The convergence of Glowinski's algorithm for elliptic problems

Vol . l l No.1 ACTA MATHEMATICAE A P P L I C A T A E SINICA Jan., 1995

THE CONVERGENCE OF GLOWINSKI'S

ALGORITHM FOR ELLIPTIC PROBLEMS*

C H U DELIN (/[~ ~ ; ~ [ ) H U XIANCHENG (~/~ :~¢.~)

(Department of Applied Mathematics, Tsing-Hua University, Be2jing 100084, China)

A b s t r a c t

Recently, Bourgat et al. {3] gave a domain decomposition algorithm which can be implemented in parallel, and many numerical experiments have illustrated its efficiency. In this paper, we make a detailed theoretical analysis about this algorithm.

Key words. Domain Decomposition, trace average operator, preconditioner, convergence, condition number

1. I n t r o d u c t i o n

Domain decomposition techniques are powerful i terative methods for solving linear systems of equations tha t arise from finite element problems. The discrete approximation to a part ial differential equation is obtained iteratively by solving problems associated with each ~ubdomain and passing information between neighbors.

Domain decomposition algorithms are designed to take advantage of parallel computers. The domain is decomposed into overlapping or non-overlapping subdomains. In the former case the algorithms are often referred to as Schwarz methods, in the latter, they are called iterative substructur ing methods. Their distinction is not always clear. For instance, Dryja and Widlund [6] have shown that the basic iterative substructuring algorithm can be analyzed as a Schwarz method. For a discussion of the relationship between the overlapping and non- pverlapping schemes, see also [7] and [8].

Domain decomposit ion methods are efficient iterative methods for solving linear systems tha t arise from finite element problems. In each i teration step, a number of smaller linear systems, which correspond to the restriction of the original problems to subdomains, are solved instead of the large original system. The number of subproblems can be large and these methods are therefore promising for parallel computing.

Recently, Bourgat et [31 proposed a domain decomposition algorithm, which can be implemented in parallel; and for selfadjoint elliptic problems, many numerical examples have shown tha t it is very efficient. In the present paper, we prove its convergence and est imate the condition number of the preconditioned system.

Received May 11, 1991. *This work is supported by the National Natural Science Foundation of China.

18 ACTA MATHEMATICAE APPLICATAE SINICA Vol.l l

In this paper, C and c will be generic constants independent of r, h, H, see Section 2 and Section 3.

2. C o n t i n u o u s C a s e w i t h T w o S u b d o m a i n s

Let ~ be an open, bounded region with boundary 0~, and the elliptic operator L has the form

L~ = - ~ ~ + ~(x)~. i,j=l

Assume matrix {a~j(x)} is symmetric and uniformly positive definite, aij(x) , d(x) are bounded and piecewise smooth in ~, d(x) _> do > O, or d(x) _= O, f E L2(~). For the sake of simplicity, we study only the case d > do > O. Consider the homogeneous Dirichlet boundary value problem

{ Z ~ = :, in ~, (2.1)

ulo~ = O,

Let ( - , - ) be the L 2 inner product and I[ " II or II "tlL 2 be the corresponding norm. The weak form of equation (2.1) is

Find u e H0t(~)such that a(u ,v ) = ( f , v ) , V v e H ] ( ~ ) , (2.2)

where the bilinear form a(u, v) is defined by

a(~,v) = ~ a~(~)0~, i,j=l

[3] solvers the equation (2.2) by Domain Decomposition method. ~ is decomposed as in Figure 1.

~2 \ r

f12

Figure 1.

The algorithm given in [3] can be described as A l g o r i t h m 1.

t Given )o E H~0(r) Step 1. Compute u~ (i = 1,2) by solving equation

{ a,(u~,v) = ( f ,v)~, , Vv e H01(~/),

uTlo~ \ r = o.

where 2

J~k i , j= l U X i OXj - - - - + d(x)uv, (f'v)f~k = S ~ fv.

k

No.1 GLOWINSKI'S ALGORITHM 19

Step 2. Compute ¢~ (i = 1, 2) by solving equation

{ ai(¢~,v) = ~l[j=l ~ (aj(u~, Tr f l (v) ) - (f, Trfl(v))gG.], V v E Hol,r(~),

Ct lo~ , \ r = 0,

where Hol,r(ai) = { v e gl(Fti), vfo,\r , = 0 }.

Tr'f 1 represents any one set Tj(v) which is defined by

Tj = { w e H I ( ~ j ) , wlo~ = vfo.~ }.

Step 3. ~ + 1 = ~x ~ - p ( ¢ ~ + ¢ ~ ) , o n r ,

where p is a positive constant. R e m a r k 1. Obviously, it is true that

aj(u~],wl) - (f, wi)~j = aj(u~,w2) - (f, w2)~ , Vw,, w2 e Tj,

so Step 2 has a unique solution. It is easy to see that Algorithm 1 can be implemented in parallel. In the following, we

analyse its convergence. Let

V = { (vl,v2) : v, e HI(~t,), v, loe,no~t = O, vl]r = v2tr, a~(vi,x) = O, Vx 6 Hol(~ti), i = 1 ,2}.

[5] studies the properties of V and gives L e m m a 1. There exist two positive constants o', ~- > 0, such that

al(vl ,vl) <. ~ra2(v2,v2), a2(v2,v2) < Tal(Vl,Vl), VV = (Vl,V2) E V.

In fact, we have

] I , , r --< &lv2l~/,(a~) < o'a2(v2,v2), (2.3)

and C, ~, a are independent of v = (v~, v2). (2.3) gives the first inequality of Lemma 1. The second inequality can be obtained in a similar way.

L e m m a 2. Assume f - 0, {u~, u~} are generated by Algorithm 1, then

2 2

_< i=1 i=1

Proof. An elementary calculation yields

2 2 2

E ai(u~,u~) = E a~(C'~,u~) <_ E a~(¢'~,¢~)½ a~(u'~,u$)½ i= 1 i= I i= I

< ~ ( ¢ i ~, ¢? ~ (u$ , ~ , i=i i=i


o r 2 2

/----i i=1

Denote by u* the solution of equation (2.2), Q = u i - I~. We know that Q satisfies i) (~',~) ~ y. ii) At Step 2 of Algorithm 1, u~ can be replaced by e~ and f = 0. iii) Let I1¢~, I '~ n n ,~ , 2¢1 satisfy (I1¢2, ¢2) E V, (¢1, I2¢1) E V, then

~+~ ~ ~ ~;+1 ~ - p ( ¢ ; + I ~ ¢ D . = ~i - p ( ¢ ~ + I1¢2 ), = ~2

Because

2

a l ( e l , ¢ ~ + I l C ~ ) + a 2 ( E ~ , ¢ 2 + 2 ¢ 1 ) = 2 a '~ '~ i = 1

2

a l ( ¢ r , I 1 ¢ ? ) + 2 ( ¢ 2 , 2 ¢ 1 ) a n

i = 1

Using Lemmal, Lemma 2, we have

2

i - -1

2 2

= E ai(ar, e•) - 40 E ai(Cr, Or) i = i i = 1

I 1 a n + p2 a ~ i(¢~ ,¢r) + al(-rl¢~, z1¢~) + a2(hCL ~¢~) i = 1

2 2 a ~,, .n. _< ~ ~(~:~, ~r) - (4p - Ca + ~ ) . 2 ) ~ ~(¢i, ¢~ ),

i = 1 i = 1

where ~ = max (cr, r, 1). If 4p - (3 + ~)p2 _> 0, Lemma 2 gives

2 2

E a/(~r+t ,~r+l) -< ( 1 - ( 4 - (3 + f l)p)p) E ai(6:,~'~). i = 1 i = l

A simple discussions shows that

4 0 < k = l - ( 4 - ( 3 + Z ) p ) p < l , if 0 < p < 3+-----~'

4 2 kmin = 1 - 3 +-----~' if Popt = 3 +----~"

Merging the above analysis, we obtain 4 T h e o r e m 1. Assume {u~, u~} are generated by Algorithm 1. If 0 < p < ~-~, then

{u~, u~} converge to the solution of equation (2.3), the convergence factor and optimal convergence factor being

4 k = 1 - (4 - (3 + ~)P)P, k~,i~ = 1 - 3 +----~"


3. D i s c r e t e C a s e w i t h T w o S u b d o m a i n s

Let 12 be a polygonal region. We approximate equation (2.2) by a Galerkin conforming finite element method. For simplicity, we consider only continuous, piecewise linear, trian- gular elements, and s0h(12) C H01(~) is the corresponding finite element space, where h is the parameter of finite element. The discrete form of equation (2.2) is

Find u e S0h(12) such that a(u,v) = (f,v), Vv E SUo(12). (3.1)

Decompose 12 into two subdomains as in Figure 1. F coincides with the finite element meshes [4]. Define S0h(F), sh(12i), sh(f~i) by

So"(r) = so~(~)lr, s " (a , ) = So~(~)l~,, So~(a,) = s " (a , ) n H3(~,) .

Similarly to Algorithm 1, [3] establishes the following algorithm A l g o r i t h m 2. Given A0 E S0h(F). Step 1. Compute u~' (i = 1, 2) by solving equation

{ ~,(~,~) = (:,~)a,, v ~ e s0~(n,) , i 1,2. u? E Sh(a~ ) , u~lr = ~x " ,

Step 2. Compute ¢~' (i = 1, 2) by solving equation

1 2 ai(¢ '~,v)- ~ j--~l [aj (u~, Tr'~l(v))- (f, Tr; l ( v ) )~] ,

¢? E sh(~) ,

Vv E Sh(a~),

where Tr~ 1 (v) can be chosen for any function ~ satisfying

E sh(n~), ~lonj = vlonj.

Step 3. ~n+l = ~= _ p(¢i~ + ¢ ~ ) on r .

Considering that Lemma 1 and Lemma 2 remain true in the discrete case, and moreover, (r, r are independent of h [5], so we have

T h e o r e m 2. Let cr, r be two constants of Lemma 1 in the discrete case, {u~, u~'} 4 then {u~, u~} converge to the solution of be generated by Algorithm 2. If 0 < p < 3--~,

equation (3.1), the convergence factor and the optimal convergence factor are

k = l - ( 4 - ( 3 + ~ ) p ) p , k m i n = 1 - - - 3+/3"

4 . D i s c r e t e C a s e w i t h M u l t i - S u b d o m a i n s

In this section, we consider the domain decomposition algorithm to solve (2.4) with multi-subdomains. It is well known that the key problem of this case lies in the handling of cross-points. A point is cross-point if it is the common vertex of three or more than

22 ACTA MATHEMATICAE APPLICATAE SINICA Vol. l l

three subdomains and it is a inner point of ft. [3] proposed a new idea for handling the cross-points.

~2 m The vertices of { ~}i=1 will be labeled vj, F~ 5 will denote the open straight line segment with endpoints vi and vh, Fi5 A O~ = 0, £ = U'~=lOfi\O~ , £i5 C F. Let S0U(F), Sh(f~),

s h ( f i ) to be

sO(r ) = So"(~)l~, s"(rz,) = So"(f) l~, , so"(~,) = s " ( f , ) n H~(~,).

[3] defines a trace function. In this paper, we consider only a simple case, others are similar.

For any i, the trace function ai is defined by V8 • SoU(F), hi(O) • S0(F) and

~,(e)(=) = ~e(=),

~,(e)(=)=Ne(=), ,~,(e)(=) = o,

Vnode x on ¢9fi, x being not a vertex of fli,

Vvertex x of fli, x being a common vertex of N subdomains,

V node x on (F\0f l i ) U (0ill A Off).

Obviously,

~,(o) = T~(O), i = l

where Tr(9) is the trace of 8 on F. Now, we describe the algorithm given in [3]. A l g o r i t h m 3. Given )~o • SoU(F). Step 1.

Step 2.

V8 • soh(f), (4.1)

Compute u~ (i = 1,- . . , m) by solving equation

{ ai(u'~,v) = (g,v)~,, Vv • s0h(ni),

Compute ¢~ (i = 1, . . - , m) by solving equation

j----1

¢? ~ sh(n~),

where T r ; 1 (hi(v)) can be chosen any function @ satisfy

Vv e Sh(f~),

Step 3. m

~ . + 1 = ~ o _ p ~ 5 ( ¢ 7 ) on r. j----1

In the remainder of this section, we analyse the convergence of Algorithm 3. Let u h be the solution of equation 13.1), ¢'~ = u'~ - uhta.. Then ¢~ satisfies

{ a~(~, v) = O, V v e S ~ ( ~ 2 , ) ,

(i) ~? e sh(f~), ~ Io~,no~j = ~5 Io~,no~.

N o . 1 G L O W I N S K I ' S A L G O R I T H M 23

m

(ii) a,(¢'~,v) = E ai(¢2 ' Tr j~(~i(v) ) ) ' j = l

m ~

L e m ~ 3. E ~(~r, ~?) <- E ~ ( ¢ 7 , ¢?)- {=I 4=I

Proo£

Vv e S~(e~).

17% m

i=i i=l

rI% ~n

o ~ (o )) i=l i=l j=l

(5 ,>) = Z ~ ~2, T~; ~ ,~,(~z j=l i=l

j = l

So

i = 1

= ~(¢~ ,~7) - < ~ (¢i ,¢~ )a~ (~ , e i ) i=l 4=i

°

ai i , a i ¢ i , ¢ i ,

i=I i=i

or

L e m m a 4 .

m m

E E ° ° a~(e~,~?) < ~i(¢~ ,¢~ )-

{=1 i = l

Let ¢$+1 _ - - Q - p ~ ' . T h e n

m m

i=I i=l

Proo£

i=I j = l i:i

E a n n i ( ¢ i , ¢ i ) - Tr 1 i----I j----I i=i - -

So

i=I i=l

Remark 2 . In Lemma 3 and Lemma 4, (4.1) is used. We note that [4] proposes a lot of inequalities related to s0h(f~) in order to show the

efficiency of its preconditioner. By means of [4], we give some lemmas.


L e m m a 5 [4] .

moreover,

L e n t m a 6.

then

moreover,

Let vj be a vertex of ~2~. Then

(~b~(vj)) 2 _< &ai(¢~, %b~);

Assume Fit C Of~4. If w e sh(~'~4) satisfies

{ ~,(~,~) = 0, v~ e S~o(~,), ~(y) = ¢?(y)/2, rhode y of rj~, w = 0, on O~2i\F~,

then

< O ((1 + H-2) (1 + In ( H ) ) ) •

Lemma 6 can be proved similarly to Lemma 3.5 of [4], here we omit it. L e m m a 7. Let F4j = 0f~i N 0~j , F4j being an open segment. If w E sh( f~) satisfies

2 ~(y) = ¢; ( y ) / , V,~ode y of r,~, w = 0, on O~i\Fi~,

a4(~,~) < ~ (¢~ ' , ¢~ ) ,

{ a~(~,x) = 0, vx e s0~(~j), ~(y) = ¢~(y)/2, Vnode y of Fit,

= 0, on 0fl~\F4j.

The trace theorem, Lemma 6 and Lemma 3.2 of [4] give

n 2 n n

On the basis of the above Iemmas, we now prove our main theorem. T h e o r e m 3. For Algorithm 3, there exists a constant ct

1 < o ' < O ( I + H -2) l + I n

such that if p < 2/0, then

m rt~

~"~a {6 nrbl a n - b l ~ -- - -

4=1 4----1

where ~ is the same as in Lemma 6. Proof Define ~ E sh(f~j) by


m a ~, n Proof. First, we analyse the relation between )-~i=i i(~i ,.~i ) ( for ~i see Lemma m a n n 4) and ~-~i=1 ~(¢~, ¢i )" Choose any subdomain ~Jo" without loss of generality, assume the

relation between ~io and other subdomains as in Figure 2.

F4 P l

q k ......... Fs

Figure 2.

It is easy to see that

vj~o =(~o ~ + ~ + ~ + ~o' + ~ + ~ + ~o ~ + ~o ~) + ( ~ + ~ + ~ + ~ ) + ( ~ + ~i + ~ ) + ( ~ + ~,' + ~ ) + ( ~ + ~ + ~ ) + (w~ + ~ + ~ ) ,

where letting V be the nodal set of OI2jo\{p, q, k, l }, w~ is defined by

ajo (ws r, x) = O, 1 n

~ ; ( ~ ) = N ¢ ~ ( ~ ) ,

~;(y) = o, (~, 8) • vl = { (k, 0), (/,4),

{ a jo (~ , x) = o, 1 ,~ ~,:(u) = ~¢, (,-),

~;(u) = o, (~, 8) • v~ = { (2, 0),

vx • So~(njo),

r being the common vertex of N subdomains

and inner point of ~2,

v y • v u (k,q,p,t } \{~} , (l, 0), (p, 0), (q, 0), (k, 1), (k, 2), (l, 2), (1, 3)

(2, 4), (p, 5), (/9, 6), (q, 7), (q, 6), (q, S), (k, 8) },

v~ • so~(njo),

Vnode y of Fr,

V, node of 0ftjo\F~ , (2, 2), (4, 0), (4, 4), (6, 0), (6, 6), (8, 0), (8, s) }.

Lemma 5, Lemma 6 and Lemma 7 give

ajo(~:, ~ ) < C(¢ : ( r ) ) ~ < C~a , (¢L ¢$),

aJo(~s, ~%) < "~s(¢~ ,¢~ ),

v(r , s ) e Vl, (r, 8) • v~.

Furthermore, the triangle inequality gives

8

ajo (qa/no, qajno) < Z ( ~ + ~)aj , (¢j'~, Cj'~). i=0

26 ACTA MATHEMATICAE APPLICATAE SINICA Vol.ll

Considering an arbitrary flJ0, we obtain

m m

E a i (~ , , ~i ) --< cr i = l i = l

(4.2)

where

because

( f t tt 1 < o ~ < O ( I + H -2) l + l n

m ~ m

• • . a n n

i = l i = l i = l i = l

By using Lemma 4, (4.2) and Lemma 3, it is true that if p < 2/or, then

rn m rn m n n p 2 ~ ÷ ~ ~ r + ~ ) - E ~ ( ~ , , ~ ) - 2 . Z . . ~ , ~(¢~,~)+ E ( ~ , ~ : ~)

/ = 1 / = 1 i = I i = i

m m

< a n n __ o'p ) E a i ( ¢ i ,•?) - Z i ( ~ , ~ ) ( 2 p - ~ i = 1 i = 1

m

<(1 - 2p + ~p~) y~ a~(~L E~'). i = l

(4.3)

This completes the proof of Theorem 1. A simple calculation yields

0 < 1 - 2 p + c ~ p 2 < 1,if 0 < p < 2 / c r

and 2

inf ( 1 - 2 p + a p 2 ) = 1 - - . O < p < 2 / a O"

From this, we have T h e o r e m 4. Let {u~}iml be generated by Algorithm 3. There exists a positive

constant or: ~r <_ O((1 + H-2)(1 + ln(H/h)) 2) such that {u~}~= 1 converge to the solution of equation (3.1), the convergence factor and optimal convergence factor being

2 = 1 - 2p -~ o-p 2, kmin = 1 - -- k

O"

5. T h e A n a l y s i s o f P r e c o n d i t i o n e r

It is well known that the conjugate gradient method (CG) is an efficient technique for solving symmetric positive-definite systems. Given a symmetric positive definite algebraic equation, CG is prior to simple iteration methods. The main problem of using preconditioned conjugate gradient method (PCG) or accelerating a given simple iteration method by CG is finding out the preconditioner contained in a given simple iterative method and to study the condition number of the preconditioned system.


In Sections 1-4, we have analyzed Algorithm 2 and Algorithm 3, which are two simple iterative method. In [3], equation (3.1) is also solved by PCG with the preconditioner contained in Algorithm 3; many numerical experiments show that it is very reliable. Our purpose in this section is to analyze this preconditioner and to estimate the condition number of the preconditioned system.

[5] points out that Q = S( I - P ) - I is the preconditioner in the iterative scheme x n+i = P x n + q for solving the system Sx = b. From the view point of parallel computation, preconditioner Q should satisfy

1) Q - i b should be easy to obtain by parallel computers. 2) The condition number of Q - I S should not be large. For equations (3.1), its algebraic form can be represented by

AU = F. (5.1)

Order the nodes of ~ according to the order of ~ i , ~22,-.. ,fire, and F, A, U, F have the forms

A = [ATr A r t ' F r ' Ur "

The block forms of/7, U coincide with that of A and

AH-~ (a(f~i,flj)), A r t -- (a(fli,flj)), Azr = (a(fli,~j)),

where fli, ~j are standard basis functions corresponding to nodes of o~m__i~i and F, respec- tively.

Define S = A r t - ATrAT~AIr. (5.2)

R e m a r k 3. S is called the F-capacitance matrix of matrix A. The system (5.1) can be solved as follows Step 1. Solve the capacitance system

SUr Fr T -~ = - AzrA H F z . (5.3)

Step 2. Obtain Ux by solving the equation,

{ u~ • Sh ( f t~ ) ,

ai(ui,x) =- ( f , x ) ,

U i ~ UF~

v x e

o n Ofli\Oft.

Because of the block-diagonalization of the matrix AH, Step 2 can be finished by solving the independent subproblem on subdomalns in parallel, so the key of solving system (5.1) is in Step 1. [3] solves (5.3) by PCG. Let

A,i = (ai(flj,/~t)), At,r,---- (ai(flj,fll)), Air , -~ (ai(flj,fl~)),

where ~3j, ~z are standard nodal basis functions corresponding to ~i and O~21\Ofl, respec- tively. Define

T --1 S~ = Ar, r, - A i r , Aii Air, .

R e m a r k 4. Si is called the Ofli\Ofl-capacitance matrix of A in ~i-


Assume that Ri is the matrix form of trace function ai (see Section 4), P is the iterative matrix of the iterative scheme from A" to A "+1 in Algorithm 3. It can be verified that

( ~ R S ~ I R T ) S P = I - p i • i = l

From [5], Q has the form: m

Q-1 = P Z RiS71RT" (5.4) i----1

(5.4) shows that Q satisfies the property 1). In order to show that Q satisfies the property 2), we prove the following theorem. T h e o r e m 5. Let S, Q be defined by (5.2) and (5.4), then there exists a positive

constant C, which is independent of h and H, such that

P<- (SA, A) <Cp(l+H -~) l + l n -~- , VA;

equivalently, the condition number of matrix Q-1S satisfies

x(Q-IS)<O ( I + H -2) 1 + l n

Proo£ have

Let ei, ¢i and ~oi be generated by Algorithm 3 with f = 0 and A '~ = A. We

(SQ-1SA, A) = p a,:(ei, ~pi), (SA, A) = ~ ai(si, ¢i). i = l i = l

Lemma 3 and Lemma 4 give

m

(SQ-'S~, ~,) ~=,E: a~(¢,,¢,) ( s ~ , ~ ) = p = -> p (5 .5)

Z ai (el, si) i=1

On the other hand, from Lemma 4 we know

i=-i i=l i=1 L i=1

(4.2) gives

ai(¢i, ¢2i) <_ C(1 + H -21 1 + In Z ai(si, ei) i=1 i= l

or (SQ-lS~, ),)

(s~, ~) <_ Cp(I + H-2) (I + ln ( H ) ) 2.

(5.5) and (5.6) complete the proof of Theorem 5.

(5 .6)


Theorem 5 shows that the matrix Q satisfy property 2) and Q is a efficient preconditioner of the matrix S.

6. C o n c l u s i o n

In Sections 1-5, we analyse the convergence of the algorithm given in [3] and estimate the condition number of the preconditioned system. Our theoretical results coincide with the numerical experiments of [3]; and show that the algorithm proposed in [3] is very efficient for partial differential equation.

References

[1] The First International Symposium on Domain Decomposition Methods for Partial Differential Equa-

tions. R. Glowinski et al. (eds), SIAM Publ., 1988.

[2] The Second International Symposium on Domain Decomposition Methods for Partial Differential Equa-

tions. T. Chan et al. (eds), SIAM Publ., 1989.

[3] J.F. Bourgat et al. Variational Formulation and Algorithms for Trace Operator in Domain Decompo-

sition Calculation. In [2].

[4] J.H. Bramble et al. The Construction of Preconditioners for Elliptic Problems by Substructuring I.

Math. of Comp., 1986, 175: 103-134.

[5] Zhang Sheng. Decomposition-parallel Algorithms for Elliptic Problems. Ph. D Thesis, Computing

Center, the Chinese Academy of Sciences, 1989.

[6] M. Dryja and O. Widlund. Some Domain Decomposition Algorithms for Elliptic Problems, In Iterative

Methods for Large Linear Systems. San Diego, CA., Academic Press, 1989.

[7] P.E. Bjcbrstad and O. Widlund. To Overlap or Not to Overlap: A Note on a Domain Decomposition

Methods for Elliptic Problems. SIAM d. ScL Star. Comput., 1989, 10(5): 1053-1061.

[8] T. Chan and D. Goovaerts. Schwarz Schur: Overlapping Versus Nonoverlapping Domain Decomposi-

tion. Technical Report CAM 88-21, UCLA, Los Angeles, CA, 1988.

Documents

The convergence of Glowinski's algorithm for elliptic problems