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g
and (3.78)
I
(lp I
- g)’ dx
=
IPI,Q
( s i n c e gradu = 0 a t i o n s ) . Hence SUP P
n
I grad u 1’ dx
=
1
5
I grad u 12 dx
IPOQ
i n t h e r e g i o n e x c l u d e d from t h e i n i t i a l i n t e g r -
inf &(u, w , p) 2 U. w
5I
-
f o r t h e c h o i c e ( 3 . 7 6 ) of p and t h e r e f o r e from ( 3 . 7 8 ) :
divp = 1 i n $2 and t h e i n t e g r a l i s e x t e n d e d t o t h e r e g i o n where I p ( x ) 12 g
( I ) The f o l l o w i n g s h o u l d be u n d e r s t o o d :
.
(SEC. 3 )
Infinite-dimensiona2 approximation
sup inf A ( u , w , p) 2 - 3 a(u, u) P
=
29
J(u) ,
h w
which, in combination with (3.71),proves the Theorem. Remark 3.11. In view of the ent subject matter to subsequent example of the above method. We find a vector u = { u , , ...,q,} and a (3.79)
- P A U = f - gradp,
(3.80)
div u = 0 ,
(3.81)
u = 0 on
0
extreme importance of the presarguments we shall give afirther consider the Stokes problem: scalar p such that:
r.
We are thus dealing with a problem of equations in which (3.80) is to be considered as a constraint. We introduce (3.82)
V = {ulu~(Hi(G?))”, d i v u = O in a }
Thus the problem (3.79),(3.80), (3.81) is equivalent to seeking : (3.84)
J(u) = 3 a(u, u )
inf J(u) , VE
v
- (f,u)
o r , equally, to seeking U E V with
(3.85)
U(U,U) =
(Ju)V U E V .
Taking (3.80)as a constraint, we introduce, by analogy with the above : (3-86)
1
M ( u . P ) = ~ a ( vU),
- (f,u) -
p(div u) dx
J”
where (3.87)
uE
W
=
(Hi(a))”,p E L z ( s l ) .
It may now be verified directly that (3.88)
sup inf N(U, p) P
V
=
J(u) .
Approximation of steady-state i n e q u a l i t i e s
30
inf m u , p ) , U E W
We c a l c u l a t e
u
bound i s a t t a i n e d f o r
V
.
( Cm.1)
For a given p , t h e lower
= V ( p ) , t h e solution of
- p A u = f - gradp,
(3.89)
u=O
on f ,
and t h u s
inf -4% p ) = - 4 a(u, u ) . v
We t h e n have: - inf
(3.90)
4 a(u, u ) = J(u)
P
where v = ~ ( p i)s t h e s o l u t i o n o f (3.89) (l). Algorithms f o r t h e approximation o f (3.90) a r e given l a t e r i n t h i s volume. rn The above methods a r e c l e a r l y not confined t o Remark 3.12. 2nd order o p e r a t o r s . I n Chapter 4 we s h a l l meet some examples o f t h e i r a p p l i c a t i o n f o r c e r t a i n 4 t h o r d e r o p e r a t o r s which appear i n t h e theory of t h i n p l a t e s .
Remark 3.13. A s w e have j u s t s e e n , t h e i d e a s o f d u a l i t y can l e a d t o a f o r m u l a t i o n which - i n appearance d i f f e r s widely from t h e i n i t i a l f o r m u l a t i o n . Moreover, more g e n e r a l l y , s t a r t i n g w i t h a problem i n t h e c a l c u l u s o f v a r i a t i o n s
-
inf J(u) v
f o r which the lower bound i s not necessarily reached (e.g. t h e theory o f m i n i m surfaces) w e can i n t r o d u c e d u a l i t y problems SUP @(PI P
suc h t h a t
(.i)
sup @@) = inf J(u) ; P
v
( i i ) t h e r e e x i s t s p which g i v e s t h e sup o f @ . For minimum s u r f a c e s , s e e T6mam /l/. We t h u s have a relaxat-
ion process
(2).
( l)
We a g a i n have a " d i s t r i b u t e d - s y s t e m o p t i m a l c o n t r o l ' ' type problem; s e e Remark 3.9.
(2)
Here, t h e t e r m " r e l a x a t i o n " i s t o be t a k e n i n a completely d i f f e r e n t sense from t h a t of t h e c l a s s i c a l relaxation methods o f numerical a n a l y s i s which, i n c i d e n t a l l y , we s h a l l a l s o use l a t e r .
Infinite-dimensional approximation
(SEC. 3)
31
Also, see Remark 3.17 later.
Various relaxation processes, particularly for nun convex functionals, are encountered in optimal control theory: see Ekeland /1/ and Bidaut /l/.
3.6
Duality (111)
Another possibility is offered by the use of conjugate functions; this classical idea, due originally to Legendre, is developed in Fenchel /l/, Mandelbrojt /l/, Hijrmander /l/, Moreau /l/, and Rockafellar / 2 / . As we shall now see, this does not assume that the form a(u,O) is symmetric, as was the case in the two previous sections. rn Over a Hilbert space V we say that a convex function u + # ( u ) is proper if:
(i) (ii)
is lower semi-continuous, with values in
@ @
is not identical to
+
00.
1- a,+Q)] ;
rn
Example 3.4. If K is a closed convex subset of the function given by
V, K # 0 ,
is a proper convex function. rn Using this concept, we see that the inequality (3.2) is equiualent to seeking u in V such that
(3.92)
U(U,
u - U) - (f, u -
U)
+ GK(u)- @ & ) > O ,
VUEV.
We also note that the function @ = j introduced in Section 2.1 is a proper convex function, so that all the inequalities mentioned so far come within the following general framework:
(3.93)
{
find U E V w i t h a(u, u - u)
- (f,u - u)
+ @(u) - @(u) 2 0
Vu E V .
In order to simplify notation in the following, we introduce the operator A where: A E U ( V ;V ' ) , (3.94) u(u, u) = (Au, u) Thus (3.93) is equivalent to
(3.95)
( ~ u - f , ~ - u ) + @ ( u ) - @ ( u ) , OVUEV.
We shall now transform (3.95), using sub-differentials. If the function @ is differentiable, then, from the convexity:
Approximation of steady-state inequalities
32
(3.96)
@(u)
- @(u) - (@'(u),u - u) 2
0
v.
VU€
We now i n t r o d u c e , i n a g e n e r a l manner, the set o f that
(3.97)
@(u) - Y u )
- ( L o - u) 2
(CHAP. 1)
< E V ' such
0 VUEV,
a concept which i s v a l i d whether @ i s d i f f e r e n t i a b l e a t u o r not. The s e t o f 5 such t h a t ( 3 . 9 7 ) i s s a t i s f i e d i s denoted by a@(u) and c a l l e d t h e sub-differential of @ a t u ( s e e Moreau /I/). We o f t e n w r i t e
(3.98)
-
@u) - @(u) - (ayu),
- u) 2
0 vv E
t h e c o r r e c t e x p r e s s i o n being ( 3 . 9 7 ) w i t h Clearly, i f @ i s d i f f e r e nt i a bl e at u :
(3.99)
a@(,)
=
{€a@@) .
{ @ '(4 1.
Using t h i s c o n c e p t , we s e e t h a t ( 3 . 9 5 ) is equivalent to - (h- f)E
(3.100)
a@@).
Remark 3.14. We observe t h a t ( 3 .loo) may be viewed a s an equ a t i o n r e l a t e d t o a multivalued operator (sometimes c a l l e d a setvalued operator). We s h a l l now t r a n s f o r m (3.100) u s i n g t h e concept o f a conjug-
ate convex function. If @ i s a p r o p e r convex f u n c t i o n , we d e f i n e the conjugate fun-
ction @*on t h e space V' by (3.101)
@*(u*) = sup [(u*, u ) - * u ) ]
.
1)
It may be shown ( s e e Moreau /l/) t h a t p i s a proper convex function.
Example 3.5. (3.102)
We t a k e V = Hd(Q),
K given by
K = { u I u E V ,u 2 0 8.e. in Q } . @=@I given by (3.91)
.
We i d e n t i f y L'(i2) with i t s dual; t h u s V c L'(Q) c where H-'(i2) i s th,e s p a c e ' o f d i s t r i b u t i o n s o f t h e form
Now t h e c o n j u g a t e f u n c t i o n o f @*(u*) = SUP (u*,U) ,
and hence
uEK
i s given by
V' = f f - ' ( Q ) ,
(SEC. 3 )
(3.103)
Infinite-dimensional approximation
@*(u*) =
{ + co
ifu*O
(3.113)
VUEK,
and hence i s e q u i v a l e n t t o f i n d i n g the projection of $ , i n the sense of the inner product ( 3 . 1 1 1 ) , on the cone of Hd(62) of functions 2 0. We p u t : A-' = G = Green's o p e r a t o r i n problem. Thus, from (3.105) and by n o t i n g t h a t is equivalent t o
(3.114)
{ '*'(Gu*- +,
(3.115)
On
u*
" - u*) 2
0 Vu*
O ,
VUEV,
we immediately g e t : Theorem (4.17)
uh
4.1.
Using the hypotheses (4.1), (4.2) ue have: i n V as h
+ 1(
+ 0.
It may be deduced from ( 4 . 1 5 ) t h a t
Proof.
11 uh 11'
Q
= (f,'3
dUh9
d 11 f
IIY'
11 uh 11
and hence (4.18)
1 IIuhII d;Ilf 1 1 ~ ' .
We can e x t r a c t from uh a sequence, a l s o denoted by u,, such that (4.19)
u,
weakly i n V as h + 0.
+w
For U E Y , w e t a k e oh s a t i s f y i n g ( 4 . 2 ) ; i n V , for t h i s c h o i c e o f l]h we have:
a(uh, v&
+
)'
and hence, i n t h e l i m i t , a(w, u) =
s i n c e uh + u Strongly
(f,u )
( 4 . 1 5 ) gives
vu E Y ;
s i n c e Y i s dense i n V we deduce t h a t w = u and hence t h a t uh + u weakly i n V without having t o e x t r a c t a subsequence. It now remains t o show t h a t u h + u strongly i n V. For t h i s we consider xh =
a(Uh - u, uh
- u) .
From (4.15) , x h = (f,U& - a(% U, - u) - a(uh, u) + (f,u) - a(& u) = 0, and t h e r e f o r e . 11 uh - u (1 + 0 . rn
(SEC. 4)
41
Interior approximations
The problem is now to extend these considerations to variation-
a I inequa Zities.
4.3
Approximations of K
We consider, for each h, a set Kh with (4.20)
(4.21)
K,= closed convex subset of V,, Kh "approximates" K in the following sense: (i) VUE K, we can find uhek;, with u h + u in V ; (Ti) If u h e K h , u,+uweakly in v , then U E K . 8
I
ExampZe 4.3.
We consider the situation as in Example 4.2 and
let (4:22)
K = { u ~ u E Vu ,2 0
a.e. in P }
Thus if we take
where V, is defined by (4.12), then (4.21) is satisfied.
8
Remark 4.1. We can weaken (4.21) by introducing K n Y and assuming K n Y to be dense in K.
4
In practical applications, the construction of Remark 4.2. is one of the difficulties presented by inequality problems.
We shall give numerous examples i'n the following chapters. (We shall also use exterior approximations, see Section 5 below). 8
Remark 4.3. A difficulty in terminology should be pointed out. The approximation V, of V is interior since V h c V ; the approximation Kh of K has an interior character since KhtV but it is not assumed that K,cK.
8
For a general study of "the approximation of Remark 4.4. convex sets", we refer the reader to the works of Mosco /2/, Joly /l/. See also Aubin /l/, Tr6moliDres 141.
4.4.
Approximation schemes for the initial problem
We now consider the inequaZity probZems (4.24)
(utK9
a(U,u - U) 2
(J - U) VUEK,
Approximation of steady-state inequazities
42
(CHAP. 1)
The approximation schemes u s i n g t h e above concepts are:
respectively. We have Theorem 4.2. We assume ( 4 . 1 ) , ( 4 . 2 ) to be satisfied (and for problem ( 4 . 2 6 ) , the conditions ( 4 . 2 0 ) , ( 4 . 2 1 ) ) . Then if u,, is the soZution of (4.26) ( r e s p . ( 4 . 2 7 ) ) we have (4.28)
in V,
uh+u
where u is the soZution of ( 4 . 2 4 ) ( r e s p . ( 4 . 2 5 ) ) . Proof. For f i x e d u i n K , we t a k e uh s a t i s f y i n g (4.21) ( i ) ; with t h i s choice of vh i n (4.26) we g e t : (4.29)
a(uh,
(4.30)
11 uh 11
d a(uh,
- (f,v h - uh)
O a.e. in Q }
and in order to simplify things slightly we shall restrict our attention to the case where the bilinear form a is defined by (4.47)
a(u, u) =
I
grad u.grad u dx .
Let f E L Z ( R ) ; the variational inequality
(4.4)
[
~ ( uu,
- u) 2
In
f ( 0 - u) dx Vu E K
then admits one and only one solution and we recall (see BrCzisStampacchia /2/ ) that
with (4.50)
11 I I ~ ~G ( c~ iifnL2,m )
where, in (4.50), C is independent of f and u. rn We shall now approximate the solution of (4.48) by means of a
(SEC.
4)
47
I n t e r i o r approximations
t h e n o t a t i o n i s as i n S e c t i o n 4.1, Example I n o r d e r t o s i m p l i f y t h e argument we s h a l l assume t h a t C2 4.2. i s a convex polygon, s o t h a t r e l a t i o n s (4.49), (4.50) s t i l l h o l d , and we s a y t h a t Y h must s a t i s f y ,
f i n i t e element method;
U T=Z.
(4.51)
T E Y h
as w e l l as
(4.7)-(4.10).
We i n t r o d u c e
C, = { M I M ~ d l ,
(4.52)
M i s a vertex of T € . T h }
and we d e f i n e t h e "approximation" Vh o f Hi@) by (4.53)
v h
{ l)h 1 V h E H k ( 0 )
=
co(dl)z,, U h l E~
V T Er
h
}
where uhIT d e n o t e s t h e r e s t r i c t i o n o f vh t o T and Pi d e n o t e s t h e We t h e n space o f p o l y n o m i a l s i n two v a r i a b l e s o f d e g r e e < 1 . d e f i n e Kh, a n a p p r o x i m a t i o n o f K , by
& = v , ~ K ={ u , I ~ , E v ~ , v , 2 0 o n 51;
(4.54)
K,, i s a closed convex subset of
vh and
it i s immediate, s i n c e
belongs t o P I , t h a t
ohlT
(4.55)
Kh
={
vh
I uh E
vh,
vh(M) 3 0 V M E
Ch
}
*
We t h e n d e f i n e t h e approximate problem by
uhEKh
(4.56) admits one and only one solution. I n connection with t h e approximation e r r o r s h a l l now p r o v e
and
11 u h - u IIH;(0)
we a s s m e t h a t the, angles of P r o p o s i t i o n 4.1. below, uniformly i n h , by e0 > 0; we then have (4.57)
, we
F,, are bounded
11 uh - I( 11 d 11 f IIL'(f2) &
where C i s independent o f h,u,f and where we r e c a l l that /I = max(Area of T)
.
TEYh
Proof.
I n t h e following, C w i l l denote v a r i o u s co n s t an t s ;
we have U(U, uh
- U) 2 Jn
a(uh, uh
- uh) 2
f(Uh
f(vh
- U) dx
- uh) dx v o E K h
9
48
Approximations of steady-state inequalities
(CHAP. 1)
and hence by a d d i t i o n
We have u E HA(62) n H 2 ( @
- Au - f E L2(Q)
and hence
and i f we
Put (4.60)
I =
- AU - f ,
we g e t , u s i n g ( 4 . 5 0 ) , (4.60’)
c II f { IICI independent llL2(D)
Q
II”(f2)
of f .
From (4.60) w e deduce t h a t (4.61)
a(u, u) =
.
and from (4.591, 1
3 11 ‘h vuh E K b
I, +
(f I ) u dx
(4.61)
lliA(f2)
Vu E H h W ) ,
1
3 11 uh -
IlfrA(f?)
+
s.
I(Uh
- u) dx
9
and hence from (4.60’) (4*62)
f 11
uh
-
< 11 uh -
lli1(f2)
llil(f2)
+ 11 f llL’(f2) 11 uh -
llL2(f2)
Vu, E Kh . To e s t i m a t e uh - u w e s h a l l u s e ( 4 . 6 2 ) , choosing a s u i t a b l e u, : f i r s t we d e f i n e t h e interpolation o p e r a t o r nh: Hh(62) n Co(62)3 Vh by
(SEC. 5 )
Exterior approximations
{ nh
n,oE
(4.63)
v,
H;(o) n co(CSa, VME z, , vu
U ( M )=
49
V(M)
E
and, from ( 4 . 5 3 1 , ( 4 . 5 5 ) we have
nhu E K ,
(4.64)
vv
EK n
co(5).
I n ( 4 . 6 2 ) we s h a l l t a k e vh = n,u ; t h i s i s meaningful owing t o t h e f a c t t h a t , a s R i s a @ o - d i m e n s i o n a Z domain w i t h a L i p s c h i t z boundary, we have H'(62) c Co(0) a n d , from (4.64 ) nhu E K , Vh We r e c a l l t h a t h = m a x ( A r e a o f T ) ; t h u s from, f o r example,
.
T E F ~
Fix-Strang /l/, C i a r l e t - R a v i a r t /l/, we have t h e r e s u l t t h a t unde r t h e s p e c i f i e d c o n d i t i o n on t h e a n g l e s o f 9,we have
(4'65)
11 nh
- u llLZ(f2) < c 11 u IIH'(f2)h
(4.66)
11 nh
-
11
IIHA(f2)
IIHqI?)
9
fi
w i t h C independent o f h , u . The e s t i m a t e ( 4 . 5 7 ) t h e n f o l l o w s t r i v i a l l y from ( 4 . 5 0 ) , ( 4 . 6 2 ) w i t h v,=n,,u , ( 4 . 6 5 ) and ( 4 . 6 6 ) .
Remark 4.12. The r e s u l t s and t h e p r o o f o f P r o p o s i t i o n 4 . 1 a r e e a s i l y e x t e n d e d t o t h e c a s e where C2 i s a convEx polyhedron of R3 s i n c e i n t h i s c a s e t h e i n c l u s i o n H'(62) c Co(62) s t i l l h o l d s . Remark 4.13. I n Chapter 2 , S e c t i o n 5 we s h a l l s t u d y a problem similar t o problem ( 4 . 4 8 ) , u s i n g a f i n i t e d i f f e r e n c e method..
5.
EXTERIOR APPROXIMATIONS
5.1
An example
I n a p p l i c a t i o n s n o t o n l y a r e i n t e r i o r methods u s e d , b u t a l s o
exterior methods. B e f o r e g o i n g on t o t h e " g e n e r a l axioms", which a r e somewhat i n v o l v e d , ,we s h a l l p o i n t o u t t h e b a s i c i d e a s u s i n g a n example which h a s been chosen t o be as s i m p l e as p o s s i b l e . We c o n s i d e r t h e Dirichlet p r o b l e m :
(5.1) (5.2)
+ u =f, u=0 on r . -
Au
We i n t r o d u c e
(5.3) and
v = HA(62)
f
g i v e n i n L2(sZ),
Approximation of steady-state inequalities
50
(5.4)
a(u, u) =
I,
(uu
(CHAP. 1)
+ grad u.grad u ) dx .
The problem (5.1), ( 5 . 2 ) i s then equivalent t o (5.5)
a(u, 11) =
( A 1)) ,
vu E
v.
Our a i m i s t o approximate u using characteristic functions, i . e . f u n c t i o n s which do not belong to HI@); t h e use of t h e s e c h a r a c t e r i s t i c f u n c t i o n s corresponds t o finite difference methods. More s p e c i f i c a l l y : l e t h E W " ; we a s s o c i a t e with it t h e : grid
R , = { M I M E R " , M = { m, h l,..., m m h , } , m i E E } .
(5.6)
With each node M o f
R,, we a s s o c i a t e the panel with c e n t r e M :
and the cross ( w i t h c e n t r e M ) :
where ei denotes t h e i t h u n i t v e c t o r i n We t h e n d e f i n e (5.8)
a h
(5.9)
= =
{MIm,'(M) c a
}
R"
.
9
c h a r a c t e r i s t i c f u n c t i o n of m:(M)
,
and (5.10)
v,=
space generated by
e, M e a h .
The f u n c t i o n s #, are not i n H'(Q), so t h a t V, is not a subspace of H'(Q). Hence, we cannot d e f i n e a ( u , O ) on V h and we must t h e r e f o r e modify a(u,v). For t h i s we r e p l a c e t h e d e r i v a t i v e s by d i f f e r e n t i a l q u o t i e n t s . I n g e n e r a l , we put
Then f o r u, u, E V, we define
(l)
The i n d i c e s "0" and "1" which appear i n t h i s n o t a t i o n correspond t o t h e d i f f e r e n t orders o f t h e approximation; w e can introduce "crosses" o f a r b i t r a r i l y high o r d e r . See Aubin / 2 / .
Exterior approximations
(uh
'h
+ i=1
6,'h)
dx
it can be s e e n immediately t h a t ( 5 . 1 3 ) admits a unique s o l u t i o n . The fundamental r e s u l t i s t h e n Theorem 5 . 1 .
As
h
+ 0
we have:
We can t h u s e x t r a c t a subsequence, a l s o denoted by u h , such that (5.18)
uh + w
,
6,uh + wi
weakly i n L'(62)
.
We s h a l l now v e r i f y t h e two f o l l o w i n g p r o p e r t i e s : aw axi
-
(5.19)
wi =
(5.20)
w E Hi@).
I n f a c t , f o r q ~ B ( 0 and ) for sufficiently s m a l l
( l)
1 h ( we
We can i n f a c t d e m o n s t r a t e s t r o n g convergence. (5.15) i s a d e q u a t e f o r our p u r p o s e s .
have:
However,
Approximation of steady-state inequalities
52
and p r o c e e d i n g t o t h e l i m i t :
(wi, cp) =
( C H A P . 1)
-
strongly i n L z ( a ) ) , and hence (5.19). I n - o r d e r t o v e r i f y ( 5 . 2 0 ) , we c o n s i d e r Eh = e x t e n s i o n o f u, t o W" by 0 o u t s i d e a ; s i n c e ( 5 . 8 ) i s t r u e , t h e n :
(a%& 6j, a,(&)
and Ch -+
= -+
6i(ch)
weakly i n L z ( a )
6ji
(5.19), it
However, as for hence
=)ELZ(W") axi
.
may be v e r i f i e d t h a t w"
i
a --(q - ax,
and
vi,
t h e r e f o r e w " = O on r , which g i v e s ( 5 . 2 0 ) . It now r e m a i n s t o show t h a t w = u For t h i s we t a k e U E V and define
.
It may be v e r i f i e d t h a t :
r, u + u ,
(5.22)
Then, t a k i n g gives a(w, u) =
dirk u uh
=
( Au)
rh
-+
u
ao axi
, we
vu E
strongly i n have uh(uh,uh)
-+
LZ(a),
a(w, u)
vi .
and hence ( 5 . 1 3 )
v,
and t h e r e f o r e w = u , which p r o v e s t h e theorem. We s h a l l now " a x i o m a t i s e " t h i s p r o o f ( l ) , by a l s o i n t r o d u c i n g "approximations" o f K and j . E x t e r i o r approximations f o r V , a, K, j .
5.2
Approximation of V. We c o n s i d e r a H i l b e r t s p a c e F and a n o p e r a t o r a w i t h
(5.23)
,
u E U(V ;F ) ,
a i n j e c t i v e from V into F.
For example, i n t h e c o n t e x t o f S e c t i o n 5 . 1 , V = and a i s d e f i n e d by (5.24)
(l)
ow =
{ ,: u,-,
H,'(a), F = (Lz(a)y+1,
"> .
..., ax,
I n order t o avoid r e p e t i t i o n i n t h e a p p l i c a t i o n s .
(SEC. 5 )
Exterior approximations
53
We next consider a family of finite-dimensional spaces V , , equipped with (Hilbert) norms (1 vh I ) y h . We assume prolongation operators p , E Y ( V , ;F ) to be given, with
For example, in the context of Section 5.1, V, is defined by ph is defined by
(5.10)and ( I v h ( I Y h is defined by (5.16); (5.26)
P h trh
{
=
vh,
d,vh,
..., 6mvh 1 ;
we have:
We say that V, constitutes an exterior approximation ( I ) of V if (5.27),
V,E
V,,
p h v h converges weakly towards
i n F=. ( E ~ V
and if (5.27) 2
{ 11
E
v , 30, E < c.
vh
with p , vh + a V
strongly in F and
IIYh
In the context of Section 5.1, we shall take (5.21); (5.22) then corresponds to (5.27)2.
uh=rhv
defined by
Approximation of a. We now consider a bilinear form a,(',, uh) on V, ( 2 , and we assume that
We say that the ah constitute an approximation of a if:
(') (2)
(3,
Terminology justified by the fact that V , Q v. In fact, the whole of this may be extended to cer'tain nonlinear problems. Where, in general, a*(rp, $) = a($, rp), a:(rp, $) = ah($,rp) This assumption becomes redundant in the case where a is symmetric and where we then choose a, to be symmetric (which is possible, if not essential).
.
54
Approximation of steady-state inequalities
(CHAP. 1)
and a l s o i f if
(5.31)
weakly i n F , t h e n
uv
+
h infah(v,,
2
4 0 , 0)
I n t h e c o n t e x t o f S e c t i o n 5.1, 8 have ( 5 . 3 0 ) and ( 5 . 3 1 ) .
.
8
u, i s d e f i n e d by ( 5 . 1 2 ) and we
Approximation of K . We t a k e a f a m i l y o f sets K, where
K, i s a c l o s e d convex s u b s e t o f V,.
(5.32)
& constitute an approximation of K
We s a y t h a t t h e
(5.33)
E K,,
p, v, converges weakly towards ( i n
F*
if: (E uK,
and i f f o r a l l U E K , w e can f i n d V , E & such t h a t p, v,, + m s t r o n g l y i n F and II v h (IYh d C.
(5.34)
Remark 5.1.
If
K = V, K h =
Vhy
we recover (5.27)1, (5.27)2.
Remark 5.2. The construction o f lu, and the satisfaction of ( 5 . 3 3 ) , ( 5 . 3 4 ) form one o f t h e main d i f f i c u l t i e s i n a p p l i c a t i o n s . We s h a l l g i v e some examples l a t e r ; w e s h a l l t h e n see t h a t t h e u s e of exterior approximations i s o f t e n more convenient t h a n t h a t of interior approximations. 8 Approximation of j. Let j be a p r o p e r convex f u n c t i o n ( s e e S e c t i o n 3.6) on v. a p r o p e r convex f u n c t i o n on V,. We c o n s i d e r j , We say t h a t t h e j , constitute an approximation of j i f : (5.35)
oh E V, ,
P h vh +
(5.36)
0, E v h ,
0E
uu weakly i n F
=s
lim infj,(v,) 3 j ( v ) ,
V , ph v,, + ov s t r o n g l y i n F
Some examples w i l l be g i v e n i n Chapters
j
4 and 5.
jk(vb)+ j ( v )
.
8
Approximation of the right-hand side. L e t f E V' . We c o n s i d e r t h e c o n t i n u o u s l i n e a r form X on V, and we s a y t h a t constitutes an approximation of f i f (5.37) I fh(vb I 11 vh IIVh . (5.38) Ph v, --* m weakly i n F =s fh(vh) + (f,0). 9
(SEC.
5)
55
Exterior approximations
5.3 Exterior approximation schemes
Approximation of probZem (4.24). We seek u h e K h with
Approximation of probZem (4.25).
We now have Theorem 5.2. Assume that the hypotheses (5.23) to (5.34) hold. Let uh (resp. u ) be the solutions of (5.39) o r (5.40), (resp. of (4.24) o r (4.25)). Then (5.41)
strongly in F .
phur + ou
Remark 5.3. Result (5.41) again gives (5.15).
B
Proof of Theorem 5.2. We shall perform the proof for the case of (5.39). The same reasoning applies to the case (5.40). We take uEKand uheKh which satisfy (5.34). Then, with this choice of in (5.39) we have:
ub d
(5.42)
ah(uh9 v h )
and hence using (5.281, a 11 uh Ilf Q and therefore
11 uh
- f k ( u h - uh) (5.291, (5.34) and (5.37) ( l )
c(1 + 11 uh 113
d c.
From (5.25) we then have: '(5*44)
11 ph uh Ilp
< c.
Hence, we can extract a subsequence, also denoted by u h , such that ph uh + (
weakly in F
56
Approximation of steady-state inequalities
which w i t h ( 5 . 3 3 ) shows t h a t (5.45)
= aw,
w e K and hence:
w EK.
weakly i n F,
phuh+ a w
(CHAP. 1)
However, f r o m (5.31), ( 5 . 4 5 ) gives (5.46)
lim inf ah@,,uh) 3 a(w, w ) .
Mcreover, s i n c e p h u h + a , s t r o n g l y i n F and s i n c e we have ( 5 . 4 5 ) , it r e s u l t s from ( 5 . 3 0 ) t h a t (5.47)
ah(uh, v&
+
a(w7 v)
3
and f r o m ( 5 . 3 8 ) t h a t (5.48)
- u& -* (f,
fh(uh
- w, '
Using ( 5 . 4 6 ) , ( 5 . 4 7 ) , ( 5 . 4 8 ) i n (5.42) w e deduce t h a t a(w, w) Q a(w, D)
- (A u - W)
VV E K ,
hence w = u which shows t h a t ph u, + ou. I n o r d e r t o prove ( 5 . 4 1 ) we i n t r o d u c e pkiih+ au
(
4
ah€&
with
strongly i n F
e x i s t s , f r o m ( 5 . 3 4 ) ) and w e put = ah(uh
xh
- 5,uh - s) .
W e have : = a,+(%
x h
- ah(&, u& - ah(% 4) + uh(ah* iik) .
From ( 5 . 3 9 ) we g e t xh
d
ah(ghr
uh) - fh(l)h
-
Uh)
- ah(ah$ uh) - ah(%, a&
We t h e n u s e ( 5 . 3 0 ) ( l ) and assume t h a t ( 5 . 3 4 ) . We t h e n deduce t h a t (5.49)
I),,
+
ah)
'
i s chosen as i n
lim sup xhd a(u, v ) - (J D - u) - a(u, u) Vv E K .
Taking v
=u
i n ( 5 . 4 9 ) w e deduce t h a t
Xh + 0
and hence, f r o m ( 5 . 2 9 ) , t h a t
(l)
The 2nd assumption i n (5.30) i s only involved i n t h e c a s e of strong convergence.
(SEC. 6 )
57
Exterior approximations
and hence, from (5.25), phuh- ph Ch + 0
strongly in F which gives (5.41).
Remark 5.4. In applications it is sufficient that (5.34) be satisfied on a dense subset of K , i.e. that the following property be satisfied: (5.50)
{
there exists X c K , X dense in K , such that Q V E X we can find vheKhwith ph + ov strongly in F and 11 vh (Ih < C .
In fact (5.34) results from (5.50); that is, V E K a n d
(Pzc.f
with Ilcpe-vll
,<E,
E + O .
From (5.50) there exists h, and p h , ~ Kwith:
11 Ph. (Ph. -
IIF
o
if
>0.
T
S i n c e B~ x B~ i s compact in R~ x F IP and s i n c e J ’ i s of c l a s s f o r a g i v e n T t h e r e e x i s t s a p a i r uo, u o € B M x BM such t h a t
II uo - uo II
=
‘5
,
- J‘(UO),
(J’(U0)
Uo
cO,
- uo) = 6&(‘5).
We s h a l l now show t h a t t h e f u n c t i o n 6; i s s t r i c t l y i n c r e a s i n g . Consider T I , ~2 w i t h O
0
+ r(u2 - U Z ) ) -
- uq), 0 d I < 1
.
However i f we i n t r o d u c e w by w
- u2
=
- (u2 - u 2 ) , ‘51
72
we have
II w
- uz
II
=
71
and i f , i n (1.91,we t a k e I = rl/t2 we g e t T2
6&(‘5,) > - (J’(w) - J ’ ( U , ) , w
- u2) >
71
> (J‘(w) - J ’ ( U 2 ) , w - u 2 ) 2 2 inf (J’(u) - J’(u), u - u) = 6&(r,) , IIu-uIl=r~
I~UEBH
which p r o v e s t h a t 6;
is s t r i c t l y increasing.
From t h e d e f i n i t i o n o f (1.10)
(J’(0)
u2).
i s s t r i c t l y monotone,
t h e r ef o r e (1.9)
-
S& i n (1.8) we have
- J‘(u), u - u) 2 s;(l
u -u
11).
if
0 0 is the smallest eigenvalue of A. If u is the solution minimising J over K, we have (1.47)
( J ' ( u " + ~ )U," + I
- U) 2 A, 11 U"+' - u 112 .
We shall now prove that the left-hand side of this i.nequa1it.y tends to zero, which will give
11 U"+' - u 11
-+
0
which will prove the theorem. Let, be the minimum over Ki of the functional ui
and
-+
+I J(u;+', ..., 41 , ui, U l + l , ..., 4) 1
This Page Intentionally Left Blank
Relaxation methods
(SEC. 1) (1.49)
=
71
P , ( ( l - a,,)#+ w , # + " ~ )
I f we assume t h a t
(1.50) 0 < g l Q w, Q 2 - g 2 , ei f i x e d . we a g a i n have t h e r e s u l t o f Theorem 1.3 ( w i t h t h e same p r o o f ) .
Remark 1.6. C l e a r l y , e v e r y t h i n g we have s a i d can be extended t o t h e c a s e of block relaxation (as i n S e c t i o n 1.3); s e e Glowinski 141, Cea and Glowinski 121. Remark 1.7. The p r o o f s g i v e n , which a r e based on estimates of the energy, can be extended t o the case o f Hilbert spaces, see
141,
Glowinski
Cea and Glowinski
121.
Remark 1.8. I n t h e u n c o n s t r a i n e d c a s e , Theorem 1.3 i s proved by means o f d i f f e r e n t methods i n Varga 111. Remark 1.9. I n t h e c o n s t r a i n e d c a s e , Auslender proposed t h e scheme:
-4"
(1.51)
where
8
= (1
0 1 ( t h i s can be s e e n g e o m e t r i c a l l y ) .
1(;+'"
Remark 1.10. C e r t a i n v a r i a n t s o f t h e above a l g o r i t h m s a r e u s e f u l i n t h e s o l u t i o n o f systems o f e q u a t i o n s . See S c h e c h t e r 111, 121, / 3 / and O r t e g a and R h e i n b o l d t /l/, 121.
1.5
A c l a s s o f n o n d i f f e r e n t i a b l e f u n c t i o n a l s which can be minimised u s i n g r e l a x a t i o n
We have s e e n (Counter-Example 1.1) t h a t , i n general, t h e r e l a x a t i o n a l g o r i t h m i s n o t s u i t a b l e f o r t h e m i n i m i s a t i o n of nondifferentiable functions. We s h a l l however prove a r e s u l t which i s i n f a c t v a l i d f o r t h e c l a s s o f f u n c t i o n a l s o f t h e form: N
(1.52)
J(U) = J,(u)
+ C ai I ui I ,
cr, >/ 0 ,
1=1
where J ,
is
cl,
s t r i c t l y convex, J,(v)
+
+ a, if
11 D 11 + a,.
@timisation
72
Theorem
algorithms
(CHAP. 2 )
1.4. If J is given by (1.52), the point relaxation (1.5) converges to the solution u of problem ( 1 . h ) .
algorithm
W e introduce supplementary variables ( ) yf, i = 1, we put:
c
..., N,y
= { y, } ;
N
-
Jib, u) = Jo(4
(1 53)
+ i=1 a, Yi
9
which d e f i n e s J 1 , (not s t r i c t l y ) convex and of c l a s s C1 on R z N ; we introduce t h e closed convex subset of R2N: ' N
(1 .54)
inf J,(u, y ) = Jl(u, I u I) = J(u) .
(1.55)
L
We introduce :
( i ) The sequence d' (n >, 0)corresponding t o t h e r e l a x a t i o n algorithm f o r J ;
( i i )The sequence z " , corresponding t o t h e ( b l o c k ) r e l a x a t i o n algorithm, with r e s p e c t t o
It can be shown t h a t i f w e s t a r t from
then
and hence
(1.59)
J l ( f ) = J(U")
.
Since
J(P)>, J(u"+') (this does not a s m e that J is d i f f e r e n t i a b l e ) ,
('1 ( 2,
We Put Jl(u,Y) = J1(ulrYlruZ,YZr...ruN,~N) = J 1 ( Z 1 , Z 2 r - . . , Z N ) . SO as t o transform an "unconstrained nondifferentiable" problem i n t o a "constrained differentiable" problem.
Relaxation methods
(SEC. 1)
aJ"(u;+1,
au,
i=l
3 aui
( s i n c e we have
(u;+l,
73
..., u;+l,u;+l,..., G)(u;-u;+I)t
..., q+',U;+,,..., 4) (4- 4") +
- fl") 3 0
from t h e d e f i n i t i o n of ). From ( 1 . 6 5 ) and ( 1 . 6 4 ) we deduce t h a t :
(1.66)
u"+1-
d + O .
.
It now remains t o show t h a t u " + u The r e a s o n i n g i s t h e same as i n t h e d i f f e r e n t i a b l e c a s e ( s e e ( 1 . 2 4 ) ) ; we p u t I* =
{u,IuIJ
and w e s t a r t from
(J;(Z"+') - J;(z*), 9'' - Z*)R>N = (Ji(un+') -
JA(u), u"+' - u ) R N 2 6lu(ll fl+' - u 11).
S i n ee (J;(z*), f"
=
- z*)R>N 3 0 ,
74
@timisation algorithms
(CIIAP. 2 )
we deduce
which, by w r i t i n g out t h e left-hand s i d e e x p l i c i t l y , g i v e s
From t h e d e f i n i t i o n o f { l(;+',y;+' } and s i n c e { u,, I u, 1 } E KI , t h e second term on t h e left-hand s i d e of (1.69) i s G 0 , and hence
Since
*+'- # - 0 ,
we deduce t h a t
If+u.
Remark 1.11. Theorem 1.4 supplements a result due t o Auslender 111, / 2 / where t h e convergence of t h e algorithm t o wards a " c r i t i c a l point" ( l ) , which can be d i s t i n c t from t h e s o l u t i o n ?A, i s shown.
Remark 1 . 1 2 . The algorithms given i n t h i s s e c t i o n a r e o f t e n considered as coming w i t h i n t h e c l a s s of so-called " d i r e c t " methods, i . e . t h o s e not u s i n g t h e expression f o r t h e d e r i v a t i v e of t h e f u n c t i o n a l - a t l e a s t i n the writing o f t h e algorithm. There a r e many o t h e r " d i r e c t " methods: t h e c o o r d i n a t e r o t a t i o n method, Rosenbrock's method, t h e method of l o c a l v a r i a t i o n s or t h e Hooke and Jeeves method; see, f o r example, Cea 121. These algorithms can be modified i n such a way Remark 1.13. t h a t ' t h e y can be used on p a r a l l e l processors ( s e e Morice 111).
(l) .
"Blockage point" i n t h e sense o f t h e Counter-Examples 1.1 and 1 . 2 .
Gradient and gradient p r o j e c t i o n methods
(SEC. 2 )
75
SYNOPSIS
W e s h a l l now g i v e a b r i e f review of t h e methods which use t h e d e r i v a t i v e o f t h e f u n c t i o n a l t o be minimised.
2.
METHODS OF THE GRADIENT AND GRADIENT PROJECTION TYPE
Here we s h a l l only p r e s e n t t h e algorithms. For t h e proofs w e refer t h e r e a d e r t o Cea / 2 / ( s e e , i n p a r t i c u l a r , pp. 70-109 and pp. 118-156) and t h e bibliography t h e r e i n . 2.1
General remarks
Assuming zi" t o hsve been c a l c u l a t e d , we seek tP+'using:
(2.1)
= If
- P,,w",
where pn 3 0 i s t o be chosen, as i s t h e d i r e c t i o n w " . Assuming J t o be d i f f e r e n t i a b l e and u s i n g formal reasoning we have :
(2.2)
- pn w?
J(tP
- p,(J'(@,
= J(u?
w")
+ ...
so t h a t it would be o f advantage t o t a k e
(2.3)
(J'Ww") 3 0 .
The most n a t u r a l choice i s t o t a k e
(2.4)
W"
= J'(@
and t h i s l e a d s t o methods of t h e g r a d i e n t type.
2.2
Methods of t h e g r a d i e n t t y p e (unconstrained c a s e )
- Optimal s t e p method. We choose p,, so t h a t
(2.9
J(d
- p,, J'(u")) = infJ(tP - pJ'(u3). P
Assuming
(2.6)
J s t r i c t l y convex, o f c l a s s t h e method i s convergent.
- Fixed s t e p method. W e choose
c',
J(u)+ 8
+
a)
if
11 u 11 + a,
76
Uptimisation algorithms
(2.7)
(CHAP. 2)
p,, = p f i x e d .
The method i s c l e a r l y e a s i e r t o apply, but t h e c o n d i t i o n s f o r convergence a r e f a i r l y s t r i c t ; f o r example, i f w e assume J t o be of c l a s s ~2 and if we put
t h e method converges i f : 0 < 6 G p G 2 ~ -1 6 ,
(2.9)
PI
G PI.
- Variable step method. We seek p a p r i o r < i n t h e form
i
(2.10)
P =dp0. k E Z , po > 0 , a > O and k t o be a d j u s t e d i n t h e course of t h e i t e r at ions.
We choose p of the form ( 2 . 1 0 ) such t h a t J(u" - P J ' W
(2.11)
< JW)
J'(u9 # o ). The method i s again convergent
( w e assume
-
if po is sufficiently small.
Divergent series method.
We consider a sequence p,, such t h a t p,, > 0, p. + 0 and
p,, = n= 0
+ a).
We t h e n d e f i n e
The choice of a s u i t a b l e sequence pn i s very t r i c k y , and t h i s method g e n e r a l l y converges very slowly.
2.3
Methods of t h e conjugate g r a d i e n t t y p e (unconstrained case)
Consider t h e quadratic case
(2.13)
J(u) = ~ ( A uU), - (f,U) ,
where A i s a symmetric p o s i t i v e - d e f i n i t e (N,N) matrix. The d i r e c t i o n s { w " , . . . , # - I } of dy are s a i d t o be conjugate with
Gradient and gradient projection methods
(SEC. 2 )
77
respect t o A i f (2.14)
( A d , d )= 0 if
i#J,
# 0
otherwise.
S t a r t i n g from y o , we assume d ' t o have been c a l c u l a t e d and search f o r d'+' by dispzacement i n t h e d i r e c t i o n w " , t h u s (2.15)
uR+' =
U" + h " ,
1 being c a l c u l a t e d i n such a way a s t o minimise J(d'+l)
= ad')
+ 1 2 (Aw",w 3 + Y(AU",w") - ( A w")) , 7 J
and hence
A f t e r N i t e r a t i o n s , we thus a r r i v e ' a t
However a d i r e c t c a l c u l a t i o n shows t h a t , Vg'BE RN
, we
have
so t h a t (2.17) g i v e s (2.19)
d" = A - ' f
= u = s o l u t i o n o f t h e problem.
Hence t h e algorithm converges i n a f i n i t e number o f i t e r a t i o n s . The problem now becomes, f i r s t o f a l l , t o discover how t o generate t h e conjugate d i r e c t i o n s , wo, ..., d-l. S t a r t i n g from u o , w e t a k e as t h e f i r s t d i r e c t i o n t h e d i r e c t i o n of s t e e p e s t descent a t u o , bence (2.20)
wo =
- J'(U0)
i.e. i n t h e q u a d r a t i c c a s e (2.21)
wO'=
- AuO + f .
We t h e n seek u1
= uo
+ 1wo ,
c a l c u l a t e d s o a s t o minimise J(uo + form (2.22)
w1 =
1wy
; we t h e n seek w' i n t h e
- J'(U') + p w 0 ,
where p i s chosen i n such a way t h a t w'and
w1 are conjugate, i.e.
Optimisation algorithms
(CHAP. 2 )
(w', AwO) = 0 ,
and sc on. We hence arrive at the foZZowing algorithm: ro= -J'(uo), w o = r o ) We assume u", In, w" t o be known ( uo a r b i t r a r y , and w e d e f i n e :
then
(2.24)
In"
=
- J'(u"+'),
I n t h e non-quadratic c a s e , t h e algorithm i s adapted as follows. We d e f i n e :
(2.26)
u"+' = u " + K w " ,
where
i s chosen so as t o minimise J(u" p+' =
+ pw"), F E W
; th'en
- J'(u"+').
I n choosing w"" We put
it i s necessary t o t a k e c e r t a i n p r e c a u t i o n s .
We check whether t h e v e c t o r s In+' and P+'form an angle with s u f f i c i e n t l y s m a l l 6 > 0 . If so, we put
(2.28)
-II - 6, 2
w"+' = 3"
otherwise, t h e d i r e c t i o n 9""
(2.29)
Q
must be discarded and we t h e n t a k e
w"+' = I n + ' .
The algorithm i s convergent, by v i r t u e of t h e c l a s s i c a l assumptions (1.2), (1.3) and with J of c l a s s C1. From a numerical p o i n t of view, it i s advantageous t o t a k e t h e following a d d i t i o n a l precaution: i f , f o r N successive i t e r a t i o n s , j = i , i + l , ..., i + N - 1 , we have
d#r' we put w'+N= # + N .
Remark 2.1. The method described above i s t h e so-called conj u g a t e g r a d i e n t method. There a r e v a r i o u s methods which allow
79
Gradient and gradient projection methods
(SEC. 2 )
t h e c o n s t r u c t i o n of N conjugate d i r e c t i o n s . I n p a r t i c u l a r , w e mention t h e method o f F l e t c h e r and Powell which a l s o allows t h e i n v e r s e m a t r i x o f J r r ( u )t o be obtained. An a p p l i c a t i o n of t h e so-called conjugate d i r e c t i o n method t o t h e problem of t h e minimisation of
J(u) = M
u , v ) - (f,0)
s u b j e c t t o t h e c o n s t r a i n t s Bv = b , w i l l be given l a t e r ( s e e Remark 4.10). Constrained case
2.4
We now consider t h e system
(2.30)
infJ(u),
UEK,
K = closed convex subset of RN.
Here, w e s h a l l adapt t h e above methods by using t h e operator
Pr p r o j e c t i n g onto K. - Point projection method
S t a r t i n g from uo E K u"+' = PJu"
(2.31)
, we
d e f i n e u"+' from u" using
- P.J'(u"))-
Theorem 2.1. Consider the case (2.13) (quadratic functional). We can now choose po and p1 so that with
(2.32)
0 < Po Q
Pn
Q
PI
the method (2.31) converges t o the solution u of problem ( 2 . 3 0 ) . Proof. (2.33)
If u i s t h e s o l u t i o n of t h e problem we have:
u = PA.
- p,J'(u))
vp,
s i n c e (2.33) i s equivalent t o (u
- p, J'(u) - u, u .-
By p u t t i n g w" = u"
u) Q 0 Vu E K , i.e. (J'(u), u
- u and u s i n g t h e f a c t t h a t Pr is
c t i o n we deduce from ( 2 . 3 1 ) , (2.33) t h a t (2.34)
- u) 2 0
11 w'+' (1 G 1 w" - p,(J'(u") - J'(u))
1
Vu E K .
a contra-
80
@timisation However, J'(u")
(2.36)
aZgorithms
- J'(u) = A(u" - u) = Aw"
II w"+' 11' d
(1
- 2 ap,
(CHAP. 2 )
so t h a t (2.35) g i v es
+ CpX) 11 w" [ I z .
We c a n choose po and p1 i n s u c h a way t h a t ( 2 . 3 2 ) g i v e s
1 - 2ap,
+ CpX Q B < 1
and t h e n ( 2 . 3 6 ) shows t h a t
11 w" \I
+ 0.
Remark 2 . 2 . The above p r o o f i s r e a d i l y e x t e n d e d t o t h e nonquadratic case with Lipschitz J ' .
- Gradient
projection method.
I n t h e u n c o n s t r a i n e d i t e r a t i o n method o f ( 2 . 3 1 ) , t h e p o i n t s We a r e now g o i n g t o project the gradients. were p r o j e c t e d o n t o K. We assume:
(2.37)
t h e convex s e t K i s bounded and d e f i n e d by a f i n i t e number o f a f f i n e l i n e a r c o n s t r a i n t s
K = { u I Gj(u) = 0, 1 and t h a t f o r a l l p o i n t s CJu)
Qj Q
I, C,(u) Q 0, 1
+ 1 Q k Q rn }
U E K ,t h e v e c t o r s
, j E I(u) , I(u) = ( j I 1
Q j Q rn, Cj(u) = 0 } ,
are l i n e a r l y i n d e p e n d e n t . Then t h e p r o j e c t i o n o n t o K o f t h e s t r a i g h t l i n e p r o c e e d i n g from , i . e . t h e s e t o f p o i n t s w such t h a t
v i n t h e d i r e c t i o n - J'(u) (2.38)
w = PJu
-~J'(u)),
p 30
i s a segmented l i n e , t h e f i r s t segment b e i n g d e n o t e d by [u, u + ] The a l g o r i t h m i s t h u s as f o l l o w s : having o b t a i n e d ~ E E K w, e define by: (2.39)
J(u"+ ') Q J(u) , Vu E [u", (u")']
.
.
The d i r e c t i o n [ u , u + ] i s o b t a i n e d u s i n g a n o p e r a t o r which p r o j e c t s onto an a f f i n e l i n e a r manifold. If t h e m a n i f o l d i s d e f i n e d by
s=
{uIAu=b}
t h e p r o j e c t i o n o f a v e c t o r a o n t o S i s g i v e n by
P&)= [ I - A * ( A A * ) - ' A ] u + A * ( A A * ) - ' b . where A* i s t h e t r a n s p o s e o f t h e m a t r i x A . The method i s c o n v e r g e n t ( e . g . u s i n g t h e assumption ( 2 . 3 7 ) and w i t h J i s s t r i c t l y convex and of c l a s s Cl) ( s e e Rosen /l/, /2/ and Canon, C u l l u m and Polak /l/).
(SEC. 3 )
Penalisation methods
81
Remark 2.3. Apart from t h e gradient projection method, it i s a p p r o p r i a t e t o mention t h e reduced gradient method introduced by Wolfe /1/ i n t h e context o f t h e minimisation o f convex functions s u b j e c t t o linear constraints, by e x t e n s i o n of t h e simplex method. This method has been g e n e r a l i s e d t o t h e c a s e of nonlinear constraints by Abadie-Carpentier /1/ ( s e e a l s o F l e t c h e r 111). A method somewhat similar t o t h e above i s t h a t of FrankWolfe 111. 3.
PENALTY METHODS AND VARIANTS
3.1
General remarks
The i d e a of p e n a l i s a t i o n ( a l r e a d y encountered i n Chapter 1, Sect i o n 3.2; s e e i n p a r t i c u l a r Remark 3.4) c o n s i s t s of “approxima t i n g ” t h e problem,
inf J(u) , ueK, K = a c l o s e d convex subset of RN, by unconstrained problems ( a s w i l l be t h e c a s e i n t h e exterior method d e s c r i b e d i n S e c t i o n 3.3 below), or by problems with passive constraints, i . e . where t h e i n f i s a t t a i n e d in the interior o f t h e set o f c o n s t r a i n t s (as w i l l be t h e c a s e i n t h e interior method ( S e c t i o n 3.2 below) and i n t h e method of c e n t r e s with v a r i a b l e t r u n c a t i o n described i n W Section 3.4 below) (I). I n t h e followi’ng w e s h a l l assume t h a t K i s defined by
(3.2)
{ G,=
K={ulG,(u)>O, I G j G m } , a concave continuous f u n c t i o n on
RN.
We s h a l l assume t h a t J s ’ a t i s f i e s :
(3.3) 3.2
J i s d e f i n e d and continuous on
w,
s t r i c t l y convex.
I n t e r i o r methods
We i n t r o d u c e t h e f u n c t i o n a l
We make t h e assumptions:
(l)
C l e a r l y it then becomes necessary t o choose an algorithm f o r the solution of the penalised problem.
8
@timisation algorithms
82
{ KK
(3.5)
0
, w i t h a nonempty = { u I GLu) > 0, j = 1, ..., m 1. i s bounded ( I )
(CHAP. 2 )
interior
i , given
by
This l a s t c o n d i t i o n i s a s s u r e d when t h e r e e x i s t s uo such t h a t G,(uo) > 0 ,
(3.6)
-
j = 1, ..., m .
The $unction u + I(u, E) i s t h e r e f o r e n o t i d e n t i c a l t o +a or t o on K. I n f a c t ,
a)
Lema 3.1. Using the hypotheses ( 3 . 2 ) , (3.31, ( 3 . 5 ) , there e x i s t s a unique u, such t h a t : (3.7).
U,EK,
(3.8)
Z(U,, E ) Proof.
< I(u, 8)
Vu E
2.
With t h e element u,chosen as i n ( 3 . 6 ) , we i n t r o d u c e
the set
(3.9)
S, = { u I u E K, Z(u,
E)
< Z(u0, E) } ,
which i s nonempty s i n c e u,ES,
.
It can be shown t h a t S, i s
closed (and hence compact s i n c e S, c K , where K i s bounded); s i n c e I(u,E) i s bounded on S, t h e Cju) remain > 0 on S, and I(u,&) i s continuous on S,., which g i v e s t h e r e q u i r e d r e s u l t . Furthermore, 0
(3.10)
S, c
K, rn
and hence t h e lema.
W e can now s t a t e and prove:
Under the assumptions (3.21, ( 3 . 3 ) , ( 3 . 5 ) i f u, Theorem 3.1 i s the element defined by (3.71, ( 3 . 8 ) we have (3.11)
u,+u
i n RN
where u i s the soZution of (3.12)
uEK,
J(u) = i d J(u) . veK
Remark 3.1. T h e o p e n a l t y method t h u s d e f i n e d i s s a i d t o be interior since u , E K . (I)
The assumption " K bounded" i s i n t r o d u c e d o n l y w i t h a view t o s i m p l i f y i n g t h e p r o o f s . I n p r a c t i c e , it i s s u f f i c i e n t t o assume t h a t t h e m i n i m u m o f J ( V ) o v e r K i s reached f o r a p o i n t u a t a f i n i t e d i s t a n c e , which i s t h e c a s e if, f o r example, J s a t i s f i e s ( 1 . 2 ) .
(SEC. 3 )
Penalisation methods 0
83
Since u,EKand K i s bounded, we can e x t r a c t a subsequence, a l s o denoted by u , , such t h a t
Proof.
(3.13)
wsK
u,4winW,
We have
which, i n t h e l i m i t , g i v e s
J(w)
< J(u)
vu E
i,
and hence
J(w) d J(u) Vu E K and s i n c e J i s s t r i c t l y convex, w = u , g i v i n g t h e r e q u i r e d result.
Remark 3.2.
We have t h e following a d d i t i o n a l property: The f u n c t i o n E -D J(u3 i s decreasing towards J(u) as E + 0. To show t h i s , l e t 0 < q < E . We p u t :
From t h e d e f i n i t i o n s of u, and u,, we have:
J(u,)
+
1 E-
and
G(u3
Q J(u$
+E
1
(3.15)
so t h a t , by a d d i t i o n ,
and hence
1 ---GO.
G(u3
1
G(u,)
Thus (3.15) t h e n g i v e s
3.3
E x t e r i o r methods
We introduce t h e f u n c t i o n a l
(3.16)
E(u,E )
= J(u)
+ -1 G(u)E
G
(4
84
Gptimisation algorithms
(CHAP. 2 )
where by d e f i n i t i o n
(3.17)
G(D)- =
2 Gj(u)- 2 sup(- Gl(u),0). =
]= 1
j= 1
We n o t e ( c f . Chapter 1, S e c t i o n 3 . 2 ) t h a t
G(D)- = 0 0 D E K .
(3.18)
We do not make any assumption analogous t o ( 3 . 5 ) . Since, i n p a r t i c u l a r , K i s n o t assumed t o be bounded, it i s n e c e s s a r y t o assume t h a t (3.19)
if 1 1 u I I + c o ,
J(D)-,+co
u€RN.
Then, s i n c e
(3.20)
E(D, E ) 2
J(D),
w e h a v e , a f o r t i o r i , E(D,E ) +
+ co
if
11 D 11 + co and hence, w i t h f l u , & )
s t r i c t l y convex i n V ,
t h e r e e x i s t s a u n i q u e u, i n @ s u c h t h a t = inf E(o, E ) .
(3.21)
V€RN
We t h e n have: Theorem 3.2.
fied.
Then, if
(3.22)
u,+ u
We asswne t h a t ( 3 . 2 ) , ( 3 . 3 ) , ( 3 . 1 9 ) are s a t i s is d e f i n e d by (3.21) and u by (3.12) we have:
u,
i n RN.
Remark 3.3. The element a p p r o x i m a t i n g u, is not n e c e s s a r i l y hence t h e t e r m i n o l o g y : e x t e r i o r method. rn in K; Proof. (3.23)
We have:
J(uJ
< flu,,
E)
< uinf E(D,E ) sK
= inf J(u) = J(u) V € K
from which it r e s u l t s t h a t u, l i e s i n a bounded s u b s e t o f RN. We can , t h e r e f o r e e x t r a c t a subsequence, a l s o d e n o t e d by u, , such t h a t u, + w i n RNand w e deduce from ( 3 . 2 3 ) t h a t
(3.24)
J(w) < J(u) .
Moreover from ( 3 . 2 3 )
1 ;G(u,)-
d J(u) - J(u,)
therefore G(w)- = limG(uJ- = 0 t-0
Penalisation methods
(SEC. 3 )
85
and hence ( s e e ( 3 . 1 8 ) : W E K which, along with (3.24) shows t h a t w = U. w
Remark 3.4. I t o s h o u l d be emphasised t h a t od s t i l l holds i f K = 0 . This allows t h i s f o r t h e s o l u t i o n of problems involving linear ints: t h e p e n a l i s a t i o n term a s s o c i a t e d with Gj(u) = 0 t h e n t a k e s t h e form
Remark 3.5. G(v)- =
c
[SUP(-
t h e e x t e r i o r methmethod t o be used
equality constraa constraint
I n (3.17) we can a l s o d e f i n e G(u)- by G,(u), 0)F with
Q
3 1.
j= 1
Remark 3.6. I n t h e context of e x t e r i o r methods, t h e following v a r i a n t can a l s o be used: with each c o n s t r a i n t Gku) 2 0 ( j = 1,m) we a s s o c i a t e a v a r i a b l e ti >/ 0 ( c a l l e d a s l a c k v a r i a b l e ) from which we g e t t h e equivalent c o n s t r a i n t p a i r { G,(u) - ti = 0, ti 2 0 ) and t h e p e n a l i s e d f u n c t i o n a l
F(u,t , E )
= J(u)
+ -1 1(Gj(U) "I
tj)'
E j=l
where t = (II, ..., 1,) E R ; t ; t h e p e n a l i s e d problem t h e n c o n s i s t s of minimising f l u , t , E ) over R" x R;t . The advantage of t h i s approach over t h a t described above i s t h a t t h e o r d e r of d i f f e r e n t i a t i o n of J and of t h e Cj i s conserved a t t h e c o s t of i n t r o d u c i n g t h e v e c t o r t ( t h e c o n d i t i o n t h a t t h e t j must be p o s i t i v e does not i n t r o d u c e any a d d i t i o n a l p r a c t ical difficulties). The r e s u l t s of S e c t i o n 3.3, concerning f l u , & ) as defined by (3.1-6), a r e e a s i l y adapted ( s e e , f o r example, Cea / 2 / ) . An a p p l i c a t i o n r e l a t i n g t o t h e model e l a s t o - p l a s t i c problem defined i n Chapter 1, Section 1 . 2 i s given i n Chapter 3, Section 8.2.2. 3.4
Method of c e n t r e s with v a r i a b l e t r u n c a t i o n
General remarks. We i n t r o d u c e , a g a i n f o r problem ( 3 . 1 ) : (3.25)
K(I)=(ulu~K,J(~)brl}
which d e f i n e s a nonempty c l o s e d convex s e t i f :
I B J(u) ( and K(J(u)) = { u
1).
The formal i d e a i s t h e n t o c o n s t r u c t an algorithm which determines : .
86
@I timisation a Zgorithms
(CHAP. 2 )
( i ) a d e c r e a s i n g sequence of & (- J(u)) ( i i ) a " c e n t r e " of K ( & ) a s s o c i a t e d w i t h e a c h & ( i n f a c t An+] i s chosen u s i n g t h e " c e n t r e " of K(A,J). rn
"Centre of a conuex set". The c o n c e p t o f t h e " c e n t r e " o f a convex s e t is not intrinsic, and numerous b a s i c a l l y e q u i v a l e n t c o n c e p t s c a n be d e f i n e d . Here w e s h a l l g i v e t h e two c o n c e p t s which seem t o be t h e most n a t u r a l and t h e most u s e f u l . I n g e n e r a l , l e t 9 b e a c l o s e d convex s u b s e t o f R N d e f i n e d by
(3.26)
9 = { v I LAu) 3 0, 0 Q j Q m }
where, V j :
(3.27) 9
Lj i s a c o n t i n u o u s concave f u n c t i o n .
We t h e n c o n s i d e r t h e " c e n t r e " o f 9 t o be either t h e p o i n t i n , i n f a c t i n g o , where
(3.28)
fi LAv)
is maximum
1-0
or t h e p o i n t i n 9 , i n f a c t i n Y o , where 1
i s minimum.
8
AppZication of the concepts (3.28), (3.29) to K ( h ) . We u s e t h e above c o n c e p t s f o r
K(X), w i t h
Lo(v) = A - J(v) , t , ( v ) = Gkv) , 1 Q j Q rn
.
I n o r d e r t o s i m p l i f y t h e n o t a t i o n , we p u t :
n G,(v) , $1
(3.30)
D(u, A) = (1 - J(v))
I= 1
rn
(3.31)
AZgorithm.
We c o n s i d e r
VoEK
.
We t a k e :
A, = J ( v 0 ) . Then we d e f i n e vl t o be t h e " c e n t r e " of (I), i.e.: (l)
K(A,) u s i n g method (3.28)
If w e u s e (3.29), u1 ( d i f f e r e n t from t h e above) i s d e f i n e d by
Penalisation methods
(SEC. 3 ) (3.32)
D(u,, 1,) =
SUP
87
D(u, 11). ( ')
v e 4111
We next d e f i n e
(3.33)
J(w,) d
w, E K ( A , )
( s e e Remark 3.7 below) such t h a t
J(U1)
and then d e f i n e A2 by (3.34)
1,
=
1, - pl(l, - A w l ) )
0 < p1 < 1
Assuming un-, and 1, t o be known w e d e f i n e v, by (3.35)
D(U., 1,) = sup D(u, 1"). u E 41.1
We d e f i n e
(3.36)
K(AJ such t h a t
W,E
d
J(WJ
J(Vn)
and w e choose (3.37)
A,,
1
=
1, - P.[A. - J(wJ1
where (3.38)
0aIIu1l2,
a>O,
a c o n t i n u o u s b i l i n e a r form on V sat-
VUEV
(where IIuII = t h e norm o f 2, i n V ) , and where i n ( 4 . 1 ) a c o n t i n u o u s l i n e a r form on V. We a l s o t a k e :
(4.3) (4.4) (4.5)
M
{
u+(Ju)
is
= c l o s e d convex s u b s e t o f V.
L = H i l b e r t s p a c e , a n d Q a f u n c t i o n from V M -+ L , l i n e a r or otherwise.
-+
L or from
A = c l o s e d convex s u b s e t o f L .
If ( , )L d e n o t e s t h e i n n e r p r o d u c t i n L , w e assume f o r a l l functions q E L t h a t : (4.6)
{
u + (q, @(u))~ i s lower semi-continuous and convex on V .
We t h e n c o n s i d e r t h e problem r
Remark 4.1. Thus, by d e f i n i t i o n , w e c o n s i d e r t h e problem i n a form a d a p t e d t o d u a l i t y ( s e e Chapter 1, S e c t i o n 3.4).- W e s h a l l s e e i n S e c t i o n 4 . 2 below how a number of i m p o r t a n t problems come w i t h i n t h e framework o f ( 4 . 7 ) .
Remark 4.2.
The f u n c t i o n
Qptimisation algorithms
90
(CHAP. 2 )
i s convex, l o w e r semi-continuous f o r t h e weak t o p o l o g y o f V ; t h i s i s t h e r e f o r e a l s o t h e c a s e for J d e f i n e d by (4.8)
4.2
40) = Jo(u) + SUP (q, “ ( 1 ) ) ) ~ . ~ E A Examples
Example 4 . 1 .
Using t h e n o t a t i o n
V = H;(Q), a(u,u)=
J‘,
A4
=
(1.18), Chapter 1, we t a k e :
V,
gradu.gradudx,
(f,u)=~,jiudx, feL2(Q),
L = (L2(Q))”, #u = grad u hence # E U ( V ;L ) , A = { q I q E L , Iq(x)l Q g a . e . i n Q } . We have ( s e e C h a p t e r 1, S e c t i o n 3.4 a b o v e ) : r
Then ( 4 . 7 ) becomes the flow problem c o n s i d e r e d i n Chapter 1, S e c t i o n 1 . 4 , which h a s a l r e a d y been c o n s i d e r e d from t h e p r e s e n t s t a n d p o i n t i n Chapter 1, S e c t i o n 3.4.
Example 4 . 2 .
We t a k e V,M,a, (f,v) a s i n Example
L = L2(s2), A , and we d e f i n e
G1 : V + L
Gl(u) = I grad u I ’
= [q
I q E L,
4.1:
q2 0
by:
- 1.
The f u n c t i o n #, i s L i p s c h i t z on V. T h i s comes w i t h i n t h e c o n t e x t o f S e c t i o n
4.1 and we have:
I n t h i s example, ( 4 . 7 ) i s t h u s t h e e l a s t o - p l a s t i c i t y problem c o n s i d e r e d i n Chapter 1 , ’ S e c t i o n 1.2. i s nondifferentiabze, and s i n c e t h i s i s a d i s W e note t h a t a d v a n t a g e ( f r o m t h e n u m e r i c a l p o i n t o f view ( l ) ) w e i n t r o d u c e G2 : H{(s2) -+ L’(s2) and A2 d e f i n e d by
(I)
E a s i l y surmountable:
s e e Chapter
5 , S e c t i o n 8.
(SEC. 4 )
Duality methods
91
indeed we t h e n have: sup
q@'(u)dx =
4eAz
0 if I g r a d u l d 1 a.e.
+
oc)
otherwise.
but, amongst o t h e r d i f f i c u l t i e s , t h e Hilbert space s e t t i n g of Section 4 . 1 i s not a p p r o p r i a t e f o r a2, i n t h e infinite dimensional case ( ) , s i n c e aZ(u) E~'(0). We n o t e t h a t @' is i n d e f i n i t e l y F r e c h e t - d i f f e r e n t i a b l e (and hence l o c a l l y L i p s c h i t z ) on Hh(62). 4.3
A saddle-point s e a r c h algorithm
Let u s now i n t r o d u c e t h e Lagrangian (4.9)
U(V9
r
4) = J O ( d
+ (4. @(U))L .
We make t h e following assumptions: (4.10)
t h e r e e x i s t s a saddle p p i n t of P(u, q) on M x A, i . e . a point { u , p } ~ M x A such t h a t U(u,q) 0 i s t o be chosen a p p r o p r i a t e l y ( s e e below).
rn
Remark 4.3. The f u n c t i o n u + Jo(u) + (p", @ ( u ) ) ~i s s t r i c t l y convex and it is " i n f i n i t e at i n f i n i t y " ; i n f a c t , from ( 4 . 1 1 ) where we t a k e u t o be f i x e d a r b i t r a r i l y i n M , we have
I1 @@IIIL Q C(ll v II + 1) and hence
(where t h e
c
denote v a r i o u s c o n s t a n t s ) .
Remark 4.4. (Motivation of t h e a l g o ri t h m ) . saddle point then
If { u , p }
is a
which i s e q u i v a l e n t t o (4.18)
.
p = pA(p +
P @( U) )
VP >
and (4.18) e x p l a i n s ( 4 . 1 3 ) .
Remark 4 . 5 . I n f a c t t h e a l g o r i t h m i s n o t completely d e f i n e d s i n c e i n ( 4 . 1 5 ) we have the choice of the algorithm f o r calculating u". I n S e c t i o n 4 . 4 we s h a l l meet a v a r i a n t i n which t h i s choice is sp ec if ie d. rn Convergence df t h e algorithm. We s h a l l now prove:
...,
We assume t h a t ( 4 . 3 ) , (4.6) are t r u e , along Theorem 4.1. with (4.10) and ( 4 . 1 1 ) ( o r ( 4 . 1 1 ' ) ) . Thus t h e algorithm defined by (4.12), ( 4 . 1 3 ) is convergent i n the sense t h a t
(SEC. 4 )
DuaZity methods
(4.19)
U"
93
strongly i n V ,
+u
where u i s the solution of (4.71, when (4.20)
0 < a. d p. d aI ,
with suitabZe a. and
1 a1 ( )
P r o p e r t i e s (4.12) , (4.16) a r e e q u i v a l e n t t o
Proof.
- U") + (p". @(u) - @(U"))L 3
(4.21)
(Jh(u"), u
(4.22)
(Jd.(u), u - u )
+ (p. @(u) - @(u));
0 Vv E M
3 0 vu E
9
;
t a k i n g u = u ( r e s p . u = u" ) i n (4.21) ( r e s p . ( 4 . 2 2 ) ) we t h e n deduce (4.23)
O(ff
- U, U" - U) + (p" - p, @(u") - @(t())L d
0.
W e put (4.24)
p"
-p
= r".
From (4.13) , (4.18) and t h e f a c t t h a t P A i s a contraction, w e deduce t h a t (4.25)
I1 r"" 1; d
11 r" + p,,(@b") - @(u))1;
+ 2 PAP, @(fi- @(&
+ PI I
11 r" 1; + Yu") - @(4Ilt ' =
However, u s i n g (4.23) and ( 4 . 1 1 ) ( o r ( 4 . 1 1 ' ) ) , we deduce from ( 4.2 5 ) t h a t
We t h e n choose a. and a1 i n such a way t h a t (4.20) implies t h a t (4.27)
2 ap,
- C: Pf 2 b > 0 .
Then ( 4 . 2 6 ) g i' v e s (4.28)
11 r"" 1;
+ fl 1I U" -
11' d 11 r" 1.;
From (4.28) we g e t t h a t t h e sequence n + 11 r" ;1 and hence t e n d s t o a l i m i t , s o t h a t BllU"-ull2+O
is decreasing
and hence (4.19)
Remark 4.6. For some g e n e r a l r e s u l t s on t h e e x i s t e n c e of saddle p o i n t s , s e e Ekeland-Temam /l/. (l)
E s t i m a t e s o f aoand a1 are s u p p l i e d i n t h e proof which follows.
Optimisation aZgorithms
94
4.4
(CHAP. 2 )
A second saddle-point search algorithm
We shall now supplement Remark 4.5, by assuming that
Then in (4.12)u" is defined by
JiY)+ @*p" = 0 , i.e. ( A E ~ ( V V? , (4.30)
Au"
is defined by a(u,u) = ( A u , ~ ) ) :
+ @*p" - f = 0
and if we introduce an iterative algorithm for the solution of (4.30) we naturally arrive at the following algorithm: with u" and p" assumed to have been calculated, we define #+'by ( l ) : (4.31)
I.4""
= u" - p1 S-'(Au"
+ @*p" - f)
where: S = identity if V ' = V = finite-dimensional space ( o r alternatively S = arbitrary symmetric positive-definite matrix) , S = the duality operator from V + V' if V is infinite-dimensional ( 2 ) , and we then definep"+'by: (4.32)
p"" = P,(p"
+ p2 @ d + ' ) ,
where in (4.31) and (4.32) p 1 and p 2 are two parameters > 0 to be suitably chosen. We have:
.
Theorem 4.2. We assume that ( 4 . 3 ) ,. . ,(4.6), (4.29) are true. we can then choose p 1 , p 2 > 0 ( 3 ) i n such a way that algorithm (4.31), (4.32) i s convergent i n the sense t h a t (4.33)
u" + u
strongly i n V
where u i s the solution o f (4.7).
Proof. (4.34)
If (u,p} denotes a saddle point of 9 ( v , q) we have:
u =u
- p1 S-'(Au
+ @*p
- f),
(l)
This is essentially an algorithm of the Arrow-Hurwicz type /l/.
(2)
If we know that the solution u of the problem is in a reflexive space W c V , we can take an isomorphism from W + W ' for S.
(3)
Estimates of p 1 and p 2 result from the proof below.
(SEC. 4 ) (4.35)
IiuaZity methods
+
p = P,(p
p2
95
@u) .
P u t t i n g w " = u " - u and using t h e n o t a t i o n (4.24) we g e t :
- p 1 S-'(Aw" + @ *
(4.36)
w"+'
(4.37)
I1 r"+' Ilf c 1I P 1;
= W"
r"),
+ 2 p2(r", @w"+')' + p;
11 @W"+' 1.;
We deduce from (4.36) t h a t SW"+' =
sw" - p,(Aw" + @* P )
and hence, t a k i n g t h e i n n e r product o f both s i d e s with w"+' i n t h e d u a l i t y between V ' and V :
11 w"+' 112 = ((S - p' A ) w", W " + ' ) - p l ( @ * P, w " + ' ) . Since
A * = A , (Au, u ) 2
OT
)I v 112,
w e have:
Thus :
so t h a t (4.39)
, if
0 0 : indeed, the proof of Theorem 4.1 shows that An is bounded in L 2 ( P ) * m is bounded in L2(P) j u " is bounded in H2(P) n V , which gives the required r e s u l t . For the definition of t h e spaces Hs(R) , for noninteger s , see Lions-Magenes /l/.
Qptimisation algorithms
108
a;,+
(5.63)
=
max (0, ail - p. &+l ) ,
w i t h Remark 5.2 h o l d i n g f o r ( 5 . 6 2 ) . Under a c o n d i t i o n on pm of t h e t y p e ( 5 . 5 7 ) ,
4 = { u"u I M r i s k converges t o u,
5.7
, the
s o l u t i o n o f t h e problem ( P d ( l ) .
A n a l y s i s of t h e n u m e r i c a l r e s u l t s
All t h e c a l c u l a t i o n s were c a r r i e d o u t f o r :
(5.64)
f(Xl,Xz)
=4XzSh12Xxl,
u s i n g t h e C I I 10070 computer a t I R I A .
Nmerical values of the parameters
5.7.1
For b we u s e d t h e t h r e e v a l u e s :
b = 0.2,
0.34,
0.69:
h i = 1/10,
1/30,
1/50;
f o r hl:
With t h e t e r m i n a t i o n c r i t e r i o n cho-sen as:
1,
(5.65)
I 4 + ' - & I < E
Mki EQh
we t o o k : E
=
f o r b = 0.69, E
E
= lo-' f o r b = 0 . 2 .
= 5.10-'
rn
f o r b = 0.34,
(CHAP. 2 )
(SEC. 5)
Application of relaxation and duality methods
5.7.2
Lo9
Solution using overrelaxation
The results in Table 5 . 1 correspond to the solution of the problem (Pk)using algorithm (5.38) with: w = 1.6 for h, = 1/10,
1.7for h, = 1/30,
1.8 for h , = 1/50,
(5.66)
u:=o,
the parameters h,, h,, b, E having the values given in 5.7.1, and the termination criterion being given by (5.65). b
03
0.34
0.69
h,
Number of iterations
CII 10070 computation time in minutes
1/10
17
0.025
1/30
25
0.08
1/50
30
0.19
1/10
17
0.025
1/30
30
0.10
1/50
35
0.22
1/10
17
0.025
1/30
35
0.12
1/50
50
0.31
Table 5.1
Fig. 5.2 shows, for b = 0.69 and 321 = 1/50, the number of iterations as ti function of w , the termination criterion in all cases being given by (5.65); also shown on this figure is the curve giving the number of iterations as a function of w (again usin same termination criterion) for the unconstrained problem ( 1,the i.e.
?
(5.67)
(l)
I
-&=f
in a ,
In fact for the discretised form of (5.67).
@ t i m i s a t i o n algorithms
110
t
( C H A P . 2)
Number of iterations
\
Fig. 5.2.
0 = 10, I[ x ]0,0.69[ 1 0.69 h, = h2 - 10 A -
Influence of the parameter w on the rate of convergence.
We can therefore make the following remarks:
For a given w, the-over-relaxationmethod is Remark 5.5. more rapid, in terms of the number of iterations, for the constrained problem than for the unconstrained problem. H It may be noted that the maximal rate of convergRemark 5.6. ence is achieved for values of w much greater than 1; this justH ifies Remark 1.9 of Section 1.4. In the constrained case, the optimal value of w Remark 5.7. is a function of f (in contrast to the unconstrained case). More precisely, if we write 0 + = [XI XED, ~ ( x > ) 01, the value of w is essentially that which corresponds to the approximate solution of the Dirichlet problem: - Au = f on a + , the discretisation parameters
(SEC. 5 )
Application of r e h a t i o n and d u a l i t y methods
111-
h l , h2 r e m a i n i n g t h e same.
C l e a r l y f2+ i s n o t known a p r i o r i , so t h a t t h e above remark i s of t h e o r e t i c a l i n t e r e s t o n l y . rn 5.7.3
Solution using Uzawa’s algorithm
The p r o b l e m ( P & w a s s o l v e d u s i n g t h e a l g o r i t h m ( 5 . 6 1 ) ~ ( 5 . 6 2 ) , ( 5 . 6 3 ) w i t h a constant p a r a m e t e r p and 2; = 0 , t h e t e r m i n a t i o n c r i t e r i o n a g a i n b e i n g g i v e n by ( 5 . 6 5 ) . The approximate D i r i c h l e t sub-problems ( 5 . 6 2 ) were s o l v e d by p o i n t o v e r - r e l a x a t i o n w i t h w = 1 . 6 f o r h l = 1/10, 1.7 f o r h1 = 1/30 and 1.8 f o r hl = 1 / 5 0 . T a b l e 5.2 shows t h e number o f i t e r a t i o n s and t h e computation t i m e s f o r a l g o r i t h m (5.611, ( 5 . 6 2 ) , ( 5 . 6 3 ) , t h e p a r a m e t e r p having i t s optimal value (determined experimentally). F i g u r e 5 . 3 shows, f o r hl = 1 / 5 0 and b = 0.69, t h e v a r i a t i o n o f t h e number o f i t e r a t i o n s as a f u n c t i o n o f p , u s i n g t h e t e r m i n a t i o n c r i t e r i o n ( 5 . 6 5 ) ; t h e f o l l o w i n g remarks may b e made: The t o t a l number o f i t e r a t i o n s r e q u i r e d t o s o l v e Remark 5.8. (P& , w i t h a g i v e n p , i s more o r l e s s i n d e p e n d e n t o f t h e a c c u r a c y w i t h which t h e approximate D i r i c h l e t sub-problems
( 5 . 6 2 ) are solved.
rn
Remark 5.9.
For b = 0 . 6 9 , we have 2 ~ ~ l b ~ 1 . 4 2 ;. 5 t h e e s t i m a t e (5.60) ( I ) i s t h u s very r e a l i s t i c f o r t h e problem (Pd s i n c e f o r p = 42.5 convergence i s a t t a i n e d i n 522 i t e r a t i o n s , f o r p = 44 i n 698 i t e r a t i o n s , and f o r p = 45 t h e s o l u t i o n a c t u a l l y d i v e r g e s .
Remark 5.10. I n view o f t h e above r e s u l t s , it a p p e a r s t h a t i n t h e c a s e o f problem ( P ) t h e o v e r - r e l a x a t i o n method i s q u i c k e r t h a n t h e d u a l i t y method, a t l e a s t when t h e D i r i c h l e t sub-problems ( 5 . 6 2 ) are s o l v e d by o v e r - r e l a x a t i o n . It h a s a l s o been e s t a b l i s h e d t h a t t h e a d j u s t m e n t o f p t o o b t a i n s a t i s f a c t o r y convergence i s more c r i t i c a l t h a n t h a t o f w. It may be n o t e d , moreover, t h a t t h e d u a l i t y method i s more demanding i n t e r m s o f s t o r a g e r e q u i r e m e n t s t h a n t h e o v e r - r e l a x a t i o n method, s i n c e it i s n e c e s s a r y t o s t o r e t h e components o f 2, The rate o f convergence o f t h e d u a l i t y method c o u l d c l e a r l y be improved by s o l v i n g t h e D i r i c h l e t problems ( 5 . 6 2 ) by a method which i s q u i c k e r t h a n p o i n t o v e r - r e l a x a t i o n f o r t h i s t y p e o f problem ( a l t e r n a t i n g d i r e c t i o n , b l o c k o v e r - r e l a x a t i o n ) o r by a d i r e c t method ( e . g . Gauss o r C h o l e s k i , e t c ) ; however , t h i s would a l s o g e n e r a l l y i n c r e a s e t h e computer s t o r a g e r e q u i r e m e n t . The above i s a l s o v a l i d i f we r e p l a c e t h e c o n s t r a i n t
.
....
(I)
R e l a t i n g t o (P) r a t h e r t h a n (Ph)
.
(CHAP. 2 )
@timisation algorithms
112
Optimal
b
0.2
0.34
0.69
h,
P
Number of iterations
C I I 10070 computation time i n minutes
1/10
45
30
0.045
1/30
41
35
0.105
1/50
30
50
0.3
1/10
16
30
0.04
1/30
13
40
0.12
1/SO
10
50
0.3
1/10
13
40
0.09
1/30
8
70
0.27
1/50
5
100
0.6
Table 5.2
of t Number iterations
36 Fg 5.3.-
I n f l u e n c e of t h e parameter p on t h e r a t e of convergence.
(SEC.
6)
113
Discussion Shape of the f r e e surface
5.7.4
We have shown i n F i g u r e 5 . 4 , f o r b = 0.69 and u s i n g t h e s o l u t i o n o f problem (P,,)f o r hl = 1 / 5 0 , h, = 0.69/10, t h e r e g i o n where u = 0, a p a r t o f t h e r e g i o n where u > O ( i . e . a + ) and t h e boundary between t h e s e two r e g i o n s which w a s one o f t h e unknown o f ( P ) cons i d e r e d as a free boundary problem. We r e c a l l t h a t i n a + , we have - Au = f. W
0.69
0.345
0 0.5
0.75
1
Pip. 5.4. The h a t c h e d r e g i o n c o r r e s p o n d s t o u = 0 .
6.
DISCUSSION
We s h o u l d emphasise t h e f a c t t h a t t h i s Chapter d o e s not a i m t o For give an exhaustive i n v e s t i g a t i o n of minimisation algorithms. more comprehensive s t u d i e s of t h i s s u b j e c t , t h e f o l l o w i n g may be consulted :
- f o r unconstrained optimisation: Osborne
111, Spang
D a n i e l /2/, Kowalik and
Ill, Vainberg /l/;
- for constrained optimisation: Abadie /l/, /2/, B a l a k r i s h n a n and Neustadt /l/, Box 111, B r a m and S a a t y /l/, Cannon, Cullum and Polak / l / , . C C a /2/, F l e t c h e r /l/, Graves and Wolfe /l/, Hadley /l/, Kuhn and Tucker 111, Kunzi, K r e l l e and O e t t l i Ill, Kunzi, Tzschach and Zehnder /l/, Lasdon /I/, L e v i t i n and Polyak /l/, Luenberger /l/, Polak /l/, Tr6moliSres / 3 / , V a r a y i a /l/, Zangwill /l/, Z o u t e n d i j k
Ill. I d e a s similar t o t h o s e i n t r o d u c e d i n S e c t i o n 1 have been i n v e s t i g a t e d by a number of a u t h o r s : Crouzeix /l/, E l k i n Ill, C6a and Glowinski / 2 / , Glowinski /4/, /5/, Mezlyakov 111, M i e l l o u 111, Motzkin and Schoenberg /l/, O r t e g a and R h e i n b o l d t /l/, 121, 131, O r t e g a and Rockoff 111, P e t r i s c h y n 111, 121, S c h e c h t e r /l/, 121, 131, S o u t h w e l l /l/, S t i e f e l /l/, Comincioli Ill, Varga 111, Cryer
I l l , /2/*
@timisation
114
algorithms
(CHAP.
2)
- f o r various properties of convex functions, s e e L e v i t i n and Polyak 111. The c o n c e p t s i n t r o d u c e d i n S e c t i o n 2 a r e o f a c l a s s i c a l n a t u r e . For t h e u n c o n s t r a i n e d c a s e we r e f e r t h e r e a d e r t o Blum 111, Cauchy 111, C r o c k e t t and Chernoff 111, Curry Ill, G o l d s t e i n Ill, 121, Greenstadt 111, Polyak I l l . The convergence o f t h e method of d i v e r g e n t series, due t o Ermo l y e v , i s g i v e n i n Polyak /2/ and f o r t h e g e n e r a l u n c o n s t r a i n e d c a s e i n Tr6moliSres 121. A t r a n s p o s i t i o n o f t h i s method t o v a r i a t i o n a l i n e q u a l i t i e s i s made i n Auslender-Gourand-Guillet 111. The s o - c a l l e d c o n j u g a t e g r a d i e n t method, due t o Hestenes 111, has been s t u d i e d by Antosiewicz and Rheinboldt 111, Beckman 111, Durand 111, F l e t c h e r and Reeves 111, Hestenes and S t i e f e l 111. This method has been g e n e r a l i s e d t o i n f i n i t e dimensions by Daniel 111, 121. For methods o f t h e same t y p e , c o n s u l t Davidon 111, F l e t c h e r and Powell 111, Pearson 111. The e x t e n s i o n o f t h e method of c o n j u g a t e d i r e c t i o n s t o t h e c a s e of c o n s t r a i n e d m i n i m i s a t i o n has been c a r r i e d o u t by Goldfarb 111, Goldfarb and Lapidus Ill, Luenberger /2/ and TremoliPres 141. For methods o f t h e p o i n t p r o j e c t i o n o r g r a d i e n t p r o j e c t i o n t y p e s e e Altman Ill, Demyanov 111, G o l d s t e i n 111, L e v i t i n and Polyak I l l , 121, Polyak 131, 141, Rosen 111, 121, Sibony 111. The p e n a l i s a t i o n of c o n s t r a i n t s of t h e t y p e G ( V ) = 0 u s i n g t h e fu n c t i o n 1 J(u) ;G(u)’ , E + 0 ,
+
w a s i n t r o d u c e d by Courant /1/.
I n t h e case of i n e q u a l i t y const1 r a i n t s , t h e e x t e r i o r p e n a l t y f u n c t i o n J(u) ; ma[- G,(u),O]
+ f
l =1
i s due t o Ablow and Brigham /l/. The i n t e r i o r p e n a l t y f u n c t i o n
The method o f c e n t r e s u s i n g t h e d i s t a n c e - f u n c t i o n
w a s i n t r o d u c e d by Huard /l/ and extended t o t o p o l o g i c a l s p a c e s by Bui-Trong-Lieu
and Huard
111.
The second d i s t a n c e f u n c t i o n
1
+
,zl qq “
1
i s t h a t o f Fiacco
and McCormick Ill. The i d e a o f v a r y i n g t h e t r u n c a t i o n e r r o r s and t h e p o s s i b i l i t y o f a c c e l e r a t i n g t h i s method by o v e r - r e l a x a t i o n comes from Tr6moliSres 111. For a g e n e r a l p r e s e n t a t i o n o f p e n a l t y methods, s e e Fiacco and McCormick /1/ and Lootsma 111. The c o n c e p t s of S e c t i o n 4 a r e less c l a s s i c a l i n n a t u r e . For t h e g e n e r a l theorems on t h e e x i s t e n c e o f s a d d l e - p o i n t s s e e Berge 111, Browder 111, Ky-Fan 111, Sion Ill, / 2 / , Ekeland-T&nam 111. For d u a l i t y theorems i n mathematical programming s e e Arrow, Hurwi c z , and Uzawa 111, Bensoussan, Lions and Temam Ill, Mangasarian /1/, R o c k a f e l l a r / 3 / and Varayia 121.
(SEC. 6 )
Discussion
115
Some i n v e s t i g a t i o n s of t h e two p r i n c i p a l d u a l i t y algorithms may be found i n Arrow, Hurwicz and Uzawa /l/, Cga, Glowinski and NCdelec / 2 / ¶ C C a and Glowinski /1/ and i n Trgrnoli8res / 3 / . I n o r d e r t o b r i n g t h i s c h a p t e r t o a c l o s e , w e should mention t h e augmented Lagrangian methods which a r e obtained by combining penalisation and d u a l i t y ; we s h a l l meet some a p p l i c a t i o n s of t h i s i n Chapter 3 , Section 10 and i n Chapter 5, Section 9. This approach appears p a r t i c u l a r l y i n t e r e s t i n g for two fundame n t a l reasons :
1) t h e p o s s i b i l i t y of a c c e l e r a t i n g t h e convergence of t h e 2)
d u a l i t y methods, t h e p o s s i b i l i t y of improving t h e conditioning of t h e functi o n a l t o be minimised.
For t h e t h e o r e t i c a l a s p e c t s , we r e f e r t o Hestenes /2/, Powell / l j , Rockafellar /4/, / 5 / , e t c . . . Also, some a p p l i c a t i o n s of t h i s methodology t o t h e s o l u t i o n of nonlinear boundary-value problems a r e given i n Glowinski-Marrocco /l/, 121, Mercier /l/.
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Chapter 3 NUMERICAL ANALYSIS O F T H E PROBLEM O F THE ELASTO-PLASTIC TORSION O F A CYLINDRICAL BAR INTRODUCTION I n t h i s c h a p t e r we s h a l l m a i n l y s t u d y t h e a p p l i c a t i o n of t h e r e s u l t s and methods o f C h a p t e r s 1 and 2 t o t h e n u m e r i c a l analysis o f t h e model e l a s t o - p l a s t i c problem c o n s i d e r e d i n Chapter 1, Section 1.2.
1.
STATEMENT OF THE CONTINUOUS PROBLEM. SYNOPSIS 1.1
PHYSICAL MOTIVATION.
S t a t e m e n t of t h e c o n t i n u o u s problem
We a g a i n adopt t h e formalism o f Chapter 1, S e c t i o n 1 . 2 ; w i t h 52 a bounded open domain i n w", w i t h boundary r , w e i n t r o auc e
(1.1)
H ; ( L 2 ) = { u ( u E H ~ ( n )y, u = O } ,
which i s a c l o s e d v e c t o r s u b s p a c e o f H'(L2) and a H i l b e r t space f o r t h e i n n e r p r o d u c t and t h e norm d e f i n e d by
(1.3)
II u II =
(1',
I grad u 1' dx)',',
t h e norm ( 1 . 3 ) b e i n g e q u i v a l e n t t o t h e norm induced by H ' ( f 2 ) .
We t h e n i n t r o d u c e
a cZosed convex s e t i n Hi@), and t h e f u n c t i o n a l
Elasto-plastic torsion o f a cylindrical bar.
118
(CHAP. 3 )
where L i s a c o n t i n u o u s l i n e a r form on H b ( Q ) . We t h e n c o n s i d e r t h e v a r i a t i o n a l problem ( P O )d e f i n e d by: (1.6)
(Po) Min J(u) . US&
S i n c e t h e f u n c t i o n a l J i s c o n t i n u o u s and s t r i c t l y convex on lim J(v) = + a , Theorem 2 . 1 o f Chapter 1,
Hb(Q), w i t h
-
IIvII + m
S e c t i o n 2 . 1 a p p l i e s , and t h i s i m p l i e s t h e e x i s t e n c e and uniquen e s s o f a s o l u t i o n for ( P O ) . We d e n o t e t h i s s o l u t i o n by u which i s a l s o t h e u n i q u e s o l u t i o n of t h e v a r i a t i o n a l i n e q u a l i t y
which i s e q u i v a l e n t t o ( P O ) . C e r t a i n p r o p e r t i e s of t h e s o l u t i o n u , f o r particular forms L , are d e t a i l e d i n S e c t i o n 2. 1.2
Physical motivation
With t h e set fi a simply connected, bounded open domain i n R 2 , w e c o n s i d e r a c y l i n d r i c a l b a r o f c r o s s - s e c t i o n fi, made o f an e l a s t i c - p e r f e c t l y p l a s t i c m a t e r i a l f o r which t h e t h r e s h o l d o f p l a s t i c i t y (b. t h e y i e l d s t r e s s ) i s d e f i n e d by t h e von Mises c r i t e r i o n ( s e e eg. Mandel 111). T h i s bar i s t h e n s u b j e c t e d t o a n i n c r e a s i n g t o r q u e , s t a r t i n g from a n u n s t r e s s e d i n i t i a l s t a t e , t h e t o r s i o n b e i n g c h a r a c t e r For e a c h v a l u e o f i s e d by t h e t w i s t a n g l e p e r u n i t l e n g t h C. C, and w i t h a s u i t a b l e system o f u n i t s , we c a n , u s i n g t h e HaarKarman p r i n c i p l e , r e d u c e t h e d e t e r m i n a t i o n o f t h e stress f i e l d t o t h e s o l u t i o n o f t h e v a r i a t i o n a l problem (where V d e n o t e s a "stress p o t e n t i a l " ) :
t h i s i s a s p e c i a l c a s e o f ( P O ) i n which t h e form L i s t h u s g i v e n by :
(1.9)
uu)= c
In
U(X)
bc
.
Remark 1.1. I n t h e c a s e o f a multiply-connected s e c t i o n R it i s c o n v e n i e n t t o employ a m o d i f i e d v e r s i o n o f f o r m u l a t i o n (1.8)j s e e Glowinski-Lanchon /l/, H. Lanchori 121. I n t h e c a s e where t h e i n i t i a l s t a t e i s n o t s t r e s s - f r e e , and/ or t h e t o r q u e i s n o t i n c r e a s i n g , f o r m u l a t i o n (1.8) i s i n c o r r e c t f o r t h e p h y s i c a l problem c o n s i d e r e d ; a s u i t a b l e approach i s
Statement of continuous problem
(SEC. 1)
119
then t o use time-dependent, or more p r e c i s e l y q u a s i - s t a t i c , mathematical models ( s e e Duvaut-Lions 111, Chapter 5 , f o r a 8 d e t a i l e d study of q u a s i - s t a t i c models i n p l a s t i c i t y ) .
Remark 1.2. We can supplement Remark 1 . 4 of Chapter 1, Section 1 . 2 by p o i n t i n g o u t t h a t s e v e r a l a u t h o r s , i n p a r t i c u l a r Shaw ill, have c a r r i e d out t h e numerical s o l u t i o n of t h e problem of t h e e l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l b a r , considered as a f r e e bouridmy problem, by using t h e p o i n t formu l a t i o n given i n t h e above-mentioned remark, ie. : (1.10)
- Au
(1.11)
Igradul = 1
=
C
Sa,
in in
a,,
with
(1.12)
R e = [xIxEP,Igradu(x)l
, 2 .
( 2 ) We refer t o BrCzis-Stampacchia,
loc c i t . , f o r t h e
d e f i n i t i o n of a r e g u l a r i t y condition f o r s u f f i c i e n t t o ensure (2.2)
r
which i s
Properties of solution of ( P o )
(SEC. 2)
L(u) = C
(2.3)
121
I,
u(x)dx,
then t h e s o l u t i o n of ( P O )i s a l s o a s o l u t i o n o f ( P I ) , d e f i n e d i n (1.15), and o f t h e e q u i v a l e n t v a r i a t i o n a l i n e q u a l i t y :
( u E K I =[ulu~Ho'(R), Iu(x)I 0, K 1 can t h e r e fore be r e p l a c e d , i n ( 1 . 1 5 ) and ( 2 . 4 ) , by t h e even s i m p l e r convex s e t
K ; = [u 1 u E Hi(R),0 Q u(x) Q 6(x, f)
a.e.1.
Remark 2.3. From t h e numerical p o i n t of view, t h e s o l u t i o n of ( P I )i s e a s i e r t h a n t h a t o f ( P O ) ; t h e numerical a n a l y s i s of ( P I ) ,which i s a v a r i a n t of t h a t of t h e v a r i a t i o n a l problem of Chapter 2 , S e c t i o n 5, w i l l form t h e s u b j e c t o f S e c t i o n 3. Some p a r t i c u l a r c a s e s where t h e s o l u t i o n i s known(')
2.3 2.3.1.
A f i r s t example
We t a k e R = p, 1[ and L(u) = C i s d e f i n e d by:
with u ' = du/dx
v(x)dx w i t h C > 0 , so that ( P O )
.
I t can be v e r i f i e d t h a t t h e s o l u t i o n u o f ( P ~ ) ( a n o df ( P I ) ) i s given by: (2.6)
(')
C
u(x) = ?x(l
- x)
if C < 2
We r e s t r i c t o u r a t t e n t i o n h e r e t o very simple examples, designed t o a c t a s t e s t c a s e s for t h e v e r i f i c a t i o n o f t h e For v a r i o u s e x p l i c i t methods developed i n t h i s c h a p t e r . s o l u t i o n s s e e Mandel /I/.
E la s t o - p l as t i c t o r s i o n o f a cyZindrica2 bar
122
( C H A P . 3)
and i f C 2 2 by:
if 0 Q x Q
21 -
1
F i g u r e 2 . 1 shows t h e s o l u t i o n of ( P o ) (and ( P I ) )corresponding t o C = 4 (we t h u s have 112 - 1 / C = 114).
Fig. 2.1. 2.3.2
S o l u t i o n of ( P O )f o r C =
4.
A second example
We t a k e n = 2 ,
0=
[X
1x
L(u)=C
= (x1, xZ),
I
u(x)dx,
X:
+ 4 < R Z ],
c>o;
t h e s o l u t i o n of ( P O )(and ( P I ) ) is t h e n given by:
(2.8)
u(x) = (C/4)( R 2 -. r’)
where
r = (2.9)
u(x) =
C 6 2/R,
Jm, and i f C 2 2/R by
{ --
if R ’ G r G R Crz/4 + ( R - 1/C) i f 0 G r Q R’ ,
with
(2.10)
if
R‘ = 2 / C . rn
Remark 2.4. We can now supplement Remark 2.1; i n t h e two. above examples, f o r which t h e d a t a are very r e g u l a r , we have, f o r s u f f i c i e n t l y l a r g e C, u E c1(Sl> n~ ' ( 0 n )H ~ ( B ) ('1
but
.$CZ(E),
u#H3(62).
NUM3RICAL ANALYSIS OF PROBLEM ( P i 2
3.
3.1
Synopsis
I n t h i s s e c t i o n we s h a l l apply t h e r e s u l t s of Chapters 1 and 2 t o problem ( P I ) , with t h e open domain fi simply-connected i n R2 ; we s h a l l consider only t h e exterwr approximation of (Pi) (i.e. of t h e " f i n i t e d i f f e r e n c e ' ' t y p e ) and no a d d i t i o n a l d i f f i c u l t i e s a r i s e , r e l a t i v e t o t h e v a r i a t i o n a l problem of Chapter 2, Section 5, o t h e r t h a n t h e convergence of t h e approximate s o l u t i o n as h -+ 0, which i s r a t h e r more awkward t o prove. With regard t o algorithms, we s h a l l r e s t r i c t our a t t e n t i o n t o t h e over-relaxation method, which appears t o be t h e b e s t s u i t e d t o t h e t y p e of c o n s t r a i n t s m e t i n t h i s problem. 3.2 3.2.1.
E x t e r i o r approximation of problem ( P i )
Formulation of the approximate problem
We r e c a l l t h a t ( P I ) i s defined bv
Suppose h > 0, intended t o approach zero; proceeding as i n Chapter 1, Section 5 . 1 and Chapter 2 , S e c t i o n 5 , w e d e f i n e the g r i d k b y :
(3.2)
R,= { M i j ( M i j E R Z M , il= {ih,jh}, i,jEZ).
With each node Mil of R,we a s s o c i a t e t h e panel with c e n t r e Mi,
(3.3)
('1
m,O(Mtj) = ](i
- 4) h, (i + 4) h[
x
1(i - 4) h, (i+ 4) h [ ,
I n f a c t i n t h e case of t h e s e two examples w e have r a t h e r more, s i n c e av a2v ' ax,' ax,ax,
Elasto-plastic torsion of a cyZindrical bar
124
(3.8)
&j
= c h a r a c t e r i s t i c function of
(CHAP. 3)
mf(Mij).
Having p u t (as i n C h a p t e r 1, S e c t i o n 5 . 1 and i n C h a p t e r 2 , S e c t i o n 51, f o r k = 1, 2
we "approximate" J by
Jh
: Vh+ R
d e f i n e d by
and w e t a k e a s t h e a p p r o x i m a t e problem
M b Jh(vd
(P13[
(3.11)
9
Uh 6 K l h
Klh
=
{ uh I uh
vh,
I ufj I 6 q M f j ,
r),V M f j E Oh } .
It may be n o t e d t h a t Klh i s bounded and c l o s e d i n V,. rn 3.2.2
SoZvabiZity af ( P l h )
We proceed as i n C h a p t e r 2 , S e c t i o n 5 . 3 . 2 :
(l)
t h e mapping
We s h a l l u s e t h e same n o t a t i o n f o r t h e mapping V h + L z ( n ) d e f i n e d by u h - r r e s t r i c t i o n t o O . o f qhuh.
(SEC. 3 )
Numerical analysis o f ( P I)
125
d e f i n e s a norm over V, denoted by 11 0, Ilk; t h e f u n c t i o n a l Jh i s t h e r e f o r e continuous and s t r i c t l y convex on Vh and s i n c e K,, i s bounded and c l o s e d i n V, we have t h e e x i s t e n c e and uniqueness of an optimal s o l u t i o n f o r (Plh), say uI.
.
E x p l i c i t formulation of
3.2.3.
(Plh)
We can w r i t e ( 3 . 1 0 ) i n e x p l i c i t form as follows:
Remark 3.1. Mpq#%
3.3
'
I n ( 3 . 1 2 ) it i s necessary t o t a k e u r n = 0 i f
Convergence of t h e approximate s o l u t i o n as h
Preliminary remark.
-+
0
I n t h i s s e c t i o n we s h a l l prove t h a t
where u i s t h e optimal s o l u t i o n o f ( P I ) . This convergence r e s u l t can be proved d i r e c t l y by proceeding as i n Chapter 2 , Section 5 , though t h e r e a r e some a d d i t i o n a l t e c h n i c a l d i f f i c u l t i e s ; however, w e have p r e f e r r e d t o u s e t h e more g e n e r a l approach o f Chapter 1, S e c t i o n 5 even though t h i s makes t h e working somewhat more complicated. 3.3.1.
Reduction o f (P,d t o an equivalent variational inequality i n K , ,
I n o r d e r t o g e t back t o t h e s e t t i n g of Chapter 1, S e c t i o n 5 , it i s convenient t o formulate (P,) and (PI,,) i n terms of v a r i a tional inequalities;
t h i s gives
gradu.grad(u - u)dx 2 C
(u
- u)dx V U E K ,
Elasto-plastic torsion of a cylindrical bar
126
(CHAP. 3 )
I n ( 1 . 2 ) w e have p u t
In
grad u.grad u dx = a(u, v )
vh+ R
and we d e f i n e n, : Vh x (3 * 5 )
ah(uh, Oh)
3.3.2.
=
kil In
'k
qh
by
uh ' k
q h vh dx .
Exterior approximation f o r HA(Q), u, K,, L
We s h a l l now v e r i f y p o i n t by p o i n t t h a t t h e p r o p e r t i e s r e q u i r e d i n Chapter 1, S e c t i o n 5.2 a r e s a t i s f i e d :
Approximation of
HA(Q).
I n view o f C h a p t e r 1, S e c t i o n
(3.16)
V = HA(Q)
(3.17)
F = (L2(Q))'
and
a : V +F
5.1, we t a k e
i n j e c t i v e , d e f i n e d by:
I n S e c t i o n 3.2.1 we d e f i n e d a f a m i l y o f f i n i t e - d i m e n s i o n a l s p a c e s vh which are H i l b e r t s p a c e s f o r t h e norms 11 I(h d e f i n e d by:
.
we r e c a l l (see Cea / 3 I y L i o n s 131) t h a t t h e norm t h e d i s c r e t e Poincare inequality:
11. llh s a t i s f i e s
F i n a l l y , we d e f i n e a f a m i l y o f prolongation operators p h E 9 ( V h ;F) d e r i v e d from t h e o p e r a t o r qh : Vh+ L2(f2), d e f i n e d i n S e c t i o n 3.2.1, r e l a t i o n (3.7), by:
(3.21)
Ph uh = { q h %r
'1
qh h
r '2
q h uh
1
3
(SEC. 3 )
Numerical analysis of (PI)
127
and i n view of ( 3 . 2 0 ) , we indeed have:
This being t h e c a s e , w e s h a l l now prove:
The f a m i l y (V,), c o n s t i t u t e s an exterior
P r o p o s i t i o n 3.1. approximation of V.
Proof. It i s necessary f o r us t o v e r i f y t h a t ( 5 . 2 7 ) of ChapWe d e f i n e r, : V - r V, by: t e r 1, S e c t i o n 5 is s a t i s f i e d . (3*23)
(rh
v)Ml,
=
‘I
4x) dx,
VMij
E
4.
mO(Mij)
Thus ( s e e Chapter 1, Section 5 . 1 )
,
VvE
v=
and hence, t a k i n g account of ( 3 . 1 9 ) , (3.25)
(I r, u 1, 6 c
independent of h .
Approximation of a The form a, : V, x V, -r W d e f i n e d i n ( 3 . 1 5 ) s a t i s f i e s , f i r s t l y
(3-26)
I ah(% v h ) I
(3 * 27)
d V h ,
vk) =
d 11 uh 1, 11 uh
11 uh 1:
Ilk
9
and secondly ( t h i s may be proved without d i f f i c u l t y ) , (5.30) and ( 5 . 3 1 ) of Chapter 1, S e c t i o n 5.2, so t h a t we have: P r o p o s i t i o n 3.2.
The f a m i l y (a,&, c o n s t i t u t e s an approx-
imation of a. Approximation of K, With t h e family K,, d e f i n e d as i n (3.11) , w e have: P r o p o s i t i o n 3.3.
The family K , , c o n s t i t u t e s m appro*
mation of K , . Proof. It i s necessary t o prove (see ( 5 . 3 3 ) and ( 5 . 3 4 ) Chapter 1, S e c t i o n 5.2): (3.28)
if
vh
EK,,with limp, h-0
U, =
r
weakly i n F, t h e n { E O K ,
128
Elasto-plastic torsion of a c y l i n d r i c a l bar
(CHAP. 3 )
and
Verification of ( 3 . 2 8 ) .
[,,t2).
We p u t [ = (to, obtain that
from Chapter 1, S e c t i o n 5 . 1 w e t h e n
Moreover, s i n c e t h e f u n c t i o n x + &x, I') i s c o n t i n u o u s on it i s uniformly c o n t i n u o u s and t h u s , by d e f i n i t i o n O f Klh, gives
(3.32)
a,
lim q(h) = 0 ,
q(h) independent of x,
h-0
+
is, V E > 0 , S i n c e t h e set [u I UEL'(Q), 1 v(x) I Q 6(x, r ) E a.e] convex and c l o s e d i n L * ( Q ) , and t h e r e f o r e weakly c l o s e d , we have from ( 3 . 3 1 ) and ( 3 . 3 2 )
and we indeed have
0 .
~ ~ (E0K) ,
We then d e f i n e a subset X i of K , by
W e have
(3.38)
= K,
+
I n f a c t (u - 8)' ( r e s p . (u 8) - ) t e n d s s t r o n g l y towards u+ ( r e s p . u - ) i n H'(f2) as E -+ 0, t h e r e f o r e T ~ ( u )+ u s t r o n g l y i n H'(f2).
Variant. The above procedure shows t h a t t h e s e t of f u n c t i o n s V of K,, which a r e zero i n a neighbourhood of r a n d bounded above i n modulus by d(x,r) - ,&, i s dense in K l . This f a c t may a l s o be demonstrated by proceeding a2 follows : l e t U E K , and l e t 0, be a sequence of f u n c t i o n s of C'(n), (O,(x) Q 1, 0, = 0 i n t h e neighbourhood of r and 0,(x) + 1 uniformly over all compact s u b s e t s of f2,
I
0 s u f f i c i e n t l y s m a l l
0
dx) I & Q 6(Mf,,r ) = a,,
v, E K,,
Vh > 0, s u f f i c i e n t l y s m a l l .
From (3.24) and ( 3 . 2 5 ) , p , r , u (=Phu&+m s t r o n g l y i n F and t h u s c o n d i t i o n (3.29) i s indeed s a t i s f i e d i n X1.
Elasto-plastic torsion of a cylindrical bar
130
(CHAP. 3 )
Remark 3.2. Following Raviart /l/, Chapter 0, we can v e r i f y (3.28) by u s i n g t h e f a c t t h a t i f p h v , + 5 weakly i n F, t h e n we have f o r a subsequence a l s o denoted by vh
(3.40)
q,
D,
-+
to s t r o n g l y i n L2(Q) and a . e . i n 62.
Remark 3.3. Basing our argument on ( 3 . 3 5 ) , w e s h a l l now prove t h a t 9(Q) n K, = K,;s i n c e t h e s e t xt c K, i s dense i n K , it w e s h a l l proceed by P e g u h . F i s s u f f i c i e n t t o show t h i s f o r xl;
isation : Consider then a r e g u l a r i s i n g sequence (p,,),
lR2
p,, 3 6(0),
p,(x)
dx
= 1, support (p,,) + 0); i f
DE
(i.e. pn E 9(R2), pa 2 0,
x,,
we extend v t o 8
ifcH'(W2), and w e d e f i n e i f , , ~ g ( R ' ) by
by 0 o u t s i d e f2;
r
We put
(3.42)
v, = r e s t r i c t i o n of
5, t o
a.
We then have
(3.43)
if,,+
if
strongly i n H'(R2)
and hence, by r e s t r i c t i o n t o 0 ,
(3.44)
v,+
v
s t r o n g l y i n H'(bl).
From (3.35) and f o r a s u f f i c i e n t l y l a r g e n , w e have
(3.45)
support (u;) c
n * 0, E 9(n),
It now remains t o show t h a t u , E K , f o r a s u f f i c i e n t l y l a r g e n . Suppose x E Q; we t h e n have:
we p u t :
(3.47) then x
(3.48)
o,(x) = [ y I y
E
R2, x - y E support @,,)I,
E w,,(x) and
diameter (a&)) = diameter support (p,,) -+ 0 as n
+
+
00.
Numerical analysis of (PI)
(SEC. 3 )
W e t h e r e f o r e have,
131
Vx'xELa,
I
(3.49)
It can r e a d i l y be deduced from (3.35) t o g e t h e r with t h e defi n i t i o n of 5 and (3.48) and ( 3 . 4 9 ) , t h a t
1 v,(x) I Q 6(x, r )
(3.50)
VX E Q provided
n i s sufficiently large ,
and hence vmsK1provided n i s s u f f i c i e n t l y l a r g e .
8
Approximation of L. I n t h e case considered, where
L(v) = cln4x)dX
, the
v e r i f i c a t i o n of (5.37), (5.38) of Chapter 1, Section 5.2 i s immediate f o r L , : Vh+W defined by: Lh(uk) =
In
4h
'h
dX .
A r e s u l t concerning strong convergence
3.3.3.
I n view of t h e p r o p e r t i e s demonstrated i n Section 3.3.2 f o r e x t e r i o r approximations of Hi(@, a, Kl,L we can apply t o problem (P,&, and more p r e c i s e l y t o t h e e q u i v a l e n t formulation ( 3 . 1 4 ) , Theorem 5.2 of Chapter 1, Section 5.3; t h i s g i v e s :
Let uh be the solution of problem ( P l h ) ; as h
Theorem 3.1.
-f
we have :
where u i s the solution o f problem ( P l ) . 3.4
8
Solution of t h e approximate problem by p o i n t overr e l a x a t i o n with p r o j e c t i o n .
We use algorithm ( 1 . 4 1 ) , (1.42) o f Chapter 2 , Section 1 . 4 ; t a k i n g account of (3.11), (3.12) and i t e r a t i n g with i n c r e a s i n g i, j t h i s t a k e s t h e form: I
(3.52)
u ; € K l h given uy"2=(l-w) ~ , + -0( ~ + l j + t r ; j + l + u ; - + : j + ~ ~ l l + h C) z 4
max (-
un i j+ l
=
Mi,
Oh .
a, min (#,+ ' I 2 ,
0,
Elasto-plastic torsion of a c y l i n d r i c a l bar
132
(CHAP. 3)
We again use t h e convention ukr = 0 i f Mk,#fZh. Since Theorem 1 . 3 of Chapter 2 , Section 1 . 4 a p p l i e s t o (P,,,) we have : Theorem 3.2. o < w < 2 .
The sequence
4 tends towards
uh
as n
+
+ oc,
if
Remark 3.4. If u: = 0, and i f 0 < w 6 1 , it r e s u l t s from (3.52) t h a t t h e sequence (Gj)" i s i n c r e a s i n g V M , , E f Z , ; t h e convergence i s t h e r e f o r e monotone and t h e coordinates uij of u, are approached from below. Remark 3.5. We could a l s o solve(P,,J by o t h e r methods and i n p a r t i c u l a r by a d u a l i t y method similar t o t h a t used i n Chapter 2 , Section 5.6; although t h e a p p l i c a t i o n o f such a method t o problem(P,& does not p r e s e n t any d i f f i c u l t i e s , w e have been content i n t h i s example, t o use t h e over-relaxation method, which seems t h e b e s t adapted t o t h e problem considered. 3.5.
Applications. Example 1.
We t a k e fZ = lo,1[ x ]0,1[ with C = 10.
3.5.1.
Numerical values o f the parameters
Mesh s i z e : h = 1/40. I n i t i a l i s a t i o n of algorithm (3.52)
: u: = 0.
Termination c r i t e r i o n f o r algorithm (3.52)
1
(3.53)
i.;,+l
:
- 4 1
1 i n algori t h m ( 3 . 5 2 ) ; moreover, s i n c e t h e p l a s t i c region i s l a r g e r Tor
Elasto-plastic torsion of a cylindrical bar
136
(CHAP. 3 )
C = 10 than f o r C = 5, more c o n s t r a i n t s a r e s a t i s f i e d with equality i n t h e former c a s e , which e x p l a i n s t h e faster rate of convergence of algorithm ( 3 . 5 2 ) . Figure 3.5 shows t h e e l a s t i c and p l a s t i c regions corresponding
to
c
= 10.
X,
Fig. 3.5.
4.
Representation of t h e e l a s t i c and p l a s t i c regions f o r C = 10 ( t h e p l a s t i c regions a r e shown h a t c h e d ) .
INTERIOR APPROXIMATIONS OF ( P a ) Synopsis.
I n t h e previous s e c t i o n w e s t u d i e d problem (Pi); we s h a l l now consider problem ( P O ! , f o r which t h e d i s c r e t i s a t i o n of t h e funct i o n a l J i s treated zn the same way whereas t h e a p p l i c a t i o n of t h e c o n s t r a i n t s (membership of K O ) poses new problems which vary In t h i s and widely with t h e mode o f approximation used f o r J . t h e following s e c t i o n s w e s h a l l t h e r e f o r e c a r r y out a systematic study of t h e approximation of J and the corresponding approxi-
mation of the convex s e t KO. I n t h i s s e c t i o n we s h a l l formulate two types of i n t e r i o r approximation i n t h e sense of Chapter 1, Section 4. I n Section 4 . 1 w e s h a l l consider a f i n i t e element method with approximation by a f f i n e l i n e a r f u n c t i o n s over t r i a n g l e s ( s e e Chapter 1, Section 4, Example 4.2), and i n Section 4 . 2 , a Galerkin approximation ( s e e Chapter 1, S e c t i o n 4 , Example 4 . 1 ) using t h e eigenfunctions of t h e o p e r a t o r - A i n H&2) as a b a s i s The convergence o f t h e s e approximations w i l l be s t u d i e d i n
(SEC.
4)
I n t e r i o r approximations o f (Po)
137
S e c t i o n 6.
4.1.
A f i n i t e element method
Triangulation of a.
4.1.1
Definition of
v h
We now go back t o Example 4.2 o f Chapter 1, S e c t i o n 4 ; suppose Y hi s a triangulation g e n e r a t e d by a f i n i t e number of t r i a n g l e s with t h e following p r o p e r t i e s : (4.1)
TEY~=-TC;
(4.2)
{ T,
T ' e Y h+ T n T' = 4
o r T and T' have a common s i d e
o r T and T' have a common v e r t e x ;
for rp ( s e e r e l a t i o n ( 4 . 6 ) o f Chapter 1, S e c t i o n we take
(4.3)
4, Example
4.2)
r p ( h ) = h = Max (Measure of T ) TcJ
h
which s a t i s f i e s t h e r e l a t i o n s S e c t i o n 4.
(4.6), ( 4 . 7 )
of Chapter 1,
We d e f i n e t h e s e t s (4.4)
-
a, =
u
T
f.Ch
(4.7)
z h = set
(4.8)
zh=[[M)MEzh,
of v e r t i c e s of yh
0
bf$r,,l.
If ME,$;, w e denote by M ( s e e Figure 4 . 1 ) ;
P,,...,Pk t h e n e i g h b o w i n g v e r t i c e s of
Fig. 4.1.
Elasto-plastic torsion of a cylindrical bar
138
(CHAP. 3 )
We d e f i n e :
(4.9)
[
w, = a f f i n e continuous f u n c t i o n over each t r i a n g l e with v e r t e x M, such t h a t w,(M) = 1, W A P , ) = 0, i = 1, ...,k and with w, zero on Q o u t s i d e t h e t r i a n g l e s with v e r t e x M,
then : (4.10)
vh
= t h e space spanned by t h e w,,
ME&,
and w e have: Proposition 4.1.
vh
i s a subspace of HA(Q) with
d im( V&=N=Card(&).
rn
Remark 4.1. Since t h e f u n c t i o n s u h e V h a r e a f f i n e over T , gradu, i s c o n s t a n t over T , V T E Y , . rn Definition of the approximate problem
4.1.2.
For ( P o ) , w e r e s t r i c t our a t t e n t i o n t o t h e case where (4.11)
L(u) = sD/udxdy, f ~ L ' ( n )
grad u.grad u dx dy ,
and hence, w i t h u(u, u ) = (4.12)
1
J(u) = 2 a(u, u ) - lnfu dx dy
and
I
Min J(u)
(Po)
OE=
KO =
[uI
U E H A ( ~lgradv ),
I < 1 aeJ.
We then d e f i n e t h e approximate problem (P,,,,) by: (4.13)
(Pd
Min
J(uJ.
rn
uhc&lnvh
Remark 4.2. The condition u , € K o n vh*Igradvh1' put ( s e e ( 4 . 1 0 ) ) : (4.14)
0,
=
,V
w,,
V,E
d 1 a.e. ; i f we
R
bf€&
then i n view of Remark 4 . 1 t h e approximate problem (4.13) reduces t o an o p t i m i s a t i o n problem i n RN with N ' ( = Card(YJ) q u a d r a t i c c o n s t r a i n t s with r e s p e c t t o t h e uy. rn
(SEC. 4 )
I n t e r i o r approximations of (PO )
139
Solvability of the approximate problem
4.1.3.
Since t h e f u n c t i o n a l J has t h e r e q u i r e d p r o p e r t i e s ( s e e Chapter 1, S e c t i o n 2 . 1 ) we have:
The approx$mate problem (4.13) admits a Proposition 4.2. unique solution u, characterised by : (4.15)
I
- ub) >/
a(uh,
u,
EK, n V,.
1.
f (0, - Uh) dX dy
VU, E KO A
vh,
rn
4.1.4. E x p l i c i t formulation of the approximate probtem L e t MI, M,,M3 be t h e v e r t i c e s o f a t r i a n g l e T e Y , with coordinates (xi,yi), i = 1, 2 , 3; t h e a f f i n e l i n e a r function defined by:
(4.16)
WJ
= vMl, i =
4
1,2,3
and 3
(4.19)
6=
z r i ,
I 6 I = 2 Measure (Triangle MI M 2M 3 ) . I n t h e following w e s h a l l use t h e n o t a t i o n m ( T ) = measure of f g r s i m p l i f i . c a t i o n , uMl = ui. If w e consideru,EVk,we have (see ( 4 . 1 4 ) ) :
T, and, (4.20)
q(x,
v) =
c
OM
wM(x9
Y).
M.PL
u s i n g t h e charIt i s a l s o p o s s i b l e (and u s e f u l ) t o d e f i n e a c t e r i s t i c f u n c t i o n s o f t h e t r i a n g l e s T E Y which, ~ t a k i n g account of (4.17) and (4.19) and u s i n g t h e obvious n o t a t i o n , gives:
(4.21)
Elasto-plastic torsion of a cylindrdcal bar
140
(CHAP.
3)
with (4.22)
Or = c h a r a c t e r i s t i c f u n c t i o n of T .
From t h i s w e deduce
which a l l o w s t h e approximate problem (Po& t o be expressed i n terms o f t h e u,.
Remark 4.3.
I,
In
(4.26) t h e i n t e g r a l
Ax, Y )W d X 9 Y )
dr
must a c t u a l l y be e v a l u a t e d on t h e s u p p o r t of w,.
Remark 4.4. I n t h e terminology o f f i n i t e element methods, t h e symmetric p o s i t i v e - d e f i n i t e m a t r i x r e l a t e d t o t h e q u a d r a t i c p a r t o f J(u& i s c a l l e d t h e s t i f f n e s s matrix a s s o c i a t e d w i t h t h e approximation considered; i n t h e approximate problem it w i l l p l a y t h e r o l e of -A i n t h e continuous problem.
4.1.5.
On the use of f i n i t e element methods of order greater than 1.
I n S e c t i o n s 4.1.1, 4.1.2, 4.1.3 an approximation w a s conside r e d which u s e s polynomials of maximum degree e q u a l t o u n i t y
(SEC. 4 )
I n t e r i o r approximations o f ( P o )
141
over each T EY h; t h i s g i v e s a constant g r a d i e n t over T, V T E Y , . The use of polynomials o f maximum degree k(> 2) over T, V T E . T , would, i n g e n e r a l , produce over T, V T E Y , a g r a d i e n t which i s a polynomial f u n c t i o n of x , y of degree k - 1 2 1 ( a n d t h u s not
constant).
The approximate f o r m u l a t i o n of t h e c o n d i t i o n I grad u(x, y ) I Q 1 a.e. could t h e n be c a r r i e d out i n various ways, i n p a r t i c u l a r ( t h i s l i s t i s not exhaustive): (4.28)
(4.31)
I { G,=
I
Brad Uh(Gr) Q 1 VT E Y h , c e n t r e of g r a v i t y of t h e t r i a n g l e T,
Igradu,(x,y))dxdy
0, V m ) ,
hence
Vm E N , Vm E N .
We r e c a l l t h a t : (4.35)
I. In
w,(x) wm(x)cix = 0
if
m
+ n;
w e s h a l l assume t h a t : (4.36)
I w,,,(x) 1’
dx = 1 Vm E N .
I n p a r t i c u l a r s i n c e t h e family ( w , ) , . ~ c o n s t i t u t e s a Galerkin b a s i s f o r H;(R), w e approximate H ; ( R ) by t h e family ( V J M e N defined by:
(’)
The p r i n c i p a l motivation behind going from k = 1 t o k = 2 i s simply t o reduce t h e number o f t r i a n g l e s r a t h e r t h a n t o i n c r e a s e t h e accuracy.
(SEC. 4 )
VM = subspace of ' H i ( R ) generated by
(4.37)
143
I n t e r i o r approximations of (Po)
4.2.2.
(W,,,)~(,,,~Y.
Approximation of KO
Since i n p r a c t i c e it i s not p o s s i b l e t o use t h e convex s e t
KO n VM w e s h a l l be content i f U E VM t o s a t i s f y t h e condition Jgradul g 1 on a f i n i t e s e t o f p o i n t s of 0 ( o r of s), say ah.. (resp. by
ah);
f o r KO t h i s g i v e s t h e approximation K t . ( ' ) defined
K ~ = [ u I u E V M Iwdv(P)I< ; 1 VPER,,].
(4.38)
We g e t a similar d e f i n i t i o n i f we r e p l a c e
nh by Zh.
Remark 4.5. The method f o r approximating KO appears t o be of t h e colZocation t y p e ; a n a t u r a l method (which w i l l be employed i n Section g), i n s p i r e d by J . B . Rosen 131, c o n s i s t s of d e f i n i n g
R, = [x I x
(4.39)
=
(xl,xz) E W, xi = ih, xz
= jh,
i, j E Z] ,
where h > 0 i s given (and intended t o approach z e r o ) . S i m i l a r l y we can d e f i n e 3, = 0, u r h where r,. i s a f i n i t e subset of p o i n t s of r f o r which a t least one of t h e coordinates is an i n t e g e r m u l t i p l e of h.
4.2.3.
Definition of the approximate problem
With t h e f u n c t i o n a l J
:HA@)+ W once again o f t h e form:
with f EL'(^), t h e approximate problem i s defined by:
M.in J(u).
(4.41)
UE%Y
4.2.4.
Solvability of the approximate problem
As t h e f u n c t i o n a l J and K& have t h e r e q u i r e d p r o p e r t i e s (see Chapter 1, S e c t i o n 2 . 1 ) we have:
(l)
This approximation w i l l be j u s t i f i e d i n S e c t i o n
6.
144
Elasto-plastic torsion o f a cylindrical bar
(CHAP. 3)
P r o p o s i t i o n 4.3. The approximate problem ( 4 . 4 1 ) admits one and o n l y one solution, #, characterised by:
Explicit formulation o f the approximate problem
4.2.5. If
UE
V , w e use t h e n o t a t i o n :
.U. (4.43)
v=Cpiwi,
pi€R V i = 1 ,
..., M ,
i=l
and t a k i n g i n t o account t h e orthonormality of t h e ( w , ) , . ~ we have:
Moreover :
P u t t i n g p = (pl,p2,...,pM)and t a k i n g account of ( 4 . 4 4 ) , ( 4 . 4 5 ) , t h e approximate problem ( 4 . 4 1 ) can be w r i t t e n e x p l i c i t l y as
Remark 4.6. In view of ( 4 . 4 4 ) t h e m a t r i x a s s o c i a t e d with t h e q u a d r a t i c p a r t of J , considered as a f u n c t i o n of p , i s diagonal; t h e r e f o r e , from t h i s p o i n t of view t h e s i t u a t i o n i s optimal i n comparison with any o t h e r t y p e of approximation_; on t h e o t h e r hand t h e c o n s t r a i n t s lgrad.v(P) < 1, V P E ~( o r O h ) expZicitly involve a l l t h e c o o r d i n a t e s of p ( t h i s i s n o t t h e case f o r t h e i n t e r i o r approximation considered i n Section 4 . 1 o r i n t h e e x t e r i o r approximations s t u d i e d i n Section 5 ) and t h i s f a c t w i l l considerably l i m i t t h e a p p l i c a b i l i t y of t h i s t y p e of approximation.
I
(SEC. 5 )
5.
Exterior approximations of (Po
145
EXTERIOR APPROXIMATIONS OF ( P o )
5.1. Approximations o f J
5.1.1.
A f i r s t exterior approximation o f J
We r e t a i n t h e d e f i n i t i o n s and t h e formalism of S e c t i o n 3.2.1;
hence i f f EL'(P), t h e approximation o f J(v) = i s d e f i n e d by: J: : Vh+ W
t h e e x p l i c i t form o f ( 5 . 1 ) b e i n g given by:
with
t h e sets O h , z h are as d e f i n e d i n S e c t i o n 3.2.1 and Remark 3.1 a l s o holds f o r ( 5 . 2 ) . 5.1.2.
A
second exterior approximation of J
I n t h e f o l l o w i n g w e s h a l l a l s o u s e t h e approximation o f J d e f i n e d by
Before making ( 5 . 4 ) e x p l i c i t w e d e f i n e t h e grid Qh as f o l l o w s :
146
Elasto-plastic torsion of a cylindrica2 bar
(CHAP. 3 )
and w e s h a l l use t h e n o t a t i o n ( s e e Figure 5.1):
Fig. 5.1.
We d e f i n e & by: Zh = {
(5.7)
M 1 MEQ,, m&U)
a t l e a s t one of t h e
belonging t o
We can now express ( 5 . 4 )
J~(uJ=
(5.8)
hZ
c
4 v e r t i c e s of
ah1.
e x p l i c i t l y i n t h e form: x
Mi+iiai+iii6Ih
the
Lj 5.2.
s t i l l being defined by (5.3) and with u, = 0 i f
MPq$Dk
E x t e r i o r approximations of KO
It seems reasonable t o approximate KO by
+
t h e function I b,q, vh 1’ IbZq, v h 1’ being c o n s t a n t over square no. 1 ( r e s p . 2, 3, 4) of s i d e h / 2 , i n Figure 5.2, and having t h e value
(SEC. 5 )
Exterior upprodmations of (Po)
147
t h e e x p l i c i t form of ( 5 . 9 ) i s given by
with, i n (5.10), 0,=0 if M,#8,. rn It r e s u l t s from (5.10) t h a t t h e use of K& imately 4 Card (L?,,) q u a d r a t i c c o n s t r a i n t s . t h e number of c o n s t r a i n t s t o around Card(f2,J of t h e c o n s t r a i n t s (5.10) a t s u i t a b l y chosen which gives:
leads t o approxI n o r d e r t o reduce w e c8n take t h e mean p o i n t s of 4 or Qb
where t h e r e l a t i o n d e f i n i n g K& can be i n t e r p r e t e d as an approximation a t M i + l 1 2 j + l 1 2of t h e c o n s t r a i n t I grad u l2 < 1 ; K& can a l s o be defined as:
148
Elasto-plastic torsion of a cyZindrica2 bar
(CHAP. 3)
Before d e f i n i n g a t h i r d e x t e r i o r approximation o f K O we i n t r o duce t h e s e t Oh d e f i n e d by:
We now c o n s i d e r t h e e x t e r i o r approximation
The e x p l i c i t formulation of t h e membership of d i s t i n g u i s h between t h r e e c a s e s : I f M i j E Qh we have :
K & l e a d s us t o
with urn = 0 i f M,$Q,. If Mi, E rhw i t h a s i n g l e neighbour i n ah( i f t h i s neighbour i s for example M i + , , , s e e F i g u r e 5 . 3 ) we have
Fig. 5.3.
The e x t e r i o r o f O h i s shown hatched.
and, i n , f a c t , a s uij = 0, we have (5.17)
1
5; I “ + I , I d
1
. ah
If M,,Erk w i t h two neighbours i n ( i f t h e two neighbours a r e , f o r example M i + , , , Mi,-,, s e e F i g u r e 5 . 4 ) we have
Exterior approximations of (Po)
Fig. 5.4.
149
The e x t e r i o r of O h i s shown hatched.
and as uij = 0, we have (5.19)
2
2
+
# + + l j
d 1.
The r e l a t i o n s d e f i n i n g K& can be i n t e r p r e t e d as an approximation at M i j c z ho f t h e c o n s t r a i n t I grad u 1' Q 1. w We s h a l l a l s o use t h e e x t e r i o r approximation of KO defined by:
(5.20)
+
d1 a.e)
,
L
which i s expressed e x p l i c i t l y by:
Remark 5.1. Obviously t h e r e are o t h e r p o s s i b l e e x t e r i o r approximations f o r K O ; f o r example, i n o r d e r t o approximate I g d u 1 2 < 1 a t Mij we can use
Elasto-plastic torsion of a c y l i n d r i c a l bar
150
(CHAP. 3)
however, without going i n t o t h e d e t a i l s , we can s t a t e t h a t (5.22) p r e s e n t s s e r i o u s problems from t h e p o i n t of view o f numerical H stability.
5.3.
Formulation of t h e approximate problem.
With t h e f u n c t i o n a l s J l ( I = 1, 2) and t h e convex s e t s KG (m = 1, 2, 3,,4)defined as i n Sections 5 . 1 and 5.2, w e s h a l l t a k e as approxi m a t e problems : (5.23)
(P&
:
Mh
Ji(uh)
.
H
mlSG
5.4.
S o l v a b i l i t y of t h e approximate problem.
The reasoning of Chapter 2 , S e c t i o n 5.3.2 (or Chapter 3, Section 3.2.2) remains v a l i d i f we n o t e t h a t t h e mappings:
d e f i n e norms over V, and t h a t t h e convex sets KG (m = 1, 2, 3,4) are closed i n V,; we t h e r e f o r e have both t h e e x i s t e n c e and uniqueness of an optimal s o l u t i o n f o r (Pa),,,,. The convergence, as h -+ 0, of t h e s o l u t i o n of t h e approximate problem t o t h a t o f t h e continuous problem w i l l be i n v e s t i g a t e d i n Section 6.
6.
CONVERGENCE OF THE INTERIOR
AND EXTERIOR APPROXIMATIONS
Synopsis. I n t h i s s e c t i o n we s h a l l demonstrate t h e convergence of t h e approximations considered i n S e c t i o n s 4 and 5; w e s h a l l refer t o t h e methods of Chapter 1, S e c t i o n 4 f o r t h e i n t e r i o r approximations and o f Chapter 1, Section 5 f o r t h e e x t e r i o r approximations. The proof of t h e convergence of t h e v a r i o u s approximations H considered i s based e s s e n t i a l l y on a d e n s i t y r e s u l t .
(SEC. 6 )
151
Convergence of approximations
6.1.
A lemma concerning d e n s i t y
We s h a l l now prove:
6.1.
Lemma
We have
O(62)n KO = K O .
(6.1)
Proof.
We proceed (as i n Section 3.3.2, Remark 3.3) using
truncation mrd regularisation; w e t h u s have t h e following two stages :
1) L e t U E K , ; we d e f i n e u, = t,(u)(as T,(u) = U, = (U
(6.2)
i n ( 3 . 3 4 ) ) by:
- E)' - (U + 8)- ,
where E > 0 i s given; a . e . on n , and s i n c e :
w e t h u s have r,(u)EH1(f2) with Igradr,(u)
I v(x) I
KO c K , = { u I u E If&!),
Q 6(x,
r)
I< 1
a. e. }
we have (as i n (3.35)) : (6.3)
o if 6(~,r ) < & otherwise Q 6(x, r ) - E
u,(x) =
and hence T,(u) However (6.4)
T,U +
u
E
KO, V& > 0. i n H'(f2) as
E -+
0,
so t h a t :
{ it i s i s
s u f f i c i e n t t o approximate a function u c K o
(6.5)
which
i d e n t i c a l l y zero i n a neighbourhood of
r.
2 ) We extend v ( u ~ K a ) t oi7 i n Rz by 0 o u t s i d e n ; we have ~ E H ' ( Rand ~ ) (graddl < 1 a.e. on W2. I f @,Jn i s a regularising sequence, w e define C R ~ O ( R 2by: ) (6.6)
(li..)=
IR2 4~)-
4 = v"*
Pn
iTn+ i7
s t r o n g l y i n H'(Rz)
PAX
V ) dV) ;
we have : (6.7)
W e use t h e n o t a t i o n : (6.8)
u, = r e s t r i c t i o n of i7,, t o f2 ;
thus i f v satisfies ( 6 . 5 ) and provided n i s s u f f i c i e n t l y l a r g e ,
Elasto-plastic torsion of a cylindrical bar
152
(CHAP. 3 )
w e have : support (En) c
(6.9)
a=.
1)"
E w-4
s t r o n g l y i n Hi@) as n-+
v,+v
+ 00
We s h a l l now show t h a t v, E K O ;i n f a c t (6.10)
gradG,,=p,rgradE,
and hence
Remark 6.1. S i n c e t h e f u n c t i o n s of a (indeed equi-Lipschitz with constant
KO are equicontinuous on 1) and s i n c e KO i s bounded i n Lm(f2) , KO i s strongly compact i n Lm(Q (from A s c o l i ' s theorem), and hence i f v, E KO Vn w i t h v, -+ v i n 9'(62) then , v E KO and v, -+ v strongly i n Lm(61). ( i e . u n i f o d y ) . 8 6.2.
Convergence of t h e i n t e r i o r approximations.
6.2.1.
Convergence of the f i n i t e element method
Using t h e r e s u l t s o f Chapter 1, S e c t i o n s 4.3 and 4 . 4 , we s h a l l now i n v e s t i g a t e t h e convergence of t h e i n t e r i o r approximation o f S e c t i o n 4.1; w e r e c a l l t h a t h w a s d e f i n e d by:
cp(h) = h = Max (measure of 2').
(6.14)
TErh
Moreover l e t us s t a t e t h a t lim
(a -
= 0 if for all K c
a, K
h-0
compact, we have 0, 3 K provided h i s s u f f i c i e n t l y s m a l l . t h e n have :
We
Theorem 6.1. I f La - Q, -+ 0 with 0 2 go > 0 Vh, f o r any angle 0 of T , V T E Y,,, we have u, 4 u strongly i n Hi@) n Lm(Q)where uh i s the solution of the approximate problem (Po,,) defined i n Section 4.1.2, relation' ( 4 . 1 3 ) , and u i s the solution of (Po)
.
Proof. We s h a l l apply Theorem 4.2 o f S e c t i o n 4.4, Chapter 1. For t h i s it i s n e c e s s a r y t o show t h a t (i) V v E Hi@), t h e r e e x i s t s vh E V, such t h a t oh + v s t r o n g l y in
ma). (ii) Vv E K , t h e r e e x i s t s v4 E KO n v h such t h a t v, v s t r o n g l y i n of t h e s e p o i n t s it i s s u f f i c i e n t t o t a k e v i n a dense subset which i s i n H i ( 0 ) f o r ( i ) ,and i n KO f o r ( i i ) (see ( 4 . 2 ) and Remark 4 . 1 , Chapter 1). -+
H,@h f o r each
(Sec.
6)
Convergence of approximations
I n a g e n e r a l manner we t a k e
U E ~ ( Q )and
VT = { triangle M,M, M, } E g
153
introduce
h ,
t h e a f f i n e l i n e a r f u n c t i o n GT on T , such t h a t
and d e f i n e u, by (6.16)
in T .
uk = 8,
It can r e a d i l y be shown, using Taylor s e r i e s expansions of order 2 ( ' ) t h a t (6.17)
I
(6.18)
I grad @Ax) - grad u(x) I d c,(u) 4 Vx E T ,
@T(x)
-
Nx) I d
cI(u)
h VX E T ,
.
where t h e c o n s t a n t s c,(u), c2(u) depend on u From t h e s e estimates and from t h e d e f i n i t i o n (6.16) w e deduce s t r a i g h t away t h a t (6.19)
uh + u
i n H,'(Q),
which gives ( i ) (on 9(Q) dense i n H,'(Q) 1. I n o r d e r t o show ( i i ) ,w e use Lemma 6.1, which allows us t o take V i n K,,nB(Q) ; s i n c e (6.20)
u =
lim(Au) i n H,'(Q), A c 1 , A + 1 ,
we can confine our a t t e n t i o n t o u with (6.21)
u E 9(Q),
I grad u(x) 1 d A < 1 .
"hen, i n a l l T s Y h we have:
and hence (6.23)
uh E KO f o r s u f f i c i e n t l y s m a l l h (such t h a t A
which, i n conjunction with F i n a l l y , we have (6.24)
uh+ u
+ cZ(u) 4Q 1 )
(6.19)~ gives ( i i ).
s t r o n g l y i n Lm(Q)
(') For a systematic study see Ciarlet-Wagschal /l/.
154
Elasto-plastic torsion of a cylindrical bar
from Remark
(CHAP. 3 )
6.1.
Remark 6.2. Falk /1/ gives an estimate of t h e approximation e r r o r as a function of h ; we should p o i n t out t h a t t h e condit i o n s of v a l i d i t y ( r e l a t i n g t o GI and Y h ) of t h i s estimate are more r e s t r i c t i v e than t h e assumptions of Section 4.1.1 and of H, Theorem 6.1. Convergence of the i n t e r i o r approximation method using the eigenfunctions of -A i n H&?)
6.2.2.
We s h a l l now study t h e convergence of t h e i n t e r i o r a?proximation 4. defined i n S e c t i o n ( 4 . 2 ) and u s i n g t h e same n o t a t i o n . We assume t h a t t h e c o n s t r a i n t 1 grad u l Q 1 i s s a t i s f i e d on t h e set defined i n Section 4.2.2, Remark 4.5, by r e l a t i o n s (4.39), (4.40). We a l s o assume t h a t t h e spectrum (BT)meN of -A has been arranged i n o r d e r of i n c r e a s i n g eigenvalue, 1.e.:
Then, with u denoting t h e s o l u t i o n o f (Po) , w e have:
AsM++co'and h + O , with
Theorem 6.2. (6.26)
hS;,
+0
, real r > 1
we have : (6.27)
t#+u
strongly i n H&?).
Proof. We apply Theorem 4.2 of Chapter 1, Section 4.4, t a k i n g t h e only p o i n t s account of Remark 4 . 1 of Chapter 1, S e c t i o n 4.3; which are not immediate are t h a t we must v e r i f y t h a t under t h e s t a t e d conditions f o r M a n d h w e have: V V E Xc K O ,
X=
K O , there exists
(6.28)
#-+u
# E K$
such t h a t
s t r o n g l y i n H,'(bl)
and (6.29)
if
# EKZ, #
4
v
weakly i n H&2)
, then
V EKO.
Verification of (6.28) We proceed as i n t h e proof of Theorem 6.1, d e f i n i n g X (6.30)
x=
by:
U L{K0n9(Q)}. 0 0, L2
(6.87) p r o v e s Lemma 6 . 2
w
Approximation of a : We have; Proposition 6.2.
The families
(a:),
and (at), constitute exterior
approximations of .a. For t h e b i l i n e a r form a: as d e f i n e d i n ( 6 . 5 0 ) , and Proof. under t h e c o n d i t i o n s ( 6 . 5 2 ) , ( 6 . 5 3 ) , . , ( 6 . 5 6 ) , t h i s h a s alr e a d y been proved ( u s i n g d i f f e r e n t n o t a t i o n ) i n S e c t i o n 3.3.2, Proposition 3.2; it t h e r e f o r e o n l y r e m a i n s t o prove it f o r under t h e c o n d i t i o n s ( 6 . 5 7 ) , ( 6.58) , ( 6.59), ( 6.61) We c l e a r l y hav’e:
...
.
4
S6.91)
{
(6.92)
i f pzh vh + u2 v weakly i n Fz w e have lim infai(uh,u& 2 a(v, D)
as p z h uh + uz D weakly i n F2 ,p2hwh -P u2 w. strongly in Fz w e have at(vh, w&
+ U(D,w )
,
.
S i n c e t h e form
(6.93)
{ ( f i , f 2 ) 3 (el, ~
1
2 )
+
s.,
(fi 61 + fz
S J dx
i s b i l i n e a r c o n t i n u o u s o v e r ( L 2 ( R 2 ) r we deduce, from ( 6 . 5 1 ) ,
( 6 . 5 8 ) , ( 6 . 5 9 ) , t h a t under t h e a s s u m p t i o n s o f c o n d i t i o n (6.91) have :
we
164
Elasto-plastic torsion of a cylindrical bar
lima:(v,, w 3 =
(6.94)
k=l
h-0
I ~
%dx z
(CHAP.
3)
=
~
~
k
(6.91). Regarding (6.92),t h i s comes from (6.511, (6.58), (6.59)and t h e weak lower semi-continuity o f t h e convex
which p r o v e s
.
and strongly continuous f u n c t i o n a l on (L2(R2))’ d e f i n e d by
(6.95)
(fl,
f2)
I
-,
(f: + fz)dx.
R’
Exterior approximations of KO. We s h a l l d i s t i n g u i s h two c a s e s depending on whether we u s e t h e norm II 111, o r t h e norm 11 (12hover V,, , t h e a p p r o x i m a t i o n o f H;(Q> We s h a l l s t a r t w i t h t h e former c a s e which i s s l i g h t l y t h e s i m p l e r o f t h e two; i n t h i s c a s e we have
.
P r o p o s i t i o n 6.3. Under conditions (6.52), (6.53), ... , (6.56), the convex sets KG ( m = 1, 2, 3 , 4) defined in Section
5.2 constitute exterior approximations of K,. It i s n e c e s s a r y t o prove ( s e e Chapter 1, S e c t i o n
Proof. f o r m = 1,
(6.96)
5.2),
2, 3, 4:
i f v,
E KG
w i t h limp,, v, = h-0
t
weakly i n Fl
, then
ce a, K O ,
and
(6.97)
VPE K O , 3 v , ~KG such t h a t p,,u,,
11 vh
IIlh
a, v s t r o n g l y i n F, and
Q c.
Verification of (6.96). We p u t
5.1, t h a t :
t = (ro,tl,t2) ;
it t h e n r e s u l t s from Chapter 1, S e c t i o n
(SEC. 6 )
Convergence of approximations
and hence 5: E u1 Hi@) . It now remains t o prove t h a t m = 1; i n f a c t :
and from
{ E U ~KO
;
165
t h i s i s immediate f o r
(6.96), (6.98):
which, i n t h e l i m i t , s i n c e t h e convex s e t
[(fl,fz)lfl,~z~~z(Q f:) ,+f2 6 1 i s closed i n (LZ(Q))’, g i v e s :
a.4
and hence, w i t h (6.98), { E u l K o . We s h a l l now t r e a t simultaneously t h e c a s e s where m = 2, 3, noting t h a t from (5.14),S e c t i o n 5.2:
4,
(6.102)
If t h e f u n c t i o n to denotes t h e element o f H1(Rz) obtained by extending to by 0 o u t s i d e Q and qhub 6fihuh (k = 1,2) are considered t o be elements of L’(R’) ( l ) we have from (6.96), (6.98),
at o ax,
weakly i n L2(Rz) (k = 1,2).
We can t h u s apply
(6.72), (6.73), which g i v e s :
(6.103) 6&h
(6.104)
uh
-b
h-o lh ?k i ( f h / Z ) skzqh uh
=
Since t h e s e t s [(fl,fz) [(&1.2.3,4
IAELW),
at o ax,,
weakly i n L2(R’).
I fl, f2EL’(R2), f: i=1,2,3,4,
+ f:
6 1 a.e] resp..
1
$ f i 6 2 a.e.
I= 1
are convex and closed i n (Lz(Rz))2(resp. (L2(R’))q , we have i n t h e limit, t a k i n g account of (5.12), (6.1021, (5.201, (6.60), (6.104): (1)
we r e c a l l t h a t s i n c e t h e supports of qhu&, 6&4 (k = 1,2) are i n Q, t h e s e f u n c t i o n s a r e considered t o be elements of L2@) or L’(R’) , as t h e c a s e may be.
166
EZasto-plastic torsion of a cyZindricaZ bar
and hence, w i t h ( 6 . 9 8 ) ,
(CHAP. 3 )
(EulKo.
Verification of ( 6.97 ) . I t i s s u f f i c i e n t ( s e e Chapter 1, S e c t i o n 5.3, Remark v e r i f y t h i s for
5.4) t o
U A{ KO n %Q) }
(6.106)
OCIl(uio+u,-J so t h a t ~ b - l - u b - 2 3 u , - u i o - l = h ; now U ~ E Kso~ t h a t u,O-l - uia-z = h , and so on u n t i l i = 2 . A similar argument i s used f o r t h e o t h e r family of r e l a t i o n s . P r o p o s i t i o n 7 . 5 . Let iM (2 < ,i < N) be such t h a t uiU = max u , ; 1S I G N + % then f o r h s u f f i c i e n t l y small we have: 0 Q uy, (7.61)
uiU
Proof. (7.62)
or : (7.63)
- uiy-l < h ,
- Ui,+l < h , = gh'fiU+ 4 , +
0 d uiy
zi,
1
ui,
- 1) *
With t h e n o t a t i o n of P r o p o s i t i o n 7.3 w e have: =I
Wfi,+ ui,-1 + uil+J
9
186
Elasto-plastic t o r s i o n o f a c y l i n d r i c a l bar
(CHAP. 3 )
putting e,
= ]
ution, i n
Remark 8 . 2 . w;*4(ra)
rn
I t may b e shown t h a t t h e n o n l i n e a r o p e r a t o r -+
p:
w - '.4'3(ra)
d e f i n e d by:
(8.6)
(B(u), w ) =
J'. (1 - I grad
vu, w E
u
1')-
grad u.grad w dx
Wi*4(ra)
i s monotone ( b e i n g t h e g r a d i e n t of a convex f u n c t i o n ) , l o c a l l y L i p s c h i t z , and p(u) = 0 V V E KO ; p i s t h u s a penalisation operato r a s s o c i a t e d w i t h KO, i n t h e s e n s e o f Chapter 1, S e c t i o n 3 . 2 . rn I n c o n n e c t i o n w i t h t h e b e h a v i o u r o f u, a s E -+ 0 , we have: Theorem 8.1.
(8.7)
When
E
+ 0
strongZy i n
u,-u
wherae u i s the s o l u t i o n of
Proof. 3.2: For
(8.8)
we have: Hi(ra>
(Po).
T h i s i s a v a r i a n t o f Theorem 3.1, Chapter 1, S e c t i o n
u a s o l u t i o n o f (Po), w e have (1 - I grad u 1')J,(u> =S J,(u) Q J(u) V& > 0 .
We t h u s have:
=0
so t h a t :
S o l u t i o n of ( P o )by p e n a l t y ' methods
(SEC. 8 )
199
so t h a t :
(I u, llq(m Q C
(8.10)
VE > 0
I
There t h e r e f o r e e x i s t s u * ~ H t ( n )and a subsequence, a l s o deno t e d by u, , such t h a t :
(8.11)
lim u,
= u*
weakly i n
Hi(@
*+O
.
Relation (8.8) implies t h a t :
jn
(8.12)
{ (1
- I grad u, 12)-
}' dx Q 4 &Mu) - J(u3)
so t h a t , t a k i n g account of ( 8 . 1 0 ) ,
11 u, I I ~ ~ . Q. (c~ ,
(8.13)
- I grad u, 12)-
{ (1
(8.14)
}z dx = 0 .
For t h e above subsequence w e t h u s have:
limu, = u*
(8.15)
weakly i n Wt*'(Q).
,+O
From ( 8 . 9 ) , ( 8 . 1 1 ) , (8.14), semi-continuity :
(8.16)
J(u*) Q J(u) ,
(8.17)
1.((1 -
(8.15) we deduce by weak lower
(gradu*1*)-}'dx=O.
R e l a t i o n (8.17) i m p l i e s t h a t # * € K O ; we t h u s have, from (8.16) and t h e uniqueness of u, u* = u . We deduce from ( 8 . 9 ) , (8.11) .that for t h e above subsequence we have :
J(u) Q lim i d J ( u 3 Q lim sup J(u3 Q J(u) , so t h a t : (8.18)
lim J(u,) = J(u) , r+O
which implies :
Elasto-plastic torsion of a cylindrical bar
200
(CHAP. 3 )
There i s t h u s convergence o f t h e norm, and hence s t r o n g convergence s i n c e we a l r e a d y have weak convergence. I n view o f t h e uniqueness o f u , we have l h u , = u w i t h o u t r e s t r i c t i n g o u r s e l v e s t o a subsequence. '+'
A variant
8.2.2
The p e n a l t y method o f S e c t i o n 8 . 2 . 1 i n t r o d u c e s a truncation which has t h e e f f e c t o f l i m i t i n g t h e d i f f e r e n t i a b i l i t y o f t h e i n o r d e r t o g e t round t h i s d i f f f u n c t i o n a l J,, t o b e m i n h i s e d ; i c u l t y w e i n t r o d u c e a s l a c k f u n c t i o n 3 0 ( s e e Remark 3.6, Chapter 2, S e c t i o n 3 . 3 ) ; i n f a c t , t h e r e i s e q u i v a l e n c e between:
(8.20)
Igradul
< 1'
a.e.
and
(8.21)
(graduI2+q= 1
q,O
a.e.
and hence w e o b t a i n t h e problem a l i s i n g r e l a t i o n (8.21)
(n,), a
a.e.
(P8)., by pen-
v a r i a n t of
It may b e noted t h a t t h e p e n a l i s e d f u n c t i o n a l a p p e a r i n g i n I n view o f ( 8 . 2 2 ) is not convex w i t h r e s p e c t t o t h e p a i r ( u , q ) t h i s we have:
.
P r o p o s i t i o n 8.3. The problem (nJa h i t s a unique solution (u8,pJ where us i s the solution of (P3and where p8 is defined by:
(8.23)
p8 = SUP (0,l
Proof.
- I grad us 1')
(1
- I grad U, 12)' .
We w r i t e
and, g i v e n u , we d e f i n e qv by:
(8.24)
4. = SUP (0,l
- 1 grad u 1)' = (1 - I grad u 12)'
I
Solution o f ( P o )b y penalty methods
(SEC. 8 )
201
so t h a t with ( 8 . 2 3 )
i . e . t h e r e s u l t which w a s t o be proved. From Remark 8.1 and P r o p o s i t i o n 8.3, it i s easy t o deduce:
The p a i r ( u , , p ~ ,t h e s o l u t i o n of (II,), i s char-
P r o p o s i t i o n 8.4.
acterised by :
I
i z a
- Au, -; C -(I k=l
(8.27)
gradu, l2 + p a -
Application t o t h e s o l u t i o n o f t h e approximate problems (I). Formulation of the penalised approximate problem.
8.2.3
There i s ( i n p r i n c i p l e ) no d i f f i c u l t y i n FreZhinary remark. applying t h e p e n a l t y method of S e c t i o n s 8.2.1, 8.2.2 t o t h e varWe s h a l l i o u s a p p r d x h a t i o n s o f (Po) d e f i n e d i n S e c t i o n s 4 and 5 . c o n f i n e o q a t t e n t i o n t o t h e approximation (Pd13of (Po) ( s e e S e c t i o n s 5.1, 5.2, 5 . 3 ) , t h i s being t h e o n l y c a s e which has given r i s e t o numerical a p p l i c a t i o n s ; t h e method used i s t h e f i n i t e dimensional analogue of t h a t i n S e c t i o n 8.2.2.
Penalisation of t h e problem We d e f i n e t h e space
{ qh Iq h
4 and
(Pdl3. the. convex s e t Lh+by:
(8+28)'
Lh
(8.29)
Lh+={qhIqhELh, qijao
=
=
qij ) Y , j c a r
qijE
1
9
vMijEch},
and t h e f u n c t i o n a l I,, : Vh x Lh + R
a s s o c i a t e d with
with:
(l)
See S e c t i o n 5.1.1 f o r t h e d e f i n i t i o n o f
J,'
(Pod13 by
202
Elasto-plastic torsion of a c y l i n d r i c a l bar
where ( s e e S e c t i o n 5.2, F i g u r e s 5.2, 5.3,
(8.32)
I
Bij = 1 if MI, E a,, ;Pi, = 2 i f M , , E r , w i t h ,Bl, = 3. i f M l , ~ r , w , ith
(CHAP. 3 )
5.4):
a s i n g l e neighbour i n Q,,, two n e i g h b o u r s i n Q,,
( i t i s a p p r o p r i a t e t o t a k e urn = 0 i n ( 8 . 3 1 ) i f Mrn$Q&. W h i l s t it i s p o s s i b l e t o t a k e a l l t h e q, e q u a l t o u n i t y , a more n a t u r a l c h o i c e i s :
, t h e d i s c r e t e analogue of We t h e n d e f i n e t h e problem (Ha,) Y by:
(4)
and p r o c e e d i n g as i n S e c t i o n s 8 . 2 . 1 ,
8.2.2,
it i s e a s y t o p r o v e :
P r o p o s i t i o n 8.5. The problem ( H d a d m i t s one and only one solution,(uk,Pk). i n v h x &+ and when E + 0, uk + u, where 4 i s t h e soZution of (PM)13. P r o p o s i t i o n 8.6 . The pair ( u k , p d which i s the unique soZution of (Ha&i s characterised by:
Remark 8 . 3 . The r e s u l t s of P r o p o s i t i o n 8 . 5 a r e s t i l l v a l i d i f f o r t h e au w e t a k e s t r i c t l y p o s i t i v e s c a l a r s , which may b e chosen a r b i t r a r i l y
.
Remark 8.4. system ( 8 . 2 7 ) .
System ( 8 . 3 6 ) i s t h e d i s c r e t e a n a l o g u e o f t h e
(SEC. 8 )
Solution of ( P o )by penalty methods
20 3
Application t o the s o l u t i o n o f the approximate problem (11). Solution of the penalised approximate problem.
8.2.4
I n view of P r o p o s i t i o n 8 . 6 it i s s u f f i c i e n t , i n o r d e r t o s o l v e t o s o l v e t h e ( n o n l i n e a r ) system ( 8 . 3 6 ) ; f o r t h i s , we s h a l l u s e a v a r i a n t o f t h e method o f point over-relaxation with
(nab ,
projection. I n t h e f o l l o w i n g (by way o f s i m p l i f i c a t i o n ) we s h a l l w r i t e pk=(pi,)y,,eijk and s i n c e Gi,(ub expl i c i t l y i n v o l v e s , i n a d d i t i o n t o Mi, , o n l y t h e f o u r immediate neighbours o f Mi, , we s h a l l w r i t e : uk=(ui,)y,,sa*,
Gi,("b = G*,("i-l,9
"ij-19 "1,'
"i.11,
Vij+l).
I t i s t h e n p o s s i b l e t o p u t i n e x p l i c i t form t h e r e l a t i o n are,
-(ubrpb) 84,
=0;
assuming t h e QU t o be g i v e n by ( 8 . 3 3 ) , we have:
(8.37)
with upI = 0 i f
bY
M,,,#c1,
.
We d e n o t e t h e l e f t - h a n d s i d e o f (8.37)
Elasto-plastic torsion of a cylindrical bar
204
(CHAP. 3 )
and it may b e s e e n t h a t ( 8 . 3 7 ) c o r r e s p o n d s t o t h e d i s c r e t i s a t i o n , a t M f J c f & , o f t h e f i r s t r e l a t i o n i n ( 8 . 2 7 ) by means o f a 13p o i n t scheme, as shown i n F i g u r e 8.1:
Fig. 8.1.
Description o f t h e over-relaxation algorithm I n t h e f o l l o w i n g , it i s assumed t h a t t h e i t e r a t i o n s a r e p e r formed w i t h i n c r e a s i n g i, and t h e n w i t h i f i x e d and i n c r e a s i n g j. We t h u s u s e t h e a l g o r i t h m :
(8.38)
u,",p,"
given a r b i t r a r i l y ,
t h e n , 4.P;b e i n g assumed known, t h e c o o r d i n a t e s a r e determined u s i n g :
and having c a l c u l a t e d
(8.4)
fl;'
= MaX(O,1
#,+'
of
4"
#+' , &+' i s d e t e r m i n e d from:
- /$JG;(4+'))
vMfjEGh.
i s t h e s o l u t i o n o f a n equI t may b e noted t h a t i n ( 8 . 3 9 ) #' a t i o n o f t h e t h i r d degree ( w i t h a s i n g l e v a r i a b l e ) .
(SEC. 8 )
20 5
Solution of ( P o ) by penalty methods
Application to the solution of the approximate problems (111). Examples.
8.2.5
We s h a l l consider a g a i n t h e example of Section we thus have 51 = ]0,1[ x lo, 1[ and f = 10.
E q l e 1.
7.5;
Numerical values of the parameters Mesh size:
:
h = 1/20
Penalisation parameter:
E
= 0.625
K
Initialisation of algorithm ( 8 . 3 8 ) - (8.40) : ut = 0, p t Termination criterion f o r algorithm ( 8 . 3 8 ) - ( 8.40) :
= 0.
Solution of (8.39) : The 3rd-degree equation determining ' : 4 from (8.39) w a s solved t h i s is a by t h e Newton-Raphson method i n i t i a l i s e d with 4, j more r a p i d process than s o l u t i o n by Cardan's method.
Analysis of the numerical results: For w = 1 ( r e s p . w = 1.7) convergence i s reached i n 391 itera t i o n s ( r e s p . 186 i t e r a t i o n s ) corresponding t o a n execution time of 1 0 s e c . ( r e s p . 6.2 sec.) 011 a n IBM 360191.
E q l e 2. thus have
We a g a i n consider Example 3 o f S e c t i o n f i s given by:
a - p,l[ x P,1[ and 10
f(x)=
(8.41)
- 10 0
if
(XI,
xz) E lo,
tr x li, I[,
if ( x 1 , x Z ) ~ ] t1[, x
if
(XI, X J
7.6; w e
P, tr
x
lo,&, lo, tr u B, 1[
x
B, 1[.
Once again, we t a k e h = 1/20, E = 0.625 x u(: = 0 , = 0 and t h e same termination c r i t e r i o n f o r algorithm (8.38)
-
.40) as i n t h e preceding example. For w = '1, convergence i s then reached i n 560 i t e r a t i o n s , corresponding t o an execution time of 1 4 . 5 sec on a n IBM 360/91, : 4 c a l c u l a t e d , as above, by t h e Newton-Raphson method. m with '
.
Remark 8.5. The approximate s o l u t i o n s t h u s c a l c u l a t e d coinci d e i n both of t h e above examples ( t o w i t h i n b e t t e r than 1%) with t h e s o l u t i o n s of t h e same approximate problems, c a l c u l a t e d by w means of t h e o t h e r methods described i n t h i s c h a p t e r . Remark 8.6. ing
E;
The r a t e o f convergence may be improved by varya r e l a t i v e l y l a r g e v a l u e i s used t o s t a r t with, and E i s
Elasto-plastic torsion of a c y l i n d r i c a l bar
206
(CHAP. 3)
then reduced i n t h e course of t h e i t e r a t i o n s of algorithm (8.38) ( 8 . 4 0 ) ( s e e S e c t i o n 8 . 3 . 5 below)
.
Remark 8 . 7 .
Ji
V a r i a n t s of
(n,J
-
may b e d e f i n e d u s i n g t h e approx-
of J ( s e e S e c t i o n 5 . 1 . 2 ) a n d / o r t h e a p p r o x i m a t i o n s K& (m = 1,2,4) o f KO ( s e e S e c t i o n 5 . 2 ) ; t h e u s e o f K;;k (m = 1,2,4) would l e a d t o a v a r i a n t of ( 8 . 3 7 ) e x p l i c i t l y i n v o l v i n g t h e v a l u e s of urn c o r r e s p o n d i n g t o t h e 9 p o i n t s shown o n F i g u r e 8 . 2 and 1 2 ( r e s p . 4 ) s l a c k v a r i a b l e s for rn = 1 ( r e s p . rn = 2 , 4 ) . imation
Fig. 8.2.
We r e c a l l ( s e e S e c t i o n 5 . 2 ) t h a t f o r rn = 2, 3 , 4 t h e number of c o n s t r a i n t s i s a p p r o x i m a t e l y e q u a l t o t h e number o f v a r i a b l e s ui,, so t h a t t h e r e s h o u l d b e a p p r o x i m a t e l y t h e same number o f s l a c k v a r i a b l e s as of v a r i a b l e s u,, ; c o n v e r s e l y for m = 1 t h e r e would b e a p p r o x i m a t e l y f o u r t i m e s as many s l a c k v a r i a b l e s as v a r i a b l e s Uij.
8.3 8.3.1
A second p e n a l t y method
Principle of the method
We w r i t e :
There i s t h e n e q u i v a l e n c e between:
(Po) M h J(v) , US&
and
Solution of ( P o )by penalty methods
(SEC. 8 )
207
o r indeed:
We s h a l l p e n a l i s e o n l y t h e r e l a t i o n :
(8.45)
q = gradu,
so t h a t , f o r
(no)l,
and, f o r
We denote by fl,( r e s p . 9% ) t h e f u n c t i o n a l d e f i n e d by (8.46) ( r e s p . ( 8 . 4 7 ) ) and .Y = If;@) x (L2(0))' ; t h e f o l l o w i n g p r o p o s i t i o n s may t h e n e a s i l y b e proved: P r o p o s i t i o n 8.6.
we have the foZZowing properties f o r
ft,(k = 1, 2): (i) (ii) (iii)
fh i s continuous and strictZy convex on V , lim
II @,bII* - + O D
f,(u,q)
=
+ 00,
& i s & t e a m d i f f e r e n t i a b z e (and even Frechet sense) on V .
C" i n the
8
P r o p o s i t i o n 8 . 7 . The penalised problems (n,),(k = 1, 2) each a h i t one and only one soZution on If,'@) x A , say (.;",& , of which the respective characterisations arc:
Elasto-plastic torsion of a c y l i n d r i c a l bar
208
(CHAP. 3 )
for k = 1
and, f o r k = 2,
- Aui + divpi = ef
There i s e q u i v a l e n c e between t h e l a s t - t w o r e l a t Remark 8.8. i o n s o f ( 8 . 4 8 ) , (8.49) a n d , r e s p e c t i v e l y :
(8.50)
p,' = PA(gradUf),
where PAd e n o t e s t h e o r t h o g o n a l p r o j e c t i o n o p e r a t o r from L'(Q) x L'(Q) + A r e l a t e d t o t h e s t a n d a r d norm o f L'(Q) x L'(Q)
;
e x p l i c i t l y , w e have:
(8.52)
The u s e o f p e n a l i s a t i o n i s j u s t i f i e d by: Theorem 8 . 2 .
(8.53) (8.54)
When
E .+
4 - c ~ strongly -+
gradu
0 we have, Vk = 1 , 2 :
in H;(Q)
strongZy i n L2(C2) x L2(Q)
where u i s the soZution of (Po). T h i s i s a v a r i a n t o f t h e proof o f Theorem 3 . 1 , o f ChaProof p t e r 1, S e c t i o n 3 . 2 ; we s h a l l c o n s i d e r o n l y t h e c a s e k = 1, t h e c a s e k = 2 b e i n g t r e a t e d i n a similar manner.
Solution of ( P o ) by penalty methods
(SEC. 8 )
L e t u b e t h e s o l u t i o n o f (PJ and p = g r a d u ; (u,p) E H,'(n) x A and:
209
we have
J(u:) Q #1,,(uf,pf) < Yl,,(U,p)= J(u) V& > 0 .
(8.55)
We t h u s have
J(u:)
(8.56)
< J(u)
VE > 0
so t h a t
11 U:
(8.57)
(I,gdcn,
0 .
Moreover, pf ~ A * p f bounded i n L2(Q) x L'(Q) , so t h a t t h e r e e x i s t s a subsequence, a l s o d e n o t e d by (uf,pf), u* E H,'(n) and p* E L2(n) x L2((ra)such t h a t when E + 0:
(8.58)
u: * U, weakly i n H i @ )
(8.59)
pi + p *
weakly i n L 2 ( n )x L'(l2) ( I )
and s i n c e A i s convex and c l o s e d i n L2(12) x LZ(n) ( a n d t h u s weakly c l o s e d ) w e have:
~ * E A .
(8.60) Relation
(8.55) i m p l i e s t h a t :
so t h a t , t a k i n g a c c o u n t o f (8.57) :
From ( 8 . 5 6 ) , (8.58), ( 8 . 5 9 ) , ( 8 . 6 2 ) w e deduce by (weak) lower semi-continuity
(8.63)
J(u*) < J(u) ,
(8.64)
p* = gradu*,
.
and (8.60) i m p l i e s u*eKO We t h u s have u * = u . The s t r o n g convergence o f ut and t h e convergence o f t h e e n t i r e (u:). f a m i l y a r e proved by p r o c e e d i n g as i n t h e p r o o f o f
(l)
I n f a c t w e have p: + p *
weakly" i n L"(R) x L"(n)
E l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l bar
21 0
(CHAP. 3 )
Theorem 8.1, t h e s t r o n g convergence o f p: r e s u l t i n g from t h a t o f u,'. and from ( 8 . 6 2 ) . 8.3.2
Application t o t h e s o l u t i o n o f approximate problems (I). Formulation of t h e penalised approximate problem i n t h e case of t h e f i n i t e element method of S e c t i o n
4.1. We s h a l l c o n f i n e our a t t e n t i o n t o t h e f i n i t e - d i m e n s i o n a l a n a l ogue o f (Ift)l , t h e o t h e r c a s e b e i n g t r e a t e d i n a s i m i l a r manner ( b y way o f s i m p l i f i c a t i o n , t h e i n d e x 1 w i l l b e s u p p r e s s e d i n t h e t h e notation of Section 4 w i l l b e retained. following) ; Having d e f i n e d t h e s p a c e Vh i n S e c t i o n 4.1.1, we i n t r o d u c e t h e subspace Lh o f L 2 ( 9 ) x L ' ( l 2 ) :
(8.65)
h= =
{
q = ( q 1 . q2) I E Lz(12) x L2(Q),41 =
1
q1TeT,q2
T€dh
=
1
TeTh
q2TOT;
q17142TE
where
(8.66)
OT =
c h a r a c t e r i s t i c f u n c t i o n of
T,,
then
The f i n i t e - d i m e n s i o n a l a n a l o g u e o f (lZt)l i s t h e n d e f i n e d by:
where Y, i s d e f i n e d by prove:
(8.46) ; it i s a s t r a i g h t f o r w a r d matter t o
P r o p o s i t i o n 8.8. The problem (&) a h i t s one and only one s o l u t i o n , (uk,Pk) , and when E -+ 0 we have
Solution of ( P o ) by penalty methods
(SEC. 8 )
(8.73)
211
pka+ grad u,, ,
where uh is the (unique) solution of Min J(l)h). w uh E V k n PO
The i n t e r i o r a p p r o x i m a t i o n method o f S e c t i o n 4 . 2 Remark 8 . 9 . (using e i g e n f u n c t i o n s o f - A i n HA@)) i s n o t s u i t a b l e f o r t h e p e n a l i s a t i o n t e c h n i q u e o f S e c t i o n 8 . 3 , for t h e r e a s o n s i n d i c a t e d i n Remark 4.6 o f S e c t i o n 4 . 2 . 5 .
Application to the solution of the approximate problems (11). Formutation of the penalised approximate problem in the case of the exterior approximations of Sec-. tion 5 .
8.3.3
O f t h e e x t e r i o r a p p r o x i m a t i o n s o f KO c o n s i d e r e d i n S e c t i o n 5.2, i s t h e o n l y one i d e a l l y s u i t e d t o t h e u s e o f t h e p e n a l t y method d e s c r i b e d i n S e c t i o n 8 . 3 ( f o r a d i s c u s s i o n o f t h e d i f f i c u l t i e s c o n n e c t e d w i t h t h e u s e o f t h e o t h e r a p p r o x i m a t e convex s e t s o f S e c t i o n 5.2, s e e Marrocco /l/). Once a g a i n , we s h a l l c o n f i n e our a t t e n t i o n t o t h e f i n i t e - d i m e n s i o n a l a n a l o g u e o f (l7h1( s e e ( 8 . 4 6 ) ) and we s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n 5. We r e c a l l t h a t
K&
and w e i n t r o d u c e
We n e x t d e f i n e , f o r 1 = 1 , 2 t h e f o l l o w i n g a p p r o x i m a t i o n s of
m1 :
(8.76)
(nd
4
Elasto-plastic torsion of a c y l i n d r i c a l bar
212
(CHAP. 3 )
and it i s then easy to prove the following propositions: Proposition 8.9. The problem(llJadmits one and only one solu t i o n , ( u k , p 3 , and when E -+ o we have, for 1 = 1 or 2
dk+d
(8.77)
where
4 is the soZution
(see ( 5 . 2 3 ) ) of
u i + l j + l - uij+l + ui+lj - uij
2h
V l = 1,2.
.
Solution of ( P o )by penalty methods
(SEC. 8 )
213
Remark 8.9. R e l a t i o n s ( 8 . 7 8 ) , (8.791, (8.80) a r e t h e d i s c r e t e H analogues o f (8.48). Remark 8.10.
There i s e q u i v a l e n c e between (8.80) and:
vMi+ 1/2 j + I/2
'h
and where Pij i s t h e o r t h o g o n a l p r o j e c t i o n o p e r a t o r from R2 + 5 r e l a t e d t o t h e E u c l i d i a n norm of R2, or:
8.3.4
,
Application to the solution of the approximate problems (111). Solution of the penalised approximate problem by over-relaxation with projection.
We s h a l l c o n f i n e o w a t t e n t i o n t o t h e e x t e r i o r approximations of S e c t i o n 8 . 3 . 3 .
Preliminary remark:
There i s e q u i v a l e n c e between:
Description of the over-relaxation algorithm. I n t h e f o l l o w i n g , it w i l l b e assumed t h a t t h e i t e r a t i o n s a r e performed w i t h i n c r e a s i n g i, and t h e n w i t h i f i x e d and i n c r e a s i n g we t h u s u s e t h e a l g o r i t h m : j;
(8;86)
up,pf
given a r b i t r a r i l y ,
214
E l a s t o - p l a s t i c t o r s i o n o f a c y l i n d r i c a l bar
t h e n , 4.d b e i n g assumed known, t h e c o o r d i n a t e s a r e determined u s i n g :
- u;+ l j + 1 +U;:;j+ (8.87)
0 ,
+ P n @~(tf))
J(u)
+ iInp” @&)
dx ,
9
sufficiently small.
I t may b e noted t h a t , g i v e n p” o f t h e e l l i p t i c problem
, fl
i s t h e s o l u t i o n i n Hi(l2)
The u s e o f ( 9 . 1 2 ) , ( 9 . 1 3 ) , ( 9 . 1 4 ) f o r s o l v i n g ( P o ) ( i n f a c t t h e i n t e r i o r and e x t e r i o r approximations o f ( P , ) ) w i l l a r i s e i n S e c t i o n s 9 . 2 , 9.3, 9.4. 9.1.3
A third formulation
I n t h e two p r e c e d i n g approaches t h e d i f f i c u l t y o f t h e problem ( i . e . t h e c o n s t r a i n t I grad u 1 Q 1 ) w a s “ e l i m i n a t e d ” ( a t l e a s t , f o r m a l l y ) by a s s o c i a t i n g w i t h it a Lagrange m u l t i p l i e r ( o r a Kuhn-Tucker m u l t i p l i e r ) s a t i s f y i n g a non-negativity c o n d i t i o n , s i n c e t h i s t y p e of c o n s t r a i n t does n o t p r e s e n t any d i f f i c u l t y t h e new approach i s based on t h e f o l l algorithmically s p e a k i n g ; owing v a r i a n t o f t h e c o n s i d e r a t i o n s o f S e c t i o n 8 . 3 . 1 . Once a g a i n we write: (9.16)
A = { q 1 q E L’(l2) x L2(Q),q = (ql, q,), q:(x)
(9.17)
X , = { (u, q) I u E H i @ ) , q E A, q = grad u } ;
(l)
+ q%x)
d 1
a .e. }
And s i n c e , a p a r t from t h e c a s e f = c o n s t a n t , t h e problem of t h e e x i s t e n c e o f a s a d d l e p o i n t f o r S?,, i n H d ( Q ) x A , , seems t o b e open even f o r f e L Z ( Q )( s e e Remark 9 . 2 b e l o w ) .
E l a s t o - p l a s t i c t o r s i o n o f a c y l i n d r i c a l bar
222
(CHAP. 3 )
t h e r e i s t h e n e q u i v a l e n c e between ( P o ) and
S i n c e t h e p r o j e c t i o n L'(62) x L'(62) + A d o e s n o t pose any p r a c t i c a l d i f f i c u l t i e s ( s e e ( 8 . 5 2 ) ) , we s h a l l b e c o n t e n t t o decouple u and q , 2 . e . t o e1imina;te t h e r e l a t i o n q = gradu , by a s s o c i a t i n g w i t h it a Lagrange m u l t i p l i e r ( i n S e c t i o n 8 . 3 we proceeded u s i n g penal-
isation)
(1)
We t h u s d e f i n e : (9.19)
L = L'(l2) x L'(62)
and t h e L a g r a n g i a n p3: HA@) x L x L + R (9.21)
by
Y,(u,q;p)=~
From t h e r e l a t i o n
we deduce t h a t i f Y3admits a s a d d l e p o i n t { u , p ; d } on x A x L, t h e n { u , p } i s t h e s o l u t i o n o f t h e problem (9 . 18 ) e q u i v a l e n t t o (Po); t h e algorithm (4.121, , ( 4 . 1 5 ) o f Chapter 2, S e c t i o n 4.3 t h e n t a k e s t h e form:
...
(9.23)
(9.25)
(l)
{
w i t h 1" known
(€
L ) , we d e f i n e { g , p " } as b e i n g t h e
element o f H,'(Q) x A which minimises Y 3 ( u , q ; A''),
p,, > 0 ,
s u f f i c i e n t l y small.
A method combining b o t h t h e s e methods w i l l b e found i n S e c t i o n 10.
(SEC. 9 )
Solution of ( P o ) b y d u a l i t y methods
With t h e f u n c t i o n k g i v e n i n L , l r a n d f a r e , solutions of
(9.26) (9.27)
- Ad'= 2 f
223
respectively,
+ divA",
p" = PA(A").
The u s e o f ( 9 . 2 3 ) , ( 9 . 2 4 1 , ( 9 . 2 5 ) f o r s o l v i n g ( P o ) w i l l be d i s c u s s e d i n S e c t i o n 9.5.
Remark 9.1. The method o f S e c t i o n 9 . 1 . 3 i s a method o f decomposition-coordination i n t h e s e n s e o f Bensoussan-Lions-Temam /I/, a p p l i e d t o t h e s o l u t i o n of t h e problem (9.18) e q u i v a l e n t t o (Po). 9.1.4
Remarks
Remark 9 . 2 . The problem o f t h e e x i s t e n c e of a s a d d l e p o i n t 9,) on H ; ( Q ) x 4 ( r e s p . H&2) x A2, Hi@) x A x L ) f o r Y , ( r e s p . Y2, seems t o b e open; i n t h i s c a s e it i s p o s s - i b l e , however, by u s i n g ( f o r example) Bensoussan-Lions-Tham /l/, Chapter 2, S e c t i o n 2, s u b s e c t i o n 1, t o prove t h e e x i s t e n c e o f s a d d l e p o i n t s f o r (9.28) (9.29)
(9.30) i n W$m(Q) x A x (Lm(Q))'x (Lm(Q))' where (,) d e n o t e s t h e b i l i n e a r form o f t h e d u a l i t y between Lm(Q) and i t s d u a l (Lm(Q))' w i t h
av
Vi, u Ir = 0
(9.31)
Wd*m(Q)=
(9.32)
(Lm(Q));= { q I q E (Lrn(Q))',( 0, q ) 2 0 vu E L 3 Q ))
E L "(Q),
On t h e o t h e r hand, t h e r e i s no d i f f i c u l t y i n p r o v i n g t h e e x i Y 3), s t e n c e o f s a d d l e p o i n t s f o r t h e r e s t r i c t i o n o f Y l ( r e s p . g2, t o t h e f i n i t e - d i m e n s i o n a l convex sets and s p a c e s a p p e a r i n g i n t h e i n t e r i o r approximations o f ( P o ) defined i n S e c t i o n 4 ( s e e Sections 9.2, 9.3 below); t h e same a p p l i e s f o r t h e e x t e r i o r
Elasto-plastic torsion of a cylindrical bar
224
approximations of Section
(CHAP. 3 )
5 ( s e e Sections 9.4, 9.5 below).
8
Remark 9 . 3 . I n t h e c a s e f = c o n s t a n t (which c o r r e s p o n d s t o t h e p h y s i c a l problem o f S e c t i o n 1 . 2 ) ¶ H . B r e z i s /5/ h a s demonstr a t e d t h e e x i s t e n c e o f P E A , ( = L:(L?)), w i t h p = 0 i n t h e e l a s t i c zone, such t h a t { u , p }, v h e r e u i s t h e s o l u t i o n o f (P,), i s t h e unique s a d d l e - p o i n t o f Y z i n H,'(L?) x A, ( f o r t h e i n t e r p r e t a t i o n o f t h i s r e s u l t i n c o n n e c t i o n w i t h problems i n mechanics, see DuvautLions /l/¶ Chapter 5 , S e c t i o n 6 . 6 ) . 9.2
9.2.1
A p p l i c a t i o n o f t h e d u a l i t y method o f S e c t i o n 9 . 1 . 2 t o t h e s o l u t i o n o f t h e a p p r o x i m a t e problems. Case o f t h e f i n i t e element a p p r o x i m a t i o n o f S e c t i o n 4 . 1 .
On the existence of a saddle point
We s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n 4 . Having d e f i n e d t h e s p a c e V, i n S e c t i o n 4 . 1 . 1 we i n t r o d u c e Lk, a subspace of L m ( 0 ) :
with
(9.34)
OT =
c h a r a c t e r i s t i c function of T ,
and
With t h e L a g r a n g i a n Y Z d e f i n e d by ( 9 . 9 ) , (9 . 10 ) w e t h e n have: P r o p s i t i o n 9.1. { u,,p, } On Vh X A ,
(9.36)
uh
The LagrangianP2 a h i t s a saddle point with:
the solution o f
Min
J(u,J,
01,c KO n V I
and (9.37)
PA( grad u,
1' -
1) = 0
.
Proof. S i n c e t h e s p a c e s Vhand 4 a r e f i n i t e - d i m e n s i o n a l , t h i s r e s u l t f o l l o w s immediately from R o c k a f e l l a r /3/, S e c t i o n 28, Theorem 28.2, 28.3. 8
(SEC.
9)
Solution of ( P o ) by d u a l i t y methods
225
Remark 9.4. S i n c e t h e p a i r { u h , p h ) i sa s a d d l e - p o i n t o f g z o n V, x A,, w e have PZ(th,p#,)6 Uz(Uh,pJ b h E V, s o t h a t , e x p l i c i t l y ,
= 0, w e have, t a k i n g account
and i n p a r t i c u l a r i f
Of
(9.371,
Under t h e c o n d i t i o n s f o r a p p l y i n g Theorem 6.1, S e c t i o n 6.2.1, when h - 0 w e have uh d u s t r o n g l y i n HA(Q)nL w ( f l ) , where u i s w e t h u s have, s i n c e ph 2 0, t h e s o l u t i o n of ( P o ) ;
11 ph lILl(I2) d
(9.40)
vh
'
an e s t i m a t e i n t'(Q) f o r ph.
SoZution of the approximate problem by means of a saddle-point-seeking algorithm ( I ) . Description of the algorithm.
9.2.2.
.. .
We use t h e a l g o r i t h m (4.12), , (4.15) o f Chapter 2, S e c t i o n 4.3, which i n t h i s p a r t i c u l a r c a s e t a k e s t h e form:
(9.41)
i
with of
P;: known
v,,which
(E
Ath) we d e f i n e
6
+-
Jn
minimises J(t+,)
as b e i n g t h e element
fl (I grad uh I' - 1) h,
+ p,(l grad 141' - l ) ) ,
(9.42)
P;"
(9.43)
pm> 0 ,
= SUP ( 0 , ~ ;
s u f f i c i e n t l y small.
The a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) i s a n "approximation" o f a l g o r i t h m ( 9 . 1 2 ) , ( 9 . 1 3 ) , ( 9 . 1 4 ) ; it may b e noted t h a t , given fl , J; is t h e unique s o l u t i o n o f :
i f. (1
(9.44)
+ pi) grad 4.grad uh dX =
v,,
In
fUh
dr
VUh €
v,,
which i s a variational equation in Vh. rn
Remark 9.5. I t i s p o s s i b l e t o w r i t e (9.41) ( a n d / o r ( 9 . 4 4 ) ) and ( 9 . 4 2 ) e x p l i c i t l y i n terms o f t h e parameters qr, T E C , and of
E l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l bar
226
(CHAP. 3 )
t h e "nodal" p a r a m e t e r s ui
= %(Mi) 9
=
G(Mi)
$
0
Mi 6 zh
( s e e S e c t i o n 4 . 1 . 1 , r e l a t i o n ( 4 . 8 ) ) by u s i n g t h e r e l a t i o n s o f Section 4.1.4; w e r e f e r t h e r e a d e r t o B o u r g a t /1/ f o r s u c h a n e x p l i c i t formulation. rn
S o l u t i o n of t h e approximate problem by means of a Convergence o f saddle-point-seeking algorithm ( 11) t h e algorithm.
9.2.3.
.
We have : Proposition 9.2. The sequence (4)" defined ( 9 . 4 2 ) , ( 9 . 4 3 ) converges t o t h e s o l u t i o n uh o f
by algorithm ( 9 . 4 1 ) ,
Proof. Taking a c c o u n t o f P r o p o s i t i o n 9.1 t h i s f o l l o w s from Theorem 4 . 1 o f Chapter 2 , S e c t i o n 4.3, as l o n g as t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d ( s e e r e l a t i o n ( 4 . 1 1 ' ) , Chapter 2, S e c t ion 4.3 ) : (i) (9.45)
eZh: Vh-P Lh
The mapping @zh(t+,)
= I grad
1' -
defined by:
1
i s locally Lipschitz;
(ii)
(9.46)
Vqe &,
Min Uh e v h
J(Uh)
the solution of
:b
+-
q@zh(U) dx
1
remains w i t h i n a bounded region independent of q. P o i n t ( i ) follows immediately s i n c e , vuh E V, I grad uhIz i s quadr a t i c w i t h r e s p e c t t o t h e "nodal" p a r a m e t e r s u, = uh(Mi), and as r e g a r d s ( i i ) ,t h e s o l u t i o n ulg o f ( 9 . 4 6 ) i s t h e t h u s C"; s o l u t i o n i n vh o f t h e v a r i a t i o n a l e q u a t i o n (9.47)
(1 4- q) grad u,.grad
uh dX =
In
VUh E v h9
f U h dX
so t h a t s e t t i n g uh = u, i n ( 9 . 4 7 ) and b e a r i n g i n mind t h a t
(9.48)
(I"
I grad u, IZ
dyZc f Q
II II''(rn
(7
q 3 0,
9
( ' ) C i s t h e PoincarC c o n s t a n t i n . H i ( n ) i . e . t h e r e c i p r o c a l o f t h e s q u a r e r o o t o f t h e s m a l l e s t e i g e n v a l u e o f - A i n Hi@).
(SEC. 9)
Solution of (Po)by d u a l i t y methods
which proves ( i i ) and t h e p r o p o s i t i o n .
227
rn
Remark 9.6. By e x p l i c i t l y w r i t i n g o u t t h e c a l c u l a t i o n s i n t h e proof of Theorem 4 . 1 of Chapter 2, S e c t i o n 4.3, i n t h e p a r t i c u l a r c a s e o f t h e algorithm (9.41), ( 9 . 4 2 ) , ( 9 . 4 3 ) , it may be shown t h a t t h e c o n d i t i o n f o r convergence o f t h e algorithm i s given by
where t h e c o n s t a n t C i s t h e one appearing i n (9.48) and I n p r a c t i c e , t h e e s t i m a t e (9.49) i s very y ( f & = TMin a r # , (Area of 2 ').
pessimistic, a t l e a s t f o r t h e examples considered below. Solution of the approximate probZem by a saddle-point-' seeking aZgorithm (111). PracticaZ considerations on the use of algorithm ( 9 . 4 1 ) , (9.421, ( 9 . 4 3 ) .
9.2.4.
The work i n t h i s s e c t i o n and i n t h e subsequent Section 9.2.5, follows Bourgat 111.
SoZution of ( 9 . 4 4 ) . The determination o f 4 from fi reduces, using ( 9 . 4 4 ) , t o t h e s o l u t i o n o f an (approximate) e l l i p t i c problem which i s of order 2, l i n e a r , and with c o e f f i c i e n t s which vary i n c1 and with n ; t h i s problem, which i s o f D i r i c h l e t type, has been solved by point over-relaxation t a k i n g as v a r i a b l e s t h e nodal values v,, VM&
Termination c r i t e r i a :
$ is i d e n t i f i e d t o
(@,,,.j#,
; we
then d e f i n e (9.50)
D,(n) = Max
I ul -
1,
Mt
c
(9.51)
D,(n) = Y"Ph
lul-ul-ll
c
lull
'
MtcPr
and t a k e f o r algorithm ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) , a termination c r i t e r i . o n of t h e form (9.52)
Di(n) < 8
.
Moreover, a t l e a s t a t t h e s t a r t of t h e i t e r a t i v e process, t h e r e i s ' n o p o i n t i n s o l v i n g t h e D i r i c h l e t sub-problems ( 9 . 4 4 ) f o r t h e over-relaxation algorithm applied with high p r e c i s i o n ; t o problems ( 9 . 4 4 ) , w e t h e r e f o r e use a termination c r i t e r i o n o f
-
228
Elasto-plastic torsion o f a c y l i n d r i c a l bar
t h e type (9.52) with
E
r e p l a c e d by E,
, defined
(CHAP. 3 )
by
Initialisations: I n t h e f o l l o w i n g examples, a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) has been i n i t i a l i s e d w i t h pt = 0 a n d , i n t h e d e t e r m i n a t i o n o f 4 from p ; by s o l v i n g ( 9 . 4 4 ) , t h e o v e r - r e l a x a t i o n a l g o r i t h m h a s been i n i t i a l i s e d w i t h G-'. 9.2.5
Solution of the approximate problem using a saddlepoint-seeking algorithm (IV). Examples
Example 1.
We c o n s i d e r a g a i n Example 1 o f S e c t i o n
7.4, t h a t
is L a = { x I x ~ R ' , x : + ~ < l }and f = l O , t h e notatt h e e x a c t s o l u t i o n b e i n g g i v e n e x p l i c i t l y by ( 7 . 8 1 ) ; i o n and t r i a n g u l a t i o n s used a r e t h e same as t h o s e i n S e c t i o n 7.4 (see F i g u r e s 7 . 3 and 7 . 4 ) . Using t h e i n i t i a l i s a t i o n s i n d i c a t e d i n S e c t i o n 9 . 2 . 4 , we t a k e i n (9.52), i n (9.531, pm=p=l i n ( 9 . 4 2 ) and w = 1 . 5 , where w i s t h e o v e r - r e l a x a t i o n p a r a m e t e r i n t h e s o l u t i o n of (9.44) ( s e e Section 9.2.4). The main r e s u l t s c o n c e r n i n g t h e a p p r o x i m a t e s o l u t i o n o f ( P o ) by means o f t h e above t e c h n i q u e s a r e shown i n T a b l e 9 . 1 ( t h e n o t a t i o n i s t h e same as t h a t i n T a b l e 7 . 1 ) . Table 9.2 shows, f o r t h e c a s e o f t r i a n g u l a t i o n y 2 ,t h e number o f o v e r - r e l a x a t i o n i t e r a t i o n s r e q u i r e d t o s o l v e t h e n t h problem i n ( 9 . 4 4 ) , u s i n g t h e i n i t i a l i s a t i o n s and t e r m i n a t i o n c r i t e r i o n indicated i n S e c t i o n 9.2.4. The r e s u l t s g i v e n i n t h i s s e c t i o n and i n T a b l e Remark 9.7. 7 . 1 o f S e c t i o n 7.4 would a p p e a r t o i n d i c a t e , a t l e a s t , f o r t h e p a r t i c u l a r c a s e c o n s i d e r e d (l), t h a t t h e method o f o v e r - r e l a x a t i o n w i t h p r o j e c t i o n d e s c r i b e d i n S e c t i o n 7 i s more r a p i d t h a n moreover t h e method o f S e c t i o n t h e d u a l i t y method g i v e n above; 7 o f f e r s t h e a d v a n t a g e s o f a s i m p l e r i m p l e m e n t a t i o n ( a n d hence e a s i e r programming) and o f a s m a l l e r s t o r a g e r e q u i r e m e n t ( f o r example, t h e r e i s no need t o s t o r e t h e c o o r d i n a t e s o f pi ) . I t should b e p o i n t e d o u t , however, t h a t i n t h e p h y s i c a l problem o f e l a s t o p l a s t i c t o r s i o n ( i . e . f = c o n s t a n t ) t h e f u n c t i o n p i s of g r e a t p h y s i c a l i n t e r e s t ( s e e Duvaut-Lions 111, C h a p t e r 5 ) and t h i s may j u s t i f y u s i n g a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) s i n c e
(')
T h i s remark i n f a c t i s c o m p l e t e l y g e n e r a l ( s e e b e l o w ) .
(SEC. 9 )
Solution of ( P o ) by d u a l i t y methods
229
t h i s g i v e s , as w e l l as an approximation o f t h e s o l u t i o n u of ( P o ) , an approximation o f t h e f u n c t i o n p .
n
1 2
4
6
8
10
28
24
22
7
6
Number of over-relaxation iterations
Further examples.
S t i l l keeping
f = 10, we have a l s o t r e a t e d
t h e following cases;
Example 2
R
Example 3
R i s t h e e q u i l a t e r a l t r i a n g l e w i t h u n i t side.
Example 4
R i s t h e T-shaped domain shown i n F i g u r e 9.1.
= 10, 1[ x 10, l[.
For t h e t r i a n g u l a t i o n s shown i n F i g u r e s 9.2, 9.3, 9.4, respe c t i v e l y , Table 9 . 3 summarises t h e main r e s u l t s f o r t h e s o l u t i o n of t h e above examples; w e r e f e r t h e r e a d e r t o Bourgat /1/ for
Elasto-plastic t o r s i o n of a c y l i n d r i c a l bar
230
x1
(CHAP. 3 )
t
Fig. 9.1.
Square
Number of v e r t i c e s N = Card(&)
I
225
N' = Card (yh) Pn
I
t
384
I I
432
10-3
I
10-3
169
195
=P 0
I
1.5
I &
r
5 x 10-4 10-3
Comp. time C I1 10070 Table
9.3
f u r t h e r d e t a i l s . I n a d d i t i o n t o t h e t r i a n g u l a t i o n s used, F i g u r e s 9.2, 9.3 and 9.4 a l s o show t h e approximated e l a s t i c and p l a s t i c zones, a t r i a n g l e T b e i n g t a k e n t o l i e i n a p l a s t i c zone when I grad u, I 2 0.995 on T .
Solution of ( P o ) by duality methods
(SEC. 9 )
Fig. 9.2.
Square cross-section : F i n i t e elements ('512 triangles), C = 10. P l a s t i c zones: @ I n i t i a l triangulation:
9.3
231
-
Application o f t h e d u a l i t y method of Section 9.1.2 t o t h e s o l u t i o n o f t h e approximate problems. Case of t h e i n t e r i o r approximation'by eigenfunctions of - A i n HA(Q1
Preliminary remark. I n g e n e r a l t h e eigenfunctions of - A i n H,'(Q) are not known explicitly, and t h i s s e r i o u s l y r e s t r i c t s t h e use of t h e approximation described i n S e c t i o n 4.2; w e have t h e r e f o r e confined our a t t e n t i o n t o a s i n g l e example, corresponding t o Q = ]0,1[ x ]o and f = 10, f o r which t h e above eigenfunctions and t h e corresponding eigenvalues are known e x p l i c i t l y and f o r which approximate s o l u t i o n s f o r problem ( P o ) have been
232
Elasto-plastic torsion o f a cylindrical bar
Fig. 9.3.
Fig. 9.4.
(CHAP. 3 )
F i n i t e - e l e m e n t c a l c u l a t i o n of t h e e l a s t o - p l a s t i c t o r s i o n of a c y l i n d r i c a l bar w i t h t r i a n g u l a r c r o s s section. Number of f i n i t e elements: 384.
T-section:
F i n i t e elements (432 t r i a n g l e s ) , = 10 P l a s t i c zones:
c
I n i t i a l triangulation:
-
(SEC. 9)
S o l u t i o n o f ( P o ) by d u a l i t y methods
o b t a i n e d , by o t h e r methods e a r l i e r i n t h e p r e s e n t volume.
233
rn
Statement of t h e continuous problem and s p e c t r a l decomposition of - A i n H i @ )
9.3.1
We c o n s i d e r t h e problem ( P o ) r e l a t i n g t o R = )O, I[ x ]0,1[ and f = 10, which w a s t r e a t e d e a r l i e r i n S e c t i o n 3 . 5 (making u s e o f t h e e q u i v a l e n c e r e s u l t o f S e c t i o n 2 . 2 ) and i n S e c t i o n s 7.5, 8 . 2 . 5 , 8 . 3 . 5 and 9 . 2 . 5 , by v a r i o u s methods ( s e e a l s o S e c t i o n 9.4 b e l o w ) . For t h e above domain 62, t h e e i g e n f u n c t i o n s o f - A i n H i ( @
are g i v e n by: (9.54)
wpq(xl,x z ) = 2 sin / I
KS,
sin y x.r2 ,
p, q E
N,
and t h e c o r r e s p o n d i n g eigenvalues b y :
(9.55)
,3/
= nZ(p2
+ 92) .
The e i g e n f u n c t i o n s ( 9 . 5 4 ) a r e o r t h o g o n a l i n H;(R) and o r t h o normal i n ~'(62). rn 9.3.2
-
Explicit formulation o f t h e approximate problem
For a s l i g h t l y g r e a t e r d e g r e e o f g e n e r a l i t y , it i s n o t a b s o l u t e l y n e c e s s a r y t o assume f = c o n s t a n t . With M E N , we d e n o t e by V, t h e s u b s p a c e o f H;(n> g e n e r a t e d by t h e wI1 f o r 1 < p , q C M, t h a t is : (9.56)
{
V,=
u ~ u E H A ( R U) ,=
Using t h e n o t a t i o n
w i t h (assuming f
(9.58)
f,
=
C
P,w~,~,ER}
I Sp.q Q M
EL'@)
w&)
( 9 . 5 6 ) , w e have f o r
U E
V,
):
f (4dx .
I n t h e p a r t i c u l a r c a s e i n which
f = c o n s t a n t = C, we.have:
i f p and q a r e odd
(9.58')
fp4 =
0 otherwise
so t h a t i f
VE
V,, rn
Elasto-plastic torsion of a cylindrical bar
234
(CHAP. 3 )
I n view o f t h e symmetries o f t h e p a r t i c u l a r Remark 9.8. problem under c o n s i d e r a t i o n , it i s p o s s i b l e t o t a k e i n ( 9 . 5 7 ' ) only pm with p and q odd, t h e o t h e r c o e f f i c i e n t s being zero; however, w e have d e l i b e r a t e l y chosen not t o t a k e advantage of t h i s s i m p l i f i c a t i o n i n t h e following. For t h e approximation o f K O , we proceed as i n S e c t i o n 4.2.2, Remark 4.5: given M ' i n t e g e r > 0 (which w i l l tend towards i n f i n i t y ) w e put h = 1 / M ' and d e f i n e :
(9.59)
Rk = { Pij I Pij = (ih, jh), i, j E 2 } .
I n t h e p a r t i c u l a r c a s e under c o n s i d e r a t i o n , t h e s e t may be w r i t t e n e x p l i c i t l y i n t h e form
(9.60)
= { PijIP,, = (ih,jh), 1 < i , j
0 ,
(0, P;I
d e f i n e flu t o be t h e element
minimises Ug(uM,P;),
of V,which
(9.68)
(9.12) , (9.13) ,
+ Pn(G;(U",) - I))
VPi, E 0,
9
sufficiently s m a l l .
The proof of P r o p o s i t i o n 9.2 can be adapted w i t h o u t d i f f i c u l t y , and it may t h u s be shown t h a t t h e sequence (flu). d e f i n e d by (9.67), Min J(u). (9.68) , (9.69) , converges t o t h e s o l u t i o n of AeK8
9.3.5
Practical considerations on the use of algorithm ( 9 . 6 7 ) ,
( 9 . 6 8 ) , (9.69) The terms cospxx,, (resp. cosqnx,,) and sinptxIi (resp. sinqrrx,,) which appear i n algorithm (9.67) , (9.68) , (9.69) through t h e intermedi a r y o f t h e G,, ( s e e ( 9 . 6 2 ) , ( 9 . 6 8 ) ) are obviously c a l c u l a t e d once Ah wN i n t h e c a l c u l Moreover, p u t t i n g U", = and f o r a l l . I*P.q*M
a t i o n of I&( w i t h A known), u s i n g (9.67), t h e vector (A>)15p,qs,, i s t h e s o l u t i o n of a l i n e a r system of equations whose matrix ( t h e c o e f f i c i e n t s of which vary w i t h n ) is symmetric and positived e f i n i t e ( l ) ; t h i s system has been solved by t h e Cholesky method. For t h e i n i t i a l i s a t i o n o f ( 9 . 6 7 ) , ( 9 . 6 8 ) , (9.69), p , O - 0 w a s always used.
9.3.6
Numerical r e s u l t s
Algorithm ( 9 . 6 7 ) , (9.68), (9.69) w a s , applied t o t h e particular case c o r r e s p o n d i n g t o f = 10, and, s i n c e t h e exact s o l u t i o n i s not known, t h e reference s o l u t i o n w a s t a k e n as t h e approximate solution calculated by t h e method of S e c t i o n 3 ( a n d t h u s using t h e equivalence r e s u l t of S e c t i o n 2 . 2 ) w i t h a mesh s i z e of 1 / 4 0 .
Results f o r h = 1/20: For M = 10 ( i . e . 100 eigenfunctions wpr ) , a time of 100 sec w a s required on an IBM 360/91 t o perform 10 i t e r a t i o n s ; taking (l)
And has t h e disadvantage o f being a f u l l m a t r i x .
Elasto-plastic torsion o f a c y l i n d r i c a l bar
236
(CHAP. 3 )
pm independent o f n , i . e . pm= p , t h e o p t i m a l v a l u e o f t h i s parameter i s v e r y c l o s e t o u n i t y ; f o r t h i s v a l u e t h e computed s o l u t i o n d i f f e r e d from t h e r e f e r e n c e s o l u t i o n by o n l y a b o u t ( i n a b s o l u t e v a l u e ) by t h e 5 t h i t e r a t i o n , and t o w i t h i n by t h e For 0.5 < p < 1 . 5 t h e r e f e r e n c e s o l u t i o n i s r e 7th iteration. produced i n l e s s t h a n 1 0 i t e r a t i o n s , t o w i t h i n a t worst.
For M = 5 ( i . e . 25 e i g e n f u n c t i o n s w w ) a t i m e o f 1 6 s e c . w a s r e q u i r e d on a n IBM 360/91 t o perform 20 i t e r a t i o n s ; once a g a i n t a k i n g p m = p t h e o p t i m a l v a l u e remains c l o s e t o u n i t y , and a l i m i t i s r e a c h e d , i n p r a c t i c e , which d i f f e r s from t h e r e f e r e n c e s o l u t i o n by a b o u t i n a b s o l u t e value, i n l e s s t h a n 10 i t e r a t i o n s ; f o r example, a t t h e p o i n t ( 0 . 5 , 0 . 5 ) ( r e s p . 1 / 2 0 , 1 / 2 0 ) o f Q a v a l u e o f 0.427 ( r e s p . 0 . 0 1 9 ) w a s o b t a i n e d i n s t e a d o f t h e 0.413 ( r e s p . 0.029) o f t h e r e f e r e n c e s o l u t i o n . 0 For M = 15, t h e method i s p r a c t i c a l l y u n u s a b l e , r e q u i r i n g a computation t i m e o f s e v e r a l t e n s o f seconds on a n IBM 360/91, f o r a single i t e r a t i o n , and w i t h a v e r y l a r g e s t o r a g e r e q u i r e m e n t .
Results f o r h = 1/10: For M = 10, a t i m e o f 50 s e c . on a n IBM 360/91 w a s r e q u i r e d t o c a r r y o u t 20 i t e r a t i o n s and, w i t h p n = p a g a i n h a v i n g i t s o p t i m a l v a l u e o f a b o u t u n i t y , convergence w a s i n e f f e c t r e a c h e d i n 6 ite r a t i o n s , t h e computed s o l u t i o n d i f f e r i n g by a n a b s o l u t e v a l u e o f about from t h e r e f e r e n c e s o l u t i o n . 9.3.7
Conclusions
T h e , r e s u l t s g i v e n above would a p p e a r t o i n d i c a t e t h a t t h e method o f s o l v i n g t h e problem ( P o ) b a s e d on t h e combined u s e o f approx i m a t i o n by e i g e n f u n c t i o n s o f - A i n H,@) ( s e e S e c t i o n 4 . 2 ) and of a f i n i t e - d i m e n s i o n a l v a r i a n t o f a l g o r i t h m ( 9 . 4 1 ) , ( 9 . 4 2 ) , ( 9 . 4 3 ) ( s e e S e c t i o n 9 . 3 . 4 ) i s v e r y c o s t l y i n terms o f b o t h c o m p u t a t i o n t i m e and s t o r a g e r e q u i r e m e n t . ( I t s h o u l d a l s o b e remembered t h a t i n g e n e r a l t h e e i g e n f u n c t i o n s o f - A i n Hi@) a r e n o t known explThe above drawbacks are l i n k e d t o t h e f a c t t h a t t h e icitly). a p p r o x i m a t i o n c o n s i d e r e d i s globaZ i n n a t u r e ( t h e e i g e n f u n c t i o n s admit as s u p p o r t ) whereas t h e c o n s t r a i n t I gradu I < 1 i s local; t h i s l e a d s t o f u l l m a t r i c e s o f l a r g e dimension ( v a r y i n g w i t h n ) i n t h e determination of i n (9.67). The above d i f f i c u l t i e s would a l s o b e e n c o u n t e r e d , though admite d l y t o a l e s s e r d e g r e e , i f it were r e q u i r e d t o o b t a i n a n approxb a t e s o l u t i o n o f ( P o ) , u s i n g a n i n t e r i o r a p p r o x i m a t i o n o f H&!) by f i n i t e e l e m e n t s o f o r d e r 3 2 . ( s e e S e c t i o n 4 . 1 . 5 ) .
a
S o l u t i o n of ( P o ) by d u a l i t y methods
237
A p p l i c a t i o n o f t h e d u a l i t y method o f S e c t i o n 9.1.2 t o t h e s o l u t i o n o f t h e a p p r o x i m a t e problems. Case o f t h e e x t e r i o r approximations of S e c t i o n 5.
Synopsis:
We s h a l l c o n f i n e our a t t e n t i o n t o t h e approximation t h e o t h e r c a s e s may be t r e a t e d i n a (see Section 5 ) ; v e r y similar manner and, apart from t h e c a s e m = 1 ( ’ ) , have a l most i d e n t i c a l s t o r a g e r e q u i r e m e n t s . w
Existence of a saddle-point. Description and convergence of the d u a l i t y algorithm
9.4.1
Existence of a saddle-point. We s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n s 5 and 8 . 2 . 3 ; we d e f i n e
thus
and a s s o c i a t e w i t h t h e a p p r o x i m a t e problem (Poh)13 t h e approximati o n Z z h o f Z 2 d e f i n e d by
w i t h ai,, pi, , G , d e f i n e d i n S e c t i o n 8 . 2 . 3 , by ( 8 . 3 3 ) , ( 8 . 3 2 ) , (8.35), respectively. The proof of P r o p o s i t i o n 9 . 1 c a n b e a d a p t e d w i t h o u t d i f f i c u l t y , so t h a t 9 2 h a d m i t s a s a d d l e - p o i n t , { uh,ph } , o n v h x A2h , with:
uh t h e s o l u t i o n of (POhIl3,
(9.73)
P&ij
G;(uh)
-
1) = 0 VMij E s
h
. w
Description o f the algorithm and convergence. We u s e a f i n i t e - d i m e n s i o n a l v a r i a n t o f a l g o r i t h m ( 9 . E ) , (9.13),
(9.14),namely:
{
with
A known (€A2,,), we d e f i n e 4 t o b e t h e element
(9.74)
of
which minimises
(9.75)
P;:
(9.76)
p,,
= max (0, P;, + p,,(Bij G;(G) - 1)) VMi,E s h , > 0 , sufficiently small.
(’)
vh
See S e c t i o n 5.2.
YZh(vh.pi)
Elasto-plastic torsion o f a c y l i n d r i c a l bar
238
(CHAP. 3 )
The proof o f P r o p o s i t i o n 9 . 2 c a n b e a d a p t e d w i t h o u t d i f f i c u l t y ; d e f i n e d by ( 9 . 7 4 ) , ( 9 . 7 5 ) , (9.76) converges t o t h e s o l u t i o n o f (P&3.
it may t h u s b e shown t h a t t h e sequence
(mm
Application
9.4.2
We s h a l l c o n f i n e o u r a t t e n t i o n t o t h e c a s e
=
lo,1[
x ]0,1[ and
f = 10 which has been t r e a t e d i n t h e p r e c e d i n g s e c t i o n s by means o f o t h e r methods ( f o r f u r t h e r examples, and a more d e t a i l e d analys i s o f t h e numerical r e s u l t s , see Bourgat /l/).
Mesh s i z e :
h = 1/20.
Solution of ( 9.74) :
By p o i n t o v e r - r e l a x a t i o n ,
with increas-
ing i , j . By p: = 0 f o r a l g o r i t h m (9.74), (9.751, (9.76) ( w i t h u i l = 0 ) i n t h e s o l u t i o n o f (9.74) by o v e r - r e l -
InitiaZisations: and by axation.
b-’
Termhation c r i t e r i a :
(9.78)
D,(n) = Max
14’ - “.;I
Putting
I,
MtJenh
we t a k e f o r a l g o r i t h m ( 9 . 7 4 ) , ( 9 . 7 5 ) , (9.76) t h e t e r m i n a t i o n c r i t erion: (9.79)
Q 10-3
For t h e s o l u t i o n o f t h e sub-problems (9.74) by o v e r - r e l a x a t i o n , w e u s e a t e r m i n a t i o n c r i t e r i o n o f which t h e s e v e r i t y i n c r e a s e s ’with n , and which has t h e form:
with (9.81)
E,,
= min(En-l, 0.2D,(n
- I)),
E,,
=
lo-”
where t h e i n t e g e r p i n ( 9 . 8 0 ) r e p r e s e n t s t h e number o f over-relaxation iterations.
Numerical r e s u l t s . Table 9.4 summarises t h e main results f o r t h e convergence o f a l g o r i t h m (9.74) , (9.75) , (9.76) :
methods
238.
P. = P w ~
Computation t i m e C I1 10070 'Cornputati o n time IBM 360191 -
Table
9.4
The s o l u t i o n t h u s c a l c u l a t e d coincides t o w i t h i n a n a b s o l u t e value of about LOd5 with t h a t obtained f o r t h e same approximate problem by t h e method o f over-relaxation with p r o j e c t i o n described Figure 9.5 shows t h e r e s t r i c t i o n t o i n S e c t i o n 7.5, Example 2. of t h e e q u i p o t e n t i a l s o f t h e f u n c t i o n p defined i n ]O,f[ x p,2 S e c t i o n 9.1.2, obtained using t h e approximation J$, provided by algorithm ( 9 . 7 4 ) - ( 9 . 7 6 ) . Note t h e c o n t i n u i t y of t h e function p ; t h i s i s i n agreement w i t h results obtained by H . Br6zis 151, G? being convex. Figure 9.6 shows t h e v a r i a t i o n s o f t h e number of i t e r a t i o n s r e q u i r e d f o r convergence, as a f u n c t i o n of p f o r w = 1.5 , and Figure 9.7 shows t h e number of over-relaxation i t e r a t i o n s r e q u i r e d t o s o l v e t h e same sub-problem (9.74) using t h e termination c r i t e r i o n ( 9 . 8 0 ) , (9.81) and with pm= p = 1, w = 1.7.
Fig. 9.5. E q u i p o t e n t i a l s of t h e dual f u n c t i o n p .
Elasto-plastic torsion of a cylindrical bar
240
2 3 PC Variation of no. of iterations as a function ofp ( w = 1 . 5 ) . 1
Fig.9.6.
15,
10.
1
\ 5 .
Y 0
3)
I
I
I
(CHAP.
10
m
30
4
241
Solution of ( P o )by d u a l i t y methods
(SEC. 9 )
A p p l i c a t i o n o f t h e d u a l i t y method o f S e c t i o n 9 . 1 . 3 t o t h e s o l u t i o n o f t h e a p p r o x i m a t e problems
9.5
We s h a l l c o n f i n e o u r a t t e n t i o n t o t h e f i n i t e element approxima t i o n of S e c t i o n 1 and t o t h e e x t e r i o r a p p r o x i m a t i o n s o f (Po) u s i n g t h e a p p r o x i m a t i o n K& of KO ( s e e S e c t i o n 5 . 2 ) .
Case of the f i n i t e eZement approximation of Section 4.1. (I). Formulation of the approximate problem and existence of a saddle-point.
9.5.1
We s h a l l r e t a i n t h e n o t a t i o n o f S e c t i o n s 4.1 and 8.3.2; ing:
(9.82)
X O h
=
{ (u, 4) I(u, q ) e x O . U E
vh,
q E'
h
1,
t h e r e i s t h e n equivalence between t h e problem
Min Y
With t h e Lagrangian Proposition 9.3. {uh,ph; Ah}
(9.84)
uh
on
(vh
x
9'3
d e f i n e d by (9.201,
Ph
J(o)
and
EvhnKo
( 9 . 2 1 1 , we have:
The Lagrangian 9, admits a saddle-point, x L h With:
Ah)
the s o l u t i o n of
Min Uh E
(9.85)
putt-
J(uh)
K O n vh
= grad U h
Proof. S i n c e t h e problem is f i n i t e - d i m e n s i o n a l , we may a p p l y 8 R o c k a f e l l a r / 3 / , S e c t i o n 2.8, Theorem 28.2, 28.3.
Case o f the f i n i t e element approximation of Section 4.1. (11). Description of the algorithm and convergence W e u s e a l g o r i t h m (4.12), ... , ( 4 . 1 5 ) of C h a p t e r 2, S e c t i o n 4.3, 9.5.2
E l a s t o p l a s t i c torsion of a cylindrical bar
242
(CHAP. 3 )
which i n t h i s p a r t i c u l a r c a s e t a k e s t h e form:
(9.88)
4"
(9.89)
p.
=
A:
>0,
+ p,(grad 4 - p;), s u f f i c i e n t l y small.
The algorithm ( 9 . 8 7 ) , ( 9 . 8 8 ) , ( 9 . 8 9 ) i s a n "approximation" of algorithm ( 9 . 2 3 ) , ( 9 . 2 4 ) , (9.25).
P r o p o s i t i o n 9.4. I f 0 < a Q pI Q /I< 1 the sequence { G,H} defined by (9.871, (9.881, (9.89) converges t o { u h , p L )with
9.5.3
.
Case of the exterior approximation o f Section 5 . ( I) Formulation of the approximate problem and existence of a saddle-point.
The preliminary remark o f S e c t i o n 8 . 3 . 3 s t i l l h o l d s , i n t h e sense t h a t t h e approximation K& of KO i s t h e only one which i s i d e a l l y s u i t e d t o t h e use o f t h e d u a l i t y method of S e c t i o n 9.1.3. We r e t a i n t h e n o t a t i o n o f S e c t i o n s 5 and 8.3.3, t h a t i s :
We i n t r o d u c e t h e approximation o f
There i s t h e n equivalence between:
xo defined
by
ISEC. 9)
Solution of ( P o )by duality methods
24 3
(9.98)
S i m i l a r l y t h e r e i s equivalence between: (9.99) and
( 9 . loo)
Remark 9.9. The approximation of ( P o ) defined by (9.96) was it may e a s i l y be shown not considered i n S e c t i o n s 5 and 6; t h a t t h e results ( o f convergence, i n p a r t i c u l a r ) demonstrated for t h e o t h e r approximations apply equally well t o problem ( 9 . 9 6 ) .
E l a s t o p l a s t i c torsion o f a c y l i n d r i c a l bar
244
We d e n o t e by Yi t h e f u n c t i o n a l from
vh
x
(CHAP. 3 )
d e f i n e d by
Lh+89
(9.97) i f = 1, and by (9.100) i f l = 2; we a s s o c i a t e w i t h problems ( 9 . 9 8 ) , ( 9 . 1 0 0 ) t h e f u n c t i o n a l s , from ( v h x &) x L, +R, 2'jh and
9'ih , which
a r e a p p r o x i m a t i o n s o f Y3 and a r e d e f i n e d . b y :
'Ji+lj+~
- V i j + ~+ u i + ~ j- u i j 2h
(9.101)
ui
)
- 41i+1/2j+1/2 +
+ 1j+ 1 - ui + 1j + u i j + I - Uij 2h
\
I = 1,2.
Using a v a r i a n t o f P r o p o s i t i o n 9 . 3 i t may b e shown t h a t { d,A,4} on ( v h x A,,) x L,, w i t h :
Yi,,(l=1, 2) admits a s a d d l e - p o i n t , (9.102)
u: t h e s o l u t i o n o f
(9.103)
I
P:i+ 1/2 j+I/*
A
(9.104)
=
+ 1/2 j + 1/2 =
i + 1/2 j + 1/29
(9.96),
u: t h e s o l u t i o n o f
1
4+lj+l-Jil+1 +4+lj-Ufj 2h
4+I j + I - U f + l j + 4 j + l - . i j 2h
vMi+1/2 j +112
(9.99),
ch
9
A+1/2 j+1/21 = P z A i ~+ 112 j+1/21 4 i + I/2 j+ 1/21 ( I = lg2)9
vMi+l/2j+l/2Ezh
where, i n (9.104) Pij i s d e f i n e d i n S e c t i o n 8 . 3 . 3 by ( 8 . 8 2 ) ( 8 . 8 3 )
9.5.4
H
Case o f the e x t e r i o r approximations o f Section 5 . (11). Description of the algorithm and convergence
We u s e a l g o r i t h m ( 4 . 1 2 ) - ( 4 . 1 5 ) o f Chapter 2 , S e c t i o n 4 . 3 , which i n t h i s p a r t i c u l a r c a s e i s e x p r e s s e d ( o m i t t i n g t h e upper index 2 ) :
1)
f o r l = 1, by
I-
w i t h $ known U?+Ij
+
UY-lj
(E Lh)
+ u;j+l + hZ
we d e f i n e U"-]
- 45
=
4~Vh , t h e n
2Jj
+
Ah by:
Solution of ( P o )by d u a l i t y methods
(SEC. 9 ) 2)
24 5
f o r l = 2, by
= 2J;:j
+
VMijeQh
with, f o r
(9.109)
z=
1 or 2
pn > 0 ,
sufficiently s m a l l .
The above a l g o r i t h m i s a n "approximation" o f a l g o r i t h m ( 9 . 2 3 ) , ( 9 . 2 4 ) , ( 9 . 2 5 ) and r e l a t i o n s ( 9 . 1 0 5 ) , ( 2 = 1,2) a r e t h e d i s c r e t e analogues of ( 9 . 2 6 ) . I t f o l l o w s from Chapter 2, S e c t i o n 4.3, Theorem 4 . 1 t h a t t h e sequence { & & } d e f i n e d by ( 9 . 1 0 5 ) t ( 2 = 1,2), (9.106), ( 9 . 1 0 7 ) , ( 9 . 1 0 8 ) i s c o n v e r g e n t , f o r pn s u f f i c i e n t l y s m a l l , t o { u ! , , ~ ! , } .
9.5.5
Case o f the e x t e r i o r approximations o f Section 5 . (111). A nwnericaZ example.
We c o n s i d e r t h e problem (Po) r e l a t i n g t o P = ]0,1[ x ]0,1[ and we c o n f i n e o u r a t t e n t i o n t o f = 10, which w a s t r e a t e d e a r l i e r ; t h e a l g o r i t h m used i s t h u s t h e a p p r o x i m a t i o n d e f i n e d by ( 9 . 9 6 ) ; d e f i n e d by ( 9 . 1 0 5 ) 1 ,
Mesh s i z e :
...
,
h = 1/20.
(9.109):
246
Elasto-plastic torsion of a c y l i n d r i c a l bar
I n i t i a l i s a t i o n of algorithm ( 9 . 1 0 5 ) 1,
.. .
(CHAP.
, (9.109) : A:
3)
= 0.
Determination of 4 from 4 : by p o i n t o v e r - r e l a x a t i o n , t h e a l g o r i t h m b e i n g i n i t i a l i s e d by 4-l With m d e n o t i n g t h e i t e r a t i o n c o u n t f o r t h e o v e r - r e l a x a t i o n , we t a k e as t e r m i n a t i o n c r i t er i o n :
.
c
Mfj E Rh
14" - $ 1
d
El.
Termination c r i t e r i o n f o r algorithm ( 9 . 1 0 5 )
c
. .. , ( 9 . 1 0 9 )
IG+1-lqI0
By assuming f o r Ye t h e e x i s t e n c e o f a s a d d l e p o i n t ( u , V u ; A) on H i @ ) x A x L , it may be shown t h a t under t h e c o n d i t i o n 0 < r o d pn d rl 0; t h e r e a d e r may cons u l t Duvaut-Lions /l/y Chapter 1, Section 7.3, for a t h e o r e t i c a l rn study of t h e c a s e p = 0. Remark 1 . 2 . The f u n c t i o n rp considered i n t h i s s e c t i o n w i l l be of t h e following type: if if
t
5/k
- g
if
( 0 imply t h a t U E Hz-K(Q). I t t h i s r e g u l a r i t y property
( so
t h a t j ( u ) = g [dluidr)
r rd
H1(rd) , extended by z e r o i n , a r b i t r a r i l y s m a l l , (1.83) and (1.88) would b e i n t e r e s t i n g t o know whether s t i l l h o l d s i n t h e l i m i t when k = 0
, but
t h i s problem would a p p e a r t o be
open. I n view o f (1.84)-(1.87) it would a p p e a r t h a t i f t h e r e e x i s t s a such t h a t
V = H1(Q)
with
1I u IIRI-qn)
c(ll f IIL'Ca) + 8 )
Q
C independent o f k , g , f. Using ( 1 . 8 9 ) , (1.90), it may b e proved by p r o c e e d i n g as f o r p o i n t ( i ), t h a t
(1.91)
{
11 uh - u Ilfil(a)
< cdll f 11Lqa) + a) + 'dl f IILqn) + 8)
+ C(q)(11 f 1ILqa + 8)'
x hl/z+E
h2(")/('-")
w i t h i n (1.9l),C and C(q) independent o f h, k, g, f, q > 0, arbitrari l y s m a l l ( q = 0 i f a = 1). We summarise t h e p r e c e d i n g r e s u l t s i n :
Let u be the solution of problem (1.41, (1.5) Theorem 1 . 4 . and uk the solution of the corresponding approximate problem (1.27); i f when h + 0 the angZes of y h a r e bounded beZow by a constant EI,, > 0 , wg have the foZZowing estimates for the approximation e @ m r 11 u,, - u llBl(a) ( i ) I f V = { v l u E H l ( Q ) , ulr-r,=O}we have:
ohere i n (1.92), C(&) and C(&,q) are independent of h, k , g , f and
268
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
(CHAP. 4 )
& , q are a r b i t r a r i l y small p o s i t i v e q u a n t i t i e s .
If Y = H'(R) there e x i s t s
(ii)
a, f
0 i s a r b i t r a r i l y small (q = 0 i f a = 1). Remark 1 . 6 . I f , i n t h e c a s e where Y = { u I V E H'(P), u =0) , we have , U E H"'(R) w i t h 3 Q a Q 1 (which i s n o t t r u e i n g e n e r a l ) w e may r e p l a c e e r r o r e s t i m a t e ( 1 . 9 2 ) by e s t i m a t e ( 1 . 9 3 ) . .&mark
1.7.
I f , i n s t e a d o f (1.27), w e had t a k e n as t h e appro-
ximate problem:
4%.uh - uk) + j(uh) (1.94) uk
In
- Auk) 2 f (oh -
dx
vuh
vh
vh
which i s n o t v e r y p r a c t i c a l from a numerical v i e w p o i n t , t h e n i n s t e a d o f ( 1 . 9 2 ) , ( 1 . 9 3 ) we would have o b t a i n e d , r e s p e c t i v e l y :
(l *95)
11 ub -
fA
lIk'(l2)
d
Z c(&, q) (11 f IILz(n) + 8 ) X
h W
il+8h"
-)
eW% "
II f IlLqm + 8
I n view of ( 1 . 9 2 ) , ( 1 . 9 3 1 , ( 1 . 9 5 ) , (1.96) it would a p p e a r t h a t t h e a d d i t i o n a l e r r o r i n t r o d u c e d by r e p l a c i n g j by j , i s ; we may t h e r e f o r e deduce t h a t i f , from " p r a c t i c a l l y " of t h e p o i n t o f view of r e g u l a r i t y , we have no b e t t e r t h a n U E H " ~ ( Q ) t h e n t h e above e r r o r i s o f t h e same o r d e r o f magnitude a s t h e a p p r o x i m a t i o n e r r o r a s s o c i a t e d w i t h t h e approximate problem ( 1 . 9 4 ) . it would a p p e a r , On t h e o t h e r hand i f we have u ~ H ' + " ( Q ) w i t h a > i n view o f (1.931, ( 1 . 9 6 ) , t h a t t h e p r i n c i p a l p a r t of t h e approxi m a t i o n e r r o r 11 u, - u \lalo i s t h e e r r o r i n t r o d u c e d by r e p l a c i n g i by
O(d)
4
1.5
Convergence o f t h e approximate s o l u t i o n s . (11). The c a s e q = 2.
269
Thermal control problems
(SEC. 1)
Throughout this section, C will be used to denote various constants. A lenitna
1.5.1
We shall use the following variant of Lemma 1.2 for the study of the convergence when h + 0:
where, i n ( 1 . 9 7 ) , C i s independent of k, g , h, S, % Suppose u h € vh Proof. Figure 1.6, and with I we define
, the
notation being that shown in Mi-116 I = I Mi-112 Mi-116 I = 4 I Mt-1 Mt I ;
Fig. 1.6. with mi = characteristic function of Mi-116MJ4i+116
wl-l12 = characteristic function o f
kfj-~&~-11&1-116
It may be shown that for the function that (1.99)
5
r.
$h(uh)
d r = Jr, $(qh
YO
vh) d r
+h(Uk)
defined by (1.22)
m i l a t e r a l problems and e l l i p t i c inequalities
270
so t h a t
(l. loo)
jh("h) =
$(qh YO
Oh)
dr .
Jr
From (1.11)and (1.100) we have
(1.101)
with
(1 .102)
and
(1.103) since
(1.104) we deduce from (1.102),
. .. ,
(1.104) t h a t
so t h a t
or alternatively, writing
T = (a
+ ()/2,
(CHAP.
4)
Thermal contro 2 problems
(SEC. 1)
271
(1.108)
We have
so that, from (1.108)
From (1.112) and from the definition of h we deduce by addition that (1.113)
1)
70 uh
- q h Y O o h llLZ(R) d
11 Y O Oh llHL(T)'
Relation (1.114) in combination with C independent of h, s, uh . U with (1.101) proves (1.97) and hence the lemma. 1.5.2
A convergence theorem
The following result a l s o holds if we assume k = 0 in (1.9).
W e have :
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
27 2
(CHAP.
4)
Theorem 1.5. If, when h -+ 0 , the angles of Yh are bounded below by a constant 8, > 0 , we have
(1.115)
strongly i n H ' ( a )
lim l(h = u h-0
where
u
and
uh
are respectively solutions of the problems (1.4),
(1.5) and (1.27).
With U E V , we denote by r h u t h e p r o j e c t i o n o f u on v h relati v e t o t h e norm u + d m . ( e q u i v a l e n t t o t h e s t a n d a r d norm of H'(l2)) ; w e t h u s have:
Under t h e assumptions adopted f o r g h w e have
(1.118)
strongly i n V .
limrhu = u h+O
P u t t i n g Uh=rhtrin (1.116) we deduce from (1.117), (1.118) t h a t :
(1.119)
a(uh,rhu- u) = 0 r
P r o p e r t y (1.118) i m p l i e s t h a t
I-
lim yo r, u
(1.122)
h-.o v3 E
= you
s t r o n g z y i n H'(l-)
[O, 1/21
and (1.121), ( 1 . 1 2 2 ) l e a d t o :
Therma2 contro 1 problems
(SEC. 1)
273
I t t h e r e f o r e remains t o show t h a t
(1.125)
11 Y O uh l ! H l D ( T ) Q c \I f IIL'G')
'
Taking a c c o u n t o f Lemma 1 . 3 from ( 1 . 1 2 5 ) t h a t : Ijh(uh)
-Auk) I d
cg
& 11 f
-
with s =
i n (1.97)
- we
deduce
IILz(I?)
which gives ( 1 . 1 2 4 ) ; it f o l l o w s from (1.116) w i t h from (1.118)- ( 1 . 1 2 0 ) , (1.123), ( 1 . 1 2 4 ) t h a t
vh
= rhu and
lim a(uh - u, uh - u) = 0
h-0
from which t h e theorem t h e n f o l l o w s .
Remark 1 . 7 .
Theorem 1 . 5 s t i l l h o l d s i f i n t h e f u n c t i o n a l J
d e f i n e d by ( 1 . 5 ) we r e p l a c e t h e term
by
Uv) where
In.
L i s a n element o f t h e d u a l V' of t h e s p a c e V. 1.6
Numerical s o l u t i o n o f t h e a p p r o x i m a t e problems
Synopsis I n t h i s s e c t i o n we s h a l l s t u d y t h e numerical s o l u t i o n o f t h e approximate problems of S e c t i o n 1 . 3 . 3 , namely
by means o f some of t h e o p t i m i s a t i o n methods d i s c u s s e d i n Chapter
2.
(CHAP. 4 )
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
274
F i r s t , it i s a p p r o p r i a t e t o e x p r e s s t h e f u n c t i o n a l s
Jo(uh), j,,(ud e x p l i c i t l y i n t e r m s o f t h e v a l u e s t a k e n by V, on Z h i f q = 1 , or on Z,u Z i i f q = 2 ; t h i s w i l l form t h e s u b j e c t of
S e c t i o n s 1.6.1, 1.6.2.
1.6.1
Formulation of the approximation problem.
(I), q = 1.
In ( 1 . 2 4 ) t h e f u n c t i o n a l j,, w a s e x p r e s s e d e x p l i c i t l y i n t e r m s of t h e v a l u e s t a k e n by u, on (1.127)
jh(uk)
=
Zh n r
, that
is:
4 C I Mi Mi+ i I (Jl(0i) + Jl(ui+ . 1))
In order t o express
J*
f u h d x e x p l i c i t l y , it i s c o n v e n i e n t t o
i n t r o d u c e t h e basis functions o f t h e s p a c e v,,( d e f i n e d by ( 1.19 )) of t h e same t y p e as t h o s e i n Chapter 1, S e c t i o n 4 . 1 , Example 4.2, that is:
d e n o t i n g by (1.129)
ai
t h e s u p p o r t o f wi we t h e n deduce
C
lDfuhdx=
Mi € r h
ui J D , / . i h .
Using t h e n o t a t i o n o f F i g u r e o f t h e t r i a n g l e T, w e have t,
so t h a t
Fig. 1.7.
1 . 7 , d e n o t i n g by A ( T ) t h e measure
27 5
Thermal control problems
(SEC. 1)
F i n a l l y , by u s i n g r e l a t i o n s ( 4 . 1 7 ) - ( 4 . 1 9 ) o f Chapter 3, S e c t i o n
4 . 1 . 4 , w e may prove
so t h a t
R e l a t i o n s (1.1271, ( 1 . 1 2 9 ) - ( 1 . 1 3 1 ) a l l o w t h e approximate problem ( 1 . 1 2 6 ) t o b e w r i t t e n e x p l i c i t l y as a f i n i t e - d i m e n s i o n a l o p t i m i s a t i o n problem.
Formulation o f t h e approximate problem.
1.6.2
(111, q = 2 .
I n ( 1 . 2 5 ) t h e f u n c t i o n a l j , w a s e x p r e s s e d e x p l i c i t l y i n terms on (&vz;)nr , t h a t i s
of t h e v a l u e s t a k e n by
(1.132)
I($(b)
~ h ( u h ) = ~ ~ l M i M i + l
In o r d e r t o write ion 1.6.1, that is
(1.133)
{
+ $(ui+l))*
+4$(vi+l,Z)
5.
we i n t r o d u c e , as i n S e c t -
fv,dxexplicitly,
t h e b a s i s f u n c t i o n s o f t h e s p a c e V hd e f i n e d by ( l . l 9 ) , (wi)M Wi
d e n o t i n g by
IE
(rh U rA)
€v,, ai t h e
W,(Mj) =
6,
s u p p o r t of
V M j € zh
wi
Using t h e n o t a t i o n of F i g u r e we have
, we
U
z;
;
t h e n deduce t h a t
1 . 8 , w i t h A ( T ) = measure of T,
276
h i l a t e r a l problems and e l l i p t i c inequulities
which allows
1.
I"
(CHAP.
4)
I uh ('dx to be written explicitly since
c
luhIzdx=
1nluh12dx.
T€Yh
Fig. 1.8.
Moreover, it may be proved that
(1.136)
i
+
I
u
-
1'
u
2
+ 2(u12 + u 2 3 - % l ) M . l M ,
+ u3 2,
~
3
+
u
*
~
l
-
+ 2(u23 + u31 - u 1 2 ) m 2 1' + + 1 - UI m 3 + o 2 m 1 + o3=2 + 2(u31 + u12 - u 2 3 ) 31' }
which allows
1"
3
~
+ +
+
I grad uh 12dx to be written explicitly since
I grad oh 1' dx =
1
1 1 grad uh 1'
TErh
dx.
Using relations (1.1321, (1.134)-(1.136) the approximate problem (1.126) may be put in the form of a finite-dimensional optimisation problem.
Thermal contro Z prob Zems
(SEC. 1)
277
Solution of the approximate problem by relaxation
1.6.3
Suppose Nh = dim Vh ; h e r e i n a f t e r we s h a l l d e n o t e by a v e c t o r o f W N h , i . e . oh = (ul, u2, ..., uNh) I t w a s seen i n Sections 1.6.1, 1.6.2 t h a t t h e a p p r o x i m a t e problem (1.126) i n f a c t r e d u c e s t o m i n i m i s i n g i n RNh a f u n c t i o n a l d e n o t e d by Jh which s a t i s f i e s t h e following conditions:
.
(i)
Jh i s s t r i c t l y convex
-
(ii)
lim
Jh (Oh)
= -k 00
Ilm I1 + m
( iii)
$ E C1(W).
Jh E C1(WNh)s i n c e
I n view o f t h e s e p r o p e r t i e s , it f o l l o w s from Theorem 1.1 o f Chapter 2 , S e c t i o n 1.1, t h a t t h e convergence o f t h e r e l a x a t i o n a l g o r i t h m a p p l i e d t o t h e m i n i m i s a t i o n o f Jh i n Ph i s a s s u r e d . T h i s a l g o r i t h m i s d e f i n e d as f o l l o w s :
(1.137)
uf = (uy,
w i t h Ir, known, using
(1.138)
aJh
..., u!~)
g i v e n a r b i t r a r i l y i n RNh ;
4" is
d e t e r m i n e d c o o r d i n a t e by c o o r d i n a t e ,
..., 4?,',4+l, @ + l , ..., Gh)= 0 ,
-((ul+',
0Q id
Nh.
aui
Remark 1.8. If t h e v a r i a b l e u, r e l a t e s t o a v e r t e x o f (q = 1) , o r o f Z,, u Z; (q = 2)which d o e s n o t b e l o n g t o rr , e q u a t i o n ( 1 . 1 3 8 ) i s linear.
Z,,
Remark 1.9 If t h e c o n s t a n t k i n (1.9) t a k e s v e r y s m a l l v a l u e s , a l g o r i t h m ( 1 . 1 3 7 ) , ( 1 . 1 3 8 ) i s g u a r a n t e e d a p r i o r i t o perform w e l l . I n f a c t , i n t h e l i m i t when k -+ 0, we have $(r) = g 1 I so that
cI
jh(uh)
=
2g
(l * 1 4 ) jh(h)
=
5 1I
(1*139)
I(I Di I + I v i + 1 I)
Mi
Mi M i +
1
1 (1
01
9
= 1
if
I + 4 I Oi + 111 I -k I ui+1 1)
9
if q = 2 ;
we are t h u s w i t h i n t h e c o n d i t i o n s f o r a p p l y i n g Theorem 1.4 of Chapter 2 , S e c t i o n 1.5, and t h i s a s s u r e s t h e convergence of t h e r e l a x a t i o n a l g o r i t h m - p r o v i d e d , o f c o u r s e , (1.138) (which i s n o t always m e a n i n g f u l ) i s r e p l a c e d by:
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
278
We s h a l l r e t u r n t o t h e above i n S e c t i o n 2 .
(CHAP.
4)
a
Remark 1.10. I n o r d e r t o improve t h e r a t e o f convergence, it can b e u s e f u l t o i n t r o d u c e a n under- o r over-relaxation p a r a m e t e r we t h e n o b t a i n t h e f o l l o w i n g v a r i a n t o f a l g o r i t h m ( 1 . 1 3 7 ) , w; (1.138) : (1.142)
u: g i v e n
(1.143)
-((Ul+’,u;+’,
(1.144)
u;+’
aJk
av*
1.6.4
=
...,tr; Ti, tr;+l”, 4+’, ...) = 0
tr; + oJ(u;+1’* - u;). a
Solution o f the approximate problem by a conjugate gradient method.
I n view o f t h e p r o p e r t i e s o f Jk , it i s also p o s s i b l e t o u s e i n t h e s o l u t i o n of t h e a p p r o x i m a t e problem ( 1 . 1 2 6 ) t h e v a r i a n t ( l ) ( 2 . 2 6 ) - ( 2 . 2 9 ) o f Chapter 2, S e c t i o n 2 . 3 , of t h e conjugate gradient a l g o r i t h m .
1.7
Examples
I n t h e f o l l o w i n g examples t h e approximPreliminary remark. a t e problems a r e a l l s o l v e d by t h e o v e r - r e l a x a t i o n method (1~ 4 2 ) - ( 1 . 1 4 4 ) d e s c r i b e d i n S e c t i o n 1 . 6 . 3 ; a l t h o u g h n u m e r i c a l experiments performed w i t h t h e c o n j u g a t e g r a d i e n t method have g i v e n some , v e r y s a t i s f a c t o r y r e s u l t s , t h e programming o f t h i s method. i s much l e n g t h i e r and t r i c k i e r ( f o r comparable computation t i m e s ) t h a n t h a t o f t h e o v e r - r e l a x a t i o n method; t h e corresponding r e s u l t s w i l l t h e r e f o r e n o t be p r e s e n t e d h e r e . a
1.7.1
Example 1
The problem ( 1 . 4 ) c o n s i d e r e d i s d e f i n e d by:
Geometric data: 62
= ]0,1[ x ]0,1[, fd =
space V : r, = r i m p l i e s V
r.
= H’(62).
Function f:f = 1. Parameter values:
(
p = 1,
Adapted t o non-quadratic
k
=
1, 0.1, 0.01,g = 1 and 0 . 2 .
cases.
279
Thermal control problems
(SEC. 1)
I n i t i a l i s a t i o n of the over-relaxation algorithm: ut
= 0.
Ternination c r i t e r i o n f o r the o v e r - r e l a a t i o n algorithm:
c lul" - $ I c IG"I Nh
(1.145)
*=lNh
QE,
i= 1
with E, =
E,
= if q = 1 ( f i n i t e elements o f o r d e r l), if q = 2 ( f i n i t e elements of order 2 ) .
Triangulation y h : Figures 1.9 and 1.10 show t h e t r i a n g u l a t i o n s employed, r e l a t i n g r e s p e c t i v e l y t o q = 1 and q = 2.
Fig. 1.9. (4
= 1).
Fig. 1-10. (4 = 2).
For 4 = 1 ( r e s P . q = 2 ) w e have 512 ( r e s p . 1 2 8 ) t r i a n g l e s and 289 nodes, and hence Nh = 289.
Analysis of the numerical r e s u l t s . Table 1.1 g i v e s c e r t a i n information concerning t h e s o l u t i o n we have of t h e a.pproximate problems and t h e results obtained; denoted by pl(uJ and p&J t h e following two q u a n t i t i e s :
A(Ud=MaX%(M),MEzh if 4 ' 1 , P2(uJ = Max uh(M),ME
zhn
ME&Uz,' i f 4 = 2 ,
i f q = 1,
M E ( Z h u z ; ) n r if q = 2 .
Unilateral problems and elliptic inequalities
280
(CHAP. 4 )
I
q=l
q=2
q=l
q=2
q=l
q=2
q=l
q=2
---------
1 1.7 1.7 0.256 0.255 0.206 0.206 50 149 0.1 1.7 1.7 0.095 0.094 0.029 0.029 56 53 0 . 0 1 1 . 7 1 . 7 0.073 0.072 0.003 0.003 56 ---------- 30 1 1.9 1.9 0.259 0.258 0.209 0.210 99 76 g = 0.2 0.1 1.9 1.9 0.233 0.232 0.183 0.183 116 185 0.01 1.9 1.9 0.233 0.232 0.183 0.183 125 185 g =
1
Table
1.1
A l l t h e c a l c u l a t i o n s were performed on a C I I 10070 w i t h a mean computation t i m e per iteration of 0.26 s e c . w i t h q = 1, and 0.42 s e e . w i t h q = 2. I n view o f t h e s e r e s u l t s , it would appear t h a t , f o r t h i s t y p e o f problem a t l e a s t , approximation by elements o f o r d e r 1 has t h e advantage over t h a t by elements o f o r d e r 2, s i n c e f o r t h e same p r e c i s i o n and t h e same t o t a l number o f degrees o f freedom, t h e computation times a r e approximately h a l f a s l o n g . F i g u r e s 1.11 and 1 . 1 2 show t h e e q u i p o t e n t i a l s o f t h e approximate s o l u t i o n s f o r g = 0 . 2 w i t h k = 1 and 0.1.
Fig. 1.11.
Fig. 1.12.
(Example 1 : g = 0.2, k = 1).
(Example 2 : g = 0.2, k = 0.1).
Thermal control problems
( S E C . 1)
281
F i g u r e 1.13 shows t h e graphs o f t h e f u n c t i o n s uk(x1,0) for
g = 0 . 2 and w i t h k = 1, 0.1. It may b e s e e n i n F i g u r e 1.13 t h a t t h e f a c t t h a t t h e f u n c t i o n uh(xl,O) p a s s e s beyond t h e c r i t i c a l v a l u e kg does not appear t o have any a d v e r s e r e p e r c u s s i o n s on t h e r e g u l a r i t y o f t h e t r a c e o f the solution. rn
0.10
0.05
Fig.1.13.
1.7.2
1 1
(Example 1:
g = 0.2).
R e p r e s e n t a t i o n o f u,(xl,O).
Example 2 .
The problem
(1.4) i s
d e f i n e d by:
Geometric data :
n = 10, 1[ x 10, 1[
9
rd =
{ (xi, x2) I 0.25 d
XI
d 0.75,
x2
=0).
Space V : V = ( u ( u ~ H ~ ( n ) , u = O on r - r d } . Function f : We have t a k e n f equal t o 10 t i m e s t h e c h a r a c t e r i s t i c f u n c t i o n of t h e s q u a r e ]0.375,0.625[ x ]0.375,0.625[ i . e . f ( x ) = 10 i f 3c belongs t o t h i s s q u a r e , f ( x ) = 0 o t h e r w i s e . Parameter values: We have t a k e n = 0 , which i s p e r m i s s i b l e since
d e f i n e s on V a norm e q u i v a l e n t t o t h e norm induced by H ' ( n ) , and 1, 0 . 2 .
k = 1, 0.1, 0.01, g =
282
Unilateral probZems and eZZiptic inequalities
(CHAP.
4)
Initialisation and termination criterion f o r the over-relaxation algorithm: see Section 1 . 7 . 1 . Triangulation r h : See S e c t i o n 1 . 7 . 1 ; t h e number Nh o f degrees o f freedom i s i n t h i s c a s e equal t o 234. F i g u r e 1 . 1 4 shows a, rr and t h e s u p p o r t o f f .
Fig. 1.14.
(Example 2 )
Analysis of the nwnericaZ results Table 1 . 2 shows some d e t a i l s r e l a t i n g t o t h e numerical r e s u l t s obtained. The mean computation times p e r i t e r a t i o n on a C I I 10070 are e s s e n t i a l l y t h e same a s t h o s e i n Example 1; since t h e values. o b t a i n e d a r e p r a c t i c a l l y t h e same, t h i s example once a g a i n confirms t h e advantage o f working, f o r t h i s type o f problem, w i t h f i n i t e elements o f o r d e r 1.
k
g = l
g = 0.2
1 0.1 0.01 1 0.1 0.01
overrelaxation parameter o
Number of iterations
q= 1
q=2
q= 1
q=2
1.67 1.67 1.67 1.67 1.67 1.67
1.7 1.7 1.7 1.7 1.7 1.7
48
45
42
39
40 48
37 45
42 48
39
Table
1.2
44
Thermal control problems
(SEC. 1)
283
F i g u r e s 1.15, 1 . 1 6 and 1.17 show t h e e q u i p o t e n t i a l s o f t h e approximate s o l u t i o n s f o r g = 1 and k = 1, 0.1, 0.01.
Fig.1.15. (Example 2: g = 1, k = 1).
Fig. 1.16. (Example 2: g = 1, k = 0.1).
Fig. 1.17. (Example 2 : g = 1, k = 0.01).
Geometric data : Q = { (x,, x 2 ) I x: c i r c l e (see Figure 1.18). space Y : Y
=
{ u I U E H'(Q), u
+ xi =
< f },
r, = t h e
lower semi-
o }.
Function f: f i s e q u a l t o 18 t i m e s t h e c h a r a c t e r i s t i c f u n c t i o n of t h e d i s c o f r a d i u s 0.25. Parameter values:
p = 0,
k = 1, k = 0.1, 0.01, g = 1, 0.2.
284
Unilateral problems and elliptic inequalities
(CHAP. 4 )
Fig. 1.18.
Fig. 1.19.
Initialisation and termination criterion for the relaxation 6ee S e c t i o n 1 . 7 .I algorithm: Triangulation r h : t h a t o f F i g u r e 1.19, i . e . 384 t r i a n g l e s , 217 nodes with Nh = 193. Analysis of the numerical results: We have used o n l y approximation by f i n i t e elements o f o r d e r 1 (q = 1 ) ; t h e results o b t a i n e d a r e shown i n Table 1 . 3 .
(SEC. 1)
Thermal control problems
k
g=1
g = 0.2
1 0.1 0.01 1 0.1 0.01
0
28 5
Number of iterations 42
1.76
1.76
35 35 42 48 56
The mean computation time p e r i t e r a t i o n i s about 0.33 s e c . on a C I I 10070; F i g u r e s 1 . 2 0 , 1 . 2 1 , 1 . 2 2 show t h e e q u i p o t e n t i a l s of t h e approximate s o l u t i o n s f o r g = 1 and w i t h k = 1, 0.1, 0.01.
Fig.1.B. (Example 3: g = 1, k = 1).
Fig.1.22. (Example 3:
Fig. 1.21. (Example 3: g = 1, k = 0.1).
g = 1, k = 0.01).
286
Unilateral problems and e l l i p t i c inequazities
(CHAP. 4 )
F i g u r e 1 . 2 3 shows, i n polar coordinates and f o r g = 0 . 2 , k = 1, 0.1, 0.01, t h e g r a p h s o f t h e f u n c t i o n s uh(i,B) f o r O E [ - x , O ] .
-
180
-
-
150
120
- 90 - 60
0 degrees
g = 0.2).
Fig.1.23. (Example 3:
1.7.4
- 30
Example 4 .
Geometric data:
see Section 1.7.3.
Space V : V = H'(O). Function f : f i s e q u a l t o 10 t i m e s t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e d i s c o f r a d i u s 0.25. Parameter values:
= 1, k = 1, 0 . 1 , 0.01, g = 1, 0 . 2 .
I n i t i a l i s a t i o n and termination c r i t e r i o n for the over-relaxation'algorithm: see S e c t i o n 1 . 7 . 1 . Triangulation we have
g h
:
s e e S e c t i o n 1.7.3;
Nh = 217. 8
i n t b c present case
..
Analysis of the numerical r e s u l t s . t h e mean computThe r e s u l t s o b t a i n e d a r e shown i n T a b l e 1 . 4 ; a t i o n t i m e p e r i t e r a t i o n w a s 0.33 s e c on a GI1 10070.
.
F i g u r e s 1 . 2 4 , 1 . 2 5 , 1 . 2 6 show t h e e q u i p o t e n t i a l s o f t h e approxh a t e s o l u t i o n s f o r g = 1 w i t h k = 1, 0.1, 0.01.
Thermal control probZems
(SEC. 1)
k
g=l
g =
0.2
1 0.1 0.01 1 0.1 0.01
1.91
1.91 Table
Fig.1.24.
Number of iterations
u
(Example 4: g = 1, k = 1).
91 96 97 91 111 97
1.4
Fig.1.U. (Example 4: g = 1, k = 0.1).
Fig.1.M. (Example
4: g
= 1, k = 0.01).
Unilateral problems and e Z l i p t i c i n e q m l i t i e s
288
(CHAP.
4)
FRICTION PROBLEMS
2.
2.1
Formulation of t h e problems
A c e r t a i n number o f model problems ( ' ) r e l a t i n g t o t h e d i s p l a cement of a s o l i d body a, w i t h f r i c t i o n on t h e boundary I' o f a, o r o v e r a p a r t rd of t h i s boundary, l e a d t o d e t e r m i n i n g a f u n c t i o n u(x), X E Q , t h e s o l u t i o n o f t h e variational problem
I n ( 2 . 2 ) , w e t a k e f E L2(n),p 2 0, g > 0 ; as r e g a r d s s h a l l confine our a t t e n t i o n t o t h e cases
V
(2.3)
=
{ u I u E H'(P), u = 0 on
v, we
r - r, )
or
V = H'(f2).
(2.4)
Remark 2 . 1 . To h e l p s i m p l i f y t h e d e s c r i p t i o n , we s h a l l we refer t h e reader t o assume i n t h e f o l l o w i n g t h a t p > O ; Duvaut-Lions, loc. c i t y f o r t h e s t u d y o f t h e c a s e p = 0. 2.2
E x i s t e n c e and u n i q u e n e s s r e s u l t s f o r problem ( 2 . 1 ) , ( 2 . 2 ) .
A s f o r Theorem 1.1 of S e c t i o n 1 . 2 , we may prove: Theorem 2 . 1 .
boundary 2.3
Assuming R t o be bounded and with Lipschitz ( 2 . 1 ) , ( 2 . 2 ) admits one and only one solution.
r, problem
R e l a t i o n s h i p w i t h t h e problems o f S e c t i o n 1
We s h a l l show t h a t problem ( 2 . 1 1 , ( 2 . 2 ) may be c o n s i d e r e d as a l i m i t i n g c a s e of problem ( l . h ) , ( 1 . 5 ) o f S e c t i o n 1.1, when t h e t h i s w i l l r e s u l t from: p a r a m e t e r k i n (1.9) t e n d s t o z e r o ; Proposition 2.1.
(I)
Let
u
be the solution of problem (2.1), ( 2 . 2 1 ,
See,Duvaut-Lions /I/, C h a p t e r 3, f o r t h e m o d e l l i n g and s t u d y o f f r i c t i o n problems which a r e much more c o m p l i c a t e d and r e a l i s t i c than those considered i n t h i s chapter.
D i c t i o n problems
(SEC. 2 )
and uk t h a t of probZem (1.4), ( 1 . 5 ) of Section 1.1; limu, = u k-0
Proof: (2.5) (2.6) (2.7)
then
strongly i n V .
We w r i t e :
u(u, u ) =
.
28 9
In
(puu
Av)
=gjrd
ik(u)
=
10
+ grad u.grad u ) d r Idr
/rF(u) dr
where, i n ( 2 . 7 ) , +k is t h e f u n c t i o n d e f i n e d by (1.10) and which was d e n o t e d by $ i n S e c t i o n 1. The s o l u t i o n s u and uk a r e c h a r a c t e r i s e d r e s p e c t i v e l y by
(2.8)
U(U, u
- U) + j ( ~- )j ( ~ )2
so t h a t from ( 2 . 6 ) , ( 2 . 7 ) , we g e t :
From ( 2 . 1 0 )
, (2.12)
we deduce
so t h a t (2.13)
l h a(uk k-0
since u uced by
- u, uk - u) = 0 ;
defines a norm on V equivalent t o t h a t ind+ m HI@), ( 2 . 1 3 ) i m p l i e s t h e s t r o n g convergence o f uk. rn
(CHAP. 4)
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
290
Remark 2 . 2 . The above r e s u l t i s fundamental f o r what f o l l o w s since-being a l s o t r u e f o r t h e approximate problems (for f i x e d h ) - it i m p l i e s t h a t t h e a p p r o x i m a t i o n e r r o r e s t i m a t e s , independent of k , o b t a i n e d i n S e c t i o n 1, a r e also v a l i d f o r problem ( 2 . 1 ) , ( 2 . 2 ) Remark 2 . 3 . Problem (1.4), ( 1 . 5 ) may b e c o n s i d e r e d as b e i n g o b t a i n e d from problem (2.1), ( 2 . 2 ) by regularisation o f t h e non-
differentiable t e r m
2.4
j(u) = g
1,
I u I dr.
Dependence o f t h e s o l u t i o n on g
We s h a l l , f o r t h e moment, d e n o t e by ua t h e s o l u t i o n o f ( 2 . 1 ) , (2.2); r e l a t i n g t o t h e p r o p e r t i e s o f t h e mapping g + u,, , we have :
We have :
Proposition 2.2.
The mapping g
( i)
( ii)
The mapping
(iii)
The mapping
u,, i s Lipschitz from
-1, g
I U~ 1
+ u(u,,
R,
+
V,
i s Lipschitz decreasing.
u,,) i s convex decreasing.
These p r o p e r t i e s f o l l o w from t h e more g e n e r a l r e s u l t s Proof. o f Lemma 2 . 1 below ( t h i s would a l s o a p p l y t o Chapter 5 , S e c t i o n 2.2, b u t , i n t h i s c a s e , we have p r e f e r r e d t o a r g u e d i r e c t l y ) . We u s e , i n p a r t , t h e same formalism as t h a t employed i n t h e precedin s e c t i o n s . /.tx $a.
Lkqwia 2.1.
Suppose we have:
I r i
- V a r e a l Hilbert space, - 0. : V x V + W a coercive, symmetric, continuous, b i l i n e a r fom - L : V-+ R, -
& 'V
linear continuous a seminorm on V, continuous on V,
*m-.
- g a scalar 2 0 ;
we shall denote by J the functional from V - r Rdefined by
J(4=
u)
+ gj(u) - L(v) .
The problem
then admits one and only one s o l u t i o n and the mapping possesses the following properties:
g+ua
Friction problems
(SEC. 2 ) ( i ) the mapping
g + ug i s Lipschitz from
291
V, ( i i ) the mapping g +j(ug) i s Lipschitz decreasing, ( iii) the mapping g + dug,u,) i s convex decreasing. R,
+
Proof. I n view o f t h e p r o p e r t i e s o f t h e form a , we may always assume t h a t t h e H i l b e r t s t r u c t u r e o f V i s d e f i n e d by t h e i n n e r product u,u =a(u,u), s o t h a t f o r t h e a s s o c i a t e d norm, we have ; moreover t h e r e e x i s t s ( R i e s z ’ s Theorem) f c V , IIOI( = unique, such t h a t L(u) = u) V U E V, which g i v e s
&
J(u) =
+ I1
u,
u
+ gi(u) - ( A 4.
112
Finally t h e continuity of j implies (2.15)
i(u) Q l i t (I u It
VUE
v
with
VfO
.
Proof vf ( i ) L e t gl,gz E R,, u1 = us,,u2 = uI, b e t h e correspondi n g s o l u t i o n s o f ( 2 . 1 4 ) ; we have (2.16)
(u19 0
- 4)+ g1(i(u) -i(U1)) 2 ( A 0 - UI)
and hence ( i ) .
rn
Proof of ( i t ) . (2.19),
We have, s i n c e j i s a seminorm, and from ( 2 . 1 5 1 ,
UniZateral problems and e l l i p t i c inequazities
292
Proof of ( i i i ) . We s h a l l f i r s t p r o v e t h e c o n v e x i t y ;
(CHAP.
4)
ug sat-
isfies
(2.20)
(u,, u - u,) Ue€ V ,
+ gj(u) - gj(tr,) 2 (Lu - ug)
s o t h a t by s u c c e s s i v e l y p u t t i n g
(2.21)
II u# (I2
u = 0 and
Vu E V
u = 2 u g i n ( 2 . 2 0 ) we g e t
+ 5u(Ug) = (LU J .
From ( 2 . 2 1 ) it f o l l o w s t h a t
(2.22)
J(UJ = - f 11 ug 112.
Now
J(u,) = min Cf II V € V
11'
+ d o ) - (Lu)I,
so t h a t t h e f u n c t i o n g + J ( u , ) t h u s a p p e a r s as t h e lower envelope o f a f a m i l y o f a f f i n e f u n c t i o n s (which a r e t h u s concave), and i s t h e r e f o r e i t s e l f com(Zz)e; t h i s r e s u l t i n combination w i t h ( 2 . 2 2 ) means t h a t t h e f u n c t i o n g -P 11 ue 11' i s convex. I n o r d e r t o prove t h a t g-, 11 ug )I2 is d e c r e a s i n g , we s h a l l u s e a p r o c e d u r e which i n f a c t w i l l a l s o d e m o n s t r a t e a number o f o t h e r With p r o p e r t i e s of t h e s o l u t i o n u o .
V,
=
Ker(j) = { u I U
E
V,j(u) = 0 } ,
VI i s a cZosed subspace o f V ; V: , so t h a t (2.23)
we s h a l l d e n o t e by V , t h e subspace
V = V k @ Vz
and t h e c o r r e s p o n d i n g d e c o m p o s i t i o n s o f u , J u , a r e
(2.24)
u =
01
+ uZ,
f = f i
+fz
9
+ ugz.
U, = ~ 9 1
I t may b e shown t h a t
(2.25)
j(u) =j(uz)
VUE V
and t h a t j ' i s a norm on v,. From (2.23)-( 2 . 2 5 ) , we t h e n have
and from (2.261, ( 2 . 2 7 ) w e deduce t h a t u,,,
is the solution of
Friction problems
(SEC. 2 )
(2.28)
Min { f II 1) 11’ - ( f ~4,1 U€V,
293
.
so t h a t
S i m i l a r l y , u12 i s t h e s o l u t i o n o f t h e problem
(2.29)’
Min { t II 0 11’ UEV*
+ ai(4 - (fzt 4 1
and w e t h u s have ( c f . ( 2 . 2 1 ) ) :
NOW
(2.31)
(2.30) implies
u,z 1 I S I I f z II
so t h a t (2.32)
i.(u,d
Q
2
;II fz II’.
From ( 2 . 3 1 ) , ( 2 . 3 2 ) , we t h e n deduce
(2.33)
limu,, = 0
weakZy i n V
0-W
and ( 2 . 3 0 ) , (2.33) imply
(2.34)
limu12 = 0 s t r o n g t y i n V .
We have (1 uIz /I2= 11 uI 11’- 11 fi 11’ , so t h a t t h e f u n c t i o n g + 11 uI211’ is t h e r e f o r e convex; moreover
(2.35)
lim 11 uI2 11’ = 0 ;
I--
11 uIz 11’ 3 0 vg , t h e convexity of taking account of (2.35) and of t h e f u n c t i o n g + 11 uIz 11’ implies t h a t it i s decreasing and hence t h a t g + )I uI 11’ i s decreasing, s i n c e I1 u, 11’
=
II fi 11’
+ II U,’
11’.
Proposition 2.2 may be deduced immediately from Lemma 2 . 1 by noting t h a t t h e mapping v
+
I 0 Idr
UnilateraZ probZems and e Z l i p t i c i n e q u a z i t i e s
294
(CHAP.
4)
s a t i s f i e s t h e p r o p e r t i e s r e q u i r e d of j i n t h e s t a t e m e n t o f t h e lemma. H Let Vi = { u I u E V,j ( u ) = 0 ) and l e t ti b e t h e s o l u t i o n o f
with
Jo(v) = + U ( U , u)
- L(u) ;
It may be seen t h a t by proving p o i n t ( T i ? ) o f Lemma 2 . 1 , w e have a l s o proved: Lemma 2 . 2 .
Ve have:
lim ue = ti strongZy in V .
H
#++a3
From t h i s we deduce: P r o p o s i t i o n 2.3. li t h a t of
Let
ue
be t h e s o l u t i o n o f (2.1), ( 2 . 2 ) , and
Iw = { v I U E v,v lrd = 0 } ; then lim ug = ti
s t r o n g l y in V .
H
#-+m
Remark 2 . 4 . The s o l u t i o n ti of ( 2 . 3 7 ) i s a s o l u t i o n o f t h e D i r i c h l e t problem
{
-Ali+pG==f liIr=O
i f V i s d e f i n e d by ( 2 . 3 ) , and of t h e mixed problem
-Ati+pG==f li
[~ 2.5
r, = ono r - r ,
=o
on
Duality p r o p e r t i e s
We denote by A t h e c l o s e d convex s u b s e t o f
Lz(rJ d e f i n e d by
(SEC. 2) (2.38)
Friction problems.
295
A = { q l q ~ L ~ ( r d ) , I q ( X ) J d la . e . on r d } ;
we t h e n have:
The solution u of problem ( 2 . 1 ) is characterised Theorem 2.2 by the existence of P E A , unique, such that
[ In
(2*39)
(grad u.grad u+puu) dx+g
1I
pu uE=V 1 u 1 a . e . on PEA.
I, I pv d r =
fudx
Vu E V ,
r d
Theorem 2.2 may b e proved by u s i n g a v a r i a n t o f t h e Proof. proof ( b a s e d on t h e Hahn-Banach Theorem) o f Theorem 3.3 o f Chapter however, we p r e f e r t o g i v e a more c o n s t r u c t i v e 1, S e c t i o n 3.4; p r o o f , based (amongst o t h e r t h i n g s ) on t h e regularisation p r o p e r t y Using t h e same formalism a s t h a t employof P r o p o s i t i o n 2.1. (1). ed i n S e c t i o n 2.3, w e denote by fp, t h e d e r i v a t i v e o f + k , so t h a t (cf. (1.9)):
(2.40)
rp,(t) =
g
if
t/k
if
- g
if
i
takg - kg < t d kg 0 .
chosen a r b i t r a r i l y ( f o r example, zero)
V U E V,ff E V
Writing (2.56) e x p l i c i t l y , w e o b t a i n
(2.58)
1 -Aff+M=f
1
+ gpn = 0 on
inB r d
with
(2.59)
ff
=0
on
r - rd i f
V i s given by ( 2 . 3 ) ,
(2.60)
=o an
on
r-
v = H ~ ( B ;)
r d
if
moreover
(2.61)
PA(q) = sup (- 1, inf (1,q)) Vq E L2(rd)
which allows (2.57) t o be w r i t t e n e x p l i c i t l y .
Unilateral problems and e l l i p t i c inequalities
300
Since
yd
i s a l i n e a r c o n t i n u o u s mapping from u
11 Yd u lILz(r.+) < 11 ?d 11 11
(2.62)
4.1 o f
it t h e n f o l l o w s from Theorem (2.63)
, we
+ L2(fd)
4)
have
;
VuE
IIY
(CHAP.
Chapter 2, S e c t i o n 4.3,
that:
strongly i n V
u"+u
with t h e condition (l):
(2.64)
2
O
= 4 7 + -IIV1l2.
We t h u s have
so t h a t t h e convergence o f a l g o r i t h m ( 2 . 5 5 ) - ( 2 . 5 7 ) i s a s s u r e d i f
Noting t h a t 3 x 2 / ( 6 + R 2 ) N 1.9 , we may t h e n deduce, by l o o k i n g a t Table 2.2, t h a t ( 2 . 7 6 ) g i v e s a p e s s i m i s t i c e s t i m a t e o f t h e convergence i n t e r v a l f o r a l g o r i t h m ( 2 . 6 7 ) - ( 2 . 6 9 ) . 2.8.5
Various remarks
A s p o i n t e d o u t e a r l i e r i n Remark 5.10 o f Chapter 2 , S e c t i o n
5.7, t h e o v e r - r e l a x a t i o n a l g o r i t h m i s more economical t h a n t h e d u a l i t y a l g o r i t h m ( I ) , b o t h i n t e r m s o f computation t i m e and s t o r a g e r e q u i r e m e n t . However, t h e d u a l i t y a l g o r i t h m h a s t h e advantage o f a l s o g i v i n g t h e a p p r o x i m a t i o n o f t h e ' m u l t i p l i e r ' p and a l l o w s ( 2 . 1 ) , ( 2 . 2 ) t o b e s o l v e d e a s i l y , a t t h e c o s t o f t r i v i a l m o d i f i c a t i o n s i f a program f o r s o l v i n g e l l i p t i c problems of mixed t y p e i s a v a i l a b l e ; i n f a c t t h e e s s e n t i a l s t a g e i n t h e implementation o f a l g o r i t h m ( 2 . 6 7 ) - ( 2 . 6 9 ) i s c l e a r l y t h e s o l u t i o n H of problem ( 2 . 6 8 ) . A s far as t h e t y p e o f f i n i t e e l e m e n t s u s e d i s concerned, it would a p p e a r t h a t f o r a g i v e n d e g r e e o f p r e c i s i o n t h e u s e o f e l e ments o f o r d e r one ( q = 1) i s much more economical t h a n t h a t of elements o f o r d e r tuo ( q = 2 ) ; t h e a d v a n t a g e o f e l e m e n t s o f o r d e r one becomes even g r e a t e r if t h e p r e s e n c e of curved boundari e s i n t r o d u c e s t h e need f o r isoparametric f i n i t e elements o f order two. ( 2 ) . I n f a c t , i n o u r o p i n i o n , t h e f o r e g o i n g remark a p p l i e s f o r e l l i p t i c problems of order two whenever t h e s o l u t i o n s a r e o f ZOW regularity ( s a y when t h e r e g u l a r i t y i s no b e t t e r t h a n H ' ) . H (l)
(')
For s o l v i n g (2.1), ( 2 . 2 ) a t l e a s t See C i a r l e t - R a v i a r t / 2 / f o r t h e d e f i n i t i o n and t h e o r e t i c a l s t u d y o f curved f i n i t e e l e m e n t s o f i s o p a r a m e t r i c t y p e .
Unilateral problems and e l l i p t i c inequalities
308
3.
(CHAP. 4 )
A PROBLEM W I T H UNILATERAL CONSTRAINTS AT TiE BOUNDARY
3.1
Synopsis
I n t h i s s e c t i o n we s h a l l c o n s i d e r t h e n u m e r i c a l a n a l y s i s o f t h e problem
grad u.grad (u - u) dx
+p
f ( ~ u)dx V U E K
(3.1)
w i t h f E L 2 ( Q ) and
(3.2)
K = ( U ~ U E H ~ ( Q ) , U >aO. e . on
r}.
A v a r i a n t o f t h i s problem ( w i t h p = 0 ) w a s c o n s i d e r e d e a r l i e r (from a t h e o r e t i c a l v i e w p o i n t ) i n Chapter 1, S e c t i o n s 1.1 and 2.1, t o g e t h e r w i t h a n i n d i c a t i o n o f t h e c o r r e s p o n d i n g p h y s i c a l background. 3.2
E x i s t e n c e a n d u n i q u e n e s s r e s u l t s for problem ( 3 . 1 ) . ( 3 . 2 1
We s h a l l suppose i n t h e f o l l o w i n g t h a t p > 0 ; it may b e n o t e d t h a t t h e r e i s e q u i v a l e n c e between (3.1), ( 3 . 2 ) and
Min J,(u) , (3.3)
s i n c e t h e f u n c t i o n J , i s c o n t i n u o u s and s t r i c t l y convex i n H'(Q) with
lim
II lJIl-+-
J,(u) =
+ co ,
and s i n c e K i s convex a n d c l o s e d i n H'(Q), Theorem 2 . 1 o f Chapter 1, S e c t i o n 2.1, i s a p p l i c a b l e , so t h a t t h e e x i s t e n c e and uniquen e s s of a s o l u t i o n f o r ( 3 . 3 ) (and, by e q u i v a l e n c e , f o r ( 3 . 1 ) , ( 3 . 2 ) ) , t h e n follow. 3.3
Regularity r e s u l t s
If t h e boundary r o f l-2 i s s u f f i c i e n t l y r e g u l a r , it may b e shown, ( s e e Lions /l/, Chapter 2 , S e c t i o n 8 and H . B r e z i s /2/) t h a t t h e s o l u t i o n u of ( 3 . 1 ) , ( 3 . 2 ) b e l o n g s t o K n H2(G?)with
(SEC. 3 )
3.4
Problem with unilateral constraints
309
Duality results
F i r s t , w e s h a l l prove:
The s o l u t i o n u of ( 3 . 1 ) , ( 3 . 2 ) i s character-
P r o p o s i t i o n 3.1.
ised by
a
in
(-Au+pu=j
Proof. We w r i t e : a(u, v ) =
I In
(grad u.grad v
+ p v ) dx .
A s K i s a convex cone w i t h apex 0, t h e s o l u t i o n u o f (3.1), ( 3 . 2 ) i s i n f a c t c h a r a c t e r i s e d by
[ a(u, v ) 3
we have implies
9(Q)cK
f v dx Vv E K
,
so t h a t t h e f i r s t r e l a t i o n i n ( 3 . 5 ) t h e n
so t h a t (3.6)
a(u, 0) =
and ( 3 . 6 ) l e a d s t o
(3.7)
-Au+p=f
inff.
A f t e r m u l t i p l i c a t i o n by V E K , and i n t e g r a t i o n by p a r t s , we deduce from ( 3 . 5 ) , ( 3 . 7 ) t h a t
(3.8)
lrvgdT.=o(u,u)-
I,
fvdx>O
VUEK;
r e l a t i o n (3.8) then i m p l i e s au/an>O a.e. on v . = u i n ( 3 . 8 ) , we hence deduce t h a t
r,
and p u t t i n g
310
UniZateraZ probZems and eZZiptic i n e q u a z i t i e s
(CHAP. 4 )
now u 2 0, aulan 2 0 a . e . s o t h a t u(au/an) = 0 a . e . on r, which completes t h e proof o f ( 3 . 4 ) . H Conversely, s t a r t i n g from ( 3 . 4 ) , t h e c h a r a c t e r i s a t i o n ( 3 . 5 ) may be r e c o v e r e d w i t h o u t d i f f i c u l t y . H We d e n o t e by g t h e Lagrangian d e f i n e d by g ( o , q ) = J ~ ( u)
1:
qo dT
and by A t h e p o s i t i v e cone o f
L2(r), that is,
r};
A = { q l q ~ L ~ ( r ) , q > o a . eon .
n o t i n g t h a t u E H'(61) i m p l i e s aulan E H ' / ' ( r ) c L 2 ( r ) we may t h e n prove (I), s t a r t i n g from P r o p o s i t i o n 3.1: Theorem 3 . 1 . Suppose u i s the soZution o f ( 3 . 1 ) , ( 3 . 2 ) ; then the Lagrangian Y admits { u, duldn } as unique saddZe p o i n t on
H'(61) x A .
H
The above r e s u l t s w i l l be u s e d i n . S e c t i o n 3 . 7 i n p r o v i n g t h e convergence o f a d u a l i t y a l g o r i t h m which i s a v a r i a n t o f t h a t described i n S e c t i o n 2.7.3. H 3.5
Approximation by f i n i t e e l e m e n t s o f o r d e r one a n d two
FormuZation o f the approximate probZem
3.5.1
We assume, a s i n S e c t i o n 1 . 3 , . t h a t 61 i s p o l y g o n a l ; this a l l o w s r h t o b e t a k e n as i n S e c t i o n 1 . 3 . 1 and t h e a p p r o x i m a t i o n v h o f t h e s p a c e H'(61) t o b e d e f i n e d by (1.19). We s h a l l approxi m a t e K by Kh, a c l o s e d convex s u b s e t o f V h d e f i n e d by
(3.9)'
Kh=Kn Vh={uhIuhEVh,Uh(P)>O
(3.912
Kh={
uh
I u h E V h, uh(P) > 0
VPEZhnr}
VPE(Z~UZ;)~T}
if
q=l
if q = 2 .
I t i s worth n o t i n g t h a t i f q = 2, we have Kh 9 K . The problem ( 3 . 1 ) , ( 3 , 2 ) , i s t h e n approximated by: (3.10)
(l)
i
a(uh, uh
- uh) >
I,
f(Uh
- uh) dx
vuh E
Kh
Kh
The proof o f Theorem 3 . 1 i s a d i r e c t v a r i a n t o f t h a t o f Theorem 5.2 of Chapter 2, S e c t i o n 5.6.
Problem with u n i l a t e r a l constraints
(SEC. 3 )
311
which a d m i t s one and o n l y one s o l u t i o n .
Convergence of the approximate solutions
3.5.2
We s h a l l now p r o v e :
We have
Lemma 3 . 1 .
C m ( E ) n K =K .
(3.11)
Proof. (3.12)
If
U E
H ' ( B ) t h e n u = u'
K = { u I u E H ' ( B ) , u-
E
- u- w i t h u ' , u - E H ' ( B ) and
Hi(B)} .
-
I n view o f ( 3 . 1 2 ) and s i n c e g(62)= H { ( B ) , it i s s u f f i c i e n t , i n o r d e r t o prove ( 3 . 1 1 ) , t o show t h a t V U E H ' ( O ) , u 2 0 a . e . on 61, it i s p o s s i b l e t o f i n d U , E Cm(a), u, > 0 on 61 such t h a t lim u, = u i n .-+m
H1(62)(strongly). If U E H'(61), u 2 0 a . e . t h e n u admits a n e x t e n s i o n b i n t o H'(R2) such t h a t b 2 0 a . e . o n W 2 ; i n f a c t i f 5 i s an a r b i t r a r y e x t e n s i o n ( l ) o f u i n t o H ' ( R 2 ) , u > 0 a . e . on 61 i m p l i e s t h a t 5 = I GI i s a l s o a n e x t e n s i o n o f u i n t o H'(W2). L e t (p,,), be a reguzarising sequence and l e t iJm = v ' z p , ; t h e n lim 5, = n-+m
i n H1(Rz) ( s t r o n g l y ) ; moreover, 5, &(x) =
)
p,(y) $x
E
Cm(W2)and
- y ) dy 2 0
Vx E R2
R'
s i n c e p , > O and 5 3 0 a . e . on W2. Let U, b e t h e r e s t r i c t c o n o f 3, t o 62; i t f o l l o w s from t h e prope r t i e s of &, t h a t U , E Cm(61),.u, > 0 o n 62 w i t h lim u, = u i n H'(61) m-+m (strongly). I n view o f t h i s lemma we have: Theorem 3 . 2 . If, when h + 0, the angZes of T,,are bounded below by a constant eo > 0, we have
lhl(h = u h+O
with u and
uh
strongly i n
H'(61)
solutions of ( 3 . 1 ) , ( 3 . 2 ) and
(?,.lo),
respectively.
Proof of Theorem 3 . 2 f o r q = 1. We a p p l y Theorem 4 . 2 o f Chait i s t h u s p t e r 1, S e c t i o n 4 . 3 , t a k i n g a c c o u n t o f Remark 4 . 1 ; necessary t o v e r i f y t h a t :
(I)
r
Since i s L i p s c h i t z , such a n e x t e n s i o n e x i s t s ( s e e , f o r example, N6fas /l/).
312
UnilateraZ problems and e l l i p t i c i n e q u a l i t i e s
(i) Vv E 1, = K, we can f i n d i n HI@);
(3.13)
(ii) i f v h e K k ,
Oh
4)
with vh + v s t r o n g l y
VkE
weakly i n H ' ( Q ) , t h e n
+v
(CHAP.
VEK
I n view of t h e i n c l u s i o n Kh c K p o i n t ( T i ) follows T F e d i a t e l y . To prove l i ) we can t a k e , following Lemma 3.1, x = Cm(62)nK; l e t n, : CO(62)+ Vhbe t h e i n t e r p o l a t i o n o p e r a t o r defined by (1.621, S e c t i o n 1 . 4 . 3 . If v E X t h e n
under t h e c o n d i t i o n on t h e a n g l e s of F h i n t h e statement of Theore m 3.1, we have, moreover, lim nhu = v i n H'(62) ( s t r o n g l y ) so t h a t t o h+O
prove (i) it is t h u s s u f f i c i e n t t o t a k e
x
=
~ ~ (n K5 ) and
= nhv.
Once a g a i n it i s s u f f i c i e n t Proof of Theorem 3.2 f o r q = 2. t o v e r i f y ( 3 . 1 3 ) ; p o i n t ( i ) i s t r e a t e d e x a c t l y a s f o r q = 1; it t h u s remains t o be shown t h a t ( t i ) holds. L e t p E Co(r); using t h e same n o t a t i o n as i n S e c t i o n 1 . 4 . 1 , E g u r e 1.5, w e d e f i n e t h e f u n c t i o n p,,, d i s t r i b u t e d over r, by ( 3 * 14)
ph(p) =
with (3.15)
c
dMi+1/2)
xi+l12(p)
i
= c h a r a c t e r i s t i c f u n c t i o n of
limp, = p uniformZy h+O
V
~
co(r) E and
-
MiMi+, ; @,
2
o
we have
if p 2
o
Since Simpson's r u l e i s e x a c t f o r polynomials of degree w e have
Jr (Ph
dr 2 0
QVh E
Kh , v p E
0 .
W r i t i n g ( 3 . 2 1 ) e x p l i c i t l y , we o b t a i n t h e Neumann problem -Au"+pn=
(3.23) on
[ : = A n
f
i n 51
r;
moreover
(3.24)
vq E L 2 ( r )
(PAq)) = 4'
which a l l o w s ( 3 . 2 2 ) t o b e p u t i n e x p l i c i t form. S i n c e yo i s a l i n e a r c o n t i n u o u s mapping from H ' ( Q ) + L ' ( ~ ) we have
vo
E
H'(Q) ;
from t h e theorem i n Chapter 2, S e c t i o n 4 . 3 , we may t h e n deduce that
(3.26)
u"
+
u
strongZy i n H ' ( Q )
under t h e c o n d i t i o n
(3.27)
0 < a.
< pn < a l
2
i n 62 Yl
= g,
w e can, i f g1 and g, a r e s u f f i c i e n t l y r e g u l a r , u s e t h e f o l l o w i n g g e n e r a l i s a t i o n o f a l g o r i t h m (4.4)-(4 . 6 ) :
(4.16)
lo chosen a r b i t r a r i l y i n L 2 ( r )
A%" = f
(4.17)
You" = 81 Y,,A~" = -
an
(SEC. 4 )
Fourth-order variationa 1 i n e q u a l i t i e s
3 21
where t h e convergence o f t h i s a l g o r i t h m i s proved i n e x a c t l y t h e same way, and t h e c o n d i t i o n on pn i s s t i l l ( 4 . 7 ) . L e t X,EO a n d l e t 6(xo) b e t h e D i r a c measure o f Remark 4 . 3 . i f N = 2 and s > 1 we have H'(f2) c Co(f2) w i t h continuous i n j e c t i o n , so t h a t 6(xo) i s t h u s l i n e a r c o n t i n u o u s o v e r H'(Q i f s > 1. It t h e n f o l l o w s ( s e e Lions-MagPnes /l/) t h a t i f f = 6 ( x , ) i n ( 4 . 1 ) ( ' ) , we have U E Hi(f2) n H'(O), Vs < 3, s o t h a t yo Auc L'(r), which a l l o w s - s i n c e Theorem 4 . 1 s t i l l h o l d s - a l g o r i t h m ( 4 . 4 ) (4.6) t o be applied t o t h e solution of ( 4 . 1 ) . x,;
Remark 4 . 4 . From (4.5) ( a n d t h i s a l s o h o l d s f o r ( 4 . 1 6 ) - ( 4 . 1 8 ) ) t h e d e t e r m i n a t i o n o f u" from Rn f a c t o r i s e s i n t o two D i r i c h l e t problems for - A , t h a t i s :
{ - w =R"f
(4.19)
-Ad'=
i n f2
p"
i n f2
yo u" = 0 .
yop" =
T h i s remark i s fundamental s i n c e it shows t h a t t h e s o l u t i o n o f
( 4 . 1 ) may be r e d u c e d t o t h a t o f a sequence o f D i r i c h l e t problems for - A . I t i s t h u s n a t u r a l t o a p p r o x i m a t e u" and p" i n H1(f2), and t h i s l e a d s t o a p p r o x i m a t i o n s f o r P w h i c h a r e e x t e r i o r ; we s h a l l r e t urn t o t h i s p o i n t i n S e c t i o n s 4.3, 4.4 and 4 . 5 .
Interpretation of algorithm ( 4 . 4 ) - ( 4 . 6 ) .
Duality r e s u l t s .
We s h a l l r e t a i n t h e a s s u m p t i o n f € L 2 ( f 2 ) and t h e n o t a t i o n w e d e f i n e 2' : V x L '(r ) + W , t h e Lugrangian a s s o c ; i a t e d w i t h problem ( 4 . 3 ) , as f o l l o w s :
1 = - yoAu
U(U, p)
(4.20)
=
:I
-
I AU 1' dx
-
I + I, fi
dx
py1
udr.
We may t h e n prove ( s e e Glowinski, loc. c i t . Theorem 4 . 2 .
on
vx
The pair
,
Section 1 . 4 ) :
(u,l) i s the unique saddle point o f 2'
Lz(r).
I n view o f Theorem 4 . 2 , w e may a p p l y t o Y t h e a l g o r i t h m ( 4 . 1 2 ) -
(4.13) o f C h a p t e r 2, S e c t i o n 4 . 3 , w i t h V = H'(l2) n H i @ ) , M = V, L = L 2 ( r ) A = L, @ = yl; t h e a l g o r i t h m t h u s o b t a i n e d
t u r n s o u t t o b e t h e same a l g o r i t h m as ( 4 . 4 ) - ( 4 . 6 ) . A variant of algorithm ( 4 . 4 ) - ( 4 . 6 ) . If
('1
U E H'(i2)
, we
have
aulan E H 1 I 2 ( r ) , s o t h a t it i s n a t u r a l t o
This i s i m p o r t a n t i n c e r t a i n a p p l i c a t i o n s .
UniZateraZ probZerns and eZZiptic inequuZities
322
extend t o
(4.21)
(CHAP. 4 )
V x H-'12(T) t h e Lagrangian 2 ' d e f i n e d b y ( 4 . 2 0 ) so t h a t
P(o,p) =
f~dx
+ ( p,
71
u):
i n ( 4 . 2 1 ) , ( , ) d e n o t e s t h e b i l i n e a r form o f t h e d u a l i t y between and H'12(r) ( l )
.
H-'I2(T)
With t h e o p e r a t o r S : H'/'(T) + H - ' / ' ( T ) a d u a l i t y o p e r a t o r ( 2 ) , we c o n s i d e r t h e f o l l o w i n g v a r i a n t o f a l g o r i t h m ( 4 . 4 ) - ( 4 . 6 ) :
t h e n , h a v i n g c a l c u l a t e d I " , w e s u c c e s s i v e l y d e t e r m i n e U"E V and A"+' E H - ' / ' ( r ) by means o f :
A'u" = f (4.23)
youn = 0 yo A d = -
I"
Using a v a r i a n t o f t h e p r o o f o f Theorem
4 . 1 we may prove:
Theorem 4.3. The sequence (u", I"), defined by ( 4.22)-( 4.24) converges strongZy i n V x H-'12(r)to (u, - yoAu), where u i s the soZution o f ( 4 . 1 ) , under the condition:
(4.25)
0 < ro < pn < rl < 2 4
with
Remark 4 . 5 . have :
We may p r o v e t h a t under c o n d i t i o n ( 4 . 2 5 ) , w e
(SEC. 4)
Fowlth-order v a r i a t i o n a l i n e q u a l i t i e s
wit.h ( l ) 0 < K < 1 ; the convergence of (4.22)-(4.24) first order.
323
is thus of
Remark 4.6. To prove the convergence of (4.22)-(4.24), it is sufficient to suppose that y o A U EH - ’ ‘ ’ ( T ) ; we may then weaken quite considerably the condition f E L Z ( f i ) , and Remark 4.3 h o l d s a fortiori. Remark 4.7. The practical utility of (4.22)-(4.24) is restricted by the difficult numerical use of H-”’(T), H’’’(r), S (however, see Glowinski 171, /a/. 8 Generalisation t o variationaZ i n c q u a z i t i e s
4.2.2
We shall use three simple examples to demonstrate the possibility of generalising algorithm (4.4)-(4.6) to the solution of varIn the following, a is a iational inequalities of order 4. bounded open domain of W N , with regular boundary r .
Example 1.
Statement of the problem.
Let K be the closed convex subset of ined by
K = { u I u E V ,y , u > O and let f E L’(62) .
V = H’(62)
n HA@)
, def-
a.e. }
The variational inequality (of order
4)
f ( -~ u ) d x V U E K UEK admits one and only one solution, which is also the solution of the minimisation problem J~(u)Q J~(u) V U E K (4.30) U EK
{
Regularity and d u a l i t y r e s u l t s Following the ideas of Lions 111, Chapter 2, Section 8.7.2, it may be proved that under the condition f EL’@), we have (l)
K depends, amongst other things, on ro and r ,
.
LFniZateraZ probZems and e Z Z i p t i c i n e q u a l i t i e s
324
, which
U E H'(61) n H&?)
yo
A24
(CHAP.
4)
implies
E Hl'Z(l-) c LZ(61).
From this regularity result, we deduce that u is characterised by (4.31)
Azu = f in 61,
{
U E
H2(61), u
yo(Au) 2 0 , y1 u 2 0 ,
=
0
on
yo(Au) y1 u = 0 on
r r.
L e t 9 : V x Lz(r)+ R be defined by (4.32)
P) = J ~ ( v ) Jr
=YP(U,
w1v d r
and
from the characterisation (4.31), we may deduce: Theorem 4.4. Suppose t h a t u i s t h e s o l u t i o n o f (4.29), then 9 admits (u, yoAu) a s a m i q u e saddle p o i n t on V x A . a
SoZution of (4.29) by a d u a Z i t y aZgorithm. In view of Theorem 4.4, we may apply to 9 t h e duality algorithm (4.12)-(4.13) of Chapter 2, Section 4.3; we then obtain the following variant of algorithm (4.4)-(4.6): (4.33)
A0
(4.34)
i
chosen arbitrarily in A (for example, zero) A%"
=
f
you" = 0
y o A ~ P=
(4.35)
P+l
an,
= P,t(J" - pn
ff)=
(an - Pn
d)'
9
Pn > 0 .
The conditions for convergence of (4.33)-(4.35) are exactly those of (4.4)-(4.6), i.e. conditions (4.7); Remarks 4.1, 4.3 and 4.4 still hold. a ExqZe 2
Statement of t h e probZem Let V = H'(61) n Hi(61), variational inequality
g
constant > 0 and
f €LZ(61); the
(SEC.
4)
Fourth-order variational iniqua2itie.s
- U) dx
Au A(u
+ g Jr
12
ldr - g j r
I
32 5
dT 2
(4.36)
f(u-U)dx
VUEV
U E V
admits one and only one s o l u t i o n , which i s a l s o t h e s o l u t i o n of t h e minimisation problem:
with
Lhcaliby and regularity r e s u l t s .
Let A 6 e t h e closed convex subset of L z ( r ) defined by: (4.38)
A ' = { p l p c L 2 ( r ) ,I p ( x ) I d 1
r};
a.e. on
proceeding as i n Chapter 1, S e c t i o n 1 . 3 ( o r a l t e r n a t i v e l y by r e g u l a r i s a t i o n as i n Chapter 4 , S e c t i o n . 2 . 3 and Chapter 5 , Section 6.1.4) we may prove t h e e x i s t e n c e of A E A such t h a t t h e s o l u t ion u of (4.36) i s c h a r a c t e r i s e d by
(4.39)
1 t
With (4.40)
A2u = f you = 0 yo Au =
i n GI
- grt
l y l u = l y l u l a.e.
Y : V x L 2 ( r )+ W defined by
W, p) =
;In
I
l2 d.x
it follows from (4.39) t h a t
point of. P on V x A . w UE
Jb.
dx
( --yoAu : u,
I, 1
+g
J I
dr
I
i s t h e unique saddle
regularity of u i s concerned, w e may deduce from i n f a c t , by analogy with t h e r e s u l t s V n H5/2(Q) j
As far as t h e
(4.39) t h a t
-
326
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
(CHAP.
4)
of Br6zis /2/ for problems of order 2, it is reasonable to conjecture that U E V n H3(SZ). rn
S o l u t i o n o f (4.36) by a d u a l i t y algorithm. In view of the above saddle-point result, we may apply to 2 the duality algorithm (b.12)-(b.13) of Chapter 2, Section 4.3; we then obtain the following variant of algorithm ( b . b ) -( 4.6): (4.41)
L'EA
i
A%"
(4.42)
chosen arbitrarily (for example, zero) =f
yod =0 yo Au" = -
(4.43)
A"+'
=
gl"
J'"(2+ P n 971 u")
Pn
3
>0
with
-
PA@) = sup (inf (1, p),
1) V p E L Z ( r ) .
The conditions for convergence can be defined precisely by (4.44)
0 c ro
and Remarks
O on B. 8 Remark 2.3.
We s h a l l apply t h e d u a l i t y t h e o r y due t o
( ' ) This proof does not use t h e s p e c i a l s t r u c t u r e o f U(U,U) and j(u).
Numerical analysis of Bingham f l u i d f l o w
3 50
(CHAP.
R o c k a f e l l a r /1/ t o t h e p r e s e n t problem (we a l s o r e f e r t o t h e account by Ekeland-T6mam /l/) We p u t ( 1 ) :
In
I grad u I dx - ( A u) ,
(2.9)
F(u)
(2.11)
nu= gradu
=g
(2)
B E H&2)
=
V
A E Y ( V ; Y).
The problem i s e q u i v a l e n t t o
(2.12)
inf [ f l u ) U
+ G(Au)] ;
we then have
(2.13)
inf [ f l u )
”
+ G(Au)] = sup [- F*(A* q) - G * ( -
q)],
@
where F* and G* a r e t h e c o n j u g a t e f u n c t i o n s of is t h e a d j o i n t o f A. Thus
A*q =
F and
G and A*
- divq
and
(2.15)
F*(A*q) = sup v
[-
Inudivqdx - g
lgradu Idx
1
+ cfi u) .
If w i s t h e s o l u t i o n o f
(2.16)
-Aw=f,
w~,=O,
then
u,u) = (grad w,
grad u )
and t h u s , i f we p u t : p =q
,&
+ gradw,
n > 2. A should not be confused w i t h t h e convex s e t denoted l a t e r on by t h e same term.
( l ) This a l s o h o l d s f o r ( 2 ) This
5)
Properties of so Zution of ( P o )
(SEC. 2 ) we have :
J*
F * ( A * q ) = s u" p [ b p . g r a d u d x - g
351
1
Igraduldx .
From this we deduce t h a t
IJ
[ O if F*(A* q) =
I+ a,
F*(A* q) =
p . g r a d u d x 1 < g J D Igraduldx
VU,
D
otherwise
{ + if 0
00
pEgK otherwise.
I f the dimension i s 1 w e have: K = { p , IP(X)I .
We have :
r;w).
ThUS
(2.20)
grad u = grad w
2.3.
- gP, -
Some p a r t i c u l a r cases f o r which t h e s o l u t i o n i s known
A f i r s t exwnple
2.3.1
We t a k e Q = p, 1[ and f = c > 0; t h i s corresponds t o a Bingham flow between two p l a n e s u n i t d i s t a n c e a p a r t . Problem (Po) i s t h u s defined by: (2.21) u
with
min
qcn,
[; j;
u’ = dvldx.
V”dx
+
gI;
,v’ldx -
cj;“dx],
(SEC. 2 )
Properties of solution of ( P o )
353
The s o l u t i o n of ( 2 . 2 1 ) i s g i v e n by: (2.22)
ug = 0
and i f g < c / 2
i
if
by: C
u,(x) = - x ( l 2
(2.23)
g 2 go = c / 2 ,
1 2
- x ) - gx
if 0 < x < - - 8
-1 - c-g < x 2 1 x) i f f <x if
U,(X)
C = -x(l
2
- x)
- g(1 -
+
c'
g < 2-1 + -, c
0 ; t h e
i f g 3 go = cR/2, by:
1
+ R') - 2 9
if R ' < r < R ,
1
if 0
< r < R',
s o l u t i o n of
Numerical analysis of Bingham fluid flow
354
(CHAP.
5)
with
(2.26)
r =
,/=R‘ ,= 2 g / c .
Remark on the rum-uniqueness of the s o l u t i o n of the dual problem. It can e a s i l y b e shown t h a t i n t h e c a s e o f t h e example o f S e c t i o n 2 . 3 . 1 t h e d u a l problem
admits one and o n l y one s o l u t i o n , t h e r e a s o n f o r t h i s b e i n g obvious ! I n t h e c a s e o f t h e example o f S e c t i o n 2 . 3 . 2 , w e s h a l l s e e t h a t t h e d u a l problem admits an i n f i n i t e number o f s o l u t i o n s : Consider a system of pozar coordinates i n which w e u s e t h e n o t a t i o n p = { p,, } ; we t h e n h a v e :
(2.28)
a divp = -i -(rp,) r a.
+ -r1 -.ae
Using ( 2 . 2 8 ) and t h e c h a r a c t e r i s a t i o n :
[
- Au
- gdiv1= c ,
1.grad u = I grad u I , where u ( r e s p . 1) i s t h e s o l u t i o n o f it can be shown t h a t 1, d e f i n e d by:
-1,
(2.30)
1,
(2.31)
1, = - r / R ’ ,
=
1,
=
(Po) (or t h e d u a l p r o b l e m ) ,
0 i f R’ < r c R ,
1, = b(r) if 0 < r c R ’ ,
with:
(2.32)
BeLOD(O,R’),
l B ( r ) l Q , / m
a.e.
on ( O , R ’ ) ,
i s a s o l u t i o n of t h e dual problem which t h u s admits an i n f i n i t e number of s o l u t i o n s , b u t o n l y one r a d i a l l y symmetric s o l u t i o n , o b t a i n e d by t a k i n g b = 0 i n ( 2 . 3 1 ) .
Remark 2.4. Miasnikov ring.
O t h e r e x a c t s o l u t i o n s may b e found i n Mosolovp a r t i c u l a r for t h e c a s e where 61 i s a c i r c u l a r
111, i n
Approximation o f ( P O ) by f i n i t e elements
(SEC. 3 )
355
Remark 2 . 5 . We can now complete Remark 2.1; i n t h e two examples above, i n which t h e d a t a i s v e r y r e g u l a r , we have, f o r sufficiently s m a l l g, ugE
c (SZ) n~ ' ( n 0H ) ~ ( o but )
ug4
c2(ii), ug4 ~
~ ( . 0 )
INTERIOR APPROXIMATION OF ( P o ) BY A FINITE ELEMENT METHOD
3.
I n t h e following s e c t i o n s , unless s t a t e d otherwise, t h e s o l u t i o n o f (Po) w i l l simply by d e n o t e d by u .
Synopsis. I n t h i s s e c t i o n we s h a l l d e f i n e a method o f approximating (PO) using f i r s t - o r d e r f i n i t e e l e m e n t s , i . e . u s i n g c o n t i n u o u s f u n c t i o n s which a r e p i e c e w i s e l i n e a r and a f f i n e on a t r i a n g u l a t i o n o f on 6 2 ; t h e convergence of t h i s approximation w i l l be i n v e s t i g a t e d i n S e c t i o n 5 and n u m e r i c a l methods f o r s o l v i n g t h e approximate problem w i l l be g i v e n i n S e c t i o n s 6 and 7.
a
3.1.
Triangulation of 0 .
Definition of
v,.
We p r o c e e d as i n Chapter 3, S e c t i o n 4 . 1 . 1 , whose n o t a t i o n we r e t a i n ; we can t h u s c o n s t r u c t an i n t e r i o r approximation o f HA(0) - i n t h e s e n s e o f Chapter 1, S e c t i o n 4 - u s i n g a f a m i l y v h of subspaces of Hi@). 3.2.
D e f i n i t i o n of t h e approximate problem
We d e f i n e t h e approximate problem (Po,,) by: (3.1)
(POh) min Jg(Uh) . Vh E v h
3.3.
S o l v a b i l i t y of t h e approximate problem
S i n c e t h e f u n c t i o n a l J,, i s c o n t i n u o u s and s t r i c t l y convex on vh, w i t h lim Jg(uh)= co, it r e s u l t s from Chapter 1, S e c t i o n IIvh
11
-
+
+m
2 . 1 , t h a t (PO,,) admits one and o n l y one s o l u t i o n , uh s a y . approximate s o l u t i o n uh i s o b v i o u s l y c h a r a c t e r i s e d by:
3.4.
The
E x p l i c i t f o r m u l a t i o n o f t h e approximate problem
The f o r m u l a s of Chapter 3, S e c t i o n 4.1.4 a l l o w u s t o o e x p r e s s Jg(uh) as a f u n c t i o n of t h e n o d a l v a l u e s vM = uh(M),M E Z h ; t h e
(CHAP. 5 )
Numerical analysis of Bingham f l u i d flow
356
approximate problem ( 3 . 1 ) then comes down t o t h e minimisation of 0 a n o n d i f f e r e n t i a b l e f u n c t i o n a l on WN (N = dim vh = card (&))-
3.5.
On t h e use of f i n i t e elements of o r d e r g r e a t e r t h a n 1
I n t h e case of an approximation by f i r s t - o r d e r f i n i t e elements, f o r which t h e g r a d i e n t of oh i s c o n s t a n t over each t r i a n g l e of r h
,
t h e r e i s no d i f f i c u l t y i n e x p r e s s i n g j(u,,) =
I,
I grad vh I dx
as an e x p l i c i t f u n c t i o n of t h e nodal values ;,0 on t h e o t h e r hand, such a c a l c u l a t i o n i s impossible i f we use an approximation by f i n i t e elements of o r d e r k, k 2 2 ; it i s t h e n necessary t o use an approximation of t h i s i n t e g r a l ; of all t h e p o s s i b l e choices, we c i t e t h e following i n which A(T)= area of T, Gr= c e n t r e of g r a v i t y of T, i n which t h e n o t a t i o n i s as shown i n Figure 3.1: MI T
Fig. 3.1.
I n S e c t i o n 7 w e s h a l l meet an example taken from F o r t i n 111, i n which t h e s o l u t i o n o f , (Po) i s approximated by a f i n i t e element method of o r d e r 2 , t h e i n t e g r a l
b
I grad v I dx being e v a l u a t e d using
( 3 . 3 ) and ( 3 . 5 ) . I n c i d e n t a l l y , it may be noted t h a t because of t h e r e l a t i v e l y low r e g u l a r i t y of t h e s o l u t i o n , t h e advantage of u s i n g f i n i t e elements of o r d e r g r e a t e r t h a n one, f o r an equal number of degrees of freedom, is not c l e a r l y e s t a b l i s h e d .
(SEC. 4 )
4.
Exterior approximation of ( P o )
357
E X T E R I O R A P P R O X I M A T I O N S OF ( P o l
We r e t a i n t h e d e f i n i t i o n s and t h e formalism of Chapter 3 ¶ S e c t i o n s 3 . 2 . 1 and 5.
4.1.
Approximation o f Jo
We approximate
d e f i n e d by: (4.1)
JOh(vh)
=
1 2 k= 1
In
5,
I 61, q h v h I2 dx -
f q h Oh
dX
9
t h e e x p l i c i t form of ( 4 . 1 ) b e i n g g i v e n by:
[(
- M cI I E ~ ~ h2
=
JOh(uh)
+
(4.2)
~ i + l j- v i j ) 2
( O i l + Ih-
1
vij)2
fil"lj;
+
(
vpq
+
(ui-ijh-
Vij)'
+
h vij- Ih-
=
M U E nh
0
- h2
if
Mpq#ah,
with: (4.3)
L j
'I
=p
f (x) dx
V M i j
Enh
.
iJjeCM,],
The s e t s
nh,zhwere
d e f i n e d i n Chapter 3 ¶ S e c t i o n 3.2.1.
Remark 4.1. We c o u l d a l s o use t h e e x t e r i o r approximation o f J o g i v e n i n Chapter 3, S e c t i o n 5.1.2, e q u a t i o n ( 5 . 4 ) , b u t s i n c e t h i s approximation h a s n o t been u s e d i n any n u m e r i c a l c a l c u l a t i o n we r e s t r i c t our a t t e n t i o n i n t h e f o l l o w i n g t o ( 4 . 2 ) . 4.2.
j;
E x t e r i o r a p p r o x i m a t i o n s o f jl
W e s h a l l g i v e below f o u r p o s s i b l e e x t e r i o r a p p r o x i m a t i o n s f o r t h e most n a t u r a l approximation i s c l e a r l y :
(4.4)
ji(uh) =
%/I 61 q h vh 1' + I a2 q h uh l2 dx
1
which can be e x p r e s s e d i n t e r m s of t h e vlj by:
NwnericaZ a n a l y s i s of Bingham f l u i d f l o w
358
+( +( +(
(4.5)
I
Upp
(CHAP. 5 )
u i - l j - uij h ui- - uij h ui+lj
=0
- uij
h
if
Mpq#Qh.
It r e s u l t s from ( 4 . 5 ) t h a t ji(u3 i s approximately t h e sum of 4card (62,)nondifferentiable terms; t o reduce t h i s q u a n t i t y t o about card (ah), we can use t h e following approximations:
f
j:(vh)=
(4.6)
+
2
( 4 . 6 ) , ( 4 . 7 ) and ( 4 . 8 ) can r e a d i l y be expressed i n terms of t h e ui,, giving:
(4.9)
(SEC.
4)
Exterior approximation bf ( P o )
I
u,=o
Remark 4.2.
I
359
if Mpq$62h. Approximation ( 4 . 1 0 ) can be r e p l a c e d by
with: i f M i j $ Q hw i t h two n e i g h b u r s uij = 1 i f M,E Oh, uij = $12 i n ah, uij = Jz/2 i f M i j $Qh w i t h a s i n g l e neighbour i n ah
.
I n fact t h i s corresponds t o t h e s i t u a t i o n portrayed i n Chapter 3 , S e c t i o n 5.2 by F i g u r e s 5 . 3 , 5 . 4 . 4.3.
Formulation o f t h e approximate problem
4.4.
S o l v a b i l i t y o f t h e approximate problem
We can a p p l y t h e r e s u l t s o f Chapter 1, S e c t i o n 2 . 1 , by n o t i n g t h a t t h e mapping
d e f i n e s a norm on V h ; w e t h e r e f o r e h a v e - i n view o f t h e c o n t i n u i t y and c o n v e x i t y o f j,,! (I = 1, 2, 3, 4) - t h e e x i s t e n c e and uniqueiless of a s o l u t i o n o f (PO,,),, I = 1,2,3;4.
5.
CONVERGENCE OF THE INTERIOR AND EXTERIOR APPROXIMATIONS
Synopsis. I n t h i s s e c t i o n w e s h a l l p r o v e t h e convergence o f t h e approximation c o n s i d e r e d i n S e c t i o n s 3 and 4 ; we s h a l l make u s e o f t h e methods o f Chapter 1, S e c t i o n 4 f o r t h e i n t e r i o r approximations and Chapter 1, S e c t i o n 5 f o r t h e e x t e r i o r a p p r o x i m a t i o n s . The convergence p r o o f s f o r t h e above a p p r o x i m a t i o n s a r e based
Nwnerical analysis of Bingham f l u i d flow
360
e s s e n t i a l l y on t h e d e n s i t y of 9(62)i n HA(i2).
(CHAP. 5 )
W
Conveqence of t h e f i n i t e element method of S e c t i o n 3.
5.1.
We s h a l l i n v e s t i g a t e t h e convergence of t h e f i r s t - o r d e r f i n i t e element approximation of Section 3 , u s i n g t h e r e s u l t s o f Chapter 1, S e c t i o n s 4.3 and 4 . 4 ; t h e n o t a t i o n i s t h a t of S e c t i o n 3 and Chapter 3, S e c t i o n 6.2.1. We have :
Theorem 5.1. We assume t h a t i f 0 denotes an angle o f T, T c F k , there e x i s t s e0 > 0 such that 0 2 Oo V T € Y h ,Vh. We a l s o assume t h a t ?.f - f.?h + 0 ( i n the sense t h a t , f o r a l l compact s e t s E of 62 we have E c f.?h f o r h s u f f i c i e n t l y small). Let u, ( r e s p . u ) be the solution o f (Pod ( r e s p . (Po) ). Then uh + u strongly i n H&?)
Pro0f. 1) We put (5. l)
uh)
Vh
= O i n relation
+ d(%)d
( 3 . 2 ) of S e c t i o n 3.3, g i v i n g :
fuh d-X
from which w e deduce, s i n c e j(uJ 2 0
where, i n (5.2), C,,= l/&, l o b e i n g t h e smallest eigenvalue of H;(I)). 2.) It r e s u l t s from t h e proof of Theorem 6.1, Chapter 3, Section 6.2.1, and from t h e d e n s i t y of g(f.?) i n H;(f.?), t h a t V v c H i ( @ w e can c o n s t r u c t a sequence r,v such t h a t
- Ain
(5.3)
{
rhV€
vh
r,v + v
Vh, s t r o n g l y i n H;(n).
If u i s t h e s o l u t i o n of
1.1) :
( P o ) , w e t h e n have ( s e e (1.61, Section
(SEC. 5 )
Convergence of approximations
and hence (uhlh i s s t r o n g l y c o n v e r g e n t i n H&2) r,,u + u s t r o n g l y i n Hi(62) .
Remark 5.1.
361
s i n c e , from (5.31,
A f a i r l y g e n e r a l d e f i n i t i o n o f t h e convergence
of Qh t o 62 w a s g i v e n i n Chapter 3 , S e c t i o n 6.2.1; u n d e r more
r e s t r i c t i v e c o n d i t i o n s - which a r e a l m o s t always s a t i s f i e d i n p r a c t i c a l a p p l i c a t i o n s - and f o r which we r e f e r f o r example t o C i a r l e t - R a v i a r t /1/, we can c o n s t r u c t rh : H ~ ( S Z+) vh such t h a t ( )
i
Vu E HG(Q) n H2(62) w e have
11 rh 1 rh
(5.8)
-v -
llH&n)
< c,(v)h'" .
c2(u)
IIL'(f2)
,
I n p a r t i c u l a r ( 5 . 8 ) i s s a t i s f i e d i f we d e f i n e
{
(5.9)
r h u = p r o j e c t i o n of u on f o r t h e norm u
rh by:
vh
--*
f o r which c a s e it i s c l e a r t h a t
(5.10)
a@,, rh u
- u) = 0
V u E H,'(62), Vuh E Vh .
I n view o f ( 5 . 6 ) , ( 5 . 8 ) and ( 5 . 1 0 ) w e t h e n have
1I uh - u llj2,;(") d agh'"
(5.11) 5.2.
+ fib,
a and
p
independent o f g and h.
Convergence o f t h e e x t e r i o r a p p r o x i m a t i o n s
I n t h i s s e c t i o n we s h a l l i n v e s t i g a t e t h e convergence o f t h e e x t e r i o r a p p r o x i m a t i o n s d e f i n e d 'in S e c t i o n 4 , u s i n g t h e r e s u l t s of Chapter 1, S e c t i o n 5; t h e n o t a t i o n i s t h a t of S e c t i o n . 4 .
5.2.1.
Reduction o f (P,,J, t o a v a r i a t i o n a l i n e q u a l i t y
The approximate problem (Poh),,I = 1,2, 3,4 i s e q u i v a l e n t t o t h e variational inequality:
( )
We r e c a l l t h a t
h = max Area(T). T E J h
Nwnerical analysis of Bingham f l u i d flow
362
(CHAP. 5 )
with
With t h e function f we a s s o c i a t e L : H;(Q) + W defined by:
L(u)=Injudx.
(5.14) 5.2.2.
Exterior approximations f o r H;(O), a, j , L
1) Exterior approximation f o r
H,’(Q), a, L
Since t h e b i l i n e a r form ah i s i d e n t i c a l t o t h a t denoted by a: i n Chapter 3, S e c t i o n 6.3.1, w e r e f e r t o Chapter 3 , S e c t i o n 6.3.2 where it was proved t h a t vh, ah, uh
+
In
f q h uh
dx
c o n s t i t u t e e x t e r i o r approximations of H;(Q), a, L, i n t h e sense of Chapter 1, S e c t i o n 5 . 2)
respectively,
Exterior approximations o f j
We ‘have:
The functionals j i I Proposition 5.1. e x t e r i o r approximations o f j . Proof.
= 1,
2, 3,4, c o n s t i t u t e
We have t o show ( s e e Chapter 1, Section 5.2) t h a t
V e r i f i c a t i o n of (5.15), (5.16) i s immediate f o r j i ; moreover, by using t h e n o t a t i o n of Chapter 3, S e c t i o n 6.3.2 w e have:
(SEC. 5 )
Convergence of approximations
363
(5.17)
(5.18)
(5.19)
+I
7 1 ( h / Z ) 6 2 q h Oh
+ ?I(-h/2)
6 2 q h Oh
2
1’
dx,
It i s t h e n a l m o s t o b v i o u s t h a t p r o p e r t i e s (5.151, ( 5 . 1 6 ) r e s u l t from t h e t r a n s l a t i o n p r o p e r t i e s d e s c r i b e d i n Chapter 3 , Section 6.3.2 i n t h e p r o o f of P r o p o s i t i o n 6 . 1 ( s e e ( 6 . 7 2 ) , (6.73)).
A strong convergence r e s u l t
5.2.3.
I n view o f S e c t i o n s 5 . 2 . 1 and 5.2.2 we can a p p l y Theorem 5.2 of Chapter 1, S e c t i o n 5.4 t o t h e problems (POk),, I = 1,2,3,4, and more p r e c i s e l y t o t h e e q u i v a l e n t f o r m u l a t i o n s ( 5 . 1 2 ) . Hence:
Let
Theorem 5.2.
( P o ) ; when h + O
of
(5.20)
I
(Jh
JOh(&
ul, a l ( J h +
u: be the soZution of (POh),, we have:
u:. 6 2 q h
JO(u)
uL }
+
{
u,-
-
u
t h e solution
strangZy i n (L2(Q))’
9
jk4> -,i(u) Vf = 1 , 2 , 3 , 4 .
6.
IGTHODS
OF SOLUTION BY REGULARISING
j
Synopsis. I n t h i s s e c t i o n we s h a l l make u s e o f t h e r e g u h - i s a t i o n t e c h n i q u e s i n t r o d u c e d i n Chapter 1, S e c t i o n 3.3. I n S e c t i o n 6.1 w e s h a l l c o n s i d e r t h e r e g u l a r i s a t i o n of t h e c o n t i n u o u s problem (Po) and i n v e s t i g a t e t h e a p p r o x i m a t i o n e r r o r t h u s c r e a t e d as a f u n c t i o n o f t h e r e g u l a r i s a t i o n parameter E. I n S e c t i o n s 6.2 and 6.3 r e s p e c t i v e l y , we s h a l l i n v e s t i g a t e t h e r e g u l a r i s a t i o n
364
(CHAP. 5 )
Numerical analysis of Bingham f l u i d flow
of t h e approximate problems defined i n Sections 3 and 4 , and t h e numerical s o l u t i o n methods f o r t h e r e g u l a r i s e d approximate problems; t h e s e techniques w i l l be a p p l i e d t o s e v e r a l examples i n Section 6.4. Regularisation of t h e continuous problem (PO)
6.1.
6.1.1
Formulation of the regularised problem
There a r e various ways of r e g u l a r i s i n g t h e n o n d i f f e r e n t i a b l e term
I,
I grad u 1 dx, but we s h a l l r e s t r i c t our a t t e n t i o n t o t h e two
approximations defined below. We consider t h e two numerical f u n c t i o n s 2 0 , and of c l a s s C’ on R, defined by:
4’.(7)
(6.1)
=
d K z-
41Kand
4zc, convex,
E ,
With t h e s e we a s s o c i a t e (pf) , approximations which are obtained by r e g u l a r i s i n g (Po) and defined by:
6.1.2
Solvability of the regularised problems. Convergence and estimation of the regularisation error. j l s ( u ) = J ’ a Q k ( l g r a d u l ) d x , I = 1,2;
We w r i t e
we then have:
Proposition 6.1. The approximate problems (Pn,I = 1, 2, admit 4 , characterised b y : one and only one solution, (6.4)
{ U- ’A.4= +o mgjA(u’3r ,
(with j ;
(6.5)
=f m P ,
=
gradient o f j . ) and by :
44, u-u’3+gjk(U)-~h,(U’.i,) 4 E H,(P) .
2 J’,f(U-d!
dX
+
Vv E HA(Q) 9
Proof. Since t h e f u n c t i o n a l s u + Jo(u) &(u) convex, continuous and d i f f e r e n t i a b l e with lim
[J&)
11 I-+-
+ gj.(u)]
=
+ 03,
are s t r i c t l y
(SEC.6 )
36 5
S o l u t i o n by r e g u l a r i s a t i o n of j
t h i s r e s u l t s from Chapter 1, S e c t i o n s 2 and 3 . 3 , and from t h e g e n e r a l t h e o r y o f monotone o p e r a t o r s ( s e e , e.g. Lions /1/ I . The r e g u l a r i s a t i o n p r o c e s s d e f i n e d above i s j u s t i f i e d by:
6.1.
Theorem we
have, f o r
If
(I 4 - u
(6.6)
Proof. (6.7)
4 is
4)
I( 4
-u
II
{ c1
= J2
In
.fu&
+
.
dx,
- U) d
(i(4 - ilc(4>>l .
jcl,
which p r o v e s
g measureja)
C2 = J2 g(1
Remark 6.1.
6.1.3.
E
d 2 gyI measure ( 0 )E ,
with y1 = 1, y2 = (1 - 2/7t),
form u +
- silC(4) 2 fnf(u -
- j(u))
whence, by d e f i n i t i o n o f
(6.12)
t h a t of (Po),
( 1 . 6 ) and ( 6 . 5 ) ) :
- u I(ib(n)= U(U~- U, 4
d g[(i&)
(6.11)
u
4 - u) + sA4) - si(4 2 Inf(d- 4 d.x
&f. - + si&) (6.8) so t h a t , by a d d i t i o n , (6.9)
; C, independent of
6 C,
We have ( s e e
a(u,
t h e s o l u t i o n of (Pi)and
1 = 1,2 :
- 2/n)
(6.6) with
,
measure ( 0 ) .
The above p r o o f h o l d s i f t h e c o n t i n u o u s l i n e a r
i s r e p l a c e d by
LE H - ' ( P ) .
Estimates of t h e r e g u l a r i s a t i o n e r r o r i n the onedimensional case.
It i s r e a s o n a b l e t o suppose t h a t t h e e s t i m a t e ( 6 . 6 ) o f t h e I n t h i s s e c t i o n we s h a l l r e g u l a r i s a t i o n e r r o r i s not optimal. t h e r e f o r e g i v e an e s t i m a t e which i s more p r e c i s e t h a n ( 6 . 6 ) f o r t h e p a r t i c u l a r one-dimensional c a s e c o n s i d e r e d e a r l i e r i n S e c t i o n 2.3.1; we s h a l l r e s t r i c t our a t t e n t i o n t o r e g u l a r i s a t i o n by t h e f u n c t i o n 41a.
Numerical analysis of Bingham f l u i d flow
366
(CHAP.
5)
The problem (Po) i s d e f i n e d by:
(6.13)
(Po)v Emill Hb(0. I ) [ ~ f ~ i i , ' d x + g l 0' l i l d x - c f ~ " d x ] ,
and t h e r e g u l a r i s e d problem (6.14)
(P,) min v E H&O. I )
[i 1;I
u'('dx
(Pb by
+ 81:
, / m d x
-c
u&].
Problems (Po) and (Pb admit unique s o l u t i o n s , denoted i n t h e following by uo and u, r e s p e c t i v e l y . We s h a l l now prove: Theorem 6.2.
If
c>2g>O
(6.15)
II uo - u, IIp(0.l) d
(6.16)
11 uo - u,
llHA(O,])
we have:
C1E, Cl independent of
E ,
CZ independent of
d Cz E =/,E,
8 .
Proof. This i s an e x e r c i s e i n c a l c u l u s ; by v i r t u e of symmetry w e may r e s t r i c t our a t t e n t i o n t o t h e i n t e r v a l . 0 d x d f We p u t po = uh and p, = 4 ; on t h e one hand w e have ( s e e S e c t i o n 2.3.1).
and on t h e o t h e r hand
poand p, are shown g r a p h i c a l l y i n F i g u r e
1
Fig. 6.1.
6.1.
(SEC. 6 )
367
S o l u t i o n by r e g u l a r i s a t i o n of j
Proof of (6.15). Twice t h e area o f t h e h a t c h e d r e g i o n i n Figure 6 . 1 i s e q u a l t o \Iu:. Hence, i n view o f (6.171,
(6.18):
(6.19)
I1 u: - 4 IlL1(0.1)
Q
'' -
10'"
1 - J 7 ) Pd p = 3 c . E2 p
(
From ( 6.19) and from u,(x) - uo(x) =
which p r o v e s
["(ul(
+ ch2 4
d4 =
2 c(e-*c - Arctg e-*.)
89 1 +3 (eZ*c + 1)" :
and u s i n g t h e f a c t t h a t e*= 'Y *,-'I3
which g i v e s
(6.40)
I2 d zgc 2 [ 1
+ a(&)]
lha(&) = 0 . 8
with
r-0
From ( 6 . 3 5 ) and ( 6 . 4 0 ) w e immediately deduce t h a t
(6.41)
11 uz - uO
IlH&O.l) %
J2g/3c
which p r o v e s Theorem 6.2.
Remark 6 . 2 .
&
Jx& 1
8
Consider t h e problem
r e g u l a r i s e d by
If P i s a d i s c o f r a d i u s R, it can be proved by p r o c e e d i n g as in t h e p r o o f o f Theorem 6 . 2 t h a t
(CHAP. 5 )
Numerical analysis of Binghwn f l u i d flow
370
(6.42)
11 u, - uo [ILao(")
(6.43)
11 u, - uo I H&",
< CI&,C , independent < C2E d-,
of
E
,
C2independent o f
E
,
(Po) ( r e s p . ( P e ) ) .
rn
where uo ( r e s p .
u, ) i s t h e s o l u t i o n o f
Remark 6.3. with f E H - ' ( P ) :
S i n c e t h e domain 0 i s a r b i t r a r y i n @ we c o n s i d e r ,
If uo and u, are t h e r e s p e c t i v e s o l u t i o n s of ( 6 . 4 4 ) and ( 6 . 4 5 ) , we c o n j e c t u r e on t h e b a s i s o f p a r t i c u l a r e x p l i c i t c a s e s t h a t :
(6.46)
1I u,
6.1.4.
- u0 IIH;(0)
< C E " ~C, independent
of
E
. rn
Application of the regularisation process fu obtaining duality results.
We s h a l l u s e t h e above r e g u l a r i s a t i o n p r o c e s s t o g i v e an elementary d e m o n s t r a t i o n o f t h e ( " d u a l i t y " t y p e ) c h a r a c t e r i s a t i o n (3.46) - ( 3 . 5 1 ) of Chapter 1, S e c t i o n 3.4; we assume i n t h e following t h a t ftz H - ' ( 0 ) . We f o r m u l a t e ( P d i n t e r m s of t h e e q u i v a l e n t v a r i a t i o n a l inequality
and t h e c o r r e s p o n d i n g regularised problem by
(6.48)
I
+ I grad v I' dx -
,/e2
2 (f, u - u, ) Vv E H;(G?),
By p u t t i n g that
(6.49)
v = 0 and v = 2 u
i n (6.47) it can r e a d i l y be shown
u); a(u, U) + d(u) = (f,
moreover t h e s o l u t i o n u, o f (6.48) i s c h a r a c t e r i s e d by
.
grad u, grad u
(6.50)
JE'
+ I grad u,
dx= (2
( X U ) VVEH;(Q),
(SEC. 6 )
Solution by regularisation of j
L e t A = { P I P = ( P ~PJ , E L2(Q) x LZ(Q),d ( x ) w e d e f i n e A,EA by
(6.51)
1, =
+ P%X)
d 1 a.e} ;
grad u,
+ I grad u, 1'
'
There t h e n e x i s t s ~ E and A an e x t r a c t e d subsequence, a g a i n denoted by (A& , such t h a t
(6.52)
1, + 1 weakly" i n Lm(Q) x Lm(Q).
From (6.48) w e have
(6.53)
a(u,,u) +gInl,.gradudx = ( L o ) V U E H & ~ ) ,
(6.54)
due, uK) + g
In
',.grad
' K
dx
=
(A
' K
)9
and because of t h e strong convergence o f u, t o u ( s e e Theorem 6.1) w e have, i n t h e l i m i t i n ( 6 . 5 3 1 , ( 6 . 5 4 ) :
(Lu),
(6.55)
o(u,u)+gInA.gradudx=
(6.56)
o(u,u)+gInA.gradudx= ( L u ) .
From ( 6 . 4 9 ) , ( 6 . 5 6 ) , w e deduce t h a t :
(6.57)
InA.gradudx=
I,
(graduldx.
We a l s o have t h e e q u i v a l e n c e
1.grad u d x = I I grad u I dx] (6.58)
{
A.grad u= Igrad u 1 AEA
which, w i t h ( 6 - . 5 5 ) , g i v e s
- Au - g d i v 1 = f , UIr = 0 ,
(6'60)
1.grad u = I grad u I , {LEA,
i. e. t h e c h a r a c t e r i s a t i o n r e q u i r e d .
6.2. R e g u l a r i s a t i o n o f t h e approximate problems 6.2.1.
Case of the interior approximation of Section 3
The n o t a t i o n is t h a t o f S e c t i o n s 3 and 6.1.1,
6.1.2;
the
(CHAP. 5)
Numerical a n a l y s i s o f Bingham f l u i d f l o w
372
r e g u l a r i s e d problem c o r r e s p o n d i n g t o t h e approximate problem ( 3 . 1 ) o f S e c t i o n 3.2 i s d e f i n e d by
1
a(d!, vh - d h ) + ddUh) -adUL)2
(6.61) with
E vh
1= 1,2.
I,f
( h- &)
voh E
vh
9
9
I t i s simple t o prove:
Theorem 6.3. Problem (6.61) admits one and o n l y one s o l u t i o n , E + 0 we have, V l = 1 , 2
and when (6.62)
d!
+
u,,
where uh is t h e s o l u t i o n o f t h e approximate problem (Pd ( s e e (3.11, S e c t i o n 3 . 2 ) . Remark 6.4.
The f o l l o w i n g convergence r e s u l t s can be r e a d i l y
proved :
(6.63) (6.64)
{
lim UL = 4 strongly in H @ ) , e:re
u’, i s t h e s o l u t i o n o f (Pc’) ( s e e (6.3) S e c t i o n 6 . 1 . 1 ) ,
lim d! = u strongly in H&2), u b e i n g t h e s o l u t i o n o f (Po).
r.h-0
I n f a c t , under t h e c o n d i t i o n s f o r which t h e estimate ( 5 . 1 1 ) was o b t a i n e d ( s e e S e c t i o n 5 . 1 , Remark 5 . 1 ) it can be shown t h a t i f u i s t h e s o l u t i o n o f (PO),
(6.65)
{ I1 p,- G
u Ili:(n, Q ah + g(Bh”z + 1’1 4 , a, yl independent o f h and E ; a, j3 independent o f 1 .
From t h e p o i n t o f view o f t h e nwnerical soh.4tion o f (6.611, we s h a l l i n f a c t u s e t h e e q u i v a l e n t f o r m u l a t i o n , g i v e n i n v a r i a t i o n a l form by
t h e c a s e 1 = 2 b e i n g a l i t t l e more c o m p l i c a t e d t o w r i t e explicitly. There i s i n p r i n c i p l e no d i f f i c u l t y i n e x p r e s s i n g ( 6 . 6 6 ) , u s i n g t h e f o r m u l a s o f Chapter 3, S e c t i o n 4.1.4 i n t h e form o f a system o f N( l ) n o n l i n e a r e q u a t i o n s whose unknowns a r e t h e v a l u e s ( ) N = dim v h = card (&).
(SEC. 6 )
Solution by regularisation of j
of dd a t t h e nodes ME&, of t h e t r i a n g u l a t i o n
373
g h .
Case of the e x t e r i o r approximation of Section 4
6.2.2.
The n o t a t i o n i s t h a t o f S e c t i o n s 4 and 6.1 .1, 6 . 1 . 2 ; the approximations a r e d e f i n e d by r e g u l a r i s i n g t h e f u n c t i o n a l s A k = 1, 2, 3, 4, o f S e c t i o n 4 . 2 u s i n g , r e s p e c t i v e l y
(6.68)
\.
\
(6.69)
(6.70)
(6.71)
with urn = 0 i f M r n $ Q h ; i n (6.70) we t a k e ui, = 1 i f 3 i s given by (4.10), or t h e v a l u e s g i v e n i n ( 4 . 1 0 ’ ) i f j i , g i v e n by t h i s l a t t e r equation, i s r e g u l a r i s e d . The r e g u l a r i s e d approximate problems o f ( 4 . 1 2 ) c o r r e s p o n d i n g t o (6.68) ( 6 . 7 1 ) a r e g i v e n by
...
It i s s i m p l e t o p r o v e :
Numerical a n a l y s i s of Bingham f l u i d flow
374
(CHAP.
5)
Theorem 6.4.
&,
Problem ( 6 . 7 2 ) a h i t s one and o n l y one s o l u t i o n , and when E - 0 we have,Vl= 1,2,
4-4,
(6.73)
where 4 i s the s o l u t i o n of the approximate problem (Po& defined by ( 4 . 1 2 ) i n Section 4.3. The problems ( 6 . 7 2 ) a r e e q u i v a l e n t t o t h e system of N ( ' ) nonl i n e a r e q u a t i o n s i n Nunknowns g i v e n by
w i t h I& = (uI,)Y,,~~~. I t i s c l e a r t h a t t h e n u m e r i c a l s o l u t i o n of ( 6 . 7 2 ) w i l l be c a r r i e d o u t on t h e e q u i v a l e n t f o r m u l a t i o n ( 6 . 7 4 ) .
Remark 6.5. For k = 1,2,4 ( r e s p . k = 3 ) system ( 6 . 7 4 ) corresponds t o a d i s c r e t i s a t i o n of t h e r e g u l a r i s e d continuous problem ( 6 . 4 ) by a 9-point ( r e s p . 13-point) f i n i t e d i f f e r e n c e scheme; s e e Chapter 3, S e c t i o n 8 . 2 . 5 , F i g u r e 8.2 ( r e s p . S e c t i o n 8 . 2 . 4 , F i g u r e 8.1) for t h e l o c a t i o n o f t h e p o i n t s i n q u e s t i o n . 6.3.
S o l u t i o n o f t h e r e m l a r i s e d approximate problem ( I ) . Method of p o i n t o v e r - r e l a x a t i o n
Description and convergence of t h e method
6.3.1.
The v a r i o u s r e g u l a r i s e d approximate problems d e f i n e d i n S e c t i o n 6 . 2 are of t h e f o l l o w i n g t y p e ( w e u s e t h e n o t a t i o n o f Chapter 2 , S e c t i o n 1):
min J(u) ,
(6.75)
V€RN
with J :
W N + W c l a s s C', s t r o n g l y convex lim J(u) = + 00 ; '
( 2 ) and s a t i s f y i n g
llVIl'+~
t h u s f o r t h e a c t u a l s o l u t i o n of ( 6 . 7 5 ) we c a n u s e t h e r e l a x a t i o n a l g o r i t h m d e s c r i b e d i n Chapter 2, S e c t i o n 1.1, whose convergence f o l l o w s as a r e s u l t o f Theorem 1.1 o f t h e same s e c t i o n ; t h a t i s :
(6.76)
uo = { uy,
...,u$ } g i v e n ;
w i t h 't known, we c a l c u l a t e u"", c o o r d i n a t e by c o o r d i n a t e , u s i n g
(SEC. 6)
SoZuti.on by r e g u l a r i s a t i o n o f j
{
(6.77)
G'', @+'. ...) i = 1 ,..., N .
J(ul+', ..., G?:, VuiER,
37 5
< J W ' , ..., G?:, vi, G+',...I
There i s a w e l l known e q u i v a l e n c e between ( 6 . 7 7 ) and
aJ
(6.78)
..., @?:,@+',@+I,
-((u;+',
80,
...) = 0 .
We can s e e k t o improve t h e speed o f convergence o f t h e a l g o r i t h m by i n t r o d u c i n g an under- or o v e r - r e l a x a t i o n p a r a m e t e r ( f o r t h e c a s e o f l i n e a r e q u a t i o n s s e e Chapter 2 , S e c t i o n 1.4), g i v i n g t h e f o l l o w i n g two p o s s i b l e v a r i a n t s of (6.761, ( 6 . 7 7 ) :
First algorithm. uo = { uy,
(6.79)
..., u i } g i v e n
with u" known, w e c a l c u l a t e
aJ aoi
c o o r d i n a t e by c o o r d i n a t e , u s i n g
..., ul+' - +' ,-1.4 ,G+l , . . . ) = 0 ,
(6.80)
-(u:+',
(6.81)
#+' = @
with
ff",
+ fl+' - @),
i = 1, 2, ..., N.
Second aZgor it hm. With ff known, we c a l c u l a t e u " + ' , c o o r d i n a t e by c o o r d i n a t e , u s i n g
..., ,@ ;: (6.82) =
@+I,#+',
aJ (1 - w)-(u;+', avi
...) =
..., q?;,q,q+', ...),
w i t h i = , l , 2, ..., N. I n t h e examples s o l v e d n u m e r i c a l l y by r e g u l a r i s a t i o n and overr e l a x a t i o n we have u s e d t h e r e g u l a r i s a t i o n o f
jn,/&'+
(gradul'dx
.
J"
Igradoldx
by
Under t h e s e c o n d i t i o n s t h e f u n c t i o n a l J o f
t h e c o r r e s p o n d i n g problem (6.75) is C" and i t s Hessian m a t r i x a t t h e p o i n t v, J"(u) i s p o s i t i v e d e f i n i t e V U E W ~ ; from S c h e c h t e r 111, 121, 131, t h e s e two p r o p e r t i e s a r e s u f f i c i e n t t o e n s u r e t h e convergence o f ( 6 . 7 9 ) , ( 6 . 8 0 ) ~(6.81) u n d e r t h e c o n d i t i o n :
(6.83)
0 <w
0,
independent of
v1 ,
U*EW,
and under t h e c o n d i t i o n (6.99)
0 < rl d pn d r2 < 2 aa/B2,
where Q i s t h e s m a l l e s t eigenvalue of S, we have, V u o ~ R N , convergence t o t h e s o l u t i o n u of ( 6 . 7 5 ) ( l ) o f the sequence U" defined b y ( 6 . 9 3 ) , ( 6 . 9 6 ) . Proof. T h i s i s a v a r i a n t o f t h e s t a n d a r d p r o o f o f t h e convergence o f t h e g r a d i e n t method; we p u t 111 u (I(= (Sv, u)'''. S i n c e t h e s o l u t i o n u o f (6.75) s a t i s f i e s
(6.100)
u
=
u - p.
S-I
J'(u) V ~ I ,
and, by s u b t r a c t i n g t h i s f r o m (6.96) ( w r i t i n g ii" = u"
(6.101)
i?'+
=
-u),
we have
2' - pn S-'(J'(d') - J ' ( u ) ) ,
and hence, by s c a l a r s q u a r e s ,
(6.102)
((1 Z + 'I([' = (11 ii" (I(' - 2 ~ , ( J ' ( u " ) J'(u), i") +
+ pi(S-l(J'(u") - J'(u)), J'(U") - J'(u)) ,
o r , i n view o f ( 6 . 9 7 ) , (6.98) and u s i n g t h e f a c t t h a t we have
1) S-I 11
=
l/a,
( l ) T h i s "u" h a s n o t h i n g t o do ( a t l e a s t f o r t h e moment) w i t h t h e s o l u t i o n o f (Po) ( d e f i n e d by (1.5)).
380
(6- 103)
Numerical a n a l y s i s of Bingham f l u i d f l m
III p 1112 - III
5)
(CHAP.
1112 2 pn(2 a - pn B2 / g) II En 112 ,
and hence convergence of g u n d e r t h e condition (6 .99) .
rn
Remark 6.8. Using t h e conventional n o t a t i o n p(S) = 11 SII it i s easy t o show, s t i l l using ( 6 . 9 7 ) , ( 6 . 9 8 ) , ( 6 . ~ 1 3 ) ~
giving t h e geometric convergence of condition
111 u" - u'III
t o zero under t h e
which i s more r e s t r i c t i v e than (6.99) (except when S = Z ).
rn
Remark 6.9. An i n v e s t i g a t i o n of t h e convergence of algorithms of t h e t y p e ( 6 . 9 3 ) , (6.96) can be found i n Brezis-Sibony /2/ and Sibony 111, under assumptions which a r e somewhat weaker t h a n t h o s e of Theorem 6.5, with a p p l i c a t i o n t o t h e s o l u t i o n of nonrn l i n e a r boundary-value problems. Remark 6.10. The r e a d e r may check t h a t algorithm (3.20) of Chapter 1, Section 3.1, Remark 3.2 i s an infinite-dimensional v a r i a n t (given i n v a r i a t i o n a l form) of algorithm ( 6 . 9 3 ) , (6.96). The g r a d i e n t of t h e r e g u l a r i s e d approximate f u n c t i o n a l s of Section 6.2 c l e a r l y s a t i s f i e s t h e assumptions f o r t h e a p p l i c a t i o n of Theorem 6.5, t h e L i p s c h i t z constant B being o f t h e form (6.106)
8 = 2B ( 1 + W E ) ) , &
6.4.2.
limO(E)=O.
rn
C-0
An example o f a p p l i c a t i o n
We consider t h e p a r t i c u l a r
62=]0,1[ x ]0,1[,
(Po) problems defined by
p= 1,f=
l O , g = 1 and 1.6.
Type o f d i s c r e t i s a t i o n : By f i n i t e d i f f e r e n c e s ( e x t e r i o r approximation) t h e n o n d i f f e r e n t i a b l e term being approximated by t h e v a r i a n t ( 4 . 1 0 ' ) of 4 and by 1: (see equation ( 4 . 1 1 ) ) . Type of r e g u l a r i s a t i o n : Mesh s i z e :
By t h e f u n c t i o n (6.2)).
(')
(see relations
h = 1/20.
( l ) This approximation of 7 + 1 7 1, although more complicated than t h a t using 7 + ,/-, i s widely used ( a t l e a s t , i t s f i r s t and second d e r i v a t i v e s ) f o r t h e t r a n s i e n t a n a l y s i s of e l e c t r i c a l networks, s i n c e it i s known how t o produce it physically.
S o l u t i o n by r e g u l a r i s a t i o n of 3'
(SEC. 6 )
Regularisation parameter :
E
381
=4~10-~.
Implementation o f the algorithm ( 6 . 9 3 ) , ( 6 . 9 6 ) : I n view o f t h e e x t e r i o r a p p r o x i m a t i o n s used and w i t h t h e corresponding n o t a t i o n , a l g o r i t h m ( 6 . 9 3 ) , (6.96) a p p l i e d t o t h e s o l u t i o n o f t h e r e g u l a r i s e d approximate problems t a k e s t h e form (6.107)
{ " given
U;+' = U; - pn Fh(G), where Fh i s t h e o p e r a t o r from vh*vh d e f i n e d by t h e l e f t - h a n d s i d e o f ( 6 . 7 4 ) ( w i t h 1 = 2 and k = 3, 4 ) and s h t h e d i s c r e t i s e d We i n i t i a l l y t o o k form o f a s u i t a b l y chosen e l l i p t i c o p e r a t o r . S = S, = I b u t w i t h such a c h o i c e t h e convergence proved t o be far t o o slow; we t h e r e f o r e t o o k , as s u g g e s t e d by Godunov-Prokopov /1/ i n c o n n e c t i o n w i t h l i n e a r e l l i p t i c problems,
t h e boundary c o n d i t i o n s b e i n g o f D i r i c h l e t t y p e , homogeneous w i t h respect t o each f a c t o r . Another s u i t a b l e c h o i c e i s S3 = - A f o r homogeneous D i r i c h l e t c o n d i t i o n s . For a g i v e n t e r m i n a t i o n c r i t e r i o n ( s e e (6.108) below) t h e two methods a r e l a r g e l y e q u i v a l e n t i n t e r m s o f t h e number o f i t e r a t i o n s ; however, t h e p o s s i b i l i t y , w i t h Sz , o f s o l v i n g t h e a u x i l i a r y problem at each i t e r a t i o n by d i r e c t methods o f t h e Gauss o r Choleski t y p e ( i n view of t h e tridiagonal s t r u c t u r e o f t h e d i s c r e t i s e d forms o f I - a2/ax:, I - a2/ax:) l e a d s f o r t h e examples c o n s i d e r e d t o a method about t e n times as f a s t , i n t e r m s of computation t i m e , as a method u s i n g S3 = - A w i t h s o l u t i o n o f t h e a u x i l i a r y D i r i c h l e t problems by p o i n t o v e r - r e l a x a t i o n w i t h o p t i m a l p a r a m e t e r .
Numerical r e s u l t s . With t h e termination c r i t e r i o n f o r t h e i t e r a t i o n i n t h e form: (6.108)
I&" -
R" =
I
0 , 1"" = PA@" + p2 grad u"") , p2 > 0 , =
t a k i n g S d e f i n e d by (9.4)
s=
(I - ;12) (I
-
2) '
w i t h homogeneous D i r i c h l e t boundary c o n d i t i o n s on r, d i d n o t l e a d ( 2 , t o a computation t i m e l e s s t h a n t h o s e o b t a i n e d w i t h t h e f i n i t e - d i m e n s i o n a l v a r i a n t s o f a l g o r i t h m (7. ( 7 . 9 ) , (7.10)of S e c t i o n 7.1. H TO conclude t h i s c h a p t e r , we would l i k e t o mention a n o t h e r a l g o r i t h m f o r s o l v i n g (1.5); t h i s i s a v a r i a n t o f a l g o r i t h m ( 1 0 . 5 ) - ( 1 0 . 7 ) o f Chapter 3, S e c t i o n 1 0 , t h e n o t a t i o n o f which we s h a l l use here. We t h u s w r i t e :
a),
L = L2(R) x LZ(s2)
t h e r e i s t h e n e q u i v a l e n c e between (Po) and
('1
Of Arrow-Hurwicz t y p e (see C h a p t e r 2 , S e c t i o n 4 . 4 ) ( 2, For r e c t a n g u l a r domains and f i n i t e - d i f f e r e n c e approximations, which a r e t h e o n l y p r a c t i c a l c a s e s f o r which it i s p o s s i b l e (and worthwhile) t o apply algorithm ( 9 . 1 ) , ( 9 . 2 ) , (9.3) with s d e f i n e d by ( 9 . 4 ) .
404
Numerical a n a l y s i s o f Bingham f l u i d flow
(CHAP.
5)
L e t u be t h e s o l u t i o n o f ( P o ) . Then (u;Vu) i s t h e unique s o l u t i o n o f ( 9 . 5 ) ; a Lagrangian n a t u r a l l y a s s o c i a t e d w i t h ( 9 . 5 )
is U(V,
9 ; P ) = AV, d
+
J', p . W
- 4) dx.
With p and q f i x e d , s i n c e piis non-coercive i n u, w e s h a l l a g a i n p e n a l i s e Vu q = 0 , giving
-
It can be shown t h a t admits a x L x L, of t h e form (u,Vu;A) (Po) and 1 t h e s o l u t i o n o f t h e d u a l Chapter 1, S e c t i o n 3.5. We s h a l l a p p l y U z a w a ' s a l g o r i t h m
u n i q u e s a d d l e p o i n t on where u i s t h e s o l u t i o n of problem o f (Po) d e s c r i b e d i n
Hi@)
(9.6)
A0€L
t o pC,g i v i n g
given;
w i t h An known, we s u c c e s s i v e l y d e t e r m i n e u",p",An+' by
(9.7)
{ YCW, P")
(9.8)
An+' = 1 . + P.(Vu"
p" ;p") d
(u",
Ye(V,
E Hi@) x
q ;P 9
V(U,
4) 6 H i m x L
L
- p") ,
P. > 0 .
I n view o f t h e e x i s t e n c e o f a s a d d l e p o i n t o f Pe on H i @ ) x L x L, under t h e c o n d i t i o n
(9.9)
0 < ro d Pn d ri < 2 / E
w e have
lim (u",~') = (u,Vu) ( s t r o n g ) i n
x L
n-+m
It may be n o t e d t h a t t h e d e t e r m i n a t i o n o f (u",p")u s i n g ( 9 . 7 ) i s an i n f i n i t e - d i m e n s i o n a l problem o f t h e m i n i m i s a t i o n o f a
nondifferentiable functional;
however, b e c a u s e of t h e v e r y s p e c i a l s t r u c t u r e of t h i s f u n c t i o n a l , we can u s e t h e r e l a x a t i o n methods d e s c r i b e d and s t u d i e d i n C6a-Glowinski 121 t o s o l v e ( 9 . 7 ) . W e r e f e r t h e r e a d e r t o Glowinski-Marrocco /3/ f o r a d e t a i l e d s t u d y of t h e a p p l i c a t i o n o f (9.6)-(9.8) t o a n a p p r o x i m a t i o n of (Po) u s i n g f i r s t - o r d e r f i n i t e e l e m e n t s .
Chapter 6 GENERAL METHODS FOR THE APPROXIMATION AND SOLUTION
OF TIME-DEPENDENT VARIATIONAL INEQUALITIES
INTRODUCTION In this chapter we study time-dependent problems. We shall consider the general subject of variational i n e q u a l i t i e s , exarnining the appro&mation and s o h t i o n of these inequalities in some detail. Three fundamental types of inequalities will be investigated: parabolic inequalities of type I, parabolic inequalities of type I1 and time-dependent inequalities of the second order in time. Each of these types of problem will be considered separately, the emphasis being laid on concrete examples which bring out the nature of the general formulation used. This general formulation will enable us to make an in-depth study of the finite difference approximation ( l ) and then for each type of inequality we shall investigate the numerical solution of several typical problems. To help identify the nature of the problems in question, let us say that it is natural from both the physical and mathematical viewpoints to extend the general formalism of steady-state inequalities : (Au - J o
- u) + j ( o )
- j ( u ) 2 0 Vo E K,u E K ,
to the case in which the solution evolves in time. an examination of problems of the following types:
This leads to
- Parabolic i n e q u a l i t i e s o f type I: find a function r-)u(r) for r E [0, T I , with values in K c V such that
Vo E K,u(r) E K,u(0) = u,,
given.
- Parabolic i n e q u a l i t i e s o f type 11: find a function t r E [0, TI , with values in V such that
+
u(r)
for
(’)
All of what follows is valid (and is somewhat simpler) for i n t e r i o r approximations of the “finite element” type; moreover, it is an interior method which we shall use in Section 11.
Time-dependent variational inequalities
406
( CHAP.
6)
- Hyperbolic inequalities (or w e l l posed i n t h e P e t r o w s k i t E [ O , T ] , with values i n V sense): find a function t+u(t) f o r such t h a t
1.
BACKGROUND
Following on from t h e i n i t i a l c o n c e p t s o u t l i n e d i n C h a p t e r 1, S e c t i o n 1.1, we g i v e h e r e a b r i e f r e v i e w o f t h e p r i n c i p a l c o n c e p t s o f f u n c t i o n a l a n a l y s i s which w i l l b e o f u s e l a t e r . The fundamenta l a s p e c t s may b e found, f o r example, i n Duvaut-Lions /l/, Chapter 1. More advanced developments are g i v e n i n Lions-MagPnes /l/, Schwartz /1/ and Sobolev /1/.
1.1
Spaces o f v e c t o r - v a l u e d
d i s t r i b u t i o n s and f u n c t i o n s
+
Given a n i n t e r v a l [O, T ] c R ( i n g e n e r a l T < 00) a n d a Banach s p a c e X w i t h norm (I. IIx , we d e n o t e by Lp(O,T ; X ) t h e s p a c e o f ( c l a s s e s o f ) f u n c t i o n s t + f ( t ) which a r e measurable from [0, TI -+ ( f o r t h e measure dt) s u c h t h a t
The s p a c e s Lp(o,T ; X ) a r e Banach s p a c e s f o r t h e f i r s t norm i f p # 00, , and f o r t h e second norm i f p = + 00. If X i s a H i l b e r t s p a c e equipped w i t h a n i n n e r p r o d u c t (,)x t h e n t h e s p a c e L2(0, T ; X ) i s a l s o a H i l b e r t s p a c e f o r t h e i n n e r
+
product
X
Background
(SEC. 1)
407
We d e n o t e by 9 ( ] 0 , T [ ; X ) t h e s p a c e o f i n d e f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s w i t h compact s u p p o r t i n p , T [ and v a l u e s i n X, and by LY(]O, T [ ; X )t h e s p a c e o f d i s t r i b u t i o n s on 10, r[ w i t h v a l u e s i n X , d e f i n e d by
WP,T [ ; X ) = U ( 9 0 0 , Ti) ; X ) , where i n a g e n e r a l manner Y ( M ; N ) d e n o t e s t h e s p a c e o f l i n e a r c o n t i n u o u s mappings from M h N . I n p a r t i c u l a r we can a s s o c i a t e w i t h f E Lp(O,T ; X ) a d i s t r i b u t i o n 7 : 9(]0,T [ )+ X d e f i n e d by
m
=
p(r)
W) dt
4 E 9(10, Ti) .
>
0
4+f(#) i s i n d e e d l i n e a r c o n t i n u o u s from Moreover, f-f i s a n i n j e c t i o n and it is appropr i a t e t o i d e n t i f y f and f Also f + O i n Lp(O,T ; X ) -?+ 0 i n T(4) --+ 0, V 4 E 9(]0,T[). We t h u s have, a l g e b r a i c 9’(]0, T [ ;X ) , i e a l l y and t o p o l o g i c a l l y The mapping
9(]0, T [ )+ X .
..
LP(0, T ; X ) c LY(]O, T [ ; X ) .
For
Y E B’(]O,T [ ;X ) , we
d e f i n e d’ffldt = f ( k ) by
Pk)(4)= (- Okf(4(lr))v4 E q10, TI), and hence, i n p a r t i c u l a r ,
1.2
ftkJ
E
9’(]0, T [ ;X ) , Vk. rn
Functional s e t t i n g
Whereas i n steady-state problems we have used one H i l b e r t s p a c e V ( o r , i n c e r t a i n c a s e s , one Banach s p a c e ) , i n t h e c a s e o f time-dependent problems w e must make u s e o f two HiZbert spaces ( a n d i n c e r t a i n c a s e s , two Banach s p a c e s ; however, we s h a l l r e s t r i c t our a t t e n t i o n t o t h e c a s e of H i l b e r t sp aces ). Thus l e t V and H be two H i l b e r t s p a c e s w i t h : (1 ’ 1)
V c H,
t h e i n j e c t i o n o f V i n H i s c o n t i n u o u s , V dense i n H .
We d e n o t e by ((J) ( r e s p . (,)) t h e i n n e r p r o d u c t i n V ( r e s p . and by 1) )I ( r e s p . 1 I ) t h e c o r r e s p o n d i n g norm. We t h u s have (1.2)
1 u I < c I1 u II
VUE
v.
We i d e n t i f y H w i t h i t s dual. Then, i n t h e i d e n t i f i c a t i o n c o m p a t i b l e w i t h t h e above, we have: (1.3)
V c H c V’
( V = dual of V ) .
H)
Time-dependent variational inequalities
408
( CHAP.
6)
We equip V' w i t h t h e "dual" norm
II w II+ = sup (w, u ) , w i t h V€V
11 t' 11
= 1
.
We d e f i n e t h e space W ( 0 , T ) by
du
u I u E L2(0, T ; V),u' = -E L2(0,T ;V') dr With t h e i n n e r product
W(0, r ) i s a H i l b e r t space. Every V E W(0, T ) i s , a f t e r p o s s i b l e m o d i f i c a t i o n on a s e t of measure z e r o , continuous from [0, TI-, H. We can t h e n d e f i n e a c l o s e d a f f i n e subspace of W(0,T) by Wo(O, T ) = { u I u E W(0, T ) , 40) = uo,
uo given i n
H}.
INTRODUCTION TO PARABOLIC TIME-DEPENDENT INEQUALITIES OF TYPE I
2.
I n t h i s s e c t i o n we s h a l l i n t r o d u c e by means o f two examples t h e g e n e r a l formalism o f p a r a b o l i c time-dependent i n e q u a l i t i e s o f type I. 2.1
Examples o f p a r a b o l i c i n e q u a l i t i e s o f t y p e I
Example I
2.1.1
I n t h i s f i r s t example we a g a i n c o n s i d e r t h e d i f f u s i o n problem given as a f i r s t example i n Chapter 1, S e c t i o n 1.1. Here, however, we s h a l l c o n s i d e r t h e g e n e r a l c a s e i n which w e seek t o c a l c u l a t e t h e evolution of t h e f l u i d p r e s s u r e when a p r e s s u r e d i f f e r e n c e i s a p p l i e d a t t h e o u t e r s u r f a c e o f t h e t h i n semi-permeable w a l l . Thus, i n s t e a d o f a t t e m p t i n g t o f i n d what t h e p r e s s u r e w i l l b e when e q u i l i b r i u m i s reached, w e seek t o determine t h e e v o l u t i o n o f t h e p r e s s u r e a t each p o i n t o f t h e e n c l o s u r e 9, s t a r t i n g from the i n i t i a l instmrt a t which a p r e s s u r e d i f f e r e n c e i s e s t a b l i s h e d a t t h e u n t i l a given i n s t a n t T ( i f T i s s u f f i c i e n t l y boundary r o f l a r g e , t h e p r e s s u r e must evolve towards t h e s t e a d y - s t a t e s o l u t i o n ) . This problem l e a d s t o t h e f o l l o w i n g formalism: f i n d a f u n c t i o n 4 x . r ) f o r r E [ O , TI, X E B such t h a t
n,
(2.1)
au
ar - Au
=f,
f o r (x, r ) E Q = p, T [ x 9 ,
w i t h the boundary conditions ( h ( x ) b e i n g t h e e x t e r n a l p r e s s u r e applied a t x c r ) :
(SEC. 2 )
(2.2)
Parabolic inequalities of type 1
824 u(x,t) > h(x)=s-(x,r) an
= 0,
409
x E r , t fixed
( w i t h t h e i n t e r n a l pressure higher than t h e ex t e r n al pressure, t h e semi-permeable w a l l p r e v e n t s any exchange o f f l u i d ) ,
(2.3)
U(X,t ) G h(x) =.
au (x, I ) 2 0 ,
xE
r,
(with t h e e x t e r n a l pressure higher than t h e i n t e r n a l pressure, t h e semi-permeable w a l l a l l o w s a p o s i t i v e f l o w from t h e o u t s i d e towards t h e i n s i d e o f Q; t h u s by v i r t u e o f c o n t i n u i t y , t h e o n l y p o s s i b l e c a s e i s u(x, t ) = h(x) ) The initial condition f o r t h e problem i s :
.
(2.4)
u(x, 0 ) = uo(x) = g i v e n ,
X E
l2 (uo 2 h ) ,
( i t is c l e a r t h a t t h e e v o l u t i o n o f t h e p r e s s u r e i s dependent on the pressure a t the i n i t i a l i n s t a n t ) . We s h a l l pose t h i s problem i n t h e form o f a time-dependent variational inequality. To t h i s end we i n t r o d u c e
,
(2.5)
V
(2.6)
u(u, u ) =
(2.7)
(hd =
(2.8)
K = {uluEV,u2honr},
= H'(R)
H
= Lz(12),
I, 1,
grad u grad u dx ,
fgdx,
and d e n o t e by u(f)the f u n c t i o n problem :
x+u(x,t);
we t h e n c o n s i d e r t h e
f i n d u ( t ) ~ K ,t t r a v e r s i n g t h e i n t e r v a l [0, TI, such t h a t
Applying G r e e n ' s formula i n ( 2 . 9 ) we o b t a i n
AU - J
D
- U)
+Ir
$(u
- U) dr 2 0 , Vu E K , u E K .
Using t h i s i n e q u a l i t y it may r e a d i l y b e shown t h a t t h e problems (2.1) - ( 2 . 4 ) a n d ( 2 . 8 ) , ( 2 . 9 ) a r e e q u i v a l e n t . An e q u i v a l e n t formalism i s as f o l l o w s :
Time-dependent variational inequalities
410
(CHAP. 6 )
We p u t :
and c o n s i d e r t h e problem d e r i v e d from ( 2 . 9 ) by i n t e g r a t i o n :
I (2*12)
I 1;
find u e X 0
( d ,u
- u) dr
such t h a t
+
I t can be shown without d i f f i c u l t y t h a t ( 2 . 1 2 ) i m p l i e s ( 2 . 9 ) . 2.1.2
Example I1
I n t h e p r e v i o u s example we i n t r o d u c e d a time-dependent inequali t y r e l a t i n g t o a convex s e t . I n t h e example which f o l l o w s we s h a l l i n t r o d u c e a time-dependent i n e q u a l i t y which i n v o l v e s a nondifferentiable term. This i n e q u a l i t y w i l l b e o b t a i n e d from t h e time-dependent f o r m u l a t i o n o f a thermal control problem ( c f . Duvaut-Lions 111, Chapter 1, 2 . 3 . 1 ) . . The n o t a t i o n w i l l be a s f o l l o w s : Q = domain i n s i d e t h e enclosure, f =boundary o f Q, u(x, 1) = temperature a t x E Q a t t h e i n s t a n t 1. A i r - c o n d i t i o n i n g u n i t s a r e p r e s e n t whose f u n c t i o n i s t o i n j e c t h e a t f l u x e s a c r o s s f when t h e temperature u(x,t), x e f , l i e s o u t s i d e an i n t e r v a l [hl, h,]. When u(x, t ) 9 [hl, h,l,
x Ef ,
w e assume t h a t we can produce a h e a t f l u x whose v a l u e i s p r o p o r t i o n a l t o t h e d i f f e r e n c e between u(x,t), x e f and t h e n e a r e r o f t h e two numbers hl, h, The problem is formulated as f o l l o w s : f i n d u(x,t) f o r r e [0, TI, x e Q such t h a t :
.
(2.13)
au - Au = f at
for ( x , t ) e Q = Q x ] O , T [ ,
w i t h t h e boundary conditions
where &u) i s a f u n c t i o n , which i s n o t d i f f e r e n t i a b l e everywhere, d e f i n e d by:
Parabolic i n e q u a l i t i e s of t y p e 1
(SEC. 2 )
gl(A - h , ) (2.15)
&A)
=
gz(l
- h2)
if
10.
lIoT I j(u)dt
(4.6)
- ( f " + ' , U - u " + l ) +i""(u) -j"+yu"+l)a 0, VUEV, ~"+'EVu , o = u o given.
It is easy to see that this inequality is equivalent to the problem
1 u 1' dx (4.7)
- k(f"",
u)
- (u, u")
+
+ kj"+'(u),
or, because of the special form of j(u) ent to solving
(see Chapter
4),
equival-
We then discretise i n space, so that (dividing by h , noting that the measure o f the last interval is h / 2 , and putting G+'=O):
(4.9)
We then write the conditions for optimality: and p such that:
u;"-u; (4.10)
k
1
- -(u;+': - 2 G + I + u ? + : ) - f l + l hz
i = 1, ..., M (4.11)
&+I
- ffM
k
+ h22
-(&+I
find
= 0,
- 1, -
U&+_l])
- f;"
2 + -pg"+I h
=0,
4",...,u"L'
(SEC.
4)
Numerical solution of i n e q u a l i t i e s o f type I
443
It is then possible to refer to a semi-implicit dual method to define the following algorithm which, for U;: known (recall that is given) , allows us to determine U;:" as follows (we deuf = 0 note by g"(" the ith iterate in the search for u ; " ) : - take any arbitrary
PO,
suitable interval p > O
for example p o ; then
- solve (4.10), (4.11) for - generate p' = P [ p o on [O, 11 Y
+ pg""
ffy+'(oq
- solve (4.10), (4.11), for p - generate
p = po
= pl
=
sign (ffy) x 1 , and a
, which
defines
fly+'('),
where P is the projection (which defines
p z = ~ [ p +' pg"+' fly+'(')], etc
G+'()',)
.
A multiplicity of variants of these schemes may be defined: for example, in (4.9) we can replace
this quantity corresponding to the square of the "quadratic" approximation of the derivative at x = 1 . 4.2.2
Obtaining the f u l l y e x p l i c i t scheme
Although this has not been justified theoretically, we can replace (4.10), (4.11)y(4.12) by
G+'- G -j-++11
(4.13)
-2g+G-')-fl+I
k
i = 1,
(4.15)
..., M - 1 ,
p" = sign (ffM) x 1
4.2.3
=o,
.
Numerical r e s u l t s
With the aim of assessing the merits of these schemes and of the approximation, we define the following quantities:
- the discrete normal derivative of the exact solution at x = l
and t = n k ,
Time-dependen t varia tiona 2 inequalities
444
V,G
I F 1
=
~(1,nk)- 41
h
(CHAP.
6)
- h,nk) ,
- the discrete normal derivative of the approximate solution at x = 1 and at t = n k , - then V= I=
U
=
max ....N = Ilk
1.
max n= 1.
....N
IVN-Vkunklx=l,
1 u(ih, nk) - u,(ih, nk) 1 ,
I = 1.....M
- and finally the difference between the exact solution and the calculated solution at t = nk ,
The times The two schemes were investigated on a BULL-GE-265. are quoted in sixths of a second, including compilation time. Table 4.5 gives the results of the explicit scheme and Table 4.6 those of the implicit scheme. For this scheme equations (4.10), (4.11) were solved by the Gauss-Seidel method with the values oop(.cxp, of the over-relaxation parameter obtained experimentally (see Table 4.7) and the parameter p fixed at 0.1. Table 4.8 shows Uand A in terms of h and k for the explicit and implicit schemes.
4.2.4
Conclusions of 4.2
With regard to the model problem of time-dependent friction considered here, we have:
1.
Fully e x p l i c i t scheme
-
danger of instability
2.
Implicit. scheme
stability appears to correspond to k/h2 Q 112.
- high accuracy, practically independent of h -
and k (but the exact solution is "quadratic"), f o r the same precision, much slower than the explicit scheme.
ECARTl
h
k
r=k
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
The
~
0.004 18 106 lo*] 1016 1021 IO~J 1030 1035 ... ... 1/500 0.OOO 1 0.001 5 0.001 2 O.OOO5 0.001 4 0.0033 0.0056 0.0083 0.011 1 0.0142 0.0174 111 OOO O.OOO045 O.OOO78 O.OOO64 O.OOO25 O.OOO70 0.001 67 0.00284 0.004 15 0.00558 0.007 10 0.00869 1/2 OOO O.OOOO11 O.OOO39 O.OOO32 0.OOO 12 O.OOO35 O.OOO83 0.001 42 0.00207 0.00276 0.00353 0.00434
145 247 463
0.006 5 lo* 1020 1032 ... 1013 1030 ... 1/500 0.0026 U20 1/1 OOO O.OOOO66 O.OOO77 O.OOO64 O.OOO24 O.OOO68 0.001 64 0.00280 0.004 10 0.00552 0.00702 0.00859 1/2 OOO O.OOOO16 O.OOO38 O.OOO32 O.OOO12 O.OOO34 O.OOO82 0.001 40 0.00205 0.00276 0.00351 0.00425
427 802
1/100
Ui0
1/100
]/I00 1/500
...
...
... ...
1/1 OOO 0.000081
1/2 OOO O.OOOO20 O.OOO38 O.OOO32 0,OOO 11 O.OOO33 O.OOO82 0.001 39 0.00204 0.002 74 0.00349 0.00427
Table
4.5.
Time-dependent
friction:
e x p l i c i t scheme
1548
ECARTZ h
k
t = k o . l
0.3
0.2
0.5
0.4
0.6
0.8
0.75 (')
0.9
1.0
Time ~
1/100 ~ O % I O - ~ 81MO-* 1 4 ~ 1 0 - ~19M0-3 26x10-' 2 8 ~ l O - ~31x10-' 3 2 ~ 1 0 - ~3 3 ~ 1 0 - ~3 4 ~ 1 0 - ~ 210 1/10 1/500 36X10-6 16~10-* 29x10-' 39xlO-* 47x10-* 53x10-* 58x10-* 63x10-* 64X1O-* 67x10-* 68x10-* 506 1/1 OOO 91XlO-' 81X10-5 l&lO-* 19xlO-* 23XlO-* 26x10-* 29XlO-* 31x10-* 32x10-* 3k10-* 35x10-* 840 1/100 ,88x10-5 79x10-* 14X10-3 19x10-3 2 2 ~ 1 02~5 ~ 1 0 - 2~ 8 ~ 1 0 - 3ot10-3 3 1 ~ 1 0 - ~3at10-3 3 3 ~ 1 0 - ~ 586 ~ 1/20 1/500 36X10-6 l6xlO-* 2@lO-* 38x10-* *lo-* 52X10-* 5&10-* 61X10-* 63X10-* 6%10-* 67%10-* 1349 ~ l&lO-* 19xlO-* 2?xlO-* 26x10-* 28xlO-* 3(k10-* 31x10-* 3&10-* 33x10-* 2 029 1/1 OOO ~ O X I O - 80X10-5 1/100 82x10-' 78x10-* 14x10-3 18xlO-' 22X10-3 2 5 ~ 1 0 - ~28x10-' 3(k10-' 3 1 ~ 1 0 - ~3 2 ~ 1 0 - ~3 3 ~ 1 0 - ~1183 1/30 1/500 3 6 ~ l O - ~15~10-* 28x10-* 38xlO-* 45x10-* 51x10-* 56x10-* 61x10-* 62xlO-* 6&lO-* 66XlO-* 2786 1/1 OOO S O X ~ O - ~80x10-* 19x10-* 19xlO-* 2klO-* 25x10-* 28xlO-* 3ot10-* 31x10-* 3&10-* 33x10-* 4 333
Table (l)
4.6.
Time-dependent
friction:
i m p l i c i t scheme
For t h i s v a l u e of t t h e s o l u t i o n i s zero and j ( v ) i s " n o n d i f f e r e n t i a b l e " .
h
k "0pl.crp
iterations
I
1/10
1/20
~
1
1 1.4
/
5 1.1 6
0
1/30
~
~~
1/1 0 OOO
1/100
1/500
1/1 OOO
1/100
1/500
1/1 OOO
1.1
1.45 20
1.15 9
1.1 7
1.5 28
1.25 13
1.12 10
5
Table
4.7.
h
a
v
(SEC. 4 )
NwnericaZ sozution o f i n e q u a l i t i e s of type I
1 h
E x p l i c i t scheme
U
k
Divergence and overflow
1/100 1/500 1/1 000 1/2 000
1/1°
2x10-3 1.5A0-3
1/1 000 1/2 000
4.3
3x10-’ 2x10-3
1/100 1/500 1/1 OOO 1/2 OOO
Table 4.8.
10-2 6~10-~ 5~10-~
Divergence and overflow
1/100 1/50
1/30
A
5x10-’ 2x10-3
447
I m p l i c i t scheme
U
A
8~10-~ 2~lO-~ 2 ~ 1 0 - ~ 4x10-5 10-3 2x10-5
not considered 8x10-3 2x104
10-4 2x10-5
10-3
10-5
not considered
Divergence and overflow
8 ~ 1 0 - ~ 5x10-’ 2x104 10- 5 10-3 5x10-6
10-3
n o t considered
2x10-3
R e l a t i v e accuracy of t h e schemes.
A model problem of t h e deformation o f a membrane
We consider a membrane f i x e d over a h o r i z o n t a l r e c t a n g u l a r frame under a constant t e n s i o n F and s u b j e c t t o a ( p o s i t i v e or negative) load q(x,t) The deformations of t h i s membrane a r e r e s t r i c t e d by a f i x e d h o r i z o n t a l plane s e t a t a d i s t a n c e u from t h e frame. We a r e r e q u i r e d t o determine t h e d e f l e c t i o n u(x, 1) o f t h e membrane. Let Q be t h e domain bounded by t h e frame and Q,the region i n which t h e membrane i s i n c o n t a c t with t h e h o r i z o n t a l plane. The problem can be expressed i n t h e form: f o r ~E[O,r] with u(x, 0) = uo(x) given, f i n d u(x, t ) such t h a t
.
I
u(x, t )
=u
on 0,.
448
Time-dependent variational i n e q w d i t i e s
(CHAP. 6)
With f = q/F, a = 0 ( l ) , we arrive at the following problem for t traversing the interval [O,T] , solve the inequality:
For this application we shall take R = ]0,1[ x )0,1[ We put a(r) = f + t sin (2 nt) ,
(2 ) :
and T = 1.
and for the exact solution we take (putting x1 = x, xz = y ) :
with the initial condition:
Corresponding to this solution we have
f
(4.17)
Figure
Y = f,
%
XE
={
11
- AU, x
Q 0
S a(t),
arbitrary ,x > a(t)
4.4 shows a cross-section of the exact solution for In this figure t = 0,0.1,0.2, ..., 1
[O, 11 for the instants
.
a double arrow CI represents the zones of R where u = 0 (i.e. the zones where the constraints are active, for which we say the convex set K is "saturated"). The spatial discretisation is the standard discretisation of (see Chapter 3, Section 3 ) H;(CI) h = h, = h, = 1/20 or 1/41,
(thus with Nh = l / h
- 1,
we have Ni variables) ,
(')
Here we have arbitrarily assumed that the membrane rests on the horizontal plane.
(2)
A time-dependent variant of the model problem considered in Chapter 2, Section 5.
(SEC.
4)
Numerical solution of i n e q u a l i t i e s o f type I
u ~ =, 1 ~ u,(M)@',
449
i s t h e c h a r a c t e r i s t i c function
where
M En k
o f t h e panel v h
G:(M),
= s p a c e o f s t e p f u n c t i o n s on
oh,
z e r o a t t h e edge,
a n d c o n s t a n t on t h e p a n e l s G:(M), M E oh, Kh
=
1 uh E
{
vh, vh
2 0 on
oh } .
For t h e t i m e d i s c r e t i s a t i o n we t a k e
k
= S X ~ O - ~ ,lo-',
~xIO-',
The approximation o f f i s , f o r
f&
=
c
5xlO-' r E
[&,(PI + 1) k[,
f(M.(n + 1) k) 0,M .
MEDh
Moreover, s i n c e f o r x > a(t) t h e c h o i c e o f f < 0 i s a r b i t r a r y , we c a r r i e d o u t t e s t s w i t h f = 0, f = - %x - a(t)), f = 4(x - Or(t)), f = - lOO(x - a(r)) f o r x > a ( t ) . For a l l t h e schemes used, t h e numerical r e s u l t s are p r a c t i c a l l y i d e n t i c a l ( s e e Tr6moli8res /4/). The numerical r e s u l t s g i v e n below r e l a t e t o t h e c a s e f=-~x-a(t)). The approximation o f A i s g i v e n by
-
Ah
=
c
uh(x
+ hzi) - 2 uh(x) + uh(x - hzi)
i = 1.2
4.3.1
h2
Approximation schemes
The schemes i n v e s t i g a t e d a r e as f o l l o w s :
Time-dependent variationa l i n e q u a l i t i e s
(CHAP.
Parabolic inequality Cross-sections of t h e exact solution, f o r y = f , a t t h e instants t = o , o . 1 , 0 . 2 )...) 1 . (u'
- Al4 - J v - u) 2 0, VVEK,UE X
I
t = 0.1, = 0.4
I +saturatim
t = 0.7 t = 0.8
t
4
4
zone of K at
X
t = o
*
0.2 t = 0.3
I =
t =
4
t = 0.6 t
*
= 0.7
t = 0.8
4
Fig. 4.4.
c ~
0.4
t = 0.5
4
-
*
t = 0.1
4
+
t+
rn
+
+ c
6)
4)
(SEC.
Numerical solution of i n e q u a l i t i e s o f type I
451
5 3 - Crank-Nicholson scheme ( l )
Sb
- Fully implicit scheme
4.3.2
Numerical r e s u l t s
Except i n t h e c a s e o f t h e e x p l i c i t scheme, t h e c a l c u l a t i o n of i s c a r r i e d o u t by t h e Gauss-Seidel method w i t h t r u n c a t i o n ( 4 2 0 ) ; t h e r e l a x a t i o n p a r a m e t e r w b e i n g chosen i n a c c o r d a n c e w i t h t h e ,formula ( s e e Varga /l/) ( 2 )
4"
3 L
w =
nh2/2)4 h2 +
(l - [ ( l i +
2h2/k2
>'I
I"
'
The t e r m i n a t i o n c r i t e r i o n u s e d i n t h e c a l c u l a t i o n o f Gauss-Seidel method w i t h p r o j e c t i o n i s EO
=
c
1 d+l (rn+ l)(M-) u;+""'(M) I Q
4"
by t h e
10-6,
ME&
th where 4+'('") i s t h e mth i t e r a t i o n v e c t o r a n d 4+1(m+1) t h e ( m + 1) (One i t e r a t i o n c o r r e s p o n d s t o a complete sweep o v e r t h e c o o r d i n ates). The r e l a t i v e a c c u r a c y o f t h e schemes w i l l be measured a t each i n s t a n t t = n k i n terms of t h e q u a n t i t y
('1
A v a r i a n t o f t h i s scheme i s g i v e n by:
I (2)
.h"
=
uO,h
T h i s i s t h e o p t i m a l p a r a m e t e r i n t h e absence o f c o n s t r a i n t s .
.
Time-dependent variational inequalities
ECART =
(CHAP. 6 )
1 I G ( M ) - u(M ;nk) I MEQk
where U;: i s t h e s o l u t i o n c a l c u l a t e d a t t = nk and u ( M ; n k ) i s t h e exact solution a t t = nk , a t t h e p o i n t M . With scheme S1 divergence w a s observed f o r h = 1/20 and k = d u r i n g t h e i n i t i a l t h e - i t e r a t i o n s ( e r r o n e o u s r e s u l t s and overflow before t = 100 k = 0.1 , w i t h T = 1). A second t e s t w i t h h = 1/20, k = S x lo-* gave a c c u r a t e r e s u l t s . For t h e S2 scheme, s e v e r a l t e s t s showed t h a t w i t h h = 1/20 , t h e b e s t time-step appeared t o b e k = s i m i l a r l y f o r S3 and S4. Table 4.9 shows t h e b e s t r e s u l t s f o r t h e s e f o u r schemes.
Conclusions of 4.3
4.3.3
For t h e s o l u t i o n of t h e i n e q u a l i t y d i s t r i b u t e d over we have:
(4.16) w i t h c o n s t r a i n t s
E x p l i c i t scheme
1.
-
danger of divergence and overflow,
-
an a c c u r a t e s t a b i l i t y e s t i m a t e i s k/h2 < i, f o r t h e same p r e c i s i o n , twice as f a s t as t h e o t h e r schemes.
ECART a c t e r k t i c s r = O . l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 S1
s2 S3 s4
h= 20 Ik=0.5)~10-~0.28 0.29 1 h = -20 k=10-3 0.27 0.43
Table 2.
-
4.9.
1
11 The
0.51 0.78 0.74 0.38 0.12 0.04 0.39 0.08 0.16 0.28 l 0 h 5 0.530.81 0.740.380.120.040.040.080.160.2924~ 0.540.820.740.380.130.040.040.070.160.2824~
0.70 0.78 0.56 0.27 0.10 0.03 0.03 0.09 0.17 0.43 26 min
Comparison o f t h e v a r i o u s schemes
The other schemes. The " i m p l i c i t i n equation", " s e m i - i m p l i c i t i n p r o j e c t i o n " and " f u l l y i m p l i c i t " schemes appear t o be s i m i l a r i n t h i s case. W
(SEC. 5 )
5.
PmaboZic inequaZities of type 11
453
INTRODUCTION TO PARABOLIC INEQUALITIES OF TYPE I1
To s t a r t w i t h we s h a l l g i v e two examples a r i s i n g i n t h e t h e o r y of thermal servo-control.
5.1
Example I
We c o n s i d e r a c o n t i n u o u s medium occupying a domain f2 w i t h boundary r a n d whose edge t e m p e r a t u r e i s servo-controlled s o as n o t t o d i m i n i s h w i t h t i m e , t h i s b e i n g a c h i e v e d by i n j e c t i n g h e a t a c r o s s r ( t h e w a l l t h i c k n e s s being n e g l i g i b l e ) . The problem may be f o r m u l a t e d as f o l l o w s ( s e e Duvaut-Lions /l/):
-
h e a t e q u a t i o n i n f2, au - Au=f. at
(5.1)
~ € 0 ~,E ] O , T [ ;
given i n i t i a l temperature, u(x, 0) = u()(x)j
- boundary c o n d i t i o n s on
x 10, T[ ( s e r v o - c o n t r o l )
,
aU
-30, at
-a Up o , -a U- =auo at an
By u s i n g G r e e n ' s formula, t h i s problem may b e e x p r e s s e d i n t h e form: f i n d u ( x , t ) s a t i s f y i n g ( 5 . 2 ) w i t h u 2 0 on r, such t h a t , a . e . t E [o, T] , we have
(5.2)
(u' - AU
- f,u - u') 2 0 ,
V v 2 0 on
r.
I n o r d e r f o r t h e problem t o be meaningful we s h a l l s e e k u such that,
(5.3)
u, 10 E L2(0,T ; H
'(a))
and such t h a t ( 5 . 2 ) i s s a t i s f i e d V V E H'(f2) w i t h v 3 0 on
r.
rn
454
Time-dependent variational inequalities
5.2
(CHAP. 6 )
Abstract formulation and existence theorem
We again use (1.1)-(1.3) and (2.21)-( 2.26) ( l ) , (2.28), ( 2 . 2 9 ) , and we assume, as is essential here ( 2 ) , that A is symmetric and independent of time ( 3 ) .
(5.4)
We put X = { u 1 U E W ( O , T ) , U ( ~ ) E Ka.e. , ~ E [ O , TI},
X, = { u I us W,(O, T ) , U ( ~ ) E Ka.e. , ~ E [ O ,T I } , X$ = { u I u E L2(0, T ; V),U' E Xo }
.
The general problem whose approximation and solution we shall study is: find (5.5)
U E
X,'
such that
+ Au - J u - d ) d t 2 0 ,
VUE X .
We recall (see Duvaut-Lions /l/, BrCzis /l/) the following existence result: Theorem 5 .l. Let f and u, be such t h a t : (5.6)
f,f'ELz(O,T;H),
(5.7)
3u,
E
u,EV,
K which s a t i s f i e s (u,
+ Au,
There then e x i s t s a unique function (5.8)
U , U ' E L ~ ( O ,VT);,
(5.9)
u(0) = u, ,
(5.10)
u'(t) E K ,
(5.11)
(u'+Au-f,u-u')>O,
(l)
(2)
c3)
a.e
.
t
E
[O,
- f (0), u - u,) u
20,
Vu E K
.
such that
TI , V U E K ,a.e. ~ E [ O , T ] ;
For simplification we shall not consider functionals j which come under the classification of parabolic inequalities of type 11. However, the proofs can be extended without difficulty to the case in which j # O . See also Remark 5.4. We require at least that the "principal part" of A be symmetric. This point is purely to simplify the proofs a little.
Parabolic i n e q u a l i t i e s of type 11
(SEC. 5 )
4 55
moreover (5.12)
u " ~ L ~ (T0;,H ) ,
(5.13)
~'(0)= u0 .
It can be shown without difficulty that the ineqRemark 5.1. ualities (5.5) and (5.11)are equivalent.
Remark 5.2. (5.14)
Assumption (5.7) is always satisfied if
K is bounded in V
o r indeed if (5.15)
- [ A u ~- f ( O ) ]
Remark 5 . 3 .
E K.
The following inequality can be established
This result enables us to conclude, when u O e x i s t s (see Remark 5.2), that u' is bounded in K for the norms L2(0,T ; v) and L"(0, T ; V) It is then easy to see that if K is not bounded, i t i s possible t o define a convex s e t
.
R sufficiently large, such that problem (5.11) defined on K,in We can then fact has the same solution as the initial problem. consider, without loss of generality, (but still under assumption (5.7)) that (5.16)
K is bounded in V .
This remark is useful for the study of the approximation.
Remark 5.4. We can likewise solve the following inequality, which is more general than (5.5): (5.17)
JOT
(u'
+ Au - 5 u - d ) df +
vu E
[j ( u ) - j(u')] df
x,u E Yo
where j is a functional of the type ( 2 . 2 7 )
.
>0,
4 56
Time-dependent variational i n e q u a l i t i e s
6.
( CHAP.
6)
APPROXIMATION OF PARABOLIC INEQUALITIES OF TYPE I1
6.1
Fundamental assumptions for the approximation
In studying the approximation we shall retain everything said in Section 3.1 with, in addition, the following assumptions: (6.1)
ah(uh, u h ) is sym??ZetriC,
I I
Kh is bounded in Vh, independently of h , i.e.
11 uh
d
c,
VUhEKh
and Vh,
h +0.
( F o r this latter assumption, see Remark 5.3).
We shall transpose the results given in Section 3.1.3 to N a n d to
xg+. In view of ( 5.6) we have f c V0(O, T;H ) and we take
as in Section 3.1.6 with f / defined by (3.27'). 6.2
Approximation scheme for parabolic inequalities of type I1
Taking account of the above assumptions, we shall now study the approximation of the inequality
We note that inequality (6.5) can be written:
(SEC. 6)
Approximation of inequaZities of type I1
457
This possesses a unique solution 6; ; thus one by one all the
up1,i = 0, 1, ..., (u;"
= u;
+ k&),
exist and are uniquely defined. Moreover we note (since Ah is symmetric) that the u;", for i=O,1, ... are uniquely defined from the respective solutions of the N minimisation problems (i = 0, ..., N - 1)
+
minimise f. I 6; ;1 f. ek(Ah $ ) h - (A,, ui - f;,6;)h subject to the constraints
where ui, (i 3 I), is given (the solution of the previous u i = uh.o We put u F 1 = ui' k6u; . problem), and
.
+
We use the following terminology:
- Explicit scheme if O = 0, - Crank-Nicholson scheme if - Implicit scheme if 0 = 1. Remark 6.1.
e=f.,
In the same spirit, the solution of the inequal-
ity (u'
+ AU -
u - u')
+ j ( ~-) j ( ~ 2' ) 0 ,
V DE
X,u E X,* ,
(where j ( u ) is defined as in ( 2.27)) leads to the schemes (omitting the suffix h )
+ A[u' + flu'+' ui+l
+j(u) - j (
-
- ui)]- fi,u -
"')2
-
,,'+I
0 , VDEK,- k
- U'
'
E
K,
o r , since A is symmetric, to the solution of the problems:
m h k i s e $ 1 6; I;+$ek(Ah &, 6;) - (Ah u; - f;, 4 ) h + j(&, subject to the same constraints as in (6.8).
4 58
Time-dependent variational ineqwzlities
( CHAP.
6)
Remark 6 . 2 . In the same way as for parabolic inequalities of type I, we can here also define decoupled schemes:
where P,is
6.3
the projection on & in the sense of the norm Convergence of the approximate inequalities
We shall now study scheme (6.5) which we rewrite (omitting the subscript h for clarity) as follows
I ( ~ ' + A U ' + ~ - ~ ~ , I I 2- ~0' ),
(6.9)
with
I uo = uo 6' =
V o E K , 6 ' E K , i = O , 1,
..., N ,
given,
- ui
ui+l
k
'
In addition we shall use the following notation: 6i+e =
6.3.1
ui
+ 1 +B
- u'
+@
k
=
6' + e [ g i + l - 6 1 ,
Stability
We shall now establish: Theorem 6.1. When (h, k ) + 0
(6.10)
i n a general fashion i f f Q 8 Q 1, under the s t a b i l i t y condition C k S W Q 1 - /3
if
0Q
we have, f o r a l l n = 0,1, ..., N, (6.11)
II u; Ilk Q c ,
e c f , (/3
fixed E]O,ID,
(SEC. 6 )
A p p r o ~ mtai o n of inequalities o f type 11
459
and
Proof. (6.14)
Writing
(6.9) for i +
+ Au""'
(6"'
-
v -
f i + ' 9
With v = 6"' in ( 6 . 9 ) ,
I 6i+l
- 6i 12
+
a'+' ) z o ,
v = 6'
- ui+B
a(ui+l+@
1 we have VVEK,~'+~EK.
in (6.14) and adding, we obtain gi+l
- 6') < ( f i + 1 - fi,
-
a'),
or, dividing by k ,
and hence, multiplying by 2, k I yi
l2
+ 2 ~ ( 6 ' ,gi+l - 6') + 2 ea(6'+l
- 6')
Q
k I gi
12,
or, since a is symmetric, (6.15)
k I y' 1'
+ ~(6"') - a($) + (2 0 - 1) ~ ( 6 " ' - 6') < k I g' 1'.
We then distinguish the following two cases:
Since the term in ( 2 0 -
1) is positive we can suppress it in i = 0 to n < N - 1 we obtain
(6.15), and by summing from k IyiIz i=O
+ ~(6"") d
k Ig'I2 i=O
+ a(6").
As a consequence of assumption (6.2)we have a(6O)G C. In view of (5.6) and (3.27') we then obtain (6.13) immediately. Using the coercivity of a(,) we deduce (6.12). The upper bound (6.11)is an immediate consequence of (6.12)and of the fact that ~(6') d C by noting that 4+1=
u:
+k
c 60. n
i=o
460
Time-dependent variationai! i n e q u a l i t i e s 2nd case : 0 Q
( CHAP. 6 )
e < 4.
We u s e t h e upper bound - 6') = h2 a(y') d Ck2 S(h)2 I y'
U(6'+'
12
,
whence
(2 e - 1) a(ai+l - 6') 2 (2 e - 1) ckz s(hl2I 7' l2 , and (6.15) i m p l i e s t h a t (6.16)
+ (2 6 - 1) CkS(h)'] I y' I' + ~(6'")
k[1
Hence i f we assume, i n t h e c a s e
1
- CkS(h)' 2 B > 0, /? f i x e d
- a(6') Q k 1 gi 1' .
2 8 - 1 < 0, t h a t €10, 1[,
it may be deduced from (6.16) t h a t
Bk I y' 1'
+
U(6'+1)
- 46') Q k I g' 12 ,
and by summation w e o b t a i n (6.11), (6.12), (6.13), as i n t h e 1st case. Thus Theorem 6 . 1 i s proved. rn
Remark 6 . 3 . I n t h e c a s e where V h , k + O , t h e i n e q u a l i t y ( 6 . 9 ) i s i n f a c t , f o r i = 0 , an equation (convex s e t n o t " s a t u r a t e d " ) ; we have 60 + AUO+O - f O = 0 , and hence
11 '6 11
=
(I
+ eks0) - f o I( Q II A U O
from which we deduce, w i t h
II + ek 11 ~6~ 11
+ 11 f o 11,
11 Auo 11 d c, 11 Ado 11 d c t h a t
(1 - ekc) II 60 II < c , that, for k sufficiently s m a l l , ' 6 i s bounded f o r t h e norm V,, independentzy o f h and k, and assumption ( 6 . 2 ) i s i n f a c t rn unnecessary. SO
h'eak convergence
6.3.2.
From Theorem 6.1, when h , k + O ( w i t h (6.10)i f 0 < 0 < 4 ) e x i s t s a subsequence, a g a i n denoted by uhc, and elements (6.17)
U E
L ~ OT ,; v),U E L ~ O T, ; v),w E ~ ' ( 0 ,T ; HI, CE
such t h a t (6.18)
ph uh,k + ou weakly" i n L"(0, T ; F) qh u , , ~+ u weakly" i n L"(0, T ; H)
v,
there
(SEC.
6)
Approximation of inequaZities of type I1
* *
(6.19)
ph Su,., + au weakly 9h 6uh.k + v weakly
(6.20)
9h S2uhsk+ W weakly i n
(6.21)
{
in in
461
L"(0, T ; F ) L"(0, T ; H )
L2(o,T ; H )
ph u: -+ at weakly i n F , 9h4 + weakly i n H .
I n t h e same s p i r i t as i n Lemma 3 . 1 it may t h e n be shown ( s e e V i aud /1/) t h a t
(6.22)
u = u',
= u(T), u ' E K ,
w = u",
s o t h a t u s a t i s f i e s (5.8), (5.9), (5.10), (5.12). t h a t u s a t i s f i e s t h e i n e q u a l i t y (5.11).
c uid .
We s h a l l show
N
Let
V E Xand vhsk=
be a sequence approximating u as i n
is0
Lenma 3.2. We t a k e u = u i i n (6.9), sum from i = O t o N - 1 and m u l t i p l y by k ; we o b t a i n ( o n c e more o m i t t i n g t h e s u f f i x h for t h e time being) N- 1
N- 1
(6.23)
1 [k(S', u') + kU(U'+@, .=
ui)
- k ( f ' , ui-
0
Si)]
2
1 k I 6'
12
+ x,
i=O
with N-1
(6.24)
x = 1 kU(Ui+@,6') i=O
Now N- I
with
Y = (2 0 - 1 ) .
c
U(Ui+'
- u')
i=O
We s h a i l d i s t i n g u i s h t h e f o l l o w i n g two c a s e s ( r e i n t r o d u c i n g the suffix h ) : 1st
(6.25)
case : f s 8 s 1.
lim inf X 2 h.t-0
f a(u(r)) - f u(uo).
462
Time-dependent variational inequalities
( CHAP.
6)
Using (6.23) and (6.31) together with the various consistency assumptions on . f , a ( , ) , f and the results (6.18), (6.19), we obtain :
IoT
(u'
+ Au - 5 v - u') dt 2 0
which is true V v e X From this we deduce that u is a solution of problem (5.5) o r (6.3).
2nd case : 0
< 0 c f.
We then have Y < 0, but the above result remains valid if we have Y - r 0. Now
Hence (6.27)
N
I YI < kC
k
< kC,
i=0
so that we indeed have Y - r O . To conclude, we note that as a consequence of the uniqueness of U , the results (6.24),. ,(6.27) are true for the entire sequence. We car1 now state:
..
Theorem 6.2. with
When h, k
< 1-p, p
CkS(h)'
we have ph uk*k
(6.28)
I
-r
P k 6uh,k
+
q h 'h.k + -r
q k 6uh.k q h 6'Uh.k ph
4
--*
u'
-r
U"
+0
cmd i n the case when 0
0). We c o n s i d e r t h e f o l l o w i n g e x a c t s o l u t i o n on Q = lo,T [ x 61 = ]0,0.1[ x lo, 1[ x 10, 1[
+
1) (x - 1)' yz[(sin 10 t - y)+I3 u(x, y , t ) = 1 + lOO(2x + 100(2y 1) ( y - 1)' xz[(sin 10 t - x)+13 - 100 XZ(1 - x)y(1 - y)Z [ ( y - sin 10 t ) + ] 3 - 100y~(1- y ) x(l - x)'[(x - sin 10 t)+]' , with, f o r a quantity g :
(7.13)
+
=g
if 9 2 0 ,
= O
if
+
gs0.
t h e problem
(where
K i s g i v e n by ( 7 . 6 ) and 3'2
i s d e f i n e d as i n S e c t i o n 5 . 2 )
(SEC. 7) Numerical solution of inequalities of type I1
471
has the exact solution (7.13)(’ ) . It may be noted, moreover, that an arbitrary constant can be added to u without modifying f (see (7.14)). Since it would therefore be possible to give results with as small a relative error as desired, we shall note that
max u(x, y , 1)
N
3.2
.
(x.y.0 E Q
1
I
I I
a
I
I
I
I
1
0 I I
II II
I
Fig. 7.3.
u on f.
I I I
a
I I
I I
Fig. 7.4.
du/dn on f
Figures 7.3 and 7.4 define the values of u and au/an on f. Figures 7.5, 7.6, 7.7 show the evolution of the error E(u) for various values of k and for the implicit, Crank-Nicholson and explicit schemes. With h = 1/20, the implicit and CrankNicholson schemes give a minimum error for k = and these schemes are almost equivalent for this value. We observe that and that in the explicit scheme overflow occurs for k > 7 ~ 1 0 - ~ gives results comparable to those obtained the value k = using the other schemes with k=10-3. The computation times (on a CII 10070), were as follows: -
-
(l)
Implicit and semi-implicit schemes with h = 1/20, k = : time N 2.5 min. Explicit scheme with h = 1/20, k = 2 ~ 1 0 :- ~time
N
6 min.
As a consequence of choosing f in accordance with (7.14), we note that the solution (7.13) also corresponds to problem (7.15) with K = H ’ ( O ) in place of (7.6). Thus the inequality (7.15)is here equivalent to an equation.
472
Time-dependent variational i n e q u a l i t i e s
E(4 4
0.01
0.05
Fig. 7.5.
0.0 1
0.0 1
t
0.1 t
Semi-implicit scheme.
0.05
Fig. 7.7.
c
Implicit scheme.
0.05
Fig. 7.6.
0.1
Explicit scheme.
0.1
t
I
(CHAP.
6)
(SEC. 7 )
7.2.
Numerical solution of inequalities of type I1
473
S o l u t i o n o f Example I1
The p a r t i c u l a r s r e l a t i n g t o t h i s example are
(7.16) Taking
V = Hd(Q), H = LZ(Q), K = { u ~ H i ( Q ) ( u ( x ) > 0a.e. Q = 10, I[ x 10, I[
XEQ}.
w e d e f i n e , w i t h Rh as i n S e c t i o n 7.1,
I
0
Fig. 7.8.
Sr, f o r h = 115.
I n t h i s c a s e t h e problem can a g a i n be w r i t t e n i n t h e form ( 7 . 5 ) and i s s o l v e d by r e l a x a t i o n , w i t h w t a k e n as i n F i g u r e 7 . 1 and u s i n g t h e t e r m i n a t i o n c r i t e r i o n g i v e n by ( 7 . 1 1 ) . We s h a l l assess t h e schemes i n terms o f E(u) g i v e n by (7.12), and w i t h
x
1 I u ' ( M ; (i + e) k ) 1' MERh
.
>"'
Here it can be shown once a g a i n t h a t S(h) = 2 f i / h . We s h a l l now i n v e s t i g a t e t h e approximate s o l u t i o n o f - t h e inequality
(where K i s g i v e n by
(7.16)and
i s d e f i n e d as i n S e c t i o n 5 . 2 ) .
474
Time-dependen t v a r i a t w n a l i n e q u a l i t i e s
(CHAP. 6 )
We take :
- Au - lOo(10 h t
(7.18)
f(x, Y, I ) =
(7 * 19)
u(x, Y. 0) = u d x , y ) ,
U'
- 9 - y')'
,
with the exact solution u(x, y ,
(7.20)
r)
=
10 sin x sin y(l - x) (1 - y ) -
-10x~l-x)(1-y)[(x~+y~-IOsinr)+]'.
Over the domain Q; =
we have u'
( X , Y ) E Q 1 10 sin t 2 x2
=0
+ y2 1,
(see Figure 7 . 9 ) .
I
Fig. 7.9.
sz
u' = 0 on Q;.
The results are shown in Figures 7.10, 7.11, 7.12, 7.13, the computation times being as follows :
- Implicit
-
and semi-implicit schemes with h = 1/20,
time * 1.2 min., Explicit scheme with h = 1/20,
k
= lob3 :
k =2
x
lo-* : time
= 3.5
For the explicit scheme, overflow occurred for k 2 7 x I O - * (see Figure 7.13). In this case the implicit scheme gives the best results.
min.
H
7.3. Conclusions In conclusion, f o r the two examples considered it was observed that f o r similar accuracy the implicit schemes were two o r three times faster than the explicit schemes, and that the latter Other examples diverge when the timestep.is not s m a l l enough. which corroborate those presented here f o r inequalities of type I1 are given in Viaud 111. H
(SEC. 7) Numerical solution of inequalities of type I1
Fig. 7.10.
Fig. 7.11.
Implicit scheme.
Semi-implicit scheme.
0.005
I I I
0
0.01
0.05
Fig. 7.12.
Explicit scheme.
0.1 t
475
Time-dependent variational i n e q u a l i t i e s
476
0
0.01
0.05
Fig. 7.13.
8.
0.1
( CHAP.
6)
i
E(u‘) f o r t h e e x p l i c i t scheme.
INTRODUCTION TO TIME-DEPENDENT INEQUALITIES OF THE SECOND ORDER I N t
8.1.
Example I
The t h e o r y o f v i s c o - e l a s t i c m a t e r i a l s w i t h edge f r i c t i o n l e a d s t o problems ( s e e Duvaut-Lions /l/) for which t h e f o l l o w i n g i s a model example: f i n d a f u n c t i o n u = u(x, t ) such t h a t a2u
au
at2
at
-- A - - A u = f
in ax]O,T[,
with t h e i n i t i a l d a t a dx, 0) = uo(x) 3
at4
5 (x, 0) = Ul(X),
and t h e boundary c o n d i t i o n s : u = 0 on one p a r t
(u”, u
(8.1)
T o x 10, r[
r
- u’) + al(u’, u - u’) + ao(u, u - u’) + j ( u ) - j(u‘) 2
B(f,u-u’)
VUEY,
40) = uo , u‘(0) = u1 , where
o f t h e boundary
U’ E V
x 10, T [ ,
(SEC. 8 )
InequaZities of order
2
in t
477
V = { u ( u E H ' ( R ) u, = O o n r , ) , ao(u, u ) = a,(u, u ) =
8.2.
1"
grad u grad u dx ,
Example I1
A number of p h y s i c a l s i t u a t i o n s ( s e e Duvaut-Lions t h e s e a r c h f o r a f u n c t i o n u such t h a t
/l/) l e a d t o
d'-Au-f>O, ~ ' 3 0 , - Au - f) = 0 i n D x 10, T [ ,
u'(u"
w i t h t h e i n i t i a l d a t a as above and w i t h t h e boundary c o n d i t i o n s g i v e n , f o r example, by u = 0 on
r
x 10, T [ .
T h i s problem can a l s o b e f o r m u l a t e d as a v a r i a t i o n a l i n e q u a l i t y o f t h e second o r d e r i n 1 , namely
(8.2)
1
(u",11 - u') + U(U, u - u') 3 (f,u - u') V U E K , u'(t) E K , ~ ( 0 )= u0 , ~ ' ( 0 = ) u1 ,
where
K = { u ( u ~ H d ( D )u ,> O
ah, u) = 8.3.
a.e. i n R } ,
1"
grad u grad u dx .
Abstract formulation
Using t h e same f u n c t i o n a l s e t t i n g as i n S e c t i o n 5 . 2 ( V c H c V') we i n t r o d u c e , w i t h K a c l o s e d convex s e t o f V, t h e convex s e t s
X = { U I U E E ( O , Tu )( ,i ) ~ K , a . e .
tE[O,T]},
x* = { u I UEL*(0,T ; V ) , U ' E x } , 30' = { 0 I u E x * , u(0) = uo, u'(0) = UI } , where uo,uI a r e g i v e n , and we s h a l l c o n s i d e r i n e q u a l i t i e s o f t h e type
(8.3)
L(u, U) =
foT (u"
AU - f, u - u') d t 3 0 , V U E X , u E Xg ,
w i t h u(0) = uo, u'(0) = u1 where A and J have t h e same p r o p e r t i e s as i n
Time-dependent v a r i a t i o n a 1 inequa t i t i e s
478
( CHAP. 6 )
S e c t i o n 5.2; i n p a r t i c u l a r A i s assumed t o be symmetric ( ’ ) (and i s c l e a r l y coercive and c o n t i n u o u s ) . The i n e q u a l i t y ( 8 . 3 ) i s more o f t e n w r i t t e n i n t h e form (u”-Au-~,v-u’)>O
(8.4)
VVEK, U E X ~ .
The e x i s t e n c e of u , t h e s o l u t i o n of ( 8 . 3 ) , i s given ( s e e Lions
111) by: Theorem 8.1.
With
(8.5)
f,S’E L2(0, T ;H),
(8.6).
A u E~ H ,
(8.7)
uIEK,
there e x i s t s a unique f u n c t i o n u s a t i s f y i n g ( 8 . 3 ) ( o r ( 8 . 4 ) ) s u c h that (8.8)
u,u’EL~(O T ,;V),
(8.9)
u ” ~ L ~ (T0;H , ).
Remark 8.1.
By i n t r o d u c i n g
X, = { v I v E L2(0, T ; V), V’ v ’ ( t ) ~ K , a.e.
E L2(0,T ;V), 40) =
u0,
~E[O,T]}
we can i n v e s t i g a t e t h e e x i s t e n c e o f weak s o l u t i o n s of t h e inequality JOT
(v”
- AU -
v’
- u’) dt 3 o ,
vv
E
ju,+ , u E
3 ~ .;
For t h i s problem w e r e f e r t h e r e a d e r t o BrCzis 121, BrCzisLions 111.
Remark 8.2. A formalism somewhat more g e n e r a l t h a n ( 8 . 3 ) i s t o consider t h e i n e q u a l i t y (8.10)
1:
(u”
+ .Au - f, v - u’) dt +
[&)
- &’)]
dr 2 0 ,
VVEX, U C X 2
(’)
It i s s u f f i c i e n t t h a t t h e p r i n c i p a l p a r t o f A be symmetric. We can a l s o consider problems i n which A i s dependent on 1 . The c a s e o f ( 8 . 1 ) may be t r e a t e d by t h e same methods.
I n e q u a l i t i e s of order 2 i n t
(SEC. 8 )
where j i s a f u n c t i o n a l o f t h e t y p e ( 2 . 2 7 ) .
479
8
Remark 8 . 3 . R e s u l t ( 8 . 8 ) a l l o w s us t o c o n s i d e r t h a t i f K i s n o t bounded t h e n it i s p o s s i b l e t o d e f i n e a convex s e t KR ( d e f i n e d as i n Remark 5.3) which i s bounded i n v, and which i s s u c h t h a t t h e problem ( 8 . 3 ) defined on KR ( r a t h e r t h a n K ) has t h e We can t h u s c o n s i d e r t h a t same s o l u t i o n as t h e i n i t i a l problem. K i s bounded i n V .
(8.11)
I n t h e same way as i n t h e p a r a b o l i c c a s e o f t y p e 11, t h i s remark i s o f u s e i n i n v e s t i g a t i n g t h e approximation. 8
9.
APPROXIMATION OF INEQUALITIES OF THE SECOND ORDER I N t
I n t h i s s e c t i o n w e embark on t h e i n v e s t i g a t i o n of t h e approxi m a t i o n o f h y p e r b o l i c i n e q u a l i t i e s o f t h e form ( 8 . 4 ) .
9.1.
Assumptions.
We s h a l l work w i t h i n t h e s e t t i n g d e f i n e d i n 6.1. s h a l l d e n o t e t h e approximate f u n c t i o n s by uhJ w i t h
A s u s u a l we
N
We s h a l l u s e t h e n o t a t i o n
For t h e approximation of uo, ul, X", u(u, u) = (Au, u), / ( I ) we make t h e following assumptions:
Approximation of the i n i t i a l values. With U ~ VE, U ,E K, we assume t h a t t h e r e e x i s t (9.1)
U0.h
(9.2)
Ul,h E K h
E
v h
Such t h a t pj, U0.h
+ 0110
Such t h a t
-B
P h U1,k
UUl
It h
c,
S t r o n g l y i n F and
(1
S t r o n g l y i n F and
I( U 1 . h [ ( h 6 c.
UOsh
6
480
Time-dependent variational inequalities
duh.k(t)
(9.3)
+
P h 0h.k
+
E
[o, r ]
u weakly
9
*
t E [o, TI , s t r o n g l y i n L2(0,T ; H ) s t r o n g l y i n L2(0, T ; F). + uu
E Kh u
9
q h 1)h.L +
p h I)h,k
(9.5)
9
i n Lm(O,T ; H ) uul weakly * i n L"(0, T ; F) p h U h , k ( T ) + UU2 and p h U h r . k ( 0 ) + UUp Weakly i n v q h d o h , , + U 3 weakly * i n L"(0, T ; H ) ( o r w i t h q h 6vh.k + u j ) P h d u h , k + UU4 weakly * i n L"(0, T ; F) ( O r w i t h q h 6Vh.k + U U 4 ) q h 6 U h , k ( T - k) + Us and q h b U h , k ( o ) + 0: Weakly i n H weakly * i n L"(0, T ; F) q h 7Vh.k + q h uh.k
uh.k(l)
(9.4)
Kh
(CHAP. 6 )
Kh i s bounded i n
vh
.
For t h i s l a s t assumption, c f . Remark
9.4.
Approximation of u(u, u). We s h a l l assume ( c f . (9.6)
IAhu0.h
Ih
(8.6)) that
G C , c o n s t a n t independent o f h .
Otherwise t h e assumptions a r e t h e same as i n S e c t i o n 6.2. p a r t i c u l a r , q , ( l ( k , u,,) i s symmetric.
In
Approximation of f. The same remarks a p p l y as i n S e c t i o n 6.2. here t h a t A f ' E
Lm(O,T ; H),
We s h a l l assume
(SEC. 9 ) Approximation of i n e q u a l i t i e s of order 2 i n t
481
and therefore f E V0([0, TI ; H ) ,
and we take N
=
fh.k(r)
i=O
such that (9.7)
9.2.
I
qhfh,k
-+
P h %,k
fi xi(r)
11 f h , k Ilk G f' in L.D(o,T ; H ) and 11 6 f h . k 1: < c .
f in
-t
L"(o,
H,
and
1
Approximation schemes
Taking account of the approximation assumptions and with
we shall
(9.8)
In th following we shall use the notation (omitting the suffix h )
find u2, u3, ..., # such that
Scheme ( 9 . 8 ) can thus be written: + Aui+@- f',u (9.9)
d'+l
(i21), VVEK,
(9.10)
uo =
~
0
U ,' = u0
-
,,i+l
- ,,i-l
2k
-
2k
ui-l
)2 0 ,
EK,
+ ku,
From the theory of elliptic inequalities, the inequality ( 9 . 9 ) ,
(CHAP. 6 )
Time-dependent variational inequazities
482
which by p u t t i n g =
- 8-l
+ k7 (Ad - BkAS'-'
can a l s o be w r i t t e n ((I 8kz A ) d'
+
- f'],
+ f',u - d') 2 0 ,
p o s s e s s e s a unique s o l u t i o n , d! T h i s e n a b l e s us one by one t o d e f i n e u n i q u e l y a l l t h e u'+l= u i - l
+ kd',
(i 2 l ) ,
w i t h t h e d a t a (9.10). S i n c e A i s symmetric, w e n o t e t h a t t h e d+' (i 3 1) can b e o b t a i n e d from t h e s o l u t i o n s d' o f t h e N 1 m i n i m i s a t i o n problems (i = 1, ..., N - 1) :
-
minimise
f I d' I'
+ f Bk'(Ad'. d') + (7,d') ,
subject t o the constraints
d'EK, where 8-', u'-', d = u'-' + k8-I a r e , f o r i 2 2 , g i v e n by t h e solutiorTt h e p r e v i o u s problem and, f o r i = 1, '6 = u1, uo = uo, u1 = uo k6'.
+
We u s e t h e f o l l o w i n g t e r m i n o l o g y :
- e x p l i c i t scheme i f 8 = 0 , - semi-impticit scheme i f 0 < 8 < 1, - implicit scheme i f 8 = 1. rn Remark 9.1.
I n t h e same s p i r i t , t h e s o l u t i o n of t h e inequa-
lity
(ff where
j(u)
+ AU - f, u - u') +Au)-j(u') 2 0 ,
VUEK,
~~2-2,
i s d e f i n e d as i n (2.27), l e a d s t o t h e scheme
-
(""1
$f+
u'-1
ui+l
+Au)-j(
+ Ad+' - f', u - d-1 2k
ui+l
2-k u i - l ) u'+l
VUEK,
)20,
+
- ui-l 2k
EK,
or, i f A i s symmetric, t o t h e s o l u t i o n o f t h e problems
minimise f 1 d' 1'
+ f 8k2(Ad',d') + (f"',d 9 + A d ' ) ,
subject t o t h e constraints
Remark 9.2.
d'EK.
rn
It i s a p p a r e n t from t h e above remark t h a t t h e
(SEC. 9 )
Approximation of i n e q u a l i t i e s of order 2 i n t
483
methods of Chapter 2 , which were o r i g i n a l l y d e s i g n e d f o r s o l v i n g e l l i p t i c variational i n e q u a l i t i e s , now a l s o p l a y a fundamental p a r t i n t h e s o l u t i o n o f time-dependent v a r i a t i o n a l i n e q u a l i t i e s .
Remark 9.3.
A v e r y l a r g e number of o t h e r schemes is a l s o Some o f t h e s e a r e mentioned below ( o m i t t i n g t h e
possible. suffix h ) :
Scheme 1 ui+l
- 2 ~ +' u i - l
(9.11)
,,i+l
VVEK,
+ Au' - f i , v
,,i+l
-
- ,,i-l 2k
).O,
- ,,i-l 2k
E K ,
with
(9.12)
I
,I
= 8-
ui-l
+ 80 ui + 8+
ui+l
8-, O0, 8+ E [O, 11
where -.
,
and
8-
+ Oo + 8+ = 1 .
Schemes 11
I
and
d
as i n ( 9 . 1 2 ) .
Schemes I11 ( d e c o u p l e d schemes)
I
ui+l =
U'
+ kP,,[
ui+l/2
-
"1
,
where ,PKh i s t h e p r o j e c t i o n on t h e (,)k norm.
& i n t h e s e n s e of
The a p p r o x i m a t i o n o f schemes I and I1 i s s t u d i e d i n TrPmo i : r e s
lbl. 9.3.
Convergence o f t h e approximate i n e q u a l i t i e s
We s h a l l now i n v e s t i g a t e t h e s t a b i l i t y and, for h, k + 0 , t h e convergence o f scheme ( 9 . 9 ) . Using t h e n o t a t i o n of S e c t i o n 9.2,
484
Time-dependent variational i n e q u a l i t i e s
( CHAP.
6)
(9.9) is expressed as follows: (yi, u - d') + u(ui+', u - d') 2 (fi, u - d') , (i 2 I ) , (9.14) scheme
VUEK, d ' E K .
Before starting the investigation we recall Gronwall's Lemma: I f pi, i = 0, I,
Lemma 9.1.
..., n, are numbers which s a t i s f y
0 Q p 1 6 Cl , constant
Z0,
n
0 Q pn+
< Cl + C2 1 ki p i ,
n = 1, ..., N ,
i=1
with C, constant 2 0 and k i 2 0 , i = 1 ,..., N , N
1 ki = T , p o s i t i v e
constant
i= 1
then
(
)
2 L(vh-u,,)
Vvh
E
Vh.
The approximate problem (2.25A) admits one and o n l y one s o l u t i o n . Remark 2.2 of S e c t i o n 2.3.1 a l s o h o l d s f o r problem
(2.25A). With r e g a r d t o t h e convergence o f ( L + )
h
when h -+ 0, we have
Let u be the solution of (1.2A), yn t h a t of Theorem 2.3: (2.25A); under the above .assumptions on V, a, L and j , and i f (Pl), (2.23A), (2.24A), (P2), (P3) are s a t i s f i e d , we have (2.26A)
(2.27A)
l i m II%-ull h+O l i m jh(\) h+O
= 0
=
,
j(u).
(SEC. 2 ) Proof:
Existence, uniqueness and approximation
551
T h i s i s a v a r i a n t o f t h a t o f Theorem 2.2 of S e c t i o n
2.3.1.
Let vo c p j t h e C . which a p p e a r below d e n o t e v a r i o u s q u a n t i t i e s which depenh onVo , b u t n o t on h ; we have, t a k i n g account o f (P,) , ( P~),
If we t a k e v = vo i n (2.29A), we deduce from (2.30A)
'lluhll which i m p l i e s t h a t 2)
2
=3
%
IIu,,II
+
cq
Y
i s bounded i n V.
Weak convergence o f ( \ )
S i n c e t h e sequence ( \ )
h
h'
i s bounded i n V, we can e x t r a c t from such t h a t
it a subsequence, a l s o d e n o t e d by ( \ ) h , (2.31A)
l i m u,, =
,*
weakZy i n V .
WO Moreover we deduce from (2.25A) t h a t
(2.32A) Using (2.3l.A)
QvE'V:
, (P,) , (P,) , (P3)
we have i n t h e l i m i t i n (2.32A),
(APP. 1)
Steady-s t a t e inequalities
552
Using the density of ’Ir in V and the continuity of j ( deduce from (2.33A)
0
, we
)
*
a(u*,v-u*)+j(v>-j(u*) ~ ~ ( v - uQ
~ E v ,
(2.34A) U * E V ,
which implies u* = u and the convergence of the e n t i r e sequence ( ) towards u. Uhh
In view of (Pl), (P2), (P ) and from the weak convergence result given above, we haveY3in the limit, in (2.35A), Vv E
Using the density of ’v in V and the continuity of j deduce from ( 2.36A) that
(
0
)
, we
Obstacle problems
(SEC. 3 )
Taking
FU
- .I
5 53
i n (2.37A) we o b t a i n
j(u) = l i m inf jh(%) = l i m ( aII%-uII
2
+jh(%))
which c l e a r l y i m p l i e s (2.26A), (2.27A). Some a p p l i c a t i o n s o f t h e above Theorem a r e given i n Chapters
4 and 5.
3.
THE OBSTACLE PROBLEM. APPROXIMATIONS
( I ) GENERAL REMARKS.
CONFORMING
Synopsis
3.1
I n Chapter 1, S e c t i o n 3.6, Example 3.5, w e considered t h e s p e c i a l c a s e o f t h e o b s t a c l e problem (1.3A) corresponding t o JI = 0 , g = 0. A v a r i a n t o f t h i s s p e c i a l c a s e which a r i s e s i n l u b r i c a t i o n t h e o r y w a s a l s o c o n s i d e r e d i n Chapter 2, S e c t i o n 5 . I n t h e p r e s e n t s e c t i o n , which follows Glowinski / l A / , w e s h a l l b p a r t i c u l a r l y concerned w i t h t h e approximation of (1.3A) by means of conforming f i n i t e element methods, and we s h a l l g e n e r a l i s e t h e r e s u l t of Chapter 1, S e c t i o n 4.7, r e l a t i n g t o t h e s p e c i a l c a s e J, = 0, g = 0. The approximation o f (1.3A), u s i n g nonconforming f i n i t e elements o f mixed type, w i l l be c o n s i d e r e d i n S e c t i o n 4. It i s a p p r o p r i a t e t o p o i n t o u t t h a t v a r i a t i o n a l problems of type (1.3A), ( 1 . 4 A ) ( t h e obstacle problem), although p a r t i c u l a r l y simple, provide a good mathematical model i n s e v e r a l important a p p l i c a t i o n s ( s e e S e c t i o n s 3.2 and 3.3 below). Moreover, amongst v a r i a t i o n a l i n e q u a l i t y problems t h e y are t h o s e f o r which t h e most d e t a i l e d t h e o r e t i c a l and numeridal r e s u l t s have been obtained. Formulation o f t h e problem.
3.2
N
Physical i n t e r p r e t a t i o n
Let R be a bounded domain i n IR , w i t h r e g u l a r boundary N we c o n s i d e r t h e f o l l o w i n g With x = {xi]i=l, V = {a}N ax. i=ly
r=aQ.
1
obstacle problt?m, which has already been formulated i n ( 1 . 3 A ) , (1.4A)
,
Steady-s t a t e i n e q u a l i t i e s
5 54
( U P . 1)
Find u c K such that (3.1A)
Vu*V(v-u)dx
f(v-u)dx
2
Vv
E
K,
n
where, i n (3.1A),
( 3 .2A)
K =
fELL(Q)
{VEH
1
and K i s d e f i n e d by
(511, v z q
a.e.
on 0 ,
vlr = g)
where J, and g a r e g i v e n f u n c t i o n s . We assume 51 c n2 ; a s t a n d a r d i n t e r p r e t a t i o n o f ( 3 . 1 A ) , (3.2A) i s t h e n t h a t i t s s o l u t i o n , u, r e p r e s e n t s t h e v e r t i c a l displacements, assumed small, o f a horizontal e l a s t i c membrane R u n d e r t h e e f f e c t o f a d i s t r i b u t i o n of v e r t i c a l forces of i n t e n s i t y f ( f i s a s u r f a c e d e n s i t y o f vertical forces). T h i s membrane i s f i x e d a l o n g i t s boundary r ( u = g), and i s constrained t o l i e above an obstacle w i t h h e i g h t g i v e n by 6 (u2JI) ; F i g u r e 3.1 below d e s c r i b e s t h i s phenomenon geometrically :
PhysicaZ interpretation:
U=$
\
u=g \
\
/
\
/
\ \
I
\
Fig. 3.1 3.3
O t h e r phenomena r e l a t e d t o t h e o b s t a c l e problem
V a r i a t i o n a l i n e q u a l i t i e s o f t h e same t y p e as ( 3 . 1 A ) , (3.2A) b u t p o s s i b l y w i t h d i f f e r e n t boundary c o n d i t i o n s a n d / o r a nonsymmetric b i l i n e a r form) a r i s e i n t h e f o l l o w i n g a p p l i c a t i o n s :
Obstacle problems
(SEC. 3 )
-
555
I
- Lubrication phenomena; see for example Cryer 111, 121, Marzulli /l/ and Chapter 2, Section 5 of the present book for solution by finite difference methods and some further references. For the modelling itself see Capriz /lA/. - Seepage o f l i q u i d s i n porous media; see in particular Baiocchi /l/, /2/, /lA/, /2A/, Comincioli /lA/, Baiocchi-BrezziComincioli /lA/, Baiocchi - Comincioli - Magsnes-Pozzi /l/, /lA/, Cryer-Fetter /lA/, Baiocchi-Cape-lo /lA/, and also their associated bibliographies. - mo-dimensiona2 potentiaZ fZoiJ of p e r f e c t f l u i d s ; see BrgzisStampacchia /l/, /lA/, B r 6 z i s /lA/, Ciavaldini-Pogu-Tournemine /lA/, Roux /lA/, and also the references therein.
- Wake probZerns; see Brczis-hvaut 111, Bourgat-Duvaut Ill. The above list is far from exf-austive; numerous other applications also exist in biomathematics, mathematical economics and in semi-conductor physics (see Hunt-Nassif /lA/, etc.)
.
3.4
Interpretation of (3.1A), (3.2A) as a free boundary problem.
Let u be the solution of (3.1A), (3.2A); we then define
no
= {xlx E n
y =
u(x) = $(x)}
an+ n aao
and finally
u+ = u(
uo =
n+
UI
no
.
A classical formulation of (3.1A), (3.2A) is then: Find u and y ( t h e f r e e boundary) such that (3.3A)
-Au = f on R
(3.4A)
u =
(3.5A)
u = g on ,'I
I/J
on R
0
4-
,
556
Steady-state i n e q u a l i t i e s
(APP. 1)
The physical interpretation of (3.3A) to (3.6A) is as follows:
(3.3A) means that the membrane is s t r i c t l y above the obstacle and (3.4A) means that on $lothe has a purely e l a s t i c behaviour; membrane is on contact with the obstacle; (3.6A) is a transmission condition at the free boundary. In fact (3.3A) - (3.6A) are not sufficient to characterise u and y and it is necessary to introduce supplementary conditions concerning the behaviour of u on y o r alternatively the global regularity of u; for example, if JI is sufficiently regular, we could require the "contiquity" of Vu on 1 y (more specifically we could impose Vu E H ($2) X H ($2)).
Remark 3.1: In Chapters 1 - 6 we have already investigated other variational inequality probfems , interpreted as free-boundary problems. 3.5
Existence, uniqueness and regularity of the solution of (3.U) .(3.2A).
With regard to the existence and uniqueness of a solution of (3.1A), (3.2A) , we can easily prove Theorem 3.1:
Suppose that
r
i s regular, and that
$ ~ H l ( $ 2 ) , gEH1'2(r) with $ I r ' g a . e . on r ; then (3.1A), (3.2A) admits one and only one solution.
Remark 3.2: We already have existence and uniqueness for functions f and $ which are much less regular than those introduced in Theorem 3.1. W With regard to the regularity of u we recall the results below, which are due to Br6zis-Stampacchia /2/ If
r i s s u f f i c i e n t l y regular and
we have f g="gr
3.6
E
LP(Q) n ( H I ($2))'
with
ZE?'~($~),
,JIE
i f for p
E
11 *+ = C
4 *P(Q)
then u ~ b ? ~ ~ ( Q ) .
Finite-element approximations of problem (3.1A), (3.2A). (I) Piecewise-linear approximations
In this section we shall consider the approximation of (3.1A),
(3.2~)by a first-order conforming finite-element method (i.e. using continuous , piecewise-affine approximations). Approximation by second-order conforming finite elements will be considered in Section 3.7.
Obstacle problems
(SEC. 3 )
-
557
1
I n t h e f o l l o w i n g we d e s c r i b e t h e results o f Brezzi- Hager- Raviart ( ' ) / l A / , t h e r e b y g e n e r a l i s i n g t h o s e o b t a i n e d i n C h a p t e r 1, S e c t i o n 4.7 f o r t h e s p e c i a l c a s e JI = 0, g = 0. We assume f o r s i m p l i c i t y t h a t n i s a bounded polygonal domain i n IR2. We a l s o assume t h a t
(3.8A)
ncO(E)
$EH'(Q)
, gEH1'2(r)
nco(r>.
The n o t a t i o n i s t h a t of Chapter 1, S e c t i o n s 4 . 1 and 4.7, e x c e p t t h a t h w i l l now r e p r e s e n t t h e l e n g t h of t h e longest s i d e of Lhe t r i a n g u l a t i o n Suppose t h e n t h a t % i s a t r i a n g u l a t i o n C h a p t e r 1, of Q which s a t i s f i e s c o n d i t i o n s (4.7) - (4.9) S e c t i o n 4.1, a n d ( 4 . 5 1 ) o f Chapter 1, S e c t i o n 4.7, i . e .
%
u
(3.9A)
T =
09
n.
TECh We t h e n i n t r o d u c e
Ch = (P
,P
E
0
ch
= {P E Ch
T
a v e r t e x of
,PB
r}
E
ch},
,
=ChnQ
and we approximate H l ( s 2 ) and K by
(3.10A)
Vh = { v ~ E 0C(Q)
( 3 . 1 1A)
5=
{vh E Vh
where i n (3.11A)
, and
v h l T e P1 v T e C h }
, vh(P)
t $(P)
VP
E
Ch, vh(P) = g ( P ) V P E Chn
h e r e a f t e r , f o r k t 0 we have:
Pk i s t h e space o f polynomials i n two v a r i a b l e s o f degree Ik. We t h e n approximate (3.1A)
If
$1
5
g
w e have
P r o p o s i t i o n 3.1:
5#
, (3.2A)
0
by (3.11A) and
and it i s t h e n c l e a r t h a t :
The d i s c r e t e o b s t a c l e problem a d n i t s one and
only one s o l u t i o n . ( ' ) H e r e a f t e r a b b r e v i a t e d t o B.H.R.
r
5 58
Steady-s t a t e i n e q u a l i t i e s
(APP. 1)
Concerning the convergence of the approximate solutions when h + 0 , see Glowinski /1A/ for the case in which the solution u is not very regular. In the following we shall estimate the approxwhen u,JI E H2(n>, a E H2(R>; we imation error Ilu, - u I I Hl(S2)
follow very closely the investigation of B.H.R. / 1 A , Section (in which a(u,v) = J R ~ u * dx ~ v + jnuv dx). For this we use:
Assme that u
Lemma 3.1:
-
(3.13A)
-Au
(3.14A)
(-Au-f)(u-@)=
Proof:
f 20
E
4/,
H2 (52) ; then
a.e. on R a.e. on a .
0
See Br6zis /2/.
Lemma 3.2: Let u and yn be t h e r e s p e c t i v e s o l u t i o n s of (3.1A), (3.2A) and ( 3.11~) ( 3.12~). We then have
VvhE
( w i t h a(v,w)
VV,WE H
dx
=
1
Kh
(SZ))
..
(3.15A)
a(yl-u,yl-u)
Proof:
We follow B.H.R.
a(yl-u,y,-u)
/1A, Theorem 2.1/.
We have Vvhc I$,
h ) = a(y,-u,vh-u)+
= a(%-u,v,-u)+a(%-u,%-v
(3.16A)
+
a(u,vh-%)
-
f(vh-yl) +
)dx -a(%,vh-%).
s2
Since u satisfies (3.12A) we have h
I,
f (vh-%)dx
- a(ylyvh-%)
I 0 Vvh
E
\,
which combined with (3.16A) implies (3.15A). We thus have 2 2 H (Q), g=glr w i t h E H (SZ),and i f the angles of %hare bounded below by eo > 0 , independent o f h, then Theorem 3.2:
where u and
If f E L2(52') ,$
% are
E
t h e r e s p e c t i v e s o l u t i o n s o f (3.1~) (3.2~)and
(3.11A1, (3.1 A).
Proof:
Once again we follow B.H.R.
/1A, Theorem 4.1/ (see
Obstacle problems
(SEC. 3) a l s o Falk
(3.18A)
/lA/).
1
559
We h a v e , from G r e e n ' s i d e n t i t y
Auv dx +
a(u,v) =
It t h e n f o l l o w s from vh-%
(3.19A)
-
E
1
H,,(Q)
Vv
jrg
v dr
E
h K h
and
1
H (Q).
(3.18A) t h a t
Let n h b e t h e V h - i n t e r p o l a t i o n o p e r a t o r on C h ' i . e . o p e r a t o r d e f i n e d by 1
V E
(-Au-f)(v h-% ) d x V V h E % '
a(u,vh-uh)-
rh : H
V
(n) nco(C)
+
the
vh ,
1
V V E H (Q)n Co(C) , V P E Ch w e have
71 v ( P ) = v ( P ) . h 2 We h a v e R c IR , and h e n c e H Jfi) c C o ( z ) w i t h c o n t i n u o u s t h e c o n d i t i o n s U E H ( Q ) , u > $ on R, u = g on r , injection; t h e n imply T ~ U E
2
\.
We t a k e vh = ?rhu i n
(3.20A)
a(%-u,%-u)
I
(3.15A), (3.19A) and h e n c e o b t a i n a(\-u,nhu-u)
+
1,
(-Au-f) (nhu-y,)dx
.
We n o t e t h a t
( 3 . 2 1 A) Let w =
'rrhu-\
= ( T ~ u - u+ ) (u-$) + ( $ T ~ $ )+ (Th$-%)
-
2 -nu - f ; w e h a v e w E L (Q) and w e deduce from (3.21A)
From Lemma 3.1 we h a v e w 2 0 a.e. and more- w(u-$) = 0 a.e.; over s i n c e E It t h e n f o l l o w s from we have . ' r h $ - ~ s O m R . ( 3.22A) t h a t
u,,
From t h e c o n d i t i o n on t h e angles of w e h a v e , s i n c e u , $ E H 2 (Q)
s t a t e d i n t h e Theorem,
560
Steady-state inequalities
(APP. 1)
where, i n (3.2bA); (3.25A), C denotes various q u a n t i t i e s which a r e independent of h,u,$. We w r i t e
1.1
=
I ,Q
(1,
then r e s u l t s from (3.20A)
(3.26A)
I uh-uI
I V V ~ ~ ~ X )and ~ ' ~ IIvII
= IIvII
,
(3.238)
- (3.25A)
j
H
1,Q
it
(Q)
that
= O W .
w e note t h a t , s i n c e i-2 i s I n o r d e r t o e s t i m a t e IIy,-uII 1 ,n bounded, we have
(3.278)
I I v I I ~ ,< ~ C
1 l ~ l ~V v, c~H o ( Q ) ,
C independent o f v . 1
It t h e n follows from (3.26A) and from Y , ' ~ ~ u E H ~ ( Qt)h a t
Comments: Concerning t h e approximation of t h e o b s t a c l e problem by f i r s t - o r d e r f i n i t e elements, t h e O(h) e s t i m a t e w a s obtained by Falk /l/, and then by Mosco-Strang /1/ ( s e e a l s o Chapter 1, Section 4.7). I n Falk /2/, / 1 A / t h e v a l i d i t y of t h i s e s t i m a t e w a s shown under q u i t e g e n e r a l assumptions on i-2 ( t h e r e s u l t s o f The problem Falk / 1 A / were considered f u r t h e r i n C i a r l e t / l A / ) . .of o b t a i n i n g L2 e s t i m a t e s of optimal o r d e r ( i . e . O ( h 2 ) ) of \-u v i a a g e n e r a l i s a t i o n of the,Aubin-Nitsche method has not y e t been completely resolved; f o r incomplete r e s u l t s i n t h i s d i r e c t i o n we r e f e r t o N a t t e r e r /1A/ and Mosco / l A / . To conclude our discussion of piecewise-affine
approximations,
it should be pointed out t h a t using adequate assumptions Baiocchi /2/ ( r e s p . Nitsche /YL/)have been a b l e t o o b t a i n f o r t h e o b s t a c l e problem
11 %-uI I
=
O(h 2 -€ ) ,
E
>O
a r b i t r a r i l y s m a l l (resp.
Obstacle problems
(SEC. 3 )
-
561
I
3.7 Finite-element approximations of problem (3.1A), (3.2A) (11) Piecewise-quadratic approximations With C
0
h
and C
h
as in Section 3.6, we define
E, P
C ' = {P E
h triangle
T E
0
z;
= {pEz;
the midpoint of a side of a
'Ch} , , P $ r), , C;I = C h U C i 0
1;: =
ChUCi
0
.
0
We then approximate Hl(0) and K by (3.29A)
Vh = { v e C 0 ( E ) , v h l T € P 2 v T e C h }
(3.30A)1
\\
(3.30A)2
E;,
2
,
0
= EvheVh,
vh(P)2+(P)
\PEE{
, vh(P)
VPEC;lnT)
= g(P)
0
VPc C i
= { v h e V h , vh(P) > $ ( P I
, vh(P)=g(P)
We remark that in the condition vh(P) > $ ( P ) required at the midpoints of the sides.
V P E C;:nl')
is only
Finally we approximate the obstacle problem (3.1A), (3.2A) by
Find
4f E
such t h a t
(3.3 1 A)
11,
i i Vu,,*V(v,-u,,)dX
where i = 1, 2. If
$Ir
s g we have
0 ,
WS'"(Q)
E
L"(Q), u,$
Vae]I
E
,+4 a d
d'O0(52 and ) if s 0 ) be t h e displacement imposed on t h e stamp i n t h e d i r e c t i o n p a r a l l e l t o t h e z-axis. Let G be t h e c o n t a c t domain, which i s not known a p r i o r i . w(x) denotes t h e displacement normal t o t h e p l a n e z = 0 and u ( x ) t h e normal component of t h e stress t e n s o r . A s i n c l a s s i c a l t h e o r y , w and u a r e r e l a t e d by
k >O
(5.2A)
1x1
=
d
x.
The c o n d i t i o n s on z = 0 d i f f e r according t o whether one i s i n s i d e G or o u t s i d e G : i n G we have
(5.3A)
w ( x ) = $ ( ~ ) - 6 and
a(x) S O
,
and o u t s i d e G we have: (5.4A)
w 0 ,
1 u [ k -*u
- (&-+)I =
I "I
0 i n IR
which i s an "obstacLe-type" problem
2
,
-
except t h a t t h e secondorder e l l i p t i c o p e r a t o r which normally appears has here been replaced by the convolution operator
(5.7A)
1
Au=k-*U.
I "I
We n o t e t h a t problem ( 5 . 6 A ) can be formulated f o r a bounded open domain ( a t l e a s t with a stamp of "reasonable shape"). I n f a c t , i f u 2 0, t h e n Au 20 and t h u s Au > 6- 4 when 6-$(x) < 0 ; ) u = 0 when 6-$(X) < 0 , t h u s , using t h e l a s t of conditions ( 5 . 6 ~ , It i s t h u s s u f f i c i e n t t o consider t h e problem analogous t o (5.6A) i n an open domain fi containing the region "6-@(x) < O " i n i t s i n t e r i o r , which i s p o s s i b l e with Q bounded i f t h e stamp z = $(x) i s "reasonable" ! We t h u s put
(5.8A)
6-6 = f
.
We f i n a l l y have t h e problem:
domain i n n2 (5 .9A)
, with
u20 ,
Au-f 2 0
Find u i n Q , a bounded open
, u(Au-f)
=
0 i n $2
where f i s given, with f < 0 i n the neighbourhood of the €owz&ry of n, and where i n (5.7A) u i s understood t o be extended by zero outside Q. 5.2
Functional formulation
We denote by 3 t h e F o u r i e r transform;
-2'ixs$ We n o t e t h a t (5.1
1
OA) =
151
formally, we have:
(x) dx
.
Stamp problem
567
so t h a t
( 5 . 1 IA)
~(Au)= k
-Q
.
I51 Consequently, i f
+
and
JI €,&(a) , we have
It i s t h e n n a t u r a l t o i n t r o d u c e :
1
E = completion of
H(n)
f o r t h e norm
t h i s space i s one of a family o f spaces o f ' f r a c t i o n a l Sobolev' t y p e , i n v e s t i g a t e d elsewhere. More g e n e r a l l y , f o r s > 0 , w e i n t r o d u c e ( s e e L. Hormander and J . L . Lions /lA/):
2(Q) = completion of (5.1 4A)
(1
1512'
&R)
f o r t h e norm
1G(5)12d5)1/2.
IR2
W e must t a k e n o t e o f t h e f a c t t h a t it i s not always p o s s i b l e t o i d e n t i f y t h i s space w i t h a subspace o f d i s t r i b u t i o n s on Q. However -1 / 2
(IR2 ) i s i d e n t i f i e d with a subspace o f
&
(5.15A)
( a n d , consequently,
i'/2(R)
8' (IR 2 ).
i s i d e n t i f i e d , f o r any Q , bounded o r
n o t , w i t h a subspace of &'(n)). To prove
j' 2 ( = 2 )
(5.15A), we c o n s i d e r a Cauchy sequence 4 n f o r ;
if
Q
4
dx =
I,
E
&lR2
),
w e have:
1C11/2 $n(lCl-1'2
$)dC
;
IR2 2 2 -1/2 s i n c e 2 )5 ) '"IRi s a Cauchy sequence-in L (lR ) and 15) 2 @ll i n L (IR ) we t h u s see t h a t @, JI dx converges. We n o t e t h a t
IlR2
.
JI i s
Steady-state inequalities
568
(5. I6A)
I
i 1 I 2 ( n )(a
E = t h e dual o f t h e space
j-I
i s g e n e r a l l y denoted by
(APP. 1)
12
space which
(n) 1
and from ( 5 . 1 1 A ) we see t h a t
A i s an isomorphism from j - ' l 2 ( s 2 )
(5. I7A)
onto B -+I '2(s2)
If ( f a $ ) denotes t h e s c a l a r product between .6-1'2(s2) - 1 12
B
(a),
(5.1 8A)
.
and
w e have:
(A$¶$) =kll$ll;
*
W e introduce
(5.19A)
K
= { v ) v ~ E v
~ iOn Sl
(9]
which d e f i n e s a nonempty c l o s e d convex s e t i n E. f i n d u E K such t h a t i s t h u s formulated:
(5.20A)
(Au,v-u)
where f i s given i n
2
(f ,v-u)
Vv
6
Problem (5.9)
K
8^'I2(n)
From (5.18A) t h i s problem comes w i t h i n t h e g e n e r a l t h e o r y of I t admits a unique solution. Variational Inequalities. 5.3
F i n i t e element approximat i o n
I n concept, any f i n i t e element method ( w i t h a s u i t a b l e approxThe d i f f i c u l t y l i e s imation o f K ) i s a p p l i c a b l e and convergent. i n t h e f e a s i b i l i t y o f calculating the c o e f f i c i e n t s . To t h i s end t h e following method i s proposed i n Bogomolnii, Eskin and Zuchowizkii /lA/.
We consider a p a r t i t i o n i n g o f RL i n t o squares of s i d e h a with edges p a r a l l e l t o t h e coordinate axes. We put
(5.21A)
$,(XI
= (l-lxJ)(l-
1x21>
i n t h e square lxll < 1 lx21 < 1 , and +o = 0 o u t s i d e t h e square. We t h e n consider t h e basis functions:
~~
(')
I n t h e sense o f d i s t r i b u t i o n s on Sl
(SEC.
6)
Nonlinear DirichZet problems
569
We assume t h a t Q i s approximated by t h e union o f s q u a r e s and t h e i r t r a n s l a t i o n s , and we approximate K by combina t i o n s of t h e 4 corresponding t o c o e f f i c i e n t s 20. This amounts t o calculating P
/xi1 < h
(5.23A)
(A($p.($q) = a p y q .
I n fact a depends o n l y on p-q: P39 (5.248) aPYq = aP = p-q where, a f t e r a simple c a l c u l a t i o n
(5.25A)
a
P
=
kh3 jj 2I.r
sin
eiP5
IR2
a
=
kh3
P
1 la1 S M
sin
-
1( 7 151
It i s p o s s i b l e t o c a l c u l a t e a (5.2618)
5, -
P
-
2 2 u s i n g an asymptotic formula
(-l)lal($a
+ O(-
D2a(L)
(PI
'
I P 12M+2
)
where we c o n s i d e r p as a continuous v a r i a b l e and t h e n t a k e p E i n t h e r e s u l t , and where t h e 4 are t h e Taylor c o e f f i c i e n t s a of t h e f u n c t i o n
2
We r e f e r t o Bogomolnii e t a l , l o c . c i t . , r e s u l t s obtained using t h i s technique.
6.
f o r numerical
SOLUTION OF NONLINEAR DIRICHLET PROBLEMS BY REDUCTION TO VARIATIONAL INEQUALITIES
6.1
Synopsis
I n t h i s s e c t i o n w e show t h a t t h e methodology of v a r i a t i o n a l i n e q u a l i t i e s can a l s o be u s e f u l i n t h e s o l u t i o n o f equations, We s h a l l i l l u s t r a t e a f a c t which h a s been used by h e 1 / l A / . t h i s u s i n g a f a m i l y of nonlinear Dirichlet probZems , c o n c e n t r a t i n g i n p a r t i c u l a r on t h e i r approximation by means o f a finite element method. We s h a l l omit most o f t h e p r o o f s , r e f e r r i n g for t h e s e t o Chan-Glowinski /lA/, Glowinski /2A/.
Steady-state inequalities
570
6.2
r
(APP. 1)
The continuous problem
L e t fi b e a bounded domain i n lRN (N t I ) = an; w i t h V = H A ( f i ) we c o n s i d e r :
w i t h a r e g u l a r boundary
-
L : V + IR, l i n e a r and continuous, i . e . L(v) = < f , v > V V E V , where f E V ' = H-I(S2) ( V ' i s t h e d u a l o f V and < * , * > i s t h e b i l i n e a r form o f t h e d u a l i t y between V' and V ) .
-
a : V X V +IR b i l i n e a r , c o n t i n u o u s , c o e r c i v e ; e x i s t e n c e of an a > 0 such t h a t
where
(1,
IIvIIv =
we t h u s have t h e
1VvI2 dx)1'2.
We do not assume a p r i o r i t h a t a ( * , * )i s symmetric.
-Q
: lR+IR,
$(t')
r$(t)
Q non-decreasing ( i . e .
QEC'(IR), Vt,t'EIR,
t ' rt),
Q(0) = 0.
We t h e n c o n s i d e r t h e nonZinear variational equation
(6.2A)
I
Find u
E
v such that Q (u)
E
L
1
(a) n V ' and
a ( u , v ) + = Vv E V .
Let A E&V,V') such t h a t a ( v , u ) = V V , W E V ; it i s c l e a r t h a t ( 6 . 2 ~ )i s equivalent t o t h e nonlinear Dirichlet problem
Au + Q(u) = f , (6.3A) U E V ,
Example: (6.4A)
Let
$(u) E L
1 '
(a).
a. ~ L - ( f l ) such t h a t
ao(x) 2 a > O a.e.
on fi
and l e t B be a constant vector i n IRN
.
We d e f i n e a ( * , * )bilinear and continuous on H 1 ( n ) x H A ( Q ) by
a(u,v) = We have
o;j
0
Vu*Vv dx +
I,
B-Vu v dx
.
Nonlinear Dirichlet problems
(SEC. 6)
571
which, combined with (6.4A) , implies a(v,v) t
\
2 lVvl dx
1 Vv e H O ( Q )
,
R i.e. the coercivity of a(*,*).
6.3
6.3.1
Existence and uniqueness results for (6.2~).(6.3A).
Introduction. A variationa Z inequa Z i t y associated with (6.2A), (6.3A). 1
If N = l we have H (Q)c C0@) with continuous injection and the proof of the exist%nce and uniqueness of a solution of (6.2~1, (6.3A) is then almost immediate. If N 2 2 , as we shall assume in the following the essential difficulty is precisely the fact that HL(Q) # Co(&). The analysis which follows can also be applied Remark 6.1: to problems in which V = H1(Q), or in which V i s a closed subspace of H1(Q). Let Q : IR-+IR be defined by t
O(t> =
;
$(T)dT 0
@
is clearly C1, convex and
20
.
We then define j ( - ) by
It is easily shown (see Chan-Glowinski /lA/, Glowinski /2A/) that : Proposition 6.1: The functional j ( - ) i s convex, proper and on L~(Q). B Assume that a( ) is symmetric; it would then be natural to associate with (6.2A), ( 6 . 3 A ) the following calculus-ofvariations problem : 2.s.c.
9 ,
Find U E V such that (6.6A) J(u) I J(v) V v
where
E
V
572
steady-state inequalities
(6.7A)
J(v)
=
(APP. 1)
T1 a(v,v)+j(v)-L(v>.
I f a ( - , * ) i s non-symmetric t h e above approach i s not f e a s i b l e , at l e a s t d i r e c t l y , b u t , on t h e o t h e r hand, w e can a s s o c i a t e with (6.2A), (6.3A) t h e E l l i p t i c Variational Inequality ( o f t y p e ( 1 . 2 A ) ) :
[ Find u ~ such v that (6.8A)
t
a(u,v-u)+j (v)-j (u) ~L(v-u) V v
E
V
which i s i n f a c t a g e n e r a l i s a t i o n o f (6.6A), (6.7A). A s regards t h e e x i s t e n c e and uniqueness o f a s o l u t i o n o f
(6.8A), i n view of P r o p o s i t i o n 6.1 w e can apply Theorem 2.1 of Section 2.2, which g i v e s :
Under the above asswnptions on a, L, 4, problem Theorem 6.1: (6.8A) admits one and only one soZution.
6.3.2
Equivalence between (6.2A), (6.3A) and (6.8A).
It i s p o s s i b l e t o prove: Theorem 6.2: The solution o f (6.8A) i s necessarily a solution of (6.2A), (6.3A). Conversely , any soZution of (6.2A), (6.3A) i s a solution o f (6.8A). The above theorem implies t h a t ( 6 . 2 ~ ) , ( 6 . M ) admits one and only one s o l u t i o n . Another consequence of t h e above equivalence i s t h a t , from t h e p o i n t of view o f t h e numerical solution, we can e i t h e r work d i r e c t l y on (6.2A), (6.3A), or a l t e r n a t i v e l y consider problem (6.8A); i n f a c t , from t h e p o i n t o f view o f t h e approxima t i o n , it i s i n our opinion simpler t o work with (6.8A).
6.4
The f i n i t e element approximation of ( 6 . 2 ~ ) , ( 6 . 3 ) and
(6.8A). 6.4.1
Synopsis
W e s h a l l now approximate problem (6.2A), (6.3A) by working d i r e c t l y on t h e e q u i v a l e n t formulation (6.8A); however, w e should p o i n t out t h a t t h e c o n s i d e r a t i o n s o f S e c t i o n 2.3.2 cannot be a p p l i e d d i r e c t l y s i n c e assumption (2.228) ( i . e . j(-) continuous on V) i s not s a t i s f i e d .
(SEC.
6)
6.4.2
Nonlinear Dirichlet problems
Triangulation of R. prob Zem.
573
Definition o f the approximate
.
2 We assume t h a t R i s a polygonal ( ’ ) bounded domain i n IR o f t r i a n g u l a t i o n s of R which s a t i s f y We i n t r o d u c e a family t h e assumptions o f S e c t i o n 3.6 and w e approximate V = H1(.Q) by
(ch)h
0
1
Vh = {vh Ivh €C0(E),vh I ’ = 0 , V h I T E P I VT E Th}. It i s t h e n n a t u r a l t o approximate ( 6 . 2 ~ )(6.3A) ~ and (6.8A), r e s p e c t i v e l y , by:
Find
u,, e V h
(6.9A)
a(u,,,v,)
+
such t h a t
I
@(u,,)v, dx = L(vh)
VV,
cVh
R
and
It i s c l e a r t h a t t h e s e two finite-dimensional
problems a r e
equiualent. From a p r a c t i c a l p o i n t o f view it i s not g e n e r a l l y p o s s i b l e t o use (6.9~)and (6.10A) as t h i s r e q u i r e s t h e c a l c u l a t i o n of i n t e g r a l s which a r e i m p o s s i b l e t o e v a l u a t e e x a c t l y . It i s t h u s n e c e s s a r y t o modify (G.gA), (6.10~)v i a t h e use o f numerical i n t e g r a t i o n procedures. I n t h e f o l l o w i n g we s h a l l b e c o n t e n t t o approximate j ( * ) . (Remark 2.2 o f S e c t i o n 2.3.1 a l s o a p p l i e s i n t h i s c o n t e x t ) . Using t h e n o t a t i o n o f F i g u r e 6.1 below, w e approximate j ( * ) by j h ( * )d e f i n e d by
I n f a c t we could a l s o c o n s i d e r j ( v ) as b e i n g t h e exact integral of a piecewise-constant f u hn c t hi o n ; more p r e c i s e l y , i f we assume t h a t t h e s e t C of nodes o f %h i s o r d e r e d from 1 t o N = Card (1 ) , t h e n w i t k each Mi E C h w e a s s o c i a t e t h e s e t $2. h
6.2) o b t a i n e d by j o i n i n g t h e centre; Of t h e t r i a n g l e s w i t h common v e r t e x M. t o t h e midpoints o f t h e edges 1 which m e e t a t Mi. ( h o r n hatched i n F i g u r e
(1)
This assumption i s n o t e s s e n t i a l .
Steady-state inequa Z i t i e s
574
( U P . 1)
M3T
F i g . 6.1
F i g . 6.2
(If. Mi
E
r,
t h e m o d i f i c a t i o n r e q u i r e d i n Figure 6.2 i s t r i v i a l ) .
W e t h e n d e f i n e a space
where
x.1
4,of
piecewise-constant
i s t h e c h a r a c t e r i s t i c f u n c t i o n of 52 i , i . e .
xi(x)
= 1 if
xiS2.
1 '
.
xi(x) = 0 if x B Q i 1 We next d e f i n e qh : Co(E)n Ho(S2) + Lh by Nh (6.13A)
f u n c t i o n s by:
qhv =
1
i=l
v(Mi)Xi
.
(SEC. 6 )
Nonlinear Dirich l e t prob Zems
57 5
It t h e n f o l l o w s from ( 6 . 1 1 ~ )- (6.13A) t h a t
and hence t h a t
an a
Find u,,eVh
such t h a t
(6.17A)
a(U,,,Vh-Uh)+jh(Vh)-jh(U,,)
'L(vh-\)
Vvh
'h
'
It c a n t h e n e a s i l y b e shown t h a t ( 6 . 1 6 ~ )and (6.17A) m e equivalent and a h i t one and only one s o l u t i o n .
Four Zemas
6.4.3.
(5)
With t h e a i m o f p r o v i n g t h e convergence o f t o u when h h + 0, we s h a l l u s e t h e f o u r lemmas below, which a r e proved i n Chan-Glowinski / 1 A / and Glowinski /2A/: Lemma 1 . 6 : (6.18A)
1
Let v e H0 ( R ) ; for n e N we d e f i n e
T v = inf(n,sup(-n,v)).
n
we then have
lim n++ 03
T v = v
n
I
s t r o n g l y i n H (R) 0
T ~ V by
576
Steady-state inequalities
( U P . 1)
( 6 . 2 ~ ) ,(6.3A)) i s The solution u of ( 6 . 8 A ) (d
Lemma 6.2:
characterised by 1
1
U E H ~ ( Q > , @(u) E L
(a),
(6.19A) ~ ~ ( v - u ~v )
a(u,v-u)+j(v)-j(u)
I
EH~(Q)
nLoo(n).
Lemma 6.3: The space 8(Q) i s dense i n Hb(Q) n LOD@) , t h i s l a t t e r space being provided with the strong topology of V and the weak* topology o f ~ ~ ( f i ) .
Lemma 6.4:
For a l l p such that 1 < p i *
we have
Lemmas 6.1, 6.2, 6.3 are d i r e c t consequences of c l a s s i c a l r e s u l t s on t h e truncation of f u n c t i o n s of H1(Q) (and H A ( f i ) ) , for which w e refer t o Stampacchia /lA/.
6.4.4
Convergence of the approximate solutions
Regarding t h e convergence of t h e approximate s o l u t i o n s when h + 0, w e have
We asswne that when h + O the angles o f c h a r e Theorem 6.3: we then have bounded below by a constant g o > 0 . (6.21A) (6.228)
lim h+O
III+,-UII~
=
0,
l i m jh(I+,) = j ( u ) h+O
where u i s the solution of (6.2A), ( 6 . 3 ) and (6.8A) and that of ( 6 . 1 6 ~ )and (6.17A).
yn i s
Proof: We proceed as i n Chan-Glowinski, l o c . c i t . , Glowinski /2A/; . t h e praof i s along t h e same g e n e r a l l i n e s a s t h a t of Theorem 2.3, S e c t i o n 2.3.2. 1)
A prior; estimates for
I+,
W e t a k e vh = 0 i n (6.17A); f a c t t h a t j,(O)
(6.23A)
I.,
= 0 1
IILII.
9
we t h e n deduce from ( 6 . 1 ~ ) and t h e
(SEC. 6 )
where
IILll*=
2)
577
Nonlinear Dirichlet problems
.
sup
v€v-Io)
Weak convergence o f ( % ) h a
I t r e s u l t s from (6.23A) and from t h e compactness of t h e i n j e c t i o n from V = H1(Q) i n t o L2(Q) t h a t w e can e x t r a c t from 0 ( L + , ) ~ a subsequence - also denoted by ( u ~ -) such ~ that = u*
(6.258) lim
weakly i n V
WO
(6.26A) lim h+O
%
= u*
strongly i n L2(Q)
(6.278)
%
= u*
a.e. on Q .
lim W O
I n view of (6.23A) s i m i l a r l y have
- (6.26A)
and Lemma
6.4
(with p = 2 ) w e
(6.28A) lim qhu,, = u* strongly i n L2(Q).
NO We t h u s have, up t o a new e x t r a c t e d subsequence
(6.296) lim qhu,, = u* a.e. on Q
N O (6.30A) lim Q(qh%) = $(u*)
a.e. on a .
h+O Let V E B(Q) j it t h e n r e s u l t s from e.g. Ciarlet /1A/ t h a t under t h e assumption on t h e angles of b given i n Theorem 6.3, h w e have
where C denotes v a r i o u s q u a n t i t i e s which are independent of v and h , and where r i s t h e interpolation operator on't: defined by
h
h
(APP. 1)
Steady-state inequalities
578 vv
n01 (a) n co(Si) we have
E
r,v(P)
6.4 (with
lim kt0 Taking vh = rhv i n
E V ~and
VP E C ~ .
= v(P)
14oreover , Lemma
rhv
p = +
m)
implies
(6.336)
\ - L(rhv-yl) From
(6.358)
(6.17A) w e
t/vE B(Q)
(6.25A), (6.26~)and
then o b t a i n
.
Proposition
6.1, w e
deduce
* *
a(u ,u )+j(u*) 2 l i m i n f (a(u,,,y,)+j(q,u,,)). h + O
I n view o f (6.24A), (6.30A) we can apply Fatou's Lema t o {$(qh\))h from which w e o b t a i n :
(6.36A) (p(u*) E L 1 ( 5 2 ) . Moreover, we have
lim h+o
@(qhrhv)dx =
n
a(u* ,u*)+j (u*)
It t h e n follows from u*
@ ( v ) d x= j ( v )
(6.35A), i m p l i e s ,
which, with
(6.378):
1,
E
v,
5
V V E B(n)
i n the l i m i t in
(6.36A), (6.37A) t h a t
@ (U*) E L
1
(6.34A) vv
a(u* , v ) + j (v)-L(v-u*)
,
E
&n).
u* s a t i s f i e s
(521,
(6.38A) a(u* ,v-u*)+j (v1-j (u*) L ~ ( v - u * ) tlv
E
an).
w e deduce from Lemma 6.3 t h e e x i s t e n c e of Let v E V n Lm(s2); a sequence {vnIn such t h a t vn E J ( Q ) V n and
vn = v strongly i n V,
(6.39A)
lim n-
(6.40A)
l i m vn = v weakty* i n L m ( Q ) . n-t+oo
(SEC. 6)
From
579
Yonlinear Dirichlet problems
(6.38A) we u* E V ,
deduce
O(U*)
EL1(,)
( 6 . 4 1 A)
~ ~ ( v ~ - uv* n. )
a(u*,vn-u*)+j(vn)-j(u*)
(6.39A)
Moreover (6.42A)
lim
a(u*,v -u*) n
n* (6.43A)
implies
lim
L(vn-u*)
=
a(u*,v-u)
= L(v-u
,
*) ,
n* and, up t o a n e x t r a c t e d subsequence (6.44A)
lim n++co
vn = v a.e. on R ,
which i m p l i e s (6.45A)
a.e. on R.
lim O(vn) = O(V) n*
(6.40) we have
I n view of
and hence Vn : (6.46A)
0 5Q(vn)
I n view of
a.e. on R.
5 const.
(6.45A), (6.46A)
we can a p p l y t h e Lebesgue dominated
convergence theorem, which g i v e s : lim n++W
In t h e l i m i t i n
u* (6.47A)
E
*
I
@(v )dx =
( 6 . 4 ~ )we
t h u s have
j(v ) = lim n n++m
v,
O(U*)
EL
1
52
@(v)dx
= j(v)
.
(a),
a(u ,v-u*)+j (v)-j(u*> z ~(v-u*) v v
E
v n Lm ( 0 )
.
5 80
Steady s t a t e i n e q u a l i t i e s
( U P . 1)
From Lemma 6.2, (6.47A) implies u*= u where u is the solution of (6.2A), (6.3A) and (6.8A). In view of the uniqueness of u it is the entire sequence ( ~ h which ) ~ converges weakly to u.
3)
Strong convergence o f
(%Ih.
From (6.17A) and from the coercivity of a(=,-), it results that v v E &a) :
Using the various convergence results from the second part of the proof, we have, in the limit in (6.48A) < lim inf jh(%) I lim inf(a
11 s - u I I v2 +jh(%))
I
slim sup(a I(\-uIIV 2 + j h U ( h) ) s
Using, as above, the density of B(n) in V n Lm($), it can be shown that ( 6 . 4 ~ )also holds for all the v E V n L (Q); with T defined by (6.18A) we then have: n 2 j ( u ) slim inf j h ( \ ) slim inf(a IIu,,-ulIv + j h (% ) ) I
(6.50A)
I
I
lim sup (a II%-uIIv 2 + j,(%))
I
a(u,-r,v-u) +j( T ~ v -L(T~v-u) )
5
v v E V, v n .
In view of the properties of T (see Lemma 6.1) we have, in the n limit in ( 6 . 5 0 ~ )when n +. + m :
(SEC. 7 )
Quasi-variational inequa Z i t i e s
581
Taking v = u i n ( 6 . 5 1 ~ )w e t h u s deduce
which proves t h e Theorem. Descriptions of various i t e r a t i v e methods which can be used t o ) ( 6 . 1 7 ~ ) can be s o l v e t h e equivalent approximate problems ( 6 . 1 6 ~ , found i n Chan-Glowinski, l o c . c i t . , which a l s o contains t h e r e s u l t s of various numerical t e s t s .
7.
INTRODUCTION TO NUMERICAL ALGORITHMS FOR QUASI-VARIATIONAL INEQUALITIES ( )
7.1
Quasi-variational i n e q u a l i t i e s
Consider, i n an open bounded ( 2 ) domain R , an o p e r a t o r
with t h e assumption of coercivity on H1( 0 ) ( 3 , :
where
(7.3A)
a(u,v) =
I '52 aij
av J
dx
+
I/
1
ai
52
zi aU v ax + /52aouv ax.
We a l s o introduce a nonZinear operator J,
+
WJ,) m
OD
from L y ( Q ) + L + ( Q ) where L + ( R ) denotes t h e s e t of f u n c t i o n s ( a . e . 2 0 ) i n L m ( R ) ; we assume t h a t t h i s o p e r a t o r M has t h e (l)
Abbreviated t o Q.V.I.
(2)
I n ' o r d e r t o c l a r i f y t h e concepts.
(3)
This assumption can be weakened and w e can consider a - H1 (n) H i l b e r t space V with HL(R) 5 V c
.
582
Steady-state inequalities
(APP. 1)
following p r o p e r t i e s :
(7.4A)
I
M i s positive increasing i .e. 0 “4J,
‘ J I , # 0 s M ( J I l ) <M(J12)
,
Under t h e s e c o n d i t i o n s , t h e r e e x i s t s one and only one function, u , which is a solution of the Q.V.I.
a(u,v-u) 2 ( f ,v-u)
v v with 0 6 v sM(u)
,
(7.6A) O 0 such t h a t a ( v , v ) ? ctllvll VVCV).
where f
It then follows from t h e above p r o p e r t i e s t h a t t h e minimisation problem
(2.4A)
I
Find J(u)
U E
S
K such that
J(v) V v e K
admits one and only one s o l u t i o n c h a r a c t e r i s e d by
(2.5A)
I
a(u,v-u) 2 ( ( f ,v-u))
VVE K.
Block o v e r r e k c a t i o n with projection
(SEC. 2)
589
Description of t h e algorithm
2.2
We n o t e t h a t
(2.6A)
J(v) = J(v I,...~N) =
T1
N
1
- 1
aij(vi,vj)
I < i ,jsN
((fi,vi))
i= 1
where f i e Vi and where t h e a i j a r e continuous b i l i n e a r forms on Vi
x
V
j'
t h e aii being symmetric and coercive (with
V v i e V i ) and, more g e n e r a l l y
aii(vi,v.)2a llv.112 1 i i
By using Riesz's theorem we deduce from (2.7A) t h e e x i s t e n c e of o p e r a t o r s A.. E k ( V . ,Vi) such t h a t
J
IJ
(2.8A)
aij(vi,vj) = ((vi,A. .v.))~ J
13
* , Aij -- *ji
9
where t h e Aii (which are s e l f - a d j o i n t ) are isomorphisms from V. onto Vi. As t h e forms aii are sylrnnetric and coercive, t h e y d e h n e a H i l b e r t s t r u c t u r e on V whose norm defined by
111 111
i'
i s equivalent t o t h e norm
II.II
we s h a l l denote by P. t h e
;
orthogonu2 projection operator from V. onto Ki according ko t h e
111 111. .
B
NOW l e t u s consider positive scalars 'to solve (2.4A) we use t h e following g e n e r a l i s a t i o n o f algorithm (1.41), (1.42) of Chapter 2:
norm w
,..:.wN;
(2.1 OA)
u
0
=
...+I
~uy),
given a r b i t r a r i l y i n K
then for n 2 0 successivezy determine ur'l, (2.11A)
'+:u
=
P.(u?-w.Ay!( 1
1
1 1 1
1
Aijuj m+l j
+
.., N, by
i = 1,.
1
j2 i
A
u?-fi)).
ij J
Remark 2.1: A v a r i a t i o n a l d e s c r i p t i o n of algorithm (2.10A) (2.11A) i s given i n Comincioli /l/. 2.3
Convergence o f algorithm (2.10A), (2.11A)
The following theorem i s proved i n Cea-Glowinski
...N,
Theorem 2.1: If 0 < wi < 2, v i = l , IunIn defined by (2.10A), (2.11A)
/2/:
we have f o r the sequence
(APP. 2)
optimisa tion algorithms
590
lim IIun-uII
=
o ,
n-
where u is the solution of (2.4A). 2.4
Various remarks
Remark 2.2: N
I n t h e case i n which dim V < +
,..
-, with
II Vi, and i n which V i - 1 .N w e have W i = w > 1 ( r e s p . i=l w = 1, w < l), algorithm (2.10A), (2.11A) coincides with t h e standard block overrelaxation ( r e s p . relaxation, underrelaxation) K = V =
algorithm a s s o c i a t e d with t h e s o l u t i o n of t h e l i n e a r system defined by Au = f ( s e e Varga 111, Young /U/) with A E L(V,V) a(u,v) = ((Au,~)) V U,VE V.
Remark 2.3: I n t h e case i n which V = lRN , we can i d e n t i f y t h e above o p e r a t o r A with an NxN symmetric p o s i t i v e - d e f i n i t e matrix. K =
If
N II 1 Ki with i=
Ki
=
{vila.+(pn,$(un+'
)-@(Un)),
=
+
n+ 1
+
(P
Since t h e v e c t o r un is a s o l u t i o n of t h e minimisation problem (3.6A)
Min d(v,pn) veM
we have (3.7A)
(J'(un) ,v-un)+(pn,$(V)-b(Un)),~
0 VV€M
0
and hence
(JA( u")
,un+' -u")
+ (pn ,$ (un+l ) -$ (u") )
which i n conjunction with (3.5A) i m p l i e s
,0 2
@timisation a Zgori thms
592
L(un+l
yp
n+l
)-l(U"YP")
2
(UP. 2 )
n+ 1 (P
or a l t e r n a t i v e l y
e(un+I ypn+l)-du n yp n ) z (3.8A) (Pn+I -pn ,Q
) -Q (u") +, (pn+I -pn
Y
0 (u")
)
,.
From t h e p r o p e r t i e s of t h e p r o j e c t i o n s , w e have
(q-pn+l ,pn+pQ(un)-pn+l), and hence i n p a r t i c u l a r which we deduce
I0
VqE A
,
(pn-pn+l ,pn+pQ(un)-pn+'), n+l
(3.9A)
S 0
from
n
Taking ( 3 . 9 A ) i n t o account , ( 3 . 8 A ) implies
(3.1 OA)
I
e(un+l
YP
n+l
)-du",p")
2
n+ 1 It now remains t o estimate (p -pnyQ(un+l)-Q(~n))L
p"+2 = P*(p n+ 1
+PQ(U"+"
; we have
Y
and hence, s i n c e t h e p r o j e c t i o n i s contractive
n+l
n
(3.11A)
or a l t e r n a t i v e l y ( 4 being Lipschitz continuous with constant C,)
Duality methods
(SEC. 3 )
593
By addition we then deduce from ( 3 . 1 0 A ) ,
(3.12A)
and hence by summation and 'dnt 0 L(U"+l
rpn+l)-yUO,p 0)
2
(3.13A) 1
n
n
"
-
2
We shall now show that under condition ( 3 . 1 A ) the sequence is convergent: in fact from ( 3 . 7 A ) we have
j =O (J:, (u") (J;(un+')
,un+' - u ~ ) ~(pn,$ + ( un+I ) -$ (u") ,un-u n+ 1 ),+(p
)
2
0
n+ 1
and hence, by addition (pn-pn+l ,$ (Un+l ) -$ (un) ) t ( J:, ( bn+l -J; (un) ,un+l -un)
(3.14A) n+l
2
4lu
which in conjunction with ( 3 . 1 1 A ) implies
and hence (by summation)
U'
n 2
IIv
Optimisation algorithms
594
(APP. 2 )
The proof o f Theorem 4.1 i n Chapter 2 , Section 4 . 3 shows t h a t under condition (3.1A) t h e sequence (pnIn i s bounded i n L; ( 3 . 1 5 8 n then implies t h a t t h e sequence u j + l - u j ;In i s convergent. 3'0 In-order t o prove t h e convergence of t h e sequence
1 11
11
f
, and hence t h a t lim IIpn+ 1 -pnlIL= 0 , it Ilpj+l-pjll2) L n n++w j=O remains t o show t h a t t h e term on t h e left-hand s i d e of t h e inequn s i n c e t h e sequence ( u In ( r e s p . a l i t y (3.13A) i s bounded; {p In) i s convergent ( r e s p . bounded) t h i s follows simply from t h e We c o n t i n u i t y of L ( * , * ) and t h e l i n e a r i t y o f q -t ( q , + ( v ) ) , . have t h u s proved ( 3 . 2 A ) . {
We s h a l l now use OpiaZ's Lennna i n o r d e r t o prove (3.3A) ; l e t M be t h e s o l u t i o n of t h e primal problem
UE
Inf SUP L(v,q) veM q e A and l e t
X
= (p E
A
, ( u , p ) is a saddle point
of d on M x A ) .
The proof of Theorem 4 . 1 i n Chapter 2 , Section 4 shows t h a t i f condition (3.1A) i s s a t i s f i e d we have convergence of t h e sequence { Ilpn-pllL)n Vp E X I n o r d e r t o be a b l e t o apply O p i a l ' s
.
Lemma, it t h e r e f o r e remains t o prove t h a t any weakly convergent subsequence {pnklk e x t r a c t e d from {pn) has i t s l i m i t i n X . We n know t h a t uX, p"k, p % + l s a t i s f y
I
(3.16A)
,"kEM,
(3.17A) 1p"k~A
.
\ Under condition (3.1A) we have lim unk = u strongly i n V , k++m
with U E M s i n c e M i s closed i n V;
lim k++m
pk
= p*
weakly i n L,
s i m i l a r l y we have
lim (p%+ip%)=o k++m
strongly i n L
(SEC. 4 )
Complementarity methods
(from (3.2A)) with p*
E
595
A s i n c e A i s convex and closed i n L.
In view o f t h e s e p r o p e r t i e s , w e have i n t h e l i m i t i n ( 3 . 1 6 ~ ,) (3.17A)
which shSjws t h a t {u,p*) i s a saddle-point of !s on M x A , and We can t h u s apply Opial's Lemma t o prove ( 3 . 3 A ) hence p E X thereby proving Theorem 3.1.
.
,
The above r e s u l t s a l s o hold for t h e sequence Rezark 3.1: {un,p In defined by algorithm ( 4 . 3 1 ) , (4.32) of Chapter 2 , Section 4.4.
4.
INTRODUCTION TO COMPLEMENTARITY METHODS
4.1
General remarks.
Synopsis.
This f o u r t h s e c t i o n c o n s t i t u t e s an i n t r o d u c t i o n t o complementarity methods, which have been widely used i n recent y e a r s f o r t h e numerical s o l u t i o n of v a r i a t i o n a l i n e q u a l i t y problems of t h e
obstacle problem t y p e ( s e e Appendix 1, Section 3 ) . The model problem f o r linear complementarity is as follows:
Given a matrix A of type N that Au-b 2 0, (4.1A)
x
N and
N , seek u eIRN such
b ER
u 2 0,
(Au-b,u) = 0 ; i%(4.1A) , ( , ) denotes t h e ordinary Euclidean i n n e r product. of IR , and i n g e n e r d
Dptimisation algorithms
596
v = {v, wtv
,. ..vN)
*
20
( U P . 2)
.
@ vi 2 0 Vi=I ,. .N,
w-vto. N
Hereafter we s h a l l use t h e n o t a t i o n Rf:= { V E R , v t 0)
.
It is c l e a r t h a t ( 4 . 1 A ) i s equivalent t o t h e variational inequ-
a l i t y problem
(4.2A) (Au,v-u)
2
N VV ER+
(b,v-u)
.
If A i s symmetric, ( 4 . 2 A ) is a necessary condition (and s u f f i c i e n t if A i s p o s i t i v e semi-definite) o f o p t i m a l i t y f o r t h e minimisation problem
(4.3A)
with
(4.4A)
1
J(u)
5
J(v)
N V v EX+
UER:
1
J(v)
(Av,v)-(b,v).
With t h e above problems we s h a l l a s s o c i a t e
a)
The least element problem
I
Find u E
(4.5A)
A
usv
where
(4.6A) b)
such t h a t
-
VVEK
N K = {vER+, Av-b
t 0)
The family o f l i n e a r p r o g r d n g problems
(4.8A)
(SEC. 4 )
Complementarity methods
5 97
where
(4.9A)
Sufficient conditions on A and p f o r all t h e above problems t o be equivalent a r e given i n Cryer-Dempster /1A/ ( s e e a l s o t h e i f t h i s equivalence can be e s t a b l i s h e d it bibliography t h e r e i n ) ; i s then p o s s i b l e t o s o l v e t h e v a r i a t i o n a l i n e q u a l i t y (4.2A) by pivoting methods i n s p i r e d by t h e simplex method of l i n e a r programming. We s h a l l show i n Section 4.2 (which i s i n s p i r e d by Cryer-Dempster, Zoc. cit.) t h a t t h e model problem f o r variationaz inequazities of
obstacle t y p e
where
(4.11A)
K
= {VC
1
H0 (521, v 2
o a.e.1
i s equivalent t o a l i n e a r complementarity problem (formally of t h e same type as (4.U))which i s i t s e l f equivalent t o an (infinitedimensional) linear programming problem. A d i s c u s s i o n of complementarity methods, t o g e t h e r with a number of r e f e r e n c e s , w i l l be given i n Section 4.3. 4.2
The o b s t a c l e problem from t h e p o i n t of view of compleme n t a r i t y methods
We follow t h e account of Cryer-Dempster /lA/. 4.2.1
Formulation of the problem.
standard results.
Let C2 be a bounded open s e t i n IRN with r e g u l a r boundary w e consider t h e variational inequality problem UE
K,
(4.12A)
vu*V(v-u)dx
2
VV E
K
r;
598
(APP. 2 )
Optimisation algorithms
where , i n ( 4 . 1 2 A ) , K i s given by ( 4 . 1 1 A ) between H-l(Q) and HA(Q) and f E H-* ( a ) .
,
,* >
denotes t h e d u a l i t y
W e know t h a t (4.12A) (which w a s i n v e s t i g a t e d e a r l i e r i n Chapter 1, Sections 3.6 and 4.7 and i n Appendix 1, Sections 3 , 4 ) admits one and only one s o l u t i o n c h a r a c t e r i s e d by
(4.13A)
1
J(u)
V V E K,
J(v)
5
UEK
(where J(v) =
I,
2 lvvl dx
-
< f ,v>) and a l s o , K being a convex
cone with vertex 0
( 4 . 1 4 A ) can also be w r i t t e n
(4.15A)
I(
V V E K,
2 0
=
0
,
UEK.
2
L 2 ( a ) , which implies t h a t U E H 2 -Au-f E L (a), w e deduce from ( 4 . 1 5 A )
If f
E
(4.16A)
]
1
(a) fl H,(n) , and
hence t h a t
-Au-f 2 0 a.e. on 52 (-Au-f)u
= 0 a . e . on 52
UEK.
Conversely, it follows from (4.15A) t h a t t h e s o l u t i o n u of
( 4 . 1 6 A ) i s a s o l u t i o n o f (4.12A). It i s c l e a r t h a t t h e infinite-dimensional problems (4.15A)
,
( 4 . 1 6 A ) a r e formally of t h e same type as ( 4 . 1 A ) and should t h u s a l s o be considered as linear complementarity problems ( i n
H p )1 The above p r o p e r t i e s extend t o v a r i a t i o n a l inequRemark 4.1: a l i t i e s which are more complicated t h a n (4.12A) and which possibly r e l a t e t o nonlinear e l l i p t i c o p e r a t o r s .
(SEC. 4 )
4.2.2
Complementarity methods
599
Other properties associated with ( 4.12A)
I n view of S e c t i o n 4 . 1 it i s n a t u r a l t o a s s o c i a t e t h e f o l l o w i n g problems w i t h t h e e q u i v a l e n t problems (4.12A) , (4.13A) , (4.15A) :
a)
The l e a s t element problem
Find
UE
fi
such that
-
(4.17A)
u l v a.e. On
a,
V V ~ K
where
k
= {vc K, -Av-f t 0)
(4.18A)
Vv-Vw dx b)
2
V W E K)
The f a m i l y o f l i n e a r problems
Find u E
such that
(4.19A) A
l
VVE K
Here a g a i n we t h e r e f o r e have a family where p E I4-l (n) i s given. of l i n e a r programming problems w i t h parameter p. The dual problem t o (4.19A) i s given by
Find X
E
;;* such
that
(4.20A)
t
VpE
ii*
By u s i n g a s u i t a b l e p o s i t i v i t y assumption on p , we s h a l l now show t h a t t h e above problems are e q u i v a l e n t . This w i l l r e s u l t from t h e f o l l o w i n g lemma, due t o Stampacchia / l A / , /2A/;
Optimisation algorithms
600
Lemma
4.1:
(APP. 2)
then
If v1 ,v2 E
l'l-v21 2
VI+V2
w = inf ( v l ' v2 ) ( = - - 2
)
also belongs t o K.
and also from
Lemma 4.2: There i s equivalence between problems (4.12A), ( b . r n ( 4 . 1 5 ~ and ) the variational inequality problem
(4.228)
IiiK
Vu*V(v-u)d%2
VVE
t
.
The proof o f Lemma 4.2 i s immediate. We are now i n a p o s i t i o n t o prove
There i s equivalence between problems ( 4.12A) , Theorem 4 . 1 : (4.13A), (4.15A) and the least element problem ( 4 . 1 7 A ) . If, i n
addition , we have
then the linear p r o g r m i n g problem (4.19A) i s equivalent t o (4.12A), (4.13A), ( 4 . 1 5 A ) , ( 4 . 1 7 A ) . Proof: 1) The equivalence between (4.12A), (4.13A) , (4.'15A) has a l r e a d y been proved elsewhere. W e s h a l l now prove t h e equivalence between (4.15A) and_(4.17A). L e t u be the solution of ( 4 . 1 5 A ) ; w e t h e n have U E K , and hence (see Lemma 4 . 1 ) : A
w = inf (u,v) which i m p l i e s , s i n c e
(4.24A)
E
K
VVE K ,
u-wzo (i.e. u-wEK),
2
0.
We a l s o have, s i n c e u-is t h e s o l u t i o n o f and s i n c e w e K c K
(4.1%)) (4.25A)
2
( 4 . 5 ) (and hence of
0.
By adding (4.24A) and (4.25A) w e t h u s have
(SEC.
601
CompZementarit y methods
4)
0 2 =
1, I
V (W-u)
I 2 dx
and hence I
VVE K ,u
=
w = inf(v,u)
which implies u < v Vv E K i.e. u is a solution of (4.17A). Conversely if u is a solution of (4.17A) we have _-
since
-Au-f t 0
U E
K
and n
v-u~OVVEK
,
and hence
-
Vu*V(v-u)dx
{ !ti,
*
=
2
0 VVEK
which shows, from Lemma 4.2, that u is a solution of (4.12A) ,
(4.15A). V-u t 0 V v
2)- If u is a solution of (4.17A) we have if p satisfies (4.23) we thus have E K ; n
2 0 V V E K which shows that ( 4.17A) implies (4.19A)
.
Conversely, if u is a solution of (4.19A) we have n
VVE K ,w
CI
=
inf(u,v)
E
K
and hence (4.26A)
2 0 .
.
However ( 4.23A) implies, since u-w 2 0 ( i e. u-w (4.278)
2
0.
By adding (4.26A) , (4.27A) we obtain
1
= 0 u-w
2
,
0
and hence w = u, from (4.23A).
E
K)
,
602
(APP. 2)
Optimisation algorithms A
CI
W e t h u s hsve U E K, u = in€ (u,v) V v e K , and hence vv E K , which proves t h a t u i s a s o l u t i o n o f ( 4 . 1 7 A ) . v2 u theorem i s t h u s completely proved.
The
Remark 4.2: There i s no d i f f i c u l t y i n f i n d i n g p E H-'(Cl) such > 0 V V E K-(0). I n f a c t it i s s u f f i c i e n t t o define
that P by
1 dx V v e Ho(Q)
(4.28A)
where C i s a positive constant. 4.3
Discussion and r e f e r e n c e s
4.3.1
Discussion
From Section 4.2 we see t h a t it i s p o s s i b l e t o reduce t h e t a s k of s o l v i n g t h e o b s t a c l e problem (4.12A) t o t h a t of s o l v i n g an equivalent l i n e a r programming problem. From a p r a c t i c a l p o i n t of view a d i s c r e t i s a t i o n i s performed using finite differences or finite elements and t h e d i s c r e t e analogues of problems (4.12A) , ( 4 . 1 3 A ) , ( 4 . 1 5 A ) , (4.17A) , ( 4 . 1 9 A ) are considered; i n particular t h e d i s c r e t i s e d form of problem (4.lgA) can b e solved by a pivoting method of t h e simplex t y p e . I n t h i s case t h e equivalence Theorem 4.1 results from t h e f a c t t h a t t h e o p e r a t o r -A a s s o c i a t e d with D i r i c h l e t boundary c o n d i t i o n s obeys t h e maximum principze ( t h i s a l s o holds f o r more complicated second-order o p e r a t o r s and f o r o t h e r forms of boundary c o n d i t i o n ; s e e Stampacchia / l A / , /2A/; i n o r d e r t h a t t h e equivalence p r o p e r t i e s of Theorem 4 . 1 can be c a r r i e d over t o t h e d i s c r e t e case it i s necessary t h a t t h e operator approximating -A obey a discrete maximum principle (see CiarletRaviart / l A / ) . I n t h e case of t h e o b s t a c l e problem (4.12A) with Q c B2, d i s c r e t i s e d by a triangular finite-element method, t h e d i s c r e t e maximum p r i n c i p l e w i l l be s a t i s f i e d i f
(i) (i5)
w e use a f i r s t - o r d e r finite-element approximation ( i . e . g l o b a l l y continuous and piecewise a f f i n e ) a l l t h e angles of t h e t r i a n g u l a t i o n are
71 2
.
If some of t h e angles are obtuse and/or f i n i t e elements o f o r d e r 2 2 are used, t h e d i s c r e t e m a x i m u m p r i n c i p l e no longer applies.
We should a l s o p o i n t out t h a t a c e r t a i n number of o p e r a t o r s which are o f fundamental importance i n p r a c t i c a l a p p l i c a t i o n s , such as t h e b i h m o n i c operator $ +. A2$ with t h e boundary conditions
$1,
= gl
,
%Ir
= g2
,
and s i m i l a r l y t h e linear
(SEC. 5 )
Minimisation of quadratic functionals
603
e l a s t i c i t y operator do not s a t i s f y a maximum p r i n c i p l e . I n view of t h e above remarks, it would appear t h a t t h e s e compl e m e n t a r i t y methods by d e f i n i t i o n have a f a i r l y l i m i t e d range of a p p l i c a t i o n , a t least as f a r as t h e s o l u t i o n of v a r i a t i o n a l inequa l i t i e s i s concerned. A f u r t h e r d i s c u s s i o n o f t h e above t o p i c s can be found i n Cryer-Dempster / l A / . 4.3.2
References
Complementarity methods have been i n v e s t i g a t e d by numerous a u t h o r s , p a r t i c u l a r l y with r e g a r d t o t h e s o l u t i o n of v a r i a t i o n a l i n e q u a l i t i e s of t h e o b s t a c l e problem type. To attempt t o g i v e a l i s t of a l l t h e r e f e r e n c e s r e l a t i n g t o t h i s c l a s s of methods would be o u t of t h e question, and w e t h e r e f o r e r e s t r i c t ourselves t o mentioning only t h e r e f e r e n c e s below, t h e b i b l i o g r a p h i e s of which a r e a l s o worth consulting. Considering only works which a r e o r i e n t e d towards t h e s o l u t i o n o f free-boundary problems of t h e o b s t a c l e problem type, w e may mention among o t h e r s : Cottle-Sacher /1A/ , Cottle-Golub-Sacher / l A / , C o t t l e / 1 A / , /2A/ , Mosco-Scarpini / l A / , S c a r p i n i /1A/ and of course Cryer-Dempster,
loc. c i t . MINIMISATION OF QUADRATIC FUNCTIONALS OVER THE PRODUCTS OF INTERVALS. USING CONJUGATE GRADIENT METHODS
5.
5.1
Synopsis
In t h i s f i f t h s e c t i o n we s h a l l supplement t h e i n v e s t i g a t i o n s of Chapter 2 , Section 2.3 ( s e e a l s o t h e discussion i n Chapter 2 , S e c t i o n 6 ) with regard t o t h e c q j u g a t e g r a d i e n t method. We consider t h e model problem i n I R
Find u c K such that (5. IA) J(u)
5
J(v)
V
V E
K,
where (5.2A)
N K = ll i=l
Ki,
Ki = [ a
bilcR
i7
( ai ,bi f i n i t e or otherwise) ,
604
Optimisation algorithms
(UP. 2 )
where A is an N x N symmetric positive-definite matrix and b eRN ; as usual (in this b ok) ( - , * ) denotes the canonical Euclidean inner product of (and more generally of mp ) . In Section 5.2 we shall describe an algorithm of conjugate gradient type, which allows an efficient solution of (5.1A) - (5.3A), and does so i n a f i n i t e number of iterations (i.e. as i n the constraintfree case of Chapter 2 , Section 2.3).
IRa
5.2
Description of the method.
Convergence results
In this section we have followed the account of D.P. O'Leary /lA/ to which we refer for further information and numerical examples. 5.2.1
x
Let u = {Al,.
General remarks
.p
1 be the solution of (5.1A) - (5.3A), and let e de ined by
.N
(5.4A)
X-Au-b;
it can easily be shown that u i s characterised by
[
V i=l,...N, X.2 O i f
(5.5A)
1
we have
ui = ai'
X.< I
Xi = 0 i f a i < u i < b i
0
.
i f u.1
=
bi
,
Remark 5.1: The above characterisation contains X as a KuhnTucker multiplier (see e.g. Rockafellar /3, Section 2 8 / ) for the problem (5.1A) - (5.3A).
...
Remark 5.2: If V i = 1 , N , we have a. = 0, bi = + w then 1 the above characterisation reduces to the linear complementarity problem
(5.6A)
I
Find u e R N such that (Au-b,u) = 0 , N Au-b eR+
, u E R+N
the solution of which was the subject of Section Appendix.
4 of
this
We shall describe Polyak's /1A/ algorithm in detail in Section 5.2.2, but we first shall give a broad outline of this algorithm
(SEC. 5 )
Mixhisation of quadratic functionaZs
605
and attempt t o e x t r i c a t e i t s g e n e r a l p r i n c i p l e s . The algorithm n n u E K , and we generates a s e uence {u } nrO such t h a t V n, a d j u s t A"(= Au -b) i n such a way as t o s a t i s f y ( 5 . 5 A ) .
4
n
Given U O E K , for n 2 0 w e t h e n c o n s t r u c t u E K, and An s a t i s f y i n g ( 5 . 5 A ) ''as well as p o s s i b l e " , using a method which i s With regard t o t h e o u t e r i t e r a t i o n loop, with i t s e l f iterative. n u known w e f i r s t d e f i n e
I"
c {1
,...N)
n n as being t h e s e t o f i n d i c e s i f o r which {ui,Ai)
(5.5A),
i s c o n s i s t e n t with
2.e. n
(5.7A)
I
=
{;I.
n
1
.
with In and Jn (={1,. .N)
n uI = {uili b,: ,:b , A: In = 11,.
..
X i > O l u l i l u ~=
= a.1 and
-
n I ) we a s s o c i a t e t h e v e c t o r s
Jn ( w e s i m i l a r l y d e f i n e and U" = {uiIi In J We t h e n reorder t h e i n d i c e s i n such a way t h a t 1;). I n view Card ( I n ) } and Jn = {l+Card (I") . .N}.
,. .
of t h e above p a r t i t i o n i n g of {l , . . . N )
N
omposition of RN (which implies R we can w r i t e t h e r e l a t i o n
(5.8A)
bi and X i < O )
and t h e corresponding dec-
-- RCard(In.)gRCard(p))
Aun-b = A"
i n t h e form
n where, i n ( 5 . 9 A ) , AII definite matrices.
n
and AJJ
a r e square, symmetric, p o s i t i v e -
The b a s i c idea behind Polyak's method l i e s in modifying n u (u; remaining f i x e d ) so as t o attempt t o make h vanish; J J l e a d s t o t h e l i n e a r system ( 5 . 1 OA)
AYJvJ = b l
this
- AnJIunI .
Since t h e m a t r i x An i s symmetric and positive definite w e can JJ apply t h e conjugate gradient method of Chapter 2 , Section 2 . 3 for t h e s o l u t i o n of (5.10A); w e can t h u s solve (5.10A) exactly
;
606
O p t h i s a t i o n algorithms
(AFT. 2 )
i n a f i n i t e number of iterations (of the inner i t e r a t i o n loop for the overall algorithm). I n p r a c t i c e , it i s u n l i k e l y t h a t t h e exact s o l u t i o n of (5.10A) w i l l s a t i s f y t h e bounds a. 0; t o s o l v e (6.1~) u s i n g t h e decomposition ( 6 . 2 ~ , ) we consider (as i n P.L. Lions-B. Mercier, loc. tit.) t h e following two algorithms : F i r s t algorithm:
(6.3A)
0
u given
n
and for n z O , u
known
Second algorithm:
(6.5A)
and f o r . (6.6A)
0
u given
n
nZO, u
known
un+'
.-
(I+AB)
-' (I+AA)-' [
(I-XB) +ABl un.
Remark 6.1: W e assume (for s i m p l i c i t y ) t h a t A,B,C are s i n g l e valued ( 2 ) o p e r a t o r s and we put
Translator's notes:
(l)
The terminology m l t i v o q u e i s sometimes used.
(2)
The terminology univoque i s sometimes used.
(SEC. 6 )
(6.7A)
611
Alternating direction methods
.
I2= (I+AA)-I (I-AB) un
U
Equation ( 6 . 7 ~ )i s equivalent t o ( I + X A ) U ~ + " ~ + X B ( U ~ ) allows us t o r e w r i t e t h e f i r s t algorithm i n t h e form
(6.8A)
u
0
n u , which
given
and f o r n z 0 , un known AA(un+l/2 )
(6.9A)
Un+1/2
(6.1 OA)
un+' +AB(u"+')
+
=
un
n+1/2 'U
-
XB(un)
,
AA( un+l I2 1 ;
l i k e w i s e t h e second algorithm i s equivalent t o
(6.11A)
u
0
given
n and f o r n z 0 , u known
(6.12A)
un+' /2+u(un+' 12)
(6. I3A)
n+ 1 u +AB(un+')
E
un-hB(un)
= un- AA(u
n+l/2
, ).
( r e s p . (6.11A)-(6.13A))algorithm has t h e appearance of an a l t e r n a t i n g d i r e c t i o n method of t h e Peaceman-Rachford /lA/ type ( r e s p . Douglas-Rachford /1A/ t y p e ) I n view of (6.8A)-(6.10A)
(6.31) , ( 6 . 4 A ) ( r e s p . (6.51)
, (6..6A))
.
I n Section 6.2 w e s h a l l g i v e some information on t h e converg( r e f e r r i n g t o P.L. Lions-B. ence of (6.3A), ( 6 . 4 A ) and ( 6 . 5 A ) , ( 6 . 6 A ) Mercier, Zoc. c i t . , and Gabay /lA/ f o r t h e p r o o f s ) . I n Section 6 . 3 w e s h a l l b r i e f l y d e s c r i b e t h e a p p l i c a t i o n o f (6.M) , (6.4A) and ( 6 . 5 ~ ,) ( 6 . 6 A ) t o t h e s o l u t i o n of t h e o b s t a c l e problem of Appendix 1, Section 3.
For f u r t h e r d e t a i l s regarding algorithms (6.3A), ( 6 . 4 A ) and ( 6 . 5 ~, ) (6.6A) , and a l s o for t h e i r implementation, w e refer t h e r e a d e r to t h e two r e f e r e n c e s c i t e d above, i n which various generali s a t i o n s a r e a l s o given. 6.2
Convergence of algorithms ( 6.3A)
, ( 6.4A)
and ( 6.5A)
, ( 6.6A)
W e follow P.L. Lions-B. Mercier /1A, Section 1/ t o which we r e f e r f o r t h e proofs o f t h e following r e s u l t s .
Optimisation algorithms
612
6.2.1
Asswnptions and additional notation.
(APP. 2)
Initialisation
We r e c a l l t h a t A,B,C are maximal (Assumption (6.2A)). We denote by D(A) t h e domain of A ( i . e . D(A) { v e H, A(v) c H, A(v) # 8 ) 1, and by R(A) t h e image of A ( i . e . R(A) = { v E H ? 3 w E H such t h a t v ~ A ( w ) } ) * We r e c a l l t h a t A i s monotone i f
-
(Y-z,u-v)
2
0
, VU E H,
y E A(u),
V V E H, z
E
A(v) ;
moreover t h e statement t h a t A i s maximal monotone i s equivalent t G saying t h a t t h e resolvent S? = (I+AA)-l ( a s i n g l e v a l u e d A o p e r a t o r ) i s a contraction defined on the whole of H. I n t h i s p r e s e n t s e c t i o n we s h a l l make t h e assumption t h a t (6.1~)admits a t l e a s t one s o l u t i o n , i . e .
(6.14A)
, 1
There e x i s t s
such t h a t
UE
H, a
E
A(u) , b E B(u) ,
a+b = 0 .
If A and B are multi-valued, it i s a p p r o p r i a t e t o d e s c r i b e algorithms (6.3A), (6.4A) and (6.5A) , (6.6A) i n r a t h e r g r e a t e r d e t a i l , i n p a r t i c u l a r with regard t o t h e i r i n i t i a l i s a t i o n . To i n i t i a l i s e (633A), (6.4~)and (6.5A), (6.6A) w e t a k e U ' E D ( B ) , then b E B(u ) , and we put
(6.15A)
v 0 = uo + Abo
A
uo = JB v
n We next d e f i n e t h e sequence { v }
n20
0
.
by
F i r s t algorithm ( ( 6 . 3 ~ (6.4A)) ~ (6.16A)
n+1 h A n v = (2JA-I)(2JB-I)v ;
Second algorithm ( (6.5Aj , ( 6.6A) ) (6.17A)
I n both cases we o b t a i n t h e sequence
(6.4A) or (6.5A), (6.6A)), s t a r t i n g from
{un} (from (6.3A), n} nrO n20' by
{V
(SEC. 6)
un = J
(6.18A)
6.2.2
613
Alternating direction vethods
p
. .
Convergence o f algorithm (6.3A), ( 6.4A)
We p u t = u+Xb
,w
= u+ha,
n n w n = 2 un- v n , b n = + -U , a We t h e n have ( s e e P.L. Lions-B. Proposition
n
n n+l w-w = r.
Mercier /1A, Section 11):
6.1: Assume (6.14A) t o be s a t i s f i e d ; we then
have (6.19A)
n n n n n the sequences u ,v ,w ,a ,b are bounded
( 6.20A)
lim re-
(bn-b,un-u)
(6.2 1A)
lim n+-
(an-a,
n+1
v
= 0
+wn
,
- u)
=
0.
The above p r o p o s i t i o n r e s u l t s i n t h e convergence p r o p e r t i e s given i n : Corollary 6.1:
I f B i s singleualued and s a t i s f i e s
For any sequence l x n j , x
(6.228)
lim
(Bx -Bx,x -x) n
D(B) Vn, such that
x weakZy and = 0, we have x = x
{Bx 1 i s bounded, xn *+Co
E
n
+
then we have (6.23A)
lim
n u = u
weakZy i n H ,
rrt+o3
where u i s the soZution of (6.1~)(unique, from (6.228)). Remark 6.2: Property (6.22A) i s s a t i s f i e d i n t h e following cases ( s e e P.L. Lions-B. Mercier, zoc. c i t . , f o r t h e ' p r o o f s ) : (i)
B i s coercive ( a l s o termed H - e l l i p t i c ) ,
i.e.
3ff
>O
such t h a t
Optimisation aZgorithms
614
(UP. 2)
(6.24A) I n t h i s case we a c t u a l l y have a s t r o n g e r c o n d i t i o n t h a t 66.23A), s i n c e ( 6 . 2 0 ~ ) md (6.24A) imply t h e strong convergence of {u 1 t o U.
(ii)
B-l
is coercive, i . e . ? B > O such t h a t
( 6.25A)
I n t h i s c a s e , i; a d d i t i o n t o
(6.23A), w e a l s o have t h e strong
convergence o f {Bu 1 t o Buy from (6.20~)and (6.258). ( i i i ) B i s s t r i c t l y monotone and weakly closed, i . e .
- v x , y ~ D ( B ) such t h a t (B(y)-B(x),y-x) = 0 we have y=x, - i f {xn is such t h a t xn ED(B) Vn, l i m xn = 'x weakly i n H and i f t h e r e e x i s t s y
n E Bxn with
i n H, then
Remark 6.3:
X E
D(B) and y = Bx.
If B i s l i n e a r , o r i f J
weakly closed.
Remark 6.4: t h a n B satisfies
If i n Corollary
n-
lim
y
= y
weakly
w n
B i s compact, t h e n B i s
6.1 w e assume t h a t A r a t h e r V"+ 1 +w"
(6.22~)~ then { 2 In converges weakly t o (6.1~). In f a c t everything s a i d i n
t h e unique s o l u t i o n of Remark
6.2 with regard t o {u"},,
6.2.3
a l s o holds f o r {
++I
+wn In 2
.
Convergence of aZgorithm ( 6.5A), (6.6A)
The n o t a t i o n i s t h a t of Sections 6.2.1 and 6.2.2; with regard t o t h e convergence of (6.5A), (6.6A), t h e following theorem i s proved i n P.L. Lions - B. Mercier /1A, Section 1.3/:
the I f (6.14A) i s s a t i s f i e d , then as n -t + Theorem 6.1: sequence {v") generated by (6.15A), (6.17A) converges weakZy t o v E H such t h a t u = J A v i s a soZutio; of (6.1~).Moreover f o r B by un = the sequence {u") defined , we have the following convergence properties:
J3
6)
(SEC.
-
I f B i s Zinear, u
- If
n
A and B are odd
A(x)
615
Alternating d i r e c t i o n .methods
= -A(-x)
converges weakly t o a soZution of (6.1~). (i.e.
VXED(A),
B(x)
V X E D ( B ) ) then {u")
= -B(-x)
converges strongZy t o a solution of (6.1~).
- If A +
B i s m d m a l monotone the weak c l u s t e r p o i n t s of
{ u n ) are soZutions of
(6.1~).
The f o l l o w i n g p r o p o s i t i o n s a r e a l s o proved i n P.L. M e r c i e r , loc. c i t .
Lions-B.
6.2: I f JBh i s weakZy closed and i f (6.14A) i s s a t i s f i e d , then u" converges weakly t o a solution o f (6.1~). Proposition
h P r o p o s i t i o n 6.3: I f JA i s compact and i f (6.14A)i s s a t i s f i e d , then un converges strongly t o a solution of (6.1~).
P r o p o s i t i o n 6.4: Assume t h a t B i s coercive and Lipschitz continuous, i. e . 3 a and M > 0 such t h a t Vx ,x E H we have 1
2
Under these conditions there e x i s t s a constant (6.28A)
Iv"-vl
0 ) t h e augmented Lagrangian dr : Hx Hx H + R defined by
(7.8A)
L,(v,q,p)
= j(v,q)
+
$11
v-ql\
+
bSv-4)
(SEC. 7 )
A.D.
619
and augmented Lagrangian methods
The f o l l o w i n g p r o p o s i t i o n may r e a d i l y be proved ( s e e e . g . Fortin-Glowinski /2A/, Glowinski /2A/) P r o p o s i t i o n 7.1:
Under the above assumptions on C , J , J A Y the augmented Lagrangian admits a unique saddle point B { u , p , A l on H x H x H and we have p = u , h = B ( u ) = -A(u), where u is the solution of ( 7 . 1 A ) , (7.2A).
%.
J
7.3
Solution of ( 7 . 1 A ) ,
(7.2A) v i a d u a l i t y a l g o r i t h m s f o r
dr
I n view o f P r o p o s i t i o n 7 . 1 it i s n a t u r a l t o e n v i s a g e s o l v i n g (7.2A) v i a t h e d e t e r m i n a t i o n o f t h e s a d d l e p o i n t s of d we shals' u s i n g t h e d u a l i t y a l g o r i t h m s o f Chapter 2 , S e c t i o n 4 ; r e s t r i c t o u r a t t e n t i o n t o Uzawa's a l g o r i t h m ( 4 . 1 2 ) , (4.13) of Chapter 2 , S e c t i o n 4 . 3 ; we t h u s have:
(7.1A),
(7.9A) then f o r An by
AOE
H given n
n 2 0, assuming that A n is known, determine u ,p
Find {un,pn)
(7.1 OA)
H x H such that V (v,q)
E
and
HxH
we have n n n dr(u ,p
(7.11A)
E
n
An+1
=
,A 1 Sdr(v,q,x") ,
An +p(u"-p")
,p>0
;
Rendering (7.10A) e x p l i c i t , we have
run+A(un)-rp"
=
- A",
rpn+B(Pn)-run
=
A".
(7.1 OA) '
The e s s e n t i a l d i f f i c u l t y i n t h e a p p l i c a t i o n o f (7.9A)-(7.11A) i s t h u s t h e s o l u t i o n o f t h e system o f two coupled e q u a t i o n s (7.lOA)'. A s p o i n t e d o u t p r e v i o u s l y i n Chapter 3, S e c t i o n 10 and Chapter 5 , S e c t i o n 9 i n a s i m i l a r c o n t e x t , it i s p o s s i b l e t o e n v i s a g e u s i n g block v a r i a n t s o f t h e relaxation methods of Chapter 2 , S e c t i o n 1 ( t h e convergence o f which h a s been e s t a b l i s h e d by Cea-Glowinski / 2 / u n d e r q u i t e g e n e r a l a s s u m p t i o n s ) t o i n t h e c a s e i n which o n l y a single r e l a x a t i o n s o l v e (7.10A)'; i t e r a t i o n i s performed, n a t u r a l l y u s i n g t h e r e s u l t s o f t h e p r e v i o u s i t e r a t i o n as i n i t i a l c o n d i t i o n s , we a r e l e d t o t h e f o l l o w i n g
Optimisation algorithms
620
( U P . 2)
algorithm (introduced by Glowinski-Marrocco 1 2 1 ) : (7.12A)
p 0 , x 1 given i n H,
then f o r n 2 1 , assuming that pn-l and An are known, determine un,pn and An" by
-
(7.13A)
n n- 1 ru + ~ ( u " )= r p
(7.14A)
rpn
(7.15A)
An+' = An + p ( un-p n) .
A*,
+ ~ ( p =~ run ) + A~,
A variant (due to Gabay /lA/, to which we also refer for the convergence results) is given by
(7.16A)
p 0 , ~ ' given i n H,
tenfor n 2 1 , assuming t h a t pn-l and An are known, detennine u ,p and by (7.17A)
run+A(un)
= rpn- 1 -An,
rpn+B(ph)
= run+An + l / 2
(7.18A) (7.19A)
9
= A
(7.20A)
n+l/2
+
p(u"-p")
.
Under "very reasonable" monotonicity and continuity assumptions for A and B y it is shown in Fortin-Glowinski / 2 A / , Glowinski /2A/ 0 that for any p ,A' E H and if (7.21A)
we have
p
E
10,
I + 6 rC 2
621
A . D. and augmented Lagrangian methods
(7.22A)
lim Ilp"-ull
\
lim
X"
=
= 0,
X weakZy in H.
TI++-
If JB is linear (or a f f i n e ) , it is proved in Gabay-Mercier /1/ that (7.22A) also holds if p E 10,ZrC
.
In fact the convergence proofs (based on energy Remark 7.1: methods) given in the above references also hold if A,B satisfy suitable monotonicity and continuity properties. without necessarily being the derivatives of convex functionals (and possibly if they are multivalued)
.
7.4
An "alternatina direction" interpretation of algorithms (7.12A)-(7.15A) and (7.16A)-(7.20A)
The case of aZgorithm (7.12A)-(7.15A): We assume that that
p =
r; it then follows from (7.14A), (7.15A)
1 = B(pn)
(and hence
)\"
= B(pn'l)
)
,
which combined with (7.13A) implies
(7.23A)
run +
A(U~)
n-1
= rp
-~(pn-1 1.
We have, from (7.14A), mn + An = rpn+B(pn), and hence, by using this in (7.13A)
(7.24A)
rpn+B(pn)
rp
n- 1 -A(un).
Putting pn-1/2 = un and r = ( 7.24A)
(7.25A)
Pn+1'2+
1 h,
XA(p n+1'2)
we finally deduce from (7.23A),
= pn-AB(pn)
,
622
(7.26A)
@timisation
pn+'+hB(pn+l)
a lgorithms
(APP. 2 )
pn-XA(p n+1/2
=
1.
R e l a t i o n s (7.25A), (7.26A) show t h a t f o r p = r , (7.12A)-(7.15A) i s i n f a c t e q u i v a l e n t t o t h e Douglas-Rachford Alternating Direction
algorithm o f S e c t i o n 6.1, Remark 6.1. The case of algorithm ( 7 . 1 6 A ) - ( 7 . 2 0 ~ ) : Here a g a i n we assume t h a t p = r; from (7.17A)-(7.2OA) t h a t 1 12 = -A(un),
An+'
it can t h e n r e a d i l y be deduced =
B(pn)
(hence
hn = B(pn-l 1) ,
and u s i n g t h i s i n (7.17A), (7.19A) g i v e s
(7.27A)
run+A(un)
= rpn'l-B(pn'l)
(7.28A)
rpn+B(pn)
= run
-
, A(un).
1 P u t t i n g p n - l I 2 = un and r = - we f i n a l l y deduce from (7.27A), A (7 . 2 8 ~
(7.29A)
rPn+1/2+A(p n+l/2
= rpn-B(pn),
(7.30A) It follows from (7.29A), (7.30A) t h a t , i f p = r , (7.16A)(7.2OA) i s i n f a c t e q u i v a l e n t t o t h e Peaceman-Rachford Alternating Direction algorithm o f S e c t i o n 6.1, Remark 6.1.
Remark 7.1: To t h e b e s t of o u r knowledge, t h e r e l a t i o n s h i p s which e x i s t between t h e alternating direction and augmented La_mangian methods were f i r s t demonstrated i n Chan-Glowinski /U/, /2A/, i n connection w i t h t h e numerical s o l u t i o n o f t h e n o n l i n e a r D i r i c h l e t problems o f Appendix 1, S e c t i o n 6.
Appendix 3 FURTHER DISCUSSION OF THE NUMERICAL ANALYSIS
OF THE ELASTO-PLASTIC TORSION PROBLEM
SYNOPSIS
1.
The aim of this appendix is to supplement Chapter 3 relating to the numerical analysis of the elasto-plastic torsion problem; we shall, in fact, restrict our attention to several supplementary results concerning the finite-element approximation of this problem and its i t e r a t i v e solution. More specifically, we shall return in Section 2 to the approximation using f i n i t e elements of order one (i.e. piecewise a f f i n e ) , previously investigated in Chapter 3, Sections 4 and 6; it will be shown that under reasonable assumptions we "almost" have = O(h1I2) (in
I I+,-uI
HL (52)
fact we have
IIuh-uII
1 Ho
= O(h) if R clR )
(a)
.
In Section 3,
which follows Falk-Mercier /lA/, it will be shown that by using an equivalent formulation it is possible to obtain an approximation error of optimal order, even if R c lR2 Finally, in Section 4, we shall give some additional information on the iterative solution of the above problem.
.
Let us conclude this first section by recalling (see Chapter
3, Section 1) that the problem under consideration is defined by Find u c K such that ( 1 .1A)
a(u,v-u)
2
L(v-u)
Vv
E
K
where 1
(1.2A)
K =
(1.3A)
a(v,w) =
( 1 .4A)
L(v) =
I V E H
0
(R),
1,
(VV~ 51
Vv-Vw dx
a.e. 1
, 1
VV,WEH (R),
1 VVEH~(R)
,f
EH-'(R)
1
= (Ho(R))'
.
624
Elas to-p l a s t i c torsion
(APP. 3 )
.
THE FINITE-ELEMENT APPROXIMATION OF PROBLEM ( 1 . 1 A ) (I) ERROR ESTIMATES FOR PIECEWISE-LINEAR APPROXIMATIONS
2.
This s e c t i o n , which follows Glowinski /2A, Chapter 2 , Section 3/ w i l l supplement t h e results of Chapter 3 , S e c t i o n 6.2. 2.1
One-dimensional case
2 Suppose t h a t n = lO,l[ and t h a t i n ( 1 . 4 A ) we have f E L (52). Problem ( 1 . 1 A ) can t h e n be w r i t t e n
( 2 . IA)
1;,
Find U E K = { V E H 01 ( O , l ) ,
du (-dv - &)dx du dx d x
2
f (v-u)dx
L e t N be an i n t e g e r > 0 and l e t h = N and f o r i = OJ,
...
e 1. =
[ X ~ - ~ , X ~ ] i, = 1 , 2
dv 1x 1
51
a.e.1 such t h a t
VV E K .
1 f; we consider xi = i h
,...N .
We t h e n approximate Hl(0,l) and K r e s p e c t i v e l y by 0
(2.2A)
Voh =
{ V h E C 0 c0,11,
vh(o)=vh(l)=o,
vhlei E P ~ , i = 1 , 2 ,
...N}
and
( w i t h , as u s u a l , P I = space o f polynomials of degree
51 ) .
The approximate problem i s t h e n defined by
1
Find \
6%
such that
Problem (2.4A.) c l e a r l y admits a unique solution. t h e approximation e r r o r IIuh-uII 1 we have H O W , 1)
Regarding
Finite-e Zement approximation
(SEC. 2) Theorem 2.1:
If
(2.5A)
II y,-uII
Proof:
Since y , r \c
f
I
625
\
Let u and
( 2 . U ) and ( 2 . h ) .
-
CE
be the r e s p e c t i v e soZutions of L2(0,1) we then have
= O(h).
K it r e s u l t s from (2.1A) t h a t
(2.6A) We deduce, by adding ( 2 . 4 A ) and (2.613) , t h a t
VvhE
$
and hence Vvhe
2 since f E L ( 0 , l ) implies t h a t Section 2 . 1 ) we have
UE
2 H (0,I)n K ( s e e Chapter 3,
1
(2.8A)
lodx !du
dx
(vh-u) dx =
We a l s o have ( t h e proof being l e f t as an exercise f o r t h e reader) :
which combined with ( 2 . 7 A ) , ( 2 . 8 ~ )implies vvhE K h
Let v r K ; we define t h e linear interpolate rhv by
Ih
r vrVoh
(2.1 OA)
(rhv) (xi)
=
v(xi)
, i=O,l,.
..N.
E Zasto-p lastic torsion
626
We have d (rhv)le. dx 1
-
X.
v(xi)-v(xi-l 1
1
h
h
(APP. 3 )
x
dv -dx, dx
i-1
and hence
d dx( rhv)
e.
1 1 since
1
dv I- dxl < 1 -
a.e. on J O , ~ C ;
we t h u s have
(2.11A)
rhVE%I
V V E K,
Replacing v by r u i n ( 2 . 9 A ) we t h e n have h h
The reguzarity property
u
E
H'(0,l)
and
imply
where C denotes v a r i o u s c o n s t a n t s independent of u and h. The e s t i m a t e (2.5A) then follows t r i v i a l l y from (2.12A)-(2.14A). 2.2
Two-dimensional case
Suppose t h a t R i s a bounded convex polygonal domain i n l R 2 with boundary r , and t h a t f E LP(n) with p > 2 (a reasonable assumption s i n c e f o r a p p l i c a t i o n s i n mechanics we have f = c o n s t . ) . we t h u s approxWe use t h e n o t a t i o n of Appendix 1, Section 3.6; imate HA(S1) and K r e s p e c t i v e l y by
(2.15A)
Voh = {vh E Co(E) ,vhIr = 0, vhlT E P I
VT
E
5)
Finite-element approximation
(SEC. 2 )
-
I
627
and
(2.16A)
Kh = K n V oh
'
g i v i n g the approximate problem
Find
\ such
I+,€
that
(2.17A) Vyl*V(vh-~)dx
2
.
VV~E
f(v,-yl)dx
Problem (2.l7A) admits a unique s o l u t i o n , and w i t h r e g a r d t o t h e convergence of t o u we have
5
Theorem 2 . 2 :
b e t a , by0,>Oas h and f we have
Suppose t h a t the angles of r h a r e bounded Then under t h e above assumptions on $2 + 0.
where u and u a r e t h e r e s p e c t i v e s o l u t i o n s of (1.1A) and (2.17A). h
Proof: (2.1 9A)
It r e s u l t s from Chapter 3, S e c t i o n 2 . 1 t h a t
u
E
d'P(Q).
By proceeding as i n t h e proof o f Theorem 2 . 1 , and u s i n g t h e fact that \ c K we o b t a i n
+a(u,vh-u)-
(2.20A)
*1 ' 71 IIVh-"Il Ho(Q>
1
(-Au-f)(vh-u)dx
Q
Vvh e
Kh'
Using t h e Halder i n e q u a l i t y we deduce from (2.20A)
\Yvhe\,
with -1+ - ' 1 P P
= 1 ( i . e . p'
=L). 1'P
L e t q be such t h a t 1 5 q + a ; t h e n suppose t h a t t h e assumptions o f Theorem 2.2 and t h a t p > 2. If
t,,satisfies
628
Ekzsto-plastic torsion
(APP. 3)
d Y p ( T ) c WISq(T) (with continuous i n j e c t i o n ) , t h e n it results from Ciarlet /1A/ and from SoboZev 's embedding theorem ( W2,P(T) c W1 ,OD(T) c Co(T) , with continuous i n j e c t i o n ) t h a t and Vv 6 WzsP(T)we have VT
E%
( 2 22A)
IIV('rrTv-v) I I
1 1 ChT1+2(--- p) IIvll$,p(T) L ~ ( T x) L ~ ( T )
I n (2.22A), I T V i s t h e linear interpolate of v at t h e t h r e e T v e r t i c e s of T, h i s t h e l e n g t h o f t h e l o n g e s t s i d e of T , and C L e t v ~ W ~ r P ( n )and l e t i s a constant inxependent of T and v. IT : H :(n) n C o ( n ) + Voh be defined by
h
s i n c e p > 2 implies ?''(a) C C o ( n ) , with continuous i n j e c t i o n , we can d e f i n e Thv( ) but i n contrast t o the one-dimensional case
we have i n general IThV
' Kh
i f
v
E
f ~ ? ' ~ ( an)K.
Since $"(a) cW""(~) with continuous i n j e c t i o n i f p > 2 , it follows from (2.22A) t h a t almost everywhere on SZ we have
Iv(Thv-v) (x)
I Irh1-2/p
vv
E
2 'qn)
from which we deduce t h a t
( (2.23A)
a h o s t everywhere on sz we have
1
t h e constant r i n (2.23A) i s independent of v and h. define rh ' Hi(Q) n?'P(Q) + Voh by
We t h e n
ITV
h
(2.24A)
(
rhv
l+rhl-2/p
We can, i n f a c t , d e f i n e lThv Vv
Y
E
K
since K
C W
1 '"(a)
C
Co(rr>.
Finite-element approximation
(SEC. 2)
-
629
I
it t h e n follows from (2.23A), (2.24A) t h a t (2.258)
rhv
Vv
%I
E
$'p(52)
n K.
Since u ~ $ ' ~ ( 5 2 )nK, it follows from (2.25A) t h a t w e can t a k e = r u i n (2.21A), g i v i n g "h h
which i m p l i e s
+ rh'-2'plI ull
2 ,p (52)
llu
II
LP ' (52) Since p > 2 we have Lp(n) C L P ' ( G ) , with continuous i n j e c t i o n ; it t h e n r e s u l t s from Strang-Fix /1/ and Ciarlet /1A/ t h a t under t h e assumptions on s t a t e d i n t h e Theorem w e have
p;n
(2*298)
I!nhu-ull 1
Ho (52)
' Chllull w29w
'
w i t h , i n (2.29A), (2.30A), C independent of h and u. (2.18A) t h e n r e s u l t s t r i v i a l l y from (2.26A)-(2.30A).
Estimate
Remadi 2.1: It follows from Theorem 2.2 t h a t i f f = constant (which i s t h e c a s e f o r a p p l i c a t i o n s i n mechanics) and i f 52 i s a convex polygonal domain, t h e n w e have an approximation e r r o r which i s l f p r a c t i c a l l y ' l of o(&) s i n c e U E w ~ , P @ )~p ( 2; t h e conditions u E W2sP (n) and u Z g on r then imply t h a t 'rrhu E Kh W e t a k e vh = IT u i n ( 2 . 8 ~ ,) (2.10A) giving h
.
1
a(%-u,yl-u)
+
I
2
a(yl-u,rhu-u)
(-Au+u-f) (vhu-yl)dx +
n
If
(rhu-%)dI'
,
which i m - l i e s , i n view of (2.5A) .(2.1 1A)
a(yl-u, yl-u) 2 a(yl-u,rhu-u)
+
(rhu-%) dr
.
It can e a s i l y be deduced from (2.11A) t h a t
It r e s u l t s from C i a r l e t / l A / , p > 2 implies
under t h e assumptions on
f o r example, t h a t
f% s t a t e d
u
with EgSp(fl)
i n t h e theorem.
I n view of (2.12A), (2.13A) it w i l l s u f f i c e t o e s t a b l i s h
(SEC. 2) Conforming and non-conforming approximations
t o prove t h e e s t i m a t e
657
(2.9A).
be i t s Vh-interpolate Let i e C 0 ( n ) be a l i f t i n g of g and l e t IT h we denote by g t h e t r a c e of IT on r . This function % on Ch; i s i n f a c t t h e i n t e r p o p a t e ( ' ) o f g f r o b t h e values taken on r n ch We then have
.
(2.15A)
gh'%
On
which combined with
It then follows from
r,
(2.6~) implies
(2.7A), (2.16~) that
Let rOh ( r e s p . r ) be t h e s u b s e t of I' which i s t h e ucion of th e t h e s i d e s of t h e t r i a n g l e s of c h c o n t a i n e d i n To ( r e s p . r+). W denote by Sh t h e g e n e r i c element of t h e s e t of s i es o f t h e 9U t r i a n g l e s of %h l y i n g on r ; i f S h C r+h., t h e n - = O ( s e e an (2.7A)); i f s h c roh then u=g on Sh, whlch i m p h e s ~h = nhu on Sh. From t h e s e r e l a t i o n s w e deduce
i
(2.18A)
'h
an c(rhu--&)-(u-g)1 aU
dr = 0
V s h c roh
" r+h
'
I n view of t h e assumption on t h e f r e e boundary s t a t e d i n t h e theorem, we have
(2.19A)
I
rOhu
I'+his less than or equal t o the number of p o i n t s on the free boundary. the number of sides sh $
ro
u r + h t h e r e e x i s t s P E sh such t h a t u ( P ) = g(P). If sh # Since g and uyr belong t o W 1 * 0 3 ( r ) , we have (from Taylor's theorem and c e r t a i n standard i n t e r p o l a t i o n r e s u l t s f o r which w e r e f e r t o C i a r l e t /lA/) ( 1 ) piecewise l i n e a r on
r
65 8
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
Since u
edYp(,) with
p > 2 implies u
E
C'
(a)we
(APP.
4)
have
(2.228) It then follows from (2.20A)-(2.22A) and from
dr = O(h) t h a t
(2.23A)
The r e q u i r e d r e l a t i o n (2.14A) then r e s u l t s t r i v i a l l y from ( 2 . 1 7 A ) , (2.18A), ( 2 . 1 9 A ) , (2.23A).
Remark 2.1: I t i s shown i n B.H.R. /1A/ t h a t t h e e s t i m a t e (2.9A) a l s o holds i f R i s a convex domain ( p o s s i b l y not polygonal) and t h e t r i a n g u l a t i o n 0: and t h e approximate convex s e t Kh a r e h s u i t a b l y chosen. 2.2.
Approximation o f t h e boundary u n i l a t e r a l problem ( l . l A ) , (1.2A) using non-conforming f i n i t e elements of mixed t y p e .
I n t h i s s e c t i o n , which follows B.H.R. /2A/, we consider t h e approximation of t h e boundary u n i l a t e r a l problem ( 1 . 1 A ) , (1.2A) using a mixed finite-element method v i a a dual formulation of ( l . l A ) , (1.2A); we adopt t h e n o t a t i o n of Appendix 1, Section 4 and we assume t h a t g , ~ 1 / 2 ( r ) . 2.2.1.
A duaZ formulation o f the bcundary unilateral problem (l.lA),
(1.2A).
I t i s c l e a r t h a t problem ( l . l A ) ,
( 1 . 2 A ) i s equivalent t o
(SEC. 2 )
Conforming and non-conforming approximations
659
With ( 2 . 2 4 A ) , (2.25A) w e a s s o c i a t e t h e Lagrangian
(2.268) and w e c o n s i d e r t h e dual problem o f ( l . l A ) ¶ (1.2A) r e l a t i v e t o k, n me l y
where
It i s q u i t e e a s y t o p r o v e : P r o p o s i t i o n 2.2:
The dual problem (2.27A) i s equivalent t o
where
(2.308)
c
(2.31A)
H(div,n)
= {q
E
q'n 2 0 on
H(div,R),
= cq
2
E
(L
,
(n))N, v * q E L2 @ ) I .
Conversely ( 2 . 2 9 ~ )(2.30A) ~ a h i t s a unique s o l u t i o n p such t h a t p = V u where u i s the s o l u t i o n of the boundary u n i l a t e r a l problem ( l . l A ) ¶ ( 1 . 2 A ) . Remark 2 . 2 : Remark 4 . 1 o f Appendix 1, S e c t i o n 4.2 a l s o h o l d s f o r (2.29A) (2.30A). Remark 2 . 3 :
Let
We t h e n have t h e f o l l o w i n g e q u i v a l e n c e
q
E
H(div,R),
(2.338)
0 Vp E A
,
660
h i Z a t e r a l problems and e l l i p t i c i n e q u a l i t i e s
4)
(WP.
where denotes t h e b i l i n e a r form of t h e d u a l i t y between H 1 i 2 ( I-) and H-1/2( r). The following saddle-point result can then e a s i l y be proved: Theorem 2 . 2 : Suppose t h a t the solution u of ( 1 . 1 A ) , ( 1 . 2 A ) belongs t o H 2 ( i 2 ) ; then l e t (2.34A)
I
=
k(q,p)
2 (lq12+(V*ql )dx + +
52
I,
f V*q dx
be the Lagrangian associated with ( 2 . 2 9 A ) , ( 2 . 3 0 A ) ; d &its unique saddle point (p,A} on H ( d i v , n ) x A such t h a t (2.35A)
p = Vu i n 51
(2.368)
X
= g-u on
Moreover Ip , A 1 (2.37A) (2.38A)
(2.39A) 2.2.2.
a
r.
i s characterised by
(p,X) EH(div,n)
XA
,
I,
[p*q+V*p V*q]dx + = - f V-q d x v q < 0 tf p E A
E
H(div,Q)
,
.
An approximation t o the dual problem (2.29A),(2.30A) using mixed f i n i t e elements.
20nce again we assume t h a t Cl i s a bounded polygonal domain i n and t h a t ?& i s a t r i a n g u l a t i o n of R. We t h e n approximate H ( d i v , n ) , H 1 I 2 ( r ) , A , C r e s p e c t i v e l y by
%=
{qh
H(div,Q) 9 l,q
T
'k+l "k+l
and
(2.40A)
V'qhlT
E
VTEPP,,
pk
qh*nl
e p k V S a s i d e of T E
ch}
(n i s a u n i t v e c t o r , normal t o S ) (2.41A)
I,,
= {%EL
2
(r), p h I S ~ p k Vscr, s
a side o f T E C ~ } ,
( SEC
-
2
(2.43A)
Conforming and non-conforming approximations
ch = (qh
'%,
qh*n ph dr s o vph €Ah)
661
-
We t h e n approximate t h e d u a l problem ( 2 . 2 9 A ) , (2.30A) by
Find ph < C h such that vqh eCh we have [ph. (qh'ph)
+v'ph
Qf g(qh7h)'n Jr
dr
v* (qh'ph)
-
1,
lax
fV* (qh-Ph)dx
9
which i s t h e d i s c r e t e analogue o f ( 2 . 2 9 A ) .
I t i s clear that (2.44A) admits one and only one solution. Let h be t h e Kuhn-Tucker v e c t o r a s s o c i a t e d w i t h (2.43A); t h e h approximate d u a l problem (2.44A) i s t h e n e q u i v a l e n t t o t h e v a r i a t i o n a l system
Find {Ph,+,)
+
( 2.45A)
1r
ph'"(ph
E H h
\r
XAh such t h a t
(Xh-g)qh*n dr =
-X h ) dr
0
1, -1,
vphEAh
[Ph '9 h + V*ph V.qh]dx f V*qh dx v q h
h
when h
%'
,
which i s t h e d i s c r e t e a n a l o g u e of (2.37A)-(2.39A), s o l u t i o n of (2.44A). Regarding t h e convergence of p /2~/!' r e s u l t s a r e p r o v e d i n B.H.R.
E
-+
ph b e i n g t h e
0, t h e following
Theorem 2 . 3 : Suppose that when h + 0 the angles o f are Then i f k = 0 ,h i f bounded below by Oo> 0, independent of h . f € H1(n) n L"(Q) and i f the s e t u E H'(Q) n w 1 '"(Q) , g E W1 '"(r), of points on the f r e e boundary ( i . e . To nT+ ) i s f i n i t e , we have
B.1I.R. /2A/ i n c l u d e s an i n v e s t i g a t i o n of t h e c a s e i n which fi F u r t h e r m o r e , Remarks 4 . 3 and 4 . 4 of Appendix i s not polygonal. 1, S e c t i o n 4.3 a l s o h o l d f o r ( 2 . 4 4 A ) , (2.45A).
Unilateral problems and e l l i p t i c inequalities
662
3.
(APP.
4)
FURTHER DISCUSSION ON THE APPROXIMATION OF FOURTH-ORDER VARIATIONAL PROBLEM USING MIXED FINITE-ELEMENT METHODS
3.1
Synopsis
The a i m of t h i s s e c t i o n i s t o supplement Section 4 o f Chapter 4 concerning t h e approximation of c e r t a i n fourth-order v a r i a t i o n a l problems. Section 3.2 gives some information on t h e convergence in of t h e mixed finite-element method of Chapter 4, Section 4.5; Section 3.3 w e b r i e f l y d e s c r i b e c e r t a i n r e s u l t s of GlowinskiPironneau /1A/ concerning t h e s o l u t i o n of t h e approximate biharmonic problem and which g e n e r a l i s e t h o s e o f Chapter 4 , S e c t i o n
4.6. F i n a l l y , i n Section 3.4, w e apply t h e above methods t o t h e s o l u t i o n of t h e fourth-order v a r i a t i o n a l i n e q u a l i t y problem The n o t a t i o n i s t h a t of Chapter 4. defined by (1.3A), (1.4A). 3.2
F u r t h e r d i s c u s s i o n of t h e converpence of t h e mixed finite-element method of Chapter 4, S e c t i o n 4.5.
Let R be a bounded domain i n I R 2 with r e g u l a r boundary; homogeneous D i r i c h l e t problem f o r A2 i s then defined by
1
U'A
the
= f i n R,
(3.1A)
au
o
on
r.
I n Chapter 4, S e c t i o n 4 . 5 w e described an approximation u s i n g mixed f i n i t e elements of order k, f o r which w e s t a t e d t h e convergence result
due t o Ciarlet-Raviart 131; t h e estimate (3.2A) supposes t h a t U E Hk+2(n) and makes various r e g u l a r i t y assumptions on t h e family ch)h , f o r which we refer t o Chapter 4 , Section 4.5. I n f a c t it results from t h e works of Scholz / l A / , /2A/, Rannacher / l A / , /2A/ and Gb-ault-Raviart /1A/ t h a t under t h e same and with s u i t a b l e (and reasonable) regulassumptions on a r i t y assumptions on u, we have 'd k ? 1 :
q),
(3.3A)
II \-uI I
€2(Q)
= O(hk-€)
(with
E
0 '
i f k22
Mixed finite-element methods
(SEC. 3 )
II Uh-4
(3.4A)
=
2 L (52)
with
0(hk+"')
663
if k 22
E: = O
,
Indeed, it i s even proved i n Scholz /3A/ t h a t f o r k l l we have ( l )
52
v'occ
(3.6A)
11 Ph-(-Au) 1 I
=
O(hk*).
We remark t h a t t h e e s t i m a t e s (3.3A), ( 3 . 4 A ) ,
(3.6A) a r e of
optimal o r quasi-optimal order. 3.3
Discussion supplementing Chapter 4 , Section 4.6 on t h e s o l u t i o n o f t h e approximate biharmonic problem (4.72)
3.3.1.
Review of t h e approximate bihamonic problem
I n Chapter spaces
4, S e c t i o n 4.5.2
we d e f i n e d t h e following d i s c r e t e
0 -
(3.7A)
Vh = {vh(vh E C (521, vhl
(3.8A)
voh
=
{vhlvh
E
vh,
vhlr =
E
VT
Pk
01
=
vhn
cCh), 1
H~(Q),
and a l s o M~
;
complement of Voh i n Vh,
(3.9A)
~ ~+ € vhlT =5 o V T E ' C ~ ,a T n r Ml-, i s defined uniquely by (3.9A).
We next consider
(1)
RoCC52
#Eocn
.
=
8
;
664
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
(UP.
4)
Problem (3.11A) admits one and only one s o l u t i o n . 3.3.2.
Solution of (3.11A) by reduction t o a variational problem i n 4 .
We have vh 3 voh @ M h ; l e t { Uhyph) be t h e Solution and l e t hh be t h e component of Ph i n % y i . e .
Of
The following theorem then r e s u l t s from Ciarlet-Glowinski
(3.11A)
12.1:
Theorem 3.1: Let {uh,ph) be t h e solution o f (3.11A) and l e e Ah be t h e component of ph i n {uhyph,hh) i s then t h e unique element o f Voh xVh x Mh such t h a t
s;
(3.14A)
1,
Vph*Vvhdx =
(3.15A)
\n
V\*Vvhdx
(3.1 6A)
n
a
vyl*vphdx
f vh dx
In
lzh%dx j$h+,
vvh
E
Voh, ph-Ah
vvh EVoh dx
\d \
3
Voh,
E
% EVoh
9
'%
z
L e t Ah€ M h a n d approximate D i r i c h l e
(3.17A)
I,
, $h
be t h e r e s p e c t i v e s o l u t i o n s of t h e problems
V%*Vvhdx
0
v vh E Voh ,
E
Vh, %-Ah
E
Voh
Mixed f i n i t e - e lernent met hods
(SEC. 3 )
V$h*Vvhdx = ln%vhdx
(3.18A)
We then define t h e b i l i n e a r form
665
vvh
y,
:
%xs
$h +
IR by
The following lemma i s proved i n Glowinski-Pironneau /lA/:
The b i l i n e a r form
Lemma 3.1:
definite.
%(*,*) i s symmetric and p o s i t i v e -
I
Now l e t uo and Poh be t h e s o l u t i o n s of t h e two agproximate D i r i c h l e t proklems
(3.20A)
(3.21A)
1
n
i,
VPoh'~hdx
I
VUoh'whdx
jKohvhdx
n
vh dx
vvh EVoh 9 poh EVoh 9
v vh EVoh,
Uoh EVoh ;
t h e fundamental r e s u l t concerning t h e s o l u t i o n of (3.11AXvia Mh, i s then Theorem 3.2: Let {uh,ph} be t h e solution of (3.11A) and i!et be t h e component o f ph zn %; Ah i s then the unique solution of t h e linear variational problem Find Ah \E such t h a t Ah
(3.22A)
%(\,%)
I:ohov%dx
-
]npoh'hdx
which i s equivalent t o a linear system with a symmetric positived e f i n i t e matrix. It can t h e n e a s i l y be shown t h a t algorithm (4.77)-(4.80) of Chapter 4, Section 4.6.1 i s a c t u a l l y a fixed-step gradient algori t h m a p p l i e d t o t h e s o l u t i o n of problem (3.22A); Glowinski-Pironneau /lA/ g i v e s ( l ) some algorithms f o r s o l v i n g (3.11A), v i a (3.22A), which a r e more e f f i c i e n t than t h e above algorithm, and i n p a r t i c u l a r some algorithms of t h e conjugate-gradient t y p e , which a r e no more complicated t o implement, each i t e r a t i o n r e q u i r i n g e s s e n t i a l l y t h e s o l u t i o n of two approximate D i r i c h l e t
( I ) See a l s o Vidrascu /1A/ for f u r t h e r d e t a i l s and numerous numerical experiments.
666
h i l a t e r a l problems mad e l l i p t i c i n e q u a l i t i e s
(APP.
4)
problems f o r - A .
3.4
Appiication t o t h e numerical s o l u t i o n of problem ( 1 . 3 A ) (1.h).
I n t h i s s e c t i o n we s h a l l consider t h e numerical s o l u t i o n of t h e problem of t h e ( f o u r t h o r d e r ) E l l i p t i c V a r i a t i o n a l I n e q u a l i t y (1.3A), ( 1 . 4 A ) . 3.4.1.
F o m l a t i o n of the continuous problem. results.
Regularity
L e t S2 be a bounded domain i n I R 2 with r e g u l a r boundary f E H - ~ ( Q ) = (HZ(L-2))' and l e t a,B E IR be such t h a t Q 5 6 .
r,
let
We t h e n consider t h e v a r i a t i o n a l problem:
Find u E K such that (3.23A) Au A(v-u) dx 2
~ V E K
where
(3.24A)
2 K = (vEH~(R)
,a
5
AvsB a.e. on SZl
and where denotes t h e s t a n d a r d b i l i n e a r form of t h e d u a l i t y between H-2(S2) and Hg(S2).
Remark 3.1: (3.258)
We have ( c f . Green's formula)
Av dx =
1, $
dr = 0
tfv
E
H,(Q). 2
Hence it follows t h a t K = @ i f Q 0 o r 5 < 0; it a l s o follows from (3.25A) t h a t K = ( 0 ) i f Q = 0 o r 5 = 0. We s h a l l t h e r e f o r e assume h e r e a f t e r t h a t
(3.26A)
a +
(A2-Al)Av dx
2 v v E H,(Q) .
u"
1"
= An- A , = un-u ; since the mapping sup ( 0 , q ) is a contraction from L ~ ( Q )+ J~:(Q), then by subtracting (3.678) from (3.61A) (with 141 = IlqlI ) ve have L2 (Q)
We+put
q + q
=
IX1 -n+l I 2
5
I
[ A-n+l 2 2 s
I
]TI2+ 2p n 13l2- 2p I T
A;"dx A?dx
I ,
+ p 2 IAu -n 2 +
p 2 IAu - nI2
,
n and hence by addition
By subtracting (3.68A) from (3.60A) and putting v = deduce
]A;"/
=
I
n
(X;-XY)AGn
dx
in,we
,
which in combination with (3.69A) implies
If p ~ 1 0 , 1 [ the inequality (3.70A) clearly implies (3.65A); as regards (3.66A), this follows from Theorem 3.1 of Appendix 2, Section 3.
674
UniZateraZ problems and eZZiptic inequazities
3.4.4
(UP.
Approximation of t h e variationaZ probZem ( 1 . 3 A ) using a mixed finite-eZement method
4)
, (1.4A)
2
W e assume i n t h e following t h a t f E L ( a ) ; w i t h t h e notation of Chapter 4 , Section 4.3 ( l ) it can e a s i l y be shown t h a t t h e r e is equivalence between (1.3A) , ( 1 . 4 A ) and
where
7
(3.72A)
j(v,q) =
(3.73A)
x = {{v,q)
1n
(q12dx
E W,
-B
-
f v dx
n
.
q 0 independent of h , we have
( 3 . 7 9A)
Proof: (3.80A)
{v,ql
E X
vhremain bounded below by
.
Since {vh,qh)
i,
V v h * V vhdx
E
=
Why
we have
lQqhvhdx
vvhEVh
*
L e t l~ E H 1 ( Q ) and l e t ph be t h e p r o j e c t i o n o f p on Vh i n t h e H 1 ( Q ) norm; we t h e n have
Under t h e assumptions on
lim h+O
IIY~-vII
h =
H1 (Q)
s t a t e d i n t h e lemma, we have
0.
676
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
(mp. 4)
From t h i s w e deduce i n t h e l i m i t i n (3.80A) t h a t
which s i n c e
(3.8 1A)
( v , q ) EH:(!J) (v,q)
E
XL2(Q)
implies t h a t
w.
We s h a l l now show t h a t define
{v,q)
E
%
.
Let $
E
Co(n) ,$ 2 0 ; we
where, i n (3.82A)
GT i s the centroid of the triangle T, and
xT
i s the c h m a c t e r i s t i c function of the triangle T.
We have (uniform continuity of 4 on
1)
0 .
Moreover we have
where, i n (3.848), t h e mi are t h e midpoints o f t h e s i d e s o f t h e t r i a n g l e T; from t h e d e k n i t i o n o f 8, ( s e e (3.75A) ) we have i n (3.8W
B+qh(miT) 2 0
tf i = 1 , 2 , 3 ,
VT
6
rh,b'{vh,qh)
EX^ ,
giving, since 4 t 0 ,
(3.85A)
(6+qh)sh$ dx
2
0
d ' $ E Co(n) ,$ 2
2 0.
We have ( 3 . 8 3 ~ )and l i m qh = q weakly i n L (Q), and hence h-to
677
Mixed finite-eZement methods
(SEC. 3 )
i n t h e l i m i t of ( 3 . 8 5 ~ )w e o b t a i n :
R
v 4 E Co(@,
(B+q) 4 dx 2 0
4 50 ,
which i s e q u i v a l e n t t o
-6
I q
a.e. on Q ;
s i m i l a r l y it can be shown t h a t q 0 such t h a t VA; E for a l l p s a t i s f y i n g
, it and
we have
where { y ~ , p ~is} t h e s o l u t i o n of ( 3 . 7 4 A ) . t h a t l i m bh = 1. h+O
It can a l s o be shown
684
UniZateraZ problems and elziptic inequalities
(Wp.
4)
Remark 3.7: The problem (3.117A) is simply an approximate mixed formulation of the biharmonic problem (3.60~); a formulation ivalent to (3.117A) is given by
For solving the approximate biharmonic problem (3.117A), (3.122A) each iteration. we refer to the discussion in Section 3.3 of this appendix and-to Chapter 4, Section 4.6; see also GlowinskiPironneau /lA/, Vidrascu /lA/, and Glowinski-Marini-Viarascu /lA/. 3.4.7
Application to the numerical solution of
a model problem
We shall now consider the particular case of the problem (1.3A), f = const., a = - 1, f3 = + 1. (1.h) in which ~ = ] O . , l [ X ] O , l [ , The triangulation used to define the approximation with finite elements of order tw is shown in Figure 1.10 of Chapter 4, Section 1.7, and consists of 128 triangles and 289 nodes; we thus have In implementing algorithm (3.116A)dim V = 289, dim Voh = 0225. (3.11$A), we have used Ah = { O , O ) ; furthermore, the optimum value for p , for various values of f, is close to 0.8. For f = 100, Figures 3.1-3.4 show the contours of pn corresponding to n = 0, 20, 40 and 240; the contours of % %e shown in Figure 3.5 (convergence is in fact virtually complete after 50 iterations).
5
Figures 3.6-3.10 show the contours of pn corresponding to n = 0, 60 and 180, for f = 1000; the conkours of % are indicated in Figure 3.11.
20, 40,
In the above figures the regions where ph ( = -Au) attains its limiting values are clearly visible; ph is equal to + 1 in the central region and - 1 in the four regions in contact with the edges. However, it should be noted that the contours of ph depend quite appreciably on especially near the free boundaries. h'
(SEC. 3 )
Mixed f i n i t e - e Zement methods
F i g . 3.1 (f = 100,
contours o f Po) h
Fig. 3.2 ( f = 100, contours o f p 1' h
686
Unilateral problems and elliptic inequalities
Fig. 3.3 (f = 100, contours of p 40) h
Fig. 3.4 (f = 100, contours
Of p
h
240
(APP.
4)
Mixed f i n i t e - e l e m e n t methods
(SEC. 3 )
\ F i g . 3.5 ( f = 100, contours o f u ) h
F i g . 3.6 ( f = 1000, contours of p:)
688
Unilateral probZems and eZZiptic inequazities
Fig. 3 . 7 (f = 1000, contours of p 20) h
F i g . 3.8 (f = 1000, contours of p ‘ 0 )
h
( U P . 4)
(SEC. 3)
Mixed finite-element methods
F i g . 3.9 (f = 1000, contours of p 6 0 ) h
F i g . 3.10 ( f = 1000, contours of p 180 h
Unilateral problems and e Z l i p t i c inequazities
690
(APP.
4)
/ F i g . 3.11 ( f = 1000, contours of
4.
y~)
NUMERICAL SIMULATION OF THE TRANSONIC POTENTIAL FLOW OF IDEAL COMPRESSIBLE FLUIDS USING VARIATIONAL INEQUALITY METHODS
4.1
Synopsis
A s we have a l r e a d y remarked i n Appendix 1, Section 6, t h e methodology of v a r i a t i o n a l i n e q u a l i t i e s can provide an e f f e c t i v e t o o l f o r t h e s o l u t i o n o f problems i n i t i a l l y formulated i n a more A f i r s t example has a l r e a d y been given i n t r a d i t i o n a l manner. Appendix 1, Section 6; i n t h e p r e s e n t s e c t i o n we s h a l l consider another example which is much more important from t h e a p p l i c a t i o n s point of view, but much more d i f f i c u l t mathematically, namely t h e numerical simulation of t h e .transonic potential f l o w of perfect compressible f l u i d s . Within t h e l i m i t e d scope o f t h i s appendix, it would be out of t h e question t o attempt t o provide a d e t a i l e d w e s h a l l t h e r e f o r e be content t o give account of t h i s s u b j e c t ; only an o u t l i n e of t h e material considered i n more depth i n BristeauGlowinski-Periaux-Perrier-Pironneau-Poirier ( ) /lA/, /2A/ (which a l s o contains a s u b s t a n t i a l bibliography on t h e s u b j e c t .
'
(1)
Abbreviated h e r e a f t e r t o B.G.4P.
4)
(SEC.
4.2 4.2.1
Transonic p o t e n t i a l f l o w
691
Mathematical formulation
The equation of c o n t i n u i t y
If R is the flow domain and r its boundary, it is shown in Landau-Lifchitz /1A/ that the flow satisfies (4.1A)
+
V*pu = 0 i n 1
with (4.2A) (4.3A)
+
u=V$,
where, in (4.1A)-(4.3A) a) b) c) d)
4.2.2
$ is the p is the
velocity potential d e n s i t y of the fluid
y is the ratio of specific heats ( y = 1.4 in air) &'is the c r i t i c a l speed of sound.
Boundary conditions
In the case of an a e r o f o i l such as B (see Figure 4.1), if we assume that the flow is uniform at infinity and tangential to the aerofoil surface r we have
B'
F i g . 4.1
692
UnilateraZ problems and elliptic inequalities
(4.4A)
(4.5A)
+
u =
+ U,
(APP. 4)
at infinity
san= O o n r B'
I n p r a c t i c e we bound R by p l a c e of (4.4A) w e use
roDas
shown i n Figure 4 . 1 and i n
It t u r n s out t h a t f o r t h e above c o n d i t i o n s t h e p o t e n t i a l 4 i s to determined only t o w i t h i n an a r b i t r a r y a d d i t i v e c o n s t a n t ; remedy t h i s s i t u a t i o n we can p r e s c r i b e a v a l u e of 4 a t some p o i n t i n R ; f o r example w e can use
(4.6A)
4.2.3
4
= 0 at the trailing edge
T.E. of B.
Lifting aerofoila and the Kutta-Joukowsky condition
The Kutta-Joukowsky c o n d i t i o n i s not p a r t i c u l a r t o t r a n s o n i c flows, as it a l s o a p p l i e s i n t h e flow of incompressible p e r f e c t f l u i d s and i n t h e subsonic flow o f compressible p e r f e c t f l u i d s . We r e f e r t o B.G. 4P /1A/ and Periaux /1A/ f o r t h e numerical treatment of t h e Kutta-Joukowsky c o n d i t i o n i n t h e c a s e of two-' and three-dimensional flows.
4.2.4
Shock-jwnp relations
The flow must s a t i s f y t h e Rankine-Hugoniot r e l a t i o n s a c r o s s a shock, namely
(4.7A)
(4.8A)
++
++
(pu*n)+ = (pu*n>-
+
(where n i s normaz t o t h e shock l i n e o r surface)
the tangential component of the ve tocity is continuous.
A s u i t a b l e v a r i a t i o n a l formulation of ( 4 .lA)-(4.3A) w i l l t a k e
(4.7A)-(4.8A)
i n t o account a u t o m a t i c a l l y .
(SEC. 4)
4.2.5
Transonic potential f l o w
693
Entropy condition
This can be formulated as follows (see Landau-Lifchitz /1A/ for more details):
There cannot be an increase i n the flow v e l o c i t y across (4.9A)
a shock as there would then be an associated decrease i n the entropy, which i s physically impossible.
The numerical implementation of (4.9A) will be discussed in Seetion 4.5.
Remark 4.1: Since the boundary between the subsonic region and the supersonic region is unknown, the problem to be solved i s in fact a free-boundary problem; the numerical treatment of this problem will take this into account. 4.3
Least-squares formulation of the continuous problem
In this section we shall not consider the practical implementation of (4.9A); we shall only discuss the variational formulation of (4.1A)-(4.5A), (4.7A), (4.8~~) and of an associated nonlinear least-squares formulation. 4.3.1
A variational formulation of the equation of continuity
We consider for simplicity the situation shown in Figure 4.2, representing a symmetric flow, subsonic a t i n f i n i t y , around a symmetric aerofoil; since the flow is symmetric, the KuttaJoukowsky condition is automaticalZy satisfied and the aerofo'il is non-li fting
.
F i g . 4.2.
694
Unilateral problems and e l l i p t i c i n e q u a l i t i e s
(UP.
4)
For p r a c t i c a l i t y ( a l t h o u g h o t h e r approaches a r e p o s s i b l e ) w e imbed t h e a e r o f o i l i n a " l a r g e " domain; u s i n g t h e n o t a t i o n o f S e c t i o n 4 . 2 , t h e c o n t i n u i t y e q u a t i o n and t h e boundary c o n d i t i o n s are
(4.1 On)
V*p($)V$
= 0 in R
with
and
(4.1 3A)
g=O on
rB,
+ +
g = prnuo3*nrn on
ra, .
We c l e a r l y have
(4.14A)
!:
p-=
g on
r
and
An e q u i v a l e n t v a r i a t i o n a l f o r m u l a t i o n o f (4.10A), (4.14A) i s
(4.15A)
The f u n c t i o n constant.
0
c a n b e d e t e r m i n e d o n l y t o w i t h i n an a r b i t r a r y
i s a n a t u r a l choice f o r 4 s i n c e Remark 4.2: The s p a c e dym(Q) physical f l o w s r e q u i r e (among o t h e r p r o p e r t i e s ) a p o s i t i v e density p; t h e r e f o r e from (4.11A) I$ h a s t o s a t i s f y
(SEC. 4)
Transonic p o t e n t i a l flow
695
(4.16A)
4.3.2
A least-squares formulation of (4.15A)
For a genuine transonic f z o w , problem (4.15A) i s n o t equivalent to a standard problem in the Calculus of Variations (as is the case for purely subsonic flows); to remedy this situation and - in some sense - "convexify" the problem under consideration, we introduce a nonlinear least-squares formulation of the transonic flow problem (4.15A), as follows:
Let X be a s e t of f e a s i b l e transonic flow s o l u t i o n s ; least-squares problem i s then (4.17A)
the
Min J(,
SEX
w i t-h (4.18A)
where, i n (4.18A) y(s) ( = y) i s a soZution o f the s t a t e equation Find y
E
HI (Q)
such t h a t
( 4 . 1 9A) p(c)V
V V ~ EVh ,
4)
Transonic p o t e n t i a l flow
(SEC. 4)
699
n = n + l , go t o (4.31A). The two n o n - t r i v i a l s t e p s o f a l g o r i t h m s ( 4 . 2 8 ~ ) - 4.35A) ( are :
( i ) t h e s o l u t i o n o f t h e single-variable m i n i m i s a t i o n problem (4.31A); p a r t ( i ) o f Remark 4 . 5 i n Appendix 3, S e c t i o n 4.3.2 s t i l l h o l d s f o r ( 4 . 3 l A ) . I t should be noted t h a t each e v a l u a t i o n of J ( 5 ) , f o r a given argument ,C, r e q u i r e s t h e s o l u t i o k o? t h e l i n e a r approximate Neumann problem (4.27A) t o o b t a i n t h e corresponding y h . n+l n+l gh from , which r e q u i r e s t h e s o l u t i o n o f two l i n e a r approx'mate Neumann problems n+f and ( 4 . 3 3 A ) ) . ( namely ( 4.27A) with 5, - +h
( i t ) the calculation
Of
Calculation o f J ' ( S n
n
I n view o f t h e importance o f and g : h h s t e p n ( i i ) , we s h a l l now d e s c r i k e i n d e t a i l t h e c a l c u l a t i o n o f n J A ( E h ) and gh ( s u p p o s i n g f o r s i m p l i c i t y t h a t p 0 = 1 ) . We have by d i f f e r e n t i a t i o n
where 6yh i s , from ( 4 . 2 7 A ) , t h e s o l u t i o n o f
6yh€Vh (4.37A)
1
and V v € V h we have h
v6yh*Vvh dx =
s2
S i n c e p(ch) w e have
=
p,(l-K(Vch(
\
p(ch)V6ch*Vvhdx + R with K =
It f o l l o w s from ( 4 . 3 6 ~-() 4 . 3 8 ~ )t h a t
6p($)V~,*Vvhdx
1 y-l y+, - and
c*
a = I/(y-l),
.
Unilateral problems and e Z l i p t i c inequalities
700
(APE'.
4)
I I n f a c t <J'( ) T) > h k ' h functional
can be i d e n t i f i e d w i t h t h e l i n e a r
It i s q u i t e e a s y t o o b t a i n gn" h
from
n+l
, using
(4.33A), (4.40A).
Remark 4.3: An e f f i c i e n t d i s c r e t e Poisson s o l v e r w i l l form a fundamental p a r t of t h e method i f t h e above a l g o r i t h m (4.28A)(4.35A) i s used.
4.4.4
Numerical solution o f a t e s t problem
W e s h a l l now apply t h e above methods t o t h e numerical s i m u l a t i o n o f t h e flow around a d i s c , t h e f l o w b e i n g uniform and subsonic a t infinity. Owing t o t h e symmetry o f t h e flow, t h e Kutta-Joukowsky condition i s s a t i s f i e d automatically. If i s sufficiently small t h e flow i s p u r e l y subsonic and a v e r y good s o l u t i o n i s o b t a i n e d i n a v e r y small number o f i t e r a t i o n s of a l g o r i t h m (4.28A)(4.35A) (=: 5 i t e r a t i o n s ) . For g r e a t e r v a l u e s o f ~~~w~~a super-
llzwll
s o n i c pocket a p p e a r s , and i f t h e computed ( l ) Mach number d i s t r i b u t i o n on t h e s u r f a c e o f t h e body i s p l o t t e d , we o b t a i n t h e d i s t r i b u t i o n i n F i g u r e 4.3 showing a r a r e f a c t i o n (or expansion) shock ; t h e l a t t e r cannot e x i s t p h y s i c a l l y . The c o r r e c t Mach number d i s t r i b u t i o n i s shown i n Figure 4.4.
('1
The convergence i s s t i n very f a s t ( = 10 i t e r a t i o n s )
(SEC.
4)
Transonic potential fZow
Expans i o n shock
Physical
F i g . 4.3
Phys ica 1 shock
M
0 and y (=y (€, ) ) the solution of the discrete variath h ional state equakion
To solve (4.48A)-(4.50A) we can use a variant of the conjugate gradient algorithm (4.28A)-(4.35A) (with Jh replaced by Jrh).
706
u n i l a t e r a l problems and e l l i p t i c i n e q u a l i t i e s
(APP.
4)
The numerical results produced by the above methods are fairly good, the sonic transition from the subsonic region to the supersonic region being very well approximated, (non-physical "expansion shocks" having been suppressed); furthermore the computed compression shocks are located accurately and are very sharp; we refer to Section 4.6 and also to B.G. 4P /2A/ and Periaux /lA/, in which numerical results produced by the above methods are presented and discussed.
A practical and very important problem concerns the proper choice of r, since for a given aerofoil the optimal value of r , of seems to be a complicated and sensitive function of the angle of attack, etc.; we shall discuss in Section 4.5.2.3 a technique which we have recently discovered, which seems to be very effective in removing (or at least reducing) this sensitivity to the choice of r.
l~mll
Remark 4.5: We may supplement Remark 4.4 as follows: since F,(Eh) appears like the integral over R of the sixth power of a discrete truncated second-order derivative of Eh, we can, by differentiation, associate with E a discrete fourth-order nonlinear operator. Since % is a convex &nctional its differential is a monotone operator. From these properties the addition of r J may be interpreted as a non-linear fourth-order h
process.
Remark 4.6: We may justify the title of Section 4.5.2 by noting that the functional E which we have introduced to regularise our problem is in fact a ]Tmore complicated) variant of the i n t e r i o r penalty f u n c t i o n a l s discussed in Douglas-Dupont /2A/ and Wheeler
/1A/. Remark 4.7: We have performed computations using exponents other than 2 and 3 in (4.45A); these two exponents in fact appear to be the optimal combination, based on the quality of the computed solutions.
4.5.2.3
A---____--___ nonlinear12---_ weighted interior Eenaltx method ___-_------------- ---___-
From the comments made in Section 4.5.2.2 regarding the sensitivity to the choice of r in the definition of J - which is rh clearly related to the strong nonlinearity of our problem - it is very tempting to control of the regularisation process associated with 3+ ) 2 . To achieve this goal, we have introduced in the above functional I$, (defined in (4.45A)) a nonlinear weighting directly related to the local value of the
Transonic potentiaZ ' f z o w
(SEC. 4 ) density; by
more precisely, instead of using E
1 3 is
where, in ( 4 . 5 1 A ) ,
707 we use
% defined
a symbolic notation defined either by
Pi 1
(4.52A)1
1
1
1
T=T(K+z) Pi
or by
with
2 1 (4.538)
Pij
=(I-
where in (4.53A)
T. Ti$ being the adjacent triangles of si e o
th which have the i
the triangulation in common.
Remark 4 . 8 : Our computer experiments indicate that a "good" value for n in (4.51A) is n = 2. rn
7 5,
Replacing by we obtain a variant of the approximate problem ( 4 . 4 8 ~- ( 4 . 5 A ) which can be solved by the same type of iterative methods; computer experiments with Eh have shown that for a NACA 0012 aerofoil the same value of r was optimal (or nearly optimal) for the following conditions
708
Unilateral problems and elliptic inequalities
Angle o f a t t a c k (degrees)
(UP.
4)
Mm
0.6 0.78 0.8
6 1 0 0
0.85 Table 4.1
The c o r r e s p o n d i n g r e s u l t s a r e d e s c r i b e d i n S e c t i o n
4 5
-2 4 *
4.6.
An_jnt_er_i~r_Ee_nal_t_~-~e_t_h_o~_usjn_g_d~_nsit_~-i~~
The i n t e r i o r p e n a l t y method d i s c u s s e d i n S e c t i o n 4 . 5 . 2 . 3 two e f f e c t s :
( i ) It p e n a l i s e s e x p a n s i o n s h o c k s v i a
[vl
combines
+
( i i ) By v i r t u e o f t h e n o n l i n e a r w e i g h t i n g t h a t we have i n t r o d u c e d , t h e r e g u l a r i s a t i o n e f f e c t i s a m p l i f i e d i n r e g i o n s where t h e Mach number i s h i g h , s i n c e i n t h e s e r e g i o n s p i s " s m a l l " . It i s t h e r e f o r e n a t u r a l t o l o o k f o r a v a r i a n t of ( 4 . 5 1 A ) which combines t h e s e two e f f e c t s more c l o s e l y ; w e s t a r t by n o t i n g t h a t f o r a p h y s i c a l shock we must have ( s t i l l f o l l o w i n g t h e s t r e a m l i n e s )
T h i s jump c o n d i t i o n (4.54A) l e a d s t o t h e f o l l o w i n g v a r i a n t of t h e entropy f u n c t i o n a l s a n d % o f S e c t i o n s 4 . 5 . 2 . 2 and 4 . 5 . 2 . 3 ( t h e n o t a t i o n o f which h a s been r e t a i n e d )
%
(4.556) The n o t a t i o n i n (4.55A) i s s e l f - e x p l a n a t o r y , which i s d e f i n e d by
e x c e p t f o r w.
1
(SEC. 4)
Transonic p o t e n t i a l fZow
709
the indices j = 1,2 corggsponding to the two adjacent triangles side of the triangulation in common. which have the i of
ch
It is then-quite easy to formulate an approximate problem in is replaced by R ; moreover the same type of ods can be applie$ to this new approximate problem. Numerical experiments have yet to be carried out to check the validity of this new approach and also to determine an optimal choice for the two exponents 6 and n in (4.55A). 4.5.2.5
Further comments
The interior penalty methods with truncation that we have discussed in the above sections have been directly inspired by some of the methods used for the numerical treatment of variational inequalities (cf. Chapter 2, Section 3 and also Glowinski /2A/, In the present case the problem to be solved definitely /?,A/). lacks those monotonicity properties which are so useful in the theory and approximation of variational inequalities; however the least-squares formulation and a l s o the formulation of an entropy c b n d i t i o n as a set of inequality relations suggest very clearly a-methodology founded on techniques which have proved successful for simpler types of inequality problems.
4.6
Numerical experiments
In this section we shall present some of the numerical results obtained using the above methods,. The results of Section 4.6.1 relate to a NACA 0012 aerofoil, and those of Section 4.6.2 to the flow around a two-component aerofoil.
4.6.1
Simulation of flows around a NACA 0012 a e r o f o i l
As a first example we consider the flow around a NACA 0012 aerofoil at various angles of attack and various freestream Mach numbers. The corresponding pressure d i s t r i b u t i o n s are shown in Figures 4.7-4.11, in which the.isomach l i n e s i n t h e supersonic region are also shown. We observe that the physical shocks are quite well-defined and also that the computed transition from the subsonic region to the supersonic region is smooth and shock-free, implying that the entropy condition has been satisfied. The above numerical results lie very close to those obtained by various authors using finite difference methods (see particularly Jameson
/MI.
710
Unilateral problems and e l l i p t i c inequalities
(APP. 4 )
NACA 0012 Aerofoil M = 0.6 a = ' 6
CF
OI
A A
'
--
* - - - - - . . . , *
-L -L
Fig. 4.7
a
L I
A
iSEC. 4)
Transonic potential flow
711
NACA 0012 Aerofoil M, = 0.78
a = l
I J
F i g . 4.8
0
712
Uni Zateral problems and e 2 l i p t i c inequalities
( APP. 4)
CI NACA 0012 Aerofoil M m = 0.8 a =
Fig. 4.9
'0
(SEC.
4) Transonic potentiaZ flow
-i
rl
713
0 7
714
UnilateraZ probZems and eZZiptic inequazities
CI
I #
#
#
.
#
#
#
#
A
NACA ,0012 Aerofoil M, = 0.85 a =
00
F i g . 4.11
#
&
(APP.
4)
(SEC.
4)
715
Transonic potential fZow
Bi NACA 0012 M, = 0.6 Q = ' 6
CI
F i g . 4.12
hailateral problems and e l l i p t i c inequalities
716
4.6.2
(AFT. 4)
Flow around a two-component aerofoil
The two-component aerofoil investigated is shown on Figure 4.12; each component is a NACA 0012 aerofoil (the upper one being component No. 1). The pressure distribution and the isomach lines are shown on Figure 4.12. We observe that the region between the two aerofoils acts as a nozzle; we also observe two supersonic regions, one between the two aerofoils and one adjacent to the upper surface of body No. 1. 4.6.3
Concluding remark
It appears from the above numerical results that the method used leads to sharp shocks and to a smooth transition from the subsonic to the supersonic region.
5.
SUPPLEMENTARY BIBLIOGRAPHY
In Section 2 of this appendix we have already given various references relating to the approximation of second-order boundaryunilateral problems using the method of conforming or non-conforming finite elements; it is also appropriate to mention, amongst other references on this and related subjects: Hlavacek /2A/, /?A/, Haslinger /lA/, Hlavacek-Lovisek /lA/, Mosco /lA/, ScarpiniVivaldi /lA/, Kawohl /lA/, Johnson / 1 A / and also the monograph ( ) of Kikuchi-Oden /1A/ on the numerical analysis of contact problems. Regarding the numerical analysis of fourth-order variational inequalities, we cite Mercier /lA/, / 2 A / , Brezzi-Johnson-Mercier /lA/, Haslinger /2A/. The numerical simulation of transonic flows has given rise to a large number of works, and we shall therefore merely cite the references on this subject contained in B.G. 4P /lA/, /2A/, Jameson /lA/, Periaw /1A/ and Hunt /lA/.
( I ) The substantial bibliography of this work will also reward
investigation.
Appendix 5 FURTHER DISCUSSION OF T H E NUMERICAL ANALYSIS OF THE STEADY FLOW OF A BINGHAM FLUID IN A CYLINDRICAL DUCT 1.
SYNOPSIS
The aim of this Appendix is to supplement Chapter 5 on the Numerical Analysis of the ppoblem of the steady flow of a Bingham We shall give some supplementary fluid in a cylindrical duct. results on the finite-element approximation of this problem and on its iterative solution.
In Section 2 we shall return to approximation by finite elements of order one (i.e. piecewise affine), previously investigated in we shall show that under reasonable Chapter 5, Sections 3 and 5; assumptions we "almost" have 11 \-ull 1 = O(h) ( l ) . In Section 3, Ho (0) which follows Falk-Mercier flA/, it will be shown that the use of an equivalent formulation allows an approximation to be obtained which is of optimal order, even if SlclR? Finally, in Section 4, we give supplementary information on the iterative in particular we describe consolution of the above problems; jugate-gradient methods similar to those in Appendix 3, Section
.
4.3. To conclude this first section we recall (see Chapter 5, Section 1) that the problem under consideration is defined by
( 1 .IA)
where
I
Find u E H
1 0
(n)
such t h a t
a(u,v-u)+gj (v)-gj (u) 2 L(V-u) Vv
1
(1.2A)
j(v) =
(1.3A)
a(v,w) = PI vv*vw dx
(1.4A)
L(v) =
n
lVvl dx
E
I H,(Q
1
Vv~H~(fi), vv,w
n
1
v v eHo(n)
,f
E
1
Ho(n)
,
1 E H - ' ( ~ ) = (H,(n))'
,
(APP. 5 )
Numerical analysis of Binghum fluid flow
718
and g and p a r e p o s i t i v e constants.
2.
FINITE-ELEMENT APPROXIMATION OF PROBLEM ( 1 . 1 A ) . ESTIMATES FOR PIECEWISE-LINEAR APPROXIMATIONS
( I ) ERROR
I n t h i s s e c t i o n , which i n p a r t f o l l o w s Glowinski / 1 A l 3 /2A3 Chapter 2, S e c t i o n 6.71 and /3A/, w e s h a l l supplement t h e r e s u l t s o f Chapter 5, S e c t i o n 5 . 1 .
2.1
One-dimensional c a s e
We t a k e n = q O , l [ and assume t h a t i n ( 1 . 4 A ) Problem ( 1 . 1 A ) can t h e n b e w r i t t e n
Find
we have f
-du 1 dx Let N be an i n t e g e r >O and l e t h = N f o r i=O,l, N and
...
,xi]
L
2
such t h a t
u e H oI ( O , I )
ei =
E
,
i=l,2
dx
w e c o n s i d e r x. = i h 1
,... N.
1 We t h e n approximate H ( 0 , l ) by 0
(2.2A)
Voh =
{Vh
€C0CO,11, vh(o)=vh(l)=o
I
,
vh ei E P ~ ,i=1,2
,...N).
(l)
The approximate problem i s t h e n d e f i n e d by
P = s p a c e o f polynomials i n one v a r i a b l e o f d e g r e e I 1. 1
(a).
Finite-element approximation - I
(SEC. 2 )
Problem (2.3A) admits a unique s o l u t i o n . approximation error, w e have
719
Concerning t h e
Theorem 2.1: Let u and u be the respective solutions of If f E k 2 ( 0 , 1 ) then we have (2.1A) and ( 2 . 3 A ) .
Proof: We r e c a l l t h a t V c HL(0.I) t h e V o h - i n t e r p o l a t e of u, i .eoh1et
cC0[O,I 1 ; l e t
h
be
r UEV h oh (2.5A) rh'(xi)
= u(xi)
Vxi,
i-I ,2
,... N.
Proceeding as i n Chapter 5 , S e c t i o n 5 . 1 it can r e a d i l y be shown that
w e deduce from (2.6A) t h a t
a(yl-u,yl-u) la(%-u,rhu-u)+g(j OF
(rhu)-j (u))+a(u,r,u-u)
i n a more e x p l i c i t form ( a f t e r u s i n
Ho(O,l) and from t h e f a c t t h a t
Schwarz's i n e q u a l i t y i n
21x11y$Sx2+y2
vx,y E I R ) :
2 The c o n d i t i o n f E L (0,l) i m p l i e s ( s e e Chapter 5, S e c t i o n 2 . 1 and B r 6 z i s 161) t h a t
with i n f a c t
Nwne&caZ m Z y s i s of Bingham f Z u i d fZow
720
(APP. 5 )
(2.9A) t h e proof o f t h i s i s l e f t as an e x e r c i s e f o r t h e r e a d e r .
By i n t e g r a t i n g by p a r t s , we deduce from (2.7A)-(2.9A)
(2.1 OA)
I
I n view of (2.8A) we have
The e s t i m a t e (2.4A) w i l l t h u s f o l l o w from (2.10A)-(2.12A) i f we can succeed i n proving t h a t j ( rhu ) S j ( u ) Vh ; w e have
However, on
J X ~ - ~ , X ~we [
have
We deduce from (2.14A) t h a t , on
]xi-,,xi[
,
(SEC. 3 )
Finite-element approximation
-
I1
and hence by i n t e g r a t i n g t h e i n e q u a l i t y (2.15A)
which i s e x a c t l y t h e i n e q u a l i t y we were t r y i n g t o p r o v e .
2.2
Two-dimensional c a s e
2
If RclR , t h e n a somewhat more c o m p l i c a t e d argument (which among o t h e r t h i n g s makes u s e o f t h e c h a r a c t e r i s a t i o n ( 3 . 4 8 ) - ( 3 . 5 1 ) of C h a p t e r 1, S e c t i o n 3.4 and ( 7 . l ) , ( 7 . 2 ) o f Chapter 5, S e c t i o n 7.1) shows t h a t i f s u i t a b l e r e g u l a r i t y assumptions ( ’ ) . f o r u and f o r t h e f r e e boundary ( 2 ) a r e s a t i s f i e d , t h e n f o r t h e f i r s t - o r d e r f i n i t e - e l e m e n t a p p r o x i m a t i o n of C h a p t e r 5, S e c t i o n 3 , and u s i n g t h e same a s s u m p t i o n s on as i n Theorem 5 . 1 o f Chapter 5 , h S e c t i o n 5.1, we have
or “ a l m o s t ” O(h); we r e f e r t o Glowinski / l A / , d e r i v a t i o n of t h e e s t i m a t e ( 2 . 1 6 ~ ) .
3.
/2A/,
/3A/ f o r t h e
FINITE-ELEMENT APPROXIMATIONS OF PROBLEM (1.1A). (11) OPTIMAL-ORDER ERROR ESTIMATES THROUGH THE USE OF AN EQUIVALENT FORMULATION
In t h i s s e c t i o n we f o l l o w t h e a c c o u n t of Falk-Mercier / l A / ; w e show t h a t t h e u s e of an e q u i v a l e n t f o r m u l a t i o n e n a b l e s us t o o b t a i n f i n i t e - e l e m e n t a p p r o x i m a t i o n s which l e a d t o e r r o r e s t i m a t e s of optimal order. I n view of t h e a n a l o g i e s which e x i s t w i t h S e c t i o n 3 s a t i s f i e d i n p a r t i c u l a r i f s1 i s a d i s c and i f f = c o n s t .
( (
2,
i . e . t h e i n t e r f a c e between
R+ = (Xe52, lVu(x)I > O )
.
no
(X
en,
IVu(x)
I
= 0) and
Numerical analysis of Bingham f l u i d fZow
722
(APP. 5 )
of Appendix 3, many of t h e r e s u l t s w i l l be s t a t e d without p r o o f .
3.1
Equivalent f o r m u l a t i o n of problem ( 1 . 1 A )
The f o l l o w i n g e q u i v a l e n c e lemma forms t h e fundamental r e s u l t i t s proof i s a v a r i a t i o n o f t h a t of of t h i s t h i r d section; Lemma 3.1 of Appendix 3, S e c t i o n 3.1, t h e n o t a t i o n of which i s retained. Lemma 3.1: Suppose t h a t n i s a simply-connected bounded domain i n IR2 ; t h e r e i s then equivalence between problem ( 1 . 1 A ) and t h e v a r i a t i o n a l problem
where I$ = {1$1,~21i s an ( a r b i t r a r y ) soZution of
The s o l u t i o n s u o f (1.U) and p of ( 3 . 1 A ) are connected by t h e relation
Remark 3.1:
W e consider t h e vector
U = {O,O,u)
i n IR3 and
, "
!
m
a
a
--}
a
{ y * a x 29 ax3
; we t h e n have w
-,
6
VXU
- , , "
5
{-,au
ax2
I n view o f (3.4A) we can t h e r e f o r e r e g a r d p as b e i n g a vectorvalued v o r t i c i t y f u n c t i o n a s s o c i a t e d w i t h t h e v e l o c i t y f i e l d d e f i n e d by u.
Finite-e lement approximation - I1
(SEC. 3 )
723
Remark 3.2: Remark 3.2 o f Appendix 3, S e c t i o n 3.1 r e l a t i n g t o W t h e c o n s t r u c t i o n of C$ from f i s a l s o a p p l i c a b l e h e r e . The r e g u l a r i t y r e s u l t s r e l a t i n g t o problem ( 1 . 1 A ) Remark 3.3: (proved i n B r & z i s /6/ and r e f e r r e d t o i n Chapter 5, S e c t i o n 2.11, t o g e t h e r w i t h r e l a t i o n (3.4A), imply t h a t i f r i s s u f f i c i e n t l y r e g u l a r , t h e n f o r t h e s o l u t i o n p o f problem (3.1A) w e have
Remark 3.4:
This remark concerns t h e e x i s t e n c e o f Lagrange
m u l t i p l i e r s a s s o c i a t e d w i t h t h e c o n d i t i o n p E H f o r problem (3.1A); it f o l l o w s from (3.1A), (3. A) t h a t a n a t u r a l Lagrangian a s s o c i a t e d w i t h problem (3.1A) i s & : L2(n))2'XH1(s2) + IR , d e f i n e d by
which l e a d s t o t h e saddle-point problem
We t h e n have t h e f o l l o w i n g p r o p o s i t i o n , t h e proof of which i s an immediate v a r i a n t of t h a t of P r o p o s i t i o n 3.1 i n Appendix 3, S e c t i o n 3.1:
There i s equivaZence between (3.7A) and
P r o p o s i t i o n 3.1:
/
{pyx)
E
HXH'
(a) ,
moreover if {p,x1 i s a soZution of ( 3 . 7 A ) , (3.8A), then p is t h e s o l u t i o n o f problem (3.1A).
724
Numerical a n a l y s i s o f gingham f l u i d f l m
(UP. 5 )
I n c o n t r a s t t o problems (3.20A), (3.21A) of Appendix 3, S e c t i o n 3.1, which a r e formally similar, t h e e x i s t e n c e of x such t h a t (?,XI i s a s o l u t i o n of problems (3.7A) , (3.8A) poses no major d i f f i c u l t y . Hence w e have P r o p o s i t i o n 3.2: Using t h notation o f Proposition 3 . 1 above, we have the existence o f x E Hf ( Q ) , such that ( p y x } i s a solution of problems (3.7A) , ( 3 . 8 ~ ) .
Proof:
I f we assume t h e e x i s t e n c e of
of the dual problem
x,
then
x
i s a solution
the converse i s also true. We denote by p(w) t h e (unique) s o l u t i o n of t h e problem
it can e a s i l y be shown t h a t ( ' )
If w e d e f i n e
371 : H 1 ($2)
+
IR by
t h e n we may deduce from (3.6A), (3.9A)-(3.12A) t h a t t h e dual problem (3.9A) admits t h e formulation
(l)
w e use t h e n o t a t i o n
t
+
= sup(O,t),
v tcIR.
Finite-e lement approximation
(SEC. 3 )
-
I1
725
i s convex ( b u t not s t r i c t l y It f o l l o w s from (3.14A) t h a t convex) and continuous on H1(Q) ; moreover we have
(3.15A)
lim
Ilwll.~'
Mw)
= + m ,
H (52) 1 where, i n (3.15A), we have used t h e n o t a t i o n ;'(a) = H (n)/IR. imply t h e e x i s t e n c e o f s o l u t i o n s of t h e The above p r o p e r t i e s o f d u a l problem ( 3 . 9 A ) , t o w i t h i n an a d d i t i v e c o n s t a n t .
3.2
Approximation o f problem (3.1A) by a f i n i t e - e l e m e n t method
We assume h e r e a f t e r t h a t R i s a bounded, convex, polygonal domain i n IR2.
3.2.1
Definition of the approximate problem
Using t h e n o t a t i o n of Appendix 3 , S e c t i o n 3.1, we approximate (3.1A) by
I t can r e a d i l y be proved t h a t P r o p o s i t i o n 3.3:
3.2.2
Problem (3.16A) a h i t s a unique sozution.
Investigation of the convergence 2
We s h a l l assume h e r e a f t e r t h a t f E L ( a ) ; with Q convex and bounded, t h i s i m p l i e s ( s e e B r & z i s /6/)t h a t t h e s o l u t i o n u of ( 1 . 1 A ) s a t i s f i e s u E H2(R) w i t h
(3.17A)
IIuII
726
Nwnerical analysis of Bingham f l u i d fZow
(APP. 5 )
where y depends o n l y on Q. W e deduce from ( 3.&A), 0 t h e s o l u t i o n p of ( 3 . l A ) s a t i s f i e s
( 3.17A) t h a t
S i m i l a r l y i f $I i s o b t a i n e d from f u s i n g t h e r e l a t i o n s o f Remark 3 . 2 i n Appendix 3, S e c t i o n 3.1, t h e n
with
(3.20A)
1144
(n))2
11 11 L2 (a
We a l s o assume t h e e x i s t e n c e o f a m u l t i p l i e r t h a t { p y x ] s a t i s f i e s (3.7A) , (3.8A).
x
E
H
2
( a ) such
H e r e a f t e r we s h a l l use TI^ t o denote t h e o r t h o g o n a l p r o j e c t i o n o p e r a t o r onto L , i n t h e norm ( (L2(C2))2; we r e f e r t o P r o p o s i t i o n s 3.3 and 3.4 of fappendix 3, S e c t i o n 3.2.2 f o r t h e v a r i o u s p r o p e r t i e s o f TI , which a r e u s e f u l i n i n v e s t i g a t i n g t h e convergence o f t h e solukion p o f (3.16A) t o t h e s o l u t i o n p o f (3.1A). More s p e c i f i c a l l y , wikh r e g a r d t o t h i s convergence w e have Theorem 3 .I ( Falk-Mercier / l A / ) : Under the above assumptions on n , f ,x and with the same assumptions on r b as i n Theorem 3 . 1 of Appendix 3, Section 3 . 2 . 2 , we have the optzmal-order error estimate
where, i n (3.21A), p i s t h e s o l u t i o n o f problem (3.1A) and ph t h a t o f t h e approximate problem (3.16A).
Proof: Proceeding as i n t h e proof of Theorem 3 . 1 of Appendix 3, S e c t i o n 3.2.2, w e o b t a i n t h e e s t i m a t e
(SEC.
4)
727
Iterative methods
The theorem w i l l be proved i f
We have
,
but
where, i n (3.25A), m ( T ) = measure of T .
We t h u s deduce
The i n e q u a l i t y (3.23A) t h e n r e s u l t s t r i v i a l l y from (3.24A), (3.26~).
4.
SUPPLEMENTARY INFORMATION ON ITERATIVE METHODS FOR SOLVING THE PROBLEM OF THE STEADY FLOW OF A BINGHAM FLUID I N A CYLINDRICAL DUCT
4.1
General d i s c u s s i o n .
Synopsis
I n Chapter 5 w e d e s c r i b e d v a r i o u s i t e r a t i v e methods f o r s o l v i n g i n t h i s c a s e , t h e most problem ( 1 . 1 A ) i n i t s d i r e c t f o r m u l a t i o n ; e f f i c i e n t c a l c u l a t i o n methods a g a i n seem t o be t h o s e b a s e d on d g o r i t h m ( 9 . 6 ) - ( 9 . 8 ) of Chapter 5, S e c t i o n 9 ( o f t h e augmented f o r f u r t h e r d e t a i l s we r e f e r t o Marrocco /lA/, Lagrangian t y p e ) ;
hk.unerica1 analysis of B ~ h g h a mf l u i d fZm
728
(APP. 5 )
Gabay-Mercier /1/, Fortin-Glowinski /lA/, /2A/, Glowinski /2A/, Glowinski-Marrocco /lA/. Furthermore, it is extremely likely that we could improve the speed of convergence of algorithm (6.93)-(6.96) of Chapter 5 , Section 6.4 (which relates to the regularised problems (6.3)) by using a variant of the conjugate-gradient type (and thus with preconditioning in the sense of Appendix 3, Section 4.3.3) ; the algorithm used would then be similar to the algorithms (4.34A)(4.42A) of Appendix 3, Section 4.3.2. In Sections 4.2, 4.3 below, we shall conclude this fourth section with an investigation of the i t e r a t i v e solution of (1.lA) via the equivalent formulation (3.lA), using duality-type algorithms based on the formulation (3.7A); in Section 4.2 we describe a Uzawa-type algorithm, and variants of the conjugate-gradient type are described in Section 4.3.
4.2
Solution of problem (3.1A) by a duality method
The material discussed in this section is similar to that in Appendix 3, Section 4.2. 4.2.1
Description of the algorithm
In view of the duality results of Remark 3.4 of Section 3.1 of this appendix we shall reduce the solution of (3.1A) (and hence of (1.1A)) to that of the saddle-point problem (3.7A), namely Find { p , ~ ) E ( L 2 ( n ) I 2
XH'(Q)
such that
(4.1A) &,w)
Ap,x)
L(q,x)
v Cq,w)
E
( L 2 ( W 2 X H 1 (Q)
with
which is justified since p is then the soZution o f (3.1A). By analogy with Appendix 3,.Section 4.2.1, we are led to the following algorithm:
(SEC. 4)
I t e r a t i v e methods
then f o r n 20,
xn+' cH1(61)
xn E H1 (52)
729
known, determine p"
E
(L2(52))
2
and
by
w i t h , i n (4.5A),
P >
0.
The practical implementation of (4.3A)-(4.5A) does not pose any major difficulties; in fact (4.4A) is equivalent (see (3.11A) of Section 3.1) to (4.6A)
pn = p(x")
=
1 u (~f$+vx*~-g)+ 0.Vx" I4+vxnI
,
In connection with (4.5A), we refer to the comments in Appendix 3, Section 4.2.1. 4.2.2
Convergence of aZgorithm (4.3A)-( 4.5A)
By proceeding as in Appendix 3, Section 4.2.2, the following theorem on the convergence of algorithm (4.3A)-( 4.5A) can be proved.
meorem 4.1:
then if
Suppose t h a t (
,
0 )
(Q
i s defined by
730
Numerical a n a l y s i s of Bingham f l u i d flo w
n
(UP. 5 )
defined by algorithm (4.3A)-(4.5A)
then f o r the sequence {p we have
.
where, i n (4.8A), p i s t h e s o l u t i o n of (3.1~) Remarks 4.1, 4.2 and 4.3 of Appendix 3, Section 4.2.2 and 4.3.2 also hold for algorithm (4.3A)-( 4.5A)
.
4.3
On conjugate-gradient type variants of algorithm (4.3A)(4.5A).
As previously mentioned in Appendix 3, Section 4.3.1 in relation to a similar problem, we might expect to accelerate the convergence of algorithm (4.3A)-(4.5A) by using a variant of conjugate-gradient type. In fact this possibility is justified by 1
The f u n c t i o n a l r/l : H (52) + R defined by (3.14A) i s convex, L i p s c h i t z continuous and G a t e a u x - d i f f e r e n t i a b l e ; i t s Gateaux d i f f e r e n t ' i a l 711' i s a l s o L i p s c h i t z continuous and i s defined by Proposition 4.1:
(4.9A)
