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1 Contemporary Mathematics Volume 00, 0000 Domain Decomposition For Linear And Nonlinear Elliptic Problems Via Function Or Space Decomposition UE-CHENG TAI Abstract. In this article, we use a function decomposition method and a space decomposition method of [5] to derive some parallel overlapping and nonoverlapping domain decomposition methods for self{adjoint linear and nonlinear elliptic problems. The function decomposition method and the space decomposition method use dierent starting points in doing the domain decomposition. 1. The function decomposition and space decomposition methods In [5], function decomposition and space decomposition methods were proposed for a general convex programming problem. Here, we shall briey show how we can use the methods for overlapping and nonoverlapping domain decomposition methods for linear and nonlinear elliptic problems. It was shown, see [5], that by suitably decomposing the energy function for an elliptic problem, we can derive the classical Alternating Direction methods, see [1, 4]. By using dierent function decompositions, we can also derive some nonoverlapping domain decomposition methods for these problems. This shows that the Alternating Direction methods and the domain decomposition methods are just dierent ways of decomposing a problem, or in other words that we can use the splitting methods to get domain decomposition methods for some linear and nonlinear elliptic and parabolic problems Mathematics Subject Classication. 65K10, 65N55, 65Y05. Key words and phrases. Parallel algorithm, domain decomposition, function decomposition, space decomposition, nonlinear problems. This paper is in nal form and no version of it will be submitted for publication elsewhere. This work is supported by the University of Bergen, Norway. It has eppeared in "Domain decomposition methods in scientic and engineering computing (Proc. of the 7th international conference on domain decomposition, Penn. State University, 1993)", D. Keyes and J. u eds, American Mathematical Society, 1995, pp c0000 American Mathematical Society /00 $ $.5 per page

2 UE-CHENG TAI The concept of space decomposition was rst introduced in a review paper [8]. There many multigrid and domain decomposition methods are presented and analyzed. It is known that the overlapping domain decomposition methods, the substructuring methods, [], and the multilevel methods, [6, 7], give some nice ways to decompose the nite element spaces. In the published papers, their convergence behaviour has been carefully analyzed for linear problems. By using the space decomposition approach of [5], we try to show that if these methods can be used for linear problems, they can also be used for some nonlinear problems. First, let us recall the results of [5]. We consider minimization (1) min vk F (v) ; K V : In case of function decomposition, we need to assume: (F1). The space V is a Hilbert space and there exist Hilbert spaces V i ; i = 1; ; ; m, such that V = \ m V i: (F). The function F : V 7! R is convex, lower{semicontinuous in V and there exist convex, lower{semicontinuous P functions F i : V i 7! R in V i ; i = m 1; ; ; m, such that F (v) = F i(v); 8v V. (F3). The subset K is closed and convex in the norm of V. There exist convex subsets K i V i ; i = 1; ; ; m, such that K = \ m K i. (F4). There exists a Hilbert space H such that V V i H; i = 1; ; ; m. In case of space decomposition, we need to assume: (S1). The space V is a reexive Banach and there exist reexive Banach spaces V i ; i = 1; ; ; m, such V = V 1 + V + + V m. (S). The subset K is closed and convex in the norm of V. There exist closed and convex subsets K i of V i such that K = K 1 + K + + K m. (S3). The function F (v) is convex, lower-semicontinuous in the norm of V and satises lim kvkv!+1 F (v) kvk V = +1. (S4). There exist constants C 0 ; C 1 such that C 0 P m v P i V m V i ; i = 1; ; ; m, and 8 >< >: 8v V; 9v i V i ; i = 1; ; ; m; such that v i = v and kv i k V C 1 kvk V : Under (F1){(F4), we nd that the minimization (1) is equivalent to () min (v1;v;v m) Q m Ki v1=v==v m F i (v i ) : kv ik V ; 8v i This is a minimization of a separable structure under the extra constraint v 1 = v = = v m : In order to use a parallel method, we need to introduce a new variable v and realize the above constraint by enforcing v i = v; i = 1; ; ; m. We will Q use augmented Lagrangian methods to deal with it. We dene L r on m H V i H m by L r (v; v i ; i ) = F i (v i ) + 1 m ( i ; v i? v) H + r m kv i? vk H :

3 FUNCTION DECOMPOSITION AND SPACE DECOMPOSITION We will seek a saddle point for L r over H Q m K i H m. We say (u; u i ; i ) is a saddle point if L r (u; u i ; i ) L r (u; u i ; i ) L r (v; v i ; i ); 8v H; v i K i ; i H : It is easy to prove that, if (u; u i ; i ) is a saddle point for L r, then u is a minimizer for (1). Under (F1){(F4), we get the following parallel algorithm for (1): Algorithm 1. Step 1. Choose initial values u 0 i K i and 0 i H (i = 1; ; m), and positive numbers r > 0, and (0; 1+p 5)r. Step. For n 1, set u n = 1 m u n?1 i + 1 rm n?1 i : Step 3. Find u n i K i; i = 1; ; ; m in parallel such that (3) F i (u n i ) + 1 m (n?1 i ; u n i ) H + r m kun i? u n k H F i (v i ) + 1 m (n?1 i ; v i ) H + r m kv i? u n k H ; 8v i K i : Step 4. Update the multipliers and go to step : n i = n?1 i + (u n i? u n ) : The following theorem (see [5]) shows the convergence: Q m Theorem 1. Suppose L r has a saddle point over H K i H m. There exists a unique solution u n i such that (3) is satised. If conditions (F1){(F4) are valid, and F i is Gateaux dierentiable with the inner product of H, then we have estimate: N (F i 0 (u n i )? F i 0 (u); u n i? u) H C m + C 3r m ; 8N > 0 : The constants C and C 3 depend only on the initial functions 0 i, u0 i and the solution u of (1). Next, we discuss the space decomposition. Under conditions (S1){(S), we can see that, if (u 1 ; u ; ; u m ) is a minimizer for (4) min (v1;v;v m) Q m Ki F (v 1 + v + v m ) ; then P m u i is a minimizer for (1). We use Jacobi method to nd a solution for the minimization (4): Algorithm. P Step 1. Choose u 0 i K m i and relaxation parameters i > 0 such that i 1. Step. For n 1, nd u n+ 1 i K i in parallel for i = 1; ; ; m such that F? m u n k + u n+ 1 i F? m u n k + v i ; 8vi K i : Step 3. Set u n+1 i as: u n+1 i = u n i + i(u n+ 1 i? u n i ), and go to step.

4 UE-CHENG TAI For this algorithm, we have the following convergence result (see [5]): Theorem. Under conditions (S1){(S4), we assume each K i is a bounded subset in V or K i = V i, function F is Gateaux dierentiable and locally uniformly convex over bounded subsets in V and F 0 is uniformly continuous over bounded subsets in V, then we have for Algorithm the convergence u n+1 = u n+1 i! u strongly in V as n! 1 : As was observed in [8], the Jacobi method may not converge for general space decomposition problems. Here, by using a suitable under relaxation, we get the sucient condition of convergence even for general minimization problems. In case that F 0 is Lipschitz continuous and coercive, an error estimate in a weak form was proved in [5], which shows the dependence of the convergence on constant C 1 =C 0.. Applications to domain decomposition Let us consider the model problem: (5) min vw 1;s 0 () Z 1 s jrvjs? fv dx We assume s. If s =, it represents a typical self{adjoint linear elliptic equation; if s 6=, it is a nonlinear elliptic equation. We will restrict our consideration only to the discrete case. As in Glowinski and Marrocco [3], if we replace the Sobolev space W 1;s 0 by a nite element space and carry out the minimization of (5) over it, the nite element solution will converge to the minimizer of (5). Assume has been partitioned into nite elements T h and the union of the nite elements form a discrete domain h. Let us dene S h as the nonconforming nite element space and V h as the conforming nite element space of k th order polynomials, i.e. S h = fv h j v h P k (e); 8e T h ; v h = 0 h g ; V h = v h j v h C 0 (); v h P k (e); 8e T h ; v h = 0 h : P We dene the inner product of S h and V h as (u; v) Sh = et h (u; v) H 1 (e). The discrete version of (5) is: 1 (6) min v hv h et Ze s jrv hj s? fv h dx : h We assume h has been partitioned into nonoverlapping subdomains 1 and, and each subdomain is the union of some elements of T h. This does not limit us to two parallel processors, because each i can again contain many disjoint subdomains. Here we will consider only the case that each i is a single connected subdomain. We will report elsewhere on the case that each i contains :

5 FUNCTION DECOMPOSITION AND SPACE DECOMPOSITION many disjoint subdomains. In order to use our algorithm, let us take m =, V 1 = V = V = S h, H = S h, K = V h, and F h;i (v h ) = et h\i Z e 1 s jrv hj s? fv h dx; 8v h S h ; i = 1; : In order to satisfy (F3), we extend each subdomain i to a larger subdomain O i ; i = 1;, and each O i is the union of some elements of T h. The subdomains i are nonoverlapping subdomains, while the subdomains O i will overlap with each other. We dene K i = v h j v h P k ; 8e T h ; v h C 0 (O i ) : If O 1 and O overlap suitably, we can have K = K 1 \ K : Thus, the assumptions (F1){(F4) are all satised and Fh;i 0 satises [3]: (F 0 h;i(v h;1 )? F 0 h;i(v h; ); v h;1? v h; ) Sh Z i jr(v h;1? v h; )j s dx; i = 1; : Therefore, we get the following convergent algorithm for (6) from Algorithm 1. Algorithm 3. Step 1. Choose initial values and constants r,. For n 1, set u n h = 1 (un?1 h;1 + un?1 h; ) + 1 r (n?1 h;1 + n?1 h; ) : Step. Solve u n h;1 H1 (O 1 ); u n h; H1 (O ) in parallel from: (7) (8) (jru n h;1j s? ru n h;1; rv h ) L (1) + 1 ( n?1 ; v h;1 h) H 1 (e) et h\o1 + r (u n h;1? u n h; v h ) H 1 (e) = (f; v h ) L (1) ; 8v h V h ; et h\o1 (jru n h;j s? ru n h;; rv h ) L () + 1 ( n?1 ; v h; h) H 1 (e) et h\o + r (u n h;? u n h; v h ) H 1 (e) = (f; v h ) L () ; 8v h V h : et h\o We obtain the value of u n h;1 in no 1, and the value of u n h; in no through: u n h;1 = u n h? 1 r n?1 h;1 in no 1 ; u n h; = u n h? 1 r n?1 h; in no : Homogeneous Dirichlet boundary conditions should be enforced for (7) and (8). Step 3. Update the multipliers as: n h;i = n?1 h;i + (u n h;i? un h ); in ; i = 1; ; and go to step. In [5] several other algorithm were also obtained for (5) and other linear selfadjoint elliptic problems. Next, we use overlapping domain decomposition for (6). As before, we assume we have partitioned into nite elements. We then decompose h into overlapping subdomains and each subdomain is the union of some elements of

6 UE-CHENG TAI T h. We assume that the subdomains can be marked by m colors, so that the subdomains with the same color do not intersect with each other. We denote the union of the subdomains with the ith color as i. Let us take V h;i = v h j v h C 0 (); v h P k ; 8e T h \ i ; v h = 0 in h n i and h : P m If the subdomains overlaps suitably, we will have V h = V h;i, and the constants C 0 ; C 1 can be explicitly estimated, see [, 9]. For simplicity, we dene for Algorithm : u n = u n i ; wn+1 i = u n+1 k + u n+ 1 i = u n? u n i + un+1 i ; 8i; n : We get from Algorithm the following overlapping algorithm for (6): Algorithm 4. P Step 1. Choose u 1 h;i V m h;i and constants i > 0 such that i 1. Step. For n 1, solve in parallel in each subdomain i the following problem: 8 >< >: (jrw n+1 h;i j s? rw n+1 h;i ; rv h ) L (i) = (f; v h ) L (i) ; 8v h V h;i ; w n+1 h;i = 0 i h ; w n+1 h;i = u n h = u n h;k i n@ h : Step 3. Set u n+1 h;i as: u n+1 h;i = u n h;i + i(wh;i n+1? u n h ) in i; i = 1; ; m; and go to the next iteration. References 1. J. Douglas and H. H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc.. 8 (1956), 41{439.. M. Dryja and O. Widlund, Multilevel additive methods for elliptic nite element problems, Parallel Algorithms for Partial Dierential Equations (W. Hackbush, ed.), Proceeding of the 6th GAMM seminar, Kiel, Jan. 19{1, 1990, Vieweg & Sons, Braunchweig, 1991, pp. 58{ R. Glowinski and A. Marrocco, Sur l'approximation par elements nis d'ordre un, et lan resolution par penalisation-dualite, d'une classe de problemes de Dirichlet non lineaires, Rev. Fr. Autom. Inf. Rech. Oper. Anal. Numer. R- (1975), 41{ D. H. Peaceman and H. H. Rachford, The numerical solution of parabolic and elliptic dierential equations, SIAM J. Appl. Math. 3 (1955), 4{ {C. Tai, Parallel function and space decomposition methods with applications to optimization, splitting and domain decomposition, Preprint No. 31{199, Institut fur Mathematik, Technische Universitat Graz (199). 6. H. Yserentant, On the multilevel splitting of nite element spaces, Numer. Math. 49 (1986), J. C. u, Theory of multilevel methods. Doctoral thesis. Cornell, Rep. AM{48, Penn. State U. (1989). 8., Iteration methods by space decomposition and subspace correction jour SIAM Rev. 34 (199). 9.. J. Zhang, Multilevel Schwarz methods, Numer. Math. 63 (199), Department of Mathematics, University of Bergen, Allegt 55, 5007, Bergen, Norway. address: Tai@mi.uib.no.