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.
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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
Numerical s o l u t i o n of t h e s t a t i c problem .
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
Numerical s o l u t i o n of t h e s t a t i c problem .
................. ...................
(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)
AUGMENTED LAGRANGIAN METHODS
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
sequence Of
(2r
and f o r a l l
po e I R M
the
- ( 2 . 3 ) converges t o the solution
Proof: {u,p}
This follows G.L.T.
i s a s a d d l e - p o i n t of
dr
[I, C h a p t e r 2 , S e c t i o n 41'.
.
Suppose
From Theorem 1.1, t h i s s a d d l e -
p o i n t i s c h a r a c t e r i s e d by
(2.5)
(A
+
t KB B ) u + Btp = b
,
(2.6)
BU
=
o
BU, 'dn.
C +
p = p + p,
h e r e i n a f t e r abbreviated t o G.L.T.
1 2 1 ( E n g l i s h t r a n s l a t i o n of G . L . T .
'
u
(1.2).
See a l s o G.L.T.
121.
( s e e a l s o Appendix 2 of G . L . T .
Ell)).
A FIRST ALGORITHM
(SEC. 2 )
W e introduce
from ( 2 . 4 )
-n p
iin = un - u ,
and ( 2 . 6 )
- p.
= p"
5
Then by s u b t r a c t i n g ( 2 . 5 )
from ( 2 . 3 ) w e o b t a i n
(2.8)
W e deduce from ( 2 . 8 )
that
and h e n c e t h a t
I t follows f r o m (2.7) t h a t
and h e n c e by s u b s t i t u t i o n i n ( 2 . 9 ) w e h a v e
If w e t h e n t a k e
O
(2.1 I )
0-
lpnl
t h e sequence
n-
lim
(2.12)
B e i n g b o u n d e d b e l o w by
is decreasing.
i s c o n v e r g e n t and h e n c e
lpn12
-
(A;~,;~)
IHn+'l2 =
+
0 ;
(2:lO)
0 , it
then implies
o,
n++m
a n d s i n c e A i s p o s i t i v e d e f i n i t e it f o l l o w s f r o m ( 2 . 1 2 ) t h a t -n u + 0 , a n d a l s o s i n c e 8" = un - u, w e h a v e lim u n = u . rn n++m Remark
2.1:
v e r g e n c e of
(2.1)
I t f o l l o w s from (2.10) t h a t w e a c t u a l l y h a v e con-
- ( 2 . 3 ) u n d e r t h e f o l l o w i n g c o n d i t i o n , which i s
less r e s t r i c t i v e t h a n ( 2 . 1 1 ) :
(CHAP. 1)
AUGMENTED LAGRANGIAN METHODS
6
where
B2
i s d e f i n e d by
(2.14)
i s the largest e i g e n v a l u e of
which i m p l i e s t h a t
.
A-'BtB
With a view t o s t u d y i n g t h e b e h a v i o u r of t h e sequence
pn
it i s
worth n o t i n g t h a t (2.15)
( I m B)'
= Ker
Bt,
hence
M = Im B fB K e r Bt.
R
(2.16)
We t h u s h a v e , f o r (2.17)
q
Denoting by
=
t h e unique decomposition
q 1 + q2
, q , e I m B , q 2 E K e r Bt.
( r e s p . P 2 ) t h e p r o j e c t o r of
P1
(resp. K e r Bt),
,
LfqERM
lRM o n t o
Im B
we t h u s have Pi
E
M M &(R ,R )
Lfi= 1 , 2 ,
(2.18)
Pi(q)
I f , additionally,
p
vqeRM
= qi
,v~=I,z.
i s a Lagrange m u l t i p l i e r f o r (1.2),
(1.4),
t h e n w e can deduce from Theorem 1.1 t h a t t h e same i s a l s o t r u e of p+q,
unique
Lfq E K e r Bt f i d m B,
.
I t f o l l o w s from t h i s t h a t t h e r e e x i s t s a
such t h a t t h e Lagrange m u l t i p l i e r s of
(1.21,
(1.4)
a r e a l l of t h e form (2.19)
p = F+q
,
q
E
Ker Bt.
h
The v e c t o r
p t h u s a p p e a r s a s t h e Lagrange m u l t i p l i e r of
with minimal norm i n
(1.2),
(1.4)
lRM.
From Theorem 2 . 1 and t h e above p r o p e r t i e s of t h e Lagrange m u l t i p l i e r s w e s h a l l now deduce:
A F I R S T ALGORITHM
2)
( S E C.
If
THEOREM 2 . 2 :
satisfies
pn
7
(2.13) t h e n t h e sequence
d e f i n e d by a l g o r i t h m (2.1) - (2.3) converges t o P;
=
P2(P011
po = 0
is t h e c o m p o n e n t o f lim
then
in
po
Ker B
t
6 + p; . In
pn
where p a r t i c u l a r if
pn=c.
n++m
The r e l a t i o n ( 2 . 3 ) i m m e d i a t e l y i m p l i e s
Proof:
Under t h e c o n d i t i o n ( 2 . 1 3 ) , w e h a v e from ( 2 . 7 ) t h a t
Since t h e quantity projection onto
2.3
B~
lim n++m
(2.21)
jBtql
pn
l i m En = 0. n++m
= 0.
d e f i n e s a norm on
ImB,
(2.21)
i m p l i e s by
that
ImB
Interpretation of algorithm (2.1) gence i f
I t then follows
pn
= p ,
and c h o i c e o f
-
(2.3).
R a t e o f conver-
r.
Algorithm (2.1)-(2.3) i s i n f a c t a g r a d i e n t t y p e algorithm applied t o t h e m i n i m i s a t i o n o f t h e d u a l f u n c t i o n a l :J : IRM +IR d e f i n e d by
J,* (4) = (2.22)
where
-
v
Mi%
e,(v,q)
A,=
-(BA-'b,q)'
+ 7 1 (A-Ib,b)
A + r B~B.
More p r e c i s e l y , by e l i m i n a t i n g re-expressed
= T(BA;lBtq,q) 1
E R
un, a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 )
can be
as
(2.23)
Remark 2 . 2 :
p0e RM,
specified arbitrarily;
The a d v a n t a g e o f f o r m u l a t i o n ( 2 . 1 ) - ( 2 . 3 )
compared
8
AUGMENTED LAGRANGIAN METHODS
(2.24)
with (2.23), explicitly;
i s t h a t w e do n o t have t o c o n s t r u c t
(CHAP.
1)
-1
Ar
i n certain applications i n p a r t i a l differential
e q u a t i o n s t h i s would i n p r a c t i c e b e u n r e a l i z a b l e s i n c e a f u l l m a t r i x of v er y l a r g e o r d e r .
Ail
The f o r m u l a t i o n ( 2 . 2 3 ) ,
w i l l be (2.24)
w i l l on t h e o t h e r hand b e v e r y u s e f u l as a t h e o r e t i c a l b a s i s f o r
studying the influence of
r
and
on t h e c o n v e r g e n c e of
p,
(2.1)-(2.3).
I n s i m i l a r f a s h i o n , by e l i m i n a t i n g
pn,
(2.1)-(2.3)
c a n b e re-
expressed as (2.25)
uo = A;l(b-Btpo)
(2.26)
"n+l = un -
, ,
P,A;IB~BU"
and Remark 2.2 h o l d s e q u a l l y f o r t h e a l g o r i t h m ( 2 . 2 5 ) , T o s t u d y t h e i n f l u e n c e of
(2.1)-(2.3),
r
w e observe t h a t i f
and
pn
pn
(2.26).
H
on t h e c o n v e r g e n c e o f (fi + P Z ( p o ) ) t h e n f r o m
= pn -
( 2 . 2 4 ) w e have
(2.201,
(2.27)
From t h e s e c o n d r e l a t i o n i n ( 2 . 2 7 ) w e d e d u c e t h a t
A - 1 B t-n+l P
(2.28)
= A-lgtpn
-
~,A-IB~BA;'B~~~.
W e have
which t o g e t h e r w i t h ( 2 . 2 8 ) i m p l i e s A-lgtpn+l = A-lBtpn - pnA-lBtB(I
(2.30)
W e put yn
yn = A
and
t h a t {Ti",
En, pn}
-1 t-n B p
pn
+
rA-l
B tB ) - I A - 1 B t-n p .
; g i v e n t h e l i n e a r r e l a t i o n s e x i s t i n g between
t h e c o n v e r g e n c e of
yn
c o n v e r g e s t o w a r d s IO,O}.
towards z e r o t e l l s u s W e deduce from ( 2 . 3 0 ) t h a t
(SEC. 2)
9
A FIRST ALGORITHM
-1 t We shall express ( 2 . 3 1 ) in a b a s i s of e i g e n v e c t o r s of A B B, but before doing so we shall first indicate, without proof, several properties of the eiqenvectors and eiqenvalues of A-'BtB:
PROPOSITION 2.1:
A-'BtB
The e i g e n v a l u e s o f
are
Z
0
and t h e
e i g e n v e c t o r s c o r r e s p o n d i n g t o two d i s t i n c t e i g e n v a l u e s a r e A-orthogona2,
i.e.
if
II with
Ai
+
Aj
A-IB~ A
-1
B
B w. =
A. w.
B
A. w.
1
t
W. =
J
1
1
J
J
then (A wi,w.) = 0. 3
PROPOSITION 2 . 2 :
If
0
rn
i s an e i g e n v a l u e o f
corresponding eigen-subspace
is
Ker B
and
A-'BtB
Im A-lBt
then the
is t h e
RN w h i c h is A - o r t h o g o n a l t o Ker B. Im A-lBt is -1 t t h u s spanned b y t h e e i g e n v e c t o r s o f A B B a s s o c i a t e d w i t h t h e e i g e n v a l u e s which are d i s t i n c t f r o m 0 . rn s u b s p a c e of
We have, of course, dim Im A-lBt
Rank Bt = Rank B.
In the following text we shall denote by N1 the rank of B and by respectively, the smallest non-zero eiqenvalue and the Am, AM, % ,: is a basis largest eiqenvalue of A-lBtB. Suppose with wi the eigenvector of A B B associated with of Im A-'Bt, =
pi}!Ll
the eigenvalue
Xi.
If
y
e
Im A-lBt
N1
y =
Z
i=l
We then deduce from
y " Im ~
that
1
v ~ = I ,. ..N ~ .
yi wi
,
A-'B~ V n 2 0
we thus have
y i E ~Vi.
,
and from (2.31)
10
AUGMENTED LAGRANGIAN METHODS
(CHAP.
I n t h e r e m a i n d e r of S e c t i o n 2 . 3 w e s h a l l assume t h a t
P,
=
P
1)
vn,
p o s t p o n i n g u n t i l S e c t i o n 3 t h e s t u d y of a l g o r i t h m s of t h e t y p e
(2.1)
- (2.3)
with variable s t e p s i z e n+l Y.
(2.33)
-
1
V - 0 ) A .i
+
xi
1 + r
W e s h a l l f i r s t examine
pn.
W e t h u s have
n Yi.
a number o f p o s s i b l e c h o i c e s f o r
p , with
g i v e n , a s t h i s w i l l a l l o w u s t o draw a number of c o n c l u s i o n s con-
r.
cerning t h e choice of (i)
r
The c a s e
=
0.
W e h e r e have (2.34)
Yi
=
(1
- pxi)y?
From a n i n s p e c t i o n o f t h e
V ~ OV,i = l , ...N 1
( c l a s s i c a l l y familiar) Figure 2 . 1
I I-PXI
P
Figure 2 . 1
t h e optimal choice for
p
(2.35)
for
p = p
opt
we t h e n h a v e
can be seen t o be
r
11
A FIRST ALGORITHM
(SEC. 2 )
(2.36)
w e deduce from t h i s t h a t t h e c o n v e r g e n c e r a t e satisfies, for
R
of t h e metho?
= popt,
p
(2.37)
Remark 2.3:
If
r
=
0 , a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) r e d u c e s t o Uzawa's
a l g o r i t h m a p p l i e d t o t h e Lagrangian
Z
d e f i n e d by (1.4), namely
specified arbitrarily;
(2.38)
p 0 c R',
(2.39)
Aun + Bt pn = b
(2.40)
pn+' = p" + p,
Bun.
The r e s u l t ( 2 . 3 7 ) i s a s t a n d a r d one i n t h e s t u d y of t h e convergence
(see f o r example, CEA 1 1 1 ,
o f f i x e d s t e p g r a d i e n t methods
MARCHOUK-KUZNETSOV 111 and t h e b i b l i o g r a p h y of t h e s e v o l u m e s ) . (ii)
The case
p
= 2r.
I n t h i s c a s e w e deduce from ( 2 . 3 3 ) t h a t
(2.41)
,+I
'i
-
A s i n d i c a t e d by F i g u r e 2 . 2 ,
1
- 'Xi y4
'hi
I +
=opt =
....N,,
Vnzo.
t h e optimal choice f o r
1
(2.42)
Vi=l
r
is
AUGMENTED LAGRANGIAN METHODS
12
(CHAP.
1)
I
Figure 2.2
r
and f o r
=
r opt,
P =
w e have
2 r opt,
W e t hereby deduce t h a t f o r t h i s c h o i c e o f
vergence rate
R
r
and
t h e con-
p
o f t h e method s a t i s f i e s
(2.44)
W e d e d u c e from ( 2 . 3 7 ) ,
5
tion r=-
1
+%
)
p=-
(2.44)
( a n d f r o m t h e b e h a v i o u r o f t h e fUnC-
t h a t algorithm (2.1)-(2.3) with 2 is i t e r a t i v e l y f a s t e r than algorithm
” t ’m n\l!”mhM
(2.38)-(2.40) with
pn = p =
2 ,
which w e r e c a l l c o r r e s p o n d s t c
’m+%
algorithm (2.1)-(2.3)
with
r = 0
and
pn
= p =
- .2
%
Am+
8
(SEC.
A FIRST ALGORITHM
2)
(iii)
The c a s e
13
r.
p =
T h i s c h o i c e i s t h e s t a n d a r d one; i n t h i s c a s e w e can deduce from (2.33) t h a t
and hence t h a t n+ 1
IYi
(2.45)
1
I
5
ynl
~
~ ,V n 2 0 .
onvergence r a t e
We deduce from ( 2 . 4 5 ) t h a t t h e (2.46)
Vi=l,...N
l + r X m
1
RS-
R
satisfies
,
I + r X
m
On i n s p e c t i o n of
(2.46)
i t appears t h a t f o r
= p =
pn
r
algorithm
( 2 . 1 ) - ( 2 . 3 ) becomes f a s t e r , i t e r a t i v e l y , a s t h e v a l u e o f larger.
r <<
I f , in particular,
1
r;-
, it
m
r
gets
f o l l o w s from ( 2 . 4 6 )
t h a t a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) w i l l i n g e n e r a l be i t e r a t i v e l y slow. 1 note t h a t i f r = - then R 3 7 . rn
We
?n
Remark 2 . 4 :
Although r e l a t i o n ( 2 . 4 6 ) a p p e a r s t o i n d i c a t e t h a t i t
i s advantageous t o work w i t h
pn
= p =
r
a s l a r g e a s p o s s i b l e , one
m u s t r e a l i s e t h a t a 2 1 o t h e r t h i n g s b e i n g e q u a l t h e d e t e r m i n a t i o n of n u i n ( 2 . 2 1 , i . e . t h e s o l u t i o n of t h e l i n e a r system (A + r BtB)un = b
(2.47)
- Bt p n ,
i s more c o s t l y ( i n computation t i m e and/or i n memory r e q u i r e m e n t s )
r.
t h e l a r g e r t h e v a l u e of
ing discussion, the matrix more i l l - c o n d i t i o n e d
I n f a c t , a s we s h a l l see i n t h e followt Ar = A + r B B becomes p r o g r e s s i v e l y
the larger
r
becomes.
Using, once a g a i n , a b a s i s of e i g e n v e c t o r s of shown t h a t t h e c o n d i t i o n number of t h e m a t r i x
restricted to t e n d s t o 1 when
-f
+
( i . e . t o t h e s u b s p a c e A-orthogonal
ImA-lBt
r
it can be
A-'BtB,
A-lBtB(I
+
m.
rA-lBtB)-'
to
Ker B )
This property c l e a r l y corresponds t o t h e
f a c t t h a t t h e t h e o r e t i c a l convergence of a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) 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 , i n t h e c a s e where
p =
r.
is
m
14
AUGMENTED LAGRANGIAN METHODS
(iv)
O p t i m a 2 c h o i c e of
w i t h given
p
From i n s p e c t i o n of F i g u r e 2 . 3 ,
(CHAP.
1)
r.
i t f o l l o w s from ( 2 . 3 3 )
F i g u r e 2.3
t h a t t h e o p t i m a l v a l u e of
p
i s t h e s o l u t i o n of t h e l i n e a r e q u a t i o n
hence 1
(2.48)
Popt
+ r(Am + ),A
=
(Am
+
),A
+
+ 2r
We deduce from ( 2 . 4 8 ) t h a t f o r
(2.49)
IYin+ 1
I I
Am A,
p = popt
lyrl
I + Am + 2rAm
and hence f o r t h e convergence r a t e
+
r ( A m + ),A
2r A
AM+(A m +A,)
2
r2
m
w e have
Vi=l,.. . N ~ , V n O ,
R, w e have
15
A FIRST ALGORITHM
(SEC. 2 )
(2.50)
W e note t h a t (2.1)-(2.3) p =
p > 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<’I/ .
of A r s a t i s f i e s
AUGMENTED LAGRANGIAN METHODS
16
(2.52)
r 11Bll2
+
1 SllArIl
5r
IIBII
2 +
(CHAP.
1)
lbll .
llA-lll
W e have moreover
t h e r e then e x i s t s
u
E
S
such t h a t
(2.53)
W e now s t u d y t h e b e h a v i o u r o f t h e f a m i l y
( u ~ when ) ~
r
+
+
m.
W e have
(2.54)
(Au,, ur)
+
rlBurlZ
I
(Av,v)
, Vv
~ K e rB n S ,
Vr
which i m p l i e s
i s bounded, w e can e x t r a c t f r o m it a subur s e q u e n c e , a l s o d e n o t e d b y u r , c o n v e r g i n g t o a n e l e m e n t u* O f IRN. Since t h e sequence
W e t h e n deduce from (2.55) t h a t
hence it f o l l o w s t h a t u
(2.57) I n v i e w of
* _Ker
B n S.
v = u
( 2 . 5 7 ) w e can t a k e
*
i n (2.54).
W e t h e n deduce
that (AU,.~,)
5 (~u,,u~)
+
r I B ~ , I 5~ ( ~ u * ,u*)
and hence t h a t
which t o g e t h e r w i t h ( 2 . 5 3 ) i m p l i e s (2.58)
lim r+-
1 = llA;lll
(AU*,
IJ*).
Vr
(SEC. 2 )
17
A FIRST ALGORITHM
I t t h e r e f o r e f o l l o w s from t h e above t h a t
I
u*
E
Ker B n S
thus
I / A- 1~ /I
lim
(2.59)
=
1 1 = -
(Au*,u*)
rt+m
'
which w i t h ( 2 . 5 2 ) i m p l i e s ( 2 . 5 1 ) .
W e have t h u s proved t h a t t h e c o n d i t i o n number of
is,
Ar
a s y m p t o t i c a l l y , p r o p o r t i o n a l t o r , which t h u s h a s t h e e f f e c t a s r i n c r e a s e s , of making i t n o r e d i f f i c u l : ,
( o t h e r things being e q u a l ) ,
t o s o l v e t h e system Ar un
(2.60)
= b
-
Bt p n .
I f w e s o l v e ( 2 . 6 0 ) by an i t e r a t i v e m e t h o d ,
t h e convergence, being
l i n k e d t o t h e c o n d i t i o n number, w i l l become s l o w e r a s t h e v a l u e of
r
i n c r e a s e s , and t h i s may l e a d t o a l a r g e number of i t e r a t i o n s t o
s o l v e ( 2 . 6 0 ) t o an a p p r o p r i a t e a c c u r a c y even i f , i n t h e obvious n n- 1 u with u
.
manner, w e i n i t i a l i s e t h e c a l c u l a t i o n o f
F u r t h e r m o r e , i f w e s o l v e ( 2 . 6 0 ) by a d i r e c t method, t h e s e n s i t i v i t y t o rounding-error
a c c u m u l a t i o n w i l l be g r e a t e r when
r
is large.
I n a l a r g e number of p r o b l e m s , t h e r e f o r e , a "good s t r a t e g y " would seem t o be t h e f o l l o w i n g :
*
Work i n " d o u b l e p r e c i s i o n "
*
With t h e p a r a m e t e r
of t h e m a t r i x definite
*
Take
r
h a v i n g a f i x e d v a l u e , as l a r g e a s
c a r r y o u t o n c e a n d f o r a l l t h e Cholesky f a c t o r i s a t i o n
possible (!),
Ar,
which w e r e c a l l i s symmetric and p o s i t i v e
. pn
= p = r.
R e m a r k 2.5:
I n t h e c a s e where an i t e r a t i v e method i s used t o
s o l v e ( 2 . 6 0 ) w e c a n , i n t h e e a r l y s t a g e s of a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) , n make do w i t h a l o w a c c u r a c y i n t h e d e t e r m i n a t i o n of u This
.
e f f e c t can be o b t a i n e d , f o r example, by c h o o s i n g t o u s e a f i x e d ( a n d " s m a l l " ) number of i t e r a t i o n s ( i n t h e s o l u t i o n of
(2.60)).
18
AUGMENTED LAGRANGIAN METHODS
(CHAP.
1)
W e s h a l l r e t u r n t o t h i s s u b j e c t i n Section 4 , i n connection with t h e method o f A r r o w - H u r w i c z .
Remark 2.6:
With r e g a r d t o t h e s o l u t i o n o f
( 2 . 6 0 ) by a n
i t e r a t i v e m e t h o d , it may b e a d v a n t a g e o u s t o u s e a p a r a m e t e r
(rn)n.
n , g i v i n g i n f a c t a sequence
which v a r i e s w i t h
( r n ) ns u c h t h a t
a u t h o r s recommend t h e u s e o f a s e q u e n c e
r
lim
=
r
Certain
+m.
n++m
The o p t i m a l c h o i c e f o r ( r n ) n seems t o b e a n open q u e s t i o n . u s e of s u c h a method combined w i t h a d i r e c t s o l u t i o n of
l i t t l e i n t e r e s t , s i n c e t h e f a c t o r i s a t i o n of
r
carried out every t i m e t h a t
>
r n-l,
would n e e d t o b e n t h i s being costly i n Ar
general.
8
VARIABLE STEP-LENGTH ALGORITHMS.
3. 3.1
The
( 2 . 6 0 ) i s of
CONJUGATE GRADIENT METHOD.
General n o t e s
W e h a v e shown
i n S e c t i o n 2.3 t h a t a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) c a n b e
i n t e r p r e t e d as a g r a d i e n t a l g o r i t h m a p p l i e d t o t h e minimisation of the dual functional
J
*
d e f i n e d by ( 2 . 2 2 ) .
With t h i s i n t e r p r e t -
a t i o n i n mind it i s n a t u r a l t o s e e k t o a p p l y , t o t h e m i n i m i s a t i o n of
J*
on
JRM, t h e s t a n d a r d i t e r a t i v e m e t h o d s f o r m i n i m i s a t i o n
of
(see f o r e x a m p l e , CEA 111, t h e r e v i e w a r t i c l e
quadratic functionals
o f MARCHOUK-KUZNETSOV C11 and t h e c o r r e s p o n d i n g b i b l i o g r a p h y f o r a In order to clarify the
thorough s t u d y o f t h e s e methods).
p r e s e n t a t i o n w e s h a l l now g i v e some review m a t e r i a l on t h e s e m e t h o d s .
c7 i s an
Suppose t h e n t h a t m a t r i x and s u p p o s e functional (3.1)
2
B
E
M
x
M
symmetric, p o s i t i v e d e f i n i t e
BM; w e a s s o c i a t e w i t h
d e f i n e d by 214) =
The m i n i m i s a t i o n p r o b l e m
1
7
(C79,q) - (B,q).
c 7
and
B
the
(SEC. 3 )
VARIABLE STEP-LENGTH ALGORITHMS
19
a d m i t s a u n i q u e s o l u t i o n which i s a l s o a s o l u t i o n o f t h e l i n e a r system a p = B .
(3.3)
To s o l v e ( 3 . 2 ) , ( 3 . 3 ) w e now c o n s i d e r d e s c e n t methods o f t h e g e n e r a l type : p0eRM,
specified arbitrarily;
p*+l
-
(3.4) = pn
pn wn
.
w w i l l i n g e n e r a l be deduced from t h e n t h e g r a d i e n t of 2 a t t h e p o i n t pn. For a given
The d e s c e n t d i r e c t i o n
d i r e c t i o n of
d e s c e n t d i r e c t i o n , w e s h a l l choose
pn
i n s u c h a way as t o o p t I n practice, we s h a l l
i m i s e a c r i t e r i o n r e l a t e d t o t h e problem. c o n f i n e o u r s e l v e s t o t h e f o l l o w i n g methods:
0
STEEPEST DESCENT METHOD: The d e s c e n t i s made i n t h e d i r e c t i o n o p p o s i t e t o t h e g r a d i e n t , hence w
(3.5)
The c h o i c e of
= gn = g r a d g ( p " )
=
dp"
-
B.
i s made by m i n i m i s i n g , w i t h r e s p e c t t o
pn
p,
the
function P
(3.6)
-+
a (P" -
P 9,)
.
W e have (3.7) pn
i s t h u s g i v e n by
(3.8)
0 MINIMUM RESIDUAL METHOD: t h e g r a d i e n t d i r e c t i o n , hence minimise, w i t h r e s p e c t t o
p,
The d e s c e n t i s s t i l l c a r r i e d O u t i n
wn
=
gn; w e choose
the residual
p,
lO(p"-pg")-BI
s o as t o
.
AUGMENTED LAGRANGIAN METHODS
20
t h e optimal
p
(CHAP.
1)
i s t h e r e f o r e g i v e n by:
(3.10)
0
C O N J U G A T E G R A D I E N T METHOD:
The c o n j u g a t e g r a d i e n t method
S
e s p e c i a l l y a t t r a c t i v e f o r s o l v i n g q u a d r a t i c problems b e c a u s e
t h e o r e t i c a 2 Z y ( i . e . i g n o r i n g r o u n d i n g e r r o r s ) it c o n v e r g e s i n a f i n i t e number of i t e r a t i o n s ( s M )
and b e c a u s e m o r e o v e r i n t h e
g e n e r a l c a s e it l e a d s t o q u a d r a t i c c o n v e r g e n c e 4 .
I t would b e
t o o l e n g t h y and i n a p p r o p r i a t e t o s t u d y h e r e t h e c o n v e r g e n c e of t
lis
method; w e i n s t e a d r e f e r t h e r e a d e r t o E . POLAK C11, J. D A N I E L 1 1 1 , CEA 1 1 1 , MARCHOUK-KUZNETSOV
111, CONCUS-GOLUB El1 e t c ..., f o r a
d e t a i l e d a n a l y s i s of t h i s a l g o r i t h m .
The a l g o r i t h m p r o c e e d s a s f o l l o w s :
1)
W e c a r r y o u t a f i r s t s t e e p e s t d e s c e n t s t e p by t a k i n g
with
g i v e n by
p0
wo = go,
(3.8),
I n t h e subsequent s t a g e s we c o n s t r u c t descent d i r e c t i o n s
2)
which a r e
Q - c o n j u g a t e s , i.e.
(0 wi, w.) = 0 V i , j , i P j . J
wn More
p r e c i s e l y , a s s u m i n g t h a t w e know t h e d e s c e n t d i r e c t i o n wn-l a n d n- 1 t h a t w e a r e a b l e t o c a l c u l a t e pn i n terms of p , we seek
wn
in the form
such t h a t
taw,,
(3.11)
W e hereby deduce t h a t
An
Wn-,)
= 0.
must t a k e t h e v a l u e
(3.12)
'
For
M
" l a r g e " q u a d r a t i c c o n v e r g e n c e becomes a g r e a t e r a t t r a c -
t i o n t h a n c o n v e r g e n c e i n a f i n i t e number of i t e r a t i o n s .
(SEC. 3 )
VARIABLE STEP-LENGTH ALGORITHMS
21
W e can f u r t h e r prove t h e orthogonality r e l a t i o n s
(ow,, w.) J
(3.13)
=
0
i f i#j,
(Pi, 8.) = 0
if i#j,
(Pi,
i f irj.
J
= 0
Wj)
By v i r t u e o f t h e s e r e l a t i o n s ,
( 3 . 1 2 ) c a n b e r e d u c e d by e l e m e n t a r y
manipulations t o
An
3)
b e i n g known, w e c a r r y o u t a n o p t i m a l d e s c e n t
wn
and t h u s
in the direction
i . e . we choose
wn, P
+
3(Prn-PWn)
t o minimise t h e f u n c t i o n
pn
.
By a c a l c u l a t i o n a n a l o g o u s t o t h a t c a r r i e d o u t i n 2 2 (3.15)
'n
--
L (gn> a wn)
( =
9 ' ,'
tff g,,q
Application t o t h e minimisation of
3.2
0
we obtain
).
m
*
J,.
W e s h a l l see h e r e how t h e methods d e s c r i b e d i n S e c t i o n 3 . 1 c a n be
applied t o t h e minimisation of
7 (BAT- I
(3.16)
J,* (9) = 1
The c o n s t a n t t e r m misation.
-
B tq , q )
(A;lb,b)
(BA;'b,q)
W e r e c a l l t h a t w e have +
$A;'b,b). 1
o b v i o u s l y p l a y s no p a r t i n t h e mini-
W e thus have, with t h e n o t a t i o n of Section 3.1
(3 .1 7 )
0
=
BA-IB~,
(3.18)
6
=
BAi'b.
The m a t r i x
*
Jr.
d
g i v e n by ( 3 . 1 7 ) i s o n l y p o s i t i v e s e m i - d e f i n i t e ,
but
i t c a n e a s i l y b e shown t h a t t h i s d o e s n o t a f f e c t t h e a l g o r i t h m s cons i d e r e d i n S e c t i o n 3 . 1 , which a l w a y s c o n v e r g e i n t h e q u o t i e n t
M IR / K e r B t .
space
of
pn
of
p
0
in *
K e r Bt
W e c a n p r o v e a s i n Theorem 2 . 2 t h a t t h e component i s i n f a c t c o n s t a n t , and t h e r e f o r e e q u a l t o t h a t
AUGMENTED LAGRANGIAN METHODS
22
(CHAP. 1)
Under these conditions the three algorithms considered in Section 3 . 1 become as follows:
0
Steepest descent algorithm:
We have (3.19)
w w
= =
gn =C7 pn-8
BA-lBtpn-BA;lb
It follows from ( 2 . 4 ) (3.20)
thus
; =
BA,'
(Btp:-b).
that we have
-
Arun = b
Btpn,
hence w
(3.21)
- Bun.
=
The optimal value of
pn
is then, from ( 3 . 8 ) ,
(3.22)
The calculation of p n thus necessitates the solutioq with respect to zn, of the linear system
hence Pn
In addition, with (3.24)
(BU",BZ,,)
pn"
Arun+' = b
-
known, we note that Btpn+l = b
-
Btpn
-
,p
u n+l
satisfies
BtBun = A run+p n Ar zn '
hence (3.25)
u
n+l
= u
n
+ pnzn.
It follows from the above formulae that at e a c h i t e r a t i o n we have to solve only a s i n g l e linear system, with matrix We can, then, Ar. re-express algorithm ( 2 . 1 ) - ( 2 . 3 ) in the following equivalent form: (3.26)
with
Po a r b i t r a r i l y s p e c i f i e d i n IRM , a n d pn, un k n o w n , d e t e r m i n e
zn, on, p
n+l
, un+l
by
uO=A-l(b-BtpO);
(SEC.
3)
VARIABLE STEP-LENGTH
z
(3.27)
= - A
-I
23
ALGORITHMS
n
t
B B u ,
(3.28)
(3.29) (3.30)
u
n+l
= u
n
+ PnZn
M i n i m u m - r e s i d u a l algorithm. W e have shown i n S e c t i o n 3 . 1 t h a t , i n t h e c a s e of t h e minimum
residual algorithm,
i s g i v e n by
pn
Pn
=
(ffgn.gn) 2
I a,l
hence t h e f o l l o w i n g v a r i a n t of a l g o r i t h m ( 3 . 2 6 ) - ( 3 . 3 0 ) :
po
(3.31)
E
BM a r b i t r a r i l y specified, a n d u D = A;'(b-BtpD),
known,
determine
(3.32)
z
=-ArBBu
(3.33)
n '
=--
(3.34)
pn+l = pn
(3.35)
u
then with
p n , un
t
-1
zn, pn,
p
n+l
,
u
n+l
by
n
( B U ~B, Z ~ )
0
n+l
I Bzn I 2
= u
'
+ p, Bun,
n
+
p,zn.
Conjugate-gradien t a l g o r i t h m .
From S e c t i o n 3 . 1 , a t e a c h i t e r a t i o n a d e s c e n t d i r e c t i o n conjugate t o w (3.36)
t h e v a l u e of
(3.37)
n-1'
h a s t o be c a l c u l a t e d ; t h u s
wn = gn + An w
An
~ =- -~Bun
b e i n g g i v e n by
+
An
Wn-]
9
wn,
24
(CHAP. 1)
AUGMENTED LAGRANGIAN METHODS
The d e s c e n t d i r e c t i o n
b e i n g known, w e t h e n need to c a l c u l a t e
wn
(3.38)
t o do t h i s we i n t r o d u c e , a s i n t h e two p r e c e d i n g a l g o r i t h m s ,
zn
such t h a t
(3.39)
z
-1
Ar
=
t
B wn,
and we a g a i n have Pn = -
(3.40)
IBunI2
mn, BZ,)
I n summary, t h e c o n j u g a t e - g r a d i e n t a l g o r i t h m can t h u s be w r i t t e n :
(3.41)
po s p e c i f i e d a r b i t r a r i l y i n
(3.42)
uo = A;l(b-BtpO);
at iteration
n
calculate the descent direction
(3.43)
w
= go = -BUO
(3.44)
w
=
-
B U ~ +
IRM;
wn
by
if n=O,
in wn-l
(3.45)
is
n31,
i f rial,
then
(3.46) (3.47)
(3.48)
pn+l = pn - pn w,,
(3.49)
u
Remark 3.1:
n+l
=
u
n
+
p,
B
zn.
I n t h e t h r e e a l g o r i t h m s d e s c r i b e d above, w e have t o
s o l v e a t e a c h i t e r a t i o n o n l y a s i n g l e l i n e a r s y s t e m , w i t h m a t r i x A,. Compared w i t h a l g o r i t h m ( 2 . 1 ) - ( 2 . 3 ) used w i t h a f i x e d a l g o r i t h m s r e q u i r e t h e a d d i t i o n a l p r e s e n c e i n memory of v e c t o r which i s of t h e same o r d e r a s un, h a v i n g
N
p,
these zn ,
a
components.
In
(SEC. 3 )
25
VARIABLE STEP-LENGTH ALGORITHMS
t h e c a s e of t h e c o n j u g a t e - g r a d i e n t a l g o r i t h m , w e a l s o need t o r e t a i n Wn-l
i n memory.
T h i s i n c r e a s e d memory r e q u i r e m e n t w i l l be j u s t -
i f i e d i f t h e a u t o m a t i c d e t e r m i n a t i o n of t h e s t e p l e n g t h
leads
pn
t o a v e r y c l e a r improvement i n t h e s p e e d of convergence compared with algorithm ( 2 . 1 ) - ( 2 . 3 )
used w i t h
pn
-
p
= r; t h i s , however,
d o e s n o t a l w a y s a p p e a r t o be t h e c a s e . Remark 3 . 2 :
a
I n t h e c a s e o f t h e problem
the conjugate-gradient algorithm, i.e.
theoretically i n rounding e r r o r s practice.
N1
( 3 . 4 1 ) - ( 3 . 4 9 ) , converges
(= r a n k B ) i t e r a t i o n s a t most.
Given t h a t
a r e p r e s e n t , t h i s r e s u l t no l o n g e r h o l d s i n
F u r t h e r m o r e , b e a r i n g i n mind t h e l a r g e s i z e of problems
a r i s i n g from t h e d i s c r e t i s a t i o n of 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 ,
it i s d e s i r a b l e t h a t , w i t h an a d e q u a t e t e r m i n a t i o n t e s t , convergence s h o u l d b e o b t a i n e d i n a number o f i t e r a t i o n s c o n s i d e r a b l y than
less
T h i s p r o p e r t y w i l l depend e s s e n t i a l l y on t h e c o n d i t i o n
N1.
number of
BA;’Bt
f o r t h d e n o t e d by
restricted to X(BA;’Bt)ImB
I m B , t h i s q u a n t i t y b e i n g hence-
.
I t can be shown t h a t
W e note t h a t
BA;’B~
(3 .5 2 )
=
B(I+~A-’B‘B)A-’B~
and t h a t t h e m a t r i x c o r r e s p o n d i n g t o t h e c a s e
r = 0
is
BA-‘Bt.
I t f o l l o w s from t h e s e p r o p e r t i e s t h a t t h e r e p l a c e m e n t of t h e Lag-
rangian
&
d e f i n e d by ( 1 . 4 ) by t h e a u g m e n t e d Lagrangian
dr.
d e f i n e d by (1.8), may b e c o n s i d e r e d as a method of p r e c o n d i t i o n i n g , i n a s e n s e cl-ose t o t h a t o f AXELSSON C l ] , being
(1+rA-’BtB);
t h e preconditioning matrix
t h i s remark i s t r u e n o t o n l y f o r t h e c o n j u g a t e -
gradient algorithm but a l s o f o r a l l t h e other algorithms studied i n the preceding s e c t i o n s .
In particular,
i n view of t h i s p r e -
c o n d i t i o n i n g , it would s e e m t h a t it s h o u l d n o t be n e c e s s a r y t o c a r r y o u t a r e i n i t i a l i s a t i o n (of t h e t y p e
w
n
=
gn)
i n t h e conjugate-
g r a d i e n t a l g o r i t h m , i n o r d e r t o c o u n t e r a c t t h e a c c u m u l a t i o n of rounding e r r o r s .
a
AUGMENTED LAGRANGIAN METHODS
26
4.
(CHAP. 1)
ON CERTAIN VARIANTS OF THE METHODS OF SECTION 2: INTRODUCTION OF A RELAXATION PARAMl2TER; METHOD OF ARROW-HURWICZ.
4.1
Synopsis
We shall show in Section 4.2 that we can i m p r o v e t h e s p e e d of c o n v e r g e n c e of algorithm (2.1)-(2.3) by utilising a r e l a x a t i o n parameter
w;
this leads u s to the algorithm
uo
(4.1)
with
un, pn
E
I R ~ po ,
arbitrarily specified;
IRM
E
known, c a l c u l a t e
u"'%,
n+: , P n ) <;e,(V,P",
2,CU
t h e n un+l
and
p
n+l
by
V"6RN,
(4.2) u
n+&
n+l
(4.3)
u
(4.4)
pn*l
N
E X ,
= u =
n
+ w(u
n++
p n + pBun+'
n
-u ) , ;
(4.2) is equivalent to (4.5)
( A + L - B ~ B ) U ~ + ' + B ~=~ b ~.
For w = 1, we once again have (with a slightly different notation) algorithm (2.1)-(2.3) with pn = p , V n 5 . In the sections following Section 4 we shall be studying algorithms of the type (4.6)
u0€RN,
(4.7)
un+'=un-w
(4.8)
arbitrarily specified;
Si'
(
(A+sBtB)un+Btpn-b
p n + ' = p n + p B u n+ 1 ,
where, in (4.7),the auxiliary operator Sr is s y m m e t r i c and positive definite. We observe that (4.6)-(4.8) is an algorithm of the A r r o w - H u r w i c z
type
see ARROW-HURWICZ-UZAWA c11, G.L.T. [l,
We can also consider parameters with n.
u
and
pn
which vary
(SEC.
4)
VARIANTS O F METHODS
C h a p t e r 2 1 , EKELAND-TEMAM C11, e t c
i s a p a r t i c u l a r case of
that (4.1)-(4.4) taking
4.2
Sr = A + r B t B
o b t a i n e d by
(4.6)-(4.8)
and c o n s t a n t v a l u e s f o r
( = Ar)
and
un
pn.
r , w e n o w s t u d y t h e convergence of a l g o r i t h m
For g i v e n and
I t s h o u l d a l s o be n o t e d
S t u d y of a l g o r i t h m ( 4 . 1 ) - ( 4 . 4 )
and i n p a r t i c u l a r t h e o p t i m a l c h o i c e of t h e p a r a m e t e r s
(4.1)-(4.4) w
...) .
27
P r o c e e d i n g as i n S e c t i o n 2 . 3 ,
p.
:
(4.9)
we put
;n+&,"n+&-,
pn
= Un-U pn-(c+p,(po)), '
yn=A-l B t-n P *
WE
(4. (4.
(4. By e l i m i n a t i o n of
;" and En+&
w e deduce f r o m ( 4 . 9 ) - ( 4 . 1 2 )
yn+l + ( p a A-' B ~ ( B I + ~ A - 'B ~ B ) - ' + - z )
(4.13)
B y m e a n s of
I)yn+ ( I -w)
r e c u r r e n t sequence.
t o t h e s t u d y of t h e convergence of a d o u b l y I n f a c t , t a k i n g i n t o account t h e r e l a t i o n s
t o z e r o i m p l i e s t h a t of
(4.9)-(4.12),
t h e convergence of
{on, pn)
{ o , a) (see S e c t i o n s 2 . 2 , 2 . 3 ) .
to
P r o c e e d i n g as i n Section 2 . 3 , ImA-'Bt
n- 1=o.
( 4 . 1 3 ) w e have reduced t h e s t u d y of t h e convergence of
algorithm (4.1)-(4.4)
of
y
yn
w e n o w express ( 4 . 1 3 )
f o r m e d f r o m t h e e i g e n v e c t o r s of
A-'BtB
on a b a s i s associated
with the s t r i c t l y p o s i t i v e eigenvalues; thus, w i t h the notation of S e c t i o n 2 . 3 ,
(4.14)
I
w e have
Vi=l,.. . ,N].
A n e c e s s a r y and s u f f i c i e n t
c o n d i t i o n of c o n v e r g e n c e f o r
(4.1)-(4.4)
28
(CHAP.
AUGMENTED LAGRANGIAN METHODS
k"Iuo,po}
E
1)
w i l l t h u s b e t h a t t h e r o o t s of t h e c h a r a c t e r -
ENx#
i s t i c equation
a s s o c i a t e d w i t h ( 4 . 1 4 ) be of m o d z d u s s t r i c t l y l e s s t h a n
Vi= I , . . .
Nl
The c o n v e r g e n c e r a t e
*
1,
w i l l satisfy
R
(4.16)
ci
Sf,
where
c:
a r e t h e r o o t s of
pxi
=
I-
0 2 (I+ I+rh,)
+
d
w
(I-
1
(4.17)
( 4 . 1 5 1 , namely
PA.
( I + --%)2 2 I+rhi
1
w-I ,
1
W e a r e now g o i n g t o s t u d y , w i t h g i v e n
15;l
t h e behaviour of
+
IciI
and
p
and w i t h
a s a f u n c t i o n of
r
still fixed, w.
The b e h a v i o u r of t h e s e two r o o t s i s p a r t i c u l a r l y s t r a i g h t f o r w a r d i n t h e c a s e where t h e y a r e imaginary ( c o n j u g a t e s ) ; i n f a c t w e t h e n have
lc;]
(4.18)
=
ItTI
=
.
W e s h a l l have t h i s s i t u a t i o n i f , i n ( 4 . 1 7 1 ,
t h e q u a n t i t y under t h e
s q u a r e - r o o t s i g n i s n e g a t i v e , which i s t h e c a s e i f (4.19)
For
w
>
*
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+
<
p
= 2
)
w
Max
> 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
AUGMENTED LAGRANGIAN METHODS
(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
AUGMENTED LAGRANGIAN METHODS
(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)
VARIANTS O F METHODS
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
AUGMENTED LAGRANGIAN METHODS
(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)
VARIANTS O F METHODS
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 <
E
< I.,
( 4 . 5 0 ) by t h e u p p e r bound
o b t a i n e d i n ( 4 . 5 4 ) and n o t i n g f r o m ( 4 . 4 8 ) t h a t (pn+l-pn12
=
p21~un12,
we o b t a i n , a f t e r r e - g r o u p i n g
various t e r m s n-l 12
4E(l-E)
+
(CHAP.
AUGMENTED LAGRANGIAN METHODS
38
Utilising (4.45)
we t h e n have
11~~112w i l l be p o s i t i v e
The c o e f f i c i e n t of
i f f o r some
E,
which remains t o be d e t e r m i n e d , w e have
(4.57)
To d e f i n e t h i s c o n d i t i o n p r e c i s e l y , w e now t r y t o f i n d , f o r g i v e n p,
t h e value of
10,lC
in
E
g i v i n g t h e b e s t bound f o r
w
t h e optimal choice f o r
is,
E
(4.58)
which t o g e t h e r w i t h ( 4 . 5 7 )
implies
(4.59)
or f u r t h e r (4.60)
If condition
(4.61)
i s s a t i s f i e d , w e have
(4.59)
c
=
(i.e.
An e l e m e n t a r y c a l c u l a t i o n shows us t h a t
the largest possible).
2pw
-
PW 2ar
4&*( 1 -&*)
Summing t h e i n e q u a l i t i e s ( 4 . 5 6 )
for
B,wE * P 2
--
> 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.
AUGMENTED LAGRANGIAN METHODS
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
VARIANTS O F METHODS
(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)
AUGMENTED LAGRANGIAN METHODS
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)
AUGMENTED LAGRANGIAN METHODS
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
AUGMENTED LAGRANGIAN METHODS
(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
STOKES & NAVIER-STOKES EQUATIONS
(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
NAVIER-STOKES EQUATIONS
(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
&
NAVIER-STOKES EQUATIONS
Figure 2.1 Triangulation of Ci
(CHAP. 2 )
(SEC.
3)
ALGORITHMS AND RESULTS
Figure 2 . 2
Streamlines
59
60
STOKES & NAVIER-STOKES EQUATIONS
(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)
specified arbitrarily;
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
ALGORITHMS AND RESULTS
(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
STOKES & NAVIER-STOKES EQUATIONS
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)
ALGORITHMS AND RESULTS
MAXIMUM VALUES OF
IE ~ .
63
AFTER 'In" ITERATIONS
1 n=15
+,-
II /
Y
Figure 3 . 1
9.6
10
P 11
STOKES
64
&
NAVIER-STOKES EQUATIONS
(CHAP.
Figure 3.2 optimal p
, theoretical and experimental, as a function of r
-
theoretical
+
experimental
2)
ALGORITHMS AND RESULTS
(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 )
NAVIER-STOKES EQUATIONS
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)
ALGORITHMS AND RESULTS
Figure 3 . 3
67
68
STOKES
&
NAVIER-STOKES EQUATIONS
Figure 3 . 4
(CHAP.
2)
(SEC.
ALGORITHMS AND RESULTS
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
<
10, however, t h e
convergence o f t h e a l g o r i t h m i s c o n s i d e r a b l y r e t a r d e d . F i g u r e 3 . 4 shows t h e v a r i a t i o n of t h e ( e x p e r i m e n t a l ) o p t i m a l v a l u e of
a s a f u n c t i o n of
p
t o diminish i f
m
m.
This demonstrates t h a t
is small ( i n our case i f
m
<
10).
p
ought
On t h e o t h e r
hand, w e o b t a i n a v e r y d i f f e r e n t p i c t u r e i f w e t r y t o v i s u a l i z e t h e e f f e c t of
m
convergence.
on t h e t o t a l number of i t e r a t i o n s n e c e s s a r y f o r To i l l u s t r a t e t h i s f a c t , F i g u r e 3 . 5 shows a s a
m t h e c u m u l a t i v e number of o v e r r e l a x a t i o n i t e r a t i o n s
f u n c t i o n of
n e c e s s a r y t o a c h i e v e c o n v e r g e n c e , t h i s number b e i n g t o a f i r s t approximation p r o p o r t i o n a l t o t h e computation t i m e .
This figure
shows t h a t t h i s c u m u l a t i v e number d e c r e a s e s a l m o s t l i n e a r l y w i t h I n t h e c a s e where
m.
( 3 . 3 ) i s s o l v e d by an i t e r a t i v e method, i t t h u s
seems t h a t a good s t r a t e g y i s t o c a r r y o u t o n l y one o r two p a s s e s a t each s t a g e .
T h i s o b s e r v a t i o n may be i m p o r t a n t i n t h e c a s e s where
t h e sub-problems a r e no l o n g e r q u a d r a t i c . c h o i c e of
r
and of
p
The problems of t h e
a r e i n t h i s c a s e open q u e s t i o n s and may be
compared w i t h t h e problem o f t h e c h o i c e of p a r a m e t e r s i n t h e Arrow-Hurwicz
algorithm.
optimal value f o r
In particular,
r , because f o r
r
t h e r e c e r t a i n l y e x i s t s an
l a r g e t h e c o n d i t i o n number of
i n c r e a s e s p r o h i b i t i v e l y , 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 convergence of t h e i n t e r n a l i t e r a t i v e method.
Ar
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
methods
These methods have proved t o be e x t r e m e l y e f f i c i e n t f o r o u r model problem; t h i s i s p a r t i c u l a r l y t r u e o f t h e c o n j u g a t e - g r a d i e n t method, t h e i m p l e m e n t a t i o n o f which i s no more d i f f i c u l t t h a n t h a t of t h e methods of s t e e p e s t d e s c e n t o r of minimum r e s i d u a l .
The p r i n c i p a l
a d v a n t a g e of t h i s t y p e of t e c h n i q u e l i e s of c o u r s e i n i t s a b i l i t y t o b e used a s a " b l a c k box" w i t h no need f o r u s e r - i n t e r v e n t i o n f o r t h e c h o i c e of p a r a m e t e r s , and w i t h o u t t h i s c h o i c e b e i n g l i n k e d t o t h e s o l u t i o n of a s p e c t r a l problem.
(SEC. 3)
71
ALGORITHMS AND RESULTS
T o i l l u s t r a t e t h e i d e a s a n d t o make c l e a r e r t h e c o m p a r i s o n b e t ween t h e v a r i o u s m e t h o d s , w e p r e s e n t i n F i g u r e 3.6 t h e d e c r e a s e o f t h e maximum v a l u e o f
lo,.g:I
t h e number o f i t e r a t i o n s .
o v e r t h e t r i a n g l e s a s a f u n c t i o n of Since t h e scale is logarithmic, t h e
s l o p e c o r r e s p o n d s t o t h e c o n v e r g e n c e r a t e of
a f i r s t - o r d e r method.
A s m i g h t b e e x p e c t e d , t h e c o n j u g a t e - g r a d i e n t method, which i s o f
second o r d e r , app ear s as t h e m o s t e f f i c i e n t t e c h n i q u e . however,
It requires,
t h e e x a c t s o l u t i o n o f p r o b l e m ( 3 . 1 1 ) , and t h i s makes it
p a r t i c u l a r l y a t t r a c t i v e i n cases where t h e s o l u t i o n i s a c c o m p l i s h e d by a d i r e c t method. A s r e g a r d s t h e o t h e r a l g o r i t h m s , t h e c o n v e r g e n c e of t h e v a r i a b l e -
s t e p l e n g t h g r a d i e n t methods i s b e t t e r t h a n t h a t o f t h e method w i t h p
f i x e d , f o r t h e o p t i m a l c h o i c e of t h e p a r a m e t e r .
p l a i n e d by t h e f a c t t h a t i n o u r t e s t s t h e
T h i s i s ex-
v a l u e s s e t t l e down t o
pn
a l i m i t c y c l e b e t w e e n two p a r a m e t e r s s i t u a t e d e i t h e r s i d e o f t h e p. I n t h e l i m i t w e t h u s have a c y c l i c v a r i a t i o n o f 'nr a s t r a t e g y w h i c h h a s a l r e a d y b e e n s u g g e s t e d i n C R O U Z E I X C11, t h e
optimal
c y c l e b e i n g d e t e r m i n e d a u t o m a t i c a l l y by t h e a l g o r i t h m .
A s might
have b e e n a n t i c i p a t e d , t h e s t e e p e s t d e s c e n t method i s s u p e r i o r t o t h e minimum r e s i d u a l method, t h e l a t t e r b e i n g more s e n s i t i v e t o t h e c o n d i t i o n number.
3.2. 4
Alaorithm with relaxation oarametsr
F i g u r e 3.6 shows a l s 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 ( 3 . 1 5 ) ( 3 . 1 8 ) , which w a s d e d u c e d f r o m Uzawa's a l g o r i t h m t h r o u g h t h e i n t r o duction of a r e l a x a t i o n parameter. been e s t i m a t e d u s i n g f o r m u l a s ( 4 . 2 2 ) ,
The o p t i m a l p a r a m e t e r s h a v e (4.23)of Chapter I.
f i g u r e shows t h a t t h e i n t r o d u c t i o n of t h e p a r a m e t e r
w
This
accelerates
t h e c o n v e r g e n c e a p p r e c i a b l y , and t h i s i s f o r a d u a l p r o b l e m which i s q u i t e w e l l condi.tioned.
The a c c e l e r a t i o n e f f e c t s h o u l d b e f e l t
s t i l l more when t h e c o n d i t i o n i n g i s w o r s e .
I n o u r model p r o b l e m ,
t h i s method i s c o m p a r a b l e t o t h e v a r i a b l e - s t e p l e n g t h algorithms.
gradient
W e have a l r e a d y n o t e d i n C h a p t e r I t h a t i n t h e c a s e o f
v a r i a b l e p a r a m e t e r s , t h i s method encompasses t h e c o n j u g a t e - g r a d i e n t method.
C o n s e q u e n t l y , s i n c e t h e d e t e r m i n a t i o n of t h e o p t i m a l
p a r a m e t e r s r e q u i r e s t h e s o l u t i o n o f a s p e c t r a l problem, t h i s a l g o r i t h m o f f e r s l i t t l e a t t r a c t i o n i n t h e case o f o u r model problem.
STOKES
12
&
NAVIER-STOKES EQUATIONS
10'
10
Id
10
10-
16
10-
10
F i g u r e 3.6
( C H A P . 2)
(SEC.
73
ALGORITHMS AND RESULTS
3)
I t s a t t r a c t i o n would be more o b v i o u s i n t h e c a s e of t h e n o n l i n e a r Navier-Stokes e q u a t i o n s .
A l g o r i t h m s of A r r o w - H u r w i c z
3.2.5
tgpe
A s w e have a l r e a d y mentioned i n S e c t i o n 3 . 1 ,
solution of
i f w e carry out the
( 3 . 3 ) by a symmetric o v e r r e l a x a t i o n ( S S O R ) method, w e
a r e a c t u a l l y u s i n g an a l g o r i t h m of t h e t y p e ( 3 . 1 5 ) - ( 3 . 1 6 ) .
This
a l g o r i t h m h a s t h e a d v a n t a g e , compared w i t h t h e p r e c e d i n g o n e s , of n o t r e q u i r i n g t h e s o l u t i o n of a l i n e a r system a t each i t e r a t i o n , t h e c a l c u l a t i o n b e i n g e n t i r e l y e x p l i c i t , and c o n s e q u e n t l y r a p i d .
The
e s s e n t i a l problem h e r e o f c o u r s e i s t o d e t e r m i n e t h e o p t i m a l p a r a meters
*
w
and
*
p
,
which minimise t h e number of i t e r a t i o n s r e q u i -
r e d t o r e a c h a s p e c i f i e d l e v e l of a c c u r a c y .
S i n c e we a r e n o t i n a
p o s i t i o n t o g i v e a t h e o r e t i c a l a n a l y s i s of t h i s q u e s t i o n , we i n s t e a d p r e s e n t h e r e some e x p e r i m e n t a l r e s u l t s , which, we h o p e , w i l l a l l o w t h e r e a d e r t o o b t a i n an i n t u i t i v e view o f t h e b e h a v i o u r of t h e a l g o rithm. F i r s t w e r e c a l l t h e r e s u l t s of C h a p t e r I , S e c t i o n 4 . 3 , where w e p o i n t e d o u t t h e a n a l o g y between t h i s a l g o r i t h m and a wave e q u a t i o n w i t h damping p r e s e n t .
I n t h i s analogy
a damping c o e f f i c i e n t and
w
h a s t h e a p p e a r a n c e of
of t h e s q u a r e o f a p r o p a g a t i o n speed.
pw
I n t u i t i v e l y w e can t h e r e f o r e e x p e c t , f o r a s p e c i f i e d v a l u e o f under-damping phenomenon f o r t h e s m a l l v a l u e s of
w,
p,
an
and an o v e r -
damping beyond some c r i t i c a l v a l u e which w i l l be a f u n c t i o n ( p r e s e n t l y unknown) o f t h e e i g e n v a l u e s of t h e v a r i o u s o p e r a t o r s which o c c u r i n t h e a l g o r i t h m .
r = 0 the
Such a phenomenon can i n f a c t b e v e r i f i e d ; f o r c r i t i c a l v a l u e a p p e a r s t o be crease f o r l a r g e r values of
w =
r.
1.
T h i s t h r e s h o l d seems t o i n -
For
r = 6 , f o r example, a t r a n -
s i t o r y p h a s e of under-damping was o b s e r v e d f o r
w =
1.1.
U n f o r t u n a t e l y i t i s n o t p o s s i b l e t o e s t i m a t e t h e o p t i m a l v a l u e of w
by making u s e of t h e c r i t i c a l damping v a l u e .
In actual fact
t h i s v a l u e may w e l l b e d i f f e r e n t f o r e a c h of t h e "components" of the solution. The main e x p e r i m e n t a l f i n d i n g s a r e a s f o l l o w s :
- The o p t i m a l s p e e d of convergence i s a l m o s t t h e s a m e f o r r = 0 and r = 1 and s u b s e q u e n t l y d i m i n i s h e s r a p i d l y a s r i n c r e a s e s because
STOKES & NAVIER-STOKES EQUATIONS
74
of t h e i l l - c o n d i t i o n i n g of
- For a given value of
w ,
(CHAP.
2)
Ar.
t h e optimal value of
p
obeys ( t o a
f i r s t approximation) t h e l a w P opt = 2 r + l .
This estimate s l i g h t l y underestimates t h e experimental value, b u t t h e speed of convergence i s n o t very s e n s i t i v e t o such an
-
underestimate
.
The c h o i c e o f
w
relax" f o r
r
i s more d i f f i c u l t .
large.
I t i s necessary t o "under-
To i n d i c a t e t h e t r e n d s , Table 3.2
summarises a few e x p e r i m e n t a l v a l u e s .
1.06
Table 3.2
I n summary, t h e b e n e f i t o f t h e augmented L a g r a n g i a n i s by n o r
means a p p a r e n t f o r s u c h a n a l g o r i t h m : t h e o p t i m a l v a l u e o f i f n o t p r e c i s e l y 0 , a t l e a s t i n t h e n e i g h b o u r h o o d of 0 .
is,
By
c o n t r a s t , t h e computational e f f o r t is competitive with t h e preceding methods i n which e v e r y i t e r a t i o n r e q u i r e s t h e s o l u t i o n o f a l i n e a r system.
The p r o b l e m o f t h e o p t i m a l c h o i c e o f t h e p a r a m e t e r s , o r o f
a n a u t o m a t i c c h o i c e s i m i l a r t o t h a t made i n t h e method o f
steepest
d e s c e n t , m e r i t s f u r t h e r s t u d y i n s p i t e of t h e d i f f i c u l t y o f t h e quest ion.
3. 2. 6
Conclusions
The methods b a s e d on t h e u s e o f t h e augmented L a g r a n g i a n are a p p a r e n t l y v e r y e f f i c i e n t f o r s o l v i n g q u a d r a t i c problems w i t h l i n e a r c o n s t r a i n t s , a n d i t i s c l e a r t h a t t h i s e f f i c i e n c y c a r r i e s across t o
a more g e n e r a l s e t t i n g .
O f a l l t h e algorithms tested,
a t t r a c t i v e a r e t h o s e i n which t h e p a r a m e t e r s a r e c h o s e n matically;
t h e most auto-
t h i s applies p a r t i c u l a r l y f o r the conjugate-gradient
(SEC. 4 )
75
NAVIER-STOKES; STEADY STATE
a l g o r i t h m i n a s s o c i a t i o n w i t h a d i r e c t method f o r t h e s o l u t i o n o f t h e l i n e a r sub-problems. The methods o f Arrow-Hurwicz
t y p e d e s e r v e a more t h o r o u g h s t u d y
as regards t h e choice of parameters.
They c o u l d p r o v e t o b e e x c e l -
l e n t f o r c a s e s i n which t h e sub-problems are n o n l i n e a r .
W e thus
h a v e h e r e a n open p r o b l e m , d i f f i c u l t m o s t c e r t a i n l y , b u t o f g r e a t practical interest. R e g a r d i n g , more e s p e c i a l l y , t h e S t o k e s p r o b l e m , t h e r e s u l t s obt a i n e d show t h a t t h e augmented L a g r a n g i a n a p p r o a c h a l l o w s u s t o r e d u c e t h e problem t o s o l v i n g s e v e r a l problems o f l i n e a r - e l a s t i c i t y type ( 3 or 4 f o r
r
s u f f i c i e n t l y l a r g e ) , which can themselves be
s o l v e d u s i n g a f a c t o r i s a t i o n c a r r i e d o u t o n c e and f o r a l l .
We
b e l i e v e t h a t t h i s method o f p r o c e d u r e may p r o v e t o b e less c o s t l y t h a n c u r r e n t l y - u s e d methods which c o n s i s t o f d i r e c t l y s o l v i n g by f a c t o r i s a t i o n t h e g l o b a l l i n e a r system i n
{u,p}.
I n t h i s case
t h e systems b e i n g d e a l t w i t h a r e a c t u a l l y l a r g e r i n s i z e , as w e l l as b e i n g s i n g u l a r , and t h e i r b a n d w i d t h i s c o n s i d e r a b l y l a r g e r .
W e can
t h e r e f o r e e x p e c t t h a t f o r l a r g e p r o b l e m s t h e augmented L a g r a n g i a n method w i l l b e f a s t e r and less s e n s i t i v e t o r o u n d i n g e r r o r s .
In
c o n c l u s i o n , w e remark t h a t t h e method h a s a l s o b e e n u s e d f o r p r o b l e m s which a r e a n a l o g o u s t o t h e S t o k e s p r o b l e m b u t which a r i s e from t h e f i e l d o f s o i l m e c h a n i c s , by M .
4.
=VIER-STOKES
4.1
SOULIE Ill.
EQUATIONS, STEADY-STATE NONLINEAR CASE
S t a t e m e n t o f t h e problem
W e now c o n s i d e r t h e case
o f t h e n o n l i n e a r Navier‘Stokes
e q u a t i o n s ; t h e s e d e s c r i b e numerous p r o b l e m s o f g r e a t p r a c t i c a l i m portance.
J u s t as i n t h e p r e c e d i n g s e c t i o n s , w e c o n s i d e r f o r simp-
l i c i t y t h e case o f a f l o w w i t h homogeneous b o u n d a r y c o n d i t i o n s i n a domain Ci o f IR2 ( o r o f i R3 ) , bounded and w i t h r e g u l a r b o u n d a r y . With t h e n o t a t i o n u s e d e a r l i e r f o r t h e v e l o c i t y v e c t o r , t h e p r e s s u r e and t h e e x t e r n a l f o r c e s , t h e p r o b l e m t o b e s o l v e d i s now
76
STOKES & NAVIER-STOKES EQUATIONS
(CHAP. 2 )
T h e s e e q u a t i o n s a r e n o n l i n e a r and t h e i r s o Z u t i o n d o e s n o t c o r r e s p o n d
t o s e a r c h i n g f o r t h e minimum o f a f u n c t i o n a l .
W e c a n however w r i t e
them i n t h e form o f a v a r i a t i o n a l e q u a t i o n i n t h e f o l l o w i n g manner. Again p u t t i n g , a s i n S e c t i o n 2 ,
and
i t c a n b e shown (see LIONS 1 2 1 , TEMAM I l l ) t h a t p r o b l e m ( 4 . 1 ) - ( 4 . 3 )
is equivalent t o seeking
u
V (see ( 1 . 1 6 ) f o r t h e d e f i n i t i o n o f V )
E
I
s a ti s f y i n g (4.6)
va(u_,v_)
Equation
+
b(u_,u_,v_) = (f_,v_)
v v _ ~ V .
i s deduced f r o m ( 4 . 1 )
(4.6)
by m u l t i p l y i n g by a f u n c t i o n
w i t h z e r o d i v e r g e n c e and t h e n i n t e g r a t i n g by p a r t s .
t e r m vanishes because w e have
1, if
v
=
;1
-
1.y dx
= 0
h a s z e r o d i v e r g e n c e and i s z e r o a t t h e b o u n d a r y .
Conversely, i f exists
1p.y dx
The p r e s s u r e
p
E
L
2
--
(4.7)
(a)
va(u,v)
Thus t a k i n g
i s a s o l u t i o n of
u "
+
(4.6)
it c a n b e shown t h a t t h e r e
such t h a t w e have
b(F,E,y)
Y E(J(R))N,
-
(p,g-y)
--
= (f,v)
Vve
I
N
(Ho(W)
.
we again obtain ( 4 . 1 ) i n t h e distribu-
t i o n a l sense. Note that
( 4 . 7 ) and
are t h e analogue of t h e o p t i m a l i t y c o n d i t i o n s f o r t h e Lagrangian (1.21)
i n the S t o k e s p r o b l e m .
W e can t h e r e f o r e c o n s i d e r (4.71,
( 4 . 8 ) as a " v a r i a t i o n a l L a g r a n g e p r o b l e m " and w e s a y t h a t a p a i r
-
{u,p}
s a t i s f y i n g t h e s e conditions c o n s t i t u t e s an equizibrium p o i n t
f o r t h i s problem.
W e s h a l l now a t t e m p t t o e x t e n d t o t h i s new t y p e
of c a s e t h e a l g o r i t h m s u s e d i n t h e p r e c e d i n g s e c t i o n s f o r t h e
(SEC. 4 )
NAVIER-STOKES; STEADY STATE
77
s o l u t i o n of t h e S t o k e s p r o b l e m . The u n d e r l y i n g i d e a i s c l o s e l y r e l a t e d t o t h e t e c h n i q u e s u s e d i n t h e p r o o f s o f e x i s t e n c e o f a s o l u t i o n , which are b a s e d on t h e s e a r c h f o r a f i x e d p o i n t f o r t h e mapping
from
T
into
V
d e f i n e d by
V
or alternatively
-
(4.10)
(4.11)
v A y + ( , v . ~ )-u+ V -p
-uIr
(4.12)
G
I
= 0. I
i s a l i n e a r , nonsymmetric v a r i a t i o n a l problem.
Problem ( 4 . 9 ) d e n o t e by
i n
R
i n
V-u = 0 _
= f
A ( -v )
t h e l i n e a r o p e r a t o r from
into
V
V'
We
( V ' : d u a l of
d e f i n e d by
V)
-
where and
-
1
b
-
I
-
VwcV,
d e n o t e s t h e b i l i n e a r f o r m o f t h e d u a l i t y between
<.,.>
I t c a n e a s i l y b e shown
V.
form
---
= va(u,w) + b ( v , u , w )
(4.13)
is anti-symmetric
V'
(see LIONS 1 2 1 ) t h a t t h e t r i l i n e a r
f o r t h e l a s t two v a r i a b l e s i f
u
E
V , i.e.
I
if v
u E
h a s z e r o d i v e r g e n c e and i s z e r o a t t h e b o u n d a r y .
V, w e h a v e
b(v,u,u) = 0 .
_ - -
-
we t h u s have
The o p e r a t o r A ( y )
is therefore
V-elliptic
Thus i f
-
w = u in (4.13),
Then, p u t t i n g
(see L I O N S C21) a n d t h e
Lax-Milgram Theorem a l l o w s u s t o i n f e r t h e 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 o f problem ( 4 . 9 ) .
is w e l l d e f i n e d .
This r e s u l t implies t h a t t h e operator T
W e f u r t h e r deduce f r o m ( 4 . 9 )
t h e a p r i o r i upper
bound
The o p e r a t o r
s e t ) of
V
T
t h u s maps a c l o s e d b a l l
( h e n c e a w e a k l y compact
into itself.
Using t h e c o m p a c t n e s s of t h e i n j e c t i o n o f
Hk(Q)
into
L 4 ( G ) , it
78
STOKES & NAVIER-STOKES EQUATIONS
can be proved t h a t t h e o p e r a t o r
t o p o l o g y of
'.
V
T
(CHAP.
i s c o n t i n u o u s f o r t h e weak Theorem can t h u s be
Schauder's Fixed-Point
Furthermore it i s c l e a r t h a t any f i x e d p o i n t of
applied.
s o l u t i o n o f t h e Navier-Stokes
T
is a
equations.
Uniqueness c a n b e proved f o r LIONS 1 2 1 ) .
2)
v
"large"
or
f
"small"
(see
Knowing t h a t t h e e x i s t e n c e of a s o l u t i o n can b e dedtheorem, it i s n a t u r a l t o t r y t o s o l v e t h e
uced from a f i x e d - p o i n t
problem by t r y i n g t o f i n d t h i s f i x e d p o i n t n u m e r i c a l l y , by means of an i t e r a t i v e method.
4.2
Basic Algorithm
I n t h e l i g h t of what h a s gone b e f o r e , w e t h u s c o n s i d e r t h e f o l l o w i n g algorithm:
specified arbitrarily;
uo
(4.16)
t h e n for
n-2 0 , w i t h
un
known, c a l c u l a t e
I
(4.17) (4.18)
- vAun+l 1 2 +(_~~-y)u~+l'~ + Vpn+l V * U ~ += ~0 ~ ~ -
(4.19)
u -
=
n+l
-
in
satisfying
s1
s1,
0 ,
by
un+l = w u n + 1 1 2 +
-
I
Actually, the case
(4.21)
in
= f
pnfl
I
n+l I 2 :
and t h e n (4.20)
-
-
-
un'%,
-
w =
(I-LO)_u"
1
defines
u
-
n+l
bY
-
un+l = T ( u n ) .
The g e n e r a l c a s e can t h u s b e w r i t t e n (4.22)
u*+'
=
w T(gn) + (I-LO):".
The c h o i c e of
w
# 1 i s c l e a r l y d e s i g n e d t o a c c e l e r a t e t h e converI t i s shown i n C R O U Z E I X C11 t h a t t h i s a l g -
gence, i f p o s s i b l e .
o r i t h m c o n v e r g e s whenever t h e s o l u t i o n i s u n i q u e .
I n c a s e s of
non-uniqueness w e may hope t o converge t o t h e s t a b l e s o l u t i o n "nearest" t o
'
-
uo,
hence c o m p a c t
(SEC. 4 )
NAVIER-STOKES;
-
-
and
79
t h e sol-
From t h e Lax-iqilgram Theorem and ( 4 . 1 4 ) ,
Remark 4 . 1 : nif of u
ution
STEADY STATE
(4.17)-(4.19)
-
By c o n t r a s t , t h e c o n v e r g e n c e i s o n l y a s s u r e d i f
un.
sufficiently near t o a solution p(T'(u)),
(where
-
-
u
i s s t r i c t l y l e s s t h a n 1.
is
uo
t h a t the spectral radius
i s t h e Frechet d e r i v a t i v e of
TI(:)
u
e x i s t s whatever t h e v a l u e s of
u - )
at
T
The i n v e s t i g a t i o n o f t h e s e " s t a b l e "
s o l u t i o n s i s a d i f f i c u l t p r o b l e m which a g a i n i s l a r g e l y l e f t o p e n . I n p r a c t i c e , t h e u s e r must b e s a t i s f i e d w i t h p r o v i n g c o n v e r g e n c e
m
experimentally. The i m p l e m e n t a t i o n o f a l g o r i t h m ( 4 . 1 6 ) - ( 4 . 2 0 )
Remark 4 . 2 :
u i r e s problem ( 4 . 1 7 ) - ( 4 . 1 9 ) t o b e s o l v ed .
req-
T h i s problem i s i n
e v e r y r e s p e c t analogous t o t h e S t o k e s problem, t h e o p e r a t o r
-
A ( u n ) = (-vA+un'V)
b e i n g l i n e a r and V - e l l i p t i c
- -
t h o u g h nonsymnietric.
(see ( 4 . 1 4 ) )
W e have a l r e a d y mentioned i n C h a p t e r I ,
S e c t i o n 5 , t h a t t h e c o n v e r g e n c e of t h e UZAWA a l g o r i t h m c a n i n t h i s
case b e p r o v e d by t h e e n e r g y methods of C h a p t e r I , S e c t i o n 2 . 1 . I n order t o state t h i s algorithm i n e x p l i c i t fashion we put4 (4.23)
Ar(yn)
- vA
=
+
un'V -
I
-
__
rV(V-)
The a l g o r i t h m i s t h e n w r i t t e n pn+lro
(4.24) t h e n for
s
2
0
,
specified arbitrarily;
with
pn+lPs
known,
compute
uni'ls
satisfying
and (4.26)
pn+l,s+l
n+l,s = P
- 'P
'F
n+l/2,s.
I t c a n b e shown t h a t t h e c o n d i t i o n f o r c o n v e r g e n c e i s i n t h i s case5
I,
V(V*v) - - .=
-
grad ( d i v v ) .
F o r c e r t a i n a p p r o x i m a t i o n s t h i s c o n d i t i o n would become 0
i).
80
STOKES & NAVIER-STOKES EQUATIONS
(4.27)
0 < p < 2 (r+v)
(CHAP.
2)
.
W e t h u s see t h a t t h e a d v a n t a g e of t h e p e n a l t y t e r m w i l l become greater as
w
g e t s smaller.
W e s h a l l r e t u r n t o t h i s remark i n
Section 5.
Remark 4.3: t h a t pnfl
I t can immediately be deduced from ( 4 . 1 7 ) - ( 4 . 1 9 )
i s a s o l u t i o n of a nonsymmetric " d u a l problem" which
does n o t i n g e n e r a l c o r r e s p o n d t o t h e m i n i m i s a t i o n of a f u n c t i o n a l and which can b e w r i t t e n
(4.28)
v*(A;I
(U_")ypn+l) = !*(A;'
(u_")f_),
an e x p r e s s i o n which i s c l e a r l y analogous t o t h a t o b t a i n e d f o r t h e L i k e w i s e w e see from ( 4 . 2 5 ) , S t o k e s problem. pn+l,s+l can be w r i t t e n
(4.29)
pn+l
9s+1
(4.26) that
.
= P n+l* s + p {~ *(A;l(_u n )~ p n +''")-1.(A;'(,")f)}
The convergence of t h e a l g o r i t h m depends on t h e s p e c t r u m of t h e
~ ~ ( f + i )L ~ ( Q ) )
o p e r a t o r (from
(4.30)
q + q
+
p!*(A,
-1
n
(_" ) y q ) .
We r e f e r t h e r e a d e r t o CROUZEIX 111 f o r a s t u d y of t h i s s p e c t r u m which i s n e c e s s a r i l y somewhat complex.
Even 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 , a s p e c t r a l a n a l y s i s of t h e convergence i s s c a r c e l y p o s s i b l e i n view of t h e non-symmetry of t h e o p e r a t o r .
Remark 4 . 4 :
I t i s a g a i n p o s s i b l e t o a p p l y h e r e c e r t a i n of t h e
v a r i a b l e - s t e p methods of C h a p t e r I , S e c t i o n 3.
A s a r e s u l t of t h e
non-symmetry of t h e problem, t h e method of s t e e p e s t d e s c e n t l o s e s i t s meaning and t h i s l e a d s u s t o f a v o u r u s i n g t h e method of minimum r e s i d u a l s , i.e.
t o determine
so a s t o m i n i m i s e
ps
ly-yn+l's+l 10.
t h i s p r o c e d u r e remaining p e r f e c t l y j u s t i f i a b l e h e r e .
Following
t h e n o t a t i o n a l r e a d y i n t r o d u c e d and f o l l o w i n g a l s o S e c t i o n 3 of t h i s c h a p t e r , t h e a l g o r i t h m can be w r i t t e n pn+l, o
(4.31)
specified arbitrarily;
cazcuzate t h e s o l u t i o n (4.32)
-
A (un)uncl/2,0
r -
On i t e r a t i o n
gn+fno o f +
J!pn+l,o
=
f. I
s , define the descent d i r e c t i o n
ws
by
(SEC.
4)
NAVIER-STOKES;
81
STEADY STATE
Then s o l v e t h e problem
and c a l c u l a t e
ps
bg
The a d v a n t a g e of s u c h an a l g o r i t h m i s o f c o u r s e t h a t t h e c h o i c e of t h e parameter
p,
always a r a t h e r d i f f i c u l t m a t t e r , i s c a r r i e d o u t
m
i n an a u t o m a t i c manner.
Remark 4.5:
I n t h e l i g h t o f t h e n u m e r i c a l r e s u l t s of S e c t i o n 3,
it would s e e m e f f e c t i v e t o employ h e r e a c o n j u g a t e - g r a d i e n t method, even though a t f i r s t s i g h t t h i s may a p p e a r d i f f i c u l t i n view of t h e non-symmetry of t h e problem.
F o l l o w i n g P A I G E 111 and GOLUB 111
it i s , however, p o s s i b l e t o c o n s t r u c t a l g o r i t h m s of c o n j u g a t e g r a d i e n t t y p e f o r a non-symmetric problem, by u s i n g a l e a s t - s q u a r e s formulation.
The a l g o r i t h m s t h u s c o n s t r u c t e d r e q u i r e a t each
i t e r a t i o n t h e s o l u t i o n of two l i n e a r s y s t e m s i n t h e symmetric c a s e of t h e S t o k e s p r o b l e m ) . t h e c a l c u l a t i o n i s t h e r e f o r e doubled. application t o (4.17)-(4.19)
( i n s t e a d of o n l y one The complexity of
A n i n v e s t i g a t i o n of t h e
of P A I G E ' s a l g o r i t h m , o r o f a v a r i a n t
b e t t e r adapted t o t h e p a r t i c u l a r case w e a r e p r e s e n t l y considering, t h u s a p p e a r s t o be a d i r e c t i o n of r e s e a r c h which o u g h t t o b e e x plored.
A s a p a r t i c u l a r r e f e r e n c e , w e may c i t e WIDLUND L 1 l .
A l g o r i t h m (4.16)-(4.20),
combined w i t h
8
(4.31)-(4.37), thus
a p p e a r s a s a method which i s s i m p l e and c o m p 2 e t e Z y a u t o m a t i c f o r c a l c u l a t i o n s i n v o l v i n g t h e Navier-Stokes e q u a t i o n s .
Unfortunately,
convergence i s n o t a s s u r e d and t h e c h o i c e o f t h e p a r a m e t e r open problem. p l a c e of
-
The u s e of t h e " a u g m e n t e d " o p e r a t o r
Ar(yn)
A(un) i s aimed a t a c c e l e r a t i n g t h e convergence o f
i s an
w
in
a2
STOKES
&
NAVIER-STOKES EQUATIONS
a t h i g h Reynolds numbers, i . e . when
(4.31)-(4.37)
N e v e r t h e l e s s t h e g l o b a l convergence of
(4.16)
(CHAP. 2 )
v
- (4.20)
by t h i s p r o c e d u r e and i s c e r t a i n l y s l o w e r f o r
w
is small. i s unchanged
small.
One
p o s s i b l e s t r a t e g y i s c l e a r l y t o c a l c u l a t e a s e q u e n c e of s o l u t i o n s a t i n c r e a s i n g Reynolds numbers, e a c h t i m e i n i t i a l i s i n g ( 4 . 1 6 ) (4.20) w i t h t h e l a s t s o l u t i o n o b t a i n e d .
I n p r a c t i c e , t h i s app-
r o a c h does a p p e a r t o g i v e good r e s u l t s i n t h e m a j o r i t y of To c o n c l u d e t h i s s e c t i o n , and b e f o r e w e e n t e r upon t h e
problems.
s t u d y of v a r i a n t s of t h e above a l g o r i t h m , w e s h o u l d p o i n t o u t t h a t (OLSON 1 1 1 , ROACHE C11) s u g g e s t t h e u s e of t h e
certain authors,
N e w t o n - R a p h s o n method, f o r s o l v i n g t h e n o n l i n e a r N a v i e r - S t o k e s
This algorithm i s w r i t t e n
problem. (4.38)
where
s p e c i f i e d a r b i t r a r i l y , '.,u"
uo
= 0 :
n t 0 , put
then for
-
6un,p"
a r e s o l u t i o n s of
= 0
,
_o
.
(4.41)
-V.6U"-
(4.42)
6_u"lr =
Problem ( 4 . 4 0 ) - ( 4 . 4 2 )
i s very s i m i l a r t o ( 4 . 1 7 ) - ( 4 . 1 9 )
a d a p t t o t h i s c a s e t h e a l g o r i t h m of Remark 4 . 4 .
and w e can
It is natural t o
e x p e c t t h a t Newton's method, i f u s e d , w i l l converge more r a p i d l y than the fixed-point
algorithm.
However, t h e o p e r a t o r a p p e a r i n g
i n ( 4 . 4 0 ) i s n o t a p r i o r i V - e l l i p t i c and t h e r e i s n o t h i n g t o a l l o w u s t o a s s e r t t h e e x i s t e n c e o f a s o l u t i o n 6 ~ ~The. u s e of Newton's method t h u s a p p e a r s more q u e s t i o n a b l e t h a n t h a t of
(4.16)-(4.20)
and i n c e r t a i n c a s e s w e may e x p e c t t o e n c o u n t e r
numerical d i f f i c u l t i e s i n t h e s o l u t i o n of Remark 4 . 6 :
(4.40)-(4.42).
I
The b a s i c a l g o r i t h m 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 r e q u i r e s t h e s o l u t i o n of a n o n s y m m e t r i c l i n e a r s y s t e m i n which t h e m a t r i x v a r i e s from one i t e r a t i o n t o a n o t h e r .
This
l a t t e r c o n s i d e r a t i o n makes t h e u s e of s u c h an a l g o r i t h m r e l a t i v e l y e x p e n s i v e ; i t t h e r e f o r e seems a t t r a c t i v e t o l o o k f o r a method i n which t h e l i n e a r p a r t w i l l have a f i x e d m a t r i x and w i l l t h u s b e s o l v a b l e i n an e f f i c i e n t manner.
The methods d e s c r i b e d i n t h e
(SEC.
5)
83
VARIANTS O F BASIC ALGORITHM
preceding s e c t i o n s allow us t o o b t a i n , very e f f i c i e n t l y , s o l u t i o n s o f S t o k e s p r o b l e m s and more g e n e r a l l y of p r o b l e m s of t h e t y p e g i v e n below
(with
-
clu
(4.43)
-
c1 2
VAU
+
I
--
V-u = 0 i n
(4.44)
0) Vp = f
in
-
a ,
a,
i t i s t h e r e f o r e n a t u r a l t o l o o k f o r i t e r a t i v e methods which e x p l o i t t h e s e p o s s i b i l i t i e s t o t h e maximum.
Algorithms o f t h i s kind are
d e s c r i b e d i n GLOWINSKI-MANTEL-PERIAUX
:11 and GLOWINSKI 1 2 , C h a p t e r
71; i n t h e above r e f e r e n c e s t h e s o l u t i o n o f t h e Navier-Stokes problem i s r e d u c e d t o t h a t of a s e q u e n c e o f p r o b l e m s o f t h e t y p e (4.43)-(4.45)
( t h e r e f o r e s o l u b l e by t h e methods o f t h e p r e c e d i n g
s e c t i o n s ) a n d o f p r o b l e m s o f n o n l i n e a r D i r i c h l e t t y p e which a r e s o l v e d by l e a s t - s q u a r e s
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.
I n p a r t i c u l a r i t s h o u l d be m e n t i o n e d t h a t t h e a l t e r n a t i n g d i r e c t i o n methods a l l o w t h e d e c o u p l i n g , i n a s i m p l e and e f f i c i e n t manner, o f t h e numerical d i f f i c u l t i e s associated with t h e incompressibility, and t h o s e a s s o c i a t e d w i t h t h e n o n l i n e a r i t y ; w e d i r e c t t h e r e a d e r t o t h e t w o r e f e r e n c e s g i v e n above f o r f u r t h e r d e t a i l s c o n c e r n i n g t h e s e methods.
5.
VARIANTS AND APPROXIMATIONS O F THE BASIC ALGORITHM O F SECTION 4 The a l g o r i t h m p r o p o s e d i n S e c t i o n 4 . 2
overall fixed-point sub-problems
r e q u i r e s , w i t h i n any one
i t e r a t i o n , t h e s o l u t i o n o f nonsymmetric l i n e a r
by methods a n a l o g o u s t o t h o s e u s e d i n t h e case o f t h e
S tok es problem.
A n a t u r a l i d e a a t t h i s p o i n t would be t o c a r r y
o u t i n i t i a l l y o n l y a n i n c o m p l e t e s o l u t i o n of t h e s u b - p r o b l e m s ;
we
h a v e a l r e a d y s e e n i n S e c t i o n 3 t h a t t h i s method of p r o c e d u r e may i n c e r t a i n cases b e a d v a n t a g e o u s and may a c c e l e r a t e t h e o v e r a l l p r o c e s s To implement t h i s i n c o m p l e t e s o l u t i o n w e c a n p r o c e e d i n t w o ways, e i t h e r by e m p l o y i n g a t e r m i n a t i o n t e s t which becomes p r o g r e s s i v e l y more s t r i n g e n t a s t h e a l g o r i t h m p r o c e e d s , o r by f i x i n g a n u p p e r bound t o t h e number o f i n t e r n a l i t e r a t i o n s . W e s h a l l be a d o p t i n g t h i s l a t t e r method h e r e ,
and w e s h a l l be
d i s c u s s i n g , w i t h i n a c o n t e x t where w e r e s t r i c t o u r s e l v e s t o a
STOKES
a4
&
NAVIER-STOKES EQUATIONS
(CHAP. 2)
single iteration, algorithms which may be viewed as nonlinear variants of the algorithm of Chapter I. We shall content ourselves with the simplest cases, which will nonetheless illustrate the fundamental approach. Variants of UZAWA type
5.1
In accordance with the general principle stated above, we here consider algorithm (4.16)-(4.20) in which at each step we carry out only a single iteration of (4.24)-(4.26) for the solution of (4.17)-(4.19). In this case the various stages can be condensed into a simplified algorithm, as follows: uo
(5.1)
with (5.2)
un
and
po
and
pn
chosen a r b i t r a r i l y ;
known, c a l c u l a t e a s o l u t i o n
A ( u ~ ) u = + "+~Op" = f r-.
t h e n knowing
-
u n ' %
of
,
untf, c a l c u l a t e
un+l
and
n+l p
blr
This algorithm can immediately be seen as the adaptation to our nonlinear problem of the algorithm of Chapter I, Section 4, in which we had introduced a relaxation parameter into the Uzawa algorithm. Note here, however, that even for w = 1 an initial value uo - of the velocity has to be specified; this corresponds to the fact that this algorithm is an approximation of the fixedpoint algorithm of Section 4.2. The choice of
can be made using the m e t h o d of minimum One could also consider using p. We should emphasise that the choice of parameters here is a very tricky open problem. pn
r e s i d u a l s described in Remark 4.4. a carefully-chosen fixed value of
Remark 5.1: The e x i s t e n c e of a solution of problem (5.2) is We assured if the operator Ar (yn) is V-elliptic (see (4.14)). have already shown that such will be the case if yn has zero divergence. This result arises from the fact that the trilinear
(SEC. 5 )
VARIANTS O F BASIC ALGORITHMS
form b ( u , v , w ) d e f i n e d i n ( 4 . 5 ) _
v -
and
f i e d by
-
w
_
if un.
85
i s anti-symmetric w i t h r e s p e c t t o
I
--
0.u = 0 .
Unfortunately t h i s condition is not s a t i s -
T o c i r c u m v e n t t h i s d i f f i c u l t y , w e can use
(see
I
TEMAM 111) t h e a n t i - s y m m e t r i s e d
form
We o b s e r v e t h a t w e c l e a r l y have i n t h i s c a s e (5.7)
Furthermore t h e supplementary t e r m does n o t p e r t u r b t h e e q u a t i o n s
.
b e c a u s e it v a n i s h e s i n t h e l i m i t when p-4 = 0 The e x i s t e n c e Of nif A r ( y n ) by A r ( u n ) where u w i l l t h e r e f o r e be a s s u r e d i f w e r e p l a c e I
I
W e s h a l l n o t a t t e m p t h e r e t o p r o v e t h e convergence of t h e s e algorithms.
The a v a i l a b l e p r o o f s
( s e e f o r example C R O U Z E I X i l 1 ) a r e
modelled on t h e p r o o f of t h e l i n e a r c a s e and a l l c u l m i n a t e i n imposing upon t h e v i s c o s i t y p a r a m e t e r a c o n d i t i o n ( v
"sufficiently
l a r g e " ) which would a p p e a r , i n t h e l i g h t of t h e e x p e r i m e n t a l r e s u l t s , t o be much t o o r e s t r i c t i v e and which f o r a l l p r a c t i c a l p u r p o s e s
l i m i t s us t o c a s e s where t h e l i n e a r S t o k e s a p p r o x i m a t i o n i t s e l f would b e a d e q u a t e .
W e t h u s have h e r e an u n r e s o l v e d t e c h n i c a l d i f f -
i c u l t y , a s s o c i a t e d w i t h o u r i n a b i l i t y t o o b t a i n an a p p r o p r i a t e upper bound f o r t h e n o n l i n e a r t e r m .
T h i s problem can a l s o be re-
l a t e d t o t h a t o f t h e non-uniqueness of t h e s o l u t i o n s .
However,
from TEMAM 111 w e c a n o b t a i n a r e s u l t c o n c e r n i n g weak convergence i n t h e mean, i . e .
M
+
+
m.
c o n v e r g e s weakly t o a s o l u t i o n when Rm;, ~ M r n Thus, l a c k i n g a p r o o f , w e s h a l l have t o make do w i t h a
heuristic discussion.
W e f i r s t a d v o c a t e t h e p r i n c i p l e t h a t any
convergence c a n o n l y come a b o u t from a d i s s i p a t i v e t e r m ,
this
p r i n c i p l e b e i n g j u s t i f i a b l e b o t h from t h e m a t h e m a t i c a l p o i n t o f view and from t h e p h y s i c a l p o i n t of view.
In the case
only d i s s i p a t i v e term i n a l g o r i t h m ( 5 . 1 ) - ( 5 . 5 ) which c o r r e s p o n d s i n ( 4 . 1 ) t o t h e t e r m
-
vhu
r
i s t h e term
.
=
0, the
- vAu"+~'* -
The n o n l i n e a r term
i t s e l f i s c o n s e r v a t i v e and would be t o t a l l y i n c a p a b l e o f producing any convergence w h a t s o e v e r towards a s t e a d y s t a t e . Now, i n t h e p h y s i c a l a p p l i c a t i o n s o f most i n t e r e s t t h e v i s c o s i t y
(CHAP. 2 )
STOKES & NAVIER-STOKES EQUATIONS
86
p a r a m e t e r i s v e r y s m a l l , which means t h a t t h e d i s s i p a t i o n c o n t a i n e d i n o u r a l g o r i t h m i s v e r y weak.
v
For
s m a l l (or f o r a l a r g e
Reynolds number, i n t h e t e r m i n o l o g y o f f l u i d m e c h a n i c s ) w e may t h e r e f o r e e x p e c t t h e convergence t o be slow.
W e t h u s see t h e
r # 0
advantage o f t h e p e n a l t y t e r m of H e s t e n e s , which f o r duces a supplementary d i s s i p a t i o n i n t o t h e algorithm.
intro-
T h i s supp-
l e m e n t a r y d i s s i p a t i o n , which d o e s n o t c h a n g e t h e s o l u t i o n , f o r c e s t h e c o n v e r g e n c e o f t h e p r e s s u r e t e r m and a d d s t o t h e n a t u r a l d i s s i p a t i o n which, however, i s a l o n e i n b e i n g a b l e t o e n s u r e t h e n convergence of u The a d v a n t a g e o f t h e method o f H e s t e n e s , 5
.
which w e p r o p o s e h e r e , o u g h t t h e r e f o r e t o become much more a p p a r e n t f o r s m a l l v a l u e s of t h e v i s c o s i t y .
I t must however b e n o t e d t h a t
i n t h i s c a s e o t h e r problems arise, such as t h e a p p e a r a n c e o f b o u n d a r y l a y e r s and p o s s i b l y t u r b u l e n c e ; s u c h p r o b l e m s r e q u i r e a n appropriate treatment, chiefly with regard t o t h e d i s c r e t i s a t i o n , and w e s h a l l n o t e n t e r upon a d i s c u s s i o n o f s u c h t o p i c s h e r e . The n u m e r i c a l s o l u t i o n o f t h e N a v i e r - S t o k e s
equations a t large
Reynolds numbers i s i n o u r o p i n i o n s t i l l a n u n r e s o l v e d p r o b l e m , and t h e methods which w e d e s c r i b e h e r e c o v e r o n l y o n e a s p e c t o f t h e question.
5.2
V a r i a n t s o f ARROW-HURWICZ
type
W e have seen i n Chapter I t h a t t h e analogue ( i n t h e l i n e a r c a s e )
of algorithm (5.1)-(5.5) t h e ARROW-HURWICZ
forms p a r t o f a f a m i l y o f a l g o r i t h m s o f A s w e have
type, with auxiliary operator.
a l r e a d y mentioned i n S e c t i o n 3 o f t h e p r e s e n t c h a p t e r , t h e s e methods c a n i n p r a c t i c e b e implemented as a v e r s i o n of i n which t h e s o l u t i o n o f
(5.1)-(5.5)
(5.2) i s c a r r i e d o u t incompletely.
The
advantage, i n c e r t a i n cases, i s t h a t t h e c a l c u l a t i o n s can b e c a r r i e d o u t i n a f u l l y e x p l i c i t manner.
The most g e n e r a l f o r m i s
written
W e o b s e r v e t h a t i n t h e case o f orithm reduces t o (5.1)-(5.5)
of course
sr
Sr = A r ( y n ) w i t h w = 1.
and
w
= 1, t h e a l g -
The s i m p l e s t case i s
= I , which g i v e s a f u l l y e x p l i c i t a l g o r i t h m .
u s e o f r e l a x a t i o n methods i n t h e s o l u t i o n o f
(5.2)
leads t o
The
(SEC.
5)
VARIANTS O F B A S I C ALGORITHM
nonsymmetric a u x i l i a r y o p e r a t o r s , Ar
87
g i v e n t h e p a r t i c u l a r form o f
(un) . The c o n v e r g e n c e o f t h i s t y p e o f a l g o r i t h m i s e v i d e n t l y e v e n
more d i f f i c u l t t o e s t a b l i s h t h a n t h a t o f t h e p r e c e d i n g case, and
it would b e o u t o f t h e q u e s t i o n t o a t t e m p t t o s t u d y i t h e r e .
In-
t u i t i v e l y , t h e b e h a v i o u r a t low Reynolds numbers o u g h t t o be s i m i -
l a r t o t h a t o f 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 i n t h e l i n e a r case. An a l g o r i t h m o f t h i s t y p e h a s b e e n u s e d ( i n t h e case w h e r e Sr = I , r = 0 ) i n FORTIN-PEYRET-TEMAM 111, up t o Reynolds numbers o f t h e o r d e r o f 1 03
v
s i d e r a b l y as
.
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 s h e s con-
d e c r e a s e s , and t h e c h o i c e o f t h e p a r a m e t e r s becomes
critical.
Numerical r e s u l t s
5.3
The t e s t s c a r r i e d o u t a r e f r a g m e n t a r y b u t n o n e t h e l e s s e n a b l e
some i n s i g h t t o b e g a i n e d i n t o t h e e f f e c t o f t h e p e n a l t y t e r m on t h e convergence of t h e a l g o r i t h m .
The n u m e r i c a l r e s u l t s w e h a v e
a v a i l a b l e are mainly f o r a l g o r i t h m ( 5 . 1 ) - ( 5 . 5 ) . The f i r s t e x p e r i m e n t a l f i n d i n g i s t h a t t h e s t u d y c a r r i e d o u t i n t h e l i n e a r case s t i l l r e m a i n s v a l i d q u a l i t a t i v e l y . value of
The o p t i m a l
i s less t h a n 1, i . e . it i s p r e f e r a b l e t o " u n d e r - r e l a x " .
w
The p r o b l e m t a k e n was a n a l o g o u s t o t h e o n e i n t r o d u c e d f o r t h e n u m e r i c a l e x p e r i m e n t s i n t h e l i n e a r case, and t h e v a l u e s o f tested were 1
and 1/10.
t w o v a l u e s of
v,
The o p t i m a l v a l u e o f
a r o u n d 0 . 1 , and t h a t o f
r = 0 , and around 7 f o r r = 5. with
v
u t i o n of
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 s h e d v.
(5.2) i s c l e a r l y m o r e d i f f i c u l t f o r
I n t h e c a s e of a l g o r i t h m ( 5 . 9 ) - ( 5 . 1 0 ) , a v a i l a b l e are v e r y i n c o m p l e t e . shes with for
v
was, for these
w a s around 1 . 5 f o r
b u t much l e s s f o r r = 5 t h a n f o r r = 0 .
seems t o b e more o r less i n d e p e n d e n t o f
r > 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
STOKES & NAVIER-STOKES EQUATIONS
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
&
NAVIER-STOKES EQUATIONS
(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
specified arbitrarily;
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 )
STOKES & NAVIER-STOKES EQUATIONS
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
known, c a l c u l a t e t h e s o l u t i o n
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 ,
&
NAVIER-STOKES EQUATIONS
(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.
This Page Intentionally Left Blank
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
DECOMPOSITION-COORDINATION
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 <
p a r t i c u l a r problem
-
m)
R ,
v * ( I v ~ ~ P - ~= vf ~ )i n
(1.23)
t o which w e s h a l l r e t u r n i n C h a p t e r V .
( F l o w of a Bingham f l u i d ) :
A p p l i c a t i o n t o Example 1 o f S e c t i o n 1.1 With t h e f u n c t i o n a l s
and
F
d e f i n e d by ( 1 . 6 ) , ( 1 . 7 ) ,
G
the
Lagrangian i s w r i t t e n (1.24)
&v,q,u)
i,
Iql 2 dx
=
g lnlq(dx
+
-
j,ivdx
+
j;-(Vv-q)dx.
s o t h a t t h e augmented L a g r a n g i a n i s P,(v.q,u)
(1.25)
= ;e(v,q,u)
We n o t e immediately t h a t
Zr
hence t h a t
is quadratic
This implies t h a t respect t o
v
on
,
with
1~
+
2
j,lVv-qI
dx.
f. i s l i n e a r w i t h r e s p e c t t o
v
and
(with positive-definite quadratic p a r t ) . and
f i x e d , w e can minimise
q
d,
V , whereas t h i s o p e r a t i o n i s i m p o s s i b l e w i t h
( t h i s drawback d i s a p p e a r s i f
F
and
G
with
2
a r e d e f i n e d by ( 1 . 8 ) , ( 1 . 9 ) ) .
I t f o l l o w s from t h i s t h a t c e r t a i n a l g o r i t h m s (see S e c t i o n 3 ) w i l l be
a p p l i c a b l e t o t h e c a l c u l a t i o n of t h e s a d d l e - p o i n t s of f o r t h o s e of
P
.
Zr
but not
We s h a l l now g i v e t h e o p t i m a l i t y c o n d i t i o n s c h a r 2 { u , p , A ] of L r on HL(R) x ( L 2 ( R ) ) 2 x (L2(f2))
a c t e r i s i n g any s a d d l e - p o i n t
We o b t a i n t h e s y s t e m o f e q u a t i o n s and ( v a r i a t i o n a l ) i n e q u a l i t i e s
(1.28)
vu-p = 0.
Note t h a t problem ( 1 . 2 6 ) i s , w i t h which i s l i n e a r w i t h r e s p e c t t o type.
p
and
A
f i x e d , a problem
u, and of a c o m p l e t e l y s t a n d a r d
The v a r i a t i o n a l i n e q u a Z i t y
(1.27) expresses t h e f a c t t h a t ,
102
DECOMPOSITION-COORDINATION
for specified
u
A,
and
p
(CHAP.
g i v e s t h e minimum o f &(u,q,X)
.
3)
W e have
h e r e a m i n i m i s a t i o n problem r e l a t i n g t o a n o n d i f f e r e n t i a b l e funct i o n a l f o r w h i c h t h e o p t i m a l i t y c o n d i t i o n i s e x p r e s s e d by a n i n -
1 1 1 , C21 and LIONS 111).
e q u a l i t y (see G . L . T .
Problem (1.27) o f f e r s
a c o n s i d e r a b l e advantage o v e r t h e i n i t i a l problem ( 1 . 4 ) .
In fact
w e c a n r e d u c e it t o a f a m i l y o f p o i n t p r o b l e m s o f t h e t y p e
where
1. I
and
,. )
(.
d e n o t e t h e norm and t h e i n n e r p r o d u c t on lR2
and where w e h a v e p u t , a t t h e p o i n t (1.30)
d(x)
=
x
R,
E
X(x) t rVu(x).
D e n o t i n g t h e s o l u t i o n of
( 1 . 2 9 ) by
t(x)
,
w e t h e n have
(1.31)
I t can e a s i l y be v e r i f i e d t h a t
s o l u t i o n of
p
d e f i n e d by
p(x
=
E(x)
is a
(1.27).
The a t t r a c t i o n o f t h e d e c o m p o s i t i o n c a r r i e d o u t i s t h u s t h a t it t r a n s f o r m s t h e " g l o b a l " n o n d i f f e r e n t i a b l e problem 1 . 4 ) i n o a f a m i l y o f l o c a l problems, c o o r d i n a t e d via t h e Lagrange m u l t i p l i e r
A.
This
t r a n s f o r m a t i o n o f c o u r s e o n l y becomes h e l p f u l i f w e c a n s o l v e t h e system (1.26)-(1.28)
by a n a p p r o p r i a t e a l g o r i t h m .
A p p l i c a t i o n t o Example 2 of S e c t i o n 1.1 ( E l a s t o p l a s t i c t o r s i o n ) : With t h e f u n c t i o n a l s
and
F
now d e f i n e d by
G
Lagrangian i s w r i t t e n : (1.32)
Z(v,q,u) =
+
jnjq12dx
Ii(q)
+
-
( 1 . 1 3 ) , (1.14), t h e
I, I, fvdx +
p-(Vv-q)dx
,
and a s i n (1.25), w e h a v e t h e augmented L a g r a n g i a n (1.33)
z,(v,q,u)
= z(v,q,U)
+
$
2 IvV-ql
-
The r e m a r k s made i n t h e p r e v i o u s example a r e w h o l l y a p p l i c a b l e a g a i n here.
Any s a d d l e - p o i n t
{u,p,A}
Of
a
~
on
HA(n)
x
(L2(iL))' x ( ~ ~ ( $ 2 ) ) ~
i s c h a r a c t e r i s e d by a s y s t e m a n a l o g o u s t o ( 1 . 2 6 ) - ( 1 . 2 8 ) , o n l y t h e
(SEC. 2 )
103
INVESTIGATION O F PROBLEM (PI
i n e q u a l i t y (1.27) being a l t e r e d i n t o r
r
W e can recognise i n (1.34)
This p r o j e c t i o n i n t o
(L
2
t h e c h a r a c t e r i s a t i o n of t h e p r o j e c t i o n o f
(n))'
c a n t h u s b e c a r r i e d o u t p o i n t by
A s i n t h e p r e c e d i n g example, t h e decomposition h a s a l l o w e d
point.
a p r o b l e m which i n c l u d e d a c o n s t r a i n t which w a s d i f f i c u l t t o manip-
u l a t e , t o b e r e d u c e d t o a f a m i l y of l o c a l p r o b l e m s , c o o r d i n a t e d v i a a Lagrange m u l t i p l i e r .
2.
INVESTIGATION OF PROBLEM ( P ) AND O F THE SADDLE-POINTS OF
2
AND
=Er-
This s e c t i o n is devoted t o review material of a t h e o r e t i c a l n a t u r e ; t h u s t h e r e a d e r who i s p r i m a r i l y i n t e r e s t e d i n t h e a l g o r i t h m i c a s p e c t s o f t h i s volume c a n a t a f i r s t r e a d i n g go f o r w a r d t o Section 3 of t h i s chapter. E x i s t e n c e and uniqueness p r o p e r t i e s f o r problem (P) .
2.1
From s t a n d a r d r e s u l t s o f Convex A n a l y s i s i n i n f i n i t e d i m e n s i o n s
(see f o r example LIONS 1 1 1 , EKELAND-TEMAM admit a s o l u t i o n i f t h e f u n c t i o n a l lim J(v)
(2.1)
= +m
when
lIv]Iv
+.
111) t h e p r p b l e m ( P )
J ( v ) = F(Bv)
+
G(v)
W i l l
satisfies
+ m ,
a s u f f i c i e n t condition f o r uniqueness being t h e s t r i c t convexity of
W e assume o f c o u r s e t h a t
J.
Remark 2 . 1 :
II'II
or
II'IIv
d e n o t e d by
Remark 2 . 2 :
+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 d e n o t e by t h e norm on
1.1
J $
.
V,
t h e norm on
Suppose t h a t w e h a v e
H
b e i n g everywhere
DECOMPOSITION-COORDINATION
104
I f the operator d e f i n e s on
I
(2.3)
is i n j e c t i v e with
B
a norm e q u i v a l e n t t o
V
lim F(Bv)
i f “VI/
-t
Im B llvll
c l o s e d i n H , then
3)
lBvl
s u c h t h a t w e have
-+ + m
+-
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
2.2
(CHAP.
k and of
2,
The u s e of t h e a l g o r i t h m s of S e c t i o n s 3 and 4 i s e s s e n t i a l l y j u s t i f i e d by t h e f o l l o w i n g : THEOREM 2 . 1 :
Suppose
{u,p,X}
is a s a d d l e - p o i n t of
2
on
t h e n { u , p , A } is a s a d d l e - p o i n t of f r , V r > 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)
<; e ( v , q , A) 5 P,(v,q.A),
H.
Suppose, c o n v e r s e l y , t h a t (2.6)
105
INVESTIGATION OF PROBLEM ( P )
i s a s a d d l e - p o i n t of
{u,p,X}
Z r ( u , p , u ) %,(u,p,A)
V{v,q}
Qr(v,q,A)
6 VXH,
VU
E
e r ;hence
H.
P r o c e e d i n g a s above, w e deduce from t h e f i r s t i n e q u a l i t y i n ( 2 . 6 ) that
p = Bu.
F u r t h e r m o r e w e have (2.7)
Lr(u,p,A)
Y !
,(v,q,A)
V {v,ql
E
VxH.
d, with r e s p e c t t o
Taking i n t o a c c o u n t t h e c o n v e x i t y of pair
{u,p}
i s , f o r given
X,
G(v) - G(u) + ( A , B ( v - u ) ) + r(Bu-p,B(v-u)) F(q)
thus, since (2.8)
-
F ( p ) - (A,q-p)
+
{v,q}, the
c h a r a c t e r i z e d by
r(p-Bu,q-p)
20
2
0 Vv
E
V,
VqEH.
p = Bu, w e f u r t h e r have
G(v)-G(u)
+ (A,B(v-u)) 2 0
The r e l a t i o n s ( 2 . 8 )
,
(2.9)
V V EV,
are equivalent t o
Furthermore ;e(U,P,U)
5
;e(U,P,X)
f o l l o w s t r i v i a l l y from Bu = p. Theorem 2 . 1 i s t h e r e f o r e c o m p l e t e l y proved.
R e m a r k 2.3:
I n infinite dimensions, t h e q u e s t i o n of t h e p o s s i b l e
DECOMPOSITION-COORDINATION
106
existence of a saddle-point,
(CHAP. 3 )
i . e . o f a Lagrange m u l t i p l i e r
A,
is
p r o b l e m a t i c a l and i s a l w a y s d e p e n d e n t on t h e p o s s i b i l i t y o f u s i n g i n some f o r m o r o t h e r t h e Hahn-Banach
Theorem.
Sufficient conditions
t o a s s u r e t h e e x i s t e n c e o f t h e m u l t i p l i e r would b e
admits a solution,
(2.10)
(P)
(2.11)
3u
E V where
(2.12)
F
i s continuous with respect t o
FDB+G
is f i n i t e , Bu
.
A p r o o f o f t h i s r e s u l t c a n b e f o u n d i n EKELAND-TEMAM
ogous r e s u l t a p p e a r s i n GABAY-MERCIER 111. (2.12)
111 and an a n a l -
In practice condition
i s t h e most d i f f i c u l t t o s a t i s f y .
In f i n i t e dimensions t h e e x i s t e n c e of a saddle-point
i s assured
s i n c e w e minimise under a l i n e a r e q u a l i t y c o n s t r a i n t ; i n t h i s c a s e
it i s s u f f i c i e n t t h a t problem
(P) a d m i t a s o l u t i o n .
I n p r o b l e m s which a r e d i s c r e t i s e d from a p r o b l e m i n i n f i n i t e dimensions w e c o u l d t h u s have
Ah
whereas t h e m u l t i p l i e r
2.3
uh
which a p p r o a c h e s
u
as
h
+
0
does n o t converge i n H.
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 A n a l y s i s
The a p p r o a c h f o l l o w e d i n S e c t i o n 1 . 2 c o n s i s t e d o f " d u a l i s i n g " t h e o r i g i n a l problem
(P) by a p r o c e d u r e which a t f i r s t s i g h t may s e e m
somewhat c o n t r i v e d .
W e now show t h a t t h i s a p p r o a c h i s no m o r e t h a n
a d i f f e r e n t way of e x p r e s s i n g t h e d u a l i t y i n t h e s e n s e o f F e n c h e l Rocka f e 1l a r
.
C o n s i d e r , t h e n , problem (2.13)
inf { F ( B v ) + G ( v ) ~
v EV
(P),
namely:
.
I n convex a n a l y s i s (see ROCKAFELLAR C41, EKELAND-TEMAM dard procedure f o r a s s o c i a t i n g a Lagrangian w i t h
(P)
[11) t h e stanis t o consider
a p e r t u r b e d f u n c t i o n a l on V x H, d e f i n e d by (2.14)
@ ( V , Z )= F(Bv-z)+G(v).
The L a g r a n g i a n a s s o c i a t e d w i t h ( 2 . 1 4 ) (2.15)
i s t h e n d e f i n e d c l a s s i c a l l y by
L(v,u) = inf { ( z , ~ ) + $ ( v , z ) } = inf { ( z , ~ ) + F(Bv-2)) Z
2
+ G(v).
(SEC. 2 )
INVESTIGATION O F PROBLEM ( P )
we can then w r i t e ( 2 . 1 5 )
Suppose
q = Bv-z;
(2.16)
L(v,lJ) = i n f {(Bv-q,u)
107
i n t h e form
+ F ( q ) j + G(v).
4
If
admits a saddle-point
L ( . ,.)
u t i o n of (2.17)
[u,X}
on H d e s i g n a t e d
H,
then
u
i s a sol-
( P ) t h e maximisation p r o b l e m
( P * ) and d e f i n e d by
sup { i n f L ( v , p ) } U
.
V
I f we express (2.19)
x
i n f sup L ( v , u ) = sup i n f L ( v , u ) . veV UEH UrH V C V
W e t h e n d e s c r i b e a s t h e d u a l problem o f
(2.18)
on V
and w e have
(P)
L(. ,.)
e x p l i c i t l y w e can w r i t e
sup i n f inf { F ( q ) + G(v) + (y,Bv-q)) I J v q
= sup
IJ
inf &(v,q,U). {v,ql
The a p p r o a c h f o l l o w e d i n S e c t i o n 1 . 2 t h u s c o n s i s t e d o f u t i l i s i n g t h e Lagrangian
L (.
,. )
i n a n e q u i v a l e n t b u t "more e x p l i c i t " form.
T h i s remark a p p l i e s e q u a l l y w e l l t o t h e augmented L a g r a n g i a n . f a c t t h e standard definition
(ROCKAFELLAR
In
[ll, FORTIN [I]) i s o b t -
a i n e d by c o n s i d e r i n g (2.20)
tJ,(V,Z)
= tJ(v,z) +
r
(21
2
,
By t h e same a r g u m e n t as above w e c a n d e r i v e t h e L a g r a n g i a n d r o f Section 1.2. I n FORTIN C11, L r ( v , u )
(2.21).
i s c a l c u l a t e d by e l i m i n a t i o n o f
with respect t o
in
v, t h i s leads t o algorithms very s i m i l a r t o those
s t u d i e d i n S e c t i o n s 3 and 4. Lr(V,p)
q
F o r t h e s o l u t i o n o f c e r t a i n p r o b l e m s which a r e n o n l i n e a r The s e a r c h f o r a s a d d l e - p o i n t o f
u s i n g a n a l g o r i t h m of t h e UZAWA t y p e (see C h a p t e r I ) r e q u i r e s
at. e a c h s t a g e t h e s o l u t i o n o f a p r o b l e m which i s nonlinear w i t h respect t o
v (see FORTIN C11 where t h e case o f Example 1 o f S e c t i o n 1.1
is d e a l t with).
The a l g o r i t h m s d e s c r i b e d i n t h e r e m a i n d e r of t h i s
c h a p t e r c a n b e c o n s i d e r e d as methods o f s o l u t i o n by d e c o m p o s i t i o n
of t h i s n o n l i n e a r p r o b l e m (see Remark 3 . 1 b e l o w ) .
m
108
DECOMPOSITION-COORDINATION
Remark 2 . 4 :
r
Suppose
can e a s i l y be shown t h a t i f then
V x H
{u,Bu,X)
2
(CHAP.
3)
0 ; by p r o c e e d i n g a s i n Theorem 2 . 1 ,
it
{u,X1
i s a saddle-point of
i s a s a d d l e - p o i n t of
fr on
V
X
on
Lr H
X
and
H,
vice versa.
Remark 2 . 5 :
An i m p o r t a n t p r o p e r t y c o n c e r n i n g t h e u s e o f t h e aug-
mented L a g r a n g i a n i s t h a t , f o r any
r
0 , t h e Lagrangians
Lr
and
t o g e t h e r w i t h t h e a s s o c i a t e d d u a l problems a r e always d i f f e r e n t -
fr i a b l e with r e s p e c t t o
u,
which i s n o t t h e c a s e i n g e n e r a l f o r r = 0.
I t i s i n f a c t proved i n FORTIN C11 t h a t f o r
r
> 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)
where
{uX,pXl
is the solution of the problem
X,V,E
H, we
110
DECOMPOSITION-COORDINATION
A d d i t i o n a l l y , w e have n o t e d i n Remark 2.5 t h a t , f o r
r
(CHAP.
3)
> 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
DECOMPOSITION-COORDINATION
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
b e i n g known, d e t e r m i n e
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.
CONVERGENCE O F A L G l
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
CONVERGENCE O F A L G l
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
--
Bun
tends t o
i n (3.5), we obtain
F (fi)-(X",fi)+r(pn-Bu?p^) t F (pn)-(An,pn)+r (pn-Bun,pn).
X"
i s bounded i n
H
and knowing t h a t
pn
-
z e r o , w e t h e r e b y deduce t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s
Bo,B1,
116
(CHAP. 3 )
DECOMPOSITION-COORDINATION
i n d e p e n d e n t of
n , such t h a t
Taking i n t o a c c o u n t assumption
i s bounded, i . e .
pn
i t f o l l o w s from ( 4 . 2 4 )
that
M > 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.
DECOMPOSITION-COORDINATION
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
DECOMPOSITION-COORDINATION
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
CONVERGENCE O F A L G l
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
DECOMPOSITION-COORDINATION
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
0
I+& ( 7 r, )
we h a v e f o r A X 2 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 (5.2)
u"
(5.3)
pn
(5.4)
hn+'-in
(5.5)
An
-t
-f
u
strongly i n V,
p
strongly i n -t
a saddle-point
i s a (weak) c l u s t e r p o i n t o f of
on
Y!?~
V
b e i n g a s a d d l e - p o i n t of
x
H
in=
W e again w r i t e
Proof:
n,
strongly i n
i s bounded i n H .
A*
Moreover i f
o
n,
Lr
on
un-u,
G(v)-G(u)+(h,B:~-u))+r(B~-p,B(v-u))
(5.7)
(F;(p),q-p)+F1
A
=
X"
pn-p,
5
0, V v
E
V, E
H,
n-p n-I ,B(v-un)) t 0, V v
E
V,
5 0,
(3.12)-(3.14)
imply
(5.10)
(F;(pn)
,q-pn)+F1 (q)-FI (pn)-(Xn,q-pn)+r(pn-Bun,q-pn)
(5.11)
A"+'
Xn+p(Bu"-pn).
and
v = u
=
n
i n (5.6),
q = pn
i n (5.7),
then
2 0,
v
=
v 9 E 8,
u i n (5.9)
q = p i n ( 5 . 1 0 ) ; h e n c e by a d d i t i o n
(in-],Bin)+
(5.12)
r I Bin I '-r
(5.13)
(FA(pn)-FA(p) ,pn-p)+rlpn12-r(pn,B;")-(Tn,~n)
Adding ( 5 . 1 2 ) ,
(5.14)
{U,P,XJ
1;
Vs
(q)-F, (p)-(X,q-P)+r(p-Bu,q-P)
G(v)-G(un)+(Xn,B(v-un))+r(Bu
W e set
A"-
=
X+p(Bu-p), V n.
In addition,
(5.9)
in =
V x H x H , w e have
(5.6)
(5.8)
i n H, { u , p , A }
A"
H.
x
(T",B;")
5
0, SO.
(5.13) 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
(FA(pn)-FA(p) ,pn-p)+r I Bin-pnl 2+(Xn,B;n-pn)+r(pn-pn-'
,B?)
S
0.
i s
124
DECOMPOSITION-COORDINATION
(CHAP.
3)
Also, s u b t r a c t i n g ( 5 . 8 ) f r o m ( 5 . 1 1 ) and t a k i n g t h e s c a l a r s q u a r e i n H, we obtain
I t t h e n f o l l o w s from ( 5 . 1 4 ) ,
(5.16)
IXn('-lXn+'l2
(5.15) t h a t
-n -n 2 -n -n-1 (2r-p) IBu -p 1 +2pr(p -p ,Bin).
>2p(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
DECOMPOSITION-COORDINATION
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
= p < 2r.
.
The proof o f t h i s r e s u l t is r a t h e r t e c h n i c a l a n d d o e s n o t s e e m t o b e e x t e n d a b l e t o t h e more g e n e r a l cases c o n s i d e r e d i n t h i s book.
(SEC. 6 )
1 27
SOME NONLINEAR PROBLEMS
Remark 5 . 2 :
I n t h e case w h e r e
is l i n e a r we note t h a t the
G
s t a g e ( 3 . 1 2 ) o f ALG2 i s a l i n e a r p r o b l e m r e l a t i v e t o t h e s y m m e t r i c t o p e r a t o r 3 B. Consequently, i n f i n i t e dimensions (and assuming B i n j e c t i v e ) i t w i l l b e c o n v e n i e n t t o S a c t o r i s e o n c e a n d f o r a1
t ( f o r example by C h o l e s k y ' s method) t h e m a t r i x B B which i n t h i s case
m
i s symmetric and p o s i t i v e d e f i n i t e .
Remark 5 . 3 :
H e r e a g a i n , as w e h a v e m e n t i o n e d i n Remark 5 . 1 f o r I
ALGl, w e c a n r e p l a c e t h e c o e r c i v i t y a s s u m p t i o n s on F o ( . ) by an 1
a s s u m p t i o n on G ( . )
On t h e c h o i c e o f
.
8
and
p
r.
W e s a w i n Section 4.3 t h a t i f t h e n t h e sequence
{u"},
F(q) =
lqI2
and i f
G
is linear,
r e l a t i n g t o ALGl c o n v e r g e s i n t w o i t e r -
a t i o n s a t t h e m o s t i f w e use
pn
= p =
w i t h t h e same a s s u m p t i o n s on
F
and
{unIn
T1
I n t h e c a s e o f ALG2,
r.
G , w e a l s o h a v e c o n v e r g e n c e of
i n t w o i t e r a t i o n s a t t h e most i f
pn
= p =
r = 1 ( f o r any
T h i s f a c t would a p p e a r t o i n d i c a t e a r a t h e r
c h o i c e of {po,X1}).
g r e a t e r d e g r e e o f r o b u s t n e s s f o r ALG1. I n g e n e r a l , f o r given choice is
p =
r.
r, experience i n d i c a t e s t h a t t h e b e s t r
The c h o i c e o f
i s more p r o b l e m a t i c a l and
i n t h i s r e s p e c t ALGZ i s more s e n s i t i v e t h a n ALG1.
Also, f o r very
" s t i f f " p r o b l e m s A L G l would a p p e a r t o b e more r o b u s t t h a n ALG2; by t h i s w e mean t h a t t h e c h o i c e of
r
i s less c r i t i c a l and t h a t t h e
c o m p u t a t i o n t i m e w i t h ALGl may b e much s h o r t e r f o r a g i v e n p r o b l e m t h a n w i t h ALG2.
6.
APPLICATION TO SOME N O N L I N E A R PROBLEMS 6.1
Introduction
The p u r p o s e o f t h i s s e c t i o n i s t o show v i a s e v e r a l e x a m p l e s t h a t t h e methods d e v e l o p e d p r e v i o u s l y a r e a p p l i c a b l e t o a v a r i e t y o f d i f f e r e n t t y p e s o f problem.
I n c e r t a i n cases, w e s h a l l r e - e n c o u n t e r
c l a s s i c a l methods f o r which t h e a l g o r i t h m s A L G l and ALG2 w i l l p r o v i d e s o l u t i o n p r o c e d u r e s which a r e o f t e n s i m p l e r t h a n t h e u s u a l t e c h n i ques.
I n o t h e r cases w e s h a l l o b t a i n d e c o m p o s i t i o n p r o c e d u r e s
w h i c h a p p e a r t o b e new and which may p o s s e s s c e r t a i n a d v a n t a g e s from t h e a l g o r i t h m i c p o i n t o f view.
Generally, it w i l l be e s t a b l i s h e d
128
DECOMPOSITION-COORDINATION
(CHAP. 3 )
t h a t t h e decomposition methods of t h i s c h a p t e r can e x t e n d t o a v e r y d i v e r s e range of a p p l i c a t i o n s ,
and t h a t t h e y c o n s t i t u t e a f l e x i b l e
means of approach t o o p t i m i s a t i o n problems, i n a manner which i s sometimes r a t h e r non-standard.
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 )
The a p p l i c a t i o n of a l g o r i t h m s A L G l and ALGZ t o t h e s e problems h a s a l r e a d y been d e s c r i b e d i n S e c t i o n 3 . 3 and w e s h a l l n o t go o v e r t h i s ground a g a i n .
W e s h a l l c o n t e n t o u r s e l v e s h e r e w i t h a few remarks.
A s r e g a r d s t h e convergence of t h e a l g o r i t h m s , we r e c a l l t h a t w e have
i n t h e c a s e of a Bingham f l u i d , and
i n t h e c a s e of e l a s t o p l a s t i c t o r s i o n . I n b o t h c a s e s , w e can w r i t e 5 2 (ql dx, t h i s f u n c t i o n a l b e i n g d i f f e r e n t i a b l e and s t r i c t l y Fo(q) = T
I,
convex.
I t can be shown w i t h o u t d i f f i c u l t y t h a t a l l t h e assurnp-
t i o n s of Theorem 4 . 1 and 5 . 1 a r e s a t i s f i e d . A s r e g a r d s t h e c h o i c e of
p
and o f
r , w e n o t e t h a t t h e s e pro-
blems a r e very c l o s e l y r e l a t e d t o t h e c a s e d e a l t w i t h i n S e c t i o n 4.3. Experimentally,
(GABAY-MERCIER
ALGZ t h e convergence of p =
r
1.
=
u
n
ill), it h a s been e s t a b l i s h e d t h a t f o r
i s f a s t e r t h a n t h a t of
pn
optimal i n t h i s c a s e 5 .
An
or
I t would c e r t a i n l y a p p e a r t h a t t h e c h o i c e of
r
=
for
1 is
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 more
d e t a i l i n Chapter V .
A n o n l i n e a r D i r i c h l e t problem
6.3
T h i s d e s c r i p t i o n f o l l o w s t h e approach of GLOWINSKI-MARROCCO C11; t h e problem d e s c r i b e d w i l l be c o n s i d e r e d i n d e t a i l i n Chapter V . For
1
<
s
<
+
T h i s assumes
and w i t h
m,
u
=
1
a bounded open s u b s e t of
in (6.1).
IRN , w e
(SEC. 6 )
129
SOME NONLINEAR PROBLEMS
consider
We wish t o s o l v e i n
where
n
t h e f o l l o w i n g n o n l i n e a r D i r i c h l e t problem
,;
f EW-I’~’(Q)
;( I + 1
= I).
I t c a n b e shown by s t a n d a r d t e c h n i q u e s ,
(see L I O N S C21), t h a t ( 6 . 4 )
p o s s e s s e s a unique s o l u t i o n which i s a l s o a s o l u t i o n of t h e minimis a t i o n problem
W e note t h a t
is n o t a H i Z b e r t s p a c e 6 .
W;”(Cl)
i n the infinite-dimensional
W e cannot t h e r e f o r e ,
c a s e , a p p l y t h e convergence theorems of
One way of s u r m o u n t i n g t h i s d i f f i c u l t y i s t o con-
A L G l and ALG2.
s i d e r a d i s c r e t i s e d problem.
We s h a l l t h e n b e i n a s p a c e of f i n i t e
dimension and it w i l l be p o s s i b l 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 b e i n g s a t i s f i e d , t o u t i l i s e A L G l and ALG2 for 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 problems.
T o a v o i d i n t r o d u c i n g new n o t a t i o n which w i l l
n e e d l e s s l y encumber t h e d i s c u s s i o n , w e s h a l l be s a t i s f i e d w i t h a
f o r m a l argument r e l a t i n g t o t h e c o n t i n u o u s problem. Thus suppose
W e then put
hence Except i f
s
=
2.
(CHAP. 3 )
DECOMPOSITION-COORDINATION
130
F i r s t w e n o t e t h a t w e c l e a r l y have (6.8)
lim Iq Is ++m
i s t h e norm o f
(91,
where
% = + a ,
q
in
( L S ( n ) ) N , i.e.
(6.9)
Moreover, f o r any CIARLET
(see f o r example GLOWINSKI-YARROCCO ill,
p,q c H
I l l ) , we have (F'(q)-F'(p),q-p)
(6.10)
if
2 alq-plz
s 22,
I n a d d i t i o n w e c a n show t h a t
(6.12)
lF'(q)-F'(p)ls,
5
B(Ip1,+1q1,)
s-2 Iq-pIs
if
s 22,
and IF'(q)-F'(p))
(6.13)
s- 1
Is,
a and s t r i c t t y positive.
The c o n s t a n t s
F o r t h e s o l u t i o n of
581q-pls
fi
if
1ssr2.
a r e i n d e p e n d e n t of
(6.4)
p
and of
q
a n d a re
o r ( 6 . 5 ) w e s h a l l u s e t h e augmented
Lagrangian:
The a l g o r i t h m A L G l c a n t h e n b e w r i t t e n :
<'A
(6.15)
for
n
2
0 , An
(~"(a))~,
-rAu" = f+V.A"-rV.p" (6.16)
a r b i t r a r i l y chosen;
b e i n g known, c a l c u l a t e t h e s o l u t i o n
in
R,
n n u , p
of
(SEC.
6)
(6.17)
SOME NONLINEAR PROBLEMS
I pnl s-2pn+,pn
=
131
rVu"+X",
then
The s y s t e m ( 6 . 1 6 ) , (3.8)-(3.10).
(6.17)
can be s o l v e d by t h e r e l a x a t i o n method
W e n o t e t h a t , by means of 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 , t h e n o n l i n e a r problem ( 6 . 1 7 ) decomposes i n t o a f a m i l y of two-dimensional problems which a r e e a s y t o s o l v e .
T o o b t a i n ALG2,
w e r e p l a c e (6.15) by (6.19)
{po,X') E H X H '
arbitrariZy c h o s e n ;
and ( 6 . 1 6 ) b y
The p r o p e r t i e s ( 6 . 8 ) and ( 6 . 1 0 ) - ( 6 . 1 3 ) a l l o w , i n f i n i t e dimensions, t h e Theorems 4 . 1 and 5 . 1 t o be a p p l i e d and t h e convergence of t h e above a l g o r i t h m s t o be t h e r e b y deduced. t h a t t h e d i r e c t s o l u t i o n of or
s
l a r g e (e.g.
s
<
(6.4) for
...)
s
i n t h e neighbourhood of
1
1 . 5 o r s > 5 ) by s t a n d a r d i t e r a t i v e methods
( c o n j u g a t e g r a d i e n t , Newton-Raphson etc
Numerical e x p e r i m e n t s show
i s very d i f f i c u l t .
,
nonlinear overrelaxation,
T o o u r knowledge,
i c i e n t methods i n t h i s c a s e a r e A L G l and ALG2.
t h e only r e a l l y e f f For more d e t a i l s
see C h a p t e r V.
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 d i r e c t i o n methods
The a l g o r i t h m s A L G l and ALG2 can a l s o be used (assuming t h e i n t r o d u c t i o n of an a p p r o p r i a t e augmented L a g r a n g i a n ) f o r t h e s o l u t i o n o f mildly n o n l i n e a r problems o f t h e t y p e (6.21)
Au+Cp(u) = f,
where, l i m i t i n g o u r s e l v e s t o f i n i t e dimensions and w r i t i n g v = { vl , . . . v N l :
132
DECOMPOSITION-COORDINATION
-
A is an
'p : IRN
N -f
X
: IR
'pi
symmetric p o s i t i v e d e f i n i t e m a t r i x ,
N
IRN i s a n o n l i n e a r mapping o f d i a g o n a l t y p e , i . e .
(V(V))~ ='pi(vi)
with
(CHAP. 3)
,
i=l,
...N,
+IR, c o n t i n u o u s and i n c r e a s i n g ( w e c a n a l w a y s assume
cpi(0) = 0 ) ,
that
- f
E
ElN.
The d i s c r e t i s a t i o n by finite differences o r finite elements of c e r t a i n m i l d l y n o n l i n e a r e l l i p t i c o r p a r a b o l i c problems l e a d s t o problems o f t h e t y p e ( 6 . 2 1 )
(some e x a m p l e s w i l l b e g i v e n i n
Chapter I V ) .
Remark 6 . 1 : s o l u t i o n of
The a l g o r i t h m s w h i c h w e s h a l l b e d e s c r i b i n g f o r
( 6 . 2 1 ) e x t e n d t o t h e case where
i s non-symmetric,
A
a n d positive definite.
t(
W e define
Since the function
$i
is
,.....N,
i = 1
'pi
i s c o n t i n u o u s and i n c r e a s i n g it f o l l o w s t h a t
a n d convex.
C1
Since the operator
A
i s symmetric it
f o l l o w s t h a t s o l v i n g (6.21) i s e q u i v a l e n t t o s o l v i n g t h e minimisa t i o n problem
(6.22)
1
J(u) S J ( v )
VvcR
N
,
ueRN,
with, i n (6.22)
where and
(.
II-II
,. )
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 i n n e r p r o d u c t on
IRN
d e n o t e s t h e a s s o c i a t e d norm.
The a b o v e p r o p e r t i e s o f a d m i t s a unique solution.
A
and
'pi
imply t h a t ( 6 . 2 1 ) ,
(6.22)
(SEC.
6)
133
SOME NONLINEAR PROBLEMS
Remark 6.2:
If
A
i s non-symmetric and p o s i t i v e d e f i n i t e , it
c a n e a s i l y b e shown t h a t ( 6 . 2 1 ) s t i l l admits a u n i q u e s o l u t i o n .
m
The p r o b l e m ( 6 . 2 2 ) i s a p a r t i c u l a r p r o b l e m ( P ) i n w h i c h , w i t h t h e n o t a t i o n of S e c t i o n 1.1, w e c a n t a k e
By v i r t u e of t h i s d e c o m p o s i t i o n w e c a n s o l v e ( 6 . 2 1 ) , a n d ALGZ
( n o t e t h a t i n t h e p r e s e n t case
Remark 6.3:
Instead of defining
G
G
( 6 . 2 2 ) by A L G l
is nonlinear).
and F
by ( 6 . 2 5 ) ,
(6.26),
we could use
W e n a t u r a l l y associate w i t h
Since t h e constraint p o i n t on
equal t o
v
-
lRN x IRN x DN.
i s l i n e a r , Z r admits a s a d d l e T h i s s a d d l e - p o i n t i s i n f a c t u n i q u e and q = 0
{u,u,Au}.
S o l u t i o n of ( 6 . 2 1 ) ,
(6.22)
I t follows from ( 3 . 4 ) ,
to t h e s o l u t i o n of (6.28)
Xo c R N ,
then f o r
n
(6.29)
(6.24) - ( 6 . 2 6 ) t h e auqmented Lagrangian
2
by ALG1; (3.5),
(6.21),
0,
r-un+rp(un)
=
f + r p n -A n ,
(6.27) t h a t t h e a p p l i c a t i o n o f ALGl
(6.22) l e a d s t o t h e f o l l o w i n g a l g o r i t h m
134
DECOMPOSITION-COORDINATION
(6.30)
(rI+A)pn = run+A",
(6.31)
A"+l
(CHAP. 3 )
= A"+pn(u"-p").
( 6 . 3 0 ) c a n b e s o l v e d by t h e b l o c k r e l -
The n o n l i n e a r s y s t e m ( 6 . 2 9 ) ,
axation method o f S e c t i o n 3.2 and w e n o t e t h a t , known, t h e c a l c u l a t i o n o f N
un
pn
and
An
being
i n (6.29) reduces t o t h e s o l u t i o n of
n o n l i n e a r e q u a t i o n s , e a c h i n a s i n g l e v a r i a b l e , which a r e i n d e p -
e n d e n t o f e a c h o t h e r and o f t h e f o r m (6.32)
rc+rpi(E) r
The p a r a m e t e r
= b
,
i=l,
being
...N . and
> 0
being
'pi
and i n c r e a s i n g ,
Co
e q u a t i o n ( 6 . 3 2 ) a d m i t s a u n i q u e s o l u t i o n which can b e c a l c u l a t e d by
(see HOUSEHOLDER C11, BRENT 1 1 1 ) .
v a r i o u s methods
un
Similarly i f
a r e known i n ( 6 . 3 0 ) , w e o b t a i n
An
and
s o l v i n g a l i n e a r system w i t h m a t r i x dent of
Assuming
rIfA.
n , it is t h u s convenient t o f a c t o r i s e
rIfA
r
pn
by
indepen-
o n c e and f o r
a l l ( b y a Gauss o r C h o l e s k y m e t h o d ) .
S o Z u t i o n of ( 6 . 2 1 ) ,
(6.22)
b y ALG2:
I t s u f f i c e s t o r e p l a c e ( 6 . 2 8 ) by " (6.33) {po,A') E R x R ,
and ( 6 . 2 9 ) by (6.34)
run+cp(un) = f+rpn-'-An.
I t t h e n f o l l o w s f r o m Theorem 5.2
(6.34),
(6.30),
(6.31) i f
Remark 6 . 4 :
Suppose t h a t
ru"+cp(u") (6.35)
O
t h a t w e have convergence of =
p < 1 +vT r.
pn = p = r
= f+rpn-'-An,
rpn+Ap" = ru"+A",
A"+'
=
An+=(un-pn)
.
I t t h e n f o l l o w s from ( 6 . 3 5 ) t h a t (6.36)
An+'
= Ap",
W e t h e n d e d u c e from ( 6 . 3 5 ) ,
(6.37)
run+cp(un) +Apn-'
(6.36) t h a t
= f+rp"-l
(6.33),
i n ALG2: w e t h e n h a v e
7)
(SEC.
NONLINEAR PROGRANNING PROBLEMS
(6.38)
rpn+Apn+cp(u")
= f+rpn-'.
Consequently t h e r e f o r e , i f un = pn-') ution of
135
pn
= p =
r , ALGZ r e d u c e s ( p u t t i n g
t o an a l t e r n a t i n g d i r e c t i o n method a p p l i e d t o t h e sol(6.21)
( 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 s I V
and IX).
Remark 6 . 5 :
W e s h a l l see i n C h a p t e r I V t h a t A L G l combined w i t h
t h e b l o c k r e l a x a t i o n method of S e c t i o n 3.2 i s more r o b u s t and more e f f i c i e n t t h a n ALGZ
if
'p
i s o f "low" d i f f e r e n t i a b i l i t y ; t h i s w i l l
be t h e c a s e i f f o r example w e u s e a f i n i t e - e l e m e n t
or f i n i t e -
d i f f e r e n c e approximation o f t h e n o n - l i n e a r e l l i p t i c problem
1 < s < 2.
with
C h a p t e r I V w i l l g i v e a number o f r e s u l t s o f num-
e r i c a l e x p e r i m e n t s r e l a t i n g t o problems o f t h e t y p e ( 6 . 3 9 ) .
7
-
APPLICATIONS TO NONLINEAR PROGRAMMING PROBLEMS I n t h i s s e c t i o n we s h a l l be presenting t w o f a m i l i e s of applic-
a t i o n s of t h e methods o f t h i s c h a p t e r t o n o n l i n e a r programming. The f i r s t , d e s c r i b e d i n S e c t i o n 7 . 1 ,
i s l i n k e d t o t h e augmented
L a g r a n g i a n methods i n t r o d u c e d by ROCKAFELLAR 111 f o r t h e c a s e of
inequality constraints.
This involves a n a t u r a l extension of t h e
method of H e s t e n e s a n d P o w e l l
s t u d i e d i n Chapter I.
I n t h i s res-
p e c t , a l g o r i t h m ALGZ can b e c o n s i d e r e d a s a new t e c h n i q u e f o r obtaining t h e required saddle-point. i o n 7.2
W e s h a l l then p r e s e n t i n Sect-
a n o t h e r a p p r o a c h w h i c h , i n somewhat more t a n g i b l e f a s h i o n ,
i s b a s e d on a d e c o m p o s i t i o n p r i n c i p l e which r e s u l t s i n a d e c o u p l i n g of t h e c o n s t r a i n t s .
F i n a l l y , i n S e c t i o n 7.3 w e s h a l l d e s c r i b e an
a p p l i c a t i o n o f t h e methods o f t h i 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 o - c a l l e d Weber p r o b l e m , i n c o n n e c t i o n w i t h w h i c h some n u m e r i c a l r e s u l t s w i l l be p r e s e n t e d .
7.1
An augmented L a g r a n g i a n i n t h e case o f i n e q u a l i t y constraints
W e consider here i n
lRN a c l a s s i c a l n o n l i n e a r programming p r o b l e m
subject t o inequality constraints:
136
DECOMPOSITION-COORDINATION
(CHAP.
3)
, v c RN ,
Inf F(v) (7.1)
gi(v) S O , i = l ,
...,M.
We s h a l l assume t h a t t h e
functions
M
a r e c o n v e x and w e w r i t e
gi
G : R~ - R ~ ,
(7.2)
(G(v))~ = gi(v)
The problem ( 7 . 1 )
i
F(v)
Inf
,
i=l,
....M.
can c l e a r l y b e w r i t t e n i n t h e form:
under the constraints
{V,d
(7.3)
qS 0
,
G(v) = q
(with q
5
0 -@ qi 5 0 Vi=l,.
..N if q
M
= {qiIi=,),
which l e a d s t o t h e augmented Lagrangian
The m i n i m i s a t i o n w i t h r e s p e c t t o
q
can be c a r r i e d o u t d i r e c t l y
(see S e c t i o n 2 . 3 ) ; i n f a c t , f o r f i x e d shown t h a t f o r i = 1,
for
q
...,PI,
5
0 t h e minimum
v
and
11,
is attained a t
i t can e a s i l y be q = p , where,
w e have
w e o b t a i n w i t h t h e n o t a t i o n of S e c t -
Substituting (7.5) i n t o (7.4), i o n 2.3
which c o r r e s p o n d s d i r e c t l y w i t h t h e form g i v e n by ROCKAFELLAR 1 1 1 . A UZAWA-type a l g o r i t h m f o r
respect t o
v, with
!J
Lr(v,u)
f i x e d , of
requires t h e minimisation with Lr(v,v).
This requires the sol-
u t i o n of a h i g h l y n o n l i n e a r problem. W e now c o n s i d e r a l g o r i t h m ALG2 f o r
that
F
and
can d e t e r m i n e
er(v,q,p).
a r e d i f f e r e n t i a b l e , f o r given
G
un
then
pn
by
X1
I f we assume and
p
0
,
we
(SEC. 7 )
137
NONLINEAR PROGRAMMING PROBLEMS
(7.7)
= 0,
F ' ( u " ) + ( G ' (u"))t[rG(un)-rpn-l+A"l
and A" -$ I ,
pn = min I0,gi(un)+
(7.8)
A"+'
,...M.
A n + l by
F i n a l l y , w e can c a l c u l a t e (7.9)
i=l
= An+p(G(u")-pn).
I n t h e case of a f f i n e c o n s t r a i n t s , Theorem 5 . 1 a s s u r e s t h e c o n v e r -
We
gence o f t h i s a l g o r i t h m u n d e r q u i t e g e n e r a l c o n d i t i o n s on
F.
have n o t t r i e d t o e x t e n d t h i s p r o o f t o t h e g e n e r a l case.
Similarly
t h e u s e o f A L G l a n d o f t h e r e l a x a t i o n method ( 3 . 8 ) - ( 3 . 1 0 ) c o n s i d e r e d as b e i n g e q u i v a l e n t
can b e
( a p a r t f r o m t h e s o l u t i o n method) t o
t h e a p p l i c a t i o n o f a UZAWA-type a l g o r i t h m t o
Lr(v,p).
Viewed i n
t h i s way, t h e u s e o f ALG2 i s a v a r i a n t i n which t h e m i n i m i s a t i o n o f Lr(v,u) with respect t o
7.2
v
i s c a r r i e d o u t i n a n i n c o m p l e t e manner.
M i n i m i s a t i o n o f a f u n c t i o n a l o v e r a n i n t e r s e c t i o n o f convex
sets 7.2.1 Let
S t a t e m e n t of V
t h e probZem
be a r e a l H i l b e r t space.
W e c o n s i d e r on
V
a funct-
W e wish t o minimise F
i o n a l F, which i s convex, p r o p e r and 1.s.c.
o v e r t h e (non-empty) c l o s e d convex s e t K , where w e h a v e
(7.10)
K =
M n i=
Ki
1
where f o r
i = 1,
...,M,
each o f t h e
i s i t s e l f convex and
Ki
c 10se d .
T h i s s i t u a t i o n o b v i o u s l y encompasses t h e c l a s s i c a l
Remark 7.1:
case of S e c t i o n 7.1.
It suffices t o put
K~
=
Iv~~~(v)~o).
W e s h a l l p r e s e n t h e r e a n o t h e r p r o c e d u r e f o r a s s o c i a t i n g a n augmented
Lagrangian w i t h t h i s problem.
Remark 7.2: quadratic,
An impoaant p a r t i c u l a r case
i.e.
is that where F is
138
DECOMPOSITION-COORDINATION
F(V) =
(7.11)
-21 a(v,v)-
v,xv =
I
(CHAP. 3 )
< A V , V > ~ - v~xv 'XV
.
In ( 7 . 1 1 ) , a(.,.) is bilinear, continuous, symmetric and V-elliptic 2
(i.e. a(v,v) 2allvllv V V E V , a > 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
specified arbitrarily;
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),
known, c a l c u l a t e t h e s o l u t i o n
{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
DECOMPOSITION-COORDINATION
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)
chosen a r b i t r a r i l y ;
{ 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)
chosen a r b i t r a r i l y ;
?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
+
(7.30) VqiEKi,
p;eKi,
i=l,
...,M.
qi-pq>
(SEC.
7)
NONLINEAR PROGRAMMING PROBLEMS
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
DECOMPOSITION-COORDINATION
(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
specified arbitrarily;
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
NONLINEAR PROGRAMMING PROBLEMS
( 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)
DECOMPOSITION-COORDINATION
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
MILDLY NONLINEAR PROBLEMS
(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.
M I L D L Y NONLINEAR PROBLEMS
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
M I L D L Y NONLINEAR PROBLEMS
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:
chosen a r b i t r a r i l y ;
(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
chosen a r b i t r a r i l y ;
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
chosen a r b i t r a r i l y ;
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 )
MILDLY NONLINEAR PROBLEMS
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
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 )
MILDLY NONLINEAR PROBLEMS
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
$'
is sing-
u(x) = 0 ; i t i s i n f a c t o b s e r v e d t h a t t h e
a s y m p t o t i c r a t e of c o n v e r g e n c e o f ALG2 i s v e r y s l o w a t t h e s e p o i n t s , and t h i s f a c t i s c l e a r l y i l l u s t r a t e d i n F i g u r e 5.2 which corresponds t o a n u m e r i c a l t e s t i n which w e h a v e i n i t i a l i s e d ALGZ f i r s t with
uo
- . A.1
= 0,
= 0 , and t h e n
uo
=
10
(=
{lo,.. . l o } ) ,
5'
=
0.
In
t h e f i r s t case t h e s o l u t i o n h a s a l r e a d y been a t t a i n e d a t t h e s i n g u l a r p o i n t s b e c a u s e it c a n e a s i l y b e s e e n t h a t t h e a l g o r i t h m w i l l l e a v e t h e v a l u e s unchanged a t t h e s e p o i n t s ;
i n t h e second c a s e t h e
s o l u t i o n h a s t o r e a c h t h e v a l u e z e r o a t t h e above p o i n t s .
uo = 0 w e h a v e a v e r y f a s t l i n e a r conve r g e n c e ; i n t h e s e c o n d case- (i.;. uo = 1,O) w e h a v e a s u b l i n e a r F i g u r e 5 . 2 shows t h a t f o r
-
(SEC. 5 )
163
NUMERICAL EXPERIMENTS
convergence.
A c l o s e e x a m i n a t i o n of t h e
results reveals that the
s l o w n e s s of t h e convergence i s l o c a l i s e d a t t h e p o i n t s where u ( x ) = 0 , t h e o t h e r v a l u e s h a v i n g a l r e a d y been o b t a i n e d t o a r e a s o n I t i s t h e r e f o r e c l e a r t h a t t h e c h o i c e of good i n -
able accuracy.
i t i a l v a l u e s c a n l e a d t o a v e r y s i g n i f i c a n t improvement; however i t
i s s t i l l v e r y i m p o r t a n t t o make t h e a l g o r i t h m more r o b u s t because i n p r a c t i c e it i s u n l i k e l y t h a t i t w i l l b e p o s s i b l e t o f i n d i n i t i a l v a l u e s which w i l l e n a b l e t h e d i f f i c u l t y a s s o c i a t e d w i t h t h e s i n g c$' 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
NUMERICAL EXPERIMENTS
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
MILDLY NONLINEAR PROBLEMS
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
$(u)
E
L1(Q) n H - I ( Q ) ,
namely:
satisfying
(6.1) a(u,v) + <$(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 <
in
AU = -Au
+ U, w e t h e n h a v e
K,
(u"-p"),
d e n o t e s t h e outward normal t o
K.
The Neumann p r o b -
l e m s ( 6 . 1 2 ) a r e i n f a c t l o c a l and are s o l v e d e l e m e n t by e l e m e n t . Remark 6.2:
I n c o n t r a s t t o t h e p r e c e d i n g s e c t i o n s , t h e de-
c o u p l i n g method n o l o n g e r n e c e s s i t a t e s s p l i t t i n g t h e o p e r a t o r i n t o a l i n e a r p a r t and a n o n l i n e a r p a r t ;
it i s t h e r e f o r e v e r y g e n e r a l
a n d c a n b e e x t e n d e d t o c a s e s m o r e complex t h a n t h o s e c o n s i d e r e d above.
The s o l u t i o n o f t h e n o n l i n e a r p r o b l e m ( 6 . 7 ) may t h e n be-
come more c o m p l i c a t e d , a n d t h e a d v a n t a g e of t h e h y b r i d f i n i t e e l e ment methods t h e n a c c r u e s e s s e n t i a l l y from t h e f a c t t h a t t h e nonl i n e a r p r o b l e m s c o n t a i n o n l y a s m a l l number o f v a r i a b l e s .
Remark 6 . 3 :
The f o r m u l a t i o n p r e s e n t e d above e n c o m p a s s e s t h e
u s u a l h y b r i d methods.
I n regard t o t h e d i s c r e t i s a t i o n of t h e
p r o b l e m by f i n i t e e l e m e n t s , however, it i s more g e n e r a l i n t h a t it a l l o w s t h e u s e of c o n f o r m i n g f i n i t e e l e m e n t s f o r t h e a p p r o x i m a t i o n of
u.
I n f a c t t h e awkward p r o b l e m o f t h e d i s c r e t i s a t i o n of t h 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 i s n o l o n g e r p r e s e n t . On t h e q u e s t i o n of t h e c o n v e r g e n c e o f a l g o r i t h m s o f t h e t y p e ALG1,
ALGZ a p p l i e d t o t h e s o l u t i o n o f
Lagrangian
(6.5),
(6.1),
v i a t h e augmented
t h e r e s u l t s o f C h a p t e r I11 a r e a p p l i c a b l e i n
X
170
(CHAP. 4 )
MILDLY NONLINEAR PROBLEMS
their entirety.
I t should also be noted t h a t t h e recoupling t h r -
ough t h e l i n e a r p r o b l e m ( 6 .
lo),
which i s g l o b a l on
Z2,
e n a b l e s us
t o a n t i c i p a t e c o n v e r g e n c e p r o p e r t i e s more o r less i n d e p e n d e n t o f t h e number of e l e m e n t s u s e d .
I.
GENERAL DISCUSSION ON CHAPTER I V
The a l g o r i t h m s d e v e l o p e d i n t h i s c h a p t e r e n a b l e u s t o s o l v e e f f i c i e n t l y a l a r g e number o f n o n l i n e a r p r o b l e m s .
I n bringing
t h e n o n l i n e a r i t y down t o a l o c a l l e v e l w e s i m p l i f y t h e p r o b l e m c o n s i d e r a b l y , and t h e r e c o u p l i n g t h r o u g h t h e s o l u t i o n o f l i n e a r s y s t e m s , w i t h m a t r i c e s which are f i x e d d u r i n g t h e i t e r a t i v e process, i s a considerable advantage.
F i n a l l y , t h e s e methods a r e
w e l l a d a p t e d t o Parallel C o m p u t a t i o n and s h o u l d prove p a r t i c u l a r l y e f f i c i e n t f o r f u t u r e g e n e r a t i o n s of computers. The f a c t t h a t w e h a v e b e e n a b l e t o o b t a i n a s p a r t i c u l a r cases a number o f a l 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 , which h a v e p r o v e d t h e i r e f f i c i e n c y elsewhere, i s t h u s a r e l i a b l e guide f o r implementation. I n Chapter I X w e s h a l l g i v e m o r e information concerning a l t e r n a t i n g d i r e c t i o n methods a n d 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 s e methods and t h e augmented L a g r a n g i a n methods
(see a l s o LIONS-MERCIER C 1 1 ) .
CHAPTER V APPLICATION
TO T H E S O L U T I O N O F S T R O N G L Y N O N L I N E A R
SECOND-ORDER BOUNDARY-VALUE
PROBLEMS
M. Fortin, R. G l o w i n s k i , A. M a r r o c c o
1.
INTRODUCTION
The aim of this chapter is to give a relatively detailed account of a number of applications of Augmented-Lagrangian methods to the solution of nonlinear second-order boundary-value problems, in which the nonlinearity relates to the gradient of the unknown function. The general setting is the same as that in Chapter 111, and we shall show how the various problems we describe can be fitted into this framework. We shall also discuss the approximation of these problems by means of finite-element methods and the consequences with regard to the iterative solution algorithms. The results of various numerical tests will serve to illustrate the behaviour of the algorithms used, and will demonstrate their efficiency. We shall show, in a relatively general context, that the framework presented permits easy implementation of the so-called ' i n t e r i o r penalty' finite-element methods.'
2.
GENERAL FRAMEWORK OF THE PROBLEMS IN CHAPTER V
This second section aims to provide a common general framework for the examples which follow: the reader should therefore not be surprised at the formal character of this framework, which is justified by our desire to unify a number of extremely diverse problems. We ought also to point out that in this chapter we shall be giving much more attention to the algorithmic aspects than to the precise mathematical formulation or to the convergence of the solutions of the approximate problems.
In the sense of DOUGLAS-DUPONT C11, WHEELER c11,
171
...
172
STRONGLY NONLINEAR B.V.P.'S
(CHAP. 5)
In this chapter, we shall thus be considering the problem of the minimisation, in appropriate functional spaces, of functionals the form (2.1)
J
of
J(v) = F(Vv) + G(v).
N in IR , and In (2.1), v denotes a function defined on a domain Vv denotes its gradient. The term F(Vv) will be written more specifically in the form' (2.2)
F(Vv)
=
i,
O(x,lVvl)dx.
In many cases the function Ip will not depend explicitly on x , and in general it will be e o n u e x with respect to /Vvl. The solutio of such a problem clearly lends itself very well to the introduction of an augmented Lagrangian of the same type as those considered in Chapter 111. Thus, using the notation of Chapter 111, the following form suggests itself:
2 In our examples, we shall have H = (L for the continuous problem, with a discrete version of this space for the approximate problems. It should be pointed out immediately that in some cases it will be possible to give a rigorous definition of d,, only in the case of the approximate problems; in particular, this will be the case for problem (3.2) of Section 3, if l < s < 2 ,and for the m i n i m a l surfaces problem of Section 6.3. In the numerical tests which will be presented, the discretisation will be performed by means of conforming finite elements of degree one , on triangles. The discretisation of the variable q which represents the gradient will thus be particularly simple since in this case the components of q will be constant over each element. Just as in Chapter IV, the discussion can be generalised to more complicated cases: quadratic finite elements and piecewise-linear gradients; with these, it will once again be necessary to make judicious use of numerical integration techniques, as we did in Chapter IV. Our approach will be centred, primarily, on the use of algorithms ALGl and ALG2 of Chapter 111, and the results of the numerical tests will serve to illustrate their convergence properties.
(SEC. 3 )
173
N O N L I N E A R D I R I C H L E T PROBLEMS
A CLASS O F NONLINEAR D I R I C H L E T PROBLEMS
3.
3.1
Formulation o f t h e aroblems.
3.1.1
Augmented L a q r a n g i a n s
continuous case
The
I n t h i s s e c t i o n w e s h a l l be considering t h e monotone n o n l i n e a r
o p e r a t o r A, defined (with
1< s
<+a)
by
Av = - V * ( \ V V ~ ~ - ~ ) V V .
(3.1)
T h i s o p e r a t o r a p p e a r s i n c e r t a i n m a t h e m a t i c a l models d e s c r i b i n g t h e m e c h a n i c a l d e f o r m a t i o n of i c e ( s e e f o r example PELISSIER 111 and t h e associated bibliography).
r ,
dary
The s o l u t i o n , i n a domain
R
w i t h boun-
of t h e n o n l i n e a r D i r i c h l e t problem
Au = f
i n R,
(3.2) u =
o o n r,
c o n s i d e r e d e a r l i e r i n C h a p t e r 111, S e c t i o n 6 . 3 ,
can b e reduced t o 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 u
Find
Variations :
V
E
(3.3)1 J(u)
5
Vv
J(v)
E
V,
where
t h e space
is, f o r
W1” 0 1 <s <
t h e norm3 (3.4)
lvls =
(a)
(I,
( d e s c r i b e d e a r l i e r i n C h a p t e r 111, S e c t i o n 6 . 3 )
a r e f l e x i v e Banach s p a c e when it i s e q u i p p e d w i t h
+m,
lVvls dx)’”.
We t h e r e f o r e h a v e h e r e , u s i n g t h e n o t a t i o n 3
i s a norm on direction i n
lRN
.
W:’S(R)
if
C?
of
(2.2),
1 s O(x,z) = - z
.
i s b o u n d e d i n a t l e a s t one
STRONGLY N O N L I N E A R B.V.P.’s
114
(CHAP.
5)
The a p p r o x i m a t e p r o b Zems
3.1.2
s = 2 , t h e space
Except f o r
i s n o t a H i l b e r t s p a c e and w e can
V
n o t t h e r e f o r e a p p l y t h e r e s u l t s of C h a p t e r I11 d i r e c t l y t o problem
( 3 . 2 ) , ( 3 . 3 ) ; f o r t u n a t e l y , however, t h i s can b e done f o r t h e a p p r o z i m a t i o n s of t h i s problem. Suppose we have a t r i a n g u l a t i o n 4 Th an
a p p r o x i m a t i o n of
R : w e c o n s i d e r on Zh
of
u s i n g conforming f i n i t e e l e m e n t s of d e g r e e
V
k=l
one o r two.
More p r e c i s e l y , f o r
where
i s t h e s p a c e of p o l y n o m i a l s of d e g r e e
Pk(K)
2 , we define
or
on
K.
Likewise, w e d e f i n e
c l e a r l y , we have (3 * 7 )
VVh E
Lkh
*h
E
Vkh‘
(3.2),
The d i s c r e t i s e d v e r s i o n of
J(*) over
d e f i n e d on
Vkh
Vkh
,
X
Lkh
X
Lkh
.
For k = l t h e c a l c u l a t i o n o f since
qh
( 3 . 3 ) t h u s c o n s i s t s of m i n i m i s i n g
l e a d i n g t o t h e augmented Lagrangian’
gr
does n o t p r e s e n t any problems
i s c o n s t a n t o v e r e a c h t r i a n g l e of
“eh
.
If
k =2, we
s h a l l use a q u a d r a t u r e f o r m u l a t o e v a l u a t e C h a p t e r I V , we s h a l l t h u s c o n s i d e r ,
i n t e g r a t i o n p o i n t s xiK,
‘
T h i s assumes
R
c
i=l,
...
J R ( q h l S dx : a s i n i n t h e element K , L numerical
,R
IR2 and b o u n d e d :
,
each b e i n g a s s i g n e d a weight
however, t h e f o l l o w i n g d i s c u s s -
i o n may r e a d i l y be e x t e n d e d t o t h e c a s e a l s o assume t h a t
R
i s polygonal.
R
c
IRN
,
N t 3.
We shall
(SEC. 3)
wi
NONLINEAR DIRICHLET PROBLEMS
175
such that we have
L
(3.9)
K
f ( x ) dx
Area (K)
1
oi f(xiK).
i=l
For k = 2 , the most appropriate choice would be to take the points x. iK as the midpoints of the sides of the triangle K (so that P.=3), each being assigned a weight 1/3 (so that w . = 1 / 3 Vi=1,2,3); this quadrature formula is in fact exact for polynomials of degree 2, and 2 this allows us to calculate the terms 2 jo)vvh-qhl dx and laph* (Vvh - qh) dx e x a c t Z y . Furthermore, again if k = 2, knowing the value of a piecewise-linear function at the midpoints of the sides of a triangle fixes this function uniquely; it then follows that the values of qlh and q2h at the midpoints of the sides of Th can be used as degrees of freedom relative to (it would be equally qh straightforward, if we so desired, to enforce matching of certain components of qh at the midpoints of the sides). Proceeding as in Chapter IV, Section 3, this leads to the discrete augmented Lagrangian
This Lagrangian is defined on Vkh xLkh x Lkh and it coincides with dr(vh,qh,uh) if k = 1 (to within the term in f vh dx ) . We have assumed for simplicity that in f v h d x has been approximated in (3.10) by the same quadrature formula as used for the other terms; however, this is not necessary. In order to gain a better understanding of the algorithms which follow, it is important to consider the numerical solution of the variational problem equivalent to the minimisation with respect to with vh and ph f i x e d . In the case of dr qh of dr and defined by (3.8) we obtain
Arb,
For For
k=l k=2
the components of ph are constant over each triangle. we would use numerical integration via the Lagrangian d
in this case we have to determine the values of
ph
rh ' at the quadrature
.
STRONGLY NONLINEAR B. V. P ' S
176
points.
(CHAP. 5 )
In both cases, we find that we have to solve, triangle-by-
triangle or point-by-point, the nonlinear system
Once lphl is known, plh and (3.12); now (3.12) implies (3.13)
lPhl
s-l
or, defining (3.14)
+
g(.)
'IPhl
by
g(q) = Irvvh+~,l
pZh
can immediately be deduced from
Irvvh+'hl,
g(q) =qS-l + rq, ;
since the function g is a homeomorphism of IR, onto IR, , (3.14) admits a unique solution which can be calculated by standard methods for nonlinear equations in one variable, although certain precautions have to be taken for extreme values of s (i.e. s close to 1, or s large)
3.2
The basic algorithm and its convergence properties
The basic algorithm for implementation of the augmented Lagranqian irh is once again algorithm ALGl of Chapter 111. The following is a brief summary of the principles of this algorithm. ALGORITHM A L G l
(3.15)
then for
(3.16)
1;
n
2 0 ,
(General form) chosen a r b i t r a r i l y i n A:
Lkh
b e i n g known, c a l c u l a t e
uh, ph
and
1;"
by
3)
(SEC.
177
NONLINEAR D I R I C H L E T PROBLEMS
I n practice, it is necessary to have a method for solving problem (3.16 ; this leads us to a more ‘operational’version of algorithm ( 3 . 1 5 - (3.17) :
ALGORITHM A L G l (i)
(Practical form)
Choose :
- a n a c c u r a c y t o l e r a n c e E, - a maximum number of i n n e r i t e r a t i o n s kmax - t h e v a l u e s by which t h e o u t e r i t e r a t i o n s a r e i n i t i a l i s e d :
(ii) F o r for
n2 0 mto m
v
6
and w i t h
:A
and
and w i t h
zm
known, s o l v e :
n-1.
z o = ph
known, p u t
p:-’
,
then
Vkh,
(3.19) (rzm-h,”)*vvh dx Vv h 6 Vkh
(note that the solution over Vkh) . Next,
m v
of (3.19) minirnises
vm
knowing t h e s o l u t i o n
of
vh
If
qh
+
drh(vm,qh,Af:) over
m+l - zm i s l e s s t h a n m + l = kmax put
E
Rrh(vh,zm,XE)
(3.19), soZue :
(actually, the ‘point’ version of these equations; ated minimises
-+
( i n the
L2
z m+l
thus calcul-
Lkh) ‘ norm, for e x a m p l e ) ,
o r if
J
(3.21)
{ = v”,
otherwise s e t
(iii) W i t h
p t = zm+l ;
m=m+l uf: and
p:
and r e t u r n t o (3.19). known, c a l c u l a t e
(triangle-by-triangZe
k = l , or a t t h e quadrature p o i n t s i f k = 2 )
i f
178
STRONGLY NONLINEAR B.V.P.'s
If Ou; - pf: is sufficiently small, t e r m i n a t e ALG1; n =n
+1
(CHAP. 5 )
otherwise set
a n d r e t u r n t o (ii).
n n The determination of {uh,ph} is thus carried out by a r e l a x a t i o n method; if we take kmax=l, we obtain algorithm ALG2 of Chapter 111. Remark
3.1: For reasons which will be given in Remark 3.2, it is m z
recommended that a termination test be used which relates to
rather than to v" ; as is pointed out in MARROCCO C11, a termination m test on v can lead to premature termination ('sticking' or pseudoconvergence) of the iterative process. In order to gain a better understanding of the behaviour of ALG1, it is instructive to study this algorithm when p = I and s = 2 ; for this value of s the problem to be solved is linear. The interesting fact is that in this case we obtain convergence in two iterations at most; this was demonstrated earlier in Chapter 111, Section 4.3, but we shall nonetheless prove it again for the particular case in question here (the proof which follows is valid for ALG1; a similar one can be given for ALG2 if p = r = l , and this value r = l will also be encountered once more when we examine the experimental convergence results). Just as in Chapter 111, it is sufficient to consider the case (so that
uh = O ) ;
(3.23)
Q
additionally, we write qh = Vvh, vhEVkh 1 (=
= {nhlshE\h,
and we denote by the norm L2(S2).
f=O
P Q
vvkh ) ,
the operator of projection from
Lkh onto Q for
It then follows from (3.19), (3.21) that (3.24)
P A" = r ( P pn Q h Q h
- v<',
and then from (3.20) that (3.25)
"+
'h
r(p"-V h
"h")
=
A.:
By projection of (3.22) onto Q and its orthogonal space, and taking account of the fact that p =r, we obtain
(SEC. 3)
NONLINEAR DIRICHLET PROBLEMS
179
We deduce from (3.24) and (3.2611 that (3.27)
so that (3.28)
P
A"+'
Q h
=
o
vn>O,
from (3.24) we have pQp;
- V<
o vnr
=
I.
By projection of (3.25) onto P
A"
Q h
We thus have we have (3.29)
= pQp;
=
Vut = 0
(I-p )A"+]
=
o
vn
for
2
n
Q
, we
then obtain
I. 2
1, so that
u
n h
=
0.
On the other hand,
-L I+r (I-Pp;.
There is thus decoupling between the convergence of i u ~ ~ n z oand that of [A:ln>o if s =2. For s * 2 , a variant of the above proof would show that PQAt always converges in a single iteration, but since it is then no longer possible to decompose (3.20) onto Q and n in a its orthogonal in Lkh , we no longer have convergence of uh finite number of iterations. We can nonetheless expect very different P A" , PQp: and (I - PQ )Anh , (1 - PQ)pt
.
rates of convergence for
R e m a r k 3.2: In view of these quite different convergence proper, IAtl,,, , it is necessties of the sequences {uEIn20 and {p:Inro ary to use a termination test somewhat more sophisticated than one which relates solely to the sequence it is f o r this reason ; that in the case of ALGl discussed earlier, we suggested terminating n the outer iterations when Vu; - p h became sufficiently small in norm.
{uE)~~~
R e m a r k 3.3: Another case, the practical importance of which will become apparent in Section 4, is the one in which the problem to be solved is linear but with a variable coefficient, i.e. of the form (with a(x) 2 a > 0 a.e.) (3.30)
Inf I$ Ina(u)IVv12 dx vcH' (a)
-
j,f.
dxl
.
We shall assume a(x) to be such that the minimisation problem (3.30) admits a unique solution. If for the solution of this problem we consider the augmented Lagrangian
180
STRONGLY NONLINEAR B . V . P . ' s
(CHAP.
then algorithm ALGl no longer converges i n t w o i t e r a t i o n s ; t h i s p r o p e r t y i s r e c o v e r e d i f i n s t e a d of
5)
however,
( 3 . 3 1 ) w e u s e t h e augmented
Lag r a n g i an
I t i s r e a s o n a b l e t o s u p p o s e t h a t t h i s p r o p e r t y c a n be u s e d t o improve t h e convergence of ALGl i n t h e n o n l i n e a r c a s e . assume t h a t w e know a n e s t i m a t e
~ ( x o) f
I n f a c t , let us
/phi s - 2 .,
we could then
c o n s i d e r an augmented L a g r a n g i a n o f t h e form
The i d e a l p r o c e d u r e would u n d o u b t e d l y b e t o u p d a t e i t e r a t i o n of A L G 1 ,
using
p:
,
n(x)
a t each
thereby o b t a i n i n g a sequence
{r-injn20
(we c o u l d even c o n s i d e r an u p d a t e d u r i n g t h e i n n e r i t e r a t i o n s ( 3 . 1 9 ) , (3.20));
however, t h i s u p d a t e m o d i f i e s , a t e a c h i t e r a t i o n , t h e m a t r i x
of t h e l i n e a r problem c o r r e s p o n d i n g t o ( 3 . 1 9 ) ;
t h i s p r o b l e m would b e
o f t h e form
Find
v", Vkh such t h a t Vvhc Vkh
we have
(3.34)
It i s a p p r o p r i a t e t o a s s e s s whether t h e a c c e l e r a t i o n of convergence t h e r e b y o b t a i n e d j u s t i f i e s t h e c o s t o f t h e new C h o l e s k y f a c t o r i s a t i o n if
(3.34)
i s s o l v e d by a d i r e c t method;
it i s c l e a r t h a t t h e u s e o f
an e f f e c t i v e i t e r a t i v e method ( 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 , m u l t i g r i d method, e t c . ) f o r s o l v i n g t h e l i n e a r p r o b l e m ( 3 . 3 4 ) would b e a n a t t r a c t i v e a l t e r n a t i v e t o a d i r e c t method which r e q u i r e s a t e a c h i t e r a t i o n a r e f a c t o r i s a t i o n of t h e matrix associated with t h e l i n e a r problem ( 3 . 3 4 ) . Note t h e o b v i o u s l i n k s which e x i s t b e t w e e n t h i s method u s i n g Newton method. { r ~ ~ ,} and ~ >a ~ W e s h a l l e n c o u n t e r a n a p p l i c a t i o n o f t h i s p r e s e n t remark l a t e r on i n S e c t i o n 4 of t h i s c h a p t e r .
H
(SEC. 3)
181
NONLINEAR D I R I C H L E T PROBLEMS
R e m a r k 3.4:
I t f o l l o w s from Remark 3 . 3 t h a t , a l l o t h e r t h i n g s
r
being e q u a l , t h e o p t i m a l c h o i c e of t h e para m e te r i c u l a r ) w i l l v a r y , i n o r d e r of magnitude, as
lphl
( i n ALG2 i n p a r t -
,. w e c a n i n
s-2
f a c t i n t e r p r e t ( 3 . 3 3 ) a s d e f i n i n g an augmented L a g r a n g i a n w i t h a $ q ( x ) , t h e optimal value r s-2 a t least) equal t o 2 /phj S i m i l a r l y , i t may b e c o n j e c t u r e d t h a t i n t h e
uariabte p e n a l i s a t i o n p a r a m e t e r e q u a l t o of w h i c h would b e ( a s y m p t o t i c a l l y i n if
were known.
ph
n
< is!
case of t h e standard penalty t e r m (i.e.
r
parameter
3.3
lBvh- qh12 d x ) t h e o p t i m a l lphl s - 2
w i l l v a r y from one p r o b l e m t o a n o t h e r a s
.
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 b e d i s c u s s i n g some o f t h e many 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 GLOWINSKI-MARROCCO 113,141 and MARROCCO 111, g i v i n g p a r t i c u l a r e m p h a s i s t o c e r t a i n p o i n t s which w e c o n s i d e r t o b e I n 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 s h a l l
e s p e c i a l l y important. d e s c r i b e , t h e domain of
s!
w i l l b e a d i s c and t h e r i g h t - h a n d
( 3 . 2 ) w i l l b e a p o s i t i v e c o n s t a n t d e n o t e d by
assumptions, t h e exact s o l u t i o n of of a d i s c of r a d i u s
C ;
side
f
under t h e s e
( 3 . 2 ) i s known, and i n t h e c a s e
c e n t r e d on t h e o r i g i n , t h i s s o l u t i o n i s g i v e n
R
by u(x) =
(3.35)
1x1
where
=
S S S s-l (s)s-l ( R s - q x ~ s - ~ ) ,
s
2
(xl
2
+
x;
)*
if
x = {xl,x2}.
W e have t a k e n
h a s b e e n t r i a n g u l a t e d by means o f a t r i a n g u l a t i o n triangles,
one;
R =0.5
i n t h e n u m e r i c a l tests which w i l l b e d e s c r i b e d .
C=O.l
t h e f i n i t e elements used being
t h e components o f
Vvh
and o f
Th
and
The d i s c
s!
containing 256
Co-conforming
and o f d e g r e e
a re t h e r e f o r e c o n s t a n t o v e r
qh
each t r i a n g l e . I n o r d e r t o e v a l u a t e t h e p e r f o r m a n c e of A L G 1 , w e h a v e t o remember t h a t i t i s n o t s u f f i c i e n t m e r e l y t o e v a l u a t e t h e number o f u p d a t e s o f
:A
via (3.17);
w i l l i n f a c t c o n v e r g e more
t h e sequence
q u i c k l y ( i n terms o f t h e number o f o u t e r i t e r a t i o n s ) a s larger;
however,
t h e increase i n
r
r
becomes
s l o w s down t h e c o n v e r g e n c e o f
t h e r e l a x a t i o n method u s e d t o s o l v e s y s t e m ( 3 . 1 6 ) ;
there is therefore
a b a l a n c e t o b e s t r u c k between t w o o p p o s i n g e f f e c t s , and t h i s i m p l i e s t h e e x i s t e n c e o f an o p t i m a l v a l u e o f
r
.
Furthermore, i n order t o
h a v e a p r o p e r e v a l u a t i o n o f t h e r e s u l t s , it i s i m p o r t a n t t o c o n s i d e r t h e t o t a l computational e f f o r t required f o r t h e s o l u t i o n of t h e
182
STRONGLY N O N L I N E A R B .V. P
.' s
(CHAP. 5 )
w e s h a l l t h e r e f o r e compare t h e c u m u l a t i v e
approximate problem;
numbers of r e l a x a t i o n i t e r a t i o n s ( 3 . 1 9 ) ,
( 3 . 2 0 ) r e q u i r e d f o r t h e con-
vergence o f t h e a l g o r i t h m t o w i t h i n a g i v e n l e v e l of p r e c i s i o n .
In
t h e c a s e o f ALG2 t h i s number c o i n c i d e s w i t h t h e number o f u p d a t e s o f
XE
3 . 2 and 3 . 3 i l l u s t r a t e t h e b e h a v i o u r of ALGl f o r
Figures 3.1,
.
t h r e e v a l u e s of
s
,
s
namely
= 1.1,
3 and 10.
Figures 3.4,
3 . 5 and
3.6 i l l u s t r a t e t h e c o r r e s p o n d i n g r e s u l t s f o r ALG2 ( u s e d i n e x a c t l y t h e same way a s A L G l w i t h p = r ) . I n Figures 3 . 1 t o 3.3
( i . e . r e l a t i n g t o ALG1) two c u r v e s a p p e a r ;
t h e c u r v e l a b e l l e d C1 shows t h e c u m u l a t i v e number o f r e l a x a t i o n i t e r -
r
a t i o n s as a f u n c t i o n of of
1;
,
t h i s number t e n d i n g t o w a r d s u n i t y a s
u r e s 3.4 t o 3 . 6 ,
s
c u r v e C 2 i n d i c a t e s t h e number o f u p d a t e s
;
c u r v e s C1 a n d C2 c o i n c i d e .
considered, the optimal
f o r A L G l and ALG2.
r
r
increases.
I n Fig-
For t h e t h r e e values of
i s t o a l l i n t e n t s a n d p u r p o s e s t h e same
Furthermore,
even f o r
a p p e a r s t o b e more e f f i c i e n t t h a n A L G 1 ;
s =1.1, a l g o r i t h m ALG2
f o r t h i s family of problems,
i t d o e s n o t a p p e a r w o r t h w h i l e t o s o l v e s y s t e m ( 3 . 1 6 ) by t h e
therefore,
r e l a x a t i o n method ( 3 . 1 9 ) , ( 3 . 2 0 ) t o v e r y h i g h a c c u r a c y b e f o r e u p d a t i n g ;
t h e same a l s o a p p l i e s f o r v a l u e s o f
than unity.
s
only s l i g h t l y greater
T h i s o b s e r v a t i o n c o n c e r n i n g t h e s u p e r i o r i t y of ALG2 over
A L G l seems t o c o n t r a d i c t what was e s t a b l i s h e d i n C h a p t e r I V ;
however,
t h i s can b e e x p l a i n e d by t h e f a c t t h a t t h e t e r m i n a t i o n t e s t u s e d i n MARROCCO 1 1 1 , from which w e h a v e t a k e n o u r r e s u l t s , r e l a t e s t o t h e
d i f f e r e n c e between
n+l uh
and
u:
now i t would o n c e a g a i n a p p e a r
;
t h a t t h e r e e x i s t s a c e r t a i n d e c o u p l i n g between t h e convergence o f and t h a t o f {piln20 or , t h i s d e c o u p l i n g becoming {uE}nzO a s w e s a w i n S e c t i o n 3.2. T h i s d e c o u p l i n g propcomplete f o r s = 2 e r t y w a s n o t p r e s e n t i n C h a p t e r I V where, i n f a c t , equal i n t h e l i m i t as
n-t
+ m
u:
and
pi
are
.
A n o t h e r o b s e r v a t i o n which a t f i r s t s i g h t i s somewhat b a f f l i n g con-
r t o t h e problem r d e p e n d s on s and a l s o on
cerns t h e high s e n s i t i v i t y of t h e optimal value of being considered.
The o p t i m a l v a l u e o f
t h e right-hand s i d e
f
of e q u a t i o n
(3.2).
L e t u s f i r s t c o n s i d e r t h e case i n which
s
i s f i x e d , a n d i n which
w e r e c a l l t h e e m p i r i c a l f o r m u l a of GLOWINSKI-MARROCCO 1 4 1 f o r t h e o p t i m a l v a l u e o f r : i f ro ( r e s p . rl ) i s t h e o p t i m a l
f=const. =C ; value of
r
for
f=Co
(resp.
f=C,),
t h e n w e have ( a p p r o x i m a t e l y )
(SEC.
3)
183
NONLINEAR D I R I C H L E T PROBLEMS
ITE3,ITIGNS
I
Figure 3 . 1
-ALGZs
S
0
Figure 3 . 4 -I&
AeMRILEO
-I*-
7
*-.
m osI
100
PARAMETERS 6-10
8-30
G.4
Figure 3 . 6
Figure 3 . 3 -I&
A.UDRe:
G.1 -1A-
A -s
I
184
STRONGLY N O N L I N E A R B.V.P.‘s
(CHAP. 5 )
2-s
(3.36)
cI
-I0 =
(+
l
o
s-l .
T h i s f o r m u l a i s t o b e v i e w e d a l o n g s i d e Remark 3.4, where i t was c o n j ectured t h a t t h e optimal
r should be proportional t o
i t f o l l o w s from (3.35) t h a t when w e p a s s from (=
I Vul )
f =C
0
i s . m u l t i p l i e d p r e c i s e l y by t h e f a c t o r (C1/Co)
consistent with
(3.36)
lphl
s-2
t o f = C1
’-*
;
.,
now
, IpI t h i s is
and c o n f i r m s t h e c o n j e c t u r e o f Remark 3.4.
The e f f e c t o f a v a r i a t i o n o f
s i s more c o m p l i c a t e d t o i n v e s t i g a t e
s i n c e w e are t h e n n o l o n g e r d e a l i n g w i t h a s i m p l e f a c t o r o f p r o p o r t ionality;
i n t h e p a r t i c u l a r c a s e o f t h e test problem c o n s i d e r e d ,
which i s i n f a c t o n e - d i m e n s i o n a l
less e a s y t o v e r i f y t h a t i f problem f o r
s
=
2
,
@
( b e i n g a x i s y m m e t r i c ) , it i s nonethe-
=On,
t h e n w e have
1 Bul
where =
I
fi i s t h e s o l u t i o n o f t h e l’(s-l)
s o l u t i o n c o r r e s p o n d i n g t o an a r b i t r a r y v a l u e o f
where
u
is the
s i n t h e range
1 i s < + - ; now, f o r s = 2 , 161 is p r o p o r t i o n a l t o 1x1 and I V U ~ ‘ - ~ w i l l t h u s b e p r o p o r t i o n a l t o ( = (x; +x: )’ )
\XI=
s-2
,
so t h a t w e h a v e J p h / s - 221
Now an o p t i m a l
-
I XI
if s
= 1.1,
r v a l u e 3x1O1l i s f o u n d f o r s = 1.1 ; 2XlO-l f o r s = 3;
and 0.9x10-2 f o r
s = 10 ;
t h e s e numbers c o r r e s p o n d e x a c t l y t o t h e
v a l u e s p r e d i c t e d by (3.37).
I t can t h e r e f o r e b e s e e n t h a t once a g a i n
t h e u s e o f a n augmented L a g r a n g i a n o f t h e t y p e (3.33), where
an e s t i m a t e o f
Ip
1 s-2 , h
rl ( x )
is
c o u l d v e r y p r o b a b l y s t a b i l i s e t h e c h o i c e o f r.
W e s h a l l e n c o u n t e r a n o t h e r a p p l i c a t i o n o f t h i s p r i n c i p l e i n S e c t i o n 6.3. F i n a l l y , w e must p o i n t o u t t h a t t h e r e s u l t s o f GLOWINSKI-MARROCCO Cll,C41 show t h e i m p o r t a n c e o f w o r k i n g i n d o u b l e p r e c i s i o n (on I B M computers a t l e a s t ) , i n p a r t i c u l a r f o r extreme v a l u e s o f s ( i . e . s c l o s e t o 1 and > > 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
(SEC. 4)
(4.5)
A MAGNETO-STATIC PROBLEM +
A =
+
Io,o,A} , j
187
= {O,O,j},
a
so that (4.4) reduces (with V = {-,
ax,
a -, ax2
a
to:
-})
ax
3
The coefficient v appearing in (4.6) depends on the material; c o p p e r and in a i r we have
in
-
in f e r r o - m a g n e t i c media we shall 0 ' assume - neglecting hysteresis effects - that v is an increasing function (in the wide sense) of 161 ; it is actually more convenient to consider v to be an increasing function of 151 and this is what we shall do in the following discussion. In the two-dimensional case we have v =v
and we can then rewrite (4.6) in the more explicit form
Finally, denoting by
$(x,a)
the function which satisfies
we see that (4.6) is the Euler equation, in an appropriate functional space V , of a problem in the Calculus of Variations, namely the minimisation of the energy functional 5 defined by (4.10)
where R is usually the median cross-section of the rotating machine. The problem
(4.11)
is clearly of the form considered in Section 2.
The choice of
V
depends on the boundary conditions; in the following we shall assume 1 that V=Ho(R), which corresponds to boundary conditions of homogene o u s D i r i c h l e t type. On the basis of laboratory measurements we have taken for v , in the ferromagnetic parts, a function of the form
STRONGLY N O N L I N E A R B.V.P.'s
188
(4.12)
(C-€)
= E +
V(E,a,C,T;U)
(CHAP.
5)
UcI ,
U"+T
t h e parameters
€,ci,C,T
being p o s i t i v e with
t h u s o b t a i n e d 6 i s monotone: t h i s smoothing f o r
v
r
.
C >
The f u n c t i o n
E.
F i g u r e 4 . 1 r e p r e s e n t s t h e r e s u l t s of
F o r r e l u c t i v i t i e s of t h e t y p e ( 4 . 1 2 ) ,
problem p o s s e s s e s a unique s o l u t i o n i f
1 V=Ho(Q).
the
Figure 4 . 1
STATOR
RESULT OF SMOOTHING ALPHA
C
T EPS
0.54192E01
0.17577E00 0.87589304 0.516363-03
,OA
!, , 0.
4.2
,,,
, , , , , , , , , , , , , , , , , , , , , , ~, 2.00
4.00
[
6.00
B""2
,,,,, ,
, , , , , , .. . ,
8.00
DO
IN MKSA
F o r m u l a t i o n u s i n g an augmented Lagrangian
Even i f w e n e g l e c t t h e n o n l i n e a r i t y , which would b e p e r m i s s i b l e f o r l o w c u r r e n t s , problem ( 4 . 8 ) h a s v a r i a b l e c o e f f i c i e n t s s i n c e t h e r e l u c t i v i t y i s n o t t h e same i n a i r a s i n i r o n .
To r e d u c e t h e problem
t o t h e form of t h e g e n e r a l c a s e , r e f e r r i n g back t o what was s a i d i n Remark 3 . 3 of S e c t i o n 3 . 2 ,
w e would u s e i n t h e g e n e r a l c a s e t h e
augmented L a g r a n g i a n 7 d e f i n e d a s f o l l o w s :
'
~~
~
Using a l e a s t s q u a r e s smoothing t e c h n i q u e . The p i n ( 4 . 1 3 ) h a s n o t h i n g t o do w i t h t h e m a g n e t i c p e r m e a b i l i t y , a q u a n t i t y which w i l l n o t a p p e a r a g a i n e x p l i c i t l y h e r e i n a f t e r .
(SEC. 4 )
A MAGNETO-STATIC PROBLEM
189
(4.13)
The o p t i m a l c h o i c e f o r
Q
i s t h e unknown f u n c t i o n
v ;
i n practice,
one a t t e m p t s t o o b t a i n a s u i t a b l e e s t i m a t e f o r t h i s f u n c t i o n . To s o l v e ( 4 . 1 1 ) w e s h a l l a g a i n u s e a l g o r i t h m s A L G l and ALG2 which we s h a l l n o t d e s c r i b e e x p l i c i t l y h e r e ;
f o r i n f o r m a t i o n on t h e s e a l g -
o r i t h m s w e r e f e r t h e r e a d e r t o t h e p r e c e d i n g s e c t i o n s and c h a p t e r s , and t o GLOWINSKI-MARROCCO 1 5 1 and MARROCCO [ll. l i n e a r problem, w e s h a l l have convergence of ALGl most two i t e r a t i o n s ( f o r
f
~
I n t h e case of a ( r e s p . ALG2) i n a t
} if ~ ri~ = ~ v a)n d i f p = r ( r e s p . p = r = 1).
t
The e x e c u t i o n of ALGl and ALG2 i n v o l v e s t h e s i m u l t a n e o u s s o l u t i o n f o r A L G 1 , and t h e s e q u e n t i a l s o l u t i o n f o r A L G 2 , o f t h e e q u a t i o n s (omitting t h e i t e r a t i o n superscripts):
which i s l i n e a r , a n d
which i s n o n l i n e a r , b u t w h i c h c a n b e s o l v e d l o c a l l y .
In fact, to
s o l v e ( 4 . 1 5 ) w e p r o c e e d as i n S e c t i o n 3 by i n t r o d u c i n g z = IpI
, which
t a k e s us f r o m s o l v i n g ( 4 . 1 5 ) t o s o l v i n g
We put
g(x;z) =
(V(X,Z
2
)+q(x))z
;
t h e d e p e n d e n c e on
x means t h a t we
have s e v e r a l t y p e s o f e q u a t i o n s d e p e n d i n g o n w h e t h e r t h e p o i n t
lies i n a i r , copper o r i r o n . (4.16) g i v e s
z
explicitly;
I n a i r and c o p p e r w e h a v e v=vo and i n iron g(x;.)
is a s t r i c t l y
x
190
STRONGLY NONLINEAR B.V.P.'s
(CHAP.
5)
i n c r e a s i n g f u n c t i o n p o s s e s s i n g a p o i n t o f i n f l e c t i o n which d e p e n d s n e i t h e r on
r
n o r on
r-
.
I f w e t a k e care w i t h t h e i n i t i a l i s a t i o n ,
w e can u s e Newton's method f o r s o l v i n g ( 4 . 1 6 ) .
For t h e d i s c r e t e
problems, t h e s o l u t i o n t a k e s p l a c e a t t h e q u a d r a t u r e p o i n t s f o r f i n i t e e l e m e n t s o f o r d e r 2 2 , and t r i a n g l e - b y - t r i a n g l e o f o r d e r 1.
f o r elements
Most o f t h e comments made i n S e c t i o n 3 r e m a i n v a l i d
f o r t h e p r e s e n t problem.
4.3
Numerical ex p er i men t s
The r e s u l t s p r e s e n t e d h e r e c o n c e r n a f o u r - p o l e c o r r e s p o n d i n g domain
rh
n
alternator;
u s e d ( a c t u a l l y a q u a r t e r o f i t ) i s shown i n F i g u r e 4 . 2 ;
t r i a n g u l a t i o n &,
the
i s t h e r e f o r e a d i s c and t h e t r i a n g u l a t i o n this
c o n t a i n s 812 t r i a n g l e s and 384 i n t e r i o r v e r t i c e s . 0
The f i n i t e e l e m e n t s u s e d a r e C - c o n f o r m i n g o f o r d e r 1.
F i g u r e 4.3
,.
shows t h e c o n t o u r s (which a r e a c t u a l l y t h e l i n e s o f m a g n e t i c i n d u c t i o n ) f o r t h e s o l u t i o n c o r r e s p o n d i n g t o a c u r r e n t of 5 ~ / m '.
Figure 4.2 T r i a n g u l a t i o n used.
Figure 4 . 3 L i n e s of m a g n e t i c i n d u c t i o n .
(SEC. 4 )
191
A MAGNETO-STATIC PROBLEM
On t h i s f i g u r e t h e r o t o r , t h e s t a t o r and t h e a i r - g a p o f t h e a l t e r n a t o r can c l e a r l y b e s e e n .
The c h o i c e of t h e f u n c t i o n n ( x ) r e q u i r e s a n
i n i t i a l estimate for v;
f o r t h e r e s u l t s p r e s e n t e d h e r e w e have used
two m e t h o d s , t h e s e c o n d o f which i s a c t u a l l y a more s o p h i s t i c a t e d v e r s i o n of t h e f i r s t : Method I:
W e know t h e m i n i m a l v a l u e s t S and
and t h e r o t o r , r e s p e c t i v e l y . q
(4.17)
Method 2:
k
= v
E~
of v i n t h e s t a t o r
For k > 0, chosen a r b i t r a r i l y , w e p u t
i n t h e a i r and t h e c o p p e r
o
qk = k
E~
Ilk = k
E
s
i n the rotor i n the s t a t o r .
Having d e f i n e d
nk
by ( 4 . 1 7 )
w e solve t h e f o l l o w i n g l i n e a r
problem ( a c t u a l l y i t s d i s c r e t e v e r s i o n )
(4.18)
I,
qk V\*vv
dx =
W e then put
rl
= v
11
= V(lVq(12)
0
i n the a i r
(4.19)
i n the rotor or the s t a t o r .
W e could consider using a systematic procedure f o r updating n ( x ) ( a f t e r a s p e c i f i e d number o f i t e r a t i o n s o f A L G l o r ALG2, f o r e x a m p l e ) . The major drawback of s u c h an u p d a t e l i e s i n t h e f a c t t h a t t h e b i l i n e a r form i n p r o b l e m ( 4 . 1 4 ) would b e m o d i f i e d v e r y f r e q u e n t l y so t h a t f o r t h e a p p r o x i m a t e p r o b l e m a new m a t r i x would h a v e t o b e f a c t o r i s e d i f t h e d i s c r e t e analogue of
(4.14)
i s s o l v e d by a d i r e c t
method.
I n p r a c t i c e , a r e l i a b l e method ( a c t u a l l y a ' c o n t i n u a t i o n ' - t y p e method) c o n s i s t s o f p r o g r e s s i v e l y i n c r e a s i n g t h e c u r r e n t d e n s i t y
j
(which i n c r e a s e s t h e n o n l i n e a r i t y o f t h e p r o b l e m ) and u s i n g t h e r e s u l t s o f t h e p r e c e d i n g c a l c u l a t i o n f o r t h e e s t i m a t i o n o f v.
The
192
STRONGLY NONLINEAR B.V.P.'s
(CHAP.
5)
r e s u l t s o f MARROCCO 111, which w i l l b e b r i e f l y r e p e a t e d h e r e , w e r e a l l o b t a i n e d u s i n g a l g o r i t h m ALG2.
W e l i s t below what w e c o n s i d e r t o be
t h e most i m p o r t a n t f e a t u r e s o f t h e s e r e s u l t s . F o r a l o w v a l u e of t h e c u r r e n t (j = 0 . 5 A/mm2)
t h e problem i s
p r a c t i c a l l y l i n e a r and v , b o t h i n t h e r o t o r and i n t h e s t a t o r , s t a y s c l o s e t o t h e minimal v a l u e s c R and c S r e s p e c t i v e l y .
For p = r = l we
o b t a i n t h e a p p r o x i m a t e s o l u t i o n w i t h e x c e l l e n t p r e c i s i o n i n two i t e r a t i o n s f o r ~ ( x c) h o s e n i n a c c o r d a n c e w i t h ( 4 . 1 7 ) w i t h k = 1. k > 1, s t i l l w i t h j = 0 . 5 A/mm2,
For
t h e value of 0 is over-estimated,
and t h i s s l o w s down t h e c o n v e r g e n c e :
h e n c e f o r k = 10 a p p r o x i m a t e l y
2 5 i t e r a t i o n s of A L G 2 are r e q u i r e d f o r c o n v e r g e n c e . For a l a r g e r v a l u e o f j ( j = 2 A/mm
2
1 , t h e i m p o r t a n c e o f t h e non-
l i n e a r i t y i n c r e a s e s a n d it i s s t i l l t h e e s t i m a t i o n o f q ( x ) which h a s t h e predominant e f f e c t on t h e r a t e o f convergence. and c o n s i d e r a n e s t i m a t e of
j = 2 A/mm2
by making
r
k
Suppose t h a t
o b t a i n e d by u s i n g ( 4 . 1 7 ) ;
q
v a r y from 1 t o 2 0 , w e o b s e r v e t h a t t h e o p t i m a l v a l u e o f
d e c r e a s e s f r o m 10 t o 1 and t h a t t h e c o r r e s p o n d i n g number o f i t e r a -
t i o n s d e c r e a s e s from 9 0 t o 50.
Figure 4.4 represents a typical 2
r e s u l t o b t a i n e d w i t h k = 2 0 a n d j = 2 A/mm
Figure 4.4
.
(SEC. 5)
POTENTIAL FLOWS OF FLUIDS
193
Method 2 (i.e. (4.18), (4.19)) is very efficient if we start with We see the optimal the estimate obtained by using (4.17) with k = 1. value of r go to 1 and the number of iterations decrease to about 25. For larger values of k the use of (4.18), (4.19) does not improve, and in fact actually decreases, the rate of convergence. 2 For j = 2 A / m , which gives a small nonlinearity, taking k = 1 amounts to estimating n(x) by solving the linear problem, and this is a natural choice to make. In contrast, for j = 7.5 A/mm 2 , which in this example corresponds to the normal rggime, the effect of the nonlinearities is larger. If we choose ~ ( x ) by method (4.17) we obtain very comparable results for k varying from 10 to 50: approximately 50 iterations to obtain the In the solution, with an optimal value of r going from 10 to 1. case of method (4.18), (4.19) we ob ain a significant improvement for k = 10, which reduces the number of iterations to about 25. For higher values, method (4.18), (4.19 can become disastrous. We can deduce from all this that the use of the augmented Lagrangian method will be effective if we succeed in obtaining a good initial The estimate of rl (x), which allows us to work (in ALG2) with r = 1. most rational choice would be to make the fullest possible use of the information provided by earlier calculations: for example, continuously increasing the current enables the nonlinear effects to be introduced gradually. It should be noted, however, that the algorithm itself is very robust; a poor choice of q(x) or of r slows down, but does not prevent, convergence. This algorithm can therefore be used with every confidence, and its performance can be improved by experiment.
5.
CALCULATION OF SUBSONIC AND TRANSONIC POTENTIAL FLOWS OF COMPRESSIBLE IDEAL FLUIDS
In this section we attempt to present a succinct examination of the possibilities offered by augmented Lagrangian methods for the numerical simulation of s u b s o n i c and t r a n s o n i c flows of compressible ideal fluids; in the t r a n s o n i c case the problem considered is difficult and has given rise to numerous publications; for further details on the numerical solution of such problems by f i n i t e e l e m e n t methods, we refer to BRISTEAU-GLOWINSKI-PERIAUX-PERRIER-PIRONNEAU-
1 94
STRONGLY NONLINEAR B.V.P.'s
(CHAP. 5 )
POIRIER Ell, 121, GLOWINSKI r2, Chapter 71, AMARA-JOLY-THOMAS 111, and to their associated bibliographies. 5.1
Formulation of the problem
In this section we consider the numerical simulation of the (and therefore i r r o t a t i o n a l ) flow of a compressible ideal fluid. Within the limited scope of this section it would be out of the question to attempt to give a detailed description of the physical assumptions underlying the model described below, and we advise the reader who is not familiar with such problems to consult one of the treatises on Fluid Mechanics, such as LANDAU-LIFCHITZ 111.
potential
Let r$ denote the v e l o c i t y p o t e n t i a l , with u = V$ the velocity field; with p denoting the density of the fluid, the mass conservation equation is then written v - p u = V.pV@ = 0.
(5.1)
By introducing certain assumptions relating to the equation of state of the fluid, we deduce from the conservation of momentum equation that
where, in (5.2): (i) (ii)
po
is the d e n s i t y of t h e f l u i d a t r e s t
y
(iii)
C,
is the r a t i o of s p e c i f i c h e a t s ; air is the c r i t i c a Z s p e e d of s o u n d .
y = 1.4 in the case of
In the case of air we thus have (5.3)
1 1 and k = 6 Combining (5.1) , (5.2) and (5.3) we obtain a nonlinear problem which is formally very similar to the magnetostatic problem studied above in Section 4. It is therefore quite natural to follow the same line of approach as in Section 4 in order to attempt to obtain a numerical in a suitably chosen system of units we have po
=
-.
(SEC.
5)
195
POTENTIAL FLOWS OF FLUIDS
s o l u t i o n o f t h e flow problems i n q u e s t i o n h e r e . W e therefore define
I
+(O)
.
Cl,l/kl
= 0.
5+I)(E2)
The f u n c t i o n
I) by
i s increasing,
The f u n c t i o n a l
convex on C0,ll and concave o n
3 d e f i n e d by
(5.5) where
Ci
i s t h e f l o w domain and
L ( * ) is
a l i n e a r f o r m which t a k e s
i n t o a c c o u n t c e r t a i n o f t h e boundary c o n d i t i o n s , i s non-convex o v e r
i t s domain of d e f i n i t i o n ;
e q u a t i o n ( 5 . 1 ) i n c o n j u n c t i o n w i t h an
adequate set o f boundary c o n d i t i o n s c h a r a c t e r i s e s t h e s t a t i o n a r y
p o i n t s of
3
, which
i n t h e p r e s e n t c a s e a r e n o t n e c e s s a r i l y minimum
I$ s a t i s f i e s lV@l s l , which c o r r e s p o n d s t o a s u b s o n i c f l o w , w e are i n t h e c o n v e x I n t h e case w h e r e t h e s t a t i o n a r y p o i n t
points. 'part'
o f t h e f u n c t i o n a l and
i s t h e unique subsonic s o l u t i o n (poss-
$
i b l y 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 ) . which
I f there e x i s t s a region i n
l O @ l > 1 , t h e n w e h a v e a g e n u i n e t r a n s o n i c f l o w and i n o r d e r t o
e n s u r e u n i q u e n e s s it i s n e c e s s a r y t o impose a s u p p l e m e n t a r y c o n d i t i o n ; t h i s c o n d i t i o n i s p r o v i d e d by t h e Second Law of
T h e r m o d y n a m i c s , and
may b e f o r m u l a t e d as f o l l o w s :
1
(5.6)
The e n t r o p y is n o n - d e c r e a s i n g a l o n g a s t r e a m l i n e orientated i n the direction of the flow.
I n t h e case o f t h e p r e s e n t p r o b l e m o f p o t e n t i a l f l o w s , t h e e n t r o p y condition (5.6)
c a n b e f o r m u l a t e d as an i n e q u a l i t y c o n s t r a i n t on t h e
p o s s i b l e d i s c o n t i n u i t i e s i n V$
,
a s follows.
Suppose t h a t
S
is a
s u r f a c e ( o r l i n e ) of d i s c o n t i n u i t y o f t h e f l o w v e l o c i t y ( i . e . a s h o c k ) ;
w e d e n o t e by by
u-
t h e v e l o c i t y i m m e d i a t e l y u p s t r e a m o f t h e s h o c k , and
u+ t h e v e l o c i t y i m m e d i a t e l y downstream;
where
n
i s t h e normal t o
S.
w e t h e n have
The c o n d i t i o n
(5.7) eliminates
e x p a n s i o n s h o c k s ( a c r o s s w h i c h t h e r e would b e a s u d d e n d r o p i n p r e s s u r e and d e c r e a s e i n e n t r o p y ) ;
it t h u s a d m i t s o n l y compression shocks.
196
STRONGLY NONLINEAR B.V.P.'s
(CHAP. 5)
It should be noted that the condition ( 5 . 7 ) depends on the solution itself (through the medium of S and n ) and that the problem thus posed falls into the context of q u a s i - v a r i a t i o n a l inequalities. The description of an i n t e r i o r penalty method for the numerical treatment of ( 5 . 7 ) and its application to the calculation of transonic flows may be found in B R I S T E A U - G L O W I N S K I - P E R I A U X - P E R R I E R - P I R O N N E A U - P O I R I E R 121, GLOWINSKI 12, Chapter 71, and GLOWINSKI-LIONS-TREMOLIERES [2, Appendix 4 1
.
5.2
Formulation via an augmented Lagranqian. a1gorithms
Solution
Since the functional 3 (see ( 5 . 5 ) ) is not convex, the problem under consideration does not (except in the purely subsonic case) fall within the general framework of Chapter 111; nonetheless, we can - formally at least - still introduce an augmented Laqrangian and consider seeking its stationary points, possibly taking account of condition ( 5 . 7 ) (by means o f an interior penalty method, for example).
We therefore introduce the augmented Laqrangian
where, in (5.8), the linear form L(*)depends on the boundary conditions of the problem. Just as in the magnetostatic problems in Section 4, we have introduced a renormalisation function TI,the optimal value of which is p ( B $ ) where $ is the required solution; the precise value of this factor, however, is not so critical in the present problem, since within the range of validity of the transonic model (which we may here take to be lV@l < 1.5, say) the density finally varies only slightly (in the sense that it remains of the order of p o ) . In order to find the stationary points of dr we shall use algorithms ALGl and ALG2; it will therefore be necessary to solve, at each iteration of ALG1, a system in {@",pn}, An being known, namely (p"-
(5.9)
A").vv dx + 7 I L(v)
y V € V,
(SEC. 5 )
POTENTIAL FLOWS O F FLUIDS
197
i s a s u b s p a c e of H 1 ( a ) , which t a k e s i n t o a c c o u n t t h e boundary c o n d i t i o n s , and where H = ( L 2 (a)I N (N = 1 ,2,3 i n a p p l i c a t i o n s )
where
V
I f w e s o l v e t h e system (5.9),
i n v o l v e s o l v i n g a n e q u a t i o n i n one v a r i a b l e of t h e t y p e
(5.11)
(p(z2)+r)z
for
r
( i f q:
1) :
= b.
W e p u t g r ( z ) = (p(z2)+r)z.
r;
(5.10) w i l l
( 5 . 1 0 ) by r e l a x a t i o n , t h e n
F i g u r e 5 . 1 shows g,
f o r v a r i o u s v a l u e s of
s m a l l t h e r e w i l l be t w o s o l u t i o n s t o t h e equation
g r ( z ) = Const.
Figure 5.1 I f i n a d d i t i o n t o An
r
w e s u p p o s e t h a t I $ ~i s known i n ( 5 . 1 0 ) , t h e n f o r
sufficiently large,
t r a s t , t h e system (5.9),
( 5 . 1 0 ) a d m i t s a u n i q u e s o l u t i o n pn; (5.10) w i t h a g i v e n An
i n con-
does not i n general
198
STRONGLY NONLINEAR B.V.P.'s
(CHAP. 5)
have a unique solution. In the case of s u b s o n i c flows, for which we remain within the domain of convexity of the functional 3 in (5.5), we can use ALGl and ALG2 without taking any particular precautions; some numerical tests for 0 c IR2 are given in MARROCCO Ell, using ALG2 with the Lagrangian dr from (5.8), in which + I has been taken; the spaces V and H in (5.9), (5.10) are approximated using Co-conforming finite elements of order 1 (i.e. piecewise affine) over a triangulation Z h of R. In the case of the nozzle shown in Figure 5.2 we have convergence of ALG2, for subsonic flows, in at most 20 iterations. optimal value of r is close to 1.
The
In the case of g e n u i n e t r a n s o n i c f Z o w s , the solution calculated by the above algorithms can depend on r and may contain expansion shocks (i.e. non-physical shocks); it is therefore necessary to incorporate into the mathematical model the condition (5.7). There are several possible ways of doing this; however, we think it useful to point out one in particular, which results directly from the decomposition associated with the relation Vv-q = 0. We have already mentioned in Section 3.4 that it is possible to impose continuity constraints on the gradient of the solution; now, the conditions (5.7), which are of inequality type, are very similar to the conA natural approach would straints on V$I considered in Section 3.4. therefore be to approximate $I (and the corresponding test functions v ) by piecewise functions of degree k 2 2, and p = V$I by piecewise if k = 2, we can impose condition (5.7), functions of degree k - 1 ; that is Ip-.nl 2 /p+.nl , at the midpoints of the element sides. It remains to investigate solution algorithms which are suitable for such a treatment of condition (5.7).
NUMBER OF NODES NUMBER OF ELEMENTS
300
490
(The f i g u r e d e p i c t s a half nozzle)
Figure 5.2
(SEC. 6 )
6.
199
FURTHER APPLICATIONS
FURTHER APPLICATIONS I n t h i s s e c t i o n w e c o n s i d e r t h r e e e x a m p l e s o f a p p l i c a t i o n s which
come w i t h i n t h e framework of S e c t i o n 2 o f t h i s c h a p t e r ; examples h a v e a l r e a d y b e e n t o u c h e d upon i n C h a p t e r 111.
t w o of these Since 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 s e p r o b l e m s i s v e r y s i m i l a r t o t h a t of t h e problems i n t h e p r e c e d i n g s e c t i o n s and s i n c e t h e n u m e r i c a l r e s u l t s obtained o n l y s e r v e t o confirm t h o s e obtained e a r l i e r , it w i l l s u f f i c e t o g i v e o n l y a r e l a t i v e l y b r i e f o u t l i n e o f them;
nonetheless, we
s h a l l a t t e m p t t o s h e d s o m e l i g h t o n t h e p a r t i c u l a r q u a l i t i e s of t h e problems c o n s i d e r e d , and t h e i r i m p a c t o n t h e n u m e r i c a l t r e a t m e n t . S e c t i o n s 6 . 1 and 6 . 2 w i l l t h u s d i s c u s s , r e s p e c t i v e l y , t h e f l o w of a Bingham fluid i n a c y l i n d r i c a l d u c t and t h e e l a s t o p l a s t i c t o r s i o n of
a cylindrical bar. 6.1
S e c t i o n 6 . 3 w i l l t r e a t a minimal s u r f a c e problem.
Flow of a v i s c o p l a s t i c Bingham f l u i d i n a c y l i n d r i c a l d u c t
F o r m u Z a t i o n of t h e probZem
6.1.1
W e c o n s i d e r t h e f l o w o f a v i s c o p l a s t i c m a t e r i a l o f Bingham t y p e i n a cylindrical duct with cross-section
p l a s t i c i t y threshold
Q.
T h i s problem i n v o l v e s a
i f t h e m e c h a n i c a l s t r e s s e s r e m a i n below
g;
t h i s t h r e s h o l d v a l u e , t h e m a t e r i a l s t a y s r i g i d , w h e r e a s beyond t h e t h r e s h o l d it b e h a v e s l i k e a n i n c o m p r e s s i b l e f l u i d w i t h v i s c o s i t y The f l o w i n t h e d u c t i s i n d u c e d by a l i n e a r p r e s s u r e d r o p
is constant over t h e cross-sections
i n practice,
1
Ho(Q) d e n o t e t h e r e q u i r e d v e l o c i t y ; 1 m i n i m i s i n g i n Ho(Q) the functional Let u
E
w i t h v > 0, g > 0 , f
E
L
2
(Q).
f
V .
which,
of t h e d u c t .
w e o b t a i n t h i s by
8
W e n o t e t h a t i f v = 0 w e a g a i n h a v e t h e p r o b l e m of S e c t i o n 3 w i t h s = 1, t h e t e r m
t o assume v > 0 problem:
It i s e s s e n t i a l n l V ~ l d xb e i n g n o n - d i f f e r e n t i a b l e . 1 f w e a r e s e e k i n g a s o l u t i o n i n H o ( Q ) f o r t h e above
t h i s problem i n t h e C a l c u l u s o f V a r i a t i o n s a d m i t s a
u n i q u e 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 v a r i a t i o n a l i n e q u a l i t y 8
2
W e h a v e assumed f E L
(a) i n
o r d e r t o s l i g h t l y widen t h e g e n e r a l i t y .
2 00
STRONGLY NONLINEAR B.V.P.'s
(CHAP. 5 )
a detailed numerical analysis of which may be found in GLOWINSKILIONS-TREMOLIERES C11, 121 and GLOWINSKI C11, C21. 6.1.2 An
S o l u t i o n b y augmented L a g r a n g i a n m e t h o d s
augmented Lagrangian associated naturally with the flow problem 1 is dr : H o ( Q ) (L2(Q))2 X (L2(Q))2
defined in Section 6.1.1 given by
+
The corresponding algorithms ALGl and ALG2 have already been described in Chapter 111, Section 3 . 3 ; however it will be worth making the effort to write out the system corresponding to the minimisation of ir with respect to q and v , with p f i x e d . The optimality condition with respect to v, at the point {u,p,X} leads to
It is important to note that problem (6.4) is in fact independent of r, in the sense that, after division by r , the bilinear form in (6.4) is {v,w} + VV-Vw dx, i.e. that associated with the homogeneous Dirichlet problem for the operator - A . Further, minimising 2 over (L the functional q + x,(u,q,X) reduces to the minimisation 2 over (L (Q))2 of the functional
and this reduces to solving a.e. in for the functional (6.6)
U+r
2 1 5 1 +~ g \ c l -
x
E
D and with 5 describing
(rvu(x)+X<x)>*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
STRONGLY NONLINEAR B.V.P.'s
(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
FURTHER APPLICATIONS
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)
FURTHER APPLICATIONS
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
FURTHER APPLICATIONS
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)
FURTHER APPLICATIONS
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
STRONGLY NONLINEAR B.V.P.'s
(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
(‘r..), 11
I
t h e p a r t o f t h e b o u n d a r y w h e r e t h e c o n t i n u o u s medium
To
i s fixed and by
v
=
{VC
(H’(Q))~, v =
o
on To}
t h e s p a c e o f admissible displacements. The o p e r a t o r B :
V
,
-f
H is defined a s follows:
av.
av.
ax: + A ax. )
( B v ) .1~. = -(
J
Bv
,
1 si,j s d
;
1
r e p r e s e n t s t h e Zinearised strain t e n s o r .
which i s p r o v e d i n DUVAUT-LIONS [l],
B
W e i n t r o d u c e a s y m m e t r i c automorphism
c1 > 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
ELASTOPLASTICITY PROBLEM
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 )
ELASTOPLASTICITY PROBLEM
, 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
ELASTOPLASTICITY PROBLEM
(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 )
ELASTOPLASTICITY PROBLEM
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 ,
<@'(v),w> =
~?I(B~)-BW
0
using
$
and d e n o t i n g
w e have
dx + G(w) , V w e V h
The g r a d i e n t method ( f o r minimising
i s e q u i v a l e n t t o mini-
.
on Vh ) i s t h e n w r i t t e n
A 2-D ELASTOPLASTICITY PROBLEM
228
uo
E
Vhand p>O
un
E
Vh
are given;
b e i n g g i v b n by r e c u r r e n c e ,
un+I = un
-
calculate
p $'(u").
I f w e choose as i n n e r p r o d u c t
<
on IRN where N i s t h e d i m e n s i o n o f $'
(CHAP. 6 )
from f o r m u l a ( 6 . 1 )
I
>
vhl
t h e natural inner product the calculation of t h e gradient However, a s a l r e a d y p o i n t e d
i s immediate.
o u t , e v e n i n t h e q u a d r a t i c case t h i s p r o c e d u r e d o e s n o t p r o d u c e good It is
r e s u l t s because o f t h e ill c o n d i t i o n i n g of t h e m a t r i x
BtSB.
a l s o p o s s i b l e t o c h o o s e a n y o t h e r i n n e r p r o d u c t on V h ,
but then we
would h a v e t o s o l v e , a t e a c h i t e r a t i o n , a l i n e a r s y s t e m w i t h m a t r i x R , where R i s t h e m a t r i x o b t a i n e d from t h e c h o s e n i n n e r p r o d u c t
W e s a y t h a t t h e m a t r i x R a c t s a s an a u x i l i a r y o p e r a t o r
conditioning operator). =
< ,>.
(or a s a pre-
W e have t e s t e d t h e f o l l o w i n g i n n e r p r o d u c t
,
(ABu)Bv dx
J,
and t h e m a t r i x R o b t a i n e d i s t h e n t h e s t i f f n e s s m a t r i x of t h e e l a s t i c p r o b l e m , which would a p p e a r a n a t u r a l c h o i c e f o r s o l v i n g t h e e l a s t o p l a s t i c problem. 6.4
Con j u g a t e - g r a d i e n t
To minimise
$
on Vh,
uo c V h
, wo
method
w e can also use t h e conjugate-gradient
method: =
$'(vo)
knowing u " m d wn( u
n+l
E
are given;
vh) ,
-- u n - p, wn,
where
= r n + l + Xnwn
,
by recurrence,
calculate minimises
p,
where
X
<'n+ 1 9 'n+ =
g ( p ) = $(u"-pw")
Crn
,
'
L e t u s s a y s t r a i g h t away t h a t t h e r e s u l t s o b t a i n e d w i t h t h i s method
w e r e less t h a n e x c e l l e n t method w a s a p p l i e d . t i o n of
pn,
W e s h a l l now d e s c r i b e t h e way i n w h i c h t h e
I n p r a c t i c e i t i s a d e q u a t e t o u s e an approxima-
and t h i s i s what was done.
It i s p o s s i b l e t o determine
a v a l u e , p o s s i b l y non-un q u e , o f t h e s e c o n d d e r i v a t i v e of
$,
since
(SEC.
$
6)
229
NUMERICAL AP P L ICAT I O N
is d i f f e r e n t i a b l e a l m o s t everywhere.
For
pn
we have chosen
t h e f i r s t i t e r a t e o f Newton's method:
which a c t u a l l y g i v e s t h e e x a c t s o l u t i o n i n t h e e l a s t i c c a s e . choice i s q u i t e adequate;
a more a c c u r a t e c a l c u l a t i o n of
This pn
would
be more e x p e n s i v e t o compute and would n o t b r i n g a b o u t any s i g n i f i c a n t improvement i n t h e convergence. o r 3N i t e r a t i o n s ;
We reinitialise
wn = rn
every
2N
once a g a i n no s i g n i f i c a n t improvement would be
e v i d e n t i f a r e i n i t i a l i s a t i o n were performed e a c h t i m e t h e i n n e r product n+l
<w
n+l
,u
n -u >
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
2-D FLOW OF VISCOPLASTIC FLUIDS
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 {{$,sl,p,A}
O < P < 2r,
w e have f o r all A o
(6.12)
Suppose that
then if
E
H
lim P" = p strongly in (L n-
2
(n)) 2 ,
246
2-D FLOW OF VISCOPLASTIC FLUIDS
lim An = n-
(6.13)
A"
weakly i n
(CHAP. 7 )
(~'(n))',
It is clear that, once again, the essential difficulty with this approach lies in the fact that system (6.8) has to be solved at each this problem can be iteration; in view of the structure of
4,
solved by a block over-relaxation method like that described in Chapter 111, Section 3.2. A s far as the choice of p is concerned, numerical experiments indicate once again that the optimal value lies close to p = r. 6.4
On variants of algorithm (6.6)
-
(6.8)
The first of these variants is algorithm ALG2 which has already been studied in some detail in Chapter I11 and used extensively in In the case under consideration here, we Chapters IV, V and VI. obtain the following: (6.14)
then, f o r n
pn,{$",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
I+& 2
r,
(SEC. 6 )
SOLUTION O F ( 4 . 1 )
W e s h a l l see t h a t t h e s o l u t i o n o f any d i f f i c u l t i e s ;
,
(6.15)
241
(5.2)
and ( 6 . 1 6 )
does not pose
i s i n f a c t a p r o b l e m o f Linear b i -
problem ( 6 . 1 6 )
h a r m o n i c t y p e w r i t t e n i n mixed f o r m u l a t i o n , a s f o l l o w s :
Find
/
{$",sn}
E
W such t h a t
(6.19)
As r e g a r d s (6.151,
T h i s p r o b l e m a d m i t s o n e and o n l y one s o l u t i o n .
it c a n e a s i l y b e shown t h a t t h i s p r o b l e m a d m i t s a u n i q u e s o l u t i o n g i v e n e x p l i c i t l y by p" =
(6.20)
71 clx"l-g, +x" Ix"l
where Xn = {Xy, X;}
A"1
X"
=
A"2 *
1x1
=
2
(6.22)
and w h e r e
i s d e f i n e d by
X" = I
(6.21)
'
+ 2rs?;l, n-' r(s"-l 22 - 1 1 )*
Once a g a i n t h e o p t i m a l c h o i c e f o r p = r;
t h e optimal choice of
r
p
a p p e a r s t o l i e close t o
i s a much more c o m p l i c a t e d p r o b l e m
s i n c e t h i s o p t i m a l v a l u e a p p e a r s t o depend on
v
and
g;
nonethe-
l e s s , t h e comments made i n Remark 6 . 1 i n C h a p t e r V , S e c t i o n 6 . 1 . 2 , a g a i n h o l d f o r t h e p r e s e n t problem. T o c o n c l u d e t h i s s e c t i o n on v a r i a n t s o f ALGl, w e s h o u l d m e n t i o n
t h e f o l l o w i n g v a r i a n t d u e t o GABAY Cll, which w e h a v e a l r e a d y m e t i n Chapter IV.
W e s h a l l r e t u r n i n Chapter I X t o t h e p r o p e r t i e s of t h i s
a l g o r i t h m which w e s h a l l r e f e r t o a
W i t h (6.141,
s
s
(6.15) a s b e f o r e , d e t e r m i n e
: An+1/2
by
24 8
2-D FLOW OF VISCOPLASTIC FLUIDS {Jln,sn} E V
then
by
by
ar.d f i n a l l y (6.25)
(CHAP. 7 )
+
in+'=
p(B{$",s"}-p").
It may be noted that in the above algorithm the variables { $ , z } and q play a symmetric role; this was not the case in ALG2, where the order chosen is governed by considerations of ellipticity, for the details of which we refer the reader to Chapter 111. Section 7 below gives the results of a number of numerical experiments in which the above algorithms (actually, finite-dimensional variants of these) have been applied to the solution of problems (3.7) and (4.1).
NUMERICAL EXPERIMENTS
7.
In this section we shall describe some of the numerical results obtained by BEGIS C21 using the methods of approximation by finite elements, and of iterative solution by augmented Laqrangian, of the preceding sections. 7.1
Formulation of the test problem
Let R = lO,lCxlO,l~ ; we consider a family of Binqham flows for which (using the notation of Section 2) we have (7.1)
v = 1,
( 7 * 2)
,f
:Ir (7.3)
=
_o, =
b
=
{bl,b21with b 2 = 0 on
b l ( 0 , ~ ~=)b l (I , x 2 ) = 0 bl(xI,l) = 0 Vxl
E
10,1[
ex2
E
r,
10,1[,
, bl(xl,O) =
1
V x l E 10,lC.
We are thus dealing with problems which can be classed as flow in a cavity with a sliding w a l l ; we note that 5 . g dI' = 0, but that
lr
(SEC. 7 )
7.2
NUMERICAL EXPERIMENTS
249
Numerical r e s u l t s
D e t a i l s of t h e p r a c t i c a l i m p l e m e n t a t i o n o f t h e methods of approxima-
t i o n a n d i t e r a t i v e s o l u t i o n d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s may b e f o u n d i n BEGIS C21.
Steady-State
Cases:
p l a s t i c regions g
(shown h a t c h e d ) and t h e v i s c o -
( i n w h i t e ) , as w e l l as t h e s t r e a m l i n e s
equipotentials of
ease w i t h
g , w e h a v e shown i n
For various values of
Figures 7.1 t o 7.5 the r i g i d regions
I$
);
(these a r i the
w e o b s e r v e t h a t t h e zones o f r i g i d i t y i n c r -
( t h e a s y m m e t r i e s which c a n b e s e e n a r e due t o t h e asym-
metries o f t h e t r i a n g u l a t i o n e m p l o y e d ) .
Unsteady Cases:
For v a r i o u s v a l u e s o f
g , we have c o n s i d e r e d t h e
case i n w h i c h t h e m a t e r i a l f i l l i n g t h e c a v i t y
( s o u(x,O)
=
0
Y xen
6 +(x,O)
i s i n i t i a l l y a t rest
Q
= 0 Y x E ~ ) , and i s t h e n s e t i n m o t i o n by
t h e s l i d i n g ( d e f i n e d by ( 7 . 3 ) ) o f t h e lower w a l l o f R.
When t h e
s t e a d y s t a t e h a s been a t t a i n e d ( t o w i t h i n a c e r t a i n p r e c i s i o n ) , t h e m o t i o n o f t h e l o w e r w a l l i s a r r e s t e d , w i t h t h e aim o f o b s e r v i n g t h e return t o the i n i t i a l state
u = 0 ( i . e . I$ = 0 ) i n I
For various values of
g
R.
I
( i n c l u d i n g g = 0 ) w e h a v e shown i n
F i g u r e 1.6 t h e b e h a v i o u r o f
i t may b e n o t e d t h a t f o r
g > 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)
NUMERICAL EXPERIMENTS
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
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 , l;? =1 1.0 a t x 2 = 0
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 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
R i g i d zones LABORIA
Figure 7.2
( g = 1)
D. B E G I S
252
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 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
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 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)
NUMERICAL EXPERIMENTS
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
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, 1.0 at 3 ' 0
1;3='l
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
0.616E-1 0.510E-3 0.100E-1
R i g i d zones LABORIA
D. B E G I S
254
2-D
FLOW O F V I S C O P L A S T I C F L U I D S
(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
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 x = 0 , l ; % = l 1 1.0 a t x 2 = 0
I N I T I A L CONDITION:
I n i t i a l velocity
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
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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
we have, when n
+ im:
s t r o n g l y in
(2.14)
un
+
u
(2.15)
p"
+
p =
(2.16)
A"
+
h weakly i n
BU
v
strongly i n H
H
where u is the solution of (P), and where is a saddle point of dr on V x H x H .
A
is such that {u,p,X}
Remark 2.5: The only nontrivial stage in the above algorithm is the solution of problem (2.11); in fact to solve (2.11), taking into consideration its very special structure, it is very convenient
262
FINITE NONLINEAR ELASTICITY PROBLEMS
(CHAP. 8)
to use block-relaxation algorithms such as those considered in CEAGLOWINSKI C11, CEA 111, 121, GLOWINSKI 12, Chapter 51 and which are also used in Chapter 111, Section 3.2. If we use these relaxation methods, and if in the calculation of [un,pn) we perform only a single r e l a x a t i o n i t e r a t i o n , s t a r t i n g f r o m obtain the algorithm described below in Section 2.5. 2.5
we then
A second algorithm for solving (P)
In this case we are in fact dealing with algorithm ALG2 of Chapter 111, Section 3.2, namely: u
(2.17)
then f o r n
-1
g i v e n i n V, 'A
u
t 0,
n- 1
g i v e n i n H,
An known, d e t e r m i n e pn,un,Xn+l s u c c e s s i v e l y by
dr(un-' ,pn,An) 5dr(un-',q,A")
Yq
E
H,
(2.18) pn
E
H,
n n k r ( u ,P ,?I
Yv E V ,
(2.19) un
E
hn+l
(2.20)
Remark 2.6:
v. =
.
A" + @(Bun-p")
Different variants of (2.17)
-
(2.20) are poss ble;
we can for example (i)
interchange the roles of
q
and
v
(see also Remark 2.7
,
(ii) update An between stages (2.18), (2.19) ; by doing this we then obtain the following variant (due to GABAY I l l ) , of (2.17) (2.20), and to which we shall return in more detail in Chapter IX: (2.21)
then f o r n
u
-I
t 0,
given i n
n- 1 u and
V, 1' g i v e n i n H,
Xn
given, calculate
kr(un-l ,pn,An) dr(un-l ,q,An) (2.22)
P"
H,
Yq
E
H,
n An+1/2 n Xn+l p , ru I by
(SEC.
2)
DECOMPOSITION O F VARIATIONAL PROBLEMS
- xn+1/2 + P(Bun-pn)
(2.25)
q
and
(2.17)
v
-
263
;
p l a y a much more symmetric r o l e i n ( 2 . 2 1 ) - ( 2 . 2 5 ) t h a n i n (2.20).
8
Remark 2.7:
-
I f algorithm (2.17)
( 2 . 2 0 ) i s u s e d , i t i s recom-
mended t h a t i n t h e second s t a g e t h e problem s o l v e d s h o u l d be t h e one which h a s t h e b e t t e r e l l i p t i c i t y p r o p e r t i e s ( s e e C h a p t e r I11 f o r t h e m o t i v a t i o n f o r such a c h o i c e ) ; used, t h i s i s n o t important
i f algorithm (2.21)
since
and
q
v
-
(2.25) i s b e i n g
t h e n p l a y a symmetric
role. A s r e g a r d s t h e convergence o f
(2.17)
-
( 2 . 2 0 ) , i t i s proved i n
C h a p t e r 111, S e c t i o n 5 , t h a t under v e r y r e a s o n a b l e a s s u m p t i o n s on F , B , G w e a g a i n have t h e convergence p r o p e r t i e s 1 +G
O
(2.26)
2.6
-
(2.16)
if
r.
Remarks on t h e c h o i c e o f
For given
(2.14)
p
r.
and
r , t h e optimal choice f o r
p
i s very close t o p = r ,
i n t h e l i g h t of t h e l a r g e number of n u m e r i c a l e x p e r i m e n t s performed w i t h t h e a l g o r i t h m s i n S e c t i o n s 2 . 4 and 2 . 5 . a much t r i c k i e r problem; gressively faster as of
r
r
t h e system ( 2 . 1 1 )
The c h o i c e o f
increases;
i n practice,
i s ill-conditioned,
becomes a c o s t l y o p e r a t i o n , s e e n t h a t when
is
f o r large values
and t o s o l v e i t a c c u r a t e l y
w h i l s t , f o r very l a r g e values of
t h e r o u n d i n g e r r o r s g e n e r a t e d can b e d i s a s t r o u s . effects arise;
r
t h e o r e t i c a l l y , t h e convergence g e t s pro-
r,
I t can t h u s be
r i n c r e a s e s , two phenomena w i t h c o n t r a d i c t o r y t h e combined e f f e c t of t h e s e two phenomena on ( 2 . 1 0 ) -
( 2 . 1 2 ) p r o d u c e s an a l g o r i t h m which i s n o t v e r y s e n s i t i v e t o t h e c h o i c e of (2.20)
r
and which i s v e r y r o b u s t .
and ( 2 . 2 1 )
-
( 2 . 3 5 ) , with
I f we use algorithms (2.17)-
p = r , t h e o p t i m a l c h o i c e of
i n g e n e r a l a v e r y d i f f i c u l t problem t o a n a l y s e .
r
is
264
2.7
2.7.1
FINITE NONLINEAR ELASTICITY PROBLEMS
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. d i s c u s sion.
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 V = H and B = I ; F
t i a l s ) A1 and A 2 ,
respectively;
and
-
we shall
admit a s d i f f e r e n t i a l s ( o r s u b d i f f e r e n -
G
(where = must be r e p l a c e d by
and A2 a r e n e c e s s a r i l y m o n o t o n e
A1
(possibly multivalued).
Suppose t h a t p = r ;
8)
Further
R e l a t i o n s between a l g o r i t h m s (2.17) - (2.20), (2.21) ( 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 m e t h o d s .
a l s o assume t h a t operators
(CHAP.
Problem
i f A1
E
(P) i s then equivalent t o
and/or A2 a r e m u l t i v a l u e d )
w e t h e n o b t a i n by e l i m i n a t i o n of
An
in (2.17)-
(2.20) : -1
(2.28)
then f o r
u
n
given;
t 0,
(2.29)
rpn+Al (p") = run- I -A2(un-'),
(2.30)
run+A2(un) = ru"-'-A,
Putting (2.31) (2.32)
u n+1/2
= pn",
run+' /'+A
1
(p")
we f i n a Z Z y o b t a i n ( un+l
12)
= run-A2 (u")
,
run+l +A2(u n+l ) = run-A (un+'/'). I
I n ( 2 . 3 1 ) , ( 2 . 3 2 ) w e can r e c o g n i s e an a l t e r n a t i n g - d i r e c t i o n m e t h o d
of D o u g l a s - R a c h f o r d
t y p e (see DOUGLAS-RACHFORD 111).
S i m i l a r l y by e l i m i n a t i o n of obtain,
( s t i l l assuming
An
An+1/2
and
p = r)
(un+l)'1
(2.33)
run+' "+A
(2.34)
n+l n+l) n+1/2-A (Un+1/2) ru +A2(u = ru 1
1
=
,
i n (2.21)
-
(2.25) w e
(SEC. 2 )
DECOMPOSITION OF VARIATIONAL PROBLEMS
265
which i s i n f a c t an a l t e r n a t i n g - d i r e c t i o n m e t h o d of Peaceman-Rachford
t y p e ( s e e PEACEMAN-RACHFORD Ill). An i n v e s t i g a t i o n o f t h e convergence o f t h e above a l t e r n a t i n g -
d i r e c t i o n methods when A1
and A2 a r e monotone o p e r a t o r s from
H
into
H I p o s s i b l y m u l t i v a l u e d , and n o t n e c e s s a r i l y t.he g r a d i e n t s o f convex f u n c t i o n a l s , may be found i n LIONS-MERCIER [ll, GABAY [11 (see a l s o Chapter I X ) .
I n t e r p r e t a t i o n of a t g o r i t h m s ( 2 . 1 7 ) - ( 2 . 2 0 ) and ( 2 . 2 1 ) ( 2 . 2 5 ) i n t e r m s of t h e n u m e r i c a l i n t e g r a t i o n of e v o l u t i o n equations
2.7.2
I t follows from S e c t i o n 2 . 7 . 1
that i f p = r , V = H and B = I , then - ( 2 . 2 5 ) can be c o n s i d e r e d a s i m p t i c i t s c h e m e s , u s i n g t h e decomposition A = A1 + A2 f o r t h e n u m e r i c a l i n t e g r a t i o n w i t h r e s p e c t t o t i m e of t h e Cauchy p r o b l e m
(2.17)
-
( 2 . 2 0 ) and ( 2 . 2 1 )
*
+ A(u) = 0,
(2.35)
u(0) =
Uo.
I n view of t h i s i n t e r p r e t a t i o n , r a p p e a r s a s t h e i n v e r s e o f a t i m e 1 2 A t , with A t = 7 (resp. A t = f o r a l g o r i t h m ( 2 . 1 7 ) - (2.20)
r)
step
(resp.
(2.21)
-
(2.25)).
A s i s shown i n BOURGAT-DUMAY-GLOWINSKI
C11,
t h i s i n t e r p r e t a t i o n o f t h e above a l g o r i t h m s , r e l a t e d t o t h e n u m e r i c a l i n t e g r a t i o n with respect t o
t
of t h e e v o l u t i o n problem ( 2 . 3 5 ) , can
b e v e r y u s e f u l f o r g i v i n g a q u a l i t a t i v e u n d e r s t a n d i n g of t h e b e h a v i o u r of these algorithms;
f o r example, it i s c l e a r t h a t t h e above a l g o r -
i t h m s w i l l become p r o g r e s s i v e l y more r e l i a b l e a s as
A t
r
increases (i.e.
becomes s m a l l e r ) .
The n u m e r i c a l i n t e g r a t i o n o f
(2.35) u s i n g a l t e r n a t i n g - d i r e c t i o n
methods i s s t u d i e d i n LIONS-MERCIER 111 under r e l a t i v e l y g e n e r a l a s s u m p t i o n s on A1 and A2. 2.7.3
Further discussion
The s o l u t i o n of problems such a s ( P ) by 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 u s i n g augmented Lagrangians seems t d b e due ( i n t h e c o n t e x t
of boundary-value problems a t l e a s t ) t o GLOWINSKI-MARROCCO [ll-C31 (See a l s o POLYAK C11 f o r a p p l i c a t i o n s i n N o n l i n e a r Programming).
For
266
(CHAP. 8 )
FIN I TE N O N L I N E A R ELASTICI T Y PROBLEMS
f u r t h e r d e t a i l s and v a r i o u s a p p l i c a t i o n s w e r e f e r t h e r e a d e r t o t h e o t h e r c h a p t e r s of t h e p r e s e n t book and t o LIONS-MERCIER C11 GABAY 111,
GLOWINSKI 111,
121
,
GABAY-MERCIER
GLOWINSKI C11, GLOWINSKI-MARROCCO 1 4 1 , 151, CHAN-GLOVJINSKI
1 1 1 and BEGIS C21
,
111, BOURGAT-DUMAY-
MARROCCO
.
[I], M E R C I E R 111,
A s f a r a s w e a r e aware, t h e a f o r e m e n t i o n e d r e l a t i o n s h i p s b e t w e e n t h e a l g o r i t h m s o f S e c t i o n 2.5
a n d a l t e r n a t i n g - d i r e c t i o n methods were
o b s e r v e d f o r t h e f i r s t t i m e by CHAN-GLOWINSKI 111,
121.
W e conclude
t h i s s e c t i o n by p o i n t i n g o u t t h a t t h e m a j o r i t y o f t h e i d e a s c o n s i d e r e d a r e d e v e l o p e d f u r t h e r i n C h a p t e r s I11 and I X .
3.
APPLICATIONS I N FINITE N O N L I N E A R ELASTICITY. ( I ) LARGE-DISPLACEMENT CALCULATION OF THE E Q U I L I B R I U M POSITIONS OF INEXTENSIBLE, FLEXIBLE PIPELINES T h i s s e c t i o n i s b a s e d o n t h e a r t i c l e by BOURGAT-DUMAY-GLOWINSKI
3.1
3.1.1
111.
Formulation o f t h e problem
G e n e r a l discussion
Flexible p i p e l i n e s play an important r o l e i n t h e e x p l o i t a t i o n of off-shore o i l f i e l d s
j
t h e e n g i n e e r s concerned a r e i n t e r e s t e d i n t h e
s t a t i c and dynamic b e h a v i o u r o f t h e s e p i p e l i n e s , i n t h e e f f e c t s of c u r r e n t s and t i d a l s w e l l s , i n t h e p r o b l e m s o f c o n t a c t a t t h e seab e d and w i t h o t h e r o b s t a c l e s ( f o r example t h e p i p e l i n e i t s e l f ) , e t c . . F i g u r e 3 . 1 below i l l u s t r a t e s t h i s t y p e o f s i t u a t i o n and shows some o f t h e n o t a t i o n which w i l l b e u s e d i n t h e f o l l o w i n g d i s c u s s i o n .
face of t
h
0 /
Figure 3 . 1
e
y
(SEC.
INEXTENSIBLE FLEXIBLE PIPELINES
3)
267
W e have:
A, B
t h e ends of t h e p i p e l i n e
:
the curvilinear coordinate; l e n g t h of t h e p i p e l i n e )
S
M(s)
s(A)
=
0, s ( B ) = L (L:
g e n e r i c p o i n t of t h e p i p e l i n e , w i t h c o o r d i n a t e s x ( s ) ,
:
Y(s).
I n t h i s chapter w e s h a l l confine our a t t e n t i o n t o t h e s t a t i c disp l a c e m e n t s of p i p e l i n e s and w e s h a l l n e g l e c t t h e e f f e c t of w a t e r currents;
f o r c a l c u l a t i o n s i n v o l v i n g c u r r e n t s , and f o r dynamic
b e h a v i o u r w e r e f e r t o BOURGAT-DUMAY-GLOWINSKI
3.1.2
C11.
Simplifying assumptions
W e s h a l l assume f o r s i m p l i c i t y ( b u t a l s o b e c a u s e t h i s s t i l l a l l o w s a number of i n t e r e s t i n g p r e l i m i n a r y r e s u l t s t o be o b t a i n e d r e g a r d i n g t h e b e h a v i o u r of t h e s e p i p e l i n e s ) , t h a t
- torsional e f f e c t s are neglected, - the pipeline i s inextensible, - t h e d i a m e t e r of t h e p i p e l i n e i s s m a l l r e l a t i v e t o i t s l e n g t h L, - we c o n f i n e o u r s e l v e s t o t w o - d i m e n s i o n a l d i s p l a c e m e n t s , - t h e p i p e l i n e is f Z e x i b l e a n d , c o n s e q u e n t l y , c a n s u p p o r t l a r g e d i s p l a c e m e n t s w h i l s t s t i l l o b e y i n g a l i n e a r c o n s t i t u t i u e law b e t w e e n t h e s t r e s s e s and t h e s t r a i n s . 3.1.3
ModeZling o f s t a t i c problems
The m a t h e m a t i c a l f o r m u l a t i o n of t h e i n e x t e n s i b i l i t y c o n d i t i o n i s g i v e n by (3.1)
x*2+y'2 = I
on TO,LI
C o n s i d e r i n g t h e p i p e l i n e a s a beam, and a l s o assuming t h a t t h e o n l y e x t e r n a l f o r c e s a r e t h o s e due t o g r a v i t y , i t 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 f i e l d s c o r r e s p o n d i n g t o s t a b l e e q u i l i b r i u m posi t i o n s a r e s o l u t i o n s of t h e f o l l o w i n g locaZ m i n i m i s a t i o n problem:
where i n ( 3 . 2 ) :
268
(CHAP. 8 )
FINITE NONLINEAR ELASTICITY PROBLEMS
(i)
E I ( > 0) i s t h e f l e x u r a l r i g i d i t y o f t h e p i p e l i n e .
( i i ) g i s t h e a c c e l e r a t i o n due t o g r a v i t y . ( i i i ) p i s t h e l i n e a r d e n s i t y ( i . e . mass p e r u n i t l e n g t h ) o f t h e p i p e l i n e ( i f t h e p i p e l i n e i s immersed i n w a t e r , i n o r d e r t o t a k e a c c o u n t of buoyancy e f f e c t s w e t a k e
P = P,-oPw.
where
and
po
r e s p e c t i v e l y are t h e i n t r i n s i c l i n e a r d e n s i t y and
0
the cross-sectional
a r e a o f t h e p i p e l i n e , and w h e r e
volumetric d e n s i t y of t h e water).
pw
is the
a
F i n a l l y , t h e l o c a l minima which w e a r e s e e k i n g a r e t o b e f o u n d i n
8 d e f i n e d by
t h e nonconvex s e t
8
= IiX,Yl/X,Y€C
1
[O,L1
,X",Y"€L
2
(O,L),
(3.3) +y"
Remark 3 . 1 : non-quadratic,
= 1
o n C O , L ] , together with appropriate boundary conditions}.
P r o b l e m ( 3 . 2 ) c a n c l e a r l y b e s e e n t o be a n o n - c o n v e x ,
n o n l i n e a r programming p r o b l e m , which i s t h u s i n h e r e n t l y
difficult.
Remark 3.2:
By u s i n g t h e n u m e r i c a l methods which w i l l b e
d e s c r i b e d b e l o w , w e s h a l l e a s i l y b e a b l e t o t r e a t c a s e s f o r which E I and/or
p
are f u n c t i o n s o f
s , a n d i n which e x t e r n a l f o r c e s a n d en-
e r g y terms, more c o m p l i c a t e d t h a n t h o s e i n ( 3 . 2 ) , a r e p r e s e n t i n t h e f u n c t i o n a l t o be minimised. 3.2
R e s u l t s on t h e e x i s t e n c e o f s o l u t i o n s f o r t h e s t a t i c p r o b l e m
The m a t h e m a t i c a l s t u d y o f problems s u c h a s ( 3 . 2 ) g o e s b a c k a t l e a s t
a s f a r as E u l e r ( t h e p r o b l e m o f t h e " e l a s t i c a " , o r e l a s t i c r o d ) . Suppose t h a t 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)
w i t h x A , y A , x B I y B g i v e n , o r by
3)
(SEC.
INEXTENSIBLE FLEXIBLE PIPELINES
269
where , i n ( 3 . 5 ) , xA, yA, x B , y B , ao, Bo, aL, BL are g i v e n w i t h 2 2 2 Bo = 1, aL + 8, = 1; t h e f o l l o w i n g t h e o r e m i s t h e n p r o v e d by
compa c t n e s s a r g u m e n t s i n BOURGAT-DUMAY-GLOWINSKI
C11.
Suppose t h a t IZBI < L and t h a t t h e b o u n d a r y or ( 3 . 5 ) are s a t i s f i e d ; then problem (3.2) admits a t l e a s t one s o l u t i o n . THEOREM 3 . 1 :
c o n d i t i o n s (3.4)
F o r a d e t a i l e d m a t h e m a t i c a l i n v e s t i g a t i o n of p r o b l e m s o f t h e above t y p e ( i n p a r t i c u l a r f o r non-uniqueness
properties) we r e f e r to
ANTMAN c11 a d ANTMAN-ROSENFELD 111.
3.3
Numerical s o l u t i o n o f t h e s t a t i c p r o b l e m . Notes
(I)
General
The 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 s i m i l a r t o ( 3 . 2 ) h a s been amongst t h e s e w e s h o u l d m e n t i o n 111 and MAIER-ANDREUZZI-GIANESSI-JURINA-TADDEI
c o n s i d e r e d by s e v e r a l a u t h o r s : HIBBIT-BECKER-TAYLOR
[11.
P r o b l e m ( 3 . 2 ) i s i n f a c t n o n t r i v i a l f r o m t h e n u m e r i c a l p o i n t of it c a n b e d e a l t w i t h by i n t r o d u c i n g a L a g r a n g i a n a s s o c i a t e d
view:
with the functional
and w i t h t h e ( n o n l i n e a r ) i n e x t e n s i b i l i t y c o n d i t i o n
Let
X
(3.1)
,
as f o l l o w s :
b e a L a g r a n g e m u l t i p l i e r a s s o c i a t e d w i t h a l o c a l minimum
i t t h e n f o l l o w s from t h e f a c t t h a t
must s a t i s f y
(3.8) +
boundary c o n d i t i o n s , 4-
EI
- -& (A
-
2)
=
-g
on IO,IT,
(3.9) +
boundary c o n d i t i o n s ,
d is
stationary
2 70
(CHAP. 8 )
FINITE NONLINEAR ELASTICITY PROBLEMS
(3.10)
~ f ~ + ; l ~=
on 1 0 , ~ ~ .
I
X
I t f o l l o w s f r o m ( 3 . 8 ) - (3.10) t h a t
can b e v i e w e d as a g e n e r a Z i s e d
e i g e n v a l u e t o which t h e r e c o r r e s p o n d s t h e g e n e r a l i s e d e i g e n f u n c t i o n {x,y}. Since t h e e s s e n t i a l d i f f i c u l t y i n problem (3.2) lies i n t h e i n e x t e n s i b i l i t y constraint transformation a n g l e between
( 3 . 1 ) , it seems n a t u r a l t o overcome i t by u s i n g t h e
x ' = cos $ Ox
,
y'
=
s i n $,
where
$
is i n fact the
and t h e d i r e c t e d t a n g e n t t o t h e p i p e l i n e a t
M = { x , y } , and where d$/ds
is t h e curvature a t
M.
This transforma-
t i o n , which i s u s e d i n a number of s i m i l a r p r o b l e m s a s w e l l as i n p r o b l e m ( 3 . 2 ) by M.O. B r i s t e a u a n d t h e s e c o n d a u t h o r ,
leads t o
problems i n v o l v i n g s eco n d - o r d er d i f f e r e n t i a l e q u a t i o n s (whereas t h e o r i g i n a l p r o b l e m i s o f f o u r t h o r d e r ) b u t i n which t h e n o n l i n e a r s t r u c t u r e i s more c o m p l i c a t e d s i n c e it i n v o l v e s t r a n s c e n d e n t a l f u n c t i o n s such a s t
s i n t and t
+
o f which can b e c o s t l y ;
cos t , t h e r e p e t i t i v e evaluation
( 3 . 5 ) , a r e expressed i n t e r m s of nonlinear i n t e g r a l rela-
type (3.4) t i o n s on
+
f u r t h e r m o r e t h e boundary c o n d i t i o n s , o f t h e
(see BOURGAT-DUMAY-GLOWINSKI
$
C1, S e c t i o n 31).
A f u r t h e r argument f o r w o r k i n g d i r e c t l y w i t h x , y i n s t e a d o f
$,
i s t h a t t h i s c o n s t i t u t e s a s t a r t i n g p o i n t f o r t h e s o l u t i o n o f much more c o m p l i c a t e d 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 c e r n i n g
incompressible materiaZs
,
f o r which t h e g e o m e t r i c domain c o n s i d e r e d
i s two- o r t h r e e - d i m e n s i o n a l
( a s p e c i f i c example of s u c h a s i t u a t i o n
w i l l be m e t i n S e c t i o n 4 ) . Numerical s o l u t i o n o f t h e s t a t i c problem.
3.4
(11) Approxima-
tion 3.4.1.
2 A p p r o x i m a t i o n of t h e s p a c e H ( 0 , L ) and t h e f u n c t i o n a l J
Since 8
2 i s a subset o f H ( 0 , L )
x
H
2
(0,L))
an i m p o r t a n t s t e p
towards t h e n u m e r i cal s o l u t i o n o f ( 3 . 2 ) w i l l b e t o d e f i n e a s u i t a b l e 2 N such t o do t h i s w e i n t r o d u c e { s i } i = o approximation of H ( 0 , L ) ; that
si
E
C0,Ll V i ,
s 2O
= 0 , sN = L and s i c
W e t h e n a p p r o x i m a t e H ( 0 , L ) by (3.11)
s ~ + Y ~i = ,O f
...,
N
- 1.
(SEC. 3)
271
INEXTENSIBLE FLEXIBLE PIPELINES
w h e r e , i n g e n e r a l , Pk i s t h e s p a c e o f p o l y n o m i a l s i n a s i n g l e v a r i a b l e of degree
2
w e h a v e Vh
k;
h = max ( s i + 1 i using
-
si ) .
+ 1).
( 0 , L ) and dim Vh = 2 ( N
c H2
I f vh
I n view o f t h e s e d e g r e e s of f r e e d o m it i s o b v i o u s t h a t
Vh
0
t o an a p p r o x i m a t i o n b y f i n i t e e l e m e n t s of H e r m i t e
0
I
I
II
I
I
I I I
I I I
I
I I I
.L
L
si
si+1/2
n
S.
Z-1
We take
Vh i t i s c o n v e n i e n t t o d e f i n e i t by
E
corresponds
t y p e (see F i g u r e 3 . 2 ) .
I
I I
-
L
+
Y
n
si+l
L
S
F i g u r e 3.2 S i n c e Vh i n (3.6)
c
H
2
2
(xi
+
x
yi2) I ~ s i
E
i+1
J introduced
xh, yh,
E
Vh
P 2 , t h e two i n t e g r a l s a p p e a r i n g
c a n b e c a l c u l a t e d e x a c t l y by u s i n g S i m p s o n ' s r u l e on e a c h sub-
J
X
furthermore s i n c e
Vh;
i n t e r v a l [ S ~ , S ~ + i~ I= , O,l, Vh
H2 ( O , L ) , t h e f u n c t i o n a l
x
(0,L)
i s d e f i n e d on Vh
implies t h a t in
Vh
x
Vh
3.4.2
...,
N - 1.
By r e s t r i c t i o n o f
J
to
w e o b t a i n a f u n c t i o n a l which d e p e n d s on 4 ( N + 1) v a r i a b l e s .
A p p r o x i m a t i o n of
6
A s w e are u s i n g an approximation o f Hermite c u b i c t y p e , t h e r e i s
no d i f f i c u l t y i n approximating boundary c o n d i t i o n s such a s ( 3 . 4 ) o r (3.5).
A s regards t h e i n e x t e n s i b i l i t y c o n d i t i o n ( 3 . 1 ) , t h e obvious
choice i s t o use
Since I x i ( s i )
N
N
li = o , {yh(si) Ii = o
c o n s t i t u t e p r e c i s e l y a subset o f
t h e d e g r e e s of f r e e d o m u s e d f o r d e f i n i n g implementation o f
xh
and
yh, t h e p r a c t i c a l
( 3 . 1 2 ) d o e s n o t p r e s e n t any d i f f i c u l t y .
w e have o b s e r v e d t h a t f o r ' s t i f f ' problems ( i . e . w i t h s t r o n g
However,
272
(CHAP. 8 )
FINITE NONLINEAR ELASTICITY PROBLEMS
v a r i a t i o n s i n t h e c u r v a t u r e ) r e l a t i v e l y i n a c c u r a t e r e s u l t s may b e o b t a i n e d by u s i n g ( 3 . 1 2 )
,
unless the d i s c r e t i s a t i o n is refined i n
t h o s e r e g i o n s where t h e c u r v a t u r e v a r i e s s t r o n g l y . can i n c r e a s e
Such a p r o c e d u r e
c o n s i d e r a b l y , and t h e r e f o r e , as f a r as a c c u r a c y i s
N
concerned, w e have found it m o r e co n v en i en t t o approximate
( 3 . 1 ) by
(3.13)
where s 1+1,2 .
=
$
( s i + si+l) (see F i g u r e 3 . 2 ) .
The n u m e r i c a l r e s u l t s p r e s e n t e d i n S e c t i o n 3 w e r e i n f a c t o b t a i n e d by using (3.13) t o approximate ( 3 . 1 ) . approximate (3.1) i n t r o d u c e s around
constraints
Using ( 3 . 1 2 ) N
(resp.
(3.13)) t o
(resp. 2N) quadratic equality
( t h e e x a c t number d e p e n d s on t h e b o u n d a r y c o n d i t i o n s ) .
In t h e following, t h e approximation o f
8 , o b t a i n e d by a p p r o x i m a t i n g B h ; it i s c l e a r t h a t
( 3 . 1 ) by ( 3 . 1 2 ) o r ( 3 . 1 3 1 , w i l l b e d e n o t e d by
i s a c l o s e d subset o f Vh
eh
3.4.3
x
Vh.
A p p r o x i m a t i o n of p r o b l e m ( 3 . 2 )
I n t h e l i g h t of S e c t i o n s 3 . 4 . 1 and 3 . 4 . 2 ,
w e a p p r o x i m a t e ( 3 . 2 ) by
where
I n r e l a t i o n t o t h e existencc? of s o l u t i o n s f o r t h e d i s c r e t e problem ( 3 . 1 4 1 , t h e f o l l o w i n g v a r i a n t o f Theorem 3 . 1 i s p r o v e d i n BOURGATDUMAY-GLOWINSKI El, S e c t i o n 5.31: THEOREM 3 . 2 :
Suppose t h a t
s u f f i c i e n t t h a t lA%I
or (3.5));
6,
i s nonempty ( f o r t h i s i t i s
< L i f t h e boundary c o n d i t i o n s a r e g i v e n by
problem ( 3 . 1 4 ) t h e n a d m i t s a t l e a s t one s o l u t i o n .
(3.4)
(SEC.
3)
3.4.4
273
INEXTENSIBLE FLEXIBLE PIPELINES
C o n v e r g e n c e 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 now i n t r o d u c e t h e n o t i o n of an i s o l a t e d s o l u t i o n f o r t h e l o c a l m i n i m i s a t i o n problem ( 3 . 2 ) :
{x,y}
Let
DEFINITION 3.1:
be a s o l u t i o n of
Ix,?} i s an i s o l a t e d s o l u t i o n of (3.2) I-
of
(3.2);
we s a y t h a t
i f t h e r e e x i s t s a neighbourhood
{ x l u l such t h a t
The convergence r e s u l t s o f t h e f o l l o w i n g theorem a r e proved i n BOURGATDUMAY-GLOWINSKI 11, S e c t i o n 5.4 1 : THEOREM 3.3: t h e d e f i n i t i o n of
Suppose t h a t t h e boundary c o n d i t i o n s a p p e a ri n g i n
6 a r e g i v e n b y (3.4) o r (3.5); s u p p o s e a l s o t h a t
i s d e f i n e d v i a (3.12).
8
We t h e n h a v e :
i f
{ x l u l i s an i s o l a t e d
(3.2), t h e n for h s u f f i c i e n t l y s m a l l t h e a p p r o x i m a t e p r o b l e m (3.14) a d m i t s a s o l u t i o n {Gh,yh} i n t h e n e i g h b o u r h o o d of { ~ , ;~ we l a l s o have s o l u t i o n of
- -
(3.17)
lim Ixh,yhl WO
--
= {x,y}
2
s t r o n g l y i n H ( 0 , ~ X)
H
2
(0,~).
S i m i l a r l y w e may p r o v e : THEOREM 3.4:
T h e o r e m 3.3. on
Suppose t h a t
8
and
4,
a r e a s i n t h e s t a t e m e n t of
T h e n if ({Xh,Gh})h i s a f a m i l y of g l o b a l minima for
J
B h l we h a v e ( a t l e a s t for a s u b s e q u e n c e )
where
{x,?}
r e a l i s e s t h e g l o b a l minimum of
Remark 3.3:
J
on
6.
H e r e , w e have n o t c o n s i d e r e d t h e 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 when
E;,
i s d e f i n e d v i a (3.13) ;
c a s e w e a r e d e a l i n g w i t h a much more d i f f i c u l t problem.
i n such a N e i t h e r have
w e c o n s i d e r e d t h e b e h a v i o u r of t h e a p p r o x i m a t e s o l u t i o n s i n t h e neighbourhood o f t u r n i n g p o i n t s o r g e n u i n e b i f u r c a t i o n p o i n t s ;
in this
d i r e c t i o n w e mention t h e works of F. K I K U C H I [ll, YAMAGUTI-FUJI1 [11, KESAVAN 111 I BREZZI-RAPPAZ-FAVIART C11- C31.
274
(CHAP. 8 )
FINITE NONLINEAR ELASTICITY PROBLEMS
3.5
N u m e r i c a l s o l u t i o n o f t h e s t a t i c problem. methods o f s o l u t i o n
(111) I t e r a t i v e
I n t h i s s e c t i o n w e f o l l o w BOURGAT-DUMAY-GLOWINSKI
11, S e c t i o n 6 1 .
General n o t e s and s y n o p s i s
3.5.1
Although i t i s c l e a r t h a t t h e c o n s i d e r a t i o n s d i s c u s s e d b e l o w s h o u l d s t r i c t l y have been developed f o r t h e approximate problems, w e have n o n e t h e l e s s chosen t o use t h e c o n t i n u o u s problem s i n c e b o t h t h e s e t y p e s o f p r o b l e m h a v e t h e same n o n l i n e a r s t r u c t u r e ;
furthermore,
t h e formalism of t h e co n t i n u o u s problem i s s i m p l e r .
The n o n l i n e a r
equality constraints (3.12),
(3.1)
( o r t h e i r finite-dimensional variants
( 3 . 1 3 ) ) c o n s t i t u t e t h e major d i f f i c u l t y t o b e surmounted i n
t h e n u m e r i c a l s o l u t i o n of t h e l o c a l m i n i m i s a t i o n p r o b l e m ( 3 . 2 ) , when w o r k i n g d i r e c t l y w i t h t h e d i s p l a c e m e n t s x , y.
Schematically, f o r
s o l v i n g ( 3 . 2 ) and i t s f i n i t e - d i m e n s i o n a l v a r i a n t s , two f a m i l i e s o f methods a r e a v a i l a b l e : Methods u s i n g m u l t i p l i e r s and p e n a l i s a t i o n
(i)
A s was s e e n i n S e c t i o n 3 . 3 ,
function
X
w e can a s s o c i a t e a L a g r a n g e m u Z t i p l i e r
with t h e e q u a l i t y c o n s t r a i n t (3.1) ;
has t o solve, with respect t o system (3.8) - (3.10) i n practice,
{x,y,A},
i n d o i n g t h i s one
the nonlinear d i f f e r e n t i a l
(actually its finite-dimensional
t h e d i s c r e t e v a r i a n t s of
(3.8)
-
variants) ;
(3.10) can b e s o l v e d
by t h e v a r i a b l e m e t r i c methods d e v e l o p e d by POWELL 121 which g e n e r a l i s e t h e w i d e l y known method o f D a v i d o n - F l e t c h e r - P o w e l l
(see a l s o MATTHIES-
STRANG I 1 1 f o r some s i m i l a r methods d i r e c t e d a t t h e s o l u t i o n o f p r o -
We h a v e t h e i m p r e s s i o n t h a t t h e
blems i n N o n l i n e a r M e c h a n i c s ) .
methods m e n t i o n e d above are t r i c k i e r t o implement t h a n t h e methods d e s c r i b e d below i n S e c t i o n 3 . 5 . 2 ,
and a l s o t h a t t h e y h a v e much
g r e a t e r computer memory r e q u i r e m e n t s when l a r g e - s c a l e
problems a r e
being treated. It i s n a t u r a l t o a s s o c i a t e with t h e Lagrangian
(3.7) t h e augmented Lagrangian
dr
A
d e f i n e d by
d e f i n e d ( w i t h r > 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 )
FINITE N O N L I N E A R ELASTICITY PROBLEMS
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
FINITE NONLINEAR ELASTICITY PROBLEMS
(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)
INEXTENSIBLE FLEXIBLE PIPELINES
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 )
FINITE NONLINEAR ELASTICITY PROBLEMS
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 )
INEXTENSIBLE FLEXIBLE PIPELINES
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
FINITE N O N L I N E A R ELASTICITY PROBLEMS
(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)
INEXTENSIBLE FLEXIBLE PIPELINES
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)
FINITE N O N L I N E A R ELASTICITY PROBLEMS
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
FINITE NONLINEAR ELASTICITY PROBLEMS
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
FINITE NONLINEAR ELASTICITY PROBLEMS
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,$,g)
=):(TI
+
5 IIVV+I-GII~ =2 I I
3
%
,I
--
U*(Vv+I-G) z z dx.
This leads to the following formulation of the elastostatic problem
1 Find
I
(4.12)
{y,E,x> -
E
W = X X Y
X
(L2(n))NXN, t h e s t a t i o n a r y p o i n t
I
/
on
W o f t h e augmented
Lagrangian
F,
where, in (4.12), y =
{GIG<
( ~ ~ ( n ) ) G-I ~ ' E~ (L2(n))NXN, ,
ii;i
4.2.2.4
det G = 1 u.e.3
.
I I
On some r e l a t i o n s b e t w e e n f o r m u l a t i o n s (4.12)
(4.6) , (4.8) and
The following results are proved in LE TALLEC C11, C21: (i) (ii) (iii)
There is equivalence between (4.8) and (4.12), Any "regular" solution of (4.6) is a solution of (4.8) and (4.12). If the functional IT is c o n v e x (which is the case for a Mooney-Rivlin material if N = 2) then any solution {u,g,i> of (4.12) is such that y (locally) minimises +Rr(y,z,?) on X ; likewise for r sufficiently large, any solution of (4.12) is such that f minimises 5 +Zr(u,$,,X) (locally on Y). I
Remark 4.2:
-
If
{u,F,A} is a solution of (4.12) the condition I
Z
.
5
2, dr(yfF,i) = 0 implies that I
2-D
(SEC. 4 )
INCOMPRESSIBLE MATERIALS
289
(4.13)
I n view of
Kirchhoff
(4.13)
,
can be seen t o b e t h e p a r t of t h e f i r s t Piola-
_X
t e n s o r corresponding t o t h e i n c o m p r e s s i b i l i t y .
It should
y i e l d s t h e stress
a l s o b e n o t e d t h a t any a l g o r i t h m s o l v i n g ( 4 . 1 2 ) field directly. 4.3
S o l u t i o n of p r o b l e m ( 4 . 1 2 )
A f i r s t a l g o r i t h m for s o l v i n g
4.3.
(4.12)
Once a g a i n , w e u s e 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 , of Section 2.4
of t h e present chapter;
and
i n t h e n o t a t i o n of S e c t i o n
4.2 we o b t a i n t h e following:
'A
(4.14)
t h e n for
n
g i v e n i n (L
2 0,
-A n
2
( a ) )NxN ,
b e i n g known, d e t e r m i n e
unlFn I
(4.16)
-.An+' I
Remark 4 . 3
=
-An
-
and
hn+l
-
I
by
p (Vun+_I-Fn), p > 0. -I
I
- z
Problem ( 4 . 1 5 ) i s e q u i v a l e n t t o t h e n o n l i n e a r system
w h i c h , when s o l v e d by b l o c k r e l a x a t i o n , l e a d s t o t h e a l g o r i t h m d e s c r i b e d i n S e c t i o n 4.3.2 4.3.2
below.
A second a l g o r i t h m f o r s o l v i n g
(4.12)
T h i s t i m e , w e employ a l g o r i t h m ALG2 o f C h a p t e r 111, S e c t i o n 3 . 2 ,
FINITE NONLINEAR ELASTICITY PROBLEMS
290
(CHAP. 8 )
and o f 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 , namely: -1
(4.19)
_xO
n- 1 n 2 0, u and
then f o r An+l by
g i v e n in
-
given i n X ,
An
b e i n g known, d e t e r m i n e
I
-
Fn,
2n
and
Itl
(4.20)
(4.21)
in+' = in
(4.22)
I
-
~(V~"+I-F") I -
Z
Z
, p > 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 )
FINITE NONLINEAR ELASTICITY PROBLEMS
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
FINITE N O N L I N E A R ELASTICITY PROBLEMS
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,
FINITE NONLINEAR ELASTICITY PROBLEMS
296
t h e n , for n
0,
2
5n- 1 and
in
b e i n g known, d e t e r m i n e
(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
FINITE N O N L I N E A R ELASTICITY PROBLEMS
(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
METHOD O F MULTIPLIERS & V.I.'s
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,
METHOD O F MULTIPLIERS & V.I.'s
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) <
(3.2)
311
I N DUALITY
+ml ,
t h e i n t e r i o r o f which w e s h a l l assume t o b e nonempty.
W e also
assume t h a t t h e f o l l o w i n g q u a l i f i c a t i o n c o n d i t i o n i s s a t i s f i e d : exists
v
i n t ( d o m ( A ) ) s u c h t h a t Bvo
E
E
there
dom(F).
We consider a variationat inequatity i n t h e following general form:
Find
such t h a t
UEV
(3 3) f
13w
E
such t h a t
A(u)
We r e c a l l t h a t i f A G : V
-f
]--,+-I
= aG,
(w,v-u)"
+ F(Bv)-F(Bu) t 0
Vv
E
V.
a G being t h e subgradient of a function
which i s c o n v e x , p r o p e r and lower s e m i - c o n t i n u o u s ,
t h e n t h e i n e q u a l i t y (3.31 i s e q u i v a l e n t t o t h e c o n v e x v a r i a t i o n a l
.
p r o b l e m (1.10) , i e.
Inf IF (BV) +G(v) VE
v
1,
i n t r o d u c e d by ROCKAFELLAR 151 t o g e n e r a l i s e F e n c h e t ' s d u a l i t y t h e o r y ( t h i s p r o b l e m i n c l u d e s i n p a r t i c u l a r t h e o r d i n a r y convex p r o g r a m i n g problems
(1.5)).
The v a r i a t i o n a l i n e q u a l i t y
a p a r t i c u l a r case o f and w i t h
(3.3) with
V = H,
with
(1.6) B
also constitutes
t h e i d e n t i t y mapping
F = IK, 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 c l o s e d convex s e t
K
w i t h nonempty i n t e r i o r d e f i n e d b y
(3.4)
I,(v)
=
i
i f v r K,
0 + m
otherwise.
T r a n s p o s i n g t h e a n a l y s i s o f MOSCO 111 i n t o t h i s f o r m a l i s m , w e a s s o c -
i a t e w i t h (3.3) t h e dual v a r i a t i o n a t i n e q u a l i t y : Find
hcn
such t h a t
(3.5) 3p
E
A~(A)
i n ( 3 . 5 ) F* : H
such t h a t
+ I-m,+-l
(~,LI-X)~+F*(U)-F*(X)
5
0 VU
E
n
;
d e n o t e s t h e convex f u n c t i o n c o n j u g a t e t o
F (see ROCKAFELLAR 1 4 1 EKELAND-TEMAM [I]) d e f i n e d o n
H
by
312
METHOD OF MULTIPLIERS & V . I . ' s
t AB(p) = { q E HI ~ V E Vs u c h t h a t
(3.7)
where
Bt
d e n o t e s t h e o p e r a t o r from
q =
H
-BV,
into
(CHAP.
9)
-B~~,A(v)},
d e f i n e d by
V
B~ = A ; ' ~ B ~ ~ A ~ .
The t i t l e o f ' d u a l i n e q u a l i t y '
i s j u s t i f i e d by t h e f o l l o w i n g
result: THEOREM 3 . 1 : A vector u E V i s a solution of the variational i n e q u a l i t y ( 3 . 3 ) i f and onZy i f t h e r e e x i s t s a s o l u t i o n A of t h e i n e q u a l i t y ( 3 . 5 ) s u c h t h a t -B t X E A ( u ) . Furthermore, u and A
a r e r e s p e c t i v e l y s o l u t i o n s of ( 3 . 3 ) and ( 3 . 5 ) i f and o n l y i f -BtA
E
A ( u ) and -Bu
and we h a v e t h e i d e n t i t y
A:(A)
E
F i r s t , w e n o t e t h a t t h e i n e q u a l i t y ( 3 . 3 ) can b e w r i t t e n
Proof:
3 w c A ( u ) such t h a t
S i n c e t h e domain o f of
H
where
F
-WE
a(F.B)(u).
h a s a nonempty i n t e r i o r , t h e r e e x i s t s a p o i n t
w e t h e r e f o r e h a v e (see
i s f i n i t e and c o n t i n u o u s ;
F
EKELAND-TEMAM 111 C h a p t e r 1, P r o p o s i t i o n 5 . 7 ) a(F0B) (u) = BtaF(Bu)
s i n c e w e have chosen t o d e f i n e t h e s u b g r a d i e n t o p e r a t o r from
of
F
i n t o subsets of
H
i s a n o p e r a t o r from
H
aF* o f t h e c o n j u g a t e f u n c t i o n
H
aF
(whereas t h e s u b d i f f e r e n t i a l
i n t o subsets of F*
a s a multivalued The s u b g r a d i e n t
H I ) .
defined i n (3.6) i s i n f a c t i d e n t i -
cal t o t h e inverse, i n t h e sense of multivalued o p e r a t o r s , of (see EKELAND-TEMAM equivalent t o q
E
111); aF* ( p )
it f o l l o w s from t h i s t h a t p
.
The i n e q u a l i t y
(3.3)
E
aF
aF(q) is
i s t h e r e f o r e equiva-
lent t o 3wcA(u), 3 A c H
such t h a t
-BtA
= w
t w e h a v e -Bu f r o m t h e d e f i n i t i o n ( 3 . 7 ) o f AB,
w i t h B u c aF*(A) ; E
At B(A)
and
A
i s such
METHOD OF MULTIPLIERS
4)
(SEC.
313
which is an equivalent formulation of the inequality (3.5). The identity (3.8) follows immediately from the property Bu E aF* ( A ) .
rn
If A = aG, the dual operator At = BoaG-lo(-Bt) is in fact the subgradient of the function G*a(-B ) (since the qualification condition is satisfied) and the dual variational inequality ( 3 . 5 ) is equivalent to the variational problem
P
the d u a l p r o b Z e m , i n t h e s e n s e of F e n c h e Z , of the variational problem (3.4).
THE METHOD OF MULTIPLIERS FOR VARIATIONAL INEQUALITIES
4.
The variational inequality Find
UEV
such t h a t
(4.1)
3 w ~ A ( u ) , {W,V-U)~+F(BV)-F(BU) 2 0
YvcV
is equivalent to the multivalued equation Find
u EV s u c h t h a t
(4.2)
T(4,
0
where since dom(F) has nonempty interior, the operator T is defined by T
(4.3)
=
A + B aF0 ~ ~B
E
since
aF(Bv') F
V
+
2
V
.
By hypothesis, A is maximal monotone. BtoaFoB; in fact, for all v E V, v' E V 1~'
:
we have
The same applies for and all E aF(Bv) a n d
(Btl-l-Btu',~-~l)V = (!J-~',BV-BV')~ 2 0
is convex and proper;
the maximal monotonicity then
METHOD OF MULTIPLIERS
314
&
V.I.'s
(CHAP. 9)
follows from the lower semi-continuity of F. The qualification condition and the property int (dom(F)) = int (dom(aF)) guarantee that T, the sum of two maximal monotone operators, is itself maximal monotone (see BREZIS C11, Corollary 2 . 7 ) . Similarly, the dual variational inequality;
is equivalent to (4.5)
0
EU(X)
where the operator U : H t
for
*
U = AB+aF
(4.6)
-t
2 H is defined by the sum
.
The subgradient aF* = (aF) is maximal monotone. Furthermore, t p and p ' E H and for all q E -4; p ) and q' E AB(p') we have (q-q',P-lJ')H
with w
=
-Btp
t
=
(v-v',w-w')v,
A(v), w'=- B p '
(q-q',v-u')H'
t
A(v'
,
so that
0,
which proves the monotonicity of .:A
In order to prove the maximal we shall
monotonicity of A; we adopt a suppiementary assumption; assume henceforth either that (4.7)
A i s coercive
(in the sense of ( 2 . 9 ) ) ,
or that (4.8)
B ~ Bi s a n i s o m o r p h i s m o f
PROPOSITION 4.1: i s satisfied. maximal mono t o n e . (4.8)
V.
Suppose t h a t one o f t h e a s s u m p t i o n s ( 4 . 7 ) Then t h e o p e r a t o r
:A
defined b y (3.7)
is
or
(SEC.
4)
METHOD OF MULTIPLIERS
315
We shall now show that for all r > 0 the multivalued
Proof:
equation
admits for all y E H a unique solution I. t exists, satisfies, by definition of AB,
The vector p
E
H, if it
y = p-rBv,
with v such that -Bt p valued equation (4.10)
-Bty
E
E
A(v) ;
v
is thus a solution of the multi-
(A+rBtB) (v)
which, for all y E H, admits a unique solution if ( 4 . 7 ) or ( 4 . 8 ) is satisfied. We then deduce the unique solution of (4.9) to be
(4.11)
U = y+rB(A+rBtB)-l (-Bty),
t and w e also deduce the maximal monotonicity of AB Proposition 2.2).
(see BREZIS 111,
Suppose that the variational inequality ( 4 . 1 ) admits a solution int (dom(A)) ; Theorem 3.1 indicates that the dual inequality t ( 4 . 4 ) possesses a solution A and X E intCdom(AB)l n dom(aF*); hence the maximal monotonicity of U ( s e e BREZIS C11). We can therefore solve the "dual" multivalued equation ( 4 . 5 ) by using the proximalpoint algorithm for U; given a nondecreasing sequence {rnl of positive numbers and an initial approximation Xo E H, we generate u
E
the sequence (4.12)
{An}
via the recurrence
X n + ~= J"~(x")
, n=0,1, ...
.
-1 By definition of the resolvent Jrn = (I+rnU) , we have U
with un+', (4.14)
pn+' -BtXn+'
such that E
A(u*+l),
316
METHOD O F MULTIPLIERS
X"+I
(4.15)
E
&
V.I.'s
(CHAP.
9)
aF(p"+l) ;
and (4.15), w e c h a r a c t e r i s e u
inserting (4.13) into (4.14)
n+l
, pn + l
a s a s o l u t i o n of
W e can t h e r e f o r e d e s c r i b e t h e a l g o r i t h m i n t h e f o l l o w i n g form, which g e n e r a l i s e s 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
(4.1)
Uzava's
a t g o r i t h m ALGl f o r t h e a u g m e n t e d L a g r a n g i a n ( 1 . 1 3 ) a s s o c i a t e d w i t h t h e v a r i a t i o n a l problem (1.10)
.
M u l t i p l i e r methods 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 ( A L G 1 ) :
Given a n o n d e creasing sequence i n i t i a l a p p r o x i m a t i o n 'X
t
H,
of p o s i t i v e n u m b e r s , and an {An) v i a t h e
{rnl
w e deduce a se q u e n c e
fottowing recurrence: (i) Given
3w""
(4.18)
f i n d {un+l,pn+l}
An,
i
E
A(un+')
E
V
x
H satisfying
such t h a t
(wn+I ,v)V+(An-rnpn+l+rnBu n+ I , B v ) ~= 0
F(pn+')-(An,pn+l)H
+
$ IBun+l-P
n+l
Vv
E
V,
I
(4.19) 5
F(q)-(?-.n,q)H
+
2 IBun+l-qlH
Vq
E
H.
(ii) U p d a t a t h e m u l t i p l i e r s :
A t each i t e r a t i o n w e t h e r e f o r e have t o s o l v e a v a r i a t i o n a l e q u a t i o n
in
v
coupled
w i t h a m i n i m i s a t i o n problem i n
q.
The convergence
of t h e method f o l l o w s from t h e convergence of 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 f o r t h e maximal monotone o p e r a t o r
U.
METHOD OF MULTIPLIERS
(SEC. 4 )
THEOREM 4 . 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
a t l e a s t one s o 2 u t i o n
,
(4.7)
317
v
E
(4.8) i s s a t i s f i e d .
i n t ( d o m ( A ) ) and t h a t one of
(4.1)
admits
t h e assumptions
T h e n t h e m e t h o d of m u l t i p l i e r s A L G l i s
w e l l d e f i n e d and g e n e r a t e s a s e q u e n c e
{An) c o n v e r g i n g w e a k l y t o
A,
a s o l u t i o n of t h e 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 ( 4 . 4 ) . If, i n A-' i s c o e r c i v e w i t h modulus y , which i m p l i e s t h a t A i s L i p s c h i t z - c o n t i n u o u s w i t h c o n s t a n t y-', t h e n t h e sequence {An) c o n v e r g e s s t r o n g l y t o A, t h e u n i q u e s o l u t i o n of ( 4 . 4 ) and
addition,
If rn
+
+
m,
Proof: 2.1,
t h e n t h e convergence of t h e sequence
{An] i s s u p e r l i n e a r .
The f i r s t p a r t of t h e theorem i s a c o r o l l a r y of Theorem
w h i l s t t h e e s t i m a t e of t h e r a t e of convergence f o l l o w s from
Theorem 2 . 2 .
Remark 4 . 1 : rezaxation,
W e can a l s o a p p l y t h e p r o x i m a 2 - p o i n t
algorithm with
( 2 . 7 ) , t o t h e dual multivalued equation (4.5) ;
a method o f m u l t i p l i e r s , i n which t h e u p d a t i n g f o r m u l a ( 4 . 2 0 )
we obtain is
changed i n t o
Remark 2.3 a l l o w s u s t o c o n c l u d e t h a t t h e method c o n v e r g e s f o r a l l p n r pn = u r n , w i t h 0 < w < 2 ;
r e s u l t o f C h a p t e r 111.
w e have t h u s g e n e r a l i s e d t h e convergence The a n a l y s i s of Remark 2 . 4
indicates that the
r a t e of convergence o f t h e method o f m u l t i p l i e r s can b e a c c e l e r a t e d by s u i t a b l y c h o o s i n g
pn
in (4.22);
assuming
r
f i x e d , an a p p r o x i -
m a t i o n of t h e o p t i m a l p a r a m e t e r i s g i v e n by p * = w * r = r + aM2 > r; t h i s r e s u l t s h o u l d b e compared w i t h t h e a n a l y s i s c a r r i e d o u t 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 .
Remark 4.2:
For t h e s o l u t i o n of
( 4 . 1 8 ) , ( 4 . 1 9 1 , w e can o b v i o u s l y
use t h e u s u a l s u c c e s s i v e r e l a x a t i o n method:
by p e r f o r m i n g o n l y a
s i n g l e i n n e r i t e r a t i o n , we o b t a i n a l g o r i t h m ALG2, t o which w e s h a l l r e t u r n i n S e c t i o n 5 below.
Remark 4.3:
W e can a l s o a t t e m p t t o a p p l y 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 r e l a t i v e t o t h e maximal monotone o p e r a t o r
T
d e f i n e d by
METHOD OF MULTIPLIERS & V. I. ' s
318
(CHAP.
9)
(4.3), t o solve the multivalued equation ( 4 . 2 ) , equivalent t o t h e inequality ( 4 . 1 ) .
Thus, g i v e n a n o n d e c r e a s i n g sequence
{ r n } of
p o s i t i v e numbers and an i n i t i a l a p p r o x i m a t i o n uo, w e d e f i n e a sequence
{un} i n V Given
un
by t h e f o l l o w i n g r e c u r r e n c e :
find
Ywn+l
un+l L *(un+'),
satisfying the variational inequality pn+l
E
such t h a t
aF(Bu"+')
(4.23) (w n+l
-1 (un+l -,P) ,v-u n+ 1 )V+(pn+l , B ~ - B ~ " + ' ) ~ Lo
vv
E
v.
Problem ( 4 . 2 3 ) i s d i f f i c u l t t o s o l v e s i n c e i t i n v o l v e s A , B and F a l l a t t h e same t i m e .
The i n t r o d u c t i o n of t h e d u a l i t y and t h e method o f
m u l t i p l i e r s e n a b l e s u s t o g e t round t h i s d i f f i c u l t y v e r y e f f e c t i v e l y .
5.
DECOMPOSITION BY MULTIPLIERS: The 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 multivalued equations ( 4 . 2 )
Find
( I ) ALTERNATING-DIRECTION
(4.1)
and ( 4 . 4 )
METHODS
are equivalent t o
and ( 4 . 5 ) , which a r e of t h e form:
X E X such t h a t
(5.1) 0E T W ,
where
T
i s a maximal monotone o p e r a t o r s u c h t h a t
T
(5.2)
= R+S,
where R and S d e n o t e maximal monotone o p e r a t o r s on X. The p r o x i m a l - p o i n t
a l g o r i t h m r e q u i r e s t h e c a l c u l a t i o n of J G which
may b e much more c o m p l i c a t e d t h a n t h a t o f J i and JE. LIONS and B.
R e c e n t l y , P.L.
MERCIER 111 have proposed t w o a l g o r i t h m s which g e n e r a l i 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 t h e m u l t i v a l u e d e q u a t i o n
form ( 5 . 1 )
when T i s g i v e n by ( 5 . 2 1 , and which i n v o l v e o n l y t h e r e s o l v e n t s o f R and S.
I n p a r t i c u l a r , t h e y a n a l y s e an a l g o r i t h m o f t h e DOUGLAS-
RACHFORD t y p e 111:
(5.3)
tn+' = Ji(2Ji-I)tn
+
(I-Ji)tn, n-0, I , . .. ,
(SEC.
5)
DECOMPOSITION ( I ) A . D .
319
METHODS
and an a l g o r i t h m o f t h e PEACEMAN-RACHFORD t y p e C11: tn+' = (2Ji-I) (2Ji-I)t"
(5.4)
, n=O, I , .
..
;
knowing an a p p r o x i m a t i o n xo of a s o l u t i o n a l g o r i t h m s must b e i n i t i a l i s e d a t
to
x
of
(5.1), these
such t h a t
xo = Ji(to),
(5.5)
that is to =
(5.6)
xo + rso, s 0 c s(x0)
We can a p p l y t h e s e a l g o r i t h m s t o t h e m u l t i v a l u e d e q u a t i o n ( 4 . 5 ) equivalent t o the dual v a r i a t i o n a l inequality ( 4 . 4 ) ; ponds t o a d e c o m p o s i t i o n o f t y p e
on t h e H i l b e r t s p a c e H .
t h i s corres-
(5.2) with
We t h u s o b t a i n two v a r i a n t s of t h e method o f v and q a r e now d e -
m u l t i p l i e r s (ALG1) i n w h i c h t h e p r o b l e m s i n coupzed.
The Douglas-Rachford v a r i a n t of t h e method of m u l t i p l i e r s : a l g o r i t h m ALG2
5.1
Ao of t h e s o l u t i o n of t h e d u a l i n e q u a z i t y to s u c h t h a t Xo = J g ( t o ) , t h a t i s
G i v e n a?? a p p r o x i m a t i o n (4.4)
we d e f i n e
w i t h po (5.9)
E
aF* (Ao),
and hence s u c h t h a t
AD E aF(po).
I n g e n e r a l we p u t (5.10)
X"
= J;(t")
;
by d e f i n i t i o n of t h e r e s o l v e n t Jg, t h e r e e x i s t s p that
n
E
aF*(Xn) such
METHOD O F MULTIPLIERS & V . I . ' s
320
(CHAP. 9 )
(5.11)
and
A"
(5.12)
E
aF (p")
.
Using t h e f o r m u l a ( 4 . 1 1 ) which c h a r a c t e r i s e s J i = ( I + r A B ) one o f t h e a s s u m p t i o n s
-'
when
(4.7), (4.8) is s a t i s f i e d , t h e recurrence
(5.3) defines tn+ 1
(5.13)
=
n+ 1 A n + rBu ,
=
(A+rBtB)-' (rBtpn-BtAn).
with (5.14)
u
n+ 1
Inserting relation
(5.11),
f o r t h e index
t h e recurrence formula f o r t h e sequence (5.15)
=
An
n + l , i n t o (5.13) we obtain
{A"}:
+ r(Bu n+l -p n+l ) ,
which i s i n f a c t i d e n t i c a l t o t h e f o r m u l a ( 4 . 2 0 )
f o r updating t h e
W e n o t e t h a t by m u l t i p l i e r s , when t h e p a r a m e t e r r i s f i x e d . n+l recurrence p E aF* (An+') and t h e r e f o r e s a t i s f i e s
An + rBun+ 1
(5.16)
E
aF(p"+')
+ rpn+l,
t h i s b e i n g a m u l t i v a l u e d e q u a t i o n which h a s t h e u n i q u e s o l u t i o n pn+l = ( F + r I ) - ](h"+rBun+l).
(5.17)
W e can t h u s use ( 5 . 1 4 ) , (5.151,
(5.17) t o d e s c r i b e algorithm
(5.3)
i n a form s i m i l a r t o t h e method o f m u l t i p l i e r s , a s below. 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
The D.R.
Given that
ho
r > 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<
(2Ji-I)tlH,
so t h a t
I tn+'-t ;1
(5.44)
5
4 I Jg( tn)-Ji( t)
;
I -
4(Ji(tn)-J:(t),
tn-t)H+
I tn-t IH.2
Using o n c e a g a i n t h e a r g u m e n t i n t h e p r o o f of P r o p o s i t i o n 2 . 1 ,
and
u s i n g c o n d i t i o n ( 5 . 2 4 ) , it c a n b e shown t h a t
(5.45)
IJi(t")-Ji(t)
I2 HS
(Ji(tn)-Ji(t),tn-t)H
-
ralXn-XIi ;
i n s e r t i n g ( 5 . 4 5 ) i n t o ( 5 . 4 4 ) and u s i n g ( 5 . 4 3 ) , t h e e s t i m a t e c a n be e s t a b l i s h e d . r* =
1, i . e .
(1 - y 3 1 / 2
The c o n s t a n t
'I2 2)
(1-
(5.41)
is m i n i m a l for
(l+yr) and i s t h u s e q u a l t o
t h e same v a l u e a s f o r A L G 2 ,
.
Comparison of
8
( 5 . 2 6 ) and ( 5 . 4 2 ) would a p p e a r t o i n d i c a t e t h a t
ALG3 i s f a s t e r t h a n ALG2.
Naturally,
t h e s e estimates are n o t
p r e c i s e enough t o allow us t o r e a c h s u c h a c o n c l u s i o n d e f i n i t i v e l y .
6.
DECOMPOSITION BY MULTIPLIERS:
(11)
PROJECTION METHODS
Once a g a i n , w e 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
(SEC.
6)
where
T
(6.2)
DECOMPOSITION
(11) PROJECTION METHODS
327
i s d e f i n e d a s t h e sum of two maximal monotone o p e r a t o r s : T = R+S.
W e now assume t h a t constant
R
is single-valued
M, and w e c o n s i d e r f o r
Remark 6 . 1 :
and L i p s c h i t z - c o n t i n u o u s w i t h
r > 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 <
converges s t r o n g l y t o
r < 2uM- 2
sequence Proof:
.
If
R-l
t h e n t h e sequence
a),
x
E
H , satisfying
If
R
x
for a l l
{xn} g e n e r a t e d b y
(6.1) f o r a l l
i s coercive ( w i t h modutus
{xnl converges weakly t o
( w i t h c o n s t a n t MI
M-'),
then the
0 < r < 2/M.
i s L i p s c h i t z - c o n t i n u o u s and c o e r c i v e , t h e n ( I - r R )
i s a s t r i c t c o n t r a c t i o n f o r a l l 0 < r < 2aM-' and w e deduce t h e f i r s t p a r t of t h e theorem by n o t i n g t h a t J; is a l s o a c o n t r a c t i o n . L e t us now assume t h a t
R-l
i s c o e r c i v e ( w i t h modulus
M-I),
i.e.
328
METHOD OF MULTIPLIERS & V . I . ' S
(CHAP.
9)
Then
I (I-rR) (x') -
(I-rR) (x)
I x2 5 /x'-xlx2 +
(Mr2-2r) (R(x')-R(x)
(6.7) =
F o r 1/M
2
r < 2/M, +
{x"} t o a s o l u t i o n
1 M,
t h e monotonicity of
0 and
\(I-rR)(x')-(I-rR)(x)
(6.8)
and ( I - r R ) Ixn+L n x
Ix
MERCIER
+
2 (Mr-I) Ix'-x/x.
w e t h e n d e d u c e , f r o m O p i a l ' s lemma, t h e weak x of ( 6 . 1 ) .
t h a t Ixn+l-xnIX
5
X
( 6 . 7 ) i m p l i e s , s i n c e Ji i s a f i r m c o n t r a c t i o n ,
convergence o f For 0 < r
x'-x)
(2-Mr)((1-rR)(x1)-(I-rR)(x),
,X'-X)~
I x2
R
l e a d s t o t h e u p p e r bound
((I-rR)(x')-(I-rR)(x),~'-x)~,
is a f i r m contraction:
w e t h e r e f o r e conclude t h a t n [ x 1 (see a l s o
0 and t h a t w e h a v e weak c o n v e r g e n c e of
-f
c 11).
We i l l u s t r a t e t h i s a l g o r i t h m by a p p l y i n g i t f i r s t t o t h e v a r i a tional inequality single-valued
( 1 . 5 ) i n t h e p a r t i c u l a r case where
o p e r a t o r from
into
V
A
is a
V, w h i c h i s monotone a n d s e m i -
c o n t i n u o u s ( a n d t h e r e f o r e maximal monotone) :
Find
u c K such t h a t
(A(u) ,V-U)"
2
0 Yv
E
K ;
the variational inequality (6.9)
(6.10)
R = A
and
is equivalent t o equation (6.1) with
s = a1
K'
JE
L E M M A 6.1: The r e s o l v e n t of S = a I K , t h e s u b g r a d i e n t of t h e i n d i c a t o r f u n c t i o n IK o f a c o n v e x s e t K w h i c h is a n o n e m p t y closed subset of
V,
is i n d e p e n d e n t of
o p e r a t o r o f p r o j e c t i o n o n t o K , i.e. (6.11)
Proof:
From Remark 2 . 1 ,
w e have
r > 0 and e q u a l t o t h e
(SEC. 6 )
DECOMPOSITION (11) PROJECTION METHODS
Jr(w)
329
= Are rnin{T 1 /v-wIv 2 + rIK(v)l
S
v t v 2
I
= Arg min v c K
l v - v l v = P,(w). m
Algorithm (6.3) t h u s l e a d s t o t h e r e c u r r e n c e (6.12)
un+ 1 = PK(un-rA(un))
n=O, 1 , .
..
;
V@ of a c o n v e x and c o n t i n u o u s l y d i f f e r e n IR , w e r e c o g n i s e i n ( 6 . 1 2 ) t h e g r a d i e n t p r o j e c t i o n m e t h o d of GOLDSTEIN C11, t h e c o n v e r g e n c e p r o p e r t i e s of which a r e o b t a i n e d a s a c o r o l l a r y of Theorem 6.1. when
A
is the gradient
tiable functional
I$ : V
+
L e t u s now c o n s i d e r t h e d u a l v a r i a t i o n a Z i n e q u a l i t y
t h i s i s t h e p a r t i c u l a r case o f mapping of
V,
and a
(3.5)
i n which
B
of ( 6 . 9 ) :
is the identity
t h e conjugate f u n c t i o n a l of IK, i s t h e
s u p p o r t f u n c t i o n ( 5 ) K L f t h e convex s e t K. i s e q u i v a l e n t t o a m u l t i v a l u e d e q u a t i o n on
The i n e q u a l i t y ( 6 . 1 3 ) V
o f t h e form (6.1).,
( 6 . 2 ) w i t h , i n t h i s case, (6.14)
R = -A
W e suppose t h a t A-1
-1
A
*(-I),
s
=
aaK.
i s c o e r c i v e ( w i t h modulus a ) which i m p l i e s t h a t
i s s i n g l e - v a l u e d and L i p s c h i t z - c o n t i n u o u s
I t c a n b e shown, a s i n Lemma 6 . 1 ,
(with constant l / a )
t h a t t h e r e s o l v e n t J S of aaK i s
i n d e p e n d e n t o f r:
a l g o r i t h m (6.3) then l e a d s t o t h e recurrence
.
33 0
METHOD O F MULTIPLIERS & V . I . ' s
(CHAP.
9)
which can b e e x p r e s s e d i n a form s i m i l a r t o t h a t of t h e method o f multipliers. Method of m u l t i p l i e r s w i t h p r o j e c t i o n ( a l g o r i t h m A L G 4 ) :
Given {un},{Xn}
r > 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.
This Page Intentionally Left Blank
REFERENCES
AMARA M., JOLY P., THOMAS J.M. ill A mixed finite-element method for solving transonic flow equations, Comp. M e t h . A p p l . Mech. Eng. (to appear). ANTMAN S.S. 111 Kirchoff's problem for nonlinearly elastic rods, Quart. A p p l . M a t h . , 32, (1974), pp. 221-240. ANTMAN S.S., ROSENFELD 111 Global behaviour of buckled states of nonlinearly elastic rods, SIAM R e v . , 20, (1975), pp. 513-566. ARROW K.J., HURWICZ., UZAWA H. 111 S t u d i e s i n n o n l i n e a r p r o g r a m m i n g , Stanford University Press, 1958. AXELSSON O., 111 A class of iterative methods for finite-element equations, Comp. M e t h . A p p l . Mech. E n g . , 9, (1976), pp. 123-137. C2l A generalized SSOR method, BIT, 13, (1972), pp. 443-457. Error bounds for finite-element methods, Numer. BABUSKA I. [11 M a t h . , 1 6 , (1971), pp. 322-333. BALL J.M. C11 Convexity conditions and existence theorems in nonlinear elasticity, A r c h . R a t . Mech. A n a l . , 63, (1977), pp. 307403. BEGIS D. 111 E t u d e n u m e ' r i q u e du c o m p o r t e m e n t d ' u n f l u i d e de B i n g k a m , IRIA-Laboria Report No. 42, December, 1973. 121 E t u d e n u m e ' r i q u e d e 1 ' 6 c o u l e m e n t d ' u n f l u i d e v i s c o p l a s t i q u e d e B i n g h a m p a r u n e me'thode d e l a g r a n g i e n a u g m e n t ; , IRIA-Laboria Report No. 355, June 1979. BENSOUSSAN A., LIONS J.L., TEMAM R. [ll Sur les mgthodes de dgcomposition, de dgcentralisation, de coordination et applications, in Sur Zes m e ' t h o d e s n u m e ' r i q u e s e n s c i e n c e s p h y s i q u e s e t e ' c o n o m i q u e s , J.L. Lions, G.I. Marchouk eds., Dunod-Bordas, Paris, 1974, pp. 133-257. BERCOVIER M. I11 R z g u l a r i s a t i o n d e s p r o b l a m e s v a r i a t i o n n e l s m i x t e s . A p p l i c a t i o n aux e'ze'ments f i n i s m i x t e s e t e x t e n s i o n ?qiu e l q u e s p r o b l ' e m e s n o n l i n e ' a i r e s , ThBse d'Etat, Universitg de Rouen, 1976. BERTSEKAS D.P., 111 Multiplier methods : A survey, A u t o m a t i c a , 12, (1976), pp. 133-145. [2] Approximation procedures based on the method of multipliers, J . O p t i m i s a t i o n T h . A p p l . , 2 3 , (1977), pp. 487-510. BOURGAT J.F., DUMAY., GLOWINSKI R. [11 Large-displacement calculations of flexible pipelines by finite elements and nonlinear programming methods, SIAM J . S c . S t a t . Comp., 1, (19801, pp. 34-81. BOURGAT J.F., GLOWINSKI R., LE TALLEC P., [11 Decomposition Of variational problems. Applications in Finite Elasticity, in 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 in E n g i n e e r i n g and A p p l i e d S c i e n c e s , R.L. Sternberg ed., Marcel Dekker, New York, 1980, pp. 445-480. 333
334
REFERENCES
BRENT R.P. 111 A l g o r i t h m s f o r m i n i m i z a t i o n w i t h o u t d e r i v a t i v e s , Prentice-Hall, Englewood Cliffs, N.J., 1973. BREZIS H., 111 O p e ' r a t e u r s Maximaux M o n o t o n e s , North-Holland, Amsterdam, 1973. BREZZI F. [l] On the existence, uniqueness and approximation of saddle-point problems arising from Laqrangian multipliers, Revue F r a n q a i s e d ' A u t o m a t i q u e I n f o r m a t i q u e R e c h e r c h e O p e ' r a t i o n n e l l e , A n a l y s e Nume'rique, 8, R - 2 , (1974), pp. 129-151. BREZZI F., RAPPAZ J., RAVIART P.A. [1l Finite-dimensional approximation of nonlinear problems, Part I : Branches of nonsingular solutions, Num. Math., 36, (1980), pp. 1-25. 121 Finite-dimensional approximation of nonlinear problems, Part 11: Limit points, Num. M a t h . , 37, (1981), pp. 1-28. 131 Finite-dimensional approximation of nonlinear problems, Part I11 : Simple bifurcations, Num. Math., 38, (1981), pp. 1-30. BRISTEAU M.O., GLOWINSKI R. , PERIAUX J. , PERRIER P., PIRONNEAU O . , POIRIER G., [l] Application of Optimal-Control and FiniteElement Methods to the calculation of Transonic Flows and Incompressible viscous Flows, in N u m e r i c a l Methods i n A p p l i e d F l u i d D y n a m i c s , B. Hunt ed., Acad. Press, London (1980), pp. 203-312. [21 Transonic flow simulations by finite-element and least-squares methods, in F i n i t e E l e m e n t s i n F l u i d s , Vol. 4, R.H. Gallagher, D.H. Norrie, J.T. Oden, O.C. Zienkiewicz eds., John Wiley, Chichester (1982), pp. 4 5 3 - 4 8 2 . CEA J. [l] O p t i m i s a t i o n : T h e ' o r i e e t A l g o r i t h m e s , Dunod, Paris, 1971. [2] O p t i m i s a t i o n : T k e o r y and a l g o r i t h m s , Lectures on Mathematics and Physics, Vol. 53, Tata Institute of Fundamental Research, Bombay, 1 9 7 8 (distributed by Springer-Verlag, Berlin). CEA J., GLOWINSKI R. [l] Sur des mgthodes d'optimisation par relaxation, Revue F r a n p a i s e d ' A u t o m a t i q u e I n f o r m a t i q u e R e c h e r c h e O p g r a t i o n n e l l e , AnaZyse Nume'rique, R - 3 , (1973)I pp- 5-32. CHAN T. , GLOWINSKI R. [l] F i n i t e - E l e m e n t a p p r o x i m a t i o n and i t e r a t i v e s o l u t i o n of 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 e q u a t i o n s , STAN-CS-78-674, Computer Science Department, Stanford University, 1978. 121 Numerical methods for a class of mildly nonlinear elliptic equations, in A t o s Do Decirno P r i m e i r o C o l o q u i o B r a s i l e i r o de M a t e m a t i c a , Vol. I, Impa, Rio de Janeiro, 1978, pp. 279-318. CHORIN A.J. [I] A numerical method for solving incompressible flow problems, J . Comp. P h y s i c s , 2, (1967), pp. 12-26. CIARLET P.G. [l] The f i n i t e - e l e m e n t m e t h o d for e l l i p t i c p r o b l e m s , North-Holland, Amsterdam, 1978. CONCUS P., GOLUB G.H. [l] A generalized conjugate-gradient method for nonsymmetric systems of linear equations, in Computing Methods i n A p p l i e d S c i e n c e s and E n g i n e e r i n g , R. Glowinski, J.L. Lions eds., Lecture Notes in Economics and Mathematical Systems, Vol. 134, Springer-Verlag, Berlin, 1976, pp. 56-65. CONCUS P., GOLUB G.H., O'LEARY D.P. [ll Numerical Solution Of nonlinear partial differential equations by a generalized conjugategradient method, C o m p u t i n g , 19, (19771, pp. 321-340.
REFERENCES
COOPER L . ,
KATZ I . N .
111
Maths. w i t h A p p l s . ,
7,
335
The Weber p r o b l e m r e v i s i t e d , Comp. and ( 1 9 8 1 ) , pp. 225-234.
C R O U Z E I X M. C11 E t u d e d ' u n e mgthode d e l i n g a r i s a t i o n . R6solution numgrique d e s E q u a tio n s d e S t o k e s s t a t i o n n a i r e s . Applications aux Qquations d e Navier-Stokes s t a t i o n n a i r e s , i n Approximations
e t Me'thodes I t e ' r a t i v e s de R e ' s o l u t i o n d ' l n e ' q u a t i o n s V a r i a t i o n n e l l e s e t de ProblBmes Non L i n e ' a i r e s , C a h i e r d e I ' I R I A , N o . 1 2 , 1 9 7 4 , pp.
139-244.
C R O U Z E I X M . , RAVIART P.A. C11 C o n f o r m i n g a n d n o n - c o n f o r m i n g f i n i t e el em ent methods f o r s o l v i n g t h e s t a t i o n a r y S t o k e s e q u a t i o n s ,
Revue F r a n q a i s e d ' A u t o m a t i q u e I n f o r m a t i q u e R e c h e r c h e Ope'rationn e l l e , A n a l y s e Nume'rique, 7 , R - 3 , 1 9 7 3 , p p . 33-76. DANIEL J . C11 The a p p r o x i m a t e m i n i m i z a t i o n of f u n c t i o n a l s , H a l l , Englewood C l i f f s , N . J . , 1970.
Prentice
111 I n t e r i o r p e n a l t y p r o c e d u r e s f o r e l l i p t i c DOUGLAS J . , DUPONT T . a n d p a r a b o l i c G a l e r k i n m e t h o d s , i n Computing Methods i n A p p l i e d S c i e n c e s , R. G l o w i n s k i , J . L . L i o n s , e d s . , L e c t u r e N o t e s i n P h y s i c s , V o l . 5 8 , S p r i n g e r - V e r l a g , B e r l i n , 1 9 7 6 , p p . 207-216. 111 On t h e n u m e r i c a l s o l u t i o n o f h e a t DOUGLAS J . , RACHFORD H.H. c o n d u c t i o n p r o b l e m s i n t w o o r t h r e e s p a c e v a r i a b l e s , T r a n s . Math. S o c . , 8 2 , ( 1 9 5 6 1 , pp. 421-439. DURAND E .
111
M a g n e ' t o s t a t i q u e , Masson, P a r i s , 1 9 6 5 .
DUVAUT G . , LIONS J . L . 111 Dunod, P a r i s , 1 9 7 2 .
L e s i n 6 q u a t i o n . s e n Me'canique e t e n P h y s i q u e ,
EKELAND I . , TEMAM R. C11 AnaZyse Convexe e t ProbLZmes Dunod, G a u t h i e r V i l l a r s , P a r i s , 1 9 7 4 .
Variationnels,
FLETCHER, R. 111 Methods r e l a t e d t o L a g r a n g i a n f u n c t i o n s , C h a p t e r V I I I of N u m e r i c a l Methods for C o n s t r a i n e d O p t i m i z a t i o n , P.E. G i l l , W. Murray e d s . , Academic P r e s s , London, 1 9 7 4 , p p . 219-239. FORTIN M. 111 M i n i m i z a t i o n o f some n o n - d i f f e r e n t i a b l e f u n c t i o n s by t h e augmented L a g r a n g i a n method o f H e s t e n e s and P o w e l l , A p p l . Math. O p t . , 2 , ( 1 9 7 6 ) I p p . 236-250. 1211 U t i l i s a t i o n d e l a m g t h o d e d e s d l g m e n t s f i n i s e n Mgcanique d e s F l u i d e s , I , C a Z c o l o , 4 , ( 1 9 7 5 ) , pp. 4 0 6 - 4 4 1 . 1212 U t i l i s a t i o n d e l a m6thode d e s QlQments f i n i s e n M g c a n i q u e d e s F l u i d e s , 11, C a l c o l o , 1, ( 1 9 7 6 ) , p p . 1-20. C31 L e p o n s s u r 1 ' A n a l y s e Convexe e t l ' a p p r o x i m a t i o n d e s problames de p o i n t s - s e l l e , Orsay Mathematical P u b l i c a t i o n , U n i v e r s i t g P a r i s - S u d , D e p a r t m e n t of M a t h e m a t i c s , 1 9 7 8 . 1 4 1 An a n a l y s i s o f t h e c o n v e r g e n c e o f m i x e d f i n i t e - e l e m e n t m e t h o d s , Revue F r a n q a i s e d ' A u t o m a t i q u e I n f o r m a t i q u e R e c h e r c h e O p e ' r a t i o n n e l l e , A n a l y s e Nume'rique, 1 1 , R - 3 , ( 1 9 7 7 ) , p p . 341-354. FORTIN M . , PEYRET R . , TEMAM R. 111 R e ' s o l u t i o n n u m g r i q u e d e s E q u a t i o n s d e N a v i e r - S t o k e s p o u r un f l u i d e i n c o m p r e s s i b l e , J o u r n a l de Me'canique, 10, ( 1 9 7 1 ) , 3 , p p . 357-390.
[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-
111 Mixed f i n i t e - e l e m e n t m e t h o d s f o r i n FORTIN M . , THOMASSET F. c o m p r e s s i b l e f l o w p r o b l e m s , J . of Comp. P h y s i c s , 3 1 , ( 1 9 7 9 ) , p p . 113-145. GABAY D . 111 Me'thodes nume'riques p o u r l ' o p t i m i s a t i o n n o n l i n e ' a i r e , T h k s e d ' E t a t , U n i v e r s i t 6 P i e r r e e t Marie C u r i e , 1 9 7 9 .
GABAY D . , MERCIER B. c 1 1 A d u a l a l g o r i t h m f o r t h e s o l u t i o n o f nonl i n e a r v a r i a t i o n a l problems v i a f i n i t e - e l e m e n t approximations, Comp. and Math. w i t h A p p l i c a t i o n s , 2 , ( 1 9 7 6 ) , 1, pp. 17-40.
336
REFERENCES
GERMAIN P. I 1 1 Me'canique d e s m i l i e u x c o n t i n u s , Vol. 1, Masson, Paris, 1 9 7 3 . GILL P.E., MURRAY W. C11 (eds.) N u m e r i c a l Methods f o r C o n s t r a i n e d O p t i m i z a t i o n , Acad. Press, London, 1 9 7 4 . GIRAULT V., RAVIART P.A. 111 F i n i t e - e l e m e n t a p p r o x i m a t i o n of N a v i e r S t o k e s e q u a t i o n s , Lectures Notes in Math., Vol. 7 4 9 , SpringerVerlag, Berlin, 1 9 7 9 . GLOWINSKI R. I l l N u m e r i c a l m e t h o d s f o r n o n l i n e a r v a r i a t i o n a l p r o b l e m s , Lectures in Math. and Physics, Vol. 65, Tata Institute of Fundamental Research, Bombay, 1 9 8 0 (distributed by SpringerVerlag, Berlin). 1 2 1 NumericaZ m e t h o d s for n o n l i n e a r v a r i a t i o n a l p r o b Zems (second edition) , Springer-Verlag, (in preparation) . GLOWINSKI R., LANCHON H. 111 Torsion Glasto-plastique d'une barre cylindrique de section multiconnexe, J o u r n a l de Mgcanique, Vol. 1 2 , NO. 1, ( 1 9 7 3 ) , pp. 1 5 1 - 1 7 1 . GLOWINSKI R., LE TALLEC P. [17 Numerical solution of problems in incompressible finite elasticity by augmented Lagrangian methods. (I) Two-dimensional and axisymmetric problems, SIAM J . A p p l . Math. Vol. 4 2 , NO. 2 , ( 1 9 8 2 ) , pp. 4 0 0 - 4 2 9 . C21 Numerical solution of problems in incompressible finite elasticity by augmented Lagranqian methods. (11) Three-dimensional problems (to appear) GLOWINSKI R., LE TALLEC P., RUAS V. [11 Approximate solution of nonlinear problems in incompressible finite elasticity, in N o n l i n e a r F i n i t e E l e m e n t A n a l y s i s i n S t r u c t u r a l M e c h a n i c s , W. Wunderlich, E. Stein, K.J. Bathe eds., Springer-Verlaq, Berlin, 1 9 8 1 , pp. 666-695.
.
GLOWINSKI R. , LIONS J.L., TREMOLIERES R. I 1 1 A n a l y s e Nume'rique d e s I n g q u a t i o n s V a r i a t i o n n e l l e s , Vol. 1 and 2 , Dunod-Bordas, Paris, 1976. [ 2 1 N u m e r i c a l A n a l y s i s of V a r i a t i o n a l I n e q u a l i t i e s , North-Holland, Amsterdam, 1 9 8 1 . GLOWINSKI R., MANTEL B., PERIAUX J. C11 Numerical Solution Of the time-dependent Navier-Stokes equations for incompressible viscous fluids by finite-element and alternating-direction methods, in N u m e r i c a l Methods i n A e r o n a u t i c a l F Z u i d Dynamics, P.L. Roe ed., Academic Press, London ( 1 9 8 2 ) , pp. 3 0 9 - 3 3 6 . GLOWINSKI R., MARROCCO A. C11 Sur l'approximation par GIGments finis d'ordre un et la r6solution par penalisation-dualite d'une classe de problemes de Dirichlet non lingaires, Revue F r a n p a i s e d ' A % t o m a t i q u e I n f o r m a t i q u e Recherche Ope'rationne Z l e , A n a l y s e n u m e r i q u e , R-2, ( 1 9 7 5 ) , pp. 4 1 - 7 6 . C 21 Sur l'approximation par Clgments finis d'ordre 1, et la r6solution par penalisation-dualit&, d'une classe de probldmes de Dirichlet non lingaires, C . R . A c a d , S c . Paris, 2 7 8 A , ( 1 9 7 4 ) , pp. 1 6 4 9 - 1 6 5 2 C31 On the solution of a class of nonlinear Dirichlet problems by a penalty-duality method and finiteelements of order one, in O p t i m i z a t i o n T e c h n i q u e s , IFIP T e c h n i c a l C o n f e r e n c e , G.I. Marchouk ed., Lecture Notes in Computer Sciences, Vol. 2 7 , Springer-Verlag, Berlin 1 9 7 5 , pp. 3 2 7 - 3 3 3 . C41 S u r Z ' a p p r o x i m a t i o n p a r e'le'ments f i n i s d ' o r d r e un e t l a re'solution par pe'nalisation-dualite' d'une c l a s s e d e p r o b l 2 m e s d e D i r i c h l e t non Z i n e ' a i r e s , Laboria Report 115, 1 9 7 5 .
REFERENCES
337
[51 Numerical solution of two-dimensional magnetostatic problems by augmented Lagrangian methods, Comp. Meth. A p p l . Mech. Eng., 12, (1977), pp. 33-46. GOLDSTEIN A.A. Ell Convex Programming in Hilbert Spaces, B U Z Z . A.M.S., 70, (1964), pp. 709-710. GOLUB G.H. [l] Sparse matrix computations : eigenvalues and linear equations, in A n a l y s e e t C 6 n t r o l e d e s S y s t z m e s , IRIA Seminars, 1974, pp. 117-140. HESTENES M. [ll Multiplier and gradient methods, J . O p t i m i z a t i o n T h e o r y and A p p l i c a t i o n s , 4, (1969), pp. 303-320. HIBBITT H.D., BECKER E.B. , TAYLOR L.M. 111 Nonlinear analysis of some slender pipelines, Comp. Meth. A p p l . Mech. Eng. 17/18, Part I, (1979), pp. 203-225. HOUSEHOLDER A.S. [ll The n u m e r i c a l t r e a t m e n t o f a s i n g l e n o n l i n e a r e q u a t i o n , McGraw-Hill, New York, 1970. JOHNSON C., THOMEE V. El1 Error estimates for a finite-element approximation of a minimal surface, Math. Comp., 29, (1975), pp. 343-349. JOURON, C. [l] Rgsolution numgrique du problsme des surfaces minima, A r c h . R a t i o n . Mech. A n a l y s i s , V o l . 59, No. 4, (19751, pp. 311342. KESAVAN S. 111 La mgthode de Kikuchi appliquge aux Qquations de Von Karmann, Numer. Math., 32, (1979), pp. 209-232. KIKUCHI F. Ell Finite-element approximation of bifurcation problems, in F u n c t i o n a l A n a l y s i s and N u m e r i c a l A n a l y s i s , Japan-France S e m i n a r , T o k y o and K y o t o 1 9 7 6 , H. Fujita ed., Japan Society for the Promotion of Science, 1978, pp. 203-222. KORT B.V. , BERTSEKAS D.P. [l] Combined primal-dual and penalty methods for convex programming, S i a m J . C o n t r o l and O p t i m i z a t i o n , 14, (1976), pp. 268-294. LANDAU L., LIFCHITZ E. [I1 Me'canique d e s f l u i d e s , Editions de Moscou, 1953. LE TALLEC P. [l] N u m e r i c a l A n a l y s i s o f E q u i l i b r i u m P r o b l e m s i n I n c o m p r e s s i b l e N o n Z i n e a r E l a s t i c i t y , Ph.D. Thesis, The University of Texas at Austin, 1980. [2] L e s p r o b l a m e s d ' g q u i l i b r e d ' u n c o r p s h y p e r d l a s t i q u e i n c o m p r e s s i b l e e n g r a n d e s d e ' f o r m a t i o n s , These d'Etat, Universite Pierre et Marie Curie, 1981. LE TALLEC P. , ODEN J.T. [l 3 Existence and characterization of hydrostatic pressure in finite deformations of incompressible elastic bodies, J o u r n a l of E l a s t i c i t y , 1 1 , (1981), 4 , LICHNEWSKY A. El1 Une mgthode de gradient conjugu6 sur des vari6t6s. Application 'a certains problsmes de valeurs propres non lingaires, Num. F u n c t . A n a l y s i s and O p t i m i z a t i o n , 1, (1979), 5, pp. 515-560. LIONS J.L. [l] C o n t r b Z e o p t i m a l d e s s y s t z m e s g o u v e r n e ' s p a r d e s d q u a t i o n s aux d e ' r i v e ' e s p a r t i e l l e s , Dunod-Gauthier Villars, Paris, 1968. C21 Q u e l q u e s me'thodes de r e ' s o t u t i o n d e s p r o b l a m e s aux l i m i t e s non l i n e ' a i r e s , Dunod-Gauthier Villars, Paris, 1969. LIONS J.L. , MAGENES E. [1 I Non-homogeneous b o u n d a r y - v a l u e p r o b l e m s , I, Springer-Verlag, New York, 1972. LIONS J.L.
, STAMPACCHIA G.
[11 Variational Inequalities, Comm. Pure
338
REFERENCES
A p p l i e d Math.,
20, (1967), pp. 493-519.
LIONS P.L., MERCIER B. El1 Splitting Algorithms for the Sum of two Nonlinear Operators, SIAM J. Num. A n a l . , 1 6 , (19791, pp. 964-979. LYNESS J.N., JESPERSEN D. 111 Moderate-Degree Symmetric Quadrature Rules for the triangle, J . I n s t . M a t h s . A p p l i c s . , 1 5 , (1975), pp. 19-32. MAIER G., ANDREUZZI F. , GIANESSI F., JURINA L., TADDEI F. [11 Unilateral contact, elasto-plasticity and complementarity with reference to off-shore pipeline design, Comp. Meth. A p p l . Mech. Eng., 17/18, Part 11, (1979), pp. 469-496. MALKUS D.S., HUGHES T.J.R. Ill Mixed finite-element methods. Reduced and selective integration techniques: A unification of concepts, Comp. Meth. A p p l . Mech. E n g . , 15, (1978), pp. 63-81. MARCHOUK G.I., KUZNETSOV Y.A. [ll Mgthodes itgratives et fonctionnelles quadratiques, in S u r l e s Methodes Nurne'riques e n S c i e n c e s P h y s i q u e s e t E c o n o m i q u e s , J.L. Lions, G.I. Marchouk eds., Dunod, Paris, 1974, pp. 1-132. MARROCCO A. 111 E s p e ' r i e n c e s nume'riques s u r d e s p r o b l ' e m e s non Z i n e ' a i r e s r e ' s o l u s p a r e'le'ments f i n i s e t l a g r a n g i e n augmente', Laboria Report No. 309, May 1978. 121 Analyse Num6rique de problsmes d'Electrotechnique, Ann. Sc. Math. Que'bec, Vol. 1 , No. 2, (1977), pp. 271-296. MARTINET B. 111 R6gularisation d'in6quations variationnelles par approximations successives, R e v . F r a n c a i s e I n f . Rech. O p e r . , (i'970), pp. 154-159. 121 DGtermination approchee d'un point fixe d'une applica__ tion pseudo-contractante, C. R. A c a d . S c i . P a r i s , 274A, 11972). pp. 163-165. MATTHIES H., STRANG G. equations, I n t . J. MEIJERINK J.A., VAN DER for linear systems M-matrix, Math. of
[13 The solution of nonlinear finite-element Num. Meth. E n g . , 14 (1979), pp. 1613-1626. VORST H . A . rll An iterative solution method of which the coefficients matrix is a symmetric C o m p . , 31, (1977), pp. 148-162.
MERCIER B. [ll T o p i c s i n f i n i t e - e l e m e n t s o l u t i o n of e l l i p t i c p r o b l e m s , Lectures in Math. and Physics, Vol. 63, Tata Institute of Fundamental Research, Bombay, 1979 (distributed by Springer-Verlag, Berlin). [21 Approximation par Gl6ments finis et r6solution par un algorithme de p&alisation-dualitg d'un problsme d'blastoplasticitg, C.R. A c a d . S c . P a r i s , 280A, (19?5),, pp. 287-290. 131 S u r La T h e ' o r i e e t 1 ' A n a l y s e Numerzque d e Probl'emes de P l a s t i c i t g , Thsse d'Etat, Universitg Paris VI, 1977. MINTY G.J. [11 Monotone (nonlinear) operators in Hilbert space, Duke Math. J . , 2 9 , (1962), pp. 341-346. MIYOSHI R. 111 A finite-element method for the solution Of fourthorder partial differential equations, Kunamoto J. S c i ( M a t h . ) , 9, (1973), pp. 87-116. MOREAU J.J.
111
Proximith et dualit6 dans un espace Hilbertien,
BUZZ. S o c . Math. F r a n c e , 93, (1968), pp. 273-299. MOSCO U. 111 Dual Variational Inequalities, J . Math. A n a l . A p p l . , 40, (19721, pp. 202-206.
MUNRO E. 111 Some techniques and applications of the finite-element method for solving magnetic-field problems, P r o c e e d i n g s o f COMPUMAG C o n f e r e n c e , April 1976, Oxford, pp. 35-46.
REFERENCES
339
111 L e s me'thodes d i r e c t e s e n t h 6 o r i e d e s e ' q u a t i o n s e l l i p t i q u e s , Masson, P a r i s , 1 9 6 7 .
NECAS J.
RIP-Methods f o r S t o k e s i a n flows, i n F i n i t e Elements 4 , R.H. G a l l a g h e r , D.H. N o r r i e , J . T . Oden, O.C. Z i e n k i e w i c z e d s . , J o h n W i l e y , C h i c h e s t e r , ( 1 9 8 2 ) p a g e s 305 - 318.
ODEN J . T .
111
i n FZuids, Vol.
OLSON M . D . , TUANN S.Y. 111 N e w f i n i t e - e l e m e n t r e s u l t s f o r t h e s q u a r e c a v i t y , C o m p . F l u i d s , 7 , ( 1 9 7 9 ) , p p . 123-135. OPIAL, Z . 111 Weak c o n v e r g e n c e o f t h e s u c c e s s i v e a p p r o x i m a t i o n f o r n o n - e x p a n s i v e m a p p i n g s i n Banach S p a c e s , B u l l . A . M.S., 7 3 , ( 1 9 6 7 ) , pp. 591-597. 111 I t e r a t i v e s o l u t i o n of n o n l i n e a r e q u a t i o n s i n s e v e r a l v a r i a b l e s , Academic P r e s s , New Y o r k , 1 9 7 0 .
ORTEGA J . , RHEINBOLDT W.C.
PAIGE C.C. 111 B i d i a g o n a l i z a t i o n of matrices a n d s o l u t i o n of l i n e a r e q u a t i o n s , SIAM J . Num. A n a l . , 11, ( 1 9 7 4 ) , 1, p p . 197-209. PARLETT B . N . 111 The S y m m e t r i c E i g e n v a l u e Problem, Englewood C l i f f s , N e w - J e r s e y , 1 9 8 0 .
Prentice H a l l ,
PEACEMAN D . H . , RACHFORD H . H . 111 The n u m e r i c a l s o l u t i o n o f p a r a b o l i c a n d e l l i p t i c d i f f e r e n t i a l e q u a t i o n s , SIAM J . A p p l . M a t h . , 3 , ( 1 9 5 5 ) , pp. 24-41. PELISSIER M.C. 111 S u r q u e l q u e s p r o b l z m e s non t i n z a i r e s e n glaciologic, O r s a y M a t h e m a t i c a l P u b l i c a t i o n , U n i v e r s i t e P a r i s - S u d , D e p a r t m e n t o f M a t h e m a t i c s , F a s c . 75-24, No. 110, 1 9 7 5 . POLAK E. [ l ] Computational methods in O p t i m i z a t i o n , N e w York, 1971.
Academic P r e s s ,
111 On t h e B e r t s e k a s m e t h o d f o r m i n i m i z a t i o n o f composPOLYAK B . T . i t e f u n c t i o n s , i n I n t e r n a t i o n a l Symposium o n S y s t e m s O p t i m i z a t i o n and A n a l y s i s , A. B e n s o u s s a n , J . L . L i o n s e d s . , L e c t u r e N o t e s i n C o n t r o l and I n f o r m a t i o n S c i e n c e s , V o l . 1 4 , S p r i n g e r - V e r l a g , B e r l i n , 1 9 7 9 , p p . 179-186. POWELL M . J . D . 111 A m e t h o d f o r n o n l i n e a r c o n s t r a i n t s i n m i n i m i z a t i o n p r o b l e m s i n O p t i m i z a t i o n , R. F l e t c h e r e d . , Academic P r e s s , London, 1969, Chapt er 19. 121 Variable metric methods f o r c o n s t r a i n e d o p t i m i z a t i o n , i n Computing Methods i n AppZied S c i e n c e s and E n g i n e e r i n g , 1 9 7 7 , I , R. G l o w i n s k i , J . L . L i o n s e d s . , L e c t u r e Notes i n Mathema t i c s , Vol. 7 0 4 , S p r i n g e r - V e r l a g , B e r l i n , 1 9 7 9 , p p . 62-72. ROACHE P . J . El1 C o m p u t a t i o n a l F l u i d D y n a m i c s , 1972. Albuquerque, N.M.,
Hermosa Publishers,
[ l ] The m u l t i p l i e r m e t h o d of H e s t e n e s a n d P o w e l l ROCKAFELLAR R.T. a p p l i e d t o convex programming, J . O p t . T h e o r y A p p l . , 1 2 , ( 1 9 7 3 ) , p p . 555-562. 1 2 1 Augmented L a g r a n g i a n s a n d a p p l i c a t i o n s t o 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 i n c o n v e x p r o g r a m m i n g , Math. O p . R e s e a r c h , 1, ( 1 9 7 6 ) , p p . 97-116. 1 3 1 Augmented L a g r a n g e m u l t i p l i e r f u n c t i o n s a n d d u a l i t y i n n o n - c o n v e x p r o g r a m m i n g , S i a m J . C o n t r o t , V o l . 1 2 , No. 2 , ( 1 9 7 4 ) , p p . 268-285. [ 41 Convex A n a l y s i s , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , N. J. , 1 9 7 0 . 1 5 1 E x t e n s i o n of F e n c h e l ' s d u a l i t y t h e o r e m , Duke Math. J . , 3 3 , ( 1 9 6 6 1 , p p . 81-89. 161 On t h e M a x i m a l i t y o f sums of n o n l i n e a r monot o n e o p e r a t o r s , T r a n s . Amer. Math. Sac., ( 1 9 7 0 ) , p p . 75-88.
340
REFERENCES C71 A dual approach to solving nonlinear programming problems by unconstrained optimization, Math. P r o g . , 5, (1973), pp. 354-373. C81 Monotone operators and the proximal-point algorithm, SIAM J . C o n t r o l a n d O p t i m i z a t i o n , 14, (1976), pp. 877- 898.
RUAS V. 111 A class of asymmetric simplicia1 finite-element methods for solving incompressible elasticity problems, Comp. M e t h . A p p l . Mech. E n g . , 27, (1981), 3, pp. 319-343. SEGAL A. 111 On the numerical solution of Stokes equations using the finite element method, Comp. M e t h . A p p Z . Mech. E n g . , 19, (19791, pp. 165-185. SOULIE M. 111 Private communication. STEWART G.W. ill I n t r o d u c t i o n t o M a t r i x C o m p u t a t i o n s , Acad. Press, New York, 1973. STRANG, G., FIX G. 111 An a n a l y s i s of t h e f i n i t e - e l e m e n t m e t h o d , Prentice Hall, Englewood Cliffs, N.J., 1973. TARTAR L. 111 T o p i c s i n N o n l i n e a r A n a l y s i s , Orsay Math. Publication, Universitg Paris-Sud, Department of Mathematics, 1978. TAYLOR R.L., ZIENKIEWICZ O.C. 111 Complementarity Energy with Penalt Functions in Finite-Element Analysis, Chapter 8 of E n e r g y M e t h o d s i n F i n i t e E l e m e n t A n a l y s i s , R. Glowinski, E.Y. Rodin, O.C. Zienkiewicz eds., J. Wiley and Sons, Chichester, 1979, pp. 153174. TEMAM R. 111 N a v i e r - S t o k e s e q u a t i o n s , North-Holland, Amsterdam, 1977 THOMAS J . M . 111 S u r Z ' a n a l y s e n u m k r i q u e d e s m e ' t h o d e s d ' e ' Z k m e n t s f i n i s h y b r i d e s e t m i x t e s , Thkse d'Etat, Universitg Pierre et Marie Curie, 1977. THOMASSET F. C11 I m p l e m e n t a t i o n of F i n i t e E l e m e n t M e t h o d s f o r N a v i e r - S t o k e s e q u a t i o n s , Springer-Verlag, New York, 1981.
111 An elliptic Collocation/Finite-Element Method with WHEELER M.F. Interior Penalties, S I A M J . Num. A n a Z . , 15, (1978), pp. 152-161. WIDLUND 0. 111 A Lanczos method for a class of nonsymmetric Systems of linear equations, S i a m J . Num. A n a l . , 15, (1978), 4, pp- 801812. WILKINSON J.H. 111 T h e a l g e b r a i c E i g e n v a Z u e P r o b l e m , Oxford University Press, 1965. WILKINSON J.H., REINSCH C. 111 Handbook f o r A u t o m a t i c C o m p u t a t i o n , VoZ. 11: L i n e a r A l g e b r a , Springer-Verlag, Berlin, 1971.
[11 On numerical deformation of singularities YAMAGUTI M., FUJI1 H. in nonlinear elasticity, in C o m p u t i n g M e t h o d s i n A p p l i e d S c i e n c e s and E n g i n e e r i n g , Part I, R. Glowinski, J.L. Lions eds., 1977, Lecture Notes in Mathematics, Vol. 704, Springer-Verlag, Berlin, 1979, pp. 267-281.
