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Numer. Math. 65, 95-108 (1993) Numerise 9 Springer-Verlag 1993

A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods*

J. Douglas, Jr. 1, P.J. Paes Leme 2, J.E. Roberts 3, and Junping Wang 4 1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA 2 Instituto Polit6cnico do Rio de Janeiro, 28600 Nova Friburgo, and Department of Mathematics, Pontificia Universidade Cat61ica do Rio de Janeiro, 22453 Rio de Janeiro, Brazil 3 INRIA, Rocquencourt, France, and Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA 4 Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA

Received June 20, 1992

Summary. A parallelizable iterative procedure based on domain decomposition techniques is defined and analyzed for mixed finite element methods for elliptic equations, with the analysis being presented for the decomposition of the domain into the individual elements associated with the mixed method or into larger subdomains. Applications to time-dependent problems are indicated.

Mathematics Subject Classification (1991): 65N30

1. Introduction

Our objective is to discuss an iterative procedure related to domain decomposition techniques based on the use of subdomains as small as individual elements for mixed finite element approximations to second order partial differential equations in two or three space variables. Analogous techniques apply in an almost unaltered fashion when larger subdomains are employed; however, the discussion below will be concentrated on the case in which the subdomains are elements. The iterative technique applies directly to coercive elliptic problems and provides a time- stepping procedure for implicit methods for parabolic or hyperbolic equations. The motivation for the procedure is that it can be very naturally and easily imple- mented on a massively parallel computer by assigning each subdomain (i.e., each element) to its own processor.

Our iterative procedure is very closely related to and based on one introduced by Despr6s [9] for a Helmholz problem and extended to another Helmholz-like problem related to Maxwell's equations by Despr6s et al. [10, t t]. As in these references, we shall make very strong use of the hybridization of mixed finite element methods introduced by Fraeijs de Veubeke [19, 20] more than twenty-five

* The research of Douglas was supported in part by the NSF and the AHPCRC and that of Paes Leme in part by the CNPq and the FINEP. Correspondence to: J. Douglas, Jr.

96 J. Douglas et al.

years ago and analyzed very carefully by Arnold and Brezzi [1]; see also [3, 5, 6]. The convergence proofs in [9, 10, 11] are given for the differential problems in strong form; only numerical results are presented to validate the iterative proced- ures for the discrete case in these papers. Another related procedure, applicable to a Helmholz-like problem in elasticity, has been introduced by Feng and Bennethum [18].

The elliptic case will be treated in detail first, since the time-stepping applica- tions are essentially corollaries of the results in the elliptic case. While the practical goal is the treatment of mixed finite element methods for the elliptic problem, the domain decomposition procedure can be considered at the differential level and the iteration applied to a mixed formulation of the differential problem. The conver- gence proof for the iteration covers the discrete case rigorously; but, since there is a technical difficulty arising from the nonlocal nature of the Sobolev space of order - 89 on the boundary of a subdomain, the proof would be only heuristic for the

mixed differential case. Our proof of convergence would also be valid for the strong form of our coercive differential case; however, Despr6s [9] has already indicated this argument. The analysis would also cover a collection of cell-centered finite difference methods and finite volume methods.

Parabolic and hyperbolic problems will be treated after the elliptic problems. Different domain decomposition procedures for mixed finite element approx-

imations have been considered by Cowser, Ewing, Glowinski, Kinton, Wang, and Wheeler (see [8, 16, 17, 21, 22]).

An outline of the paper is as follows. In Sects. 2 and 3 the domain decomposi- tion is defined and a mixed formulation of the differential problem is recalled; then, the iterative procedure is illustrated for the differential problem. In Sect. 4 the mixed finite element procedure is introduced, the corresponding iteration defined, and a convergence argument given under minimal hypotheses on the partition into subdomains. In Sect. 5 it is shown that the spectral radius of the iterator for the mixed finite element procedure is less than one; in the next section, we show that this spectral radius has a bound of the form 1 - ch for quasiregular partitions. If, instead, the decomposition of the domain is fixed and the partition for the finite element procedure is compatible with the decomposition, then this bound is improved to 1 - cx/h. The final section contains a brief treatment of the very effective application of this iterative procedure to time-dependent problems.

2. The domain decomposition

Let f2 c IR a, d = 2 or 3, be a bounded domain with a Lipschitz boundary c3f2. Let {f2 i, j = 1 . . . . . M} be a partition of f2:

(2.1) ~=~JJ~=l~j: ~ j n ~ = ~ , j , k .

Assume that OOi, j = 1 , . . . , M, is also Lipschitz and that g?j is star-shaped. In practice, with the exception of perhaps a few f2Ss along 0f2, each f2j would be convex with a piecewise-smooth boundary. Let

(2.2) F = c~f2, Fj = F ~ ~ 2 , Fik = F~j -- c~f~j n c~f2~.

Parallel iterative procedure 97

3. The mixed formulation of the differential problem

Consider the Dirichlet problem

(3.1.i) - V . ( a V u ) + cu = f , x~f2 ,

(3.1.ii) u = g, x~gf2 ,

and assume that the coefficients a(x) and c(x) satisfy the bounds

O < a o < a ( x ) 1 for reasonable f and 9 are assured. Let the flux be denoted by

(3.2) q = - aVu ,

and set e(x) = a(x) - 1. Under reasonable hypotheses, the Dirichlet problem (3.1) is equivalent to its following (global) mixed formulation:

(3.3.i) c~q + Vu = 0, x e f 2 ,

(3.3.ii) divq + cu = f , x~f2 ,

(3.3.iii) u = g, x~Of2.

The weak formulation of (3.3) is given by seeking {q, u}~H(div, ~) x L2(f2) = V x W such that

(3.4.i) (~q, v)~ - (u, divv)~ = - (9, v . V ) r , w V ,

(3.4.ii) (divq, w)o + (cu, w)e -- ( f w)a, we W .

Let us consider decomposing (3.3) or (3.4) over {f2j}. In addit ion to requiring {qj, u j } , j = 1 , . . . , M, to satisfy

(3.5.i) ~q~ + Vu~ = O, x ~ f 2 j ,

(3.5.ii) divq~ + cui = f xEf2~ ,

(3.5.iii) u~ = g, x ~ F j ,

it is necessary to impose the consistency conditions

(3.6.i) u~ = Uk, xeF~k ,

(3.650 q~ ' v i + qk'Vk = O, XSFjk ,

where v i is the unit outer normal to O~. It is more convenient [9, 10] to replace (3.6) by the Robin boundary condit ion

(3.7.i) - f l q j . v j + u j = f l q k " V k + U k , x ~ l ' j k ~ 63Q j ,

(3.7.ii) --flqk" Ilk "~ IAk = flqj" vj + u j , x6l-jk ~ 0~'-~ k ,

98 J. Douglas et al.

where fl is a positive (normally chosen to be a constant) function on U_Fjk. Now, move toward a new weak formulation by testing (3.5.i) against a vector v~ V i = n(div, f2j):

(3.8) (o~qj, V)aj -- (U j, divvj)~j + (u~, v. v)o~j = O, ve Vj.

Apply (3.5.iii) and (3.7.i) to (3.8) to obtain (3.9.i) below, and test (3.5.ii) against we ~ = L2(f2j) to obtain the second equation in the system below. Thus, the weak mixed formulation of (3.1) over the partition {I2j} is given by the seeking of {q j, u j}~ Vj x Wj, j = 1 . . . . . M, such that

(3.9.i) (ctq~,v)oj - ( u j , divv)oj + ~ ( f l ( q j . v j + qk 'Vk) + Uk, V 'Vi)r jk k

= - ( o , ~ ' v j ) r ~ , v e V ~ ,

(3.9.ii) (divqj, w) + (cuj, w) = ( f w), we Wj .

There is a technical difficulty with (3.9.i); if vje Vj and Vke Vk, it is not necessarily the case that the product of their normal components is integrable on Fjk. Also, the meaning of the restriction of an L2-function on ~2k to F~k is not clear. Thus, (3.9) is properly viewed as motivation for the treatment of the discrete case, and the remainder of the remarks in this section must be treated as heuristic.

The objective of a domain decomposition iterative method is to localize the calculations to problems over smaller domains than f2. Here, it is feasible to localize to each (2j by evaluating the quantities in (3.9) related to f2j at the new iterate level and those in (3.9) related to neighboring subdomains Ok such that F~k :~ f25 at the old level. Specifically, the algorithm in the differential case would be as follows:

(3.10) Select{q~ u o } e Vj x W j, j = 1 . . . . . M, arbitrarily ;

then recursively compute { qy, u7 } by solving

(3.11.i) (ctq~., v)~, - (u~., div v)a, + ~ (flq~" v j, v. v~ )r~ k

2~ Pqk " Vk k

+ u ' U ~ , v . v j ) j k -- ( g , v . v j ) ~ , v e Vj ,

(3.11.ii) (divq~., w)a, + (cu~, w)a, = ( f w)a,, w~ Wj.

4. The mixed finite element problem

We shall treat the case in which {O j} is a partition of f2 into individual elements (simplices, rectangles, prisms), though an inspection of the argument would indi- cate that larger subdomains are permissible. Let V h x W h be a mixed finite element

P a r a l l e l i t e r a t i v e p r o c e d u r e 9 9

space over {f2j}; any of the usual choices is acceptable: [-3, 5-7, 24-26]. Each of these spaces is defin