A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM FOR A NONLINEAR SINGULARLY PERTURBED PARABOLIC PROBLEM

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volue 3, Nuber 2, Pages c 2006 Institute for Scientific Coputing and Inforation A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM FOR A NONLINEAR SINGULARLY PERTURBED PARABOLIC PROBLEM IGOR BOGLAEV (Counicated by Lubin Vulkov) Abstract. This paper deals with discrete onotone iterative algoriths for solving a nonlinear singularly perturbed parabolic proble. A block onotone doain decoposition algorith based on a Schwarz alternating ethod and on a block iterative schee is constructed. This onotone algorith solves only linear discrete systes at each tie level and converges onotonically to the exact solution of the nonlinear proble. The rate of convergence of the block onotone doain decoposition algorith is estiated. Nuerical experients are presented. Key Words. singularly perturbed parabolic proble, block onotone ethod, nonoverlapping doain decoposition, onotone Schwarz alternating algorith. 1. Introduction We are interested in onotone Schwarz alternating algoriths for solving the nonlinear reaction-diffusion proble (1) µ 2 (u xx + u yy ) + u t = f(p, t, u), P = (x, y), (P, t) Q = (0, T, = 0 < x < 1, 0 < y < 1}, f u (P, t, u) 0, (P, t, u) Q (, ), (f u f/ u), where µ is a sall positive paraeter. The initial-boundary conditions are defined by u(p, t) = g(p, t), (P, t) (0, T, u(p, 0) = u 0 (P ), P, where is the boundary of. The functions f(p, t, u), g(p, t) and u 0 (P ) are sufficiently sooth. Under suitable continuity and copatibility conditions on the data, a unique solution u(p, t) of (1) exists (see 6 for details). For µ 1, proble (1) is singularly perturbed and characterized by the boundary layers of width O(µ ln µ ) at the boundary (see 1 for details). In the study of nuerical solutions of nonlinear singularly perturbed probles by the finite difference ethod, the corresponding discrete proble is usually forulated as a syste of nonlinear algebraic equations. A ajor point about this syste is to obtain reliable and efficient coputational algoriths for coputing the solution. In the case of the parabolic proble (1), the iplicit ethod is usually in use. On each tie level, this ethod leads to a nonlinear syste which requires soe kind of iterative schee for the coputation of nuerical solutions. Received by the editors June 20, 2004 and, in revised for, March 24, Matheatics Subject Classification. 65N06, 65N50, 65N55, 65H

2 212 I. BOGLAEV A fruitful ethod for the treatent of these nonlinear systes is the ethod of upper and lower solutions and its associated onotone iterations (in the case of unperturbed probles see 8, 9 and references therein). Since the initial iteration in the onotone iterative ethod is either an upper or a lower solution, which can be constructed directly fro the difference equation without any knowledge of the exact solution (see 2 for details), this ethod eliinates the search for the initial iteration as is often needed in Newton s ethod. This eliination gives a practical advantage in the coputation of nuerical solutions. In 3, we proposed a discrete iterative algorith which cobines the onotone approach and the iterative doain decoposition ethod based on the Schwarz alternating procedure. In the case of sall values of the perturbation paraeter µ, the convergence factor ρ of the onotone doain decoposition algorith is estiated by ρ = ρ + O(τ), where ρ is the convergence factor of the onotone (undecoposed) ethod and τ is the step size in the t-direction. The purpose of this paper is to extend the onotone doain decoposition algorith fro 3 in a such way that coputation of the discrete linear subsystes on subdoains which are located outside the boundary layers is ipleented by the block iterative schee (see 12 for details of the block iterative schee). A basic advantage of the block iterative schee is that the Thoas algorith can be used for each linear subsyste defined on these subdoains in the sae anner as for one-diensional probles, and the schee is stable and is suitable for parallel coputing. For solving nonlinear discrete elliptic probles without doain decoposition, the block onotone iterative ethods were constructed and studied in 10. In 10, the convergence analysis does not contain any estiates on a convergence rate of the proposed iterative ethods, and the nuerical experients show that these algoriths applied to soe odel probles converge very slowly. In contrast, a nuerical algorith based on a cobination of the doain decoposition approach and the block iterative ethod applied on subdoains outside the boundary layers converges ore quickly than the original block iterative ethod. The structure of the paper is as follows. In Section 2, we consider a onotone iterative ethod for solving the iplicit difference schee which approxiates the nonlinear proble (1). In Section 3, we construct and investigate a block onotone doain decoposition algorith. The rate of convergence of the block onotone doain decoposition algorith is estiated in Section 4. The final Section 5 presents results of nuerical experients for the proposed algorith. 2. Monotone iterative ethod On Q introduce a rectangular esh h τ, h = hx hy : hx = x i, 0 i N x ; x 0 = 0, x Nx = 1; h xi = x i+1 x i }, hy = y j, 0 j N y ; y 0 = 0, y Ny = 1; h yj = y j+1 y j }, τ = t k = kτ, 0 k N τ, N τ τ = T }. For a esh function U(P, t), we use the iplicit difference schee (2) L h U(P, t) + 1 τ U(P, t) U(P, t τ) = f(p, t, U), (P, t) h τ, U(P, t) = g(p, t), (P, t) h τ, U(P, 0) = u 0 (P ), P h,

3 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 213 where L h U(P, t) is defined by L h U = µ 2 ( D x +D x + D y +D y ) U. D x +D x U(P, t), D y +D y U(P, t) are the central difference approxiations to the second derivatives D x +D x U k ij = ( xi ) 1 ( U k i+1,j U k ij) (hxi ) 1 ( U k ij U k i 1,j) (hxi 1 ) 1, D y +D y U k ij = ( yj ) 1 ( U k i,j+1 U k ij) (hyj ) 1 ( U k ij U k i,j 1) (hyj 1 ) 1, where U k ij U (x i, y j, t k ). xi = 2 1 (h xi 1 + h xi ), yj = 2 1 (h yj 1 + h yj ), Reark 1. In this paper, we use the standard backward Euler approxiation to the first derivative u t. Our analysis can be extended to higher order schees in tie 4. Now, we construct an iterative ethod for solving the nonlinear difference schee (2) which possesses the onotone convergence. Represent the difference equation fro (2) in the equivalent for (3) LU(P, t) = f(p, t, U) + τ 1 U(P, t τ), LU(P, t) L h U(P, t) + τ 1 U(P, t). We say that on a tie level t τ, V (P, t) is an upper solution of the difference proble (4) LV (P, t) + f(p, t, V ) τ 1 V (P, t τ) = 0, P h, if it satisfies V (P, t) = g(p, t), P h, LV (P, t) + f ( P, t, V ) τ 1 V (P, t τ) 0, P h, V (P, t) = g(p, t), P h. Siilarly, V (P, t) is called a lower solution on a tie level t τ with a given function V (P, t τ), if it satisfies the reversed inequality and the boundary condition. Additionally, we assue that f(p, t, u) fro (1) satisfies the two-sided constraints (5) 0 f u c, c = const. The iterative solution V (P, t) to (2) is constructed in the following way. On each tie level t τ, we calculate n iterates V (n) (P, t), P h, n = 1,..., n using the recurrence forulas LZ (n+1) (P, t) + c Z (n+1) (P, t) = LV (n) (P, t) + f (P, t, V (n)) (6) τ 1 V (P, t τ), P h, Z (n+1) (P, t) = 0, P h, n = 0,..., n 1, V (n+1) (P, t) = V (n) (P, t) + Z (n+1) (P, t), P h, V (P, t) V (n ) (P, t), P h, V (P, 0) = u 0 (P ), P h, where an initial guess V (0) (P, t) satisfies the boundary condition V (0) (P, t) = g(p, t), P h. The following theore gives the properties of the iterative algorith (6).

4 214 I. BOGLAEV Theore 1. Let V (0) (P, t) be an upper or lower solution in the iterative ethod (6), and let f(p, t, u) satisfy (5). Suppose that on each tie level the nuber of iterates n satisfies n 2. Then the following estiate on convergence rate holds ax V (t) U(t) t τ h C(ρ)n 1, where U(P, t) is the solution to (2), ρ = c / ( c + τ 1), constant C is independent of τ and W (t) h ax P h W (P, t). Furtherore, on each tie level the sequence V (n) (P, t) } converges onotonically. The proof of this theore can be found in 3. Reark 2. Consider the following approach for constructing initial upper and lower solutions V (0) (P, t) and V (0) (P, t). Suppose that for t fixed, a esh function R(P, t) is defined on h and satisfies the boundary condition R(P, t) = g(p, t) on h. Introduce the following difference probles (7) LZ (0) q (P, t) = q LR(P, t) + f(p, t, R) τ 1 V (P, t τ), P h, Z (0) q (P, t) = 0, P h, q = 1, 1. Then the functions V (0) (P, t) = R(P, t)+z (0) 1 (P, t), V (0) (P, t) = R(P, t)+z (0) 1 (P, t) are upper and lower solutions, respectively. The proof of this result can be found in 3. Reark 3. Since the initial iteration in the onotone iterative ethod (6) is either an upper or a lower solution, which can be constructed directly fro the difference equation without any knowledge of the solution as we have suggested in the previous reark, this algorith eliinates the search for the initial iteration as is often needed in Newton s ethod. This eliination gives a practical advantage in the coputation of nuerical solutions. Reark 4. In fact, it is sufficient that the condition (5) holds true only between the upper and lower solutions defined by (7). Reark 5. We can odify the iterative ethod (6) in the following way. The onotone convergent property of the upper and lower sequences in Theore 1 still holds true if the coefficient c in the difference equation fro (6) is replaced by c (n) (P, t) = ax f u (P, t, U), V (n) (P, t) U(P ) V (n) (P, t), t = fixed. To perfor the odified algorith we have to copute two sequences of upper and lower solutions siultaneously. But, on the other hand, this odification increases the rate of the convergence of the iterative ethod. Reark 6. The iplicit two-level difference schee (2) is of the first order with respect to τ. Fro here and since ρ c τ, one ay choose n = 2 to keep the global error of the onotone iterative ethod (6) consistent with the global error of the difference schee (2). 3. Monotone doain decoposition algoriths We consider decoposition of the doain into M nonoverlapping subdoains (vertical strips), = 1,..., M: = x (0, 1), x = (x 1, x ), x 0 = 0, x M = 1, Γ b = x = x 1, 0 y 1}, Γ e = x = x, 0 y 1},

5 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM = Γ b, Γ b = Γ e 1, = 2,..., M. Thus, we can write down the boundary of as = Γ b Γ e Γ 0, Γ 0 =. Additionally, introduce (M 1) interfacial subdoains ω, = 1,..., M 1: ω = ω x (0, 1), ω x = ( x b, x e ), ω 1 ω =, x b < x < x e, = 1,..., M 1. The boundaries of ω are denoted by γ b = x = x b, 0 y 1 }, γ e = x = x e, 0 y 1}, γ 0 = ω. Figure 1 illustrates the x-section of the ultidoain decoposition. x 1 x 1 x x b 1 xe 1 xb xe ω x 1 ω x x x +1 Figure 1. We now introduce eshes on, = 1,..., M and on ω, = 1,..., M 1. Suppose that the following set of esh points belongs to esh h then Γ hb,e,0 x b, x, x e } M 1 =1 hx, h = h, ω h = ω h, = Γ b,e,0 h, γ hb,e,0 = γ b,e,0 ω h Stateent and convergence of onotone doain decoposition algorith. Consider a parallel doain decoposition algorith for solving proble (2). On each tie level t τ, we calculate n iterates V (n) (P, t), P h, n = 1,..., n. To find V (n), firstly, we solve probles on the nonoverlapping subdoains h, = 1,..., M with Dirichlet boundary conditions passed fro the previous iterate. Then Dirichlet data are passed fro these subdoains to the interfacial subdoains ω h, = 1,..., M 1, and probles on the interfacial subdoains are coputed. Finally, we piece together the solutions on the subdoains. Step 0. Initialization: On the esh h, choose an upper or lower solution V (0) (P, t), P h satisfying the boundary condition V (0) (P, t) = g(p, t) on h. For n = 1 to n do Steps 1-3 Step 1. For = 1 to M do On the subdoain h, copute the esh function Z (n) (P, t), satisfying the difference schee LZ (n) (P, t) + c Z (n) (P, t) = LV (n 1) (P, t) + f (P, t, V (n 1)) (8) τ 1 V (P, t τ), P h, and denote Z (n) (P, t) = 0, P h, V (n) (P, t) = V (n 1) (P, t) + Z (n) (P, t), P h.

6 216 I. BOGLAEV Step 2. For = 1 to M 1 do On the interfacial subdoain ω h, copute the difference proble (n) L Z (P, t) + c Z(n) (P, t) = LV (n 1) (P, t) + f (P, t, V (n 1)) (9) τ 1 V (P, t τ), P ω, h and denote Z (n) (P, t) = 0, P γ h0 Z (n) (P, t), P γ hb Z (n) +1 ; ; (P, t), P γhe, Ṽ (n) (P, t) = V (n 1) (n) (P, t) + Z (P, t), P ω h. Step 3. Copute the solution V (n) (P, t), P h by piecing the solutions on the subdoains (10) V (n) V (n) (P, t), P (P, t) = h \ (ω 1 h ωh ), = 1,..., M; Ṽ (n) (P, t), P ω h, = 1,..., M 1. Step 4. Set up (11) V (P, t) = V (n ) (P, t), P h. Reark 7. We note that the original Schwarz alternating algorith with overlapping subdoains is a purely sequential algorith. To obtain parallelis, one needs a subdoain coloring strategy, so that a set of independent subprobles can be introduced. The proposed odification of the Schwarz algorith is very suitable for parallel coputing. Algorith (8)-(11) can be carried out by parallel processing, since the M probles (8) for V (n) (P, t), = 1,..., M and the (M 1) probles (9) for Ṽ (n) (P, t), = 1,..., M 1 can be ipleented concurrently. Reark 8. Reark 2 on the initial iteration holds true for algorith (8)-(11), i.e. an upper or lower solution can be constructed directly fro the difference equation without any knowledge of the solution. On esh 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, consider the following difference probles: (12) LΦ s (P ) + c Φ s (P ) = 0, P h, Φ s (P ) = 1, P Γ hs, Φ s (P ) = 0, P h \Γ hs, s = 1, 2, 3, 4, where L fro (3) and Γ hs is the s-th side of the rectangular esh h. We suppose that Γ h1 = x = x a ; y = y j, 0 j N y }, Γ h2 = x = x b ; y = y j, 0 j N y }, Γ h3 = x = x i, 0 i N x; y = 0}, Γ h4 = x = x i, 0 i N x; y = 1}. Introduce the notation b = 2 1 ( h b + h b+ ), e = 2 1 ( h e + h e+ ),

7 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 217 where h b, h b+ are the esh step sizes on the left and on the right fro point x b, respectively, and h e, h e+ are the esh step sizes on the left and on the right fro point x e, respectively, µ 2 κ b = (c + τ 1 ) b h b+, κ e = q b = Φ II, q e = Φ I γ hb+ γ he µ 2 (c + τ 1 ) e h e, W γ hb+,he ax P γ hb+,he W (P ), γ hb± = x = x b ± h b±, 0 y 1 }, γ he± = x = x e ± h e±, 0 y 1 }, where Φ II (P ) is the solution to (12) on ϑ hb = h ω h with s = 2 and Φ I (P ) is the solution to (12) on ϑ he = h +1 ωh with s = 1. The following theore contains the properties of algorith (8)-(11), and the proof of this theore can be found in 3. Theore 2. Let V (0) (P, t) be an upper or lower solution in the doain decoposition algorith (8)-(11), and let f(p, t, u) satisfy (5). Suppose that on each tie level, the nuber of iterates n satisfies n 2. Then the following estiate on convergence rate holds (13) ax V (t k ) U(t k ) h 1 k N τ C (c + ν) (ρ + λ) n 1, λ = ax κ b q; b κ e q e } (, ν = c + τ 1) λ, 1 M 1 where U(P, t) is the solution to (2), ρ = c /(c + τ 1 ) and constant C is independent of τ. Furtherore, on each tie level the sequence V (n) (P, t) } converges onotonically. Reark 9. We ention here that the onotone doain decoposition algorith (8)-(11) is a particular case of the block onotone doain decoposition which will be constructed in the next section. In Reark 15 and Section 4.2 below, we present sufficient conditions to guarantee the inequality ρ + λ < 1 required in Theore 2, and in Section 4.2 we study how constant C depends on the sall paraeter µ and the space esh. Reark 10. Rearks 4 and 5 to Theore 1 hold true for algorith (8)-(11) in Theore Block onotone doain decoposition algorith. On tie level t k, write down the difference schee (2) at an interior esh point (x i, y j ) h in the for d ij Uij k l ij Ui 1,j k r ij Ui+1,j k b ij Ui,j 1 k t ij Ui,j+1 k = f ( x i, y j, t k, Uij k ), +τ 1 U k 1 ij + G k ij, d ij = l ij + r ij + b ij + t ij + τ 1, l ij = µ 2 ( xi h x,i 1 ) 1, r ij = µ 2 ( xi h xi ) 1, b ij = µ 2 ( yj h y,j 1 ) 1, t ij = µ 2 ( yj h yj ) 1, where G k ij is associated with the boundary function g(p, t k). Define vectors and diagonal atrices by ( ( Ui k = Ui1, k..., Ui,N k y 1), G k i = G k i1,..., G k i,n y 1), Fi k ( ) ( U k i = fi1 k ( ) ( U k i1,..., f k i,ny 1 Ui,N k y 1)), L i = diag ( l i1,..., l i,ny 1), Ri = diag ( r i1,..., r i,ny 1).

8 218 I. BOGLAEV Then the difference schee (2) ay be written in the for A i Ui k ( L i Ui 1 k + R i Ui+1 k ) ( ) = F k i U k i + τ 1 U k 1 i + G k i, i = 1,..., N x 1, with the tridiagonal atrix A i d i1 t i1 0 b i2 d i2 t i2 A i = b i,ny 2 d i,ny 2 t i,ny 2 0 b i,ny 1 d i,ny 1 Matrices L i and R i contain the coupling coefficients of a esh point respectively to the esh point of the left line and the esh point of the right line. Since d ij, b ij, t ij > 0 and A i is strictly diagonally doinant, then A i is an M- atrix and A 1 i 0 (cf. 12). Introduce two nonoverlapping ordered sets of indices M α = kα 1α,..., Mα }, α = 1, 2, M 1 + M 2 = M, M 1, M 1 M 2 =, M 1 M 2 = 1,..., M}. Now, we odify Step 1 in algorith (8)-(11) in the following way. Step 1. On subdoain h, M 1, copute V (n) (P, t k ) = satisfying the difference schee (14) V (n),i (P, t k), 0 i i }, M 1, A,i V (n),i (t k) + c V (n),i (t k) = L,i V (n 1),i 1 (t k) + R,i V (n 1) +c V (n 1),i (t k ) F,i k,i+1 (t k) ( ) V (n 1),i (t k ) +τ 1 V k 1,i + G k,i, 1 i i 1, V (n),i (t k) = V (n 1),i (t k ), i = 0, i, where i = 0 and i = i are the boundary vertical lines, and G k = G k,i, 1 i i 1}, V k 1 = V k 1,i, 1 i i 1}, V (n 1) (t k ) = V (n 1),i (t k ), 0 i i } are the parts of g (P, t k ), V (P, t k 1 ) and V (n 1) (P, t k ), respectively, which correspond to subdoain h. On subdoain h, M 2, copute esh function V (n) (P, t k ), M 2 satisfying (8). Reark 11. Algorith (14) ay be considered as the line Jacobi ethod or the block Jacobi ethod for solving the five-point difference schee (8) on subdoain h, M 1 (cf. 12). Basic advantages of the block iterative schee (14) are that the Thoas algorith can be used for each subsyste i, i = 1,..., i 1 and all the subsystes can be coputed in parallel. On h = hx hy, we represent a difference schee in the following canonical for (15) d(p )W (P ) = e(p, P )W (P ) + F (P ), P h, P S(P ) W (P ) = W 0 (P ), P h,

9 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 219 and suppose that d(p ) > 0, e(p, P ) 0, c(p ) = d(p ) P S (P ) e(p, P ) > 0, P h, where S (P ) = S(P ) \ P }, S(P ) is a stencil of the difference schee. Now, we forulate a discrete axiu principle and give an estiate on the solution to (15). Lea 1. Let the positive property of the coefficients of the difference schee (15) be satisfied. (i) If W (P ) satisfies the conditions d(p )W (P ) e(p, P )W (P ) F (P ) 0( 0), P h, P S(P ) W (P ) 0( 0), P h, then W (P ) 0( 0), P h. (ii) The following estiate on the solution to (15) holds true (16) W h ax W 0 ; F/c h h. The proof of the lea can be found in 11. Theore 3. Let V (P, t τ) be given and V (0) (P, t), V (0) (P, t) be upper and lower solutions corresponding to V } (P, t τ). Suppose that f(p, t, u) satisfies (5). Then the upper sequence V (n) (P, t) generated by (8)-(10), (14) converges onotonically fro above } to the unique solution V(P, t) of the proble (4), and the lower sequence V (n) (P, t) generated by (8)-(10), (14) converges onotonically fro below to V(P, t): V (0) (P, t) V (n) (P, t) V (n+1) (P, t) V(P, t), P h, V(P, t) V (n+1) (P, t) V (n) (P, t) V (0) (P, t), P h. Proof. Introduce the notation Z (n) (P, t k ) = V (n) (P )(t k ) V (n 1) (P, t k ), P h, M 1. Consider the case of the upper sequence, i.e. V (0) (P, t k ) is an upper solution. For n = 1 and M 1, by (14) ( A,i Z (1),i (t k) + c Z (1),i (t k) = A,i V (0),i(t k ) L,i V (0),i 1(t k ) ) ( ) +R,i V (0),i+1(t k ) + F,i k V (0),i(t k ) (17) τ 1 V k 1,i G k,i 0, 1 i i 1, where we have taken into account that V (0) (P, t k ) is the upper solution. Since A 1,i 0 then (A,i + c I ) 1 0, where I is the (i 1) (i 1) identity atrix. Thus, we conclude that Z (1) (P, t k ) 0, P h, M 1. By (8) LZ (1) (P, t k ) + c Z (1) (P, t k ) = LV (0) (P, t k ) + f (P, t k, V (0)) (18) τ 1 V (P, t k 1 ) 0,

10 220 I. BOGLAEV P h, Z (1) (P, t) = 0, P h, M 2. By the axiu principle in Lea 1, we conclude that Z (1) (P, t k ) 0, P h, M 2. Thus (19) Z (1) (P, t k ) 0, P h, = 1,..., M. By (9) (20) (1) L Z (P, t k ) + c Z(1) (P, t k ) = Z (1) (P, t k ) = LV (0) (P, t k ) + f (P, t k, V (0)) τ 1 V (P, t k 1 ) 0, P ω, h 0, P γ h0 Z (1) ; (P, t k ), P γ hb ; Z (1) +1 (P, t k), P γ he. Using the nonpositive property of Z (1) (P, t k ), by the axiu principle in Lea 1, we conclude that (21) Z(1) (P, t k ) 0, P ω h, = 1,..., M 1. (19) and (21) show that V (1) (P, t k ) V (0) (P, t k ), P h. By induction, we prove that V (n) (P, t k ) V (n 1) (P, t k ), P h for each n 1. Now we verify that V (n) (P, t k ) is an upper solution for each n. Fro the boundary conditions for V (n) (P, t k ) and Ṽ (n) (P, t k ), it follows that V (n) (P, t k ) satisfies the boundary condition in (2). Represent (14) in the for (22) ΛV (n) (t k ) where we have introduced the notation ΛV (n) ΛV (n) (P, t k ) i i = L,i Z (n),i 1 (t k) R,i Z (n),i+1 (t k) ( ) + c V (n 1),i (t k ) F,i k V (n 1),i (t k ) ( ) c V (n),i (t k) F,i k V (n),i (t k), (P, t k ) = LV (n) (P, t k ) + f By the ean-value theore and (5) ( P, t k, V (n) ) τ 1 V (P, t k 1 ), ( ) = A,i V (n),i (t k) L,i V (n),i 1 (t k) + R,i V (n),i+1 (t k) ( ) +F,i k V (n),i (t k) τ 1 V k 1,i G k,i. c W f(w ) c Z f(z) = c (W Z) F (n) u (W Z) 0, whenever W Z. Using this property, (19) and L,i 0, R,i 0, we conclude ΛV (n) (P, t k ) 0, P h, M 1. Fro (8) for M 2, by the ean-value theore, (5) and (19), we have ( ) (23) ΛV (n) (P, t k ) = c f u (n) Z (n) (P, t k ) 0, P h, M 2. Siilarly, we prove (24) ΛṼ (n) (P, t k ) = ( c f (n) u ) Z(n) (P, t k ) 0, P ω h.

11 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 221 Thus, by the definition of V (n) in (10), we conclude that ( LV (n) (P, t k ) + f P, t k, V (n)) τ 1 V (P, t k 1 ) 0, P h \ γ h, γ h = M 1 =1 γ hb,e. To prove that V (n) (P, t k ) is an upper solution of (4), we have to verify only that the last inequality holds true on the interfacial boundaries γ hb, γ he, = 1,... M 1. We check this inequality in the case of the left interfacial boundary γ hb, since the second case is checked in a siilar way. Introduce the notation W (n) (P, t k ) = V (n) (P, t k ) Ṽ (n) (P, t k ). Fro (14) and (9), we have (25) A,i W (n),i (t k) + c W (n),i (t k) = L,i Z (n),i 1 (t k) R,i Z (n),i+1 (t k), P ϑ hb, ϑ hb = h ω h, M 1. In view of (A,i + c I ) 1 0, L,i 0, R,i 0 and (19) which holds true for each n 1, W (n) (P, t k ) 0, P ϑ hb, M 1. Fro (8), (9) and (21), we conclude (26) LW (n) (P, t k ) + c W (n) (P, t k ) = 0, P ϑ hb, W (n) (P, t k ) = 0, P ϑ hb \ Γ he, In view of the axiu principle in Lea 1, W (n) Thus (27) V (n) Fro (2), (9), (10) and Ṽ (n) W (n) (P, t k ) 0, P Γ he. (P, t k ) 0, P ϑ hb, M 2. (P, t k ) Ṽ (n) (P, t k ) 0, P ϑ hb, = 1,..., M 1. (P, t k ) = V (n) (P, t k ), P γ hb, µ 2 D+D y V y (n) (P, t k ) = µ 2 D+D y V y (n) (P, t k ), P γ hb. Fro (2), (10) and (27), we obtain µ 2 D+D x V x (n) (P, t k ) µ 2 D+D x V x (n) (P, t k ), P γ hb. In the notation fro (22), we conclude ΛV (n) (P, t k ) ΛV (n) (P, t k ) 0, P γ hb. This leads to the fact that V (n) (P, t k ) is an upper solution of proble (4). By (19), (21), sequence V (n) } is onotone decreasing and bounded by a lower solution. Indeed, if V is a lower solution, then by the definitions of lower and upper solutions and the ean-value theore, for δ (n) = V (n) V, we have Lδ (n) (P, t k ) + f u δ (n) (P, t k ) 0, P h, δ (n) (P, t k ) = 0, P h. In view of the axiu principle in Lea 1, it follows that V V (n), n 0. Thus, li V (n) = V as n exists and satisfies the relation V (P, t k ) V (n+1) (P, t k ) V (n) (P, t k ) V (0) (P, t k ), P h. Now we prove the last point of this theore that the liiting function V is the solution to (4). Letting n in (8), (9) and (14) shows that V is the solution of (4) on h \ γ h. Now we verify that V satisfies (4) on the interfacial boundaries

12 222 I. BOGLAEV γ hb, γ he, = 1,..., M 1. Since V (n) V (n) (P, t k ), P Γ he, we conclude that li V (n) n Fro here it follows that (P, t k ) = li n (P, t k ) Ṽ (n) (P, t k ) = V (n 1) (P, t k ) Ṽ (n) (P, t k ) = V (P, t k ), P ϑ hb. li ΛV (n) (n) (P, t k ) = li ΛV (P, t k ) = 0, P γ hb, n n and hence, V (P, t k ) solves (4) on γ hb. In a siilar way, we can prove the last result on γ he. Under the condition (5), proble (4) has the unique solution V (see 3 for details), hence V = V. This proves the theore. Reark 12. The proposed algorith (8)-(10), (14) can be applied for solving unperturbed probles of for (1), i.e. in the case of µ = O(1). Furtherore, the block iterative schee (14) can be applied on the whole coputational doain h. However, as we show below, this algorith can be ost efficiently used at sall values of µ and in the case of the location of subdoains h, M 1, outside the boundary layers. Reark 13. Rearks 4 and 5 to Theore 1 hold true for algorith (8)-(10), (14) in Theore Convergence of algorith (8)-(11), (14) We now establish convergence properties of algorith (8)-(11), (14) Convergence analysis of algorith (8)-(11), (14). If we denote Z (n) (P, t k ) = V (n) (P, t k ) V (n 1) (P, t k ), P h, then fro (8)-(11), (14), it follows that on h, = 1,..., M, Z (n) can be written in the for Z (n) Z (n) 1 (P, t k), x 1 x x e 1; (P, t k ) = Z (n) (P, t k ), x e 1 x x b ; Z (n) (P, t k ), x b x x, where for siplicity, we indicate the discrete doains only in the x-variable, i.e. x 1 x x e 1 eans x 1 x x e 1, 0 y 1}, and assue that for = 1, M, the corresponding doains x 0 x x e 0 and x b M x x M are epty. Introduce the notation l = ax M 1 r = ax M 1 ax 1 i i 1 ax 1 i i 1 l()i }, l ()i = L,i, L,i ax 1 j N y 1 l ij, } r()i, r ()i = R,i, R,i ax ij, 1 j N y 1 l ()e = L,i e 1, r()b = R,i b, ( ) κ = l()e ; r ()b, 1 c + τ 1 ax 1 M 1 where the indices i e 1, i b correspond to x e 1 and x b, respectively, and i = 0, i = i correspond to x and x +1, respectively.

13 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 223 Theore 4. For the block onotone doain decoposition algorith (8)-(11), (14) the following estiate holds true (28) Z (n) Z (t k ) (ρ + λ) (n 1) (t h k ), h λ = l + r l + r c + κ ax 1; + τ 1 c + τ 1, where Z (n) = V (n) V (n 1), ρ = c / ( c + τ 1). Proof. Let V (0) be an upper solution. induction, we get for n 1 A,i Z (n),i (t k) + c Z (n),i (t k) = L Then siilar to (17), (18) and (20), by ΛV (n 1) (P, t k ), 1 i i 1, M 1,,i LZ (n) (P, t k ) + c Z (n) (P, t k ) = ΛV (n 1) (P, t k ), P h, M 2, (n) Z (P, t k ) + c Z(n) (P, t k ) = ΛV (n 1) (P, t k ), P ω, h = 1,..., M 1. Using (16) with c = c + τ 1, we get the following estiates on Z (n) (29) Z (n) 1 ΛV (n 1) (P, t k ) (tk c + τ 1 ), P h, h (n) 1 ΛV (n 1) Z (P, t k ) ax (tk c + τ 1 ) Z (n) (t k ) ; Z (n) +1 (t k) γ hb γ he ; ω h and, P ω h. Fro (10), (22), (23) and (24), on h, = 1,..., M, we have (c (n 1) f u ) Z ΛV (n 1) 1 (P, t k), x 1 x < x e 1; (P, t k ) = (c f u ) Z (n 1) (P, t k ), x e 1 < x < x b, M 2 ; (c (n 1) f u ) Z (P, t k ), x b < x x, ΛV (n 1) (P, t k ),i = (c f u ) Z (n 1),i ) +R,i Z (n 1),i+1 (t k) ( (t k ) L,i Z (n 1),i 1 (t k) Z (n), i e 1 < i < i b, M 1, where the indices i e 1, i b correspond to x e 1 and x b, respectively. Fro here and (5) and taking into account L,i 0, R,i 0 and (19), we get 1 ΛV (n 1) Z (30) (P, tk c + τ 1 ) ρ (n 1) 1 (t k ), P h h \ (γ 1 he γ hb ), where ρ 1 = ρ + (l + r)/ ( c + τ 1). Now, we prove the following estiates 1 ΛV (n 1) (31) (tk c + τ 1 ) γ hb 1 ΛV (n 1) (tk c + τ 1 ) γ he 1 ρ 1 + ρ 2r ()b c + τ 1 Z (n 1) (t k ) h, ρ 1 + ρ Z 2l ()e (n 1) c + τ 1 (t k ), h

14 224 I. BOGLAEV where ρ 2 = ax 1; (l + r)/ ( c + τ 1). Using (10), on the boundary γ hb, we can write the relation ΛV (n 1) (P, t k ) = ΛV (n 1) (P, t k ) µ2 b h b+ (Ṽ (n 1) ( ) ( ) ) P b+, t k V (n 1) P b+, t k, P = ( x b, y ) γ hb, P b+ = ( x b + h b+, y ) γ hb+. Fro (22) and (23), we can represent ΛV (n 1) (P, t k ) on γ hb in the for ( ΛV (n 1) (t k ) = (c f u ) Z (n 1),i (t k ) L,i Z (n 1),i 1 (t k) i ) +R,i Z (n 1),i+1 (t k), i = i b, M 1, where the index i b corresponds to γ hb, ΛV (n 1) (P, t k ) = (c f u ) Z (n 1) (P, t k ), P γ hb, M 2. Thus, we conclude the estiate 1 ΛV (n 1) Z (P, tk c + τ 1 ) ρ (n 1) 1 (t k ) + µ 2 h (c + τ 1 ) b h b+ V (n 1) ( ) ( ) P b+, t k V (n 1) P b+, t k, P = ( x b, y ) γ hb, P b+ = ( x b + h b+, y ) γ hb+. Applying (16) to (25) and (26), and taking into account that Ṽ (n 1) (P, t k ) V (n 1) (P, t k ) = V (n 1) (P, t k ) V (n 2) (P, t k ), P Γ he, we get the estiates Ṽ (n 1) (t k ) V (n 1) Z (t k ) ρ (n 1) hb 2 (t k ), M 1, ϑ Γ he Ṽ (n 1) (t k ) V (n 1) (t k ) Z (n 1) (t k ), M 2. hb ϑ Γ he Thus, we prove (31) on γ hb. Siilarly, we can prove (31) on the boundary γ 1. he Fro (29), (30) and (31) and using the definition of Z (n) (P, t k ), we prove the theore. Theore 5. Let V (0) (P, t) be an upper or lower solution in the doain decoposition algorith (8)-(11), (14) and let f(p, t, u) satisfy (5). Suppose that on each tie level, the nuber of iterates n satisfies n 2. Then the following estiate on convergence rate holds (32) ax 1 k N τ V (t k ) U(t k ) C (c + ν) (ρ + λ) n 1, ν = ( c + τ 1) λ, where U(P, t) is the solution to (2), ρ and λ are defined in Theore 4, and constant C is independent of τ. Furtherore, on each tie level the sequence V (n) (P, t) } converges onotonically. Proof. Denote W (P, t) = U(P, t) V (P, t). Using the notation fro (22) and (2), we can write the following difference proble for W (P, t) (33) LW (P, t) + f u (P, t)w (P, t) = ΛV (n ) (P, t) + τ 1 W (P, t τ), P h, W (P, t) = 0, P h.

15 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 225 Fro here, (30), (31) and using (16), we obtain the estiate W (t k ) h τ (c + ν) Z (n ) (t k ) + W (t h k τ) h. Using (28), we prove by induction the estiates ( k ) W (t k ) C l τ (c + ν) (ρ + λ) n 1, k = 1,..., N τ, (34) Z (1) (t l ) h l=1 ( LV τ (0) (t l ) + f V (0)) τ 1 V (t l τ) C h l, where all constants C l are independent of τ. Since N τ τ = T, we prove the estiate in the theore with C = T C 0, where C 0 = ax 1 l Nτ C l. Reark 14. We entioned in Reark 12 that the block onotone iterative schee (14) can be applied on the whole coputational doain h, i.e. in the case of M 1 = 1} and M 2 =. As follows fro the proofs of Theores 4 and 5, these theores hold true with λ = (l + r)/ ( c + τ 1). Reark 15. In the case of M 1 =, as follows fro the proofs of Theores 4 and 5, these theores hold true with λ = κ, κ = ax κ b ; κ e }, 1 M 1 where we use the notation fro (13) and (28). Since the solutions to (12) are bounded by 0 Φ s (P ) 1, then in (13), 0 q b,e 1. Thus, the last estiate on λ follows fro (13) Estiates on the rate of convergence of algorith (8)-(11), (14). Here we analyze a convergence rate of algorith (8)-(11), (14) applied to the difference schee (2) defined on a piecewise equidistant esh of Shishkin-type and on its odification. On the Shishkin esh, the difference schee (2) converges µ-uniforly to the solution of (1) (see 7 for details). The piecewise equidistant esh of Shishkin-type is fored by the following anner. We divide each of the intervals x = 0, 1 and y = 0, 1 into three parts each 0, σ x, σ x, 1 σ x, 1 σ x, 1, and 0, σ y, σ y, 1 σ y, 1 σ y, 1, respectively. Assuing that N x, N y are divisible by 4, in the parts 0, σ x, 1 σ x, 1 and 0, σ y, 1 σ y, 1 we use a unifor esh with N x /4 + 1 and N y /4 + 1 esh points, respectively, and in the parts σ x, 1 σ x, σ y, 1 σ y with N x /2 + 1 and N y /2 + 1 esh points, respectively. This defines the piecewise equidistant eshes in the x- and y-directions condensed in the boundary layers at x = 0, 1 and y = 0, 1: x i = y j = ih xµ, i = 0, 1,..., N x /4; σ x + (i N x /4)h x, i = N x /4 + 1,..., 3N x /4; 1 σ x + (i 3N x /4)h xµ, i = 3N x /4 + 1,..., N x, jh yµ, j = 0, 1,..., N y /4; σ y + (j N y /4)h y, j = N y /4 + 1,..., 3N y /4; 1 σ y + (j 3N y /4)h yµ, j = 3N y /4 + 1,..., N y, h x = 2(1 2σ x )Nx 1, h xµ = 4σ x Nx 1, h y = 2(1 2σ y )Ny 1, h yµ = 4σ y Ny 1, where h xµ, h yµ and h x, h y are the step sizes inside and outside the boundary layers, respectively. We choose the transition points σ x, (1 σ x ) and σ y, (1 σ y ) in Shishkin s sense (see 7 for details), i.e. σ x = in 4 1, υ 1 µ ln N x }, σy = in 4 1, υ 2 µ ln N y },

16 226 I. BOGLAEV where υ 1 and υ 2 are positive constants. If σ x,y = 1/4, then N 1 x,y are very sall relative to µ, and in this case the difference schee (2) can be analyzed using standard techniques. We therefore assue that σ x = υ 1 µ ln N x, σ y = υ 2 µ ln N y. In this case the eshes hx and hy are piecewise equidistant with the step sizes (35) N 1 x < h x < 2N 1 x, h xµ = 4υ 1 µn 1 x ln N x, Ny 1 < h y < 2Ny 1, h yµ = 4υ 2 µny 1 ln N y. The difference schee (2) on the piecewise unifor esh (35) converges µ- uniforly to the solution of (1): (36) ax (P,t) h τ U(P, t) u(p, t) K ( (N 1 ln N ) 2 + τ ), N = in N x ; N y }, where constant K is independent of µ, N and τ. The proof of this result can be found in 7. Consider algorith (8)-(11), (14) on the piecewise unifor esh (35) with the subdoains h, M 1 and the interfacial subdoains ω h, = 1,..., M 1 located in the x-direction outside the boundary layer, where the step size h x fro (35) is in use. In this case, l, r and κ in (28) are bounded by τµ 2 /h 2 x, and we estiate λ in (28) by (37) λ 2τµ2 h 2 x + τµ2 h 2 x ax 1; 2τµ2 Thus, if µ h x and 2τ 1, then λ 3τ, hence, the right hand side in (32) is estiated by (38) C (c + ν) (ρ + λ) n 1 C (ρ + 3τ) n 1, where constant C is independent of τ. Reark 16. We ention that the iplicit difference schee (2) is of the first order with respect to τ and ρ = c / ( c + τ 1) c τ. Thus, to guarantee the consistency of the global errors in the difference schee (2) and in the block onotone doain decoposition algorith (8)-(11), (14) it is enough to choose n = 2. Reark 17. Without loss of generality, we assue that the boundary condition g(p, t) = 0. This assuption can always be obtained via a change of variables. Let on each tie level the initial function V (0) (P, t) be chosen in the for of (7), i.e. V (0) (P, t) is the solution of the following difference proble (39) LV (0) (P, t) = q f(p, t, 0) τ 1 V (P, t τ), P h, h 2 x. V (0) (P, t) = 0, P h, q = 1, 1, where R(P, t) = 0. Then the functions V (0) (P, t), V (0) (P, t) corresponding to q = 1 and q = 1 are upper and lower solutions, respectively. Fro here and (34), it follows that Z (1) LV (t l ) τ (0) (t h l ) + f (P, t h l, 0) τ 1 V (t l τ) h 2τ f (P, t l, 0) τ 1 V (t l τ) h 2τ f (P, t l, 0) h + 2 V (t l τ) h C l. To prove that all constants C l are independent of the sall paraeter µ, we have to prove that V (t l τ) h are µ-uniforly bounded. For l = 1, V (P, 0) = u 0 (P ),

17 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 227 where u 0 is the initial condition in the differential proble (1), and, hence, C 1 is independent of µ and τ. For l = 2, we have Z (1) (t 2 ) 2τ f (P, t h 1, 0) h + 2 V (t 1 ) h C 2, where V (P, t 1 ) = V } (n ) (P, t 1 ). As follows } fro Theore 3, the onotone sequences V (n) (P, t 1 ) and V (n) (P, t 1 ) are µ-uniforly bounded fro above by V (0) (P, t 1 ) and fro below by V (0) (P, t 1 ). Applying (16) to the proble (39) at t = t 1, we have V (0) (t1) τ f(p, t h 1, 0) τ 1 u 0 (P ) h K 1, where constant K 1 is independent of µ and τ. Thus, we prove that C 2 is independent of µ and τ. Now by induction on l, we prove that all constants C l in (34) are independent of µ, and, hence, constant C = T ax 1 l Nτ C l in (32) is independent of µ and τ. Thus, if µ h x and 2τ 1, then fro (32), (38) and (36), we conclude that the onotone doain decoposition algorith (8)-(11), (14) converges µ- uniforly to the solution of the differential proble (1). Now we odify the piecewise equidistant esh of Shishkin-type in the x-direction. Let the nuber of esh points N xµ and the step size h xµ in the boundary layers be chosen in the for (40) N xµ = α ln(1/µ), h xµ = υµ, where α and υ are positive constants. In this case, the transition points σ x and (1 σ x ) are defined by σ x = h xµ N xµ = (αυ) µ ln ( µ 1). We note that, in general, the difference schee (2) on the odified piecewise equidistant esh (35), (40) does not converge µ-uniforly to the solution of (1). Consider algorith (8)-(11), (14) on the odified esh (35), (40) and assue µ h x. Using (37) with the step size h xµ, we estiate λ in the boundary layers in the for λ 2υ 2 τ + υ 2 τ ax 1; 2υ 2 τ. If 2υ 2 τ 1, then λ 3υ 2 τ, and fro here and (37), the right hand side in (32) is estiated by C (c + ν) (ρ + λ) n 1 C (ρ + rτ) n 1, r = 3 ax 1; υ 2, where constant C is independent of τ. We ention that Reark 16 holds true for the block onotone doain decoposition algorith (8)-(11), (14) on the odified esh (35), (40). 5. Nuerical experients Consider proble (1) with f(p, t, u) = (u 4)/(5 u), g(p, t) = 1 and u 0 (P ) = 1, which odels the biological Michaelis-Menton process without inhibition 5. This proble gives V (P, t 1 ) = 4, x h ; 1, x h, V (P, t 1 ) = 0, x h ; 1, x h, where V (P, t 1 ) and V (P, t 1 ) are the upper and lower solutions on the tie level t 1 = τ corresponding to V (P, 0) = 1, P h. Suppose that we initiate our algoriths with V (0) (P, t 1 ) = V (P, t 1 ) or with V (0) (P, t 1 ) = V (P, t 1 ) and thus

18 228 I. BOGLAEV generate sequences of lower and upper solutions V (n) (P, t 1 )} and V (n) (P, t 1 )}, respectively, with respect to V (P, 0). It follows fro Theore 1 in 3 that 0 V (P, t 1 ) V (n) (P, t 1 ) V (n) (P, t 1 ) V (P, t 1 ) 4, P h, n 0. Now for k 2, let V l (P, t k 1 ) = V (n ) (P, t k 1 ), V r (P, t k 1 ) = V (n ) (P, t k 1 ). Since the boundary condition g and the function f in the test proble are independent of tie, the esh functions V (0) (P, t k ), V (0) (P, t k ) defined by V (0) (P, t k ) = V l (P, t k 1 ), V (0) (P, t k ) = V r (P, t k 1 ) are lower and upper solutions with respect to V l (P, t k 1 ) and V r (P, t k 1 ), respectively. Applying Theore 1 fro 3, one has by induction on k that 0 V (n) (P, t k ) V (n) (P, t k ) 4, P h, 0 n n, 1 k N τ. Since each of our coputed esh functions satisfies the above inequalities, we ay suppose that f u is bounded above by c = 1. On each tie level t k, the stopping criterion is chosen in the for V (n) (t k ) V (n 1) (t k ) δ, h where δ = All the discrete linear systes corresponding to set M 2 are solved by GM RES-solver. It is found that in all the nuerical experients the basic feature of onotone convergence of the upper and lower sequences is observed. In fact, the onotone property of the sequences holds at every esh point in the doain. This is, of course, to be expected fro the analytical consideration. Consider the block onotone doain decoposition algorith (8)-(11), (14) on the piecewise unifor esh (35) with N x = N y. The doain decoposition of the coputational doain h consists of the three subdoains h, = 1, 2, 3 and the two interfacial subdoains ω h, = 1, 2, such that M 1 = 2}, M 2 = 1, 3}. The interfacial subdoains ω h, = 1, 2 contain only three esh points in the x-direction and lie straightaway outside the boundary layers in the x-direction. In Table 1, for µ = 10 2, 10 3 and 10 4 and for various values of N x, we give the average (over ten tie levels) nubers of iterations n τ1, n τ2, (τ 1 = , τ 2 = 10 2 ) required to satisfy the stopping criterion. The µ-dependence of the step sizes h x and h xµ of the piecewise unifor esh (35) is tabulated in Table 2. Since for our data set we allow σ x > 0.25, the step size h xµ is calculated as (41) h xµ = 4 in 0.25, σ x} N x. Fro the data presented in Tables 1 and 2, it follows that if the condition µ h x holds true then the nubers of iterations are equal to the nubers of iterations for the undecoposed onotone iterative algorith (6), where GM RES-solver is in use on the whole doain h, i.e. M 1 =, M 2 = 1}. If we violate this condition as in the case with µ = 10 2, N x = 512, 1024 and τ 1 = , then the nuber of iterations n τ1 exceeds the nuber of iteration for the undecoposed onotone algorith. Thus, the nuerical experients confir our theoretical estiates that the onotone doain decoposition algorith (8)-(11), (14) can be ost efficiently used if the condition µ h x holds true. For µ 10 3, n τ is independent of µ, which confirs the unifor convergence results shown in Reark 17. In Table 3, we present the nuerical results fro 3 for the onotone doain decoposition algorith (8)-(11) on the piecewise unifor esh (35) corresponding

19 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM 229 Table 1. Average nubers of iterations for τ 1 = , τ 2 = 10 2 for the block onotone doain decoposition algorith (8)- (11), (14) on the piecewise unifor esh (35). N x n τ1 ; n τ ; ; ; ; ; ; ; ; ; 3.0 µ ; 10 4 Table 2. The µ-dependence of h x, h xµ. N x h x ; h xµ E-02; 1.30E E-02; 1.30E E-02; 1.30E E-03; 7.59E E-02; 7.59E E-02; 7.59E E-03; 3.91E E-03; 4.34E E-03; 4.34E E-03; 1.95E E-03; 2.44E E-03; 2.44E E-04; 9.77E E-03; 1.35E E-03; 1.35E-05 µ to the nuerical experients reported in Table 1. As follows fro Tables 1 and 3, the block onotone doain decoposition algorith (8)-(11), (14) requires at ost the sae nuber of iterations as in the onotone doain decoposition (14) to reach the given accuracy. Since in algorith (8)-(11), (14) the Thoas algorith is in use on the subdoain located outside the boundary layers, it is clear that algorith (8)-(11), (14) executes ore quickly than algorith (8)-(11). Table 3. Average nubers of iterations for τ 1 = , τ 2 = 10 2 for the onotone doain decoposition algorith (8)-(11) on the piecewise unifor esh (35). N x n τ1 ; n τ ; ; ; ; ; ; ; 3.0 µ ; 10 4 Now consider the case M 1 = 1}, M 2 =, i.e. the block onotone iterative ethod (14) solves the difference schee (2) on the whole coputational doain h. In Table 4, for τ 1 = , τ 2 = 10 2 and for various values of N x and µ, we give the average (over ten tie levels) nubers of iterations n τ1, n τ2 required to satisfy the stopping criterion. In the case of the nuerical experients presented in Table 4, we violate the condition µ h x. For µ 10 3, n τ is independent of µ, which confirs the unifor convergence results shown in Reark 17. In the contrast to the block onotone doain decoposition algorith (copare with Table 1), here n τ is a onotone increasing function in N x. For µ 10 1, the block onotone iterative ethod (14) converges very slowly. On the odified esh (35), (40), consider the block onotone doain decoposition algorith (8)-(11), (14) and the block onotone iterative ethod (14).

20 230 I. BOGLAEV Table 4. Average nubers of iterations for τ 1 = , τ 2 = 10 2 for the block onotone iterative ethod (14) on the piecewise unifor esh (35). N x n τ1 ; n τ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 13.4 µ Table 5. Average nubers of iterations on the odified esh (35), (40). ALGORITHM n τ1 ; n τ2 (8)-(11), (14) 4.0; ; ; 2.9 (14) 9.0; ; ; 3.0 N xµ µ e 4 e 8 e 12 For algorith (8)-(11), (14), we decopose each of the boundary layers in the x- direction 0, σ x and 1 σ x, 1 into M µ subdoains and solve the probles (8) on these subdoains by GMRES-solver, the proble on the subdoain σ x, 1 σ x is solved by the block iterative ethod (14). Thus, the total nuber of subdoains is M = 2M µ + 1. The interfacial subdoains ω h, = 1,..., M 1 contain only three esh points in the x-direction, and the probles (9) on the interfacial subdoains are solved by the Thoas algorith. In Table 5, for values of µ = exp( k), k = 4, 8, 12 and N xµ = 8 ln(1/µ), N x = 4N xµ, we give the average (over ten tie levels) nubers of iterations n τ1, n τ2, (τ 1 = , τ 2 = 10 2 ) required to satisfy the stopping criterion. Since for our data set we allow σ x > 0.25, the step size h xµ is calculated by (41). We ention that n τ1 and n τ2 for algorith (8)-(11), (14) are independent of the nuber of subdoains M and equal to the nuber of iterations for the undecoposed onotone iterative algorith (6), where GMRES-solver is in use on the whole doain h. For µ exp( 5), n τ in the block onotone doain decoposition algorith (8)-(11), (14) and in the block onotone iterative ethod (14) is independent of µ. Thus, the ain features of algorith (8)-(11), (14) on the piecewise unifor esh (35) highlighted in Table 1 hold true for the algorith on the odified esh (35), (40). An advantage of the odified esh (35), (40) is that the boundary layers ay be decoposed into a set of subdoains without deteriorating the convergence rate of iterations, and the probles (8) on these subdoains can be solved in parallel. In the contrast to the block onotone iterative ethod (14) on the piecewise unifor esh (35) (copare Tables 4 and 5), the algorith (14) on the odified esh (35), (40) converges µ- and N-uniforly in (32). References 1 I. Boglaev, Finite difference doain decoposition algoriths for a parabolic proble with boundary layers, Coput. Math. Applic., 36 (1998),

21 A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM I. Boglaev, On onotone iterative ethods for a nonlinear singularly perturbed reactiondiffusion proble, J. Coput. Appl. Math., 162 (2004), I. Boglaev, Monotone iterative algoriths for a nonlinear singularly perturbed parabolic proble, J. Coput. Appl. Math., 172 (2004), I. Boglaev and M. Hardy, A onotone weighted-average Schwarz iterates for solving a nonlinear singularly perturbed parabolic proble (in preparation). 5 E. Bohl, Finite Modelle Gewöhnlicker Randwertaufgaben, Teubner, Stuttgart, O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural ceva, Linear and Quasi-Linear Equations of Parabolic Type, Acadeic Press, New York, J.J.H Miller, E. O Riordan and G.I. Shishkin, Fitted Nuerical Methods for Singular Perturbation Probles, World Scientific, Singapore, C.V. Pao, Monotone iterative ethods for finite difference syste of reaction-diffusion equations, Nuer. Math., 46 (1985), C.V. Pao, Finite difference reaction diffusion equations with nonlinear boundary conditions, Nuer. Methods Partial Diff. Eqs., 11 (1995), C.V. Pao, Block onotone iterative ethods for nuerical solutions of nonlinear elliptic equations, Nuer. Math., 72 (1995), A. Saarskii, The Theory of Difference Schees, Marcel Dekker Inc., New York-Basel, R.S. Varga, Matrix Iterative Analysis, Prentice-Hall Englewood Cliffs, New Jersey, Institute of Fundaental Sciences, Massey University, Private Bag , Palerston North, New Zealand E-ail: I.Boglaev@assey.ac.nz URL: