By Michel Fortin

The aim of this quantity is to provide the rules of the Augmented Lagrangian strategy, including quite a few purposes of this system to the numerical resolution of boundary-value difficulties for partial differential equations or inequalities coming up in Mathematical Physics, within the Mechanics of constant Media and within the Engineering Sciences.

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**Example text**

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 , . . 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. 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 .

I n t h e r e m a i n d e r of S e c t i o n 2 . 3) with variable s t e p s i z e n+l Y. 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. 34) Yi = (1 - pxi)y? N 1 ( c l a s s i c a l l y familiar) Figure 2 . 1 I I-PXI P Figure 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 . 3 ) . to P r o c e e d i n g as i n Section 2 . 3 , ImA-'Bt n- 1=o. ( 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 . 14) I w e have Vi=l,.. ,N]. 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 .