d a r d

L a g r a n g i a n o f h y b r i d methods w i t h a m u l t i p l i e r on t h e i n t e r f a c e s of C o n s i d e r , i n f a c t , a p a r t i c u l a r f i n i t e ele-

the triangulation. ment

and

g

Suppose t h a t

g

K

E

H-’

( a K ) ; w e can s o l v e

t a k e s on a s i n g l e v a l u e a l o n g e a c h o f t h e i n t e r -

f a c e s ( a p a r t from a s i g n change t o t a k e a c c o u n t of t h e o r i e n t a t i o n of t h e n o r m a l ) ; it can e a s i l y b e s e e n t h a t the t e r m

(u,q),

( L I , v )=~ 0

and t h a t

r e d u c e s t o b o u n d a r y t e r m s o f t h e form

I t i s of i n t e r e s t t o s t a t e t h e o p t i m a l i t y c o n d i t i o n s o f t h e p r o b l e m :

i n v a r i a t i o n a l f o r m , t h i s amounts t o f i n d i n g

{u,p,x} E V

x

HxH

that

- (f,d-

(6.7)

a(p,q)

(6.8)

r(u-p,v)H + (X,v), = 0

(6.9)

0 ( U - P , ~=) ~

+

(,$(p),q)

Y p

E

H.

YV

(x,q),

E

+

r ( p - u , d H = 0 YqcH,

V (= H 1o ( Q ) ) ,

such

6)

(SEC.

169

HYBRID METHODS

X

For gi ven

and

u, t h e problems

(6.7)

a r e d e c o u p l e d and are

s o l v e d e l e m e n t by e l e m e n t ; t h e r e c o u p l i n g i s a c h i e v e d v i a t h e m u l t i p l i e r and v i a t h e l i n e a r problem ( 6 . 8 ) which i s of t h e form

-Au +

in R,

u = F(p,A)

(6.10)

u I r = 0,

where

F(q,A)

denotes a right-hand

s i d e d e p e n d i n g on

p

and on

I t i s v e r y s i m p l e t o a d a p t a l g o r i t h m s A L G l and ALGZ t o t h i s c a s e ;

An

note t h a t t h e updating of

(An+l-Xn,~)H

(6.11)

=

P,(U

n

-p

n

is carried out using , I I ) ~ Y v c H.

T a k i n g i n t o a c c o u n t ( 6 . 4 ) and p u t t i n g Ax"+'

= Ax"

+ p,

(6.12)

ah" +

where

nK

P,

A(u"-p")

a
> 2 ) . I n s i m i l a r v e i n , comparisons between ALG1, ALG2 and o t h e r methods o f s o l v i n g p r o b l e m (3.2) ( i n p a r t i c u l a r , nonl i n e a r o v e r r e l a x a t i o n m e t h o d s ) may b e f o u n d i n GLOWINSKI-MARROCCO 1 4 1 ; it would a p p e a r t h a t t h e a l g o r i t h m s d e s c r i b e d a b o v e a r e t h e m o s t e f f i c i e n t f o r s o l v i n g (3.2), p a r t i c u l a r l y f o r s c l o s e t o 1 o r >>2.

(SEC. 3)

3.4

185

N O N L I N E A R DIRICHLET PROBLEMS

C o n t i n u i t y c o n s t r a i n t s on t h e g r a d i e n t ; p e n a l t y methods

W e consider

Interior

( f o r s i m p l i c i t y ) t h e c a s e o f a Zinear problem

(s =2)

o f t h e t y p e ( 3 . 2 ) , s o l v e d by f i n i t e - e l e m e n t methods o f d e g r e e a t l e a s t two.

I n a s t a n d a r d a p p r o x i m a t i o n , t h e n o r m a l component o f t h e

g r a d i e n t of t h e approximate s o l u t i o n i s n o t continuous a c r o s s i n t e r element boundaries. able

f o r c e r t a i n a p p l i c a t i o n s , it may b e d e s i r -

Now,

(and u s e f u l ) t o enforce t h i s continuity.

One p o s s i b l e a p p r o a c h

f o r achieving t h i s c o n s i s t s of considering t h i s c o n t i n u i t y condition a s a s u p p l e m e n t a r y c o n s t r a i n t which c a n b e t r e a t e d e i t h e r by means o f a L a g r a n g e m u l t i p l i e r o r by means of a p e n a Z t y t e r m ;

t h i s l a t t e r app-

r o a c h i s known by t h e name o f t h e ' i n t e r i o r p e n a l t y m e t h o d ' , t h e pena l i s a t i o n b e i n g a p p l i e d t o a n y jump i n t h e n o r m a l d e r i v a t i v e o c c u r r i n g

a t t h e e l e m e n t i n t e r f a c e s , i n t h e i n t e r i o r o f t h e domain. The framework which w e h a v e j u s t d e s c r i b e d d o e s i n f a c t a d a p t q u i t e r e a d i l y t o s u c h methods:

i t i s s u f f i c i e n t t o impose m a t c h i n g c o n d i -

t i o n s on t h e f u n c t i o n s from

Lkh

and t o u s e a v a r i a n t o f t h e augmented

Lagrangian d e f i n e d earlier. A p a r t i c u l a r l y f a v o u r a b l e c a s e i s t h a t i n which,

if

k = 2 , we

e n f o r c e m a t c h i n g o f t h e components o f t h e g r a d i e n t a t t h e m i d p o i n t s of t h e el ement s i d e s degree o n e ) ;

(L2h t h e n comprises nonconforming e l e m e n t s o f

Vvh- q h = O

w e w i l l t h u s be s u b s t i t u t i n g t h e c o n s t r a i n t

f o r t h e c o n s t r a i n t o f matching t h e normal d e r i v a t i v e s .

Since the

m i d p o i n t s o f t h e s i d e s c o i n c i d e w i t h t h e q u a d r a t u r e p o i n t s which i n t e g r a t e polynomials of degree t w o e x a c t l y , t h e b a s i s of

L2h

formed by

a s s o c i a t i n g a n i n t e r p o l a t i o n f u n c t i o n w i t h e a c h node i s o r t h o g o n a l . It t h e r e f o r e follows t h a t t h e c a l c u l a t i o n of

p:

r e m a i n s a p o i n t calc-

u l a t i o n d e s p i t e t h e inter-element matching conditions.

However,

d e s p i t e t h e l i n e a r i t y of t h e p r o b l e m , w e n o l o n g e r h a v e c o n v e r g e n c e of

CuE>n20 i n a f i n i t e number o f i t e r a t i o n s ;

c o n s t r a i n t imposed i n Nonetheless,

i n f a c t , t h e matching

L2h i n v a l i d a t e s t h e p r o o f o f S e c t i o n 3 . 2 .

s i n c e t h e c a l c u l a t i o n of

pn h

v i a t h e analogue of

i s l i n e a r ( s i n c e s = 2 ) a n d l o c a Z , w e can d e t e r m i n e p: of

uE

and

(3.20)

a s a function

X i and i n s e r t t h e r e s u l t s o b t a i n e d i n t o t h e a n a l o g u e of

( 3 . 1 9 ) , and t h e r e b y e l i m i n a t e t h e i n n e r r e l a x a t i o n i t e r a t i o n s a t t h e

c o s t o f h a v i n g t o a s s e m b l e a m a t r i x which i s o n l y s l i g h t l y more compl i c a t e d t h a n t h a t i n t h e o r i g i n a l scheme.

r

as l a r g e as p o s s i b l e , i . e .

Then w e n e e d o n l y c h o o s e

a t t h e l i m i t of t h e machine p r e c i s i o n ,

i n o r d e r t o o b t a i n a correct s o l u t i o n o f t h e l i n e a r problem o b t a i n e d

186

STRONGLY NONLINEAR B.V.P.'s

(CHAP. 5)

by elimination of ph. R e m a r k 3.5: The convergence, as h + O , of such approximations with constraints on the normal derivative is often quite difficult to prove (but see DOUGLAS-DUPONT [ll, WHEELER 111 and the corresponding

bibliography, as well as FORTIN-SOULIE 111 in which a nonconforming element of degree two is considered, for which convergence can be proved).

4.

A MAGNETO-STATIC PROBLEM

4.1

Formulation of the problem

In this section we shall be discussing some of the results obtained by GLOWINSKI-MARROCCO r 5 1 and MARROCCO Ill in connection with the calculation of the magnetic state of rotating machines (motors, alternators, ) or static machines (transformers). Here, we are concerned with problems in which even the linearised case has a highly variable coefficient, and in which it will be necessary to use an augmented Lagrangian of the kind introduced in Remark 3.3 of Section 3.2.

...

With

$

= {-,

a

-,a

are written: axl ax2

(4.2)

+

-

ax3

the Maxwell equations of magnetostatics

+

B = LIH,

+-+

(4.3)

-1,a

v.B

= 0 ; t

in ( 4 . 1 ) - (4.3), 6 is the m a g n e t i c f i e l d vector, J is the c u r r e n t + d e n s i t y vector, B the m a g n e t i c i n d u c t i o n vector and 1~ the m a g n e t i c p e r m e a b i l i t y of the medium. In view of (4.3) there exists a vector potential h such that $ = $ x 1; ; the above equations therefore lead to the equation (4.4)

itx

(.$Xi)

=,;

with v = l/v. We have v =vrvo , where v is the relative magnetic reluctivity and where vo is the magnet*c.

iR2,

(SEC. 6 )

FURTHER APPLICATIONS

201

In t h e d i s c r e t e c a s e t h e s o l u t i o n i s performed t r i a n g l e by t r i a n g l e , o r a t t h e q u a d r a t u r e p o i n t s used f o r e v a l u a t i n g

in

P u t t i n g d ( x ) = rVu(x) + h ( x ) , t h e lql dx. above m i n i m i s a t i o n problem i s g i v e n by

The d e c o m p o s i t i o n a s s o c i a t e d w i t h t h e c o n s t r a i n t q

-

Q v = 0 and w i t h

t h e augmented L a g r a n g i a n ( 6 . 3 ) h a s t h u s a l l o w e d u s t o e l i m i n a t e any d i f f i c u l t y associated with t h e nondifferentiable t e r m

Remark 6 . 1 :

I f w e r e f e r back t o Remark 3 . 3 of S e c t i o n 3 , i t

would b e a d v i s a b l e , i n o r d e r t o improve t h e convergence of A L G l and

ALGZ, t o u s e a p e n a l t y t e r m o f t h e form

2ln

11 ( x ) IVv-ql

&I.

f o r m a Z t y , 11 r e p r e s e n t s an e s t i m a t e of v + n o t a p p l i c a b l e h e r e , s i n c e i t becomes i n f i n i t e ;

d x , where,

This expression i s nonetheless t h i s

l e a d s u s t o t h i n k t h a t t h e p e n a l t y term s h o u l d b e v e r y l a r g e i n t h e r i g i d z o n e s , i . e . t h o s e i n which Vu = 0.

S i n c e t h e r i g i d zones

i n c r e a s e w i t h g , we s h o u l d e x p e c t t o see t h e o p t i m a l v a l u e o f 9

( i n ALG2 p a r t i c u l a r l y ) i n c r e a s e w i t h g

Remark 6 . 2 :

r

.

I n t h e zones where g 2 l d l , which a f t e r convergence

c o r r e s p o n d t o t h e r i g i d z o n e s of t h e problem, w e h a v e , from ( 6 . 7 ) , Inserting t h i s result i n t o (6.4)

IpI = 0.

i t can r e a d i l y be s e e n

t h a t i n these regions we a r e i n f a c t solving

taking

r

t o b e l a r g e i n t h e s e r e g i o n s i n f a c t amounts t o f o r c i n g

-Au to h a v e t h e v a l u e z e r o .

6.1.3

Numerical r e s u l t s

The n u m e r i c a l r e s u l t s o f MARROCCO Cll, which w e s h a l l d i s c u s s 9

All t h i s assumes t h a t w e a r e u s i n g p = r i n ALG1, ALGZ.

202


(CHAP. 5)

very briefly in this section, were obtained in a very simple case for which the exact solution is known; with the domain as the disc of radius 1 centred at the origin, the solution of problem (6.2) is given, for f

=

C ( > 0),by u E O i f g > g

c

C 2

= -

(6.9)

approximation by Co-conforming finite elements of order 1 was used, the corresponding triangulation "eh comprising 256 triangles.

An

The calculations were performed for g = 2,5,8, the rigid zone then being the circle of radius R' = 0.2, 0.5, 0.8, respectively. If we consider ALG2 (with p

=

r) , the optimal value of

r

is 5 x

(resp. 1.0, 7.0) for g = 2 (resp. 5 , 8), the corresponding numbers of iterations being respectively 10, 25, 50 for a termination test which n relates solely to the convergence of the sequence [uh?n20. In all the cases considered, ALGl (with p = r) performs less effectively than ALG2, and this is true even for problems in which the nonlinearity is very large, i.e. g is large ( g = 8 for example). The calculations confirm that, for ALG1, the convergence of the relaxation iterations is slow in the rigid zones. Decoupling between the conand that of {p;3,,, and {Xz}n20 is also evident: vergence of [un} h ntO the value of :u depends only on the components of :p and X E in the space VVh, where

Vh

=

{v Iv EC'(E~),V~J~EP, V K c C h , v = 0 on 3%) h h h

10

,

and it would certainly appear that these components converge rapidly and do not require an accurate solution for [ u ~ , p ~to } be obtained at each iteration of ALG1. A powerful algorithm (in terms of the number of iterations) would un-

doubtedly be ALG2 with r made to increase during the course of the calculation so as to accelerate convergence in the rigid zones.

(SEC.

6)

203


Remark 6 . 3 :

I n Chapter V I I o f t h i s book w e d e s c r i b e t h e a p p l i c a -

t i o n of t h e augmented Lagrangian methods o f C h a p t e r I11 t o t h e s o l u t i o n of problems i n v o l v i n g t h e flow of Bingham f l u i d s which a r e much more c o m p l i c a t e d t h a n t h o s e c o n s i d e r e d i n t h i s s e c t i o n ;

i n f a c t , by

switching t o t h e stream function, we obtain v a r i a t i o n a l i n e q u a l i t i e s 11

of o r d e r 4

, whereas

problem

i s a v a r i a t i o n a l i n e q u a l i t y of

(6.2)

order 2.

6.2

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

6.2.1

F o r m u Z a t i o n of t h e p r o b l e m

The p h y s i c a l m o t i v a t i o n of t h e problem i s a s f o l l o w s :

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 of i n f i n i t e l e n g t h and w i t h c r o s s s e c t i o n R , made of an i s o t r o p i c elastic/perfectly-plastic m a t e r i a l ,

( i . e . t h e y i e l d s t r e s s ) b e i n g g i v e n by

t h e t h r e s h o l d of p l a s t i c i t y t h e von Mises c r i t e r i o n .

S t a r t i n g from an u n s t r e s s e d i n i t i a l s t a t e ,

an i n c r e a s i n g t o r s i o n a l c o u p l e i s a p p l i e d t o t h e b a r , t h e t o r s i o n b e i n g c h a r a c t e r i s e d by t h e a n g l e o f t w i s t p e r u n i t l e n g t h , denoted by C

i n t h e following notes. W e can t h e n reduce t h i s problem ( s e e GERMAIN C11 f o r a d e t a i l e d

a n a l y s i s ) t o seeking a function

u

,

t h e so-called s t r e s s p o t e n t i a l

( d e f i n e d 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 ) .

For s l i g h t l y g r e a t e r

g e n e r a l i t y we s h a l l assume t h a t R i s R-connected simply c o n n e c t e d ) ;

R = 3. t h e wi, (6.10)

( i f R = 0, R

W e d e n o t e by R* t h e domain o b t a i n e d by t h e union o f R and i = 1, ...R.

2

W e next d e f i n e

= IqlqE

(L’(Q*))’,

191 < I

a.e.,

q=o o n

K = {vlv

E

Hb(R*), Vv E

,

and f i n a l l y t h e f u n c t i o n a l J : Hk(R*)

-f

IR

by

(6.12)

11

i.e.

,...a ) ,

wi, i = ~

then (6.11)

is

F i g u r e 6 . 1 i l l u s t r a t e s a s i t u a t i o n i n which

r e l a t i v e t o an e l l i p t i c o p e r a t o r o f o r d e r 4.

204

(CHAP. 5 )

STRONGLY NONLINEAR B . V . P . ' s

The s t r e s s p o t e n t i a l

u

,

mentioned above, i s t h e n t h e s o l u t i o n

o f t h e f o l l o w i n g p r o b l e m i n t h e C a l c u l u s of V a r i a t i o n s

( i n some

appropriate system o f u n i t s ) :

(6.13)

J(u)

5

J(v)

Vv

E

K,

t h i s i t s e l f b e i n g 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 i n e q u a l i t y problem

(6.14)

The e s s e n t i a l d i f f i c u l t y w i t h

(6.13), ( 6 . 1 4 ) s t e m s f r o m t h e c o n s t r a i n t

of b e l o n g i n g t o K.

Figure 6.1

S o l u t i o n of

6.2.2

(6.13), ( 6 . 1 4 )

by a u g m e n t e d L a g r a n g i a n m e t h o d s

W e i n t r o d u c e t h e augmented L a g r a n g i a n

dr

: H;(R*)

x

( L ~ ( Q * )x ) (~ L ~ ( Q * )-+ ) ~IR d e f i n e d by

W e s h a l l determine HA(Q*)

x

K"

x

u

by s e e k i n g t h e s a d d l e p o i n t s o f

dr on

( L 2 ( Q * ) ) 2 ; for t h i s , w e employ a l g o r i t h m s ALGl o r ALG2

of C h a p t e r I11 whose i m p l e m e n t a t i o n f o r 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 t h e c a s e where d e s c r i b e d i n C h a p t e r 111, S e c t i o n 3.3.

i s s i m p l y c o n n e c t e d ) was

(SEC. 6)


We suppress the iteration indices;

205

the implementation of ALGl and

ALG2 requires the solution (simultaneous or sequential, depending on

the case in question) of the following equations and inequalities, with

h

fixed

:

p'(q-p)

dx

2

(6.17)

J R*

(rVu+h)*(q-p)dx

I

Y q c K,

Equation (6.17) is solved pointwise in explicit fashion since p = 0 (6.18)

P =

in wi, i = I ,

...k ,

h+rVu sup(l +r, 1 X+rVul)

in R,

so that our decomposition method has eliminated the difficulties directly related to the von Mises criterion IOU]

5

1.

In practice, (6.18) is solved at a certain number of points depending on the discretisation used. For finite-element approximations in which the functions u and v in ( 6 . 1 3 ) , (6.14) are approximated by piecewise-linear functions, (6.18) is solved triangle-by-triangle to obtain the two constant components of p In the general case the points are chosen to correspond with a quadrature formula which This is exact for integrating terms of the form fn* p - q dx corresponds to introducing an approximate convex set gh whose s u p p o r t function (see EKELAND-TEMAM 111 for this concept) approximates the support function of K , i.e. In lql dx (in the case without any holes) by the use of the chosen quadrature method.

.

.

It is not possible to apply Remark 3 . 3 of Section 3 to this example; it is, however, easy to see that in the plastic zones (where IpI = lVul = l), taking r to be large will, in view of (6.16), force -Vu + 8 - p to vanish. We can therefore expect an optimal value of r which will increase with C , since increasing Remark 6.4:

the twist angle causes an enlargement of the plastic zones.

206

STRONGLY NONLINEAR B . V . P . ' s

(CHAP.

5)

NurnericaZ r e s u l t s

6.2.3

The n u m e r i c a l r e s u l t s o b t a i n e d by MARROCCO 111 ( w i t h p = r ) conf i r m t h e r e s u l t s of S e c t i o n 6 . 1 . 3 f l u i d i n a c y l i n d r i c a l duct.

r e l a t i n g t o t h e f l o w o f a Bingham

A l g o r i t h m ALG2 i n f a c t p e r f o r m s

b e t t e r t h a n A L G l and once a g a i n t h e r e g i o n s of R where t h e convergence

i s s l o w e s t a r e t h o s e i n which t h e n o n l i n e a r e f f e c t s m a n i f e s t themt h a t i s , i n t h e c a s e of t h e t o r s i o n problem, t h e r e g i o n s

selves

R

= 10 1C

rate

The n u m e r i c a l t e s t s w e r e p e r f o r m 4 w i t h

Vul = 1.

where

-

X

10, 11 and

C = 10

and t h e s e show t h a t t h e convergence

measured by t h e number of i t e r a t i o n s

-

i s more o r l e s s independ-

t h i s i s shown by F i g u r e 6 . 2

e n t of t h e d i s c r e t i s a t i o n ;

(in relation

t o A L G 2 ) i n which t h e c u r v e s 1 , 2 , 3 c o r r e s p o n d r e s p e c t i v e l y t o a

=Ch

triangulation

w i t h 128, 512 and 2048 t r i a n g l e s .

These c u r v e s

i n d i c a t e 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 c o n v e r g e n c e , a s a f u n c t i o n of

r

.

One o f t h e consequences of t h e e x t r e m e l y weak dependence o f A L G l and ALG2 on t h e c h o i c e of optimal

r

h

i s t h a t it i s p o s s i b l e t o d e t e r m i n e t h e

on a c o a r s e mesh, and t h e n t o u s e t h e o p t i m a l

o b t a i n e d f o r c a l c u l a t i o n s on a much f i n e r mesh.

Figure 6.2

r

thus

(SEC. 6 )

201


Remark 6 . 5 :

C h a p t e r V I d e s c r i b e s t h e a p u l i c a t i o n o f A L G l and ALG2

t o t h e n u m e r i c a l s o l u t i o n o f an e l a s t o p l a s t i c i t y p r o b l e m which i s much more c o m p l i c a t e d t h a n t h a t d i s c u s s e d i n t h e p r e s e n t s e c t i o n ;

nonethe-

l e s s , t h e b a s i c p r i n c i p l e s o f s o l u t i o n u s i n g an augmented L a g r a n g i a n remain t h e same, a n d are o n c e a g a i n b a s e d on t h e g e n e r a l c o n c e p t s deve l o p e d i n C h a p t e r 111.

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 minimal s u r f a c e s problem

6.3

F o r m u l a t i o n of

6.3.1

t h e problem

I n t h i s w e w i l l be considering t h e a p p l i c a t i o n o f t h e g e n e r a l methods o f C h a p t e r I11 t o t h e s o l u t i o n o f a p r o b l e m which o n c e a g a i n falls

-

-

formally, a t l e a s t

w i t h i n t h e g e n e r a l framework d e f i n e d i n

Section 2 of t h e p r e s e n t chapter;

t h i s i s a p a r t i c u l a r l y s i m p l e (as

We

f a r a s i t s f o r m u l a t i o n i s c o n c e r n e d ) m i n i m a l s u r f a c e s problem. thus consider the contour ( w i t h boundary

r

),

m3 ,

in

d e f i n e d u s i n g a domain

Cl o f

R

2

by

C = {{x,g(x)}

(6.19)

C

eR3,

X E

r,

g(x) E R )

,

and t h e f u n c t i o n a l (6.20) The m i n i m a l s u r f a c e s p r o b l e m i s t h e n d e f i n e d by

(6.21)

where (6.22)

W e a r e h e r e d e a l i n g w i t h a n o n t r i v i a l problem s i n c e

difficulties

-

t h e space

Wl'l(C2)

is not reflexive;

-

among o t h e r

w e h a v e t o con-

s i d e r (see EKELAND-TEMAM [ l ] ) g e n e r a l i s e d s o l u t i o n s , a n d t h e c o n d i t i o n

ulr

= g

c a n n o t be s a t i s f i e d i n t h e u s u a l s e n s e , even f o r v e r y r e g u l a r

boundaries

I'

and f u n c t i o n s

g .

The t r e a t m e n t w h i c h f o l l o w s i s

t h e r e f o r e f o r m a l , and i s t o t a l l y j u s t i f i e d o n l y f o r d i s c r e t i s e d prob-

l e m s (which t h e n f a l l w i t h i n t h e c o n t e x t o f Theorems 4 . 2 and 5.2 o f C h a p t e r 111, S e c t i o n s 4 and 5 , r e s p e c t i v e l y ) .

208

STRONGLY N O N L I N E A R B . V . P . ' s

(CHAP.

5)

S o l u t i o n of p r o b l e m ( 6 . 2 1 ) b y a u g m e n t e d L a g r a n g i a n algorithms

6.3.2

W e i n t r o d u c e t h e augmented L a g r a n g i a n (6.23)

Lr(v,q,p)

=

I n d m dx + 5

I

n

IVv-ql 2 dx +

u w e s h a l l t h u s s e e k ( f o r m a l l y i n i n f i n i t e dimensions) dr on (Vg n H1(n)) x ( L 2 ( n ) ) 2 x ( L 2 ( Q ) )2 by W e s h a l l t h e r e f o r e be l e d t o a l g o r i t h m s of t h e t y p e A L G 1 , A L G 2 .

To d e t e r m i n e

t h e s a d d l e p o i n t s of

s o l v e , a t e a c h i t e r a t i o n , s i m u l t a n e o u s l y f o r A L G l and s e q u e n t i a l l y f o r ALG2, t h e f o l l o w i n g n o n l i n e a r system ( w e omit t h e i t e r a t i o n i n d i c e s ) , with

h

fixed :

(6.25)

The n o n l i n e a r e q u a t i o n can be s o l v e d p o i n t by p o i n t ; z

=

IpI

,

putting

we f i r s t have t o s o l v e t h e f o l l o w i n g n o n l i n e a r e q u a t i o n i n

one v a r i a b l e :

f o r which Newton's method may be a p p l i e d w i t h o u t d i f f i c u l t y .

Depend-

i n g on t h e t y p e of a p p r o x i m a t i o n u s e d , w e s o l v e ( 6 . 2 6 ) e i t h e r elementby-element,

o r a t q u a d r a t u r e p o i n t s , a s f o r t h e n o n l i n e a r problems

d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n s of t h e p r e s e n t c h a p t e r .

Remark 6 . 6 :

On t h e b a s i s o f Remark 3.3 of S e c t i o n 3, w e s h o u l d 2 q ( x ) /Vv - ql dx , w i t h

i n t h i s c a s e u s e a p e n a l t y t e r m of t h e form ~ ( x an ) e s t i m a t e of large.

(1+ lVuI2)-'

; this

fR

t e r m i s s m a l l when

lVul

We must t h e r e f o r e e x p e c t t h a t t h e o p t i m a l v a l u e of

r

is

for

ALG2 ( w i t h p = r ) , and i n t h e c a s e o f t h e L a g r a n g i a n (6.231, w i l l b e

less t h a n u n i t y .

6.3.3

Numerical r e s u l t s

The r e s u l t s o f MARROCCO Cll, which w e summarise b r i e f l y h e r e , rel-

a t e t o t h e case where

R i s t h e c i r c u l a r c o r o n a d e f i n e d by

:

(SEC. 6)


t h e boundary c o n d i t i o n s a r e

g(x) = O

on t h e c i r c l e of r a d i u s

on t h e c i r c l e of r a d i u s

4

and

1.

Since t h e s o l u t i o n i s a x i s y m m e t r i c i t i s e a s i l y c a l c u l a t e d , and w i t h I = (xf + x22 ) % i t i s g i v e n by

g(x) = B

(=const.)

209

XI

B

u(x) = B

4

Arg ch

),

h a v i n g t o be d e t e r m i n e d from t h e v a l u e of

the constant A that

-

u ( x ) = A ( A r g ch

(6.28)

if

IXI

=

1).

l e s s than a c r i t i c a l value

BC=- 2 . 0 7 .

If

B > Bc,

t i o n ' b r e a k s down' i n t h e s e n s e t h a t t h e c o n d i t i o n

1.

be s a t i s f i e d on t h e c i r c l e o f r a d i u s i s e by means o f

B

(knowing

This c l a s s i c a l solution e x i s t s only f o r

Co-conforming

then t h e solu-

u = B can no l o n g e r

Numerically, i f w e d i s c r e t -

f i n i t e e l e m e n t s , t h i s shows i t s e l f

F i g u r e 6 . 3 where c r o s s - s e c t i o n s

(see

o f an approximate s o l u t i o n a r e shown)

a s a v e r y l a r g e g r a d i e n t n e a r t h e boundary where t h e a f o r e m e n t i o n e d I n view of Remark 6 . 6 w e s h o u l d exp-

'breakdown' phenomenon o c c u r s . e c t , i n t h e c a s e of ALG2 decrease as since f o r

1 (resp.

B

increases;

(with

0.1);

,

t o see t h e o p t i m a l v a l u e of

r

t h i s i s c o n f i r m e d by t h e n u m e r i c a l t e s t s ,

B = 1 ( r e s p . 2.07,

0.2,

p =r )

4 ) t h e o p t i m a l v a l u e of

r

is close t o

t h e c o r r e s p o n d i n g numbers o f i t e r a t i o n s a r e

20, 30 and 5 0 , r e s p e c t i v e l y .

W e t h u s s e e t h a t Remark 3 . 3 of S e c t i o n

3 h a s e n a b l e d t h i s phenomenon t o be p r e d i c t e d even though i t r u n s c o u n t e r t o t h e n u m e r i c a l e x p e r i m e n t s of t h e e a r l i e r s e c t i o n s o f t h e p r e s e n t c h a p t e r where, o t h e r t h i n g s b e i n g e q u a l , t h e o p t i m a l v a l u e of

r

i n c r e a s e d when t h e n o n l i n e a r e f f e c t s became more s i g n i f i c a n t .

A s f a r a s A L G l i s c o n c e r n e d , it t u r n s o u t once a g a i n t o be more

e x p e n s i v e t h a n ALG2.

Remark 6.7:

JOURON [ll g i v e s a d e t a i l e d a c c o u n t of t h e approxim-

a t i o n o f t h e minimal-surfaces

problem by means of methods u s i n g con-

forming f i n i t e e l e m e n t s o f o r d e r

1 , and o f t h e i r i t e r a t i v e s o l u t i o n

by n o n l i n e a r o v e r r e l a x a t i o n methods

( s e e also JOHNSON-THOMEE [11 and

CIARLET C1, C h a p t e r 51 f o r f i n i t e - e l e m e n t imal-surfaces problem).

a p p r o x i m a t i o n s of t h e min-

2 10

(CHAP. 5 )

STRONGLY NONLINEAR B.V.P. ' s

1.00

3.60

3.20

2.60

2.10

2-00

1.60

1-20

0.80

0.u

0. 0.

1.60

3.20

6.60

6.10

00

Figure 6.3

7.

DISCUSSION ON CHAPTER V I n t h i s c h a p t e r w e h a v e a p p l i e d t h e methods o f C h a p t e r I11 t o t h e

s o l u t i o n o f problems w i t h v a r i o u s p h y s i c a l o r i g i n s .

W e have t h e r e b y

b e e n a b l e t o d e m o n s t r a t e t h e f a c t t h a t t h e d e c o m p o s i t i o n o f a nonl i n e a r p r o b l e m t h r o u g h t h e i n t r o d u c t i o n of a n augmented L a q r a n g i a n

i s a r o b u s t method,

r e a d i l y a d a p t a b l e t o numerous s i t u a t i o n s ( w e

s h a l l b e s e e i n g f u r t h e r examples o f t h i s i n C h a p t e r s V I , V I I & V I I I ) . T h i s r o b u s t n e s s and t h i s g e n e r a l i t y e n a b l e t h e augmented-Lagrangian methods t o b e u s e d f o r t h e e f f i c i e n t s o l u t i o n o f numerous t y p e s o f problems. Two phenomena w o r t h y o f o u r a t t e n t i o n h a v e become a p p a r e n t :

the

f i r s t of t h e s e c o n c e r n s t h e p a r t i a l d e c o u p l i n g b e t w e e n t h e c o n v e r g e n c e o f t h e sequence

Cun}n>O a p p r o x i m a t i n g t h e unknown f u n c t i o n

t h a t o f t h e sequence

[pn}n>O

a p p r o x i m a t i n g Vu ;

i s t o t a l i n t h e l i n e a r case i f w e p u t

p = r .

u , and

t h i s decouplinq

T h i s i s one a s p e c t o f

a l g o r i t h m s A L G l and ALG2 which would m e r i t a more d e t a i l e d i n v e s t i q ation :

i t i s t h i s d e c o u p l i n q which p a r t i a l l y e x p l a i n s t h e s u p e r i o r -

(SEC. 7 )

DISCUSSION

211

this is

i t y , f o r t h i s t y p e of problem a t l e a s t , of ALG2 o v e r A L G 1 ;

t r u e even f o r problems i n which t h e n o n l i n e a r i t y i s v e r y s t r o n g . had i n f a c t o b s e r v e d t h e o p p o s i t e s i t u a t i o n i n C h a p t e r I V ;

V of t h e

e r e n c e o b v i o u s l y r e l a t e s t o t h e f a c t t h a t t h e image u n d e r space

i n which

V

( L 2 ( Q )) N

u

We

the diff-

i s s o u g h t i s a s t r i c t , c l o s e d s u b s p a c e of

and we can have v e r y r a p i d convergence o f t h e component o f

pn which b e l o n g s t o t h i s s p a c e and y e t slow convergence o f t h e sequn ence Ip I t would b e i n t e r e s t i n g t o a n a l y s e t h i s phenomenon i n more d e t a i l , w i t h a view t o d e v e l o p i n g a l g o r i t h m s which e x p l o i t t h i s feature a s f a r as possible.

la

The second phenomenon i s r e l a t e d t o t h e r61e which c o u l d be p l a y e d

by a p e n a l t y t e r m , w i t h v a r i a b l e c o e f f i c i e n t s , of t h e form 2 11 ( x ) I Vv - ql dx , f o r a c c e l e r a t i n g convergence. A considerable advantage would l i e i n t h e f a c t t h a t f o r A L G 2 , w i t h

ably chosen, t h e optimal choice of

r)

r would b e c l o s e t o

1.

p = r

and

rl

suit-

In reality, the

r e q u i r e s an a p r i o r i knowledge of t h e s o l u t i o n .

Thus,

a s one p o s s i b i l i t y , w e can c o n s i d e r a l g o r i t h m s which i n v o l v e u p d a t i n g q(x)

d u r i n g t h e c o u r s e of t h e c a l c u l a t i o n ;

i f t h e l i n e a r systems a r e

s o l v e d by d i r e c t m e t h o d s , however, s u c h an u p d a t e would r e q u i r e t h e f a c t o r i s a t i o n of a new m a t r i x , which i s a r e l a t i v e l y e x p e n s i v e o p e r ation.

T h i s drawback would d i s a p p e a r i f p o w e r f u Z i t e r a t i v e m e t h o d s

c o u l d b e used t o s o l v e t h e s e l i n e a r s y s t e m s ;

amongst t h e methods

which can be c o n s i d e r e d , we may l i s t p r e c o n d i t i o n e d c o n j u g a t e - g r a d i e n t methods, m u l t i g r i d methods, e t c .

Secondly, t h e r e e x i s t s i t u a t i o n s

f o r which a f a m i l y of s i m i l a r problems h a s t o b e s o l v e d , d i f f e r i n g o n l y t h r o u g h t h e v a l u e s of a few p a r a m e t e r s . be p o s s i b l e t o u s e a f u n c t i o n

t-

I n such c a s e s , i t would

d e r i v e d from a mean s o l u t i o n , or t o

employ a s t r a t e g y o f g r a d u a l l y i n c r e a s i n g t h e p a r a m e t e r s , w i t h an upd a t e of

0 when t h e s o l u t i o n h a s changed s u f f i c i e n t l y .

been a b l e t o u s e t h e e x i s t e n c e of an o p t i m a l c o e f f i c i e n t c e r t a i n form - r e l a t e d t o t h e b e h a v i o u r of

Vu

-

W e have a l s o 0

having a

t o predict, a t least

q u a l i t a t i v e l y , t h e c o r r e s p o n d i n g b e h a v i o u r of t h e o p t i m a l p a r a m e t e r with t h e c o e f f i c i e n t

11

taken equal t o

r

1.

F i n a l l y , w e s h o u l d p o i n t o u t t h a t f o r t h e problems t r e a t e d i n t h e p r e s e n t c h a p t e r , t h e c h o i c e of

Ar

makes i t p o s s i b l e t o make

r vary

w i t h o u t h a v i n g t o r e f a c t o r i s e t h e m a t r i x of t h e l i n e a r system which occurs during t h e c a l c u l a t i o n ; be h e l p f u l i f w e c o u l d make

i n c e r t a i n c a s e s i t would c e r t a i n l y

r v a r y i n an e f f e c t i v e manner, though

t h e p r e c i s e means of d o i n g t h i s remains t o be d e t e r m i n e d .

212

STRONGLY NONLINEAR PROBLEMS

(CHAP.

5)

The magnetostatic problems investigated in Section 4 are of great industrial importance (transformers, rotating machines, electromagnets in particle accelerators, read/write heads for disks and magnetic tapes, etc.); since the formulation used in Section 4 is by no means the only one possible, we consider it necessary, in view of the importance of the subject, to indicate a few other formulations and to make a number of observations on the associated augmented-Laqranqian algorithms. Following, for example, MUNRO C11 we can, in magnetostatics, define the functions Uc and U by, respectively,

'0

[ (uC

: complementary magnetic energy p e r u n i t v o l u m e ) , A

[

(U : s t o r e d m a g n e t i c e n e r g y p e r u n i t v o l u m e ) t

Suppose that

+

in (4.1) ;

3 =O

then there exists $I such that

+

i.e. H derives from a s c a l a r p o t e n t i a l ; it is convenient in this case to use U c , which gives the energy functional r

ss

(7.4)

=

jn uC dx.

5

If # 6 , then H no longer derives from a scalar potential, but V * B = O implies the existence of a vector potential it such that + + + + V x A = B ; it is then more convenient to use A (this is what was done in Section 4 in the case where A = { O , O , A } ) , the energy functional to be used being defined by + +

-t

The augmented Lagrangians associated with the above two situations are defined as follows: (i)

t

+

When j = O +

+

and when the s c a l a r p o t e n t i a l

$I is used, we

obviously put H = V @ and we p e n a l i s e and d u a l i s e this linear constraint so as to obtain the augmented Lagrangian i : defined by

(SEC. 7 )

DISCUSSION

213

(7.6)

I n i m p l e m e n t i n g t h e above a l g o r i t h m s A L G l , ALG2 w e o b t a i n t h e f o l l owing e q u a t i o n s ( t h e i t e r a t i o n i n d i c e s h a v e been o m i t t e d ) :

(V

i s a subspace of

d i t i o n s , and

Vo

H

1

(R) which t a k e s i n t o a c c o u n t t h e b o u n d a r y con-

is t h e associated test-function

space, corresponding

t o homogeneous b o u n d a r y c o n d i t i o n s ) ,

A t t h e numerical i n t e g r a t i o n p o i n t s or triangle-by-triangle,

(7.8)

on t h e a p p r o x i m a t i o n u s e d , tion (in

IR’

depending

l e a d s t o t h e f o l l o w i n g v e c t o r equa-

o r IR3 d e p e n d i n g on t h e d i m e n s i o n

N

of t h e p r o b l e m ) :

+

H ERN,

(7.9)

+

+

B(H) + r q H = C,

where

+ C i s a known v e c t o r .

I n t h e case where t h e m a t e r i a l i s i s o -

t r o p i c , we can reduce t h e s o l u t i o n o f equation i n (ii)

IR+ g i v i n g When

$*

p e n a l i s e and d u a l i s e

161

6 and 6=3

X

(7.9) t o t h a t of a n o n l i n e a r

( l i k e (4.16)

i n Section 4 ) .

when t h e v e c t o r p o t e n t i a l 2 i s u s e d , w e + A ; t h i s g i v e s t h e augmented L a g r a n g i a n

which i n c l u d e s t h e augmented L a g r a n g i a n d e f i n e d by (4.13) i n S e c t i o n 4.2

as a p a r t i c u l a r c a s e .

are t h e n , r e s p e c t i v e l y ,

The e q u a t i o n s c o r r e s p o n d i n g t o ( 7 . 7 ) , ( 7 . 8 )

214

STRONGLY N O N L I N E A R B . V . P . ' s

(CHAP.

5)

(7.12)

A s before,

i s solved triangle-by-triangle

(7.12)

o r a t t h e numerical

i n t e g r a t i o n p o i n t s , depending on t h e a p p r o x i m a t i o n chosen, by s o l v i n g t h e f o l l o w i n g i n lRN :

(7.13)

H(B) + rnB

=

C.

(7.11), on t h e o t h e r hand, can pose a number o f

The s o l u t i o n of

d i f f i c u l t i e s and i t i s c o n v e n i e n t t o d i s t i n g u i s h t h e c a s e s N = 2 and N = 3;

Ti

if

{O,O,A}

=

t h i s l e a d s t o a problem i n

lR2

and, a s w e

saw i n S e c t i o n 4 , t h e r e l a t i o n

r e d u c e s t h e s o l u t i o n of

( 7 . 1 1 ) t o t h a t of a l i n e a r e l l i p t i c problem

o f second o r d e r and of s t a n d a r d t y p e .

For

N = 3

,

problem ( 7 . 1 1 )

i s i n g e n e r a l i l l - p o s e d s i n c e t h e semi-norm

i s n o t a norm on ( H 1 ( Q ) / l R )

+

vx

++

(V+Vl$) = $ X Z

;

t h i s i s due t o t h e f a c t t h a t

w$ ,

which means t h a t t h e v e c t o r p o t e n t i a l i s i n g e n e r a l d e f i n e d o n l y t o

(see DURAND [I]). It then follows t h a t t h e 1 3 g i v e n by ( 7 . 4 ) i s n o t c o e r c i v e i n ( H (,)/El)

within a gradient functional

3,

.

It

i s shown i n MARROCCO [ 2 1 t h a t a f u n c t i o n a l s p a c e a d a p t e d t o 3-dimensi o n a l m a g n e t o s t a t i c problems i s t h e f o l l o w i n g :

(SEC. 7 )

215

DISCUSSION

and t h a t i t i s s u f f i c i e n t t o a d d t o t h e f u n c t i o n Z V a t e r m o f t h e 1 ( w i t h a ( x ) t ci0 > 0) i n o r d e r t o make it type 2 a ( x ) IV.A12dx The f u n c t i o n a l Zv thus corrected admits a c o e r c i v e on W

In

.

u n i q u e minimum on W ,

-f

t h e corresponding v e c t o r p o t e n t i a l A s a t i s f y i n g

t h e Maxwell e q u a t i o n s o f m a g n e t o s t a t i c s , a s w e l l as t h e c o n d i t i o n + + d e f i n e d by I f w e a d d t h e a b o v e term t o t h e L a g r a n g i a n A: V * A = 0. ( 7 . 1 0 ) , w e o b t a i n i n p l a c e of

it is reasonable t o take a

(7.11)

= rq,

i n which case, f o r c e r t a i n g e o m e t r i e s

( i f R i s a p a r a l l e l e p i p e d , f o r example),

(7.16)

c a n b e decomposed i n t o -+

t h r e e p r o b l e m s o f D i r i c h l e t t y p e ( o n e f o r e a c h component o f A ) ; equation (7.12)

r e m a i n s unchanged.

1

The a d d i t i o n o f t h e t e r m

gci16.a12

dx

to

ZV

and J!:,

+ +

s i d e r e d as a p e n a l i s a t i o n o f t h e c o n d i t i o n V - A = 0;

may b e con-

it i s therefore

n a t u r a l t o t h i n k of a s s o c i a t i n g a L a g r a n g e m u l t i p l i e r w i t h t h i s constraint; ci =

t h i s l e a d s t o t h e augmented L a g r a n g i a n

d:

defined ( i f

r n ) by

A l l t h e a b o v e r e m i n d s u s of t h e S t o k e s p r o b l e m i n C h a p t e r 11, t h e

function

nq

playing t h e

rsle

of a pressure.

In t h e implementation

o f a l g o r i t h m s A L G 1 , ALG2 i n r e l a t i o n t o t h e L a g r a n g i a n ( 7 . 1 7 ) , e q u a t i o n ( 7 . 1 2 ) r e m a i n s unchanged;

-+

a s r e g a r d s t h e e q u a t i o n i n A , t h i s becomes

+ (7.18)

~ ( i f x ~ ) . ( $ x d~x ) + r dx +

;I

-

I,

q($.t)($*G)dx

(r$-I) (5 x G)dx -

j;q$*:

=

dx

&E

W.

From t h e p o i n t of v i e w of a p p r o x i m a t i o n by f i n i t e e l e m e n t s , t h e a f o r e m e n t i o n e d a n a l o g y w i t h t h e S t o k e s p r o b l e m s u g g e s t s u s i n g , amongst o t h e r s , n o n c o n f o r m i n g f i n i t e e l e m e n t s P1 o f t h e t y p e u s e d e a r l i e r i n

216


(CHAP. 5 )

Chapter I1 for approximating Stokes and Navier-Stokes problems (in 3 dimensions,

Th will

be a family of t e t r a h e d r a and the associated

d e g r e e s o f f r e e d o m will be the values taken at the centres of the + + + faces of these tetrahedra by the approximations Ah, vh of 2 and v).

(See MARROCCO 1 2 1 for more details on these nonconforming approximations and on the corresponding numerical experiments). We shall conclude this chapter with a few bibliographic comments. We have already pointed out that the nonlinear operator v

+

-v* ( 1

vv/ s-2vv)

of Section 3 , has appeared in mathematical models in g l a c i o l o g y and we refer the reader to PELISSIER [ 1 1 and the associated bibliography: the numerical augmented-Lagrangian treatment was introduced by GLOWINSKI-MARROCCO C11 and developed in MARROCCO C11. For the magnetostatic problem of Section 4, the reader may refer to GLOWINSKIMARROCCO C51. The potential-flow problems of Section 5 are classical, and the reader interested in the fluid-mechanical aspects of these problems may refer to LANDAU-LIFCHITZ C11. The viscoplasticity and elastoplasticity problems of Section 6 are treated, in particular, in DUVAUT-LIONS [l], and the numerical treatment of the corresponding variational inequalities is described in detail in GLOWINSKI-LIONSTREMOLIERES C11,[21 and GLOWINSKI C11,[21; the case of the torsion of a cylindrical bar with multi-connected cross-section is treated in GLOWINSKI-LANCHON [l]. In connection with minimal-surface problems we have already cited EKELAND-TEMAM C11 in which the concept of a g e n e r a l i s e d solution is discussed; once again, this is a classical problem which has given rise to numerous works.

CHAPTER V I

A P P L I C A T I O N OF A L G O R I T H M A L G Z TWO-DIMENSIONAL

TO A

E L A S T O P L A S T I C I T Y PROBLEM B.

Mercier

INTRODUCTION

W e s h a l l now c o n s i d e r a new example a r i s i n g from t h e Mechanics of

I n comparison t o t h e p r e c e d i n g examples, i n

Continuous Media.

p a r t i c u l a r Examples 1 and 2 of C h a p t e r 111, t h e s i t u a t i o n w i l l b e somewhat d i f f e r e n t :

i n f a c t the functional

(but d i f f e r e n t i a b l e ) ;

'noncoercive'

as

G

F

w i l l be n o n c o e r c i v e

i s l i n e a r , t h e problem ( P ) w i l l b e

and w e w i l l n o t be a b l e t o p r o v e t h e e x i s t e n c e o f a

s o l u t i o n i n i n f i n i t e dimensions.

I n c o n t r a s t , t h e d u a l of

which i n v o l v e s F*, t h e c o n j u g a t e of

F

,

is w e l l posed

(P),

a s F* i s

c o e r c i v e i n t h e example which w e a r e c o n s i d e r i n g , and t h i s a l s o i m p l i e s t h e d i f f e r e n t i a b i l i t y of F.

F u r t h e r m o r e , a s w e saw i n

C h a p t e r V i n c o n n e c t i o n w i t h o t h e r p r o b l e m s , i n o r d e r t o improve t h e r a t e of convergence o f a l g o r i t h m A L G Z ,

t h e r e i s an a d v a n t a g e t o be

g a i n e d i n t h i s example by c h o o s i n g a p e n a l t y t e r m i n t h e augmented Lagrangian norm on

1.

H

dr I

which i s n o t e q u a l t o t h e s q u a r e of t h e n a t u r a l

b u t which i s a s s o c i a t e d w i t h a n o t h e r q u a d r a t i c f o r m .

THE CONTINUOUS PROBLEM

W e c o n s i d e r a c o n t i n u o u s e l a s t o p l a s t i c medium h e l d f i x e d on one p a r t of i t s boundary. placement f i e l d

u

W e s e e k t h e stress f i e l d

u

and t h e d i s -

which a r e s e t up i n t h e c o n t i n u o u s medium when

i t i s s u b j e c t e d t o e x t e r n a l forces ( t h e above s i t u a t i o n i s i l l u s t r a t e d i n F i g u r e 1.1).

21 7

218

A 2-D

(CHAP. 6 )

ELASTOPLASTICITY PROBLEM

rigid part

F i g u r e 1.1

Let R

c

-

The c o n t i n u o u s medium b e f o r e a n d a f t e r a p p l i c a t i o n o f t h e external loads

( d = 2 o r 3 i n a p p l i c a t i o n s ) b e t h e open bounded domain

lRd

w i t h s u f f i c i e n t l y r e g u l a r boundary,

u

w e s e e k t h e stress f i e l d

ium;

H W e d e n o t e by

= { T =

r e p r e s e n t i n g t h e c o n t i n u o u s medi n t h e space

2 T . . E L ( a ) , T. = T . l 0 ; d p r o d u c t on IR

where

.

coefficients,

1.1

and

*

From K o r n ’ s inequality,

i s o f c l o s e d image.

A

o f IRd2 s a t i s f y i n g

d e n o t e t h e norm and t h e E u c l i d i a n i n n e r

T h i s automorphism t a k e s i n t o a c c o u n t t h e e l a s t i c i t y

s o t h a t t h e energy o f t h e c o n t i n u o u s medium i s w r i t t e n

W e d e n o t e by

G(v)

t h e f u n c t i o n e q u a l a n d o p p o s i t e t o t h e work

(SEC. 2 )

THY PROBLEM ( P )

219

v

done by t h e e x t e r n a l f o r c e s i n a d i s p l a c e m e n t uous medium ( G E = {T

E

, ];.Bv

H

V

E

of the contin-

d u a l o f V), and w e d e f i n e

V',

E

dx + G(v)

=

0

, V V E V) ,

which i s t e r m e d t h e s e t o f s t a t i c a l l y - a d m i s s i b l e s t r e s s f i e l d s . 2

F i n a l l y , w e d e n o t e by

C c IRd

t h e ( c 1 o s e d ) p Z a s t i c i t y convex s e t

and by

K

(1.2)

= {T E

H

, T(X)

E

C a.e. x ~ S 2 )

t h e set o f p l a s t i c a l l y a d m i s s i b l e stress f i e l d s .

u

states t h a t t h e stress f i e l d

Hencky's l a w t h e n

i s t h e s o l u t i o n of t h e o p t i m i s a t i o n

problem Min T1 j S 2 ( f l T ) - T dx T€WE

(1.3)

I f t h e condition

.

K n E # 0

i s s a t i s f i e d , t h a t i s , i f w e are

' b e l o w ' t h e L i m i t load,

t h i s problem a d m i t s a u n i q u e s o l u t i o n ,

(1.1).

and

(Note t h a t

depends l i n e a r l y on C

K G I

Since the set

are c Z o s e d ) .

E

from

E

i f t h e o r i g i n belongs t o t h e i n t e r i o r of

( O E I n t C ) , which w e s h a l l assume t o b e t h e case h e r e , and i f t h e

e x t e r n a l f o r c e s ( a n d h e n c e G ) are s u f f i c i e n t l y s m a l l , t h e n t h e c o n d i t i o n K n E # 0 w i l l be r e a l i s e d .

2.

THE PROBLEM ( P )

A s w e have s t a t e d above, w e s h a l l e q u i p

H

, not

with its natural

i n n e r product, but with an i n n e r product r e l a t e d t o t h e energy, namely: (p,q) = Jn(A-'p)*q

a n d w e d e n o t e by

II*II

dx

,

t h e a s s o c i a t e d norm.

With t h i s n o t a t i o n , t h e

1

energy t o be minimised i n (1.3) i s w r i t t e n as ~ I I A T I I

as : E = ITCH

so t h a t

, (AT,Bv)

+ G(v) = 0 , V V E V )

,

and t h e set

E

220

A 2-D


if

0

- (AT,Bv)}

Sup { - G ( v ) VE

v

6)

(CHAP.

T E E ,

=

otherwise,

+-

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

Min Sup {711A-rl12 - (A-r,Bv) T€K

Its dual

- G(v)I

.

V€v

( o b t a i n e d by p e r m u t a t i o n of t h e minimum and t h e supremum) i s

w r i t t e n , a f t e r a change of s i g n , a s

(2.1)

Inf { F ( B v ) vcv

+ G(v)}

KO = { q c H

, A-'qcKI

with

and (2.3)

T h i s d u a l i s c l e a r l y a problem of t h e form i n v e s t i g a t e d i n C h a p t e r 111; t h e f u n c t i o n F i s t h e c o n j u g a t e of t h e f u n c t i o n a l z1[ i q l l 2+ I K 0 ( q ) , and c o n s e q u e n t l y i t i s d i f f e r e n t i a b l e , b u t n o n c o e r c i v e i n g e n e r a l . Problem ( 2 . 1 )

t h e r e f o r e d o e s n o t always admit a s o l u t i o n i n i n f i n i t e

d i m e n s i o n s , even i f

K n E

i s nonempty (see t h e counter-example i n

MERCIER C 3 1 ) .

3.

A P P R O X I M A T I O N BY FINITE ELEMENTS

I n p r a c t i c e , w e a r e o b l i g e d t o r e d u c e t h e problem t o f i n i t e dimensions i n o r d e r t o s o l v e ( 1 . 3 ) duce a f a m i l y of t r i a n g u l a t i o n s h > 0;

with

h

given,

Gh

th ;

i n d e x e d by a p a r a m e t e r

i s a s e t of t r i a n g l e s c o v e r i n g

s a t i s f y i n g t h e following properties: t r i a n g l e s of

To t h i s end w e i n t r o -

(or (2.1)).

{ThIh,

let

T , T ' c "eh

Q

( I )

b e two d i s t i n c t

t h e n w e have

To s i m p l i f y t h e d e s c r i p t i o n we assume t h a t

Q

I

2 i s a polygon i n IR.

(SEC. 3 )

APPROXIMATION BY FINITE ELEMENTS

TnT' = @

,

T n T' = 1

o n e c o m p l e t e common e d g e

T n T' = 1

one common v e r t e x .

221

or

I n s h o r t , t h e s i t u a t i o n shown i n F i g u r e 3 . 1 i s f o r b i d d e n .

F i g u r e 3.1 : The p a r a m e t e r triangle in

h

Forbidden s i t u a t i o n

d e n o t e s , f o r example, t h e d i a m e t e r o f t h e l a r g e s t

rh .

W e t h e n d e n o t e by

Vh

c

V

t h e s p a c e of f i n i t e

e l e m e n t s c o n s t i t u t e d by p i e c e w i s e affine a n d c o n t i n u o u s d i s p Z a c e m e n t

f i e l d s o v e r e a c h t r i a n g Z e of p= W e d e n o t e by Hh c H t h e s u b s p a c e h' o f Hh composed of piecewise-constant t e n s o r s o v e r e a c h t r i a n g l e o f

c h ,so

B

t h a t t h e operator

maps

i n t o a p a r t of

Vh

Hh

.

We

then put Eh = { T ~ H C,

and t o a p p r o x i m a t e

,

(ATh,Bvh) + G(vh) = 0 , V v h E

h

-f

0

9

( 1 . 3 ) we c h o o s e t h e f i n i t e - d i m e n s i o n a l

which admits a u n i q u e s o l u t i o n when

v,}

ah

.r

problem

I t c a n b e shown t h a t

ah

-f

o

131).

(see MERCIER

The d e f i n i t i o n o f t h e d u a l o f

( 3 . 1 ) a g a i n d e p e n d s on c o n s i d e r i n g

(on KxV) t h e Lagranqian (3.2)

1

l l A ~ 1 1-~ (h~,Bv) - G(v)

where t h e d u a l v a r i a b l e

u

(uh

, i n f i n i t e d i m e n s i o n s ) i s i n t h i s case

t h e L a g r a n g e m u l t i p l i e r of t h e s t r e s s ions).

-r

E

A s t h e i n t e r i o r o f t h e convex s e t

E (T K

t

Eh

i n f i n i t e dimens-

i s empty i n

H

,

it

i s n o t p o s s i b l e t o d e d u c e t h e e x i s t e n c e o f u from t h i s remark. On K n Hh h a s a nonempty i n t e r i o r i n Hh , t h u s , from ROCKAFELLAR 1 4 , S e c t i o n 281, a s showing t h e e x i s t e n c e o f uh

t h e o t h e r hand,

222

A 2-D

l o n g as

Eh n

K

that

n Eh

( i n t K) i s nonempty

(which i s a s t r o n g e r c o n d i t i o n t h a n

b e nonempty b u t w h i c h i s t r u e i f t h e e x t e r n a l f o r c e s are

s u f f i f i c i e n t z y smaZZ).

which i s t h e d u a l o f

Incidentally,

also s a t i s f i e s

uh

( 3 . 1 ) and which i t s e l f i s e v i d e n t l y a p r o b l e m of

t h e t y p e i n v e s t i g a t e d i n C h a p t e r 111 ( n o t e t h a t

(2.1)

d i s c r e t i s e d form o f

u

(CHAP. 6 )


, but

is c l e a r l y a

(3.3)

t h e convergence of

uh

to

u

,

even i f

e x i s t s , is improbable).

APPLICATION OF ALGORITHM ALGZ

4.

The augmented L a g r a n g i a n i n t r o d u c e d i n C h a p t e r I11 i s o f t h e fc,rm:

z,(v,q,P)

= F(q)

+

G(v)

+

W e n o t e t h a t i n t h e p r e s e n t case LEMMA 4 . 1 : Let tu,p,Xl a s o l u t i o n of ( 2 . 1 ) , p = B u

where

u

G

+

:IIBv-qlI

2

.

is linear.

b e a s a d d l e p o i n t of

dr

Proof.

then

u

is

The f i r s t p a r t f o l l o w s d i r e c t l y f r o m Theorem 2 . 1 o f From S e c t i o n 2 . 3 of C h a p t e r 111, i t a l s o f o l l o w s t h a t

i s a s a d d l e p o i n t of t h e L a g r a n g i a n (see ( 2 . 1 6 1 , L(v,P)

I

I\v\l* +

IK0(v) - (~,Bv) - G(v)

u

= AT.

C h a p t e r 111)

I

( 3 . 2 ) a f t e r t h e s i m p l e c h a n g e of

which a g a i n g i v e s t h e L a g r a n g i a n variable

;

and f u r t h e r m o r e

is a s o Z u t i o n of t h e i n i t i a Z p r o b l e m (1.1).

C h a p t e r 111. {u,A}

(P,Bv-q)

H

Obviously, t h e e x i s t e n c e o f such a s a d d l e p o i n t ,

is doubtful i n the infinite-dimensional

case;

l i k e t h a t of

u

t h e problem w i l l be s o l v e d i n f i n i t e d i m e n s i o n s . I n view of t h e l i n e a r i t y of

G

,

,

however, i n p r a c t i c e ,

a l g o r i t h m ALGZ may b e w r i t t e n :

(SEC. 4)

APPLICATION OF ALG2

{po,ho]

E

HxHendP> 0 u r e g i v e n ;

Ipn,Xn}

E

HxH b e i n g g i v e n b y r e c u r r e n c e ,

(4.2),

the solution

un+'

of

r(Bu"+'

(4.2)2

the solution

pn+l

of

F'(pn+l)+rpn+l

(4.2)

An+'

3

223

= A"

+ p(Bun+I-p"+l)

The calculation of pn+l written out explicitly. we have

calculate

,Bv)+(Bv,Xn-rpn)+ =

X"

G(v) = 0 V v

+ rBun+l

E

V

,

,

.

in stage (4.2)2 of algorithm (4.2) can be In fact, in view of the definition of F ,

KO is the projection onto KO. From the definitions where n : H vo is local, and the nonlinear (2.3) and (1.2) of KO and of K , equation (4.2)2 can therefore be solved almost everywhere. It decomposes triangle by triangle for the approximate problem, since we have taken the precaution of choosing a space of piecewise-constant functions for Hh We can even solve (4.2)2explicitly with (4.3). -+

.

LEMMA 4.2 1 p"+l = --((I +r) $"

Proof.

nopn+l

.

Let

- no$n)

1

$" = I + r (Xn+rBun+l)

;

t h e n we h a v e

.

We show that @n is a convex combination of pn+l and = vopn+l, which then gives the Consequently we have

result. Stage (4.2)2 of algorithm (4.2) is linear since G is linear. finite dimensions, this consists of solving a linear system with matrix BtSB , where the matrix S is symmetric and positivedefinite relative to the inner product ( * , * ) .

In

Synopsis: We shall now prove the convergence of algorithm (4.2) Here we are in a situation which is the in the case where p = r r e v e r s e of that in Chapter 111: the problem (P) is noncoercive

.

and, in contrast, its dual ((1.3) in this case) is coercive. In Chapter 111, it was (P) which was coercive and its dual (in Examples

224

A 2-D


(CHAP. 6 )

1 and 2 o f S e c t i o n 1.1 a t l e a s t ) which w a s n o t . Conseque,itly, w e n { u ,pn} + { u , p i a n d c o u l d p r o v e only a weak c o n v e r g e n c e

proved t h a t

.

An

property f o r

Here, i n c o n t r a s t , w e s h a l l show t h a t

and t h a t w e h a v e o n l y a weak c o n v e r g e n c e p r o p e r t y f o r

An

+

{un,pni.

X We

have a somewhat a n a l o g o u s s i t u a t i o n f o r t h e a p p r o x i m a t i o n , s i n c e

uh

u

+

(isomorphic t o

regard t o

5.

.

uh

X) and s i n c e w e c a n n o t p r o v e a n y t h i n g w i t h

CONVERGENCE OF ALGORITHM ALG2

I f there e x i s t s a saddle point

THEOREM 5 . 1

of the

{u,p,Al

L a g r a n g i a n dr , t h e n a l g o r i t h m ( 4 . 2 ) c o n v e r g e s f o r p = r i n t h e { u n , p n ) r e m a i n s b o u n d e d and A n + A when following sense : n + + m .

Proof.

Xn+l =

so t h a t since

An

Fl(p"+l)

=

(5.2)

An

-

a r e l i n k e d by a s i m p l e r e l a t i o n .

and pn

A, f + l

pn +

A =

=

pn

rPa + l =

F'(p)

-

p and

x

+

11

in =

i.e.

-n+ 1 r(Bu ,Bv) =

-

un

(5.1)3.

Putting

u, we o b t a i n

r B?+'

w e also o b t a i n

-n -n (Bv,X -rp ) ,

by i n t r o d u c i n g t h e o p e r a t o r P : H

t h e image of B:

-

Furthermore,

we have

i n (5.112 i n view of

By s u b t r a c t i n g (5.1)1 f r o m ( 4 . 2 1 1

(5.3)

zr ,

i s a s a d d l e p o i n t of

{u,p,A1

S i m i l a r l y w e have xn

from ( 4 . 2 ) * w e o b t a i n ( s i n a e

By s u b t r a c t i n g ( 4 . 2 ) 3

r):

p =

+

I m B , which p r o j e c t s o n t o

6)

(SEC.

225

NUMERICAL A P P L I C A T I O N

t h e r e f o r e ( 5 . 2 ) becomes

x

-n+l

-n+l

+rp

and s q u a r i n g ( s i n c e

= (I-P)X"+r

P

and

7

P

a r e 2 orthogonal projectors)

I-P

I ~ P + ' I ~ r~( p+n + l , x n + ' ) +

I I ~ ~ I I ~

r 2 1 l p n + l l 1 5~ I I X ~ r2 ~ ~ ~ +

which p r o v e s , s i n c e by t h e m o n o t o n i c i t y o f F ' ,

( p n + l l In+') 2

0, that

A n and pn remain bounded, and t h e r e f o r e un a l s o r e m a i n s bounded i n view o f

(5.4).

-n+l Furthermore ( p since F'(q) =

, 5;nfl)

+

0 when n

+

-t

m,

which can be w r i t t e n ,

q , i n t h e form

IT

n+ 1

(TOP

-

TOP, p

n+ I

-p)

+

0

.

Now n+ 1

n+ 1

n+ I

w e have

i n view of t h e p r o p e r t i e s o f p r o j e c t i o n o n t o a convex s e t : t h u s proved t h e convergence of

6.

An

to

A

when

n

-f

+

m.

NUMERICAL APPLICATION

6.1

D e s c r i p t i o n o f t h e m e c h a n i c a l problem

W e have c o n s i d e r e d t h e problem of t h e bending o f an e n c a s t r e d beam of l e n g t h

L!

and o f t h i c k n e s s

2a,

s u b j e c t e d t o a s h e a r f o r c e Fo

(see F i g u r e 6 . 1 ) .

-

+a

lF0

0

a -a

Figure 6 . 1

-

E n c a s t r e d beam s u b j e c t t o a s h e a r f o r c e

Fo.

~

226

(CHAP. 6 )


A 2-D

W e assume t h a t t h e w i d t h o f t h e beam ( i n t h e d i r e c t i o n

orthoOx3 g o n a l t o t h e p l a n e o f t h e f i g u r e ) i s s u f f i c i e n t l y l a r g e and w e w i l l

then be j u s t i f i e d i n s t u d y i n g t h e p l a n e - s t r a i n

p r o b l e m i n s t e a d of

t h e t h r e e - d i m e n s i o n a l p r o b l e m , by a s s u m i n g t h a t t h e d i s p l a c e m e n t f i e l d d e p e n d s o n l y on

x1

strain tensor

can then be w r i t t e n :

= Bu

E

and

x2

and s a t i s f i e s

The

u 3 = 0.

W e s h a l l see t h a t t h e same d o e s n o t a p p l y f o r t h e s t r e s s t e n s o r . Assuming t h e medium t o b e i s o t r o p i c , w e h a v e

where

tr(e)

t r ( e ) 6 + 2ue

A

A-le

e,

denotes t h e trace of t h e t e n s o r

t h e Lam6 c o n s t a n t s ,

6

and

A

i s t h e Kronecker t e n s o r .

1-1

and

are

W e have

a d o p t e d t h e von Mises p l a s t i c i t y c r i t e r i o n , and t h e p l a s t i c i t y convex s e t i s t h e r e f o r e w r i t t e n

where

i s a coefficient c h a r a c t e r i s t i c of t h e material, 1 T ( T ~ T t r ( T ) 6 ) , and I*

a

t h e d e v i a t o r of t h e t e n s o r E u c l i d i a n norm of

IR9.

T~

I

is

is the

An e x p l i c i t c a l c u l a t i o n shows t h a t

r

vle

0 2 .

D

1

Zfle

fi

(leD(

I

$ ( e ) E Kotr(e) +

k A s may b e e x p e c t e d ,

F

$ ‘ ( e ) = KO t r ( e ) d

X

f

2 3

.

, +

a.e. xeS2 rnin(Z?J,

,

a33 # 0.

9)

otherwise.

w e have

where

k f i )eD -

I eDI

S i n c e i n t h e e q u i l i b r i u m state w e have general

k f i

z?J

is differentiable:

F’(e) = $‘(e(x))

where KO =

-

5

a

= @‘(E(u)),

w e see t h a t i n

The p r o b l e m i s n o n e t h e l e s s t w o - d i m e n s i o n a l ,

t h e t h i r d component o f t h e d i s p l a c e m e n t i s z e r o .

and

(SEC.

6)

6.2

NUMERICAL APPLICATION

221

Choice of c o n s t a n t s

We have chosen

1.33 , X 30

p =

we s h a l l n o t d e f i n e h e r e ) ,

( i n I m p e r i a l u n i t s which

=

a = 2

and

chosen w e r e uniform, w i t h mesh i n t e r v a l s We have chosen ml

R = 20. R ; and

1 .

The t r i a n g u l a t i o n s 2a -

(see Figure 6 . 2 ) .

m2

= 10 and m2 = 6 f o r an i n i t l a l mesh ( t h i s g i v e s

dim Vh = 1 4 0 ) and ml

=

1 4 and m2 t

With t h e s e d a t a t h e m a t r i x

+

in the e l a s t i c case ( 5 =

B SB

m)

=

8 f o r a second mesh (dim Vh = 2 5 2 ) .

of t h e l i n e a r system t o be s o l v e d

i s b a d l y c o n d i t i o n e d and it i s n e c e s s a r y

t o perform t h e c a l c u l a t i o n s i n d o u b l e p r e c i s i o n .

m

2

I

I

\

m

t

1

F i g u r e 6 . 2 - F i r s t t r i a n g u l a t i o n used (m, = 10, m2 = 6 ) . The m a j o r i t y of i t e r a t i v e methods a r e i n e f f i c i e n t f o r s o l v i n g t h i s l i n e a r s y s t e m , w i t h t h e e x c e p t i o n of t h e c o n j u g a t e - g r a d i e n t method and i t s v a r i a n t s .

Even s o , t h i s l a t t e r method o n l y s t i l l converges

i n a number of i t e r a t i o n s c l o s e t o t h e number o f v a r i a b l e s , and t h i s means t h a t it i s n o t c o m p e t i t i v e w i t h d i r e c t methods.

It is quite

p o s s i b l e t h a t t h i s c o n c l u s i o n would need t o be r e c o n s i d e r e d i f a s u i t a b l e change of t h e i n n e r p r o d u c t w e r e made, s i n c e t h i s may have a preconditioning e f f e c t .

~ have For t h e e l a s t o p l a s t i c c a s e , w e have chosen k = ~ 6 . f O - and compared a l g o r i t h m ALG2 w i t h two o t h e r s t a n d a r d a l g o r i t h m s .

6.3

G r a d i e n t method

W e put mising by (6.1)

@

@ ( v ) = F(Bv)

+

G(v);

s o l v i n g (P,)

which i s d i f f e r e n t i a b l e o v e r Vh; t h e i n n e r p r o d u c t chosen on V h ,

becomes p o s i t i v e o r s m a l l e r t h a n a s u f f i c i e n t l y - s m a l l p o s i t i v e constant.

The problem i s f u n d a m e n t a l l y i l l - c o n d i t i o n e d :

( e l a s t i c ) l i n e a r c a s e we c a n s u c c e s s f u l l y c o n s t r u c t d i r e c t i o n s and s o l v e t h e problem;

N

in the conjugate

however, w e a r e u n a b l e t o do

t h i s i n t h e ( e l a s t o p l a s t i c ) nonlinear case. I n t h e l i g h t of r e c e n t r e s u l t s (see AXELSSON C11, CONCUS-GOLUBO ~ L E A R Y [l]) it would a p p e a r t h a t t h e i d e a o f c h a n g i n g t h e i n n e r

product

improvement. methods. R

on Vh c o u l d i n t h i s c a s e l e a d t o a s i g n i f i c a n t T h i s i s t h e b a s i c i d e a b e h i n d s o - c a l l e d preconditioning

Two o p t i o n s a p p e a r a v a i l a b l e :

( a r i s i n g from t h e i n n e r p r o d u c t

I

e i t h e r t o take a s t h e matrix t h a t of t h e e l a s t i c problem,

o r t o u s e an i n c o m p l e t e Cholesky d e c o m p o s i t i o n o f t h i s m a t r i x f o l l o w i n g an i d e a due t o M E I J E R I N K - V A N DER VORST C11, which would s i g n i f i c a n t l y reduce t h e c o s t of each i t e r a t i o n .

The u s e of some

p r e c o n d i t i o n i n g i s i n o u r o p i n i o n e s s e n t i a l i f t h e performance o f t h e conjugate-gradient

method i s t o be improved.

It should be noted

t h a t t h e u s e o f an a u x i l i a r y o p e r a t o r i n t h e g r a d i e n t method, j u s t a s i n t h e p e n a l i s a t i o n - d u a l i t y methods s t u d i e d i n t h e p r e s e n t book, h a s an a n a l o g o u s e f f e c t which amounts t o changing t h e m e t r i c o f t h e s p a c e .

6.5

Choice of t h e p a r a m e t e r s f o r a l g o r i t h m ALGZ

W e have chosen f o r t h e t e r m i n a t i o n t e s t f o r a l g o r i t h m ALGZ:

When t h e f o r c e Fo i s s u f f i c i e n t l y s m a l l , t h e problem i s p u r e l y e l a s t i c and t h e c h o i c e

r

=

1

i s optimal.

I n t h e e l a s t o p l a s t i c domain, t h e

2 30

choice

r < 1

r

seem t o be

(CHAP. 0 1

ELASTOCPLASTICITY PROBLEM

A 2-D

accelerates t h e convergence: a good c h o i c e would 1 or r = I n f a c t , i n t h i s case, f r o m t h e v e r y

=

?j.

f i r s t i t e r a t i o n w e a r r i v e a t a s o l u t i o n two o r t h r e e t i m e s l a r g e r t h a n t h a t of t h e e l a s t i c p r o b l e m , a n d w h i c h i s a p p r o x i m a t e l y of t h e o r d e r

o f m a g n i t u d e of t h e e l a s t o p l a s t i c s o l u t i o n .

of

r

is not crucial.

However, t h e c h o i c e

The f o l l o w i n g t a b l e s summarise t h e r e s u l t s

obtained :

Table 6 . 1

-

0.2

0 . I01

I33

0.4

0.203

245

0.5

0.270

41 1

-

-

0.55

0.322

547

42

35

3

0.6

0.404

514

-

-

-

R e s u l t s f o r ml

"1 "2 tl,t2

=

6 , m2 = 10 ( 1 4 0 v a r i a b l e s )

v e r t i c a l d i s p l a c e m e n t o b s e r v e d a t x = R , x2=-a, 1 number of i t e r a t i o n s o f t h e c o n j u g a t e - g r a d i e n t method

:

number o f i t e r a t i o n s of ALGZ

:

r e s p e c t i v e machine t i m e s ( i n s e c o n d s on IBM 370/168).

Conjugate gradient

U

-

:

:

U

FO -

(elastic)

-

nl

I

ALG2

r a d i e n t ( w i t h auxliary operator)

I

n3 1

0.2

0.116

181

0.4

0.236

371

-

0.5

0.346

645

82

0.55

0.460

800

-

0.6

0.680

1200

215

0.65

1 .oo

-

250

T a b l e 6 . 2 - R e s u l t s for m = 8 , m2 = 1 4 (252 v a r i a b l e s ) ( c o m p a r e d w i t h T a b l e 6 . 1 we have added t h e v a l u e s of r u s e d i n ALGZ and n3 ( t h e

number of i t e r a t i o n s ) and t (machine t i m e r e l a t i v e t o t h e g r a d i e n t method w i t h a u x i l i a r y o p e r a 3 o r ) ) .

(SEC. 7 )

DISCUSS I O N

A

8o ..

number o f iterations

60

..

40

_.

20

.. 0.5

F i g u r e 6.3

7.

-

231

1.5

1.

2.

r

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 of ALG2 a s a r ( c a s e ml = m2 = 3 ) . f u n c t i o n of

DISCUSSION F or t h e problem c o n s i d e r e d h e r e ,

a l g o r i t h m ALG2 i s t w o t o t h r e e

t i m e s f a s t e r t h a n t h e g r a d i e n t method w i t h a u x i l i a r y o p e r a t o r and t e n t o t w e n t y t i m e s f a s t e r t h a n t h e c o n j u g a t e - g r a d i e n t method w i t h o u t p r e conditioning.

The good p e r f o r m a n c e o f b o t h o f t h e f i r s t two methods

may b e a t t r i b u t e d t o t h e f a c t t h a t , e v e n t h o u g h w e h a d a l i n e a r s y s t e m t with m a t r i x B SB t o s o l v e a t each i t e r a t i o n , s i n c e t h i s m a t r i x w a s

f i x e d a n d o f banded s t r u c t u r e , i t was f a c t o r i s e d o n c e and f o r a l l C h o l e s k y ' s method) a t t h e s t a r t o f t h e a l g o r i t h m i n t o a p r o d u c t LL L b e i n g l o w e r t r i a n g u l a r a n d o f banded s t r u c t u r e .

(by

t

,

A t each i t e r a t i o n

w e t h e r e f o r e h a v e t o s o l v e o n l y t w o l i n e a r s y s t e m s w i t h m a t r i x L , and t h i s i s extremely rapid.

A s regards stage (4.2)2 of algorithm (4.21,

t h i s c a n a l s o b e performed v e r y r a p i d l y s i n c e it decomposes t r i a n g l e

by t r i a n g l e . I t i s now a p p r o p r i a t e t o e x p l a i n t h e i m p o r t a n c e o f t h e c h o i c e o f

t h e i n n e r product adopted i n S e c t i o n 2. Lagrangian

dr

I n f a c t , i n t h e augmented

t h e p e n a l t y t e r m i s t h e s q u a r e o f t h e norm o n H .

A t a n y r a t e , i t w a s u n d e r s u c h an a s s u m p t i o n t h a t t h e p r o o f s i n

C h a p t e r I11 w e r e p e r f o r m e d .

However, i f w e t a k e f o r H t h e ' n a t u r a l '

i n n e r p r o d u c t o n H: (p,d

=

1,

p . 4 dx

t h e n t h e p e r f o r m a n c e of a l g o r i t h m ALG2 d e t e r i o r a t e s s i g n i f i c a n t l y . I n t h i s case t h e m a t r i x

BtSB

w i l l not, i n f a c t , be t h e matrix of

t h e u n d e r l y i n g e l a s t i c problem, b u t t h a t o f an e l a s t i c problem w i t h d i f f e r e n t c o e f f i c i e n t s , which d o e s n o t b e a r s u c h a c l o s e r e l a t i o n t o

2 32

A 2-D ELASTOPLASTICITY PROBLEM

(CHAP. 6 )

the (nonlinear) problem being treated. The same applies also for the gradient algorithm with auxiliary operator. We cannot over-emphasise for such problems the importance of the

imev ' p ~ b h m t~3 the >Pam on whit\ w~ are wX!xin>

-

thiak is, < - thy vocabulary of the gradient method, of choosing a good auxiliary operator or, in the vocabulary of the conjugate-gradient method, of choosing a good 'preconditioning'. This concept is also important for the penalisation-duality algorithms: the choice of the penalty term is at our disposal and it is necessary to take the one closest to the nonlinear problem being considered, as has already been pointed out in Chapter V.

CHAPTER VII A P P L I C A T I O N TO T H E N U M E R I C A L S O L U T I O N O F THE T W O - D I M E N S I O N A L F L O W OF I N C O M P R E S S I B L E V I S C O P L A S T I C F L U I D S D.

1.

GENERAL NOTES.

B e g i s , R.

Glowinski

SYNOPSIS

The p r e s e n t c h a p t e r i s b a s e d l a r g e l y on BEGIS C21 and GLOWINSKILIONS-TREMOLIERES C2

,

Appendix 61.

It extends S e c t i o n 6 . 1 of

C h a p t e r V r e l a t i n g t o t h e 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.

Here we s h a l l be c o n s i d e r i n g t h e much more c o m p l i c a t e d pro-

blem of t h e u n s t e a d y flow o f a f l u i d of t h e above t y p e i n a bounded two-dimensional

cavity.

W e s h a l l see t h a t t h e i n t r o d u c t i o n of a

s t r e a m f u n c t i o n e n a b l e s t h e problem c o n s i d e r e d t o b e r e d u c e d t o a

p a r a b o l i c v a r i a t i o n a l i n e q u a l i t y of o r d e r 4 w i t h r e s p e c t t o t h e s p a c e variables.

W e s h a l l t h e n examine t h e a p p r o x i m a t i o n of t h e above

problem by methods u s i n g m i x e d f i n i t e e l e m e n t s

a p p r o x i m a t i o n ) and f i n i t e d i f f e r e n c e s

(for the spatial

( f o r the approximation i n t i m e ) .

W e s h a l l t h e n show t h a t t h e s e a p p r o x i m a t e problems can be s o l v e d by

t h e augmented L a g r a n g i a n methods o f C h a p t e r 111, t h e a l g o r i t h m s t h e r e b y o b t a i n e d g e n e r a l i s i n g t h o s e of C h a p t e r V , S e c t i o n 6 . 1 , t i n g t o t h e f l o w of a Bingham f l u i d i n a c y l i n d r i c a l d u c t .

rela-

Finally,

w e s h a l l p r e s e n t some n u m e r i c a l r e s u l t s o b t a i n e d by t h e above methods, and t h i s w i l l d e m o n s t r a t e some o f t h e p r o p e r t i e s o f Binqham f l u i d s .

FORMULATION O F BINGHAM FLOWS USING THE VELOCITY AND THE PRESSURE

2.

-v

Let =

R

{vl,v2}

b e a bounded domain i n IR2 w i t h r e g u l a r boundary d e n o t i n g an IR2-valued f u n c t i o n , w e p u t

233

r.

With

234

2-D

we a s s o c i a t e with

z

FLOW OF VISCOPLASTIC FLUIDS

(CHAP. 7 )

t h e inonempty) a f f i n e space

I n t h e f o l l o w i n g d i s c u s s i o n , w e s h a l l n e g l e c t t h e e f f e c t s of

i n e r t i a ( a s s o c i a t e d w i t h t h e t r i l i n e a r form (.,.,-) of C h a p t e r 11, Section 4.1, relation ( 4 . 5 ) ) ; t h i s l e a d s us (see DUVAUT-LIONS C1, C h a p t e r V I 1 ) t o model t h e u n s t e a d y f l o w i n satisfying

u

=

5

on

r,

R

o f a Binqham f l u i d

by

(2.9)

\

~ ( 0 =)

y o € H,

_f€

L 2 (0,T;VA).

W e recall t h a t i n (2.9) v i s t h e v i s c o s i t y of t h e f l u i d , 4 i s t h e t h r e s h o l d of p l a s t i c i t y

(yield stress)

f i s a d e n s i t y of e x t e r n a Z f o r c e s .

DUVAUT-LIONS C 1 , C h a p t e r 6 1 p r o v e s 1

g :

u n i t normal v e c t o r on

r,

(for

z=

9)

t h e e x i s t e n c e and

p o i n t i n g outwards from R.

(SEC. 3 )

235

BINGHAE4 FLOWS V I A A STREAM FUNCTION

#

t h e case

uniqueness o f a s o l u t i o n o f problem ( 2 . 9 ) ;

Q, with

z

s a t i s f y i n g ( 2 . 7 ) , may b e t r e a t e d a n a l o g o u s l y . W e h a v e assumed i n t h e above t h a t 5 ( = g l r ) i s i n d e p e -

Remark 2 . 1 : n d e n t of t ;

t h e r e a r e no f u r t h e r d i f f i c u l t i e s i n t r e a t i n g t h i s case

n u m e r i c a l l y by t h e methods t o b e d i s c u s s e d s u b s e q u e n t l y .

3.

FORMULATION O F BINGHAM FLOWS USING A STREAM FUNCTION I n t h i s s e c t i o n we s h a l l adopt t h e following t w o simplifying

assumptions:

R i s simply c o n n e c t e d

(i)

(ii)

2

I

= UI

n o n e t h e l e s s Remark 2 . 1 s t i l l h o l d s f o r s i t u a t i o n s i n which ( i ) a n d / o r I f w e c o n f i n e o u r a t t e n t i o n t o two-dirnensionaZ

(ii)a r e n o t s a t i s f i e d .

flows, we can e l i m i n a t e t h e c o n d i t i o n

7-s=

by i n t r o d u c i n g a s t r e a m f u n c t i o n d e f i n e d

0 i n a natural

manner

( 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 ) by (3.1)

The c o n d i t i o n (3.2)

JI

(3.3)

-an =

W e shall take L e t v_

E

Vo

u n i q u e l y by

(3.4)

W e recall t h a t

=

2

= Q

on

Const.

on

o on

li, = 0 ;

r on

r

implies

r,

,

r,

w h i c h f i x e s t h e c o n s t a n t m e n t i o n e d above.

we associate with

v_

t h e function

@

E

2

Ho(R)

defined

2 36

2-D

I n view o f

(3.1),

FLOW OF VISCOPLASTIC FLUI.7S

( 3 . 4 ) we can r e d u c e ( 2 . 9 )

bolic variational inequaZity

(CHAP.

t o t h e f o l l o w i n g para-

(of o r d e r 4 with respect t o t h e space

variables) :

/

~ i n dii, E L2 (O,T;Ho(R)) 2 n Lm(O,T;Hb(Q))

such t h a t

(3.7)

where

(3.9)

Remark 3 . 1 : (3.10)

I n f a c t w e have

i(@, , @ 2 ) =

jt@1A@2

dx

2

W l A2 E Ho(W

.

I n t h e f o l l o w i n g w e s h a l l b e u s i n g ( 3 . 8 ) and (3.10) s i m u l t a n e o u s l y .

*

I f , i n (2.9),

( f , v ) = \,fv

dx t h e n

=

af2 af, -. ax, ax2

7)

(SEC. 4)

4.

APPROXIMATION OF STEADY-STATE PROBLEM

237

APPROXIMATION OF THE STEADY-STATE PROBLEM

4.1

Synopsis.

Formulation of the steady-state problem

Before approximating ( 3 . 7 )

-

by means of a mixed finite-element

method - we shall first study the approximation of the corresponding steady-state problem, i.e. the following elliptic variational inequality of order 4: ( 3 )

where a(*,-) and j(.) are defined by (3.8), (3.9); ( 4 . 1 ) is equivalent to the rninimisation problem

where, in ( 4 . 2 ) ,

we note that

we have

We shall assume in the following that f

H-'(n);

E

in fact there

would be no difficulty in treating the case in which ( f , $ ) = i,f

A$ dx V-$

E

H:(Q)

; f

E

L2(n).

Since the bilinear form a(-,.) is H:(O)-eZZiptic

(i.e. coercive), 2

and the functional j(.) is convex and continuous on H o ( R ) , with 4 + (f,@)Zinear and continuous, then it is a classical result (see, for example, L I O N S 113) that ( 4 . 1 1 , (4.2) admits a unique s o l u t i o n . 4.2

Approximation of ( 4 . 1 ) , method

We shall approximate ( 4 . 1 ) ,

(4.2) by a mixed finite-element

(4.2) here by a mixed finite-element

method suggested by MIYOSHI C11. The objective is to reduce the approximation to that of a problem in which we only have to perform 1 2 2 the discretisation of H ( a ) and L ( a ) instead of discretising H ( a ) 3

We shall henceforth omit the symbol

-.

238

(CHAP. 7 )

FLOW OF VISCOPLASTIC FLUIDS

2-D

which i s a much more c o m p l i c a t e d t a s k .

To d o t h i s , w e f i r s t

i n t r o d u c e a weakened v a r i a t i o n a l f o r m u l a t i o n o f o u r problem.

The

new v a r i a t i o n a l problem t h u s o b t a i n e d p o s s e s s e s a unique s o l u t i o n which c o i n c i d e s w i t h t h a t of conditions.

(4.1),

under f a i r l y u n r e s t r i c t i v e

(4.2)

For a g e n e r a l p r e s e n t a t i o n of t h i s approach, t h e r e a d e r

may r e f e r t o GIRAULT-RAVIART Thus, s u p p o s e I$

E

2

[ll.

and p u t f o r 1

Ho(Q)

5

i,j

5

2,

(4.4)

W e then have, f o r a l l v

E

H1 ( a ) ,

(4.5)

C o n v e r s e l y i f I$ $

E

2 Ho(Q)

and

E

z

1

and

Ho(Q)

and

q?

z = {zij}15i,j52 a r e r e l a t e d by ( 4 . 4 ) .

s a t i s f y (4.5)

then

Thus w r i t i n g

(4.6)

where w e h a v e a,B

E

10,lC w i t h

c1

+

B = 1, t h e n

and p u t t i n g

t h i s l e a d s u s t o r e p l a c e p r o b l e m (4.1), ( 4 . 2 )

by t h e f o l l o w i n g problem:

T h i s p r o b l e m , which i s e q u i v a l e n t t o t h e i n i t i a l p r o b l e m , o f f e r s a c o n s i d e r a b l e advantage as f a r as t h e d i s c r e t i s a t i o n i s concerned, s i n c e it r e q u i r e s o n l y t h e a p p r o x i m a t i o n o f t h e s p a c e s H 1 (Q) and

.

L2 ( Q )

(4.4).

The d i s c r e t e v a r i a b l e s are t h e n r e l a t e d by a weak f o r m o f

(SEC. 4 )

W e s h a l l assume i n t h e f o l l o w i n g t h a t

IR2 ;

let

then put

i

voh

Cl

i s a convex polygon i n

b e a s t a n d a r d f a m i l y of t r i a n g u l a t i o n s o f

{Zhlh

Vh = {V h E C o ( E ) , v h l K E P k

(4.10)

239

APPROXIMATION O F STEADY-STATE PROBLEM

=

Ivh vh,

h

=

a$h (q

VKEVh}

o on r l = Vh

aVh

7+

.

We

,

1 n Ho(n),

wh )avhd xj

R

xi

x = 0

Vvhc Vh,

1

15

i,j

5 2 )

.

I t may b e n o t e d t h a t t h e a p p r o x i m a t i o n s o f H ( Q ) and L

2

formed h e r e u s i n g t h e same s p a c e o f f i n i t e e l e m e n t s .

This procedure

(a) a r e

per-

i s w e l l a d a p t e d t o t h e p r e s e n t s i t u a t i o n , b u t it i s n o t t h e o n l y

means p o s s i b l e .

F i n a l l y t h e approximate problem w i l l o b v i o u s l y be:

T o conclude, we n o t e t h a t t h e f a c t t h a t w e a r e u s i n g

'ij

-

a2$ axiax j

a s an a u x i l i a r y v a r i a b l e means t h a t t h e p r o c e s s i s

p a r t i c u l a r l y w e l l adapted t o t h e treatment of t h e nondifferentiable

t e r m appearing i n t h e f u n c t i o n a l t o be minimised;

it is f o r t h i s

r e a s o n a l s o t h a t t h e a b o v e mixed method h a s b e e n c h o s e n . 4.3

S o l v a b i l i t y of p r o b l e m ( 4 . 1 1 )

The f o l l o w i n g t h e o r e m i s p r o v e d i n GLOWINSKI-LIONS-TREMOLIERES C2, Appendix 6 , S e c t i o n 4.4.31: The approximate problem

THEOREM 4 . 1 :

(4.11)

a d m i t s o n e and o n l y

one s o l u t i o n . The s o l u t i o n o f

(4.11)

by a l g o r i t h m s o f t h e ALGl o r ALG2 t y p e w i l l

f o r m t h e s u b j e c t of S e c t i o n 6 l a t e r i n t h e p r e s e n t c h a p t e r .

4.4

Convergence of t h e a p p r o x i m a t e s o l u t i o n s

W e 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 t o t h e cases k = 1, 2

4 . 1 below f o r k s o l u t i o n s when h

3);

2

+

(see Remark

c o n c e r n i n g t h e convergence o f t h e approximate

0, we have:

240

2-D FLOW OF VISCOPLASTIC FLUIDS

THEOREM 4.2:

S u p p o s e t h a t when h

bounded b e l o w , u n i f o r m l y i n h, b y go

+

0 the angles o f

0;

(CHAP. 7 )

‘Eh r e m a i n

suppose a l s o t h a t t h e

condition Max

h(K)

KECh

(4.12)

Min

( w h e r e h(K)

h(K)

5‘1

V%,7

i n d e p e n d e n t o f h,

l e n g t h o f t h e l o n g e s t s i d e o f KI i s s a t i s f i e d .

=

We

t h e n have (4.13)

lim

{$h,sh~ = { ~ l , s ~s t r o n g l y i n

kt0

{$h,sh}

where

~I ~ (x n( ~ ) ~(n))~,

i s 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 p r o b l e m (4.11),

$

i s t h a t o f t h e c o n t i n u o u s p r o b l e m (4.1), (4.2) and where s =

{sij’lsi, j22

with

s.. 17

=

~

a*$

axiax j

We refer the reader to GLOWINSKI-LIONS-TREMOLIERES c2, Appendix 6, Section 4.4.41 for the proof of Theorem 4.2. Remark 4.1: We have assumed above that k = 1, 2; in fact, similar convergence results could be obtained for approximations based on finite elements of order k 2 3 , but given the l i m i t e d r e g 4 2 u l a r i t y of the solutions ( $ ,d H (n) X Ho(Q) in general) the use of elements of such a high order is not justified. 4.5

Approximation using numerical integration

From a practical point of view it is necessary to use a numerical integration procedure in order to approximate the functional J ( * , * ) in (4.9), (4.11); we shall restrict our attention to the case k = 1. Let Ch denote the set of the vertices of Bh; we approximate ?on Vh the inner product induced by L2 ( a ) , i.e.

where, in (4.14), m ( P ) is the sum of the areas of the triangles which have P as a common vertex. In view of (4.14) we shall in fact use in (4.11) the functional Jh(-,-)defined (if k = 1) by

(SEC. 5 )

APPROXIMATION OF EVOLUTION PROBLEMS

where fh is an approximation of f. Wh defined by ( 4 . 8 ) , we shall, if k

=

241

Similarly, instead of using 1, use Wh defined by

Using the relations ( 4 . 1 6 ) it is easy to express z . . (P), YP E Ch llh explicitly as a function of the values taken by Qh on Ch; in fact the matrix associated with the discrete inner product (*,*)h in In the numerical solution, it is therefore possible Vh is diagonal. we refer the reader to BEGIS C21 f p to eliminate the variable zh; further details.

APPROXIMATION OF THE EVOLUTION PROBLEM (3.7)

5.

5.1

Semi-discretisation with respect to time

Let k = At ( > 0) denote one step in the time discretisation; then approximate (3.7) by the following i m p Z i c i t s c h e m e (where $n 2 $(nk) and where the have been omitted) :

-

/

f o r $" known, d e t e r m i n e Qn+l b y solving

(5.1)

The use of the above semi-discrete scheme has thus enabled us to reduce the solution of the evolution problem (3.7) to that of a sequence of elliptic variational inequalities, equivalent to the

we


242

following sequence of minimisation problems (with n

(CHAP. 7 )

2

0):

Find $n+l E Ht(a) such that (5 2)

I J"+' k ($"+I)

5

JE+l (@) W$ E Ht(0)

where

The discretisation of ( 5 . 2 ) , ( 5 . 3 ) by the mixed finite-element method of Section 4 is treated in Section 5.2 below. 5.2

Complete discretisation of (3.7)

The notation is the same as that in Section 4.2; we approximate = j l 0 by $: E Voh and the semi-discrete scheme (5.1) by the following: $O

With the function for n = O l l l .

..,

$: E voh known, obtain the minimisation problem

~ $ ~ + l , s ~ +bly } solving,

where ( j ( . ) still defined by (4.7)):

It can easily be shown that problem ( 5 . 4 ) , ( 5 . 5 ) admits a unique solution; furthermore, the comments in Section 4.5 concerning the use of numerical integration are still valid for problem ( 5 . 4 ) , (5.5). With regard to the convergence, as h, k + 0, of the above approximate solutions to the solution of problem ( 3 . 7 ) , we refer the reader to

SOLUTION O F ( 4 . 1 ) ,

(SEC. 6 )

SOLUTION O F ( 4 . 1 )

6.1

243

[21.

GLOWINSKI-LIONS-TREMOLIERES

6.

(5.2)

( 5.2)

BY AUGMENTED-LAGRANGIAN METHODS

Synopsis

I n t h i s s e c t i o n w e s h a l l show t h a t i t i s p o s s i b l e t o s o l v e t h e

steady state problem ( 4 . 1 ) , o r t h e s e q u e n c e o f p r o b l e m s ( 5 . 2 ) (obt a i n e d by t h e semi-discretisation in time o f p r o b l e m ( 3 . 9 ) ) , by means o f augmented L a q r a n q i a n methods w h i c h f a l l w i t h i n t h e g e n e r a l framework d e f i n e d i n C h a p t e r 111.

W e s h a l l confine our attention

t o t h e case which i s continuous with respect to the space variables, b u t t h e q e n e r a l i s a t i o n t o problems which are approximate i n space a n d t i m e d o e s n o t 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 y

( a p a r t from t h e

f a c t t h a t t h e f o r m a l i s m which h a s t o b e c o n s t r u c t e d i s e x t r e m e l y cumbersome). 6.2

The model p r o b l e m .

Problems ( 4 . 1 ) blem

1

and ( 5 . 2 ) 2

Find $ c H o ( R )

I n t r o d u c t i o n o f a n auqmented L a g r a n q i a n l e a d us t o c o n s i d e r t h e m i n i m i s a t i o n pro-

such that

with

and y

2

0 (y = 0 f o r t h e s t e a d y - s t a t e

a r i s e s from problem ( 5 . 2 ) ) . of

(6.1),

p r o b l e m , y = l/k i f

(6.1)

The p r i n c i p 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

( 6 . 2 ) a r i s e s f r o m t h e nondifferentiable f u n c t i o n a l

T o g e t round t h i s d i f f i c u l t y (as w e l l as t o s i m p l i f y t h e d i s c r e t i s a t -

i o n o f t h e p r o b l e m ) w e s h a l l a d o p t t h e framework o f S e c t i o n ( 4 . 2 ) c o n s i d e r a mixed v a r i a t i o n a l f o r m u l a t i o n of p r o b l e m ( 6 . 1 ) ,

(6.2).

and

244

2-D

we again put

With j ( - ) s t i l l d e f i n e d by ( 4 . 7 )

+

(CHAP. 7 )

FLOW O F VISCOPLASTIC FLUIDS

gj(z)

-

(f,@),

so t h a t it i s c l e a r t h a t ( 6 . 1 )

,

(6.2)

can be w r i t t e n

I n o r d e r t o a d a p t t h e g e n e r a l method of C h a p t e r I11 t o t h i s case,

it i s n a t u r a l t o i n t r o d u c e h e r e a s u p p l e m e n t a r y v a r i a b l e 2 q = {qiIiTl E (L2(n))2 related t o z by t h e l i n e a r e q u a t i o n s

I t i s t h i s c o n s t r a i n t ( 6 . 4 ) t h a t w e s h a l l be t r e a t i n g by p e n a l i s a -

t i o n and d u a l i t y , v i a t h e i n t r o d u c t i o n of a n a u g m e n t e d L a g r a n g i a n . S o as t o a l l o w t h e n o t a t i o n of C h a p t e r I11 t o b e u s e d h e r e , w e p u t :

W e then define f o r r > 0, { @ , z } Laqrangian

dr:

V x H x H

-+

t

v,

q

E

HI

u

E

H

t h e augmented

JR by

The s o l u t i o n o f p r o b l e m ( 6 . 1 )

,

(6.2) t h e n r e d u c e s t o s e e k i n g a

SOLUTION OF (4.1), (5.2)

6)

(SEC.

245

saddle point of dr on V x H x H . We could also have considered in the above the decomposition associated with F(q) = T BV 1,Iql' dx + 191 dx,

1

pi,

In the following sections we-shall have t o solve problems correspondon V, z and 1-1 being fixed. This ing to the minimisation of &, The minimisation leads to solving a linear m i x e d p r o b l e m in $ , z . remarks made earlier relating to the discretisation and the use of numerical integration still apply, and we can consider the solution of such a problem as being standard. Application of ALGl to seekinq a saddle point of

6.3

dr

In view of Section 6.2, it is natural to solve problem (6.11, (6.2) by using algorithm ALGl of Chapter 111; we then obtain the following:

xo 6 H

(6.7)

then, for

pn

t

H

n

then

(

( ~7 - ( n )2 ) given,

=

0, An

2

An+'

E

H being k n o w n , d e t e r m i n e {IJIn,sn}

E

V and

by

tIJI",s">E

v,

pn

E

H,

In view of the convergence results established in Chapter 111, Section 4, we have: THEOREM 6.1:

o n V XH XH; (6.10)

dr

a d m i t s a Saddle p o i n t {{$,sl,p,A}

O < P < 2r,

w e have f o r all A o

(6.12)

Suppose that

then if

E

H

lim P" = p strongly in (L n-

2

(n)) 2 ,

246


lim An = n-

(6.13)

A"

weakly i n

(CHAP. 7 )

(~'(n))',

It is clear that, once again, the essential difficulty with this approach lies in the fact that system (6.8) has to be solved at each this problem can be iteration; in view of the structure of

4,

solved by a block over-relaxation method like that described in Chapter 111, Section 3.2. A s far as the choice of p is concerned, numerical experiments indicate once again that the optimal value lies close to p = r. 6.4

On variants of algorithm (6.6)

-

(6.8)

The first of these variants is algorithm ALG2 which has already been studied in some detail in Chapter I11 and used extensively in In the case under consideration here, we Chapters IV, V and VI. obtain the following: (6.14)

then, f o r n

pn,{$",s"?

{$-I

t

, s - ' ~E V ,

AO

c H

0, {$n-l, sn-'1

and An+'

are given,

E

V, An c H b e i n g k n o w n , we d e t e r m i n e

s u c c e s s i v e l y by

(6.15)

(6.16)

(6.17)

It follows from Chapter I11 that the convergence results of Theorem 6.1 still hold if instead of (6.10) we have (6.18)

o 0 the rigid state i s attained i n a

f i n i t e t i m e w h i c h grows p r o g r e s s i v e l y s m a l l e r a s

g

becomes l a r g e r ;

t h i s a c c o r d s w i t h p h y s i c a l i n t u i t i o n and c a n b e j u s t i f i e d t h e o r e t i c ally. Methods f o r t h e n u m e r i c a l s i m u l a t i o n o f t h e t w o - d i m e n s i o n a l

flow

of Bingham f l u i d s may b e f o u n d i n FORTIN C21, BEGIS C11, t h e s e b e i n g b a s e d on d i f f e r e n t p r i n c i p l e s velocity-pressure

(including t h e d i r e c t use of t h e

f o r m u l a t i o n of S e c t i o n 2 ) ;

numerous n u m e r i c a l

r e s u l t s a r e a l s o g i v e n i n t h e s e r e f e r e n c e s , w h i c h a re i n a g r e e m e n t w i t h t h o s e p r e s e n t e d h e r e (see a l s o C h a p t e r V I o f GLOWINSKI-LIONSTREMOLIEFES Cll,

C2l).

2 50

2-D

(CHAP.

FLOW O F V I S C O P L A S T I C F L U I D S

7)

BINGHAM FLUID PARAMETERS : T h r e s h o l d of p l a s t i c i t y Viscosity E x t e r n a l force: f l ( x 1 , x 2 , t )

0.5 1.0

=O.O

,f2(x1,x2,t)=8.0

BOUNDARY C O N D I T I O N S : N o r m a l c o m p o n e n t of v e l o c i t y T a n g e n t i a l c o m p o n e n t of v e l o c i t y

0.0 0.0 a t %=O, 1;5=l 1.0 a t x = O

2

I N I T I A L CONDITION:

I n i t i a l velocity

0.0

STEADY STATE

M a x . v a l u e of s t r e a m f u n c t i o n V a l u e of s t r e a m f u n c t i o n on l i n e 1 D i f f e r e n c e b e t w e e n successive l i n e s

llllil

0.928E-1

0.510E-3 0.100E-1

R i g i d zones LABORIA

D.BEGIS

(SEC.

7)


251

BINGHAM F L U I D PARAMETERS : T h r e s h o l d of p l a s t i c i t y Viscosity fl(x1,x2,t) E x t e r n a l force:

=O.O,

1.0 1.0 f2(x1,x2,t)

=o.o


0.0 0.0 a t y = O , l;? =1 1.0 a t x 2 = 0



0.0

STEADY STATE

M a x . v a l u e of stream f u n c t i o n V a l u e of stream f u n c t i o n on l i n e 1 D i f f e r e n c e between s u c c e s s i v e l i n e s

III/I/

0.8643-1 0.510E-3 0.100E-1


Figure 7.2

( g = 1)

D. B E G I S

252

2-D

(CHAP.


7)

BINGHAM F L U I D PARAMETERS : T h r e s h o l d of p l a s t i c i t y Viscosity fl(x1,x2,t) =O.O, E x t e r n a l force:

2.5 1.0 f*(x1,x2,t)

=o.o

BOUNDARY C O N D I T I O N S : N o r m a l c o m p o n e n t of V e l o c i t y T a n g e n t i a l c o m p o n e n t of V e l o c i t y

0.0 0.0 a t y = O , 1.0 a t 3 ' 0

l;3=1



0.0

STEADY STATE

M a x . v a l u e of s t r e a m f u n c t i o n Value of s t r e a m f u n c t i o n on l i n e 1 D i f f e r e n c e b e t w e e n successive l i n e s

0.737E-1 0.510E-3

0.100E-1 LABORIA

D. B E G I S

(SEC.

7)


253

BINGHAM F L U I D PARAMETERS : T h r e s h o l d of p l a s t i c i t y Viscosity E x t e r n a l force: fl(x1,x2,t)

=O.O,

5.0 1.0 f2(x1,x2,t)

=O.O


0.0 0.0 a t %=O, 1.0 at 3 ' 0

1;3='l



0.0

STEADY STATE

M a x . v a l u e of s t r e a m f u n c t i o n V a l u e of s t r e a m f u n c t i o n on l i n e 1 D i f f e r e n c e b e t w e e n successive l i n e s

llllil

0.616E-1 0.510E-3 0.100E-1


D. B E G I S

254

2-D


(CHAP.

7)

BINGHAM F L U I D PARAMETERS : T h r e s h o l d of p l a s t i c i t y Viscosity E x t e r n a l force: fl(x1,x2,t)

10.0 1.0 =O.O,

f 2 ( x 1 , x 2 , t ) =O.O


0.0 0.0 a t x = 0 , l ; % = l 1 1.0 a t x 2 = 0



0.0

STEADY S T A T E

Max. v a l u e of s t r e a m f u n c t i o n V a l u e of stream f u n c t i o n on l i n e 1 D i f f e r e n c e b e t w e e n successive l i n e s

llllli

0.500E-1 0.500E-3 0.100E-1

Rigid zones LABORIA

D.BEGIS

(SEC.

7)

NUMERICAL E X P E R I M E N T S

4

255


CHAPTER V I I I A P P L I C A T I O N TO T H E S O L U T I O N OF F I N I T E NONLINEAR E L A S T I C I T Y PROBLEMS J.F.

1.

Bourgat, R.

GENERAL NOTES.

Glowinski, P.

Le T a l l e c

SYNOPSIS

The aim o f t h i s c h a p t e r i s t o show t h a t t h e g e n e r a l p r i n c i p l e s of d e c o m p o s i t i o n - c o o r d i n a t i o n s t u d i e d i n Chapter I11 have a range o f a p p l i c a t i o n c o n s i d e r a b l y w i d e r t h a n t h a t c o n s i d e r e d i n C h a p t e r 111, which a r o s e from C o n v e x A n a l y s i s and Monotone O p e r a t o r s ;

in fact,

w e s h a l l show i n t h i s c h a p t e r t h a t t h e s e p r i n c i p l e s and t h e a s s o c i a t e d a l g o r i t h m s w i l l l e a d t o t h e development of i t e r a t i v e methods, s t i l l r e l a t e d t o A L G 1 , ALG2

(and p o s s i b l y A L G 3 ) , which a r e e x t r e m e l y

e f f i c i e n t f o r s o l v i n g n o n - c o n v e m v a r i a t i o n a l p r o b l e m s a r i s i n g from

N o n l i n e a r E l a s t i c i t y , i n which t h e d i s p l a c e m e n t s and/or t h e s t r a i n s a r e l a r g e r e l a t i v e t o t h e more u s u a l c o n t e x t of L i n e a r E l a s t i c i t y , where t h e y a r e assumed t o be ' v e r y s m a l l ' :

t h i s f i n i t e c h a r a c t e r of

t h e d i s p l a c e m e n t s and/or t h e s t r a i n s j u s t i f i e s t h e t i t l e of t h e p r e s e n t c h a p t e r which must n e c e s s a r i l y be c o n s i d e r e d a s merely an i n t r o d u c t i o n t o a v a s t and d i f f i c u l t s u b j e c t which i s a s y e t s t i l l r e l a t i v e l y undeveloped i n t e r m s o f n u m e r i c a l methodology. The p r i n c i p l e s and methods mentioned above w i l l be a p p l i e d t o t h e s o l u t i o n of two t y p e s of problem a r i s i n g from F i n i t e N o n l i n e a r E l a s t -

icity;

namely:

( i ) I n S e c t i o n 3 , t h e l a r g e - d i s p l a c e m e n t c a l c u l a t i o n of e q u i l i b r i u m c o n f i g u r a t i o n s f o r a c l a s s of i n e x t e n s i b l e and f l e x i b l e p i p e l i n e s .

( i i ) I n S e c t i o n 4 , t h e mechanical b e h a v i o u r of i n c o m p r e s s i b l e ,

elastic

m a t e r i a l s of Mooney-Rivlin t y p e . The r e s u l t s o f n u m e r i c a l e x p e r i m e n t s w i l l be p r e s e n t e d and d i s c u s s e d f o r b o t h t h e above c a s e s . Although t h e methods used have t h e i r formal o r i g i n i n Chapter 111, 251

258

(CHAP. 8 )

FINITE N O N L I N E A R ELASTICITY PROBLEMS

i t seems d e s i r a b l e t o r e p e a t

(without proof) t h e e s s e n t i a l points

r e l a t i n g t o t h e p r i n c i p l e s and methods o f C h a p t e r 111, so a s t o

w e do t h i s i n

improve t h e r e a d a b i l i t y o f t h e p r e s e n t c h a p t e r : S e c t i o n 2 , below.

T o c o n c l u d e t h i s i n t r o d u c t i o n w e would l i k e t o p o i n t o u t t h e f o l l o w i n g s e l e c t i o n o f works i n w h i c h v a r i o u s a s p e c t s o f t h e a b o v e nonl i n e a r e l a s t i c i t y problems a r e developed:

111, GLOWINSKI-LE TALLEC 1 1 1 , 1 2 1 , LE TALLEC

BOURGAT-DUMAY-GLOWINSKI

[11, 1 2 1 , GLOWINSKI-LE TALLEC-RUAS 1 1 1 , RUAS 111 and e s p e c i a l l y TALLEC 111 o n which t h e c o n t e n t s o f t h i s c h a p t e r

BOURGAT-GLOWINSKI-LE a r e heavily based.

2.

DECOMPOSITION OF V A R I A T I O N A L PROBLEMS.

ASSOCIATED ALGORITHMS.

I n t h i s s e c t i o n w e s h a l l b r i e f l y summarise t h e v a r i o u s c o n s i d e r a t i o n s which w e r e d e v e l o p e d i n d e t a i l i n C h a p t e r 111:

t h i s w i l l allow

t h e r e a d e r who i s more p a r t i c u l a r l y i n t e r e s t e d i n t h e a p p l i c a t i o n s t r e a t e d i n t h i s s e c t i o n t o t a c k l e it d i r e c t l y without f i r s t having t o r e a d C h a p t e r I11 (which c a n t h e r e f o r e b e p o s t p o n e d t o a s e c o n d r e a d i n g ) .

2.1

A f a m i l y of v a r i a t i o n a l p r o b l e m s

I n t h e following we s h a l l restrict our a t t e n t i o n t o real H i l b e r t

w e t h u s l e t V and H b e two s u c h s p a c e s , e q u i p p e d w i t h t h e

spaces;

norms and i n n e r p r o d u c t s

11'11,

((*,*))

respectively.

proper,

IR u (2.1)

1.1,

and Let B

(',-),

d(V,H)

E

and l e t

F

{+m),

respectively; dom (G) ndom

G b e two convex,

and

l o w e r semi-continuous f u n c t i o n a l s from

H

and

V

into

w e assume t h a t

(FOB)

# 0,

where

w i t h a s i m i l a r d e f i n i t i o n f o r dom (FOB). €4,

F, G , a b o v e , t h e minimisation p r o b l e m :

W e a s s o c i a t e with V, H,

(SEC. 2)

DECOMPOSITION OF VARIATIONAL PROBLEMS

(p)

1

Find

ucV

J(u) 5 J(v)

where J

V

:

(2.2)

259

such t h a t Vv

E

V,

is defined by

+

J(v) = F(Bv) + G(v).

The functional J ( - ) and problem (P) have a very special structure; thus it is natural to think in terms of using methods which take advantage of this structure. The majority of the considerations which follow can be applied to variational problems of the form Remark 2.1:

where f

E

V' (the d u a l space of V) and where A1 (resp. A2) are mono-

t o n e o p e r a t o r s (possibly multivalued) from H into H' (the dual of H )

(resp. from V into V'); the operator A = B'oAIOB + A2 from V into V' is not in general the gradient (or subgradient ('I)of a functional J (B' denotes the t r a n s p o s e of the operator B). For numerical results relating to these generalisations we r e f e r the reader to LIONS-MERCIER C11, GABAY C11 ( s e e also Chapter IX of the present book and GLOWINSKI-LIONS-TREMOLIERES 12, Appendix 21). If we assume that in addition to (2.1) we also have (2.4)

lim

llvll++

J(v) =

+m

then ( P ) admits a solution which is u n i q u e if J is s t r i c t l y c o n v e x . Remark 2.2:

The applications to Nonlinear Elasticity in Sections 3 and 4 actually relate to n o n - c o n v e x minimisation problems. 2.2

A

decomposition principle

We shall now briefly summarise the developments of Chapter 111; See EKELAND-TEMAM C11 for this concept.

FINITE NONLINEAR ELASTICITY PROBLEMS

260

we thus define W

V

c

8)

by

H

x

(CHAP.

Problem ( P ) i s e q u i v a l e n t t o

Find

{u,pl

E

W

such t h a t

(=)

{v,qj

j(u,p) 5 j(v,q)

6

w

with

Remark 2 . 3 :

The new p r o b l e m

(IT)

c l e a r l y resembles m i x e d f o r m u l a -

t i o n s , t o t h e e x t e n t t h a t t h e r e l a t i o n Bv

-

q = 0 suggests t h e intro-

d u c t i o n o f a Lagrange m u l t i p l i e r .

Remark 2 . 4 : ring

(T)

P r o b l e m s ( P ) and

(IT)

a r e e q u i v a l e n t , b u t by c o n s i d e -

w e h a v e i n some ways s i m p l i f i e d t h e n o n l i n e a r s t r u c t u r e o f

( P ) though a t t h e c o s t o f a new v a r i a b l e (2.7)

q

and of t h e r e l a t i o n

Bv-q = 0 ;

i n f a c t , s i n c e r e l a t i o n ( 2 . 7 ) i s l i n e a r , some v e r y e f f i c i e n t t e c h -

w e s h a l l t r e a t it i n t h e f o l l o w i n g

n i q u e s e x i s t f o r t r e a t i n g it:

work by making s i m u l t a n e o u s u s e o f p e n a l i s a t i o n and L a g r a n g e m u l t i -

p l i e r methods, t h r o u g h t h e medium of a s u i t a b l y - c h o s e n a u g m e n t e d Lagrangian. 2.3 Let

(2.8)

An augmented L a g r a n g i a n a s s o c i a t e d w i t h ( I T )

r > 0;

we define

dr:

= F(q)+G(v)+

&,(v,q,p)

V x H

x

5 1 Bv-qI 2

+

I t i s shown i n C h a p t e r 111, S e c t i o n 2 . 2 ,

p o i n t of

(2.9)

ic,

I

on

v

{u,p,X}

+(U,P,!J)

x H 6

VX

5

H

+

by

(U,Bv-q)

t h a t i f {u,p,A1 i s a s a d d l e

x H (i.e. H X

H and

il,(U,P,X)

‘il,(v,q,lJ)

y {v,q.vl

E

v x H X H)

9

DECOMPOSITION OF VARIATIONAL PROBLEMS

(SEC. 2) then

{u,p} is a solution of

(with p 2.4

=

(T),

i.e.

u

261

is a solution of (P)

Bu).

A first algorithm for solving ( P )

To solve (P) and ( n ) we shall determine the saddle points of by a duality algorithm of the type considered in GLOWINSKI-LIONSTREMOLIERES [I, Chapter 21,[2, Chapter 2 and Appendix 21. Such an algorithm applied to the solution of (2.9) is algorithm ALG1, introduced in Chapter 111, Section 3.1; that is: E H , given

(2.10)

then f o r n

t 0,

Xn b e i n g known, d e t e r m i n e un,pn,Xn+l b y

I

(2.12)

=

An

.

+ p(Bu"-p")

As regards the c o n v e r g e n c e of (2.10) - (2.12), it is shown in Chapter 111, Section 4, that under very reasonable assumptions on F, B, G and if (2.13)

O 0 ) by

(3.18)

I f we replace

d.

by

Ar,

t h e conditions f o r

l e a d t o t h e following v a r i a n t of

dr

(3.8) - (3.10):

t o be s t a t i o n a r y

INEXTENSIBLE FLEXIBLE PIPELINES

(SEC. 3 )

\EI ~ ( 4 -)

a dx d - 2 - 2 ds (A z)- r - ( ( x ' +y' ds

-1)

dx ) ds

275

= 0

on lO,LC,

(3.19)

boundary c o n d i t i o n s

( +

EI ; ( 4 )

-

-

(A

-

2)- r a ( x ' ~ + Y ' ~ -2I ) ds

) =

-

pg

on IO,L[,

(3.20) +

boundary c o n d i t i o n s

x ~ ~ + ; * ~ -=I o on I O , L C ,

(3.21)

which i s c l e a r l y e q u i v a l e n t t o s y s t e m ( 3 . 8 ) - ( 3 . 1 0 ) . I t i s c l e a r t h a t t h e a b o v e a p p r o a c h , u s i n g a n augmented L a g r a n g i a n ,

f u r t h e r c o m p l i c a t e s a p r o b l e m which i s a l r e a d y c o m p l i c a t e d enough i n

i t s own r i g h t s i n c e ( 3 . 1 9 ) (3.8) - ( 3 . 1 0 ) ; than

(3.8)

-

-

( 3 . 2 1 ) i s even 'more n o n l i n e a r '

furthermore,

(3.10).

than

( 3 . 1 9 ) - ( 3 . 2 1 ) a r e 'more c o u p l e d '

I f we t a k e

A

= 0

i n (3.19),

( 3 . 2 0 ) , and i f

w e do n o t c o n s i d e r ( 3 . 2 1 ) , we o b t a i n t h e n e c e s s a r y c o n d i t i o n s o f o p t i m a l i t y f o r a p r o b l e m d e d u c e d f r o m ( 3 . 2 ) by p e n a t i s a t i o n o f t h e 2 + y V 2- 1 = 0. w condition x' Methods u s i n g d i r e c t m i n i m i s a t i o n on m a n i f o l d s

(ii)

I n s t e a d of

'relaxing'

t h e constraint (3.1), i.e. xI2 + y t 2

-

1 = 0,

by Lagrange m u l t i p l i e r s a n d / o r p e n a l i s a t i o n , w e c a n a t t e m p t t o m i n i -

mise

J

d i r e c t l y on t h e m a n i f o l d d e f i n e d by ( 3 . 1 ) , a s i s done i n

GABAY [11 a n d LICHNEWSKY 111 ( b y t h e o p t i m a l d e s c e n t o r t h e c o n j u g a t e

g r a d i e n t method).

However, a l t h o u g h t h e s e methods a r e e x t r e m e l y

e l e g a n t i n t h e i r u n d e r l y i n g p r i n c i p l e s and a r e v e r y e f f i c i e n t f o r c e r t a i n p r o b l e m s i n t h a t t h e y p e r f o r m t h e m i n i m i s a t i o n on t h e g e o d e s i c s o f t h e m a n i f o l d , t h e y are i n p r a c t i c e somewhat d i f f i c u l t t o implement i f t h e number o f c o n s t r a i n t s i s v e r y l a r g e ;

t h i s is certainly the

case f o r t h e d i s c r e t e v a r i a n t s o f ( 3 . 2 ) d e s c r i b e d i n S e c t i o n 3 . 4 . The methods which w e s h a l l d e s c r i b e i n S e c t i o n 3.5.2

differ quite

c o n s i d e r a b l y f r o m t h e t w o t y p e s o f method m e n t i o n e d a b o v e ; t h e y do

nonetheless

h a v e a c e r t a i n number o f c h a r a c t e r i s t i c s i n common w i t h them,

i n t h e sense t h a t :

(CHAP. 8 )


They a r e a l s o b a s e d o n t h e u s e o f an a u g m e n t e d L a g r a n g i a n ;

in

t h e p r e s e n t case, however, t h e c o n s t r a i n t s t o b e t r e a t e d by Lagrange m u l t i p l i e r s and p e n a l i s a t i o n a r e l i n e a r , and t h i s constitutes a substantial simplification. W e r e t a i n t h e n o t i o n o f d i r e c t m i n i m i s a t i o n on a m a n i f o l d which

( i n a certain sense) is associated with the inextensibility condition 3.5.2

(3.1).

S o l u t i o n o f p r o b l e m ( 3 . 2 ) b y an a u g m e n t e d L a q r a n g i a n method -

In s p i t e o f t h e f a c t t h a t p r o b l e m ( 3 . 2 ) i s non-convex,

t o s o l v e it

w e s h a l l a p p l y t h e methodology d e v e l o p e d i n C h a p t e r I11 a n d summarised i n Section 2 of t h e present chapter. I n t h e p r e s e n t c o n t e x t , problem (P ) i s problem ( 3 . 2 ) , t h a t i s

IX,YIE

{F

L

LOC min

8

+ pg

y dsl

,

0

d d e f i n e d by ( 3 . 3 ) .

with

I

L

!o(x"2+y"2)ds

W e t h e n have t h e f o l l o w i n g v e r y o b v i o u s

proposition : Proposition 3.1:

The p r o b l e m (P) i s e q u i v a l e n t t o t h e p r o b l e m

with

if= I { X , y , p , q l E

(3.22)

z x

2

(L (0,L))

2

, X"P,

y"q,

p2+q2 =

11,

2

i s t h e s u b s p a c e of H ( 0 , L ) x H2 ( 0 , L ) d e f i n e d b y t h e b o u n d a r y 6. c o n d i t i o n s s p e c i f i e d for I x , y 3 i n t h e d e f i n i t i o n o f

where

2

The n e x t s t e p i s t o ' r e l a x ' t h e f u n c t i o n a l r e l a t i o n between { x , y l and { p , q j by i n t r o d u c i n g ( w i t h r > 0 ) t h e f o l l o w i n g a u g m e n t e d L a g r a n g -

ian:

I N E XTEN SI BLE FLEXIBLE PIPELINES

(SEC. 3)

277

By a n a l o g y w i t h t h e convex s i t u a t i o n d e s c r i b e d i n C h a p t e r 111 and

- - - - - -

i n S e c t i o n 2 of t h i s c h a p t e r , w e suppose t h a t {x,y,p,q,X,l~} i s a

( l o c a l ) saddle point f o r

s

(3.24)

kr 2

2

,

= { ~ p , q E (~L ( 0 , ~ ) ) p2+q2 = I

it can then be proved t h a t

- -

2

(L (0,L))2, where

on Z x S x

{x,?}

E

6 ,

a.e.1 ;

x' = P I ,

that

-

-

y ' = q , and t h a t

A,~J a r e L a g r a n g e m u l t i p l i e r s f o r t h e e q u a l i t y c o n s t r a i n t s x'

y'-q

-

p = 0,

= 0.

I n view of t h e s e p r o p e r t i e s ,

dr

Lagranqian

it i s t h u s n a t u r a l t o extend t o t h e

d e f i n e d by (3.23) t h e i t e r a t i v e methods o f C h a p t e r

111, t h e d e s c r i p t i o n o f which i s r e p e a t e d i n S e c t i o n s 2.4 and 2.5;

t h e c o r r e s p o n d i n g a l g o r i t h m s a r e d e s c r i b e d i n S e c t i o n s 3.5.3 and

3.5.4 below. 3.5.3

A f i r s t i t e r a t i v e method u s i n g

dr

T h i s i s i n f a c t a l g o r i t h m A L G l o f C h a p t e r 111, S e c t i o n 3.1, a n d

of S e c t i o n 2.4 o f t h e p r e s e n t c h a p t e r .

Using t h e n o t a t i o n o f S e c t i o n

3.5.2 a b o v e , t h i s a l g o r i t h m i s w r i t t e n : (3.25)

A',~O

are given;

t h e n for n t 0 , a s s u m i n g t h a t n n n n Xn+l n+l x r~ r~ r q r P by

An

and

pn

a r e known, c a l c u l a t e

(3.26)

(3.27)

The n o n - t r i v i a l p a r t of a l g o r i t h m (3.25) s o l v i n g of p r o b l e m (3.26);

-

(3.27)

i s obviously t h e

w e c a n a g a i n p r o c e e d by b l o c k r e l a x a t i o n

(see S e c t i o n 2.4, Remark 2.5) by m i n i m i s i n g a l t e r n a t e l y w i t h r e s p e c t t o {x,y} and

we obtain

-

{p,q);

i f we confine t h i s t o a single inner i t e r a t i o n

with a s u i t a b l e i n i t i a l i s a t i o n - t h e v a r i a n t of algorithm

278


(3.25)

-

3.5.4

(3.27) d e s c r i b e d i n S e c t i o n 3.5.4

(CHAP. 8 )

below.

A second i t e r a t i v e method u s i n g

I n t h i s c a s e t h e a l g o r i t h m w e u s e i s ALG2 o f C h a p t e r 111, S e c t i o n and of S e c t i o n 2.5 o f t h e p r e s e n t c h a p t e r ; t h a t i s :

3.2,

X1 D

(3.28)

then f o r

1

0

0

,x ,y

~ J

given;

assuming t h a t x

n t 1,

IXn+l,

a t e { p n , q n } , { x n , y n } and

(3.29)

i

{pn,qn}

Find

E

S

n-1

{xn,yn} € 2

in,

un

a r e known, c a l c u l -

vn+l} by

such t h a t

er(xn-l ,yn-l ,Pn,qn,Xn,lJn)

Find

, y n-1 ,

5 dr(X

such t h a t

V {q,p} n-1

S

E

n-1 ,y ,p,q,Xn,un),

67 {x,y} € 2 ,

(3.30) Zr(X

n

n n n n n n n n n rY YP 3 9 ,X ,lJ ) SILr(X,Y,P .q ,A ,lJ )

,

(3.31)

Remark 3.4:

- (3.31) i s given i n [I, S e c t i o n 6 . 2 . 1 1 i n which u s e i s made o f a

A v a r i a n t of algorithm (3.29)

BOURGAT-DUMAY-GLOWINSKI

r e l a x a t i o n p a r a m e t e r i n t h e c a l c u l a t i o n o f {xn,yn}.

Remark 3.5:

W e could a l s o use, instead of

v a r i a n t deduced from a l g o r i t h m ( 2 . 2 1 )

-

(3.28) - (3.31)

(2.25) i n Section 2.5,

,

the

Remark

2.6. From t h e p r a c t i c a l p o i n t o f view i t i s e s s e n t i a l t o h a v e a l e n t , more e x p l i c i t , formulations c t i o n , i t may b e n o t e d t h a t

f o r ( 3 . 2 9 ) and ( 3 . 3 0 ) .

In t h i s dire-

(3.30) i s i n f a c t e q u i v a l e n t t o t h e

f o l l o w i n g f o u r t h - o r d e r boundary-value

(3.32) (+

equiv-

boundary c o n d i t i o n s ,

system:

(SEC.

3)


279

(3.33) + boundary c o n d i t i o n s .

I f t h e b o u n d a r y c o n d i t i o n s are g i v e n by ( 3 . 4 ) o r ( 3 . 5 ) , w e can t h e n solve

( 3 . 3 2 ) a n d ( 3 . 3 3 ) i n d e p e n d e n t l y o f e a c h o t h e r , and f u r t h e r m o r e

t h e i r d i s c r e t i s e d v e r s i o n s a r e l i n e a r s y s t e m s w i t h t h e same m a t r i x ; t h i s matrix is sparse,

of

n

i f

r

s y m m e t r i c , p o s i t i v e d e f i n i t e and i n d e p e n d e n t

i s fixed;

i n t h i s c a s e w e can p e r f o r m a C h o l e s k y

f a c t o r i s a t i o n o n c e a n d f o r a l l , and a t e a c h i t e r a t i o n o f (3.31) w e s h a l l have t o s o l v e o n l y systems t o determine

4

W e s h a l l now s t u d y t h e s o l u t i o n of

(3.29);

t o o b t a i n {pn,qn3 it

on C0,Ll t h e t w o - d i m e n s i o n a l

problem

(3 .34-)

+

2 q ( s ) = 1, ( 3 . 3 4 ) r e d u c e s t o

(3.35)

where, i n ( 3 . 3 5 )

, we

-

{xn,yn}.

is necessary t o solve, a.e.

2 However, s i n c e p ( s )

(3.28)

sparse, t r i a n g u l a r , well-posed

have p u t

minimisation

280

(CHAP. 8 )


Remark

3.6:

W e h a v e j u s t shown t h a t

(3.30) i s a w e l l - p o s e d

problem i f t h e b o u n d a r y c o n d i t i o n s a r e g i v e n by ( 3 . 4 ) o r ( 3 . 5 ) . Problem ( 3 . 2 9 ) i s a l s o w e l l p o s e d i f { X n ( s ) , Y n ( s )

?(s)

= Yn(s)

= 0 , t h e e n t i r e c i r c l e p2

+

1 #

{O,O};

if

q2 = 1 is a solution.

In

a c t u a l f a c t , i n a 2 1 t h e n u m e r i c a l e x p e r i m e n t s which w e h a v e p e r f o r m e d ,

w e have n o t e d t h a t t h i s p r o b l e m a t i c a l s i t u a t i o n n e v e r arose i f was s u f f i c i e n t l y l a r g e ;

Remark 3 . 7 :

r

it i s p o s s i b l e t o a c c o u n t f o r s u c h b e h a v i o u r .

I n a c c o r d a n c e w i t h Remark 2.7 o f S e c t i o n 2 . 5 ,

we

-

s o l v e t h e p r o b l e m i n { x , y ) a t t h e s e c o n d s t e p of a l g o r i t h m ( 3 . 2 8 )

as t h i s p r o b l e m i s a s s o c i a t e d w i t h a s t r o n g l y e l l i p t i c o p e r a t o r ( i n c o n t r a s t t o t h e p r o b l e m i n { p , q } which i s a s s o c i a t e d w i t h a non-mono t o n e , mu 1 t i v a t u e d o p e r a t o r ) (3.31)

I

.

3.6

Numerical e x p e r i m e n t s

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 and d i s c u s s t h e n u m e r i c a l r e s u l t s d b t a i n e d i n s o l v i n g a number o f t e s t p r o b l e m s ;

w e r e f e r t o BOURGAT-

DUMAY-GLOWINSKI C11 f o r f u r t h e r n u m e r i c a l t e s t s , a n d i n p a r t i c u l a r 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 p r o b l e m s i n which t h e r e are w a t e r c u r r e n t s a c t i n g on t h e p i p e l i n e , a n d o f dynamic p r o b l e m s

(oscillations, for

example) concerni n g t h i s p i p e l i n e . 3.6.1

D e s c r i p t i o n of t h e t e s t p r o b l e m

Me c h a n i c a 1 p a rame t e rs : E I = 7000 N m L ,

p = 7 . 6 7 Kg/m,

L = 32.6 m.

Boundary c o n d i t i o n s : x ( 0 ) = y(O), x ' ( 0 ) = 1, y ' ( 0 ) = 0, y ( L ) = 0, x ' ( L ) = 1, y ' ( L ) = 0.

x(L) = 1,2,3,4,5,6,7,8; 3.6.2

Further information concerning t h e numerical s o l u t i o n

F o r a p p r o x i m a t i n g ( 3 . 2 ) we u s e d a u n i f o r m d i s c r e t i s a t i o n of C 0 , L l w i t h h = L/50

and 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 S e c t i o n 3.4.

The

a p p r o x i m a t e p r o b l e m s w e r e s o l v e d by a d i s c r e t i s e d v a r i a n t o f a l g o r i t h m (3.28)

-

(3.31) with

=

r

=

took a d i s c r e t i s e d v e r s i o n of

5 0 000.

For t h e termination test we

(SEC. 3 )


281

(3.37)

3.6.3

P r e s e n t a t i o n o f t h e numerical r e s u l t s

(i) We show in Figure 3 . 3 , for x(L) = 2,3,4,5,6 which were obtained as follows:

the numerical results

We first calculated the solution corresponding to x(L) = 6 by initialising in ( 3 . 2 8 ) with

XI

= 0,

=

(3.38)

xo(s)

= 3(1-cosiTi),

y o ( s ) = -3

siniTSL

,

which corresponds to a s e m i c i r c l e with diameter AB; as the length of this semicircle is 3 7 ~= 9 . 4 2 4 ..., we can see that the initial L.

O D

-L

D

-6

-12.

-16o

I

I

Figure 3 . 3

I

I

(x(L) = 2,3,4,5,6)

I

1 I

282


(CHAP.

s o l u t i o n l i e s a l o n g way from t h e s o l u t i o n r e q u i r e d ;

convergence

w a s r e a c h e d i n 166 i t e r a t i o n s of a l g o r i t h m ( 3 . 2 8 ) x(L) = 5,4,3,2

-

(3.31).

8)

For

( t h i s was t h e o r d e r w e a c t u a l l y f o l l o w e d ) w e u s e d a

kind of incrementa2 m e t h o d , t h e i n i t i a l i s a t i o n o f

-

(3.28)

(3.31)

b e i n g p e r f o r m e d by u s i n g t h e r e s u l t s o b t a i n e d f o r t h e p r e v i o u s v a l u e

of X ( L ) . F o r r e a s o n s of c l a r i t y t h e s o l u t i o n s c o r r e s p o n d i n g t o x ( L ) = 6 , 4 , 2 , respectively,

a r e p i c t u r e d i n d i v i d u a l l y i n Figures 3.4,

4

- 16.-8.

Figure

12. -16.-8.

8.

-4.

3.4:( x ( L )= 6 )

-4.

-

-

-a. -

-12.

-

-16.l -8.

'

' -4.

4.

Figure 3.5:

4.

0.

-4.

'

A ' 0.

F i g u r e 3.6:

'

' 4.

'

3.5 and 3 . 6 .

' 8.

(x(L)=2)

' 12.

(x(L)=4)

12.

(SEC. 3)


2 83

T a b l e 3 . 1 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 c o n v e r g e n c e , using t h e termination test (3.37):

Number o f iterations 166

105 105

107

I05

Table 3.1

The above f i v e c a l c u l a t i o n s were p e r f o r m e d i n a s i n g Z e c o m p u t e r r u n , and r e q u i r e d t h r e e mi n u t es on a

(ii)

C I I / I R I S 80 computer.

F i g u r e 3.7 shows t h e n u m e r i c a l r e s u l t s o b t a i n e d a s f o l l o w s f o r

x(L) = 1,2,3,4,5,6,7,8

e a c h c a l c u l a t i o n h a s b e e n p e r f o r m e d by

:

i n i t i a l i s i n g a l g o r i t h m (3.28)

-

(3.31) w i t h

A'

=

u1

= 0

and {xo,yo}

w e are t h u s

c o r r e s p o n d i n g t o t h e lower semicircle w i t h d i a m e t e r A B :

4'

0.

7 t-

-4.

-

-8.

-

-12.

-

- 16. -8.

-4.

F i g u r e 3.7:

0.

4.

B.

(x(L)=1,2,3,4,5,6,7,8)

1

t

12

-16.l -8.

'

'

- 4.

'

'

0.

F i g u r e 3.8:

'

' 4.

'

' 8.

(x(L)=4, 5)

'

12

284

(CHAP. 8)


s t a r t i n g from a p o i n t f a r away f r o m t h e r e q u i r e d s o l u t i o n and w e a r e

W e o b s e r v e i n F i g u r e 3.7 two

n o t employing an i n c r e m e n t a l s t r a t e g y .

t y p e s o f form f o r t h e s o l u t i o n s c a l c u l a t e d ( t h i s c o r r e s p o n d s t o i t can a l s o be seen t h a t i f x(L) i s

d i s t i n c t branches of s o l u t i o n s ) ;

s u f f i c i e n t l y s m a l l t h e n t h e s o l u t i o n s i n F i g u r e 3 . 7 d i f f e r from t h o s e Since the criticaZ

o b t a i n e d i n ( i ) u s i n g a n i n c r e m e n t a l method.

vaZue o f x ( L ) f o r t h e above phenomenon seems t o l i e b e t w e e n 4 and 5 ,

we have s i n g l e d o u t t h e s o l u t i o n s f o r x ( L )

=

4 and 5 s e p a r a t e l y i n

F igu re 3.8. T a b l e 3 . 2 below i n d i c a t e s 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 conv e r gen ce :

I

x(~)

I

1 2 3 4

I

Number o f iterations 220

;;: 220

170 187

Table 3.2 The above e i g h t c a l c u l a t i o n s c o r r e s p o n d t o an o v e r a l l e x e c u t i o n t i m e o f 7 m i n u t e s on a C I I / I R I S 80 computer.

Further discussion

3.6.4

T a b l e 3.3 shows t h e v a l u e s t a k e n by t h e f u n c t i o n a l i n ( 3 . 6 ) ) f o r t h e s o l u t i o n s of

x(L) Incremental strategy (case ( i ) )

J

(defined

( 3 . 2 ) d e s c r i b e d i n S e c t i o n 3.6.3

8

7

6

5

4

3

2

1

-8561

-8142

-7688

-7199

-6674

-6112

-5510

-4868

-8561

-8142

-7688

-7199

-9434

-9702

-9932

-10124

mincrmta

Table 3.3

above.

2-D INCOMPRESSIBLE MATER; 3LS

(SEC. 4 )

Table 3.3

2 85

demonstrates t h e following (hardly s u r p r i s i n g ) f a c t :

by

u s i n g an i n c r e m e n t a l s t r a t e g y w e have been a b l e t o f o l l o w one b r a n c h of s o l u t i o n s , d e s p i t e t h e f a c t t h a t more s t a b l e s o l u t i o n s e x i s t f o r t h e same v a l u e s of x ( L )

4.

.

APPLICATIONS I N FINITE NONLINEAR ELASTICITY. (11) TWO-DIMENSIONAL CALCULATIONS INVOLVING LARGE DISPLACEMENTS AND LARGE STRAINS FOR INCOMPRESSIBLE MATERIALS O F MOONEY-RIVLIN TYPE

4.1

Synopsis

The aim of t h i s s e c t i o n i s t o a s s e s s t h e p o s s i b i l i t i e s o f f e r e d by t h e mthds of C h a p t e r 111 and of S e c t i o n 2 of t h e p r e s e n t c h a p t e r , 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 n o n l i n e a r problems a r i s i n g i n t h e f i e l d In t h i s s e c t i o n we

of multidimensional F i n i t e Nonlinear E l a s t i c i t y . s h a l l b e c o n c e n t r a t i n g on a r e l a t i v e l y ' s i m p l e '

s t a t i c problem,

namely t h e m e c h a n i c a l b e h a v i o u r of a two-dimensionaZ an i n c o m p r e s s i b Ze m a t e r i a l of M o o n e y - R i v l i n

type.

c u l t y i n t h i s problem i s t h e i n c o m p r e s s i b i l i t y

body made from

The major d i f f i -

c o n d i t i o n and w e s h a l l

see how t h e d e c o m p o s i t i o n - c o o r d i n a t i o n methods of C h a p t e r 111, and o f S e c t i o n 2 of t h e p r e s e n t c h a p t e r , p r o v i d e a s i m p l e and e l e g a n t means of overcoming t h i s d i f f i c u l t y .

The method d e s c r i b e d has i n f a c t

also

been u s e d s u c c e s s f u l l y f o r t h e s o l u t i o n of t h e s t a t i c e q u i l i b r i u m problem f o r t h r e e - d i m e n s i o n a l b o d i e s ;

problems of t h i s k i n d a r e much

more d i f f i c u l t , and f o r t h e i r n u m e r i c a l t r e a t m e n t by t h e methods o f t h i s book w e refer t h e r e a d e r t o GLOWINSKI-LE TALLEC I l l , LE TALLEC

4.2

I 2 1 and

[ll, 121. F o r m u l a t i o n o f t h e problem

4.2.1

Notation.

MechanicaZ a s s u m p t i o n s

A fundamental problem i n N o n l i n e a r E l a s t i c i t y i s t h e c a l c u l a t i o n of

the &formations homogeneous,

and d i s p l a c e m e n t s o f a s o l i d body c o n s i s t i n g o f a

i s o t r o p i c , h y p e r e z a s t i c and i n c o m p r e s s i b Z e m a t e r i a l , sub-

j e c t e d t o volume f o r c e s

pof

(po is t h e density i n t h e reference

c o n f i g u r a t i o n ) and t o s u r f a c e f o r c e s

so.

In a Lagrangian formulation, t h e e n e r g y f u n c t i o n a Z c o r r e s p o n d i n g t o a d i s p Z a c e m e n t f i e l d y

i s g i v e n by


286

where, i n (4.1), R

i s a domain i n

rence c o n f i g u r a t i o n ;

aR

IRN

(=an, u a n 2 )

8)

(CHAP.

Corresponding t o t h e r e f e -

i s t h e boundary o f

R , t h e body

an1:

w e h a v e d e n o t e d by u ( v ) t h e i n t e r n a Z e l a s t i c F o r a Mooney-Rivlin m a t e r i a l w e energy f u n c t i o n ( p e r u n i t m a s s ) .

b e i n g f i x e d on have

(4.2)

where, i n ( 4 . 2 ) ,

(4.3)

,

Ii i s t h e

ith i n v a r i a n t o f t h e t e n s o r FF

t

ziil

,

with

- -

F = I+Vv, ...-

(4.4)

and E l ,

E2 a r e p o s i t i v e c o e f f i c i e n t s which d e p e n d on t h e m a t e r i a l .

displacement

y

The

must a l s o s a t i s f y t h e i n c o m p r e s s i b i l i t y c o n d i t i o n ,

which i s e x p r e s s e d by det F(v) = 1 a.e.

--

(4.5)

Remark 4 . 1 :

y;

on R.

W e h a v e assumed i n (4.1) t h a t

So

is independent of

t h i s c o r r e s p o n d s t o a c l a s s i c a l s i m p l i f y i n g a s s u m p t i o n known a s

t h e dead l o a d a s s u m p t i o n ;

t h i s assumption f a c i l i t a t e s t h e presenta-

t i o n o f t h e p r o b l e m w i t h o u t c h a n g i n g i t s f u n d a m e n t a l n a t u r e , inasmuch

as

t h e e s s e n t i a l d i f f i c u l t y lies i n t h e incompressibility condition

(4.5).

W e r e f e r t o GLOWINSKI-LE TALLEC 111 and LE TALLEC [ 1 1 , 1 2 1

f o r t h e g e n e r a l i s a t i o n of t h e a l g o r i t h m s i n S e c t i o n 4.3 t o t h e case where t h e d e a d l o a d a s s u m p t i o n i s n o l o n g e r s a t i s f i e d :

a number o f

n u m e r i c a l t e s t s showing t h e e f f i c i e n c y o f t h e s e g e n e r a l i s e d a l g o r i t h m s may a l s o b e f o u n d i n t h e above r e f e r e n c e s .

4.2.2

Mathematical f o r m u l a t i o n s

In t h i s section we s h a l l describe various formulations f o r t h e

(SEC. 4 )

2-D

e l a s t o s t a t i c problem;

INCOMPRESSIBLE MATERIALS

287

demonstrating t h e i r equivalence i n t h e general

case i s s t i l l a n open m a t h e m a t i c a l p r o b l e m ( w e r e f e r t o LE TALLEC 111, c21, a s w e l l a s t o LE TALLEC-ODEN [ I 1 f o r a d i s c u s s i o n on t h e s e

see a l s o t h e d i s c u s s i o n

questions regarding equivalent formulations;

i n Section 4.2.2.1). Formulation b y minimisation of the energy functional

4.2.2.1

-u

It i s r e a s o n a b l e t o assume t h a t t h e d i s p l a c e m e n t s

corresponding

t o t h e stable equilibrium states s a t i s f y t h e f o l l o w i n g c o n d i t i o n :

5

(4.6)

locally minimises on K

the functionaz

-v

+

n(y),

w h e r e , f o r a n i n c o m p r e s s i b l e Mooney-Rivlin m a t e r i a l , w e h a v e

K =

1 N (H (n)) ,

y

on 2R1,

=

det F(v) = 1 3

-

(4.7)

a.e., and

y

-t

IT(:)

F-’ - (y) E

(L~(Q))~~~I ,

d e f i n e d by ( 4 . 1 )

,

(4.2)

,

(4.3).

The e x i s t e n c e o f s o l -

u t i o n s f o r ( 4 . 6 ) , ( 4 . 7 ) i s p r o v e d i n BALL C11.

4.2.2.2

Formulation b y equilibrium equations

The e q u i l i b r i u m p o s i t i o n s

( s t a b l e o r unstable) correspond t o t h e

s o l u t i o n s o f t h e s y s t e m of 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 e q u a t i o n s

where

DIT i s t h e d i f f e r e n t i a l o f

IT

1 N ( o n H ( Q ) ) ) and where

(4.9)

( i n ( 4 . 9 ) w e h a v e u s e d t h e c l a s s i c a l n o t a t i o n of Mechanics w i t h r e g a r d t o summation and d i f f e r e n t i a t i o n ) .

The above f u n c t i o n

p

is clearly

a Lagrange muZtipZier a s s o c i a t e d w i t h t h e i n c o m p r e s s i b i l i t y c o n d i t i o n


288

(CHAP.

8)

(4.5) and is seen to be a p r e s s u r e . 4.2.2.3

F o r m ulation by augmented Lagrangian

We proceed as in Sections 2 and 3 (and as in Chapter 111) by 'relaxing' the linear relation (4.4) simultaneously by a Lagrange multiplier and penalisation, giving the augmented Lagrangian (with r > 0): (4.1 I )

d,(y,$,g)

=):(TI

+

5 IIVV+I-GII~ =2 I I

3

%

,I

--

U*(Vv+I-G) z z dx.

This leads to the following formulation of the elastostatic problem

1 Find

I

(4.12)

{y,E,x> -

E

W = X X Y

X

(L2(n))NXN, t h e s t a t i o n a r y p o i n t

I

/

on

W o f t h e augmented

Lagrangian

F,

where, in (4.12), y =

{GIG
0.

Problem ( 4 . 2 1 ) , which i s e q u i v a l e n t t o

(4.23)

i s i n f a c t an u n c o n s t r a i n e d m i n i m i s a t i o n p r o b l e m , t h e s o l u t i o n o f which p r e s e n t s l i t t l e d i f f i c u l t y , e s p e c i a l l y i f

r

is sufficiently

large;

i f N = 2 , t h e f u n c t i o n a l i n (4.23) i s q u a d r a t i c ,

(4.21),

( 4 . 2 3 ) r e d u c e s t o s o l v i n g a l i n e a r probZem r e l a t i v e t o a n

o p e r a t o r w i t h p a r t i a l d e r i v a t i v e s of second o r d e r

(similar to the

L i n e a r E l a s t i c i t y o p e r a t o r ) which i s i n d e p e n d e n t of finite-dimensional

and s o l v i n g

n , and whose

v a r i a n t s are l i n e a r systems a s s o c i a t e d w i t h

p o s i t i v e - d e f i n i t e m a t r i c e s which a r e i n d e p e n d e n t o f ( w e t h e n u s e a p r e - f a c t o r i s a t i o n of t h e s e m a t r i c e s ) .

symmetric,

Problem ( 4 . 2 0 ) i f N = 2,

i s n o t so s t r a i g h t f o r w a r d ( i n a p p e a r a n c e a t l e a s t ) ;

(4.20) reduces (omitting t h e index

Find

i

F c

(L2 ( R ) ) 4

,G

s u c h t h a t F11F22-F12F21= 1 a . e .

+

In

[rGij-2(r(ui 2

.+Gij)-A.

,J

over the s e t o f the

'

The

n) to:

and w h i c h m i n i m i s e s t h e f u n c t i o n a l

(4.24)

6.. 13

n

i n (4.24)

(*)

.)G. . ] dx

13

1J

G E (L2 ( a ) )4 s u c h t h a t GllG22-G12G21= 1 a.e.

D

i s t h e Kronecker d e l t a .

(SEC. 4 )

291

INCOMPRESSIBLE MATERIALS

2-D

5 -

I n so f a r a s t h e r e a r e no derivatives of

and

-

i n (4.24)

( i n t h e o r y a t l e a s t ) of problems i n

s o l v e an i n f i n i t y

,

w e can

it i s t h u s n e c e s s a r y t o

s o l v e t h i s l a t t e r problem p o i n t by p o i n t ;

IR4

of t h e

type :

i

Find

{F..}cR4

such t h a t

1J

F11F22-F12F21= I Gij

minimises t h e f u n c t i o n a l

(4.25)

{IG..}€R

Over

13

4

, GllG22-G12G2, =

-+

2 rG..

ij

-

2a

and w h i c h

ijGij

I}.

The above c o n s t r a i n t i s d i a g o n a l i s e d w i t h t h e a i d of t h e new variables

-

Using b = [bi}4=1. d e f i n e d by ( 4 . 2 6 ) , problem ( 4 . 2 5 ) r e d u c e s t o

Find

c2 = c3

(4.27)

such t h a t

= -I

and w h i c h minimises

over

The s o l u t i o n s of

(4.28)

Eibi2

=

(4.27)

{ b i l e R4

where t h e s c a l a r = 2)

(4.29)

2 c.b.

b eR4

=

I ~ 4 ~ ,I ~1.~1c ?== 21~

2 , c1 = c4

5

-+

-

rci

1,

-

2z.c.

1 1

.

a r e g i v e n by

, bi

p

1 1

= zi/(r+z.p),

Yi=l,2,3,4,

( t h e Lagrange m u l t i p l i e r a s s o c i a t e d w i t h

satisfies 2 2 2 2 2 (z1+z4)/(r+p)’ = 2+(z2+z3)/(r-p)

2 Suppose t h a t z1

+

2

z4 # 0;

.

it t h e n can e a s i l y be shown t h a t ( 4 . 2 9 )

admits j u s t one s o l u t i o n i n l - r , + r [ :

furthermore, using t h e I m p l i c i t

F u n c t i o n T h e o r e m (see LE TALLEC C11, C21 GLOWINSKI-LE TALLEC 111 f o r f u r t h e r d e t a i l s ) i t can be shown t h a t t h i s s o l u t i o n of

t o I-r,+rC

( 4 . 2 9 1 belonging

i s p r e c i s e l y t h a t a s s o c i a t e d , v i a (4.28), w i t h t h e gZobaZ

292

(CHAP. 8 )


m i n i m u m o f t h e f u n c t i o n a l .c

r c2 . - 2 z .1c .1 1

-f

on

E

2 =C 2 ~a n d a l s o t h a t

~

t h e r e a r e i n f a c t n o o t h e r l o c a l o r g l o b a l minima. on I - r , + r [

i s a t r i v i a l problem;

,F

u s i n g ( 4 . 2 8 ) , and

from

Solving (4.29)

-

w e t h e n deduce b from

by u s i n g ( 4 . 2 6 ) .

i s i n t e r p r e t e d mechanically as a p r e s s u r e ;

p , by

The m u l t i p l i e r

GLOWINSKI-LE TALLEC [11 t h a t t h i s m u l t i p l i e r

LE TALLEC 1 1 1 , [ 2 1 ,

is equal t o t h e pressure

p

p

i n f a c t , i t i s shown i n

which a p p e a r s i n ( 4 . 8 )

p

(which t h e r e f o r e

j u s t i f i e s o u r u s e of i d e n t i c a l n o t a t i o n ) .

Remark 4 . 3 : I n t h e n u m e r i c a l t e s t s w e have performed, n o t once 2 2 d i d w e e n c o u n t e r t h e case z1 + z 4 = 0; i n f a c t we conjecture t h a t

r

for

s u f f i c i e n t l y l a r g e t h i s s i t u a t i o n cannot arise, i f N = 2 , f o r

problem ( 4 . 1 2 ) .

Furthermore,

t h i s condition of "r

sufficiently

l a r g e " i s f u n d a m e n t a l , as i s shown i n LE TALLEC Cll, 121 and GLOWINSKILE TALLEC C11 ( t h e s e r e f e r e n c e s e v e n go so f a r a s t o g i v e a l o w e r

bound f o r

r , t h i s bound b e i n g r e l a t e d t o c e r t a i n norms o f t h e p r e s s -

ure p ) .

4.4

Numerical tests

Suppose t h a t N = 2 ; (4.8)

,

E

Qh,

(as w e l l as problems

( 4 . 1 2 ) ) t o a f i n i t e - d i m e n s i o n a l p r o b l e m by u s i n g a finite-eleW e h a v e u s e d r e c t a n g u l a r finite e l e m e n t s

ment approximation. K

we reduce problem (4.6)

where

t h e displacement

R.

i s a q u a d r a n g u l a t i o n of

Qh

v

by

I

yh

E

Co:E)

x

co(E),

We t h e n a p p r o x i m a t e

such t h a t

(4.30)

where (4.31)

t h e i n c o m p r e s s i b i l i t y c o n d i t i o n ( 4 . 5 ) i s imposed a t t h e c e n t r e of (which i s e q u i v a l e n t t o i m p o s i n g it K E Qh a s an a v e r a g e over e a c h r e c t a n g l e ) .

each elementary r e c t a n g l e

The c o n v e r g e n c e o f t h e a p p r o x i m a t e s o l u t i o n s when h d i f f i c u l t question;

t h i s t o p i c i s t a c k l e d i n LE TALLEC

I n t h e n u m e r i c a l tests which f o l l o w ,

R

-t

0 i s a very

[I], C21.

i s a (two-dimensional)

(SEC. 5 )

APPLICATIONS TO EIGENSYSTEM PROBLEMS

bar containing a crack; further.

t h i s c r a c k i s assumed n o t t o p r o p a g a t e any

F i g u r e 4 . 1 shows t h e r i g h t - h a n d p o r t i o n o f t h e b a r , t h e

c r a c k and t h e q u a d r a n g u l a t i o n

Qh).

an2

W e suppose t h a t i n =

an

293

( a c t u a l l y t h e r i g h t - h a n d h a l f of

Qh

(4.11,

(4.2)

w e h a v e p o = 1, E l = 1,

and t h a t So c o r r e s p o n d s t o h o r i z o n t a l f o r c e s a p p l i e d t o 1

t h e e n d s of t h e b a r and t e n d i n g t o e l o n g a t e i t , t h e f o r c e s b e i n g 2 ( i n modulus).

d e n s i t y of these

The b a r t h u s s t r e t c h e s u n d e r t h e a c t i o n

o f t h e s e f o r c e s , and F i g u r e 4 . 2

shows t h e e q u i l i b r i u m p o s i t i o n o b t a i n e d ;

t h i s w a s c a l c u l a t e d by means of t h e d i s c r e t i s e d v a r i a n t of a l g o r i t h m (4.14)

-

(4.16)

,

i n i t i a l i s e d w i t h t h e c o n f i g u r a t i o n of F i g u r e 4.1.

Using p = r = 10, c o n v e r g e n c e o f

(4.14)

-

(4.16)

was attained i n

2 0 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 t i m e o f 5 s e c o n d s on a CDC 6 4 0 0 .

I t is i n t e r e s t i n g t o observe t h e behaviour o f t h e crack.

Figure 4 . 1

F i g u r e 4.2 A number of n u m e r i c a l t e s t s r e l a t i n g t o o t h e r t w o - d i m e n s i o n a l

p r o b l e m s and t o c e r t a i n a x i s y m m e t r i c a n d t h r e e - d i m e n s i o n a l p r o b l e m s may b e f o u n d i n LE TALLEC 111, 1 2 1 ,

5.

GLOWINSKI-LE TALLEC [ 1 1 , [ 2 1 .

SOME REMARKS ON THE APPLICATION OF THE ALGORITHMS OF SECTION 2 TO THE SOLUTION OF EIGENVALUE AND EIGENVECTOR PROBLEMS

The p r o b l e m s i n F i n i t e N o n l i n e a r E l a s t i c i t y c o n s i d e r e d i n S e c t i o n s 3 and 4 o f t h i s c h a p t e r a r e somewhat r e m i n i s c e n t o f e i g e n v a Z u e and

294


eigenvector problems:

(CHAP.

8)

they i n f a c t involve t h e minimisation of

(sometimes q u a d r a t i c ) o v e r s e t s d e f i n e d by n o n Z i n e a r

functionals

equality constraints.

It i s t h e r e f o r e n a t u r a l t o consider using

t h e a l g o r i t h m s of Chapter I11 and of S e c t i o n 2 of t h e p r e s e n t c h a p t e r f o r s o l v i n g c e r t a i n e i g e n v a l u e / e i q e n v e c t o r problems.

In

t h e following, w e s h a l l confine o u r a t t e n t i o n t o t h e determination of t h e s m a l l e s t e i g e n v a l u e of a s y m m e t r i c p o s i t i v e - d e f i n i t e m a t r i x o f an a s s o c i a t e d e i q e n v e c t o r ) ;

(and

it i s i n f a c t p o s s i b l e t o g e n e r a l i s e

t h e d i s c u s s i o n below t o t h e s o l u t i o n of c e r t a i n n o n l i n e a r e i g e n v a l u e problems. Let

-

be an N

A

x

N symmetric p o s i t i v e - d e f i n i t e matrix;

-

let

-

Am ( > O ) be i t s s m a l l e s t e i g e n v a l u e and l e t xm ( f 0 ) be an a s s o c i a t e d eiqenvector.

We t h u s have

and it i s a c l a s s i c a l r e s u l t t h a t

Xm

i s a (non-unique) s o l u t i o n of

t h e m i n i m i s a t i o n problem

I t i s a l s o known c l a s s i c a l l y t h a t i f w e a s s o c i a t e w i t h

t h e Lagrangian

then

Am

L : RN X

R +.

R

(5.2) - ( 5 . 4 ) ,

d e f i n e d by

i s t h e Lagranqe m u l t i p l i e r a s s o c i a t e d w i t h t h e m i n i m i s a t i o n

problem ( 5 . 2 )

and w i t h t h e Lagrangian ( 5 . 5 ) .

(SEC. 5 )

APPLICATIONS TO EIGENSYSTEM PROBLEMS

295

I n o r d e r t o apply t h e decomposition-coordination

methods of

C h a p t e r I11 and o f S e c t i o n 2 of t h e p r e s e n t c h a p t e r , w e f i r s t n o t e t h a t problem ( 5 . 2 ) i s e q u i v a l e n t t o

(5.6)

where

W e then associate with t h e l i n e a r c o n s t r a i n t

Lagrangian

dr

:

JR3N

+

y-3

=

In order t o solve (5.6)

X

S)

X

xo

then, f o r n

2 0,

dr

on

t h e a p p l i c a t i o n of a l g o r i t h m A L G l of S e c t i o n 2 . 4

IRN ;

( a n d o f C h a p t e r 111, S e c t i o n 3 . 1 ) (5.9)

t h e augmented

(and t h e r e f o r e ( 5 . 1 ) , ( 5 . 2 ) ) w e a r e

t h e r e f o r e l e d t o d e t e r m i n e ( l o c a l ) s a d d l e p o i n t s of (lRN

9,

IR d e f i n e d , w i t h r > 0 , by

leads t o the algorithm

given,

- -

b e i n g known, d e t e r m i n e { x n , p n }

-An

then

bg

and

(5.11)

X"+l = A"

I

I

+

p(p-p"),

p > 0.

Once a g a i n , w e can s o l v e ( 5 . 1 0 ) by a b l o c k - r e l a x a t i o n method and, a s b e f o r e , by r e s t r i c t i n g t h i s t o a s i n g l e i n n e r r e l a x a t i o n i t e r a t i o n ,

w e deduce from ( 5 . 9 ) - ( 5 . 1 1 ) t h e f o l l o w i n g v a r i a n t ( o f t y p e ALGZ ( c f . S e c t i o n 2 . 5 and C h a p t e r 111, S e c t i o n 3 . 2 ) ) :

(5.12)

x

-1

and

given,


296

t h e n , for n

0,

2

5n- 1 and

in


(CHAP.

pn, Y

8)

zn and

s u c c e s s i v e l y by

An''

(5.13)

(5.14)

(

,"ERN,

-

(5.15)

=

An

+

p(_xn-pn),

P>O.

I t i s c l e a r t h a t t h e m i n i m i s a t i o n problem ( 5 . 1 4 ) 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 of t h e l i n e a r s y s t e m

(rz+5)_xn=

(5.16)

S i n c e t h e m a t r i x rI

+

-

rp"-," I

.

A i s s y m m e t r i c and p o s i t i v e - d e f i n i t e ,

w e can

I

p e r f o r m once and f o r a l l a Choiesky f a c t o r i s a t i o n of t h i s m a t r i x ; hence we s o l v e , a t e a c h i t e r a t i o n , two l i n e a r s y s t e m s o f t r i a n g u l a r matrices

.

The s o l u t i o n of problem ( 5 . 1 3 ) d o e s n o t p r e s e n t any f u r t h e r d i f ficulties;

i n f a c t , t a k i n g account of t h e c o n d i t i o n

CJ

E

S , problem

(5.13) i s e q u i v a l e n t t o (5.17)

Max (g ,_Xn+rZn-l ) ,

qrs so t h a t

Taking a c c o u n t of

( 5 . 1 6 ) , ( 5 . 1 8 ) , w e can w r i t e a l g o r i t h m (5.12)

( 5 . 1 5 ) i n t h e f o l l o w i n g , more p r a c t i c a l , form: (5.19)

5-1 and lo g i v e n ,

-

(SEC. 5 )

APPLICATIONS TO EIGENSYSTEM PSOBLEMS

then, for n

and

xn-l

2 0,

An

297

b e i n g known

(5.21)

An+1

(5.22)

A"

=

p > 0.

+ p(_X"-p"),

-A n

S i m i l a r l y , by u p d a t i n g

i n (5.19)

-

( 5 . 2 2 ) between t h e s t a g e s

( 5 . 2 0 ) and ( 5 . 2 1 ) , w e o b t a i n t h e f o l l o w i n g a l g o r i t h m (of t y p e ALG3) i n which

-

and

(5.23)

x

-1

q

and

-

then, for n z 0, x

-

-

AO

n- 1

A"

pn =

(5.24)

p l a y symmetric r o l e s :

-

given,

and

-

An

being known,

+ rxn-' ."

11 An+rxn-l 11 ' I

-

(5.25)

(5.26)

(5.27)

The convergence of t h e above a l g o r i t h m s remains t o b e p r o v e d ; nonetheless,

i f w e suppose t h a t

t h e n w e have

Am W e note

=

*

( a n d it i s s u f f i c i e n t t o s e t

r i t h m s ( 5 . 1 9 ) - ( 5 . 2 2 ) and ( 5 . 2 3 )

-

p = r

(5.27)

t o show t h i s ) t h a t a l q o a r e v a r i a n t s of t h e p o w e r

298


(CHAP.

8)

m e t h o d , which i s a s t a n d a r d method f o r t h e c a l c u l a t i o n o f t h e e i g e n v a l u e s and e i q e n v e c t o r s of m a t r i c e s

(see WILKINSON E l l , WILKINSON-

REINSCH [I], STEWART 1 1 1 , PARLETT [l]). by M.O.

Numerical t e s t s c a r r i e d o u t

B r i s t e a u a t I N R I A , have d e m o n s t r a t e d t h e good c o n v e r g e n c e

p r o p e r t i e s of t h e above a l g o r i t h m s a s l o n g a s

large ( i f we put

p =

r

is t a k e n s u f f i c i e n t l y

r , which o n c e a g a i n s e e m s t o b e t h e b e s t c h o i c e ) .

W e l e a v e i t a s an e x e r c i s e f o r t h e r e a d e r t o d e r i v e v a r i a n t s of

t h e above a l g o r i t h m s which e n a b l e t h e o t h e r e i q e n v a l u e s and e i g e n v e c t o r s of

-

A

t o be c a l c u l a t e d .

CHAPTER IX A P P L I C A T I O N S OF T H E M E T H O D OF M U L T I P L I E R S TO VARIATIONAL INEQUALITIES D. Gabay

1.

INTRODUCTION

T h i s c h a p t e r e x t e n d s a n d complements some o f t h e r e m a r k s made i n t h e p r e v i o u s c h a p t e r s , i n p a r t i c u l a r i n C h a p t e r 111.

We generalise

t h e augmented L a q r a n g i a n method t o t h e case 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 a n d w e g i v e t o i t t h e m o r e a p p r o p r i a t e name o f t h e m e t h o d of

m u l t i p l i e r s s i n c e t h e s e p r o b l e m s do n o t g e n e r a l l y i n v o l v e a L a g r a n g i a n . W e s h a l l a l s o d e m o n s t r a t e t h e e q u i v a l e n c e between a l g o r i t h m A L G l and

a method of s o l u t i o n well-known i n N o n l i n e a r A n a l y s i s , namely t h e proximal-point

Finally, we reconsider i n d e t a i l t h e

algorithm.

i d e a s i n t r o d u c e d e a r l i e r , i n C h a p t e r I V , o n t h e s u b j e c t of a Z t e r n a t i n g

d i r e c t i o n m e t h o d s , and w e d e s c r i b e t h e r e l a t i o n s h i p b e t w e e n t h e s e methods and ALG2 and ALG3 (see a l s o C h a p t e r V I I I , S e c t i o n 2 ) . f a c i l i t a t e a p r o p e r p r e s e n t a t i o n of t h e problems,

To

we f i r s t recall a

number o f d e f i n i t i o n s and r e s u l t s .

1.1 Let

X

(1.1)

be a real H i l b e r t space equipped w i t h t h e i n n e r product

W e d e s i g n a t e a s a m o n o t o n e o p e r a t o r a m u l t i - v a l u e d mapping

( a , . ) .

T:

Monotone o p e r a t o r s

X

+

2'

such t h a t w e have

( z ' - ~ , x ' - x ) ~ " O Yx,x'

W e say t h a t

T

E X , YzET(x),

Yz' E T ( x ' ) .

i s a maximal monotone o p e r a t o r i f , i n a d d i t i o n , t h e

graph

i s n o t s t r i c t l y i n c l u d e d w i t h i n t h e g r a p h of a n y o t h e r monotone 299

METHOD O F MULTIPLIERS & V.I.'s

3 00

o p e r a t o r on

(CHAP. 9 )

T h e s e o p e r a t o r s o c c u r i n Convex A n a l y s i s and i n t h e

X.

s t u d y of c e r t a i n 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 ( s e e , f o r e x a m p l e , B F E Z I S C11).

I n f a c t a v e r y i m p o r t a n t s p e c i a l case a r i s e s from t h e

f i e l d o f Convex A n a l y s i s . which i s p r o p e r , convex

its subgradient

The o p e r a t o r x

Let $ :

a$, d e f i n e d a t x

+

X

+

+-I

I-m,

be a function

and lower semi-continuous E

on

i s maximal monotone, and i f x

a$(x)

of t h e m u l t i v a l u e d e q u a t i o n

and c o n s i d e r

X

X by

E

X is a solution

(*)

t h e n w e a l s o have

The m u l t i v a l u e d e q u a t i o n problem ( 1 . 5 ) ,

( 1 . 4 ) is thus equivalent t o t h e optimisation

( o r convex programming p r o b l e m ) , w h i c h i s i m p l i c i t l y a

c o n s t r a i n e d p r o b l e m s i n c e t h e s e t o f p o i n t s where

@ ( y )=

+-

is

o b v i o u s l y e x c l u d e d from t h e s e t of a d m i s s i b l e s o l u t i o n s . However, i t i s n o t a l w a y s p o s s i b l e ( i n p a r t i c u l a r i n e x a m p l e s a r i s i n g from t h e t h e o r y o f 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 ) t o a s s o c i a t e an o p t i m i s a t i o n problem w i t h a multivalued equation:

we

then consider d i r e c t l y a formulation using a v a r i a t i o n a l inequality. Given a nonempty,

c l o s e d , convex s u b s e t

tone operator

on

seek x

E

K,

A

X,

K

of

X

a n d a maximal mono-

n o t n e c e s s a r i l y d e f i n e d by a s u b g r a d i e n t , w e

s a t i s f y i n g t h e f o l l o w i n g v a r i a t i o n a l i n e q u a l i t y (see

LIONS-STAMPACCHIA C11): (1 - 6 )

There e x i s t s Z E A ( X )

such t h a t

(Z,Y-X)~LO Y y c K .

L e t u s now c o n s i d e r t h e m u l t i v a l u e d e q u a t i o n (1.7)

(*)

0 E T(x).

Translator's note:

Sometimes known u n d e r t h e name m u l t i v o q u e

equation.

(SEC.

1)

301

INTRODUCTION

Any s o l u t i o n of t h e i n e q u a l i t y ( 1 . 6 )

a l s o s a t i s f i e s ( 1 . 7 ) with

T

d e f i n e d by

(1 - 8 )

T(x) =

I

A(x)+NK(x)

if x E K,

B otherwise,

where N K ( x ) d e n o t e s t h e c o n e n o r m a l t o K a t x " ) ; h a s shown t h a t

T

i s maximal monotone.

a maximal monotone s i n g Z e - v a Z u e d convex cone i n

ROCKAFELLAR 1 6 1

W e note t h a t i f

o p e r a t o r and i f

K

A

is

is a closed

with v e r t e x 0, then t h e v a r i a t i o n a l i n e q u a l i t y

X

(1.6) i s equivalent t o the complementarity problem:

Find -A(x)

where

XEK

E

K'and

such t h a t (A(x),x),

= 0,

denotes t h e p o l a r cone

KO

( * ) o f K.

The m u l t i v a l u e d e q u a t i o n ( 1 . 7 ) of t h e type

t h u s u n i f i e s v a r i a t i o n a l problems

( 1 . 5 ) , 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 t y p e ( 1 . 6 ) and

c o m p l e m e n t a r i t y problems of t h e t y p e ( 1 . 9 )

.

W e can t h e r e f o r e t r a n s -

p o s e a l g o r i t h m s d e s i g n e d f o r s o l v i n g problems of one t y p e t o t h e s o l u t i o n of problems o f t h e o t h e r two t y p e s .

1.2

The method of m u l t i p l i e r s

The development of O p e r a t i o n a l R e s e a r c h d u r i n g t h e l a s t twenty y e a r s h a s promoted an i n c r e a s e d emphasis on t h e i n v e s t i g a t i o n of o p t i m i s a t i o n problems;

more advanced n u m e r i c a l methods and e x p e r i m e n t s

t h e r e f o r e e x i s t i n t h e f i e l d of convex programming t h a n i n t h e f i e l d s 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 and c o m p l e m e n t a r i t y problems.

The uni-

f y i n g framework p r e s e n t e d above would t h u s i n d i c a t e one p o s s i b l e methodology f o r r e d u c i n g t h e gap between t h e s e domains:

namely,

g i v e n a known a l g o r i t h m f o r t h e convex programming problem (1.5), f i n d t h e corresponding general algorithm f o r solving t h e multivalued e q u a t i o n ( 1 . 7 ) and deduce from it t h e c o r r e s p o n d i n g a l g o r i t h m f o r t h e v a r i a t i o n a l i n e q u a l i t y (1.6), which c a n b e s p e c i a l i s e d t o t h e

302

METHOD OF MULTIPLIERS

c o m p l e m e n t a r i t y p r o b l e m (1.9).

&

T h i s approach h a s been f o l l o w e d

111

i m p l i c i t l y by GLOWINSKI-LIONS-TREMOLIERES relaxation algorithms, gradient algorithms conjugate-gradient

(CHAP. 9 )

V.I.'s

,

121 f o r d e f i n i n g

(with a u x i l i a r y operator)

,

a l g o r i t h m s , d u a l i t y methods a n d p e n a l i s a t i o n

m e t h o d s , f o r 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 . The method of m u l t i p l i e r s h a s f o r a number o f y e a r s g e n e r a t e d a c o n s i d e r a b l e amount o f i n t e r e s t f o r t h e s o l u t i o n o f c o n s t r a i n e d o p t i m i s a t i o n problems, b o t h f o r i t s s i m p l i c i t y o f implementation and f o r t h e

(see BERTSEKAS 111).

a d v a n t a g e s it o f f e r s o v e r p e n a l i s a t i o n methods

P r o p o s e d i n i t i a l l y by HESTENES [11 and POWELL 111 f o r m i n i m i s a t i o n problems w i t h e q u a l i t y c o n s t r a i n t s , i t can b e i n t e r p r e t e d a s a g r a d i e n t method f o r s o l v i n g a d u a l p r o b l e m a s s o c i a t e d w i t h a n augmented L a g r a n g i a n , t h i s b e i n g o b t a i n e d by a d d i n g t o t h e o r d i n a r y L a g r a n g i a n

r > 0 (which need n o t

a p e n a l i s a t i o n t e r m d e p e n d i n g on a p a r a m e t e r tend t o i n f i n i t y ) ;

penalisation-duality

h e n c e t h e a l t e r n a t i v e name:

ROCKAFELLAR 1 1 1 , C71 d e f i n e d t h e method f o r convex program-

method.

ming p r o b l e m s and d e m o n s t r a t e d i t s g l o b a l and l i n e a r ( s u p e r l i n e a r i f

r

-t

+

m)

convergence.

I n t h i s c h a p t e r , w e p r o p o s e t o e x t e n d t h e method o f m u l t i p l i e r s

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 and t h e n t o p r o p o s e a p p r o x i m a t i o n s o f t h i s method which w i l l e n a b l e a d e c o m p o s i t i o n o f t h e c a l c u l a t i o n s t o T h i s work may b e r e c o g n i s e d a s a g e n e r a l i s a t i o n of

be effected.

C h a p t e r I11 and o f GABAY-MERCIER

111, i n which t h i s o b j e c t i v e w a s

achieved f o r p a r t i c u l a r i n e q u a l i t i e s corresponding t o convex v a r i a -

t i o n a l problems of t h e form

IF(BV)+G(V)~,

inf

( I .lo)

V€V

where

F

and

G

are f u n c t i o n s w i t h v a l u e s i n

convex, p r o p e r , l o w e r semi-continuous

real Hilbert spaces ( - , - ) H and

(. , *

)v

H

and

V,

I-m,+ml,

a n d which a r e

and d e f i n e d r e s p e c t i v e l y o n t h e

equipped with t h e i n n e r products

; it c a n b e shown ( s e e C h a p t e r 111, GABAY-MERCIER

117, FORTIN 1 1 1 ) t h a t w e c a n a s s o c i a t e w i t h (1.10) t h e r e g u l a r i s e d d u a l p r o b l e m ( o r augmented d u a l p r o b l e m )

where,

for r

2

:

0 , t h e concave f u n c t i o n a l Qr

: H

+

C-m,+m[

i s d e f i n e d by

(SEC. 1)

303

INTRODUCTION

The f u n c t i o n a l

qr

i s always d i f f e r e n t i a b l e f o r

r > 0. W e can

s o l v e problem (1.10) and i t s d u a l (1.11) by s e e k i n g on V

X H

X H a

s a d d l e p o i n t of t h e a u g m e n t e d L a g r a n g i a n d e f i n e d ( a s i n C h a p t e r 111) bY

It is possible t o obtain

d i r e c t l y from t h e v a r i a t i o n a l pro-

k,

blem (1.10) by i n t r o d u c i n g t h e v a r i a b l e

q

and t h e a r t i f i c i a l con-

s t r a i n t Bv-q = 0 , t h i s c o n s t r a i n t t h e n b e i n g p e n a l i s e d and d u a l i s e d a l o n g t h e l i n e s of t h e o r i g i n a l p r o c e d u r e of HESTENES C11. k i n g s a d d l e p o i n t s of

k,,

Uzawa's a l g o r i t h m a p p l i e d t o this,

For s e e -

we have made e x t e n s i v e u s e i n t h i s book of

dr, (ALGl), and a p a r t i c u l a r v a r i a n t of

I n o r d e r t o c l a r i f y t h e r e s t of t h e d e s c r i p t i o n , we

(ALG2).

g i v e below a b r i e f r e s t a t e m e n t of a l g o r i t h m A L G 1 ,

t h e p r o p e r t i e s of

which w e r e s t u d i e d i n C h a p t e r 111.

Xo b e i n g c h o s e n a r b i t r a r i l y , we s e e k for n = 0, 1, z 1 : Xn known, s o l u t i o n s un E V , pn E H , of (1.14)

Pr(un,pn,Xn)

t h e n we c a l c u l a t e (1.15)

An+'

5

Xn+l

d,(v.q,X")

VV

6

V,

W

6

with

H,

by

= Xn+p(Bun-pn).

We t h e n c o n s i d e r t h e augmented d u a l f u n c t i o n a l

W e h a v e , from (1.14), $,(An)

=

dr(un,pn,Xn);

Jir

d e f i n e d by

f u r t h e r m o r e , i t can

be shown ( s e e ROCKAFELLAR C21, FORTIN C 1 1 ) t h a t w e have ( I .17)

...,

METHOD OF MULTIPLIERS

304

&

V.I.'s

(CHAP. 9)

$o is in fact the functional defining the dual of problem (1.10) in the sense of Fenchel (see ROCKAFELLAR 151, EKELAND-TEMAM C11); that is

where

where F* and G* denote the conjugate functions of F and G ( 3 ) , defined on H and V respectively, and where Bt is the operator from H into V defined by t ( B v , d H = (v,B q ) ,

(1.19)

v v c V,

vqc H

(4)

In the terminology of MOREAU C11, relation (1.17) states that -rqr is the p r o x i m a l p o i n t mapping" relative to -rJio. We deduce from this (see ROCKAFELLAR C21 that the method of multipliers (algorithm ALGl with p = r) generates the same sequence of iterates {An} nzO as the p r o x i m a l p o i n t a l g o r i t h m for solving the multivalued equation (1.7) with T = - a $ o. This algorithm was introduced by MARTINET 111, [ 2 1 and generalised by ROCKAFELLAR C81 for an arbitnary maximal monotone operator T on H; it constructs a sequence An E H in accordance with the recurrence relation An+l

(1.21)

where :J

=

= J;(x")

(I+rT)-'

n=O,I,

...,

is a contracting single-valued operator called

the r e s o l v e n t of T (see BREZIS C11). This observation will guide our approach to generalising the method of multipliers to variational inequalities. After first recalling, in Section 2 below, the convergence properties of the proximal-point algorithm, we then define in Section 3 a variational inequality which generalises the variational problem (1.10) and which includes the inequality (1.6) as a particular case, and we associate with this a dual variational inequality. In Section 4 we apply the proximal-point algorithm to the representation of the

("See Section 3 for the relation between Bt and the usual adjoint B' of B.

*

Translator's note:

a p p l i c a t i o n de p r o x i m i t e ' in the original French.

(SEC. 2 )

PROXIMAL-POINT

305

ALGORITHM

d u a l v a r i a t i o n a l i n e q u a l i t y , i n m u l t i v a l u e d ("muZtivoque") form, and t h i s d e f i n e s a method of m u l t i p l i e r s f o r 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 by p e n a l i s a t i o n - d u a l i t y ,

g e n e r a l i s i n g a l g o r i t h m ALG1.

Observing t h a t t h e multivalued o p e r a t o r a s s o c i a t e d w i t h t h e dual i n e q u a l i t y h a s t h e form of t h e sum of two maximal-monotone o p e r a t o r s ,

w e i n v e s t i g a t e i n S e c t i o n 5 some a p p r o x i m a t i o n s o f t h e p r o x i m a l - p o i n t a l g o r i t h m which t a k e a d v a n t a g e of t h i s s t r u c t u r e . In particular, we a p p l y two a l g o r i t h m s r e c e n t l y p r o p o s e d by P.L.

LIONS-MERCIER 111 which

g e n e r a l i s e t h e a l t e r n a t i n g - d i r e c t i o n methods t o t h e s o l u t i o n of t h e m u l t i v a l u e d e q u a t i o n (1.7) and we t h e r e b y o b t a i n two v a r i a n t s of t h e method of m u l t i p l i e r s f o r 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 ; t h e s e v a r i a n t s p r o v i d e a d e c o m p o s i t i o n scheme c o o r d i n a t e d v i a t h e m u l t i p l i e r s (see BENSOUSSAN-LIONS-TEMAM

111).

One of t h e s e v a r i a n t s

g e n e r a l i s e s 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 t h e a l g o r i t h m ALG2 from

Cll, which can t h e r e f o r e be i n t e r -

C h a p t e r I11 and from GABAY-MERCIER

p r e t e d a s an a l t e r n a t i n g - d i r e c t i o n method a s was n o t e d i n CHANGLOWINSKI I l l , f o r a p a r t i c u l a r c l a s s of p r o b l e m s , and a s was p o i n t e d o u t i n C h a p t e r s I V and V I I I .

The o t h e r v a r i a n t comprises a l g o r i t h m

ALG3 mentioned i n C h a p t e r V I I I .

I n Section 6 we study another

a p p r o x i m a t i o n of t h e p r o x i m a l - p o i n t t i n g of t h e operator

a l g o r i t h m which employs a s p l i t -

W e then r e c o v e r t h e p o i n t p r o j e c t i o n gra-

T.

d i e n t method f o r convex programming

(GOLDSTEIN

sation t o the variational inequality (1.6); i n e q u a l i t y of

(1.6)

[ll), and i t s g e n e r a l i applied t o t h e dual

t h i s may be i n t e r p r e t e d a s a method of m u l t i -

p l i e r s with projection.

2.

THE PROXIMAL-POINT

X

Let

(.

,.)

ALGORITHM

be a r e a l H i l b e r t space equipped w i t h t h e i n n e r product

and t h e c o r r e s p o n d i n g norm

monotone o p e r a t o r on X.

Find

-

I I

,

and l e t

T

b e a maximal-

W e wish t o s o l v e t h e m u l t i v a l u e d e q u a t i o n

X E X such

that

(2.1)

0 E T(x).

For a l l x unique y (2.2)

E

E

X and a l l

X such t h a t x

E

(I+rT) (y)

.

r > 0 t h e r e e x i s t s ( s e e M I N T Y c11) a


306

(CHAP. 9 )

= (I+rT)-' i s t h u s s i n g l e - v a l u e d and d e f i n e d on t h e w e c a l l t h i s t h e r e s o l v e n t o f T. This is a contrac-

The o p e r a t o r J: whole o f X;

t i o n from

X

x,

into

i.e.

W e s h a l l show t h a t J; i s m o r e o v e r a f i r m c o n t r a c t i o n . PROPOSITION 2.1:

monotone o p e r a t o r

Proof:

r > 0 t h e r e s o Z v e n t JG of t h e maximal it s a t i s f i e s

For a l l

is a f i r m c o n t r a c t i o n , i . e .

T

We put y = J;(x),

From ( 2 . 2 ) w e h a v e

y' = J;(x').

x = y+rz

where

Z E

x' = y'+rz'

where

z' E T(y').

T(y),

W e can t h u s w r i t e 2 Iyl-yIx = ( y ' - y , ~ ~ - x )- ~r(y'-y,z'-z) 5

(y'-y,x'-x)x

i n view of t h e maximal m o n o t o n i c i t y o f T ;

Remark 2 . 1 :

If T =

a@,

i n t h e t e r m i n o l o g y of MOREAU [11 t o the functional

We note t h a t 0

E

hence t h e r e s u l t .

t h e s u b g r a d i e n t of a c o n v e x , p r o p e r , X

lower semi-continuous f u n c t i o n @ :

relative

X

J;

+

1-m,+=7Il

then

i s t h e p r o x i m a l p o i n t mapping

r@.

T ( x ) i s e q u i v a l e n t t o J g ( x ) = x.

The s o l u t i o n

of t h e m u l t i - v a l u e d e q u a t i o n ( 2 . 1 ) t h u s r e d u c e s t o s e e k i n g t h e f i x e d p o i n t s of t h e c o n t r a c t i o n mapping DEFINITION 2.2:

sequence

J;.

(see ROCKAFELLAR

C 81 ) :

Given a nondecreasing

{ r n } of p o s i t i v e n u m b e r s and an a r b i t r a r y p o i n t

t h e proximal-point a l g o r i t h m g e n e r a t e s a sequence X

according t o t h e recurrence r e l a t i o n

xo

E

XI

{x"} of p o i n t s of

(SEC.

2)

PROXIMAL-POINT ALGORITHM

Remark 2 . 2 :

307

a l g o r i t h m can b e i n t e r p r e t e d as

The p r o x i m a l - p o i n t

an i m p l i c i t d i s c r e t i s a t i o n scheme f o r t h e m u l t i v a l u e d e v o l u t i o n equation (2.6)

0

E

dt

+ T(x) with x(0) = xo,

t h e parameters OE-

rn

xn+'-x"

representing the t i m e steps for the discretisation + T(xn+I).

n

The f o l l o w i n g convergence r e s u l t s t a t e s t h a t t h e sequence c o n v e r g e s t o a s o l u t i o n of t h e s t e a d y - s t a t e e q u a t i o n 0

THEOREM 2.1:

equation ( 2 . 1 ) .

cxnl

T(x).

S u p p o s e t h a t t h e r e e x i s t s a t l e a s t one s o l u t i o n t o The p r o x i m a l - p o i n t

x

[xn} w h i c h c o n v e r g e s w e a k l y t o

Proof:

E

Using r e l a t i o n

E

a l g o r i t h m g e n e r a t e s a sequence X

such t h a t

0

E

T(x)

and

( 2 . 3 ) which e x p r e s s e s t h e f a c t t h a t

is a firm contraction, we obtain f o r a l l

r JTn

n

-x,xn-x ) X = 1

(/x"+'-xlx

2

+

lx"-\;

By a d d i n g t h e s e i n e q u a l i t i e s from

-lx"+l-x"

n = 0

Ix).

t o an a r b i t r a r y i n t e g e r N,

we arrive a t

which shows t h a t

Ixn+' - x n l X

converges t o

0

and t h a t t h e sequence

{xn} i s bounded. By h y p o t h e s i s , t h e s e t o f t h e f i x e d p o i n t s o f J G i s nonempty ( s i n c e t h e r e e x i s t s a t l e a s t one s o l u t i o n of

(2.1)) ;

O p i a Z ' s Lemma,


308

(CHAP. 9 )

see OPIAL Ell, t h u s e n a b l e s t h e weak c o n v e r g e n c e o f t h e s e q u e n c e {x"} t o b e e s t a b l i s h e d (see a l s o MARTINET I l l ,

I 2 1 a n d ROCKAFELLAR

Ill). Remark 2.3:

W e can d e f i n e t h e proximal p o i n t a l g o r i t h m w i t h

r e l a x a t i o n by t h e r e c u r r e n c e

Theorem 2 . 1 r e m a i n s v a l i d f o r any r e l a x a t i o n p a r a m e t e r 0 < w < 2 .

J

I t can i n f a c t be shown t h a t i f J w = (l-w)I+wJ

is a firm contraction,

then

i s a l s o a f i r m c o n t r a c t i o n f o r 0 < w < 1 from which

- xn I

w e d e d u c e t h a t I xn+l

0 and t h e weak c o n v e r g e n c e .

+

For

1 < w < 2 w e can e s t a b l i s h t h e i n e q u a l i t y

which a l l o w s u s t o p r o v e t h a t I xn+l If

T

x n l X again converges t o 0 .

( w i t h modulus a > O ) , i . e .

i s coercive (Z'-Z,X'-X)~ 2

(2.9)

-

ciIx'-xl~ Yx,x'

E

X, Vz

E

if

T(x), Yz' E T(x'),

t h e n i t can e a s i l y be shown t h a t

-

(2.10)

~.JC~(X')

J~rn (x)lX< (]+am)-' Ix'-xIx

JEn

which i m p l i e s t h a t

has a unique f i x e d p o i n t

unique s o l u t i o n s a t i s f y i n g 0 THEOREM 2 . 2 :

I f

T

Vx,x'

E

E

H,

x

which i s t h e

T(x).

i s coercive, the proximal-point

(2.3) generates a sequence

algorithm

{ x n l w h i c h c o n v e r g e s s t r o n g l y and

l i n e a r l y t o t h e unique s o l u t i o n

x

of

the multivalued equation (2.1).

If

rn

with

Proof: The i n e q u a l i t y ( 2 . 1 0 ) c a n b e w r i t t e n , w i t h x ' x a s o l u t i o n of ( 2 . 1 ) , as

+

+m,

t h e convergence i s s u p e r l i n e a r . =

xn and

(SEC. 2 )

PROXIMAL-POINT

309

ALGORITHM

which i m p l i e s t h e s t r o n g convergence and which g i v e s an e s t i m a t e o f t h e r a t e of convergence. linear since 1im n++m

Remark 2 . 4 :

I xn+]-x I I xn-xlx

I f rn

+

+

m,

t h e convergence i s s u p e r -

= o .

The i n t r o d u c t i o n of t h e r e l a x a t i o n mentioned i n

Remark 2.3 e n a b l e s t h e convergence of t h e p r o x i m a l - p o i n t be a c c e l e r a t e d i f t h e parameter

w

i n t h e recurrence

a b l y chosen (assuming f o r s i m p l i c i t y t h a t that

r

algorithm t o

(2.7) is suit-

is fixed).

i s c o e r c i v e ( c o n d i t i o n ( 2 . 9 ) ) and a l s o t h a t i t i s u n i f o r m l y

T

Lipschitz-continuous,

i.e.

(T then being single-valued)

w i t h , n a t u r a l l y , M t a. W e now i n t r o d u c e t h e n o t a t i o n Xn+1/2 = by d e f i n i t i o n , w e have xn = xn+1/2+rzn+1/2, zn+li2

Suppose

=

z:n'1/2),

where

and from ( 2 . 1 0 ) we have

W e can w r i t e ( 2 . 7 ) i n t h e form

and w e o b t a i n ( f o r w > 1) t h e e s t i m a t e

The c o e f f i c i e n t on t h e r i g h t i s a minimum f o r w * = l + a / r d for

r

sufficiently large);

(w* < 2

it t h e n f o l l o w s t h a t

This estimate is not very sharp

b u t , compared w i t h ( 2 . 1 1 ) ,

it demon-

s t r a t e s t h e a c c e l e r a t i o n of t h e r a t e of convergence produced by overrelaxation.

m

B e f o r e c o n c l u d i n g t h i s s e c t i o n we s h o u l d mention a r e l a t i o n which e x i s t s between t h e r e s o l v e n t s o f t h e o p e r a t o r T - I which w i l l be used l a t e r on i n t h i s C h a p t e r

T

and i t s i n v e r s e (see S e c t i o n 5 ) .

METHOD O F MULTIPLIERS & V . I . ' S

310

G i v e n r > 0 , We p u t

PROPOSITION 2 . 2 :

=

&

(CHAP.

9)

we t h e n have t h e

l/r;

relation

Proof: z

E

by d e f i n i t i o n w e have x = y + r z

We p u t y = J;x:

T ( y ) , from which w e deduce t h a t y

E

T-I

(E

(x-y) )

.

with

W e therefore

obtain EX

E

(I+ET-l

which t h e n g i v e s ( 2 14).

3.

VARIATIONAL INEQUALITIES I N D U A L I T Y

In t h e following,

and

V

d e n o t e r e a l H i l b e r t s p a c e s equipped

H

respectively with t h e inner products ( - , - ) denote t h e corresponding d u a l spaces.

and

< * , - > H , x H t h e b i l i n e a r forms of d u a l i t y between

between onto

and

H'

and o f

V'

H,

and by

H

onto

hV H'

and

hH

and

V'

W e d e n o t e by V'

H'

< * , * > v , x and v and

and

V

t h e isomorphisms of

V

d e f i n e d r e s p e c t i v e l y by

< A ~ , V => (~U ,,V ~) ~ ~Y U , V E V

with A p c V ' ,

I n many c a s e s i t it p o s s i b l e t o i d e n t i f y

w i t h i t s d u a l and w e

H

A H = I.

t h e n have

Let A : V

-f

2'

b e a ( m u l t i v a l u e d ) maximal monotone o p e r a t o r on

V , w i t h domain

Let B : V B'

-t

H b e a c o n t i n u o u s l i n e a r o p e r a t o r from

be i t s a d j o i n t (B'

< B ' g ' ,v>v, F : H

+

I-m,+ml

E

d(H',V'))

H,

let

d e f i n e d by

= < q ' , B v > ~ Vv , ~ E ~V , Vq'

E

be a lower semi-continuous,

w i t h e f f e c t i v e domain

V into

H'

and l e t

p r o p e r , convex f u n c t i o n

V.I.'s

(SEC. 3 )

dom(F) = { q l q E H, F(q)
0, E

and i n i t i a l a p p r o x i m a t i o n s

a F ( p o ) , define t h e s e q u e n c e s

recurrence :

Ao

(ALG2) :

E

H I po

Iun},Ipn},IAnl

E

H

such

by t h e

(SEC. 5 )

DECOMPOSITION ( I ) A . D .

( i ) Xn and pn b e i n g known, s e e k variational inequality

(5.18)

3wn+Ic A ( u n + l )

( i i ) u n + l b e i n g known,

such t h a t

un+l

(W"+],V)~+

321

METHODS

V

E

satisfying the

(Xn-rpn+rBun+l,Bv)H= 0

VVC V ;

of t h e m i n i m i s a t i o n

s e e k a s o l u t i o n pn"

problem:

u p d a t e t h e m u l t i p l i e r s by

(iii)

I n t h i s form, t h e method can be s e e n t o be a v a r i a n t of a l g o r i t h m A L G l of S e c t i o n 4 i n which t h e problem ( 4 . 1 8 ) ,

(4.19)

is solved i n

approximate f a s h i o n by p e r f o r m i n g o n l y a s i n g l e r e l a x a t i o n s t e p . T h i s g e n e r a l i s e s a l g o r i t h m ALG2 of C h a p t e r I11 t o t h e c a s e of Problem ( 5 . 1 8 ) i s e s p e c i a l l y s i m p l e

variational inequalities.

i s an a f f i n e s i n g l e - v a l u e d o p e r a t o r s i n c e , f o l l o w i n g an a p p r o p r i a t e d i s c r e t i s a t i o n , it reduces t o t h e s o l u t i o n of a l i n e a r s y s t e m w i t h a m a t r i x w h i c h is c o n s t a n t d u r i n g t h e c o u r s e o f t h e i t e -

when

A

rations.

Problem ( 5 . 1 9 ) c o n s i s t s of t h e m i n i m i s a t i o n o f a s t r o n g l y

convex f u n c t i o n which i s i n d e p e n d e n t of

B;

w e can t h e r e f o r e s o l v e

t h i s e a s i l y u s i n g an i t e r a t i v e method, even i f

B

is ill-conditioned;

w e have a c h i e v e d a d e c o u p l i n g of t h e d i f f i c u l t i e s r e l a t i n g t o F and t o B. F i n a l l y , we note t h a t i f F has a separable s t r u c t u r e , i.e. i f H Hi,

can b e w r i t t e n a s t h e C a r t e s i a n p r o d u c t of

....m

i = 1,

and i f

F

m

spaces

is d e f i n e d a s t h e sum

m (5.21)

1

F =

Fi

i=l

of f u n c t i o n s Fi

:

Hi

+

I-m,+ml

which a r e l o w e r s e m i - c o n t i n u o u s ,

p r o p e r and convex, t h e n p r o b l e m ( 5 . 1 9 )

p r o b l e m s o n e a c h of t h e

Hi

decomposes i n t o

of t h e form:

m

independent

METHOD OF MULTIPLIERS & V. I .

322

where t h e s u b s c r i p t THEOREM 5 . 1 :

i

d e n o t e s t h e component p e r t a i n i n g t o H i .

Suppose t h a t t h e v a r i a t i o n a l i n e q u a l i t y

a d m i t s a t l e a s t one s o l u t i o n

u

itn

d e f i n e d and c o n s t r u c t s a s e q u e n c e weakly t o

t

such t h a t

i n e q u a l i t y (4.4). Proof:

h = JE(t)

The s e q u e n c e

(4.1)

i n t ( d o m ( A ) ) and t h a t o n e o f t h e

E

(4.7) - (4.8) i s s a t i s f i e d .

assumptions

(CHAP. 9 )

' S

Then a Z g o r i t h m ALGZ i s w e l l =

hn+rpn}

which converges

s a t i s f i e s t h e dual v a r i a t i o n a l

{An} i s b o u n d e d and

IXn+l-AnlH

-t

0.

The t h e o r e m f o l l o w s from Theorem 3 . 1 a n d f r o m a n a n a l y s i s

of t h e convergence o f a l g o r i t h m ( 5 . 3 )

(see LIONS-MERCIER [I], S e c t i o n

1.3, Proposition 2 ) . W e can conclude t h a t t h e sequence

Ji or

if

JE i s a compact mapping.

X

converges weakly t o

CX"}

F i n a l l y , w e g i v e an e s t i m a t e o f

t h e r a t e of c o n v e r g e n c e i n t h e s p e c i a l case w h e r e ( a F j - l i s b o t h Lipschitz-continuous

a

2

y ) , i.e.

( w i t h c o n s t a n t y ) a n d c o e r c i v e ( w i t h modulus

i f f o r a l l q,q'

E

H and f o r a l l h

t

a F ( q ) , h'

E

aF(q'):

(5.24)

THEOREM 5 . 2 :

S u p p o s e t h a t t h e a s s u m p t i o n s o f Theorem 5 . 1 a r e

s a t i s f i e d and f u r t h e r m o r e t h a t constant

(aF)-l

y ) and c o e r c i v e ( w i t h m o d u l u s

i s Lipschitz-continuous

a

2

y).

Then t h e sequence

{An} d e f i n e d b y a l g o r i t h m ALGZ c o n v e r g e s s t r o n g l y t o

of t h e duaL v a r i a t i o n a l i n e q u a Z i t y

(with

A, a s o l u t i o n

(4.4), and

I n p a r t i c u l a r , t h e r e e x i s t s an o p t i m a l p a r a m e t e r

r*

f o r w h i c h we

get the estimate (5.26)

Proof:

S e e LIONS-MERCIER 111, S e c t i o n 1 . 3 ,

Remark 5 . 1 :

Proposition 4

Using t h e r e l a t i o n s h i p e s t a b l i s h e d i n P r o p o s t i o n 2 . 2

(SEC.

5)

DECOMPOSITION

METHODS

( I ) A.D.

323

between t h e r e s o l v e n t s of aF* and i t s i n v e r s e aF, we can r e - e x p r e s s t h e recurrence

( 5 . 3 ) d e f i n i n g a l g o r i t h m ALG2 i n t h e form

where

Using t h e e q u a t i o n ( 4 . 1 1 )

E=l/r.

shown t h a t

Etn

(5.28)

E t n+l =

where

?:

defining

a l s o s a t i s f i e s the recurrence

-

Ji((2JiF-I) (Etn))+(I-JiF)(Et"),

i s t h e c o n t r a c t i o n d e f i n e d on

H

(when ( 4 . 7 ) o r ( 4 . 8 ) i s

s a t i s f i e d ) by

The f o r m u l a t i o n ( 5 . 2 8 )

i s i n t e r e s t i n g f o r two r e a s o n s :

it expresses

a l g o r i t h m ALG2 i n t h e terms of t h e p r i m a l v a r i a t i o n a Z i n e q u a l i t y ( 4 . 1 ) and i t i n t r o d u c e s n a t u r a l l y t h e new o p e r a t o r

? :

.

I n t h e s p e c i a l c a s e where V = H and B = I , r e l a t i o n ( 5 . 2 8 )

is i n fact

i d e n t i c a l t o t h e Douglas-Rachford a l g o r i t h m ( 5 . 3 ) f o r s o l v i n g t h e multivalued equation (5.30)

0

E

Au + a F ( u ) ,

associated with t h e primal v a r i a t i o n a l i n e q u a l i t y ;

it thus generates

t h e same i t e r a t e s a s when it i s a p p l i e d t o t h e d u a l v a r i a t i o n a l i n equality

.

5.2

The Peaceman-Rachford a l g o r i t h m ALG3

v a r i a n t o f t h e method o f m u l t i p l i e r s :

W e u s e t h e same i n i t i a l i s a t i o n ( 5 . 8 ) , again put (5.31)

tn = Xn+rp"

with

w e have (5.32)

(2.J:-I) (t") = X"-rp"

X"

= J rS( t n ) ;

( 5 . 9 ) a s b e f o r e , and we

324

METHOD O F MULTIPLIERS & V . I . ' s

(CHAP.

9)

I t i s now c o n v e n i e n t t o i n t r o d u c e

An+' /'

(5.33)

n+l u

where

= J r (An-rpn) =

R

,

X"-rpn+rBun'l

i s a g a i n d e f i n e d by ( 5 . 1 4 ) .

The r e c u r r e n c e ( 5 . 4 )

is

written tn+' = Jr (A"-rp") R

(5.34)

with

pn+l

+ (Ji-I) (X"-rp")

=

A n+1/2+rBun+l

now d e f i n e d by

W e can t h e r e f o r e d e s c r i b e a l g o r i t h m ( 5 . 4 )

(5.35),

(5.36),

The P.R.

by u s i n g ( 5 . 1 4 ) ,

v a r i a n t of t h e method o f m u l t i p l i e r s (ALG3) :

r > 0 , and i n i t i a l a p p r o x i m a t i o n s Ao E H and po E H Xo E aF(po), d e f i n e t h e s e q u e n c e s {un},{pn},{An} v i a t h e

Given such t h a t recurrence.. (i)

Given

find

An,pn,

un+l

E

v

s a t i s f y i n g the variationa2

equation 3wn+l

E

A(un+')

such t h a t

(5.37) (wn+',v)V+(Xn-rpn+rBun+l , B v ) ~= 0 V v E V ;

(ii)

(iii)

(5.33),

a s below.

Update t h e m u l t i p l i e r s :

Find

p

n+l

E

H

s a t i s f y i n g t h e minimisation problem

(SEC.

DECOMPOSITION ( I ) A. D

5)

METHODS

325

2 H

1 (iv)

S

+ $IBun+]-qlH 2 Yq

F(q)-(A”+’/’,q),

E

H ;

Update t h e rnuZtipZiers

The Peaceman-Rachford Rachford v a r i a n t ( D . R . )

variant

( P . R. ) d i f f e r s from t h e Douglas-

only through t h e a d d i t i o n of t h e i n t e r m e d i a t e

update of t h e m u l t i p l i e r s ( 5 . 3 8 ) ;

it t h u s o f f e r s t h e same s e t o f

The Peaceman-Rachford v a r i a n t i s , however, less ‘ r o b u s t ’ ,

advantages.

i n t h a t it c o n v e r g e s under more r e s t r i c t i v e a s s u m p t i o n s t h a n t h e Douglas-Rachford v a r i a n t ;

n o n e t h e l e s s , a s w e s h a l l see, i f i t does

c o n v e r g e , t h e n i t s r a t e of convergence i s f a s t e r .

THEOREM 5 . 3 :

satisfied.

S u p p o s e t h a t t h e a s s u m p t i o n s of Theorem ( 5 . 1 ) a r e

Then t h e a l g o r i t h m ALG3 i s w e l l d e f i n e d and t h e s e q u e n c e s

and I t n = An+rpn} a r e bounded i n H ; there e x i s t s a n e x t r a c t e d s u b s e q u e n c e of I t n } w h i c h c o n v e r g e s w e a k l y t o t E H {Bun},{pn},{Xn}

such t h a t Proof:

X

=

Jg(t) s a t i s f i e s

t h e duaZ v a r i a t i o n a l i n e q u a l i t y

The f i r s t p a r t o f t h e theorem f o l l o w s from LIONS-MERCIER

113, S e c t i o n 1 . 2 , P r o p o s i t i o n 1. J;

We next note that, since

a r e f i r m contractions ( i n t h e sense of

r

(2Js-I)

(4.4).

Jk

and

( 2 . 3 ) ) , then, ( 2 J k - I ) and

a r e themselves c o n t r a c t i o n s , a s i s t h e i r product.

Since t h e

r s e t o f f i x e d p o i n t s o f ( 2 J i - I ) ( 2 J s - I ) i s nonempty ( s i n c e by h y p o t h e s i s t h e r e e x i s t s a t l e a s t one s o l u t i o n of t h e e q u a t i o n 0 E R ( p ) + S( p ) ) , w e c a n e x t r a c t from I t n } a subsequence which c o n v e r g e s weakly t o one of these fixed points

t.

W e s h a l l now g i v e an e s t i m a t e f o r t h e r a t e o f convergence, under t h e same a s s u m p t i o n s a s f o r a l g o r i t h m ALG2. THEOREM 5 . 4 :

satisfied.

S u p p o s e t h a t t h e a s s u m p t i o n s of T h e o r e m - 5 . , 2 a r e

Then t h e sequence

v e r g e s s t r o n g l y t o A, have

{An}

d e f i n e d b y a l g o r i t h m ALG3 con-

a s o l u t i o n of t h e d u a l i n e q u a l i t y

(4.4),

and we

326

METHOD O F MULTIPLIERS & V . I .

(CHAP. 9 )

' S

(5.41)

In particular,

t h e r e e x i s t s an o p t i m a l param e t e r

for w h i c h

r*

We

have t h e e s t i m a t e (5.42)

Proof:

A s condition (5.23) states t h a t

(aF)-'

is Lipschitz-

continuous, t h i s implies t h a t

S i n c e t h e mapping ( 2 J r - I ) R

i s a c o n t r a c t i o n , w e have

I (2Ji-I)tn -

Itn+'-tlH
0

Algorithm ( 6 . 3 )

t h e algorithm

can be i n t e r p r e t e d a s a d i s c r e t i s a -

t i o n scheme f o r t h e m u l t i v a l u e d e v o l u t i o n e q u a t i o n

which i s e x p l i c i t r e l a t i v e t o

r

and i m p l i c i t r e l a t i v e t o

R

denotes t h e time s t e p , t h e n ( 6 . 3 )

S;

if

d e f i n e s t h e s o l u t i o n of t h e

d i s c r e t i s e d equation

n+ 1 -xn O

(6-5)

E

X

+ R(xn) + S(x"+').

If

THEOREM 6 . 1 :

i s Lipschitz-continuous

R

and c o e r c i v e ( w i t h modulus (6.3) 0
0

Given

(i)

(ii)

find

An,

V,

E

de'fine

un + l , s a t i s f y i n g t h e v a r i a t i o n a l e q u a t i o n

(A(U~+')+A",V)~ = 0

(6.17)

Xo

and a n i n i t i a l a p p r o x i m a t i o n

v i a the recurrence:

YveV ;

Update t h e m u l t i p l i e r s :

An+]

(6.18)

= (I-P&An+,un+l).

Theorem 6 . 1 g i v e s t h e c o n d i t i o n s f o r convergence of t h e method. COROLLARY 6 . 1 :

Suppose t h a t t h e v a r i a t i o n a l i n e q u a l i t y (6.9)

a d m i t s a t l e a s t o n e s o l u t i o n and t h a t t h e o p e r a t o r ( w i t h modulus a sequence

0).

Remark 6 . 2 : V

X

{An) w h i c h c o n v e r g e s w e a k l y t o

dual variational inequality

if

A

i s coercive

T h e n a l g o r i t h m ALG4 i s w e l l d e f i n e d a n d g e n e r a t e s E

(6.13), for a l l 0

I f t h e convergence of

V, a s o l u t i o n o f t h e

r

2a.

{An] i s s t r o n g ( f o r example

i s of f i n i t e d i m e n s i o n ) , i t can b e shown t h a t t h e sequence

{An} c o n v e r g e s l i n e a r l y and t h a t t h e r e e x i s t s an o p t i m a l s t e p

Remark 6 . 3 :

Algorithm ( 6 . 1 7 )

,

r*.

( 6 . 1 8 ) e f f e c t s a decomposition

o f t h e v a r i a t i o n a l i n e q u a l i t y ( 6 . 9 ) w h e r e i n t h e problem r e l a t i n g t o A and t h a t r e l a t i n g t o t h e c o n s t r a i n t v

t h e i n t r o d u c t i o n of t h e m u l t i p l i e r s

7.

E

K become decoupled due t o

An.

GENERAL DISCUSSION

T h i s c h a p t e r , which i s o f a d i s t i n c t l y more a b s t r a c t c h a r a c t e r t h a n t h e p r e c e d i n g c h a p t e r s , h a s p r i m a r i l y been aimed a t d e m o n s t r a t i n g t h e l i n k s which e x i s t between t h e augmented-Lagrangian method and some of t h e well-known methods of n o n - l i n e a r

analysis.

These

(SEC. 7 )

DISCUSSION

331

c o n n e c t i o n s may s e r v e t o s u g g e s t some new a p p r o a c h e s f o r t h e s t u d y of a l g o r i t h m s and f o r t h e o p t i m i s a t i o n of t h e p a r a m e t e r s which c o n t r o l convergence.

I n p a r t i c u l a r , w e b e l i e v e t h a t t h e t e c h n i q u e s developed

i n t h i s c h a p t e r s h o u l d a l l o w an i n v e s t i g a t i o n o f t h e convergence of v a r i a n t s of a l g o r i t h m ALGl i n which a r e Z a x a t i o n p a r a m e t e r is

introduced t o a c c e l e r a t e t h e convergence o f t h e i n n e r i t e r a t i o n s . T h i s p r o c e d u r e f o r a c c e l e r a t i n g convergence h a s a c t u a l l y been used i n C h a p t e r I i n t h e s i m p l e r c o n t e x t of q u a d r a t i c f u n c t i o n a l s and l i n e a r constraints.


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[11 FORTIN M . , SOULIE M. ment on t h e t r i a n g l e

A second-order (to appear).

non-conforming

f i n i t e ele-

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Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems

Augmented Lagrangian methods

Numerical Solution of Elliptic Problems

Numerical Solution of Elliptic Problems

Numerical Methods and Applications

Numerical Methods in Bifurcation Problems

Numerical methods for exterior problems

Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics

Numerical Methods for Exterior Problems

NUMERICAL METHODS FOR EXTERIOR PROBLEMS

Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (Studies in Applied and Numerical Mathematics)

Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (Studies in Applied and Numerical Mathematics)

Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (Studies in Applied and Numerical Mathematics)

Economic Growth: Theory and Numerical Solution Methods

Methods and Theories for the Solution of Problems of Geometrical Constructions Applied to 410 Problems

Methods and theories for the solution of problems of geometrical constructions applied to 410 problems

Numerical Solution of Integral Equations

Numerical Solution of Elliptic Equations

Numerical Solution of Differential Equations

Numerical solution of initial-value problems in differential-algebraic equations

Numerical solution of initial-value problems in differential-algebraic equations

Numerical solution of two point boundary value problems

Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations

Numerical solution of two point boundary value problems

Numerical Solution of Two Point Boundary Value Problems

Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

Numerical Solution of Integral Equations

Numerical Solution of Nonlinear Equations

Numerical Solution of Elliptic Equations

Numerical Methods for Least Squares Problems

Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems