On an uniform multidomain decomposition method applied to a singularly perturbed problem with regular boundary layers

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1 Journal of Computational and Applied Mathematics 166 (2004) On an uniform multidomain decomposition method applied to a singularly perturbed problem with regular boundary layers Igor Boglaev, Vic Duoba Institute of Fundamental Sciences, Massey University, Private Bag , Palmerston North, New Zealand Received 29 July 2002; received in revised form 6 June 2003 Abstract This paper deals with an iterative algorithm for multidomain decomposition applied to the solution of a singularly perturbed convection diusion problem.uniform convergent properties of the algorithm are established.numerical results are presented. c 2003 Elsevier B.V. All rights reserved. Keywords: Singularly perturbed convection diusion problem; Regular boundary layer; Uniform convergence; Domain decomposition method; Decomposition of boundary layers; Parallel computing 1. Introduction We are interested in iterative domain decomposition methods for solving the convection diusion problem with regular boundary layers ( ) 9 2 u 9x + 92 u 9u + b 2 9y 2 1 9x + b 9u 2 + cu = f; (x; y) ; 9y u = g on 9; b 1 1 0; b 2 2 0; c c 0 on ; (1) where = {P =(x; y):0 x 1; 0 y 1}, is a small positive parameter, 1 ; 2 and c are constants and 9 is the boundary of.for 1, problem (1) is singularly perturbed and characterized by the regular boundary layers of width O( ln ) atx = 1 and y =1. Corresponding author. addresses: i.boglaev@massey.ac.nz (I.Boglaev), v.duoba@massey.ac.nz (V.Duoba) /$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi: /j.cam

2 14 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Iterative domain decomposition algorithms based on Schwarz-type alternating procedures for solving singularly perturbed problems have received much attention for their remarkable speed and parallelizability (see [1,4 7] and references cited there). In [7], for solving problem (1), the classical Schwarz alternating method and some variants of it were analysed.in the case of domain decomposition into two subdomains, a convergence rate for the continuous problem (i.e., without resort to discretization in subdomains) as a function of the perturbation parameter and an amount of overlap between two subdomains was studied. In [5], on the basis of asymptotic criteria, representations of optimal interface positions for the Schwarz alternating procedure were derived.for one dimension version of problem (1), in the case of domain decomposition into the two subdomains [0;x 1 ]; [x 2 ; 1];x 1 x 2, the interface positions x 1 ;x 2 are of order O( ln ).If the number of mesh points in each subdomain is the same, N, then this interface condition is satised when N is of order O(1=( ln )).Since the number of mesh points depends inversely on the perturbation parameter, then, in general, this approach leads to a nonuniform (in the perturbation parameter) convergent domain decomposition procedure. In [6], a two-level iterative domain decomposition method with overlapping vertical strips has been introduced.the iterative method from [6] consists of the two iterative processes: outer iterations and inner iterations.one outer iteration represents computation of dierence problems on the overlapping subdomains in serial, starting from the rst left subdomain and nishing o on the last right subdomain (according to upwind error propagation).an inner iteration consists of computation of the dierence problem on each subdomain in parallel.but, how it follows from the theoretical investigation in [6], the suggested iterative method, in general, cannot guarantee convergence if it starts from an arbitrary initial guess. In this paper, we introduce a multidomain modication of the Schwarz alternating method proposed in [3] and applied in [1] for solving singularly perturbed reaction diusion problems and in [2] for solving a singularly perturbed problem with parabolic layers.in this approach, the domain is partitioned into many nonoverlapping subdomains with interface.small interfacial subdomains are introduced across the interface, and values computed for can be used as approximate boundary values for solving problems on the nonoverlapping subdomains. We show that the algorithm converges uniformly in the perturbation parameter on the piecewise equidistant meshes of Shishkin-type [8].The piecewise uniform meshes allow us to decompose the computational domain into subdomains outside boundary layers and inside them as well, and possess load balancing.this property is very important for implementation of the iterative algorithms on parallel computers, since it avoids loss of eciency due to any processors being idle. The organization of this paper is as follows.in Section 2, we consider an undecomposed algorithm which exhibits uniform convergence in the perturbation parameter.in Section 3, we construct the iterative algorithm based on the multidomain decomposition and investigate convergence properties of this algorithm.we estimate here the convergence rate of the algorithm for two variants of decomposition of the computational domain: the balanced and unbalanced ones.finally, in Section 4, numerical results are presented. 2. Undecomposed algorithm Here for solving problem (1), we construct a dierence scheme on piecewise uniform meshes which possesses uniform convergence in the perturbation parameter.on introduce a rectangular

3 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) mesh h = hx hy : hx = {x i ; 0 6 i 6 N x ; x 0 =0; x Nx =1; h xi = x i+1 x i }; hy = {y j ; 0 6 j 6 N y ; y 0 =0; y Ny =1; h yj = y j+1 y j }: (2) For a mesh function U(P); P h, we use the upwind dierence scheme U(P)=f(P); P h ; U = g on 9 h ; (3) where U(P) is dened by U = (D x +D x + D y +D y )U + b 1D x U + b 2 D y U + cu: D+D x U(P), x D+D y y U(P) and Dx U(P), D y U(P) are the central dierence and backward dierence approximations to the second and rst derivatives, respectively, D x +D x U ij =( xi ) 1 [(U i+1;j U ij )(h xi ) 1 (U ij U i 1;j )(h xi 1 ) 1 ]; D y +D y U ij =( yj ) 1 [(U i;j+1 U ij )(h yj ) 1 (U ij U i;j 1 )(h yj 1 ) 1 ]; D x U ij =(h xi 1 ) 1 (U ij U i 1;j ); D y U ij =(h yj 1 ) 1 (U ij U i;j 1 ); xi =2 1 (h xi 1 + h xi ); yj =2 1 (h yj 1 + h yj ); where U ij = U(x i ;y j ). Now introduce a special nonuniform mesh from [8] that is adapted to the singularly perturbed behaviour of the exact solution.a piecewise equidistant mesh of Shishkin-type is formed by the following manner.we divide each of the intervals x =[0; 1] and y =[0; 1] into two parts [0; 1 x ]; [1 x ; 1], and [0; 1 y ]; [1 y ; 1], respectively.assuming that N x ;N y are even, in each part we use a uniform mesh with N x =2+1 and N y =2 + 1 mesh points in the x- and y-directions, respectively. This denes the piecewise equidistant mesh in the x- and y-directions condensed in the boundary layers at x = 1 and y =1: { ihx ; i=0; 1;:::;N x =2; x i = 1 x +(i N x =2)h x ; i= N x =2;:::;N x ; { jhy ; j =0; 1;:::;N y =2; y j = 1 y +(j N y =2)h y ; j = N y =2;:::;N y ; h x = 2(1 x )Nx 1 ; h x =2 x Nx 1 ; h y = 2(1 y )Ny 1 ; h y =2 y Ny 1 : (4) The transition points 1 x,1 y are determined by x = min{2 1 ; ( 0 = 1 ) ln N x }; y = min{2 1 ; ( 0 = 2 ) ln N y }; where 0 is positive constant.if x;y =1=2, then N 1 x; y are very small relative to.this is unlikely in practice, and in this case the dierence scheme (3) can be analysed using standard techniques.

4 16 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) We, therefore, assume that x =( 0 = 1 ) ln N x ; y =( 0 = 2 ) ln N y ; h x =2( 0 = 1 )Nx 1 lnn x ; Nx 1 h y =2( 0 = 2 )Ny 1 lnn y ; Ny 1 h x 2N 1 x ; h y 2N 1 y : (5) Theorem 1. Let 0 1 in (5). The dierence scheme (3) on the piecewise uniform mesh (4) and (5) converges -uniformly to the solution of (1): max U(P) u(p) 6 Cd(N; 0 ); P h d(n; 0 )= 0 ln N + 1 N N ; N = min{n x;n 0 1 y }; where constant C is independent of ; N. The proof of the theorem can be found in [8]. 3. Domain decomposition algorithm We consider decomposition of the domain into M nonoverlapping subdomains (vertical strips) m ; m=1;:::;m: m = x m (0; 1); x m =(x m 1 ;x m ); m = {x = x m ; 0 6 y 6 1}; m m+1 = m : Thus, we can write down the boundary of m as 9 m = m 0 m 1 m ; m 0 = 9 9 m : Additionally, we consider (M 1) interfacial subdomains! m ; m=1;:::;m 1:! m =! m x (0; 1);! m x =(xm;x b m); e! m 1! m = ; xm b x m xm; e m=1;:::;m 1: The boundaries of! m are denoted by b m = {x = xm; b 0 6 y 6 1}; e m = {x = xm; e 0 6 y 6 1}; 0 m = 9 9! m : Fig. 1 illustrates the x-section of the multidomain decomposition. On m ; m =1;:::;M;! m ; m =1;:::;M 1, we introduce meshes h m = hx m hy,! h m =! hx m hy, where hy from (2) and hx m = {x mi ; i=0; 1;:::;N mx ; x m0 = x m 1 ; x Nmx = x m ; h mi = x m;i+1 x mi }; m =1;:::;M;

5 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Fig.1.! hx m = {X mi ; i=0; 1;:::;N m! ; X m0 = xm; b X Nm! = xm; e H mi = X m;i+1 X mi }; m =1;:::;M 1: We suppose that h = h m, and the mesh points in! h m; m=1;:::;m 1 coincide with the mesh points of h. We consider the following iterative domain decomposition algorithm for solving problem (3).On each iterative step, rstly, we solve problems on the nonoverlapping subdomains h m; m=1;:::;m with Dirichlet boundary conditions passed from the previous iterate.then Dirichlet data are passed from these subdomains to the interfacial subdomains! h m; m =1;:::;M 1, and problems on the interfacial subdomains are computed.finally, we impose continuity for piecing the solutions on the subdomains together. Step 0: Initialization.On the whole mesh h, choose an initial mesh function V (0) (P); P h satisfying the boundary conditions V (0) (P)=g(P) on9 h. Step 1: On subdomains h m; m =1;:::;M, compute mesh functions v m (n) (P), m =1;:::;M (here the index n stands for a number of iterative steps) satisfying the following dierence schemes: v (n) m (P)=f(P); P h m; (6) v (n) m (P)= { g(p); P h0 m ; h0 m = 0 m h ; V (n 1) (P); P h m 1 h m; h m = m h m: (7) Step 2: On the interfacial subdomains problems:! h m; m =1;:::;M 1, compute the following dierence z (n) m (P)=f(P); P! m; h g(p); m (P)= v m (n) (P); v (n) m+1 (P); P he m ; z (n) P h0 m ; P hb m ; h0 m = 0 m h ; hb m = b m! h m; hb m = b m! h m: Step 3: Compute the continuous solution V (n) (P);P h by piecing the solutions on the subdomains { v (n) V (n) m (P); P m h \ (! m 1 h (P)=!h m); m=1;:::;m; (9) z m (n) (P); P! h m; m=1;:::;m 1: Step 4: Stopping criterion.if a prescribed accuracy is reached, then stop; otherwise go to Step 1. (8)

6 18 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Algorithm (7) (9) can be carried out by parallel processing, since on each iterative step n the M problems (7) for v m (n) (P); m=1;:::;m and the (M 1) problems (8) for z m (n) (P); m=1;:::;m 1 can be implemented concurrently. Remark 1. We note that the original Schwarz alternating algorithm with overlapping subdomains is a purely sequential algorithm.to obtain parallelism, one needs a subdomain colouring strategy, so that a set of independent subproblems can be introduced.the proposed modication of the Schwarz algorithm is very suitable for parallel computing.the computational eectiveness of algorithm (7) (9) depends on sizes of the interfacial subdomains.our theoretical analysis and numerical experiments represented below show that the small-sized interfacial subdomains are needed to essentially reduce the number of iterations Convergence of algorithm (7) (9) We now establish convergence properties of algorithm (7) (9). On mesh h = hx hy : hx = {x i ; i=0; 1;:::;N x ; x 0 = x a ;x N x = x b }; where x a x b, and hy from (2), consider the following dierence problems: w(p)+(p)w(p)=f(p); P ; h w(p)=w 0 (P); P 9 ; h (10) where = (D+D x x + D+D y y )+b 1D x + b 2 D y ; ( = c) and s (P)+ s (P)=0; P ; h s (P)=1; P hs ; s (P)=0; P 9 h \ hs ; s=1; 2; 3; 4; (11) where (P) = const 0, hs is the sth side of the rectangular mesh h.we suppose that h1 = {x a ; y j ; 0 6 j 6 N y }; h2 = {x b ; y j ; 0 6 j 6 N y }; h3 = {x i ; 0 6 i 6 N x ; y =0}; h4 = {x i ; 0 6 i 6 N x ; y =1}: Lemma 1. If w(p) and s (P), s =1; 2; 3; 4 are the solutions to (10) and (11), respectively, then we have the following estimates: w(p) h 6 max[ w 0 (P) 9 h ; F(P) h = ]; (12) [ ] 4 4 w(p) 6 s (P) w 0 (P) hs + 1 s (P) F(P) h = ; (13) s=1 where P h, and w(p) 9 h max w(p) ; F(P) h P 9 h max F(P) : P h s=1

7 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Proof. The required estimate (12) follows immediately from the maximum principle for the dierence operator +. Introduce the function W (P) satisfying the problem W (P)+ W (P)= F(P) h ; P h ; W (P)= w 0 (P) hs; P hs ; s=1; 2; 3; 4: W (P) can be written in the form [ 4 W (P)= s (P) w 0 (P) hs + 1 s=1 ] 4 s (P) F(P) h = : s=1 The correctness of this estimate can be tested by direct substitution.from a standard comparison theorem, it follows that: w(p) 6 W (P); P h : This concludes the proof of the lemma. Introduce the notation q b m = 1 m(p)+ 2 m(p) hb m ; qe m = 1 m+1(p)+ 2 m+1(p) he m ; q = max 16m6M 1 (qb m;qm); e (14) where 1 m(p); 2 m(p); m=1;:::;m are the solutions to (11) with = c on h m for s = 1 and s =2, respectively. We formulate and prove a convergence result for algorithm (7) (9). Theorem 2. Algorithm (7) (9) on the piecewise uniform mesh (4) and (5) converges to the solution u(p) of (1) with the following rate: max V (n) (P) u(p) 6 C(d(N; 0 )+q n ); P h here d(n; 0 ) and V (n) (P) are dened in Theorem 1 and (9), respectively, the contraction coecient q (0; 1) is dened in (14), and constant C is independent of, N x ; N y, and q. Proof. Introduce the mesh functions (n) m (P)=v (n) m (P) U(P); P h m; m=1;:::;m; (n) m (P)=z (n) m (P) U(P); P! h m; m=1;:::;m 1; W (n) (P)=V (n) (P) U(P); P h ;

8 20 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) where U(P) from (3).From (3) and (7) (9), we have m (n) (P)=0; P m; h (n) m (P)=V (n 1) (P) U(P); P 9 h m; m=1;:::;m; (15) (n) m (P)=0; P! m; h m (n) (P); m (P)= (n) P hb m ; (n) m+1 (P); P he m ; From (15) and (13), we have 0; P h0 m ; m=1;:::;m 1: (n) m (P) 6 1 m(p) (n) m (P) h m m(p) (n) m (P) h m ; (17) where m 1;2 (P) are the solutions to (11) with = c on domain h m for s =1; 2.Since m (n) (P)=V (n 1) (P) U(P), P m 1 h h m, then it follows the inequality: m (n) (P) 6 [m(p)+ 1 m(p)] W 2 (n 1) (P) h; P h m; m=1;:::;m: From here, (9) and the maximum principle for (15) and (16), we conclude the estimate W (n) (P) 6 q W (n 1) (P) h h: (18) Note that function m (P)=m(P)+ 1 m(p) 2 is the solution to the problem m (P)+c m (P)=0; P h m; (16) m (P)=1; P h m 1 h m; m (P)=0; P h0 m ; m=2;:::;m 1; 1 (P)=1; P h 1; 1 (P)=0; P h0 1 ; M (P)=1; P M 1; h M (P)=0; P M h0 ; Applying the strong maximum principle to this problem, we establish that m (P) 1; hb m m(p) 1; he m 1 and we prove that q in (14) belongs to the open interval (0; 1).From here, (18) and Theorem 1, we conclude the convergence property of algorithm (7) (9). Remark 2. Theorem 2 guarantees us that the domain decomposition algorithm (7) (9) converges for any initial guesses. Consider (11) with = c, and introduce notations: q m = m!(p) 2 h m + m(p) 2 ; (19) hb m where m!(p); 2 m(p) 2 are the solutions to (11) on! h m and h m, respectively.

9 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Theorem 3. Suppose that for algorithm (7) (9) a number of iterates n M 1, then the following estimate on rate of convergence holds: (n) 6 Q (n (M 1)) ; where we denote (n) = max W (n) (P) h 16m6M 1 m ; Q= M 1 m=1 q m ; with the notations from (19), W (n) (P) =V (n) (P) U(P), and U(P); V (n) (P) from (3) and (9), respectively. Proof. From (16) and (13), we have m (n) (P) 6 m!(p) 1 m (n) (P) + hb 2 m m!(p) (n) m+1 (P) ; he m P! h m; m=1;:::;m 1; where m;!(p) 1;2 are the solutions to (11) with = c on domain! h m for s =1; 2.Since (n) 1 (P)=0; P h 0, then from (17) for m = 1, we get the estimate (n) 1 (P) 6 2 1(P) (n) 1 (P) ; P h 1 1: h From here, (18) and (21) for m = 1 and taking into account that (n) 2 (P)=(n 1) 1 (P);P 1 h, we have (n) 1 (P) 6 1 h [ 2 1;!(P) h 1 + 1(P) 2 hb ](n 1) : 1 The last inequality in notations (19) is rewritten as (n) 1 (P) 6 q 1 (n 1) : (22) 1 h From (17) for m =2, (18) and (21) for m = 2, we conclude that (n) 2 (P) 6 2 h 1 2;!(P) h 2 [ 2(P) 1 hb (n) 2 2 (P) + 1 h 2 2(P) hb (n) 2 2 (P) ] 2 h + 2 2;!(P) h 2 (n 1) : Since (n) 2 (P)=(n 1) 1 (P);P 1 h and using (22) with n 1, the above inequality has the form (n) 2 (P) 6 q 2 (n 1) + q 2 h 1 (n 2) : By induction, it can be proved the following estimate: (n) m (P) h m 6 (20) (21) m q k (n (m k+1)) : (23) k=1 From (12) and (15), we conclude that (n) m (P) 6 (n 1).Using this estimate and again evaluating (16) with (12), we get max m (n) m (P) 6 max m {max[ (n) m (P) hb m ; (n) m+1 (P) he m ]} 6 (n 1) :

10 22 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Since (n) 6 max m m (n) (P), we prove that (n) 6 (n 1) ; n 1: From here and (23), theorem holds true. Remark 3. From (20), we conclude that (n) 6 (Q e ) n (0) ; Q e =(Q) 1=(M 1) ; where Q e is the eective contraction factor of the domain decomposition algorithm (7) (9) Convergence analysis of algorithm (7) (9) The interfacial subdomains outside the boundary layer: Consider algorithm (7) (9) with the interfacial subdomains! m hx ; m=1;:::;m 1 located in the x-direction outside the boundary layer, i.e., N Mx N x =2+N M 1;!, where the notations are from (4) and (6). On! m hx and m hx introduce the one-dimensional problems: x 2 m(x)=0; x! m hx ; 2 m(xm)=0; b 2 m(xm)=1; e x 2 m(x)=0; x hx m ; 2 m(x m 1 )=0; 2 m(x m )=1; x = D+D x x + 1 D : x The solutions of these problems can be written in the forms 2 m(x i )= ( x) Nm! i ( x ) Nm! ; i=0;:::;n m! ; 1 ( x ) Nm! (24) 2 m(x i )= ( x) Nmx i ( x ) Nmx 1 ( x ) Nmx ; i=0;:::;n mx ; x = ( + 1 h x ) 1 : (25) Lemma 2. The following inequalities hold true: 2 m!(p) 6 2 m(x); P! h m; m=1;:::;m 1; m(p) m(x); P h m; m=1;:::;m; (26) where m!(p) 2 and m(p) 2 are the solutions to (11) with = c, s =2, on! h m and h m, respectively. Proof. We check the rst inequality in (26) for 2 m!(p), and the inequality for 2 m(p) can be proved in a similar way.note that from the maximum principle, it follows that 2 m(x) 0; 2 m!(p) 0. From (11) and (24), we conclude that w m (P)= 2 m(x) 2 m!(p) satises the dierence problem w m (P)=(b 1 (P) 1 )D x 2 m(x)+c 2 m!(p); P! h m; w m (P) 0 on 9! h m:

11 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Prove that D x 2 m(x) 0.If on the contrary, D x 2 m takes negative values, then there exists a point x i, where 2 m(x i ) 2 m(x i 1) 0; 2 m(x i +1) 2 m(x i ) 0.It means that the left-hand side in (24) atx i is strictly positive, so we get contradiction.from 2 m!(p) 0, D x 2 m(x) 0 and (1), we conclude that the right-hand side in the dierence equation is nonnegative.by the maximum principle for the operator, it follows that w m (P)= 2 m(x) 2 m!(p) 0; P! h m.thus, we prove the lemma. From (25), we estimate 2 m and 2 m by 2 m(x i ) 6 p i Nm! ; i=0;:::;n m! ; 2 m(x i ) 6 p i Nm ; i=0;:::;n m : From here and (19), we conclude that p=1+ 1h x ; q m = 2 m(p) hb m + 2 m!(p) h m 6 p Db m =hx + p De m =hx ; D b m = x m x b m; D e m = x e m x m : (27) Now, we estimate the convergence factor Q in Theorem 3 by Q 6 (M 1) max 16m6M 1 {p Db m =hx + p De m =hx } To minimize p Db m =hx + p De m =hx, we choose x m as the middle point of the interval [x b m;x e m].thus, Q 6 Q; Q =2(M 1)p D=2hx ; D= min 16m6M 1 {xe m xm}; b (28) where D is the minimal size of the interfacial subdomains in the x-direction. Remark 4. In the case h x, say = h x; 1, we have ln p ( 1) ln( 1 =h x ).If 1 = O(1), then we approximate Q in (28) by ) D( 1) ln h 1 x Q 2(M 1) exp ( : 2h x If we suppose that all interfacial subdomains are of equal and maximal size D = N x (2(M 1)) 1 h x, then conclude the following approximation: ( Q 2(M 1) exp ( 1)N ) xln N x : 4(M 1) Thus, we conclude that in the case of the interfacial subdomains located in the x-direction outside the boundary layer, the iterative domain decomposition algorithm (7) (9) converges -uniformly. The interfacial subdomains inside the boundary layer (the balanced domain decomposition): Suppose that N x is divisible by 2M and M is even, we decompose the boundary layer [1 x ; 1] and the region outside the layer [0; 1 x ] into M=2 equal subdomains, respectively, where x from (5).We note that each of the subdomains hx m ; m =1;:::;M contains the same number of mesh

12 24 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) points 2I +1;I = N x =(2M).From (6), we have { xmi = x hx m 1 + ih x ; i=0; 1;:::;2I; m =1;:::; M; m = x mi = x m 1 + ih x ; i=0; 1;:::;2I; m = M +1;:::;M; x m 1 = { 2(m 1)Ihx ; m=1;:::; M; (1 x )+2(m M 1)Ih x ; m= M +1;:::;M; (29) where M M=2 and h x, h x are the uniform step sizes outside and inside the boundary layer, respectively.we choose the interfacial subdomains in the following forms: { Xmi = x! hx m b + ih x ; i=0; 1;:::;2I! ; m=1;:::; M 1; m = X mi = xm b + ih x ; i=0; 1;:::;2I! ; m= M +1;:::;M 1;! hx M = { X Mi = xb M + ih x; i=0; 1;:::;I! ; X Mi =(1 x)+ih x ; i= I! +1;:::;2I! ; x m I! h x ; m=1;:::; M 1; xm b = (1 x ) I! h x ; m= M; x m I! h x ; m= M +1;:::;M 1: (30) Here the interfacial subdomains! hx m ; m =1;:::;M 1 contain the same number of mesh points 2I! + 1, and the centre of the discrete interval! hx m is located at x m.we suppose 1 6 I! 6 I, such that! m 1 hx!hx m = ; m=2;:::;m 1. Now for decomposition (29) and (30), we estimate coecient Q from (29) in Theorem 4.Since! h m; m=1;:::; M 1 are located outside the boundary layer, then for q m ; m=1;:::; M 1, we can use estimate (27), i.e., q m 6 Q 1 ; Q 1 =2p I! ; p=1+ 1h x ; m=1;:::; M 1: (31) Subdomains! h m; m= M +1;:::;M 1 and h m; m= M +1;:::;M are localized inside the boundary layer, where the uniform step size h x is in use.now, we apply (27) with h x, and get q m 6 Q 2 ; Q 2 =2p I! To estimate q M ; p =1+ 1h x ; m= M +1;:::;M 1: (32) from (19), we only have to evaluate 2 2 M (P) 6 p hb I! : M Similar to Lemma 2, we can prove the inequality 2 M! piecewise equidistant mesh. 2 M (x); x!hx M 2 M (x i)= 2 M (x M )( x) I! i ( x ) I! 1 ( x ) I! ; i=0;:::;i! ; (P) M!, since from (27) it follows that: h (P) 6 2 M can be written in the form M (x); P!h M, where!hx M is the

13 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) M (x (1 2 M i)= (x M ))( x) 2I! i + 2 (x M M ) ( x) I! ; i= I! ;:::;2I! ; 1 ( x ) I! 2 M (x M )=d 1d 2 (d 1 d 2 + d 3 ) 1 ; d 1 = h x ( + 1 (h x + h x )2 1 )h x ; d 2 = ( x) I! 1 ( x ) I! 1 ( x ) I! ; d 3 = 1 x 1 ( x ) : I! Thus, we conclude the estimate 2 M! (x M ;y j) 6 d 1 d 2 (d 1 d 2 + d 3 ) 1 ; j =0; 1;:::;N y : Writing down the term d 1 d 2 in the form d 1 d 2 = d 1 h x [ + 1 (h x + h x )2 1 ] 1 ; d=( x ) I! (1 ( x ) I! ) 1 ; and taking into account that 1 h x [ + 1 (h x + h x )2 1 ] 1 6 2, we have the estimate d 1 d 2 6 2d: Since the maximum of d 1 d 2 (d 1 d 2 + d 3 ) 1 over d 1 d 2 occurs at the maximum value of d 1 d 2, it follows that 2 M! (x M ;y j) 6 Q 3 ; Q 3 =2d(2d + d 3 ) 1 ; j =0; 1;:::;N y : (33) Now, substituting this estimate and (31) and (32) in(20), we get the estimate on Q Q 6 (M=2)(Q 1 + Q 2 )+Q 3 : Remark 5. From (5) and (32), it follows that Q 2 = 2 exp( I! ln( lnn x =N x )); 0 1; then for suciently large N x, we have Q 2 2 exp( 2 0 I! ln N x =N x ); I! 6 N x (2M) 1 : From (33), it follows that Q 3 =2( x ) I! [2( x ) I! + d 3 (1 ( x ) I! )] 1, and since d 3 1 x,we conclude Q 3 6 2( x ) I! (1 x ).If we suppose that h x, then Q 3 is approximated by Q 3 2 exp( 2 0 I! ln N x =N x ); I! 6 N x (2M) 1 : For suciently large N x, h x and the maximal size of the interfacial subdomains I! = I; I = N (2M) 1, from (31), it follows that ( ) 1 h Nx=2M x Q 1 2 =o(q 2 ); and we have Q 6 Q; Q MNx 0=M : (34) Thus, for the balanced decomposition (29) and (30), the iterative domain decomposition algorithm (7) (9) converges -uniformly.

14 26 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Remark 6. Consider the limiting case of decomposition (29) and (30), where only the last subdomain h M lies in the boundary layer (the unbalanced decomposition), i.e., region [0; 1 x] outside the layer is decomposed into M 1 equal subdomains and all subdomains h m; m=1;:::;m 1 contain the same number of mesh points.if h x, we approximate Q 3 by Nx 0=2, and conclude that Q 6 Q; Q MNx 0=2 : (35) Thus, for the unbalanced decomposition, algorithm (7) (9) converges uniformly in the small parameter. Note here that getting the better convergence property of the algorithm on the unbalanced decomposition, we have lost load balancing, since the sizes of domains h M and!h M 1 for large values of M are suciently bigger then others.to keep load balancing for the algorithm on the unbalanced decomposition, we need to use the second level of parallelization for solving discrete systems on these two subdomains. 4. Numerical results As a test problem, consider problem (1) with (b 1 ;b 2 )=(2; 3); c=1; g= 0 and f(x; y) such that the function u(x; y) = 2 sin(x)(1 exp( 2(1 x) 1 ))y 2 (1 exp( 3(1 y) 1 )) is the exact solution of the problem. The stopping criterion is chosen in the form max V (n) (P) u(p) 6 ln N=N; P h where V (n) (P) from (9).In all our numerical experiments, we choose N x = N y = N, and the linear systems have been solved using GMRES solver with the diagonal preconditioner as in [9]. GMRES is a class of iterative solvers based upon Krylov subspace methods, but have the added feature that the solution has the minimal residual over the space P 0 + K n, where the former is the starting vector and the latter the Krylov subspace dened by the span of the successive descent directions, or, equivalently, the successive residuals. GMRES methods are generally stable and robust.their generic disadvantage of requiring an additional dimension to be stored per additional basis vector can be avoided by restarting the process, but with the stopping solution used as the starting solution subsequently.in the experiments, the maximum size of Krylov subspace constructed was set to 20, and a maximum of 50 restarts is permitted (for a fuller explanation of the GMRES methodology see [10]). The sizes x ; y of the boundary layers in (5) are dened by the parameter 0.For the balanced decomposition (29) and (30), the subdomains hx m ; m= M=2+1;:::;M are situated in the boundary layer [1 x ; 1], and, hence, the parameter 0 may be considered as a parameter of the domain decomposition algorithm.introduce the following notation: n k is a number of iterations with k = 0. In Table 1, we give the numbers of iterations n k ; k =2; 4 on the balanced domain decomposition with the maximal size of the interfacial subdomains at N =32; 128.Our numerical results show, that for N; M xed, n k is independent of.the uniform convergence result conrms estimate (34).For M xed, the number of iterations n k (N ) is a decreasing function of N, and for N xed, n k (M) and

15 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Table 1 Numbers of iterations for the balanced decomposition M n 2; n 4; N =32 n 2; n 4; N = ; 2 2; 2 2; 2 2; 2 2; 2 2; 2 4 4; 4 5; 4 5; 4 4; 4 4; 4 4; ; 8 12; 9 12; 9 10; 7 10; 7 10; 7 16 n.a. n.a. n.a. 30; 18 30; 18 30; Table 2 Numbers of iterations for the balanced decomposition with N = 128 M n k k Table 3 Numbers of iterations for the balanced decomposition with N = 128 M n n.a. n.a. n.a. n.a. I! the ratio n k (2M)=n k (M) are increasing functions of M.These results are in qualitative agreement with the estimate from (34). The number of iterations as a function of the parameter k is listed in Table 2.The experiments show that for M xed, n(k) is a monotone decreasing function of k which is in agreement with estimate (34).We note here that the limiting value of n(k) is of order M. In Table 3, for various numbers of M and sizes I! of the interfacial subdomains (the full width of the interfacial subdomain is 2I! +1), we give the numbers of iterations for the balanced decomposition with k=2 and N =128.The number of iterations as a function of the size of the interfacial subdomains is monotone decreasing.these functions for M xed vary very quickly for small values of I! and

16 28 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) Table 4 Numbers of iterations for the unbalanced decomposition M n 2; n 4; N =32 n 2; n 4; N = ; 3 3; 3 3; 3 3; 3 3; 3 3; 3 5 5; 4 5; 4 5; 4 5; 4 5; 4 5; 4 9 n.a. n.a. n.a. 8; 7 9; 7 9; Table 5 Numbers of iterations for the unbalanced decomposition with N = 128 M n k k relatively small sizes of the interfacial subdomains are needed to essentially reduce the numbers of iterations. Table 4 represents the numbers of iterations n k ; k=2; 4 for the unbalanced domain decomposition with the maximal size of the interfacial subdomains and N =32; 128.The main features of the algorithm on the balanced domain decomposition highlighted from Table 1 hold true for the unbalanced domain decomposition, where only the last subdmain h M lies in the x-direction inside the boundary layer.these results conrm estimate (35).In the contrast to the algorithm on the balanced decomposition, the number of iterations on the unbalanced one is a linear function in M which is in agreement with the estimate from (35).As we can see from Tables 1 and 4, the algorithm on the unbalanced decomposition converges suciently faster then on the balanced decomposition, comparing M = 4(2 + 2) from Table 1 with M = 3(1 + 2) from Table 4, and so on. Similar to Table 2 for the balanced domain decomposition, in Table 5, we give the number of iterations as a function of the parameter k for the unbalanced domain decomposition.it should be noted that for suciently large values of M, the limiting values of n are less then M, e.g., for M = 17 the limiting value is n = 11. References [1] I.Boglaev, On a domain decomposition algorithm for a singularly perturbed reaction diusion problem, J.Comput. Appl.Math.98 (1998) [2] I.Boglaev, V.Duoba, Domain decomposition for a singularly perturbed problem with parabolic layers, in: V.V. Kluev, N.E. Mastorakis (Ed.), Topics in Applied and Theoretical Mathematics and Computer Science, WSEAS Press, Cairns, 2001, pp.7 12.

17 I. Boglaev, V. Duoba / Journal of Computational and Applied Mathematics 166 (2004) [3] C.N.Dawson, Q.Du, T.F.Dupont, A nite dierence domain decomposition algorithm for numerical solution of the heat equation, Math.Comput.57 (1991) [4] P.Farrell, I.Boglaev, V.Sirotkin, Parallel domain decomposition methods for semi-linear singularly perturbed dierential equations, Comput.Fluid Dynamics J.2 (1994) [5] M.Garbey, A Schwarz alternating procedure for singular perturbation problems, SIAM J.Sci.Comput.17 (1996) [6] M.Garbey, Yu.A.Kuznetsov, Yu.V.Vassilevski, A parallel Schwarz method for a convection diusion problem, SIAM J.Sci.Comput.22 (2000) [7] T.Mathew, Uniform convergence of the Schwarz alternating method for singularly perturbed advection diusion equations, SIAM J.Numer.Anal.35 (1998) [8] J.J.H. Miller, E. O Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientic, Singapore, [9] H.-G.Roos, A note on the conditioning of upwind scheme on Shishkin meshes, IMA J.Numer.Anal.16 (1996) [10] Y.Saad, M.H.Schultz, GMRES: a generalized minimal residual method for solving nonsymmetric linear systems, SIAM J.Sci.Statist.Comput.7 (1986)