Iterative Domain Decomposition Methods for Singularly Perturbed Nonlinear Convection-Diffusion Equations

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1 Iterative Domain Decomposition Methods for Singularly Perturbed Nonlinear Convection-Diffusion Equations P.A. Farrell 1, P.W. Hemker 2, G.I. Shishkin 3 and L.P. Shishkina 3 1 Department of Computer Science, Kent State University, Kent, Ohio 44242, USA farrell@cs.kent.edu 2 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands P.W.Hemker@cwi.nl 3 Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, 16 S. Kovalevskaya Street, Ekaterinburg , Russia Grigorii@shishkin.ural.ru, Lida@convex.ru 1. Introduction We consider special numerical approximations to a domain decomposition method for a boundary value problem in the case of singularly perturbed nonlinear convection-diffusion equations, with the perturbation parameter ε. As a rule, a differential problem is approximated by nonlinear grid equations (iteration-free schemes), which are then solved by suitable iterative methods. In the case of ε-uniform robust methods (c.f. [1]) we require that the solution of the iterative scheme be ε-uniformly close (in the maximum norm) to the solution of the boundary value problem and, furthermore, that the number of iterations required for solving the discrete problem be independent of the parameter ε. Such iterative methods can be effectively solved by means of sequential and parallel domain decomposition algorithms (c.f. [2, 3]). We are interested in constructing schemes of this type which are ε-uniformly robust. In the present paper we construct and study base iterative schemes and their decompositions, i.e. sequential and parallel iterative schemes, for a singularly perturbed nonlinear boundary value problem consisting of an ordinary differential equation of convection-diffusion type. We establish necessary conditions under which the iteration-free scheme is convergent ε-uniformly and the iterative schemes are ε-uniformly robust. The schemes constructed (utilizing piecewise uniform meshes) converge ε-uniformly with the rate O(N 1 ln N) (the convergence rate is optimal or unimprovable (c.f. [4]), the required number of iterations for the iterative schemes is O(ln N), and thus, the ε-uniform amount of computation in solving these iterative schemes is O(N 1 ln 2 N). 2. Problem formulation. The objective of research On the set D, where D =(0,d), D = D Γ, (2.1) we consider the boundary value problem for the quasilinear singularly perturbed equation L (2.2) u(x) εa(x) u (x)+b(x) u (x) c(x) u(x) =F (x, u(x)), x D, (2.2) u(x) =ϕ(x), x Γ. Here a(x), b(x), c(x), f(x), x D, functions satisfying the conditions: F (x, u), (x, u) H, H D R are sufficiently smooth This research was supported in part by the Dutch Research Organisation NWO under grant N and by the Russian Foundation for Basic Research under grant N

2 a 0 a(x) a 0, b 0 b(x) b 0, 0 c(x) c 0, x D; (2.3a) ϕ(x) M, x Γ; F (x, u) M, (x, u) H; a 0,b 0 > 0; and the parameter ε takes arbitrary values from the half-interval (0, 1] 1. For simplicity, the following condition is assumed to be fulfilled: F (x, u)+c(x) 0, 2 u F (x, u) u2 M, (x, u) H. (2.3b) It ensures that the solution of problem (2.2), (2.1) exists and is unique for all values of ε (0, 1]. As the parameter ε tends to zero, a boundary layer appears in a neighbourhood of the set Γ 1 ; Γ=Γ 1 Γ2,Γ 1 and Γ 2 are the left and right sides respectively of the boundary of D. Our aim is to construct ε-uniformly convergent difference schemes for problem (2.2), (2.1). In the case of iterative schemes we also require that the number of iterations also be independent of ε. 3. Iteration-free difference scheme In this section we describe an ε-uniformly convergent difference scheme. On the set D we introduce the mesh D h, (3.1) which is a mesh with any distribution of the nodes satisfying the condition h MN 1, where h= max i h i, h i = x i+1 x i, x i,x i+1 D h, and N + 1 is the number of nodes in the mesh D h. On the mesh D h, we approximate problem (2.2), (2.1) by the difference scheme [6] Λ z(x) =F (x, z(x)), x D h, z(x) =ϕ(x), x Γ h. (3.2) Here D h = D D h,γ h =Γ D h,λ εa(x) δ x x +b(x) δ x c(x), x D h, δ x x z(x), δ x z(x) are the second (central) and the first (forward) differences; δ x x z(x) =2 ( h i 1 + h i) 1 [δx z(x) δ x z(x)], δ x z(x) = ( h i) 1 ( z(x i+1 ) z(x) ), δ x z(x) = ( h i 1) 1 ( z(x) z(x i 1 ) ), x = x i. The operator Λ is monotone [6] ε-uniformly. Applying the majorant function technique and taking into account a-priori estimates, we find u(x) z(x) Mε 2 N 1, x D h. (3.3) For uniform meshes D h (with the stepsize h = dn 1 ) (3.4) the following estimate is derived: u(x) z(x) M ( ε + N 1) 1 N 1, x D h. (3.5) The estimate (3.5) is optimal with respect to the entering values of N, ε, i.e. the estimate u(x) z(x) µ(n 1,ε), x D h, where µ(n 1,ε) 0 for N and fixed value of ε, isin general false if the following condition is satisfied: µ(n 1,ε)=o ( (ε + N 1 ) 1 N 1) for all N and ε, for which (ε + N 1 ) 1 N 1 = O(1). Thus, the difference scheme (3.2) on the meshes (3.1) and (3.4) does not converge ε-uniformly. We now consider the difference scheme (3.2) on a special mesh condensing in a neighbourhood of the boundary layer. On D we introduce the piecewise uniform mesh D h = D h (σ). (3.6) 1 Here and below M, M i (or m) denote sufficiently large (small) positive constants which do not depend on ε or on the discretization parameters. Throughout the paper, the notation L (j.k) (M (j.k),g h(j.k) ) means that these operators (constants, grids) are introduced in equation (j.k).

3 To construct the mesh (3.6), we divide the interval [0,d] into two parts [0,σ] and [σ, d], and in each part we use a uniform grid. The mesh width is equal to h (1) =2σN 1 in [0,σ] and to h (2) =2(d σ) N 1 in [σ, d]. We define σ = σ(n, ε) = min [ 2 1 d, Mε ln N ], where M = m 1, m is an arbitrary number satisfying 0 <m<m 0, m 0 = min D [a 1 (x) b(x)]. On the mesh (3.6) we have the estimate u(x) z(x) MN 1 min [ ln N, ε 1], x D h, (3.7) and also the following ε-uniform estimate u(x) z(x) MN 1 ln N, x D h. (3.8) Theorem 1 Let the data of the boundary value problem (2.2), (2.1) satisfy condition (2.3), and let a, b, c C 4+α (D), F C 4+α (H), α>0. Then the difference scheme (3.2), (3.6) is ε- uniformly convergent. The solution of the scheme satisfies estimates (3.3), (3.5), (3.7), (3.8); estimates (3.5), (3.7) and (3.8) are optimal with respect to N, ε and N respectively. 4. Base iterative difference scheme We now consider a base iterative difference scheme based on the stabilization method. a. We first introduce the relaxation scheme for the continuous problem (2.2), (2.1). Note that the solution of this problem is a stationary solution of the boundary value problem for the parabolic semidiscrete equation, which is discrete with respect to t L (4.1) u(x, t) =F (x, û(x, t)), x G h, (4.1a) u(x, t) =ϕ(x, t), (x, t) S h. (4.1b) Here G h = G G h, S h = S G h, G = D [0, ), S = G \ G, G h = D ω 0, (4.2) ω 0 is a uniform mesh on the axis t with the step-size h t =1; û(x, t)= u(x, t h t ); the operator L is defined by the relation Lu(x, t) {εa(x) 2 x 2 + b(x) } x c(x) p(x, t)δ t u(x, t), δ t u(x, t) is the backward difference derivative in t, δ t u(x, t) =h 1 t (u(x, t) û(x, t)); p(x, t) 0, (x, t) G h ; ϕ(x, t) =ϕ (2.2) (x) for t t 0 > 0. In general, the linear scheme (4.1), (4.2) is not monotone and does not converge as t. The condition p(x, t) u F (x, u), (x, t) G h, u R (4.3) is sufficient to ensure the monotonicity of scheme (4.1), (4.2); this condition is also close to necessary. By virtue of the relation (F (x, u) F (x, û)) u F (x, u)(u û) = O( u û ), equation (4.1a) can be considered as an approximation of the method of lines along x (on the mesh in t with the step-size h t = 1) of the boundary value problem for the parabolic equation L (2.2) u(x, t) P (x, t, u(x, t)) u(x, t) =F (x, u(x, t)), (x, t) G, u(x, t) =ϕ(x, t), (x, t) S, t where P (x, t, u) =p(x, t) F (x, u), (x, t) G, u R. For simplicity, we assume the u following condition to be fulfilled: u F (x, u)+c(x) c 0, p 0 P (x, t, u) p 0, (x, t) G, u R, c 0,p 0 > 0. (4.4) Taking into account this condition, which is sufficient for the monotonicity of the scheme (4.1), (4.2), we have the estimate u(x) u n (x) Mq n, x D, (4.5)

4 where u n (x) =u(x, t n ), (x, t n ) G h is the solution of problem (4.1), (4.2), t n = nh t, u(x), x D is the solution of problem (2.2), (2.1); q p 0 (p 0 + c 0 ) 1. Thus, the solution of problem (4.1), (4.2) converges ε-uniformly to the solution of problem (2.2), (2.1) as t (n ). b. We now consider an iterative method for solving the difference problem (3.2), (3.1). We define the difference scheme associated with problem (4.1), (4.2) by Here Λ z(x, t) =F (x, ẑ(x, t)), (x, t) G h, z(x, t) =ϕ(x, t), (x, t) S h. (4.6) G h = D h ω 0, (4.7) where ω 0 = ω 0(4.2), D h is one of the meshes (3.1), (3.4), (3.6), Λz(x, t) {Λ (3.2) p(x, t)δ t }z(x, t), (x, t) G h. The condition (4.3) is sufficient for the ε-uniform monotonicity of scheme (4.6), (4.7). Under condition (4.4) we find z(x) z n (x) Mq n, x D h, where z n (x) =z(x, t n ), q = q (4.5). Thus, the solution z n (x) of the iterative difference scheme (4.6), (4.7), on an arbitrary mesh (3.1) converges, as n, to the solution z(x) of the difference scheme (3.2), (3.1) ε-uniformly. But scheme (4.6), (4.7), just as scheme (3.2), on the mesh (3.1) does not converge ε-uniformly as N, n. In the case of the special mesh (3.6) we have the estimate u(x) z n (x) M [ N 1 min[lnn,ε 1 ]+q n ], x D h, (4.8) and also the ε-uniform estimate u(x) z n (x) M [ N 1 ln N + q n ], x D h, (4.9) that is the scheme (4.6), (4.7), (3.6) converges ε-uniformly as N, n. Under the condition n = n = M ln N, (4.10) on the mesh (3.6) we can obtain the estimate u(x) z n (x) MN 1 ln N, x D h ; (4.11) that is the scheme converges ε-uniformly as N. Theorem 2 Let the hypothesis of Theorem 1 hold for the solutions of the boundary value problem (2.2), (2.1), and let the data of the difference scheme (4.6), (4.7), (3.6) satisfy condition (4.4). Then the solution of the difference scheme converges to the solution of the boundary value problem ε-uniformly, as N, n. For the numerical solutions the estimates (4.8) and (4.9) hold, and also, under condition (4.10), the estimate (4.11) is valid. Estimates (4.8) and (4.9) are optimal with respect to the values of N, n, ε and N, n respectively. Remark. The number of iterations n (4.10) which guarantees the ε-uniform convergence (as N ) of the base iterative scheme (4.6), (4.7), (3.6) is independent of ε, i.e. this scheme is ε-uniformly robust. 5. Difference scheme for the domain decomposition method For the iterative difference scheme (4.6), (4.7), (3.1), we introduce the domain decomposition method on overlapping subdomains (c.f. [5]). a. Let the open subdomains D k, k =1,...,K (5.1a)

5 with boundaries Γ k = D k \ D k cover the domain D: D = K k=1 Dk. We denote the minimal overlap of the sets D k and D [k] = K i=1,i k Di by δ k, and the smallest value of δ k by δ, i.e. { min k,x 1,x ρ(x1,x 2 )=δ, x 1 D k, x 2 D [k], x 1, x 2 D k D [k]}, k =1,...,K, (5.1b) 2 where ρ(x 1,x 2 ) is the distance between the points x 1,x 2. Generally speaking, δ = δ(ε). By G(t 1 ) and G[t 1 ] we denote the intervals G(t 1 )={(x, t) : (x, t) G, t = t 1 }, G[t 1 ]= { (x, t) : (x, t) G, t = t 1 }, t1 ω 0. The interval G(t 1 ) is divided into the subdomains G k (t 1 )=G k G(t 1 ), G k = D k ω 0, k =1,...,K. Assume G k (t 1 )=G k (t 1 ) S k (t 1 ), S k (t 1 )=S0 k(t 1) S kl (t 1 ), S0 k(t 1)=G k [t 1 h t ], S kl (t 1 )=G k [t 1 ] \ G k (t 1 ), k =1,...,K. (5.1c) On the sets G k, G k (t 1 ) we construct the meshes G k h = G k G h, G k h (t 1 )=G k (t 1 ) G h, k =1,...,K; (5.1d) we assume that the boundaries of the partition subdomains {D k } pass through the nodes of the mesh D h, where D h is one of the meshes (3.1), (3.4), (3.6). Assuming that the function z(x, t), x D h is known for t = t n 1 ω 0, we find z(x, t) for t = t n ω 0. To do this, we first find the functions z k/k (x, t), (x, t) G h (t n ), by solving: ( ) Λ z k/k 0 (x, t) = 0, (x, t) G k h (tn ), (5.2a) { } z(x, t; t z k/k n ), k =1, 0 (x, t) = z (k 1)/K, (x, t) Sh k (x, t), k 2 (tn ), find z k/k 0 (x, t) for (x, t) G k h (t n ); k =1,...,K, t n ω 0, where z k/k 0 (x, t), (x, t) G k h (tn ), z k/k } (x, t) = z(x, t; t n ), k =1, z (k 1)/K, (x, t) G h (t n ) \ G k h (t n ) (x, t), k 2 (x, t) G h (t n ); k =1,...,K, t n ω 0 ;, (5.2b) Here Λ (z(x, t)) Λ (4.6) z(x, t) F (x, ẑ(x, t)), (x, t) G h. set G h [t n ] is defined by the relation The required function z(x, t) on the z(x, t) =z K/K (x, t), x D h, t n ω 0. (5.2c) In the relations (5.2a) and (5.2b), z(x, t; t n ), (x, t) G h (t n ), t n ω 0 is the extention to G[t n ]of the function z(x, t n 1 ), (x, t) G h [t n 1 ]; assume } z(x, t n 1 ), (x, t) G h (t n ), z(x, t; t n )= ϕ(x, t n, (x, t) G h [t n ]; ), (x, t) G h S (5.2d) z(x, t n 1 ), (x, t) G h [t n 1 ]; b. In the case of the condition (x, t) G h (t n ).

6 [ δ = δ (5.1b) (ε) > 0, ε (0, 1], inf ε 1 δ (5.1b) (ε) ] > 0 ε (0,1] (5.3) the solution z(x, t) of scheme (5.2), (5.1), (3.1) for n converges to the solution z(x) of scheme (3.2), (3.1) ε-uniformly z(x) z n (x) Mq n, x D h, (5.4) where q 1 m, z n (x) =z(x, t n ), t n ω 0. On the meshes (3.4) and (3.6) we obtain the estimates u(x) z n (x) M [ (ε + N 1 ) 1 N 1 + q n], x D h(3.4) ; (5.5) u(x) z n (x) M { N 1 min[ln N, ε 1 ]+q n}, x D h(3.6), (5.6) and also the ε-uniform estimate (on the mesh (3.6)) u(x) z n (x) M [ N 1 ln N + q n], x D h(3.6). (5.7) On the mesh (3.6), under the additional condition n = n = M ln N, (5.8) we have the estimate u(x) z n (x) MN 1 ln N, x D h(3.6). (5.9) Theorem 3 Let the hypotheses of Theorem 2 hold. Then the solution of the difference scheme (5.2), (5.1), (3.6) under condition (5.3) converges, as N, n, to the solution of the boundary value problem (2.2), (2.1) ε-uniformly; the condition (5.3) is neseccary and sufficient for ε- uniform convergence. The numerical solutions satisfy the estimates (5.4) (5.7) and, under the additional condition (5.8), the estimate (5.9). Estimates (5.5), (5.6) and (5.7) (estimate (5.9)) are optimal with respect to the values of N, n, ε and N, n (the value of N). Remark 1. For the base iterative scheme (4.6), (4.7), (3.1) under conditions (4.4), (4.10) and for the sequential scheme (5.2), (5.1), (3.6) under conditions (4.4), (5.3), (5.8), the amount of computational work does not exceed O(N 1 ln 2 N), but it is no less than O(N 1 ln N). Remark 2. Similar to the construction of the sequential scheme (5.2), (5.1), (3.6), an ε-uniform robust parallel scheme can be constructed (c.f. [5]). References [1] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O Riordan and G.I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and Hall/CRC, Boca Raton, [2] G.I. Marchuk, Methods of Numerical Mathematics, 3rd edn., Nauka, Moscow, 1989 (in Russian). [3] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin, [4] G.I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations, Ural Branch of Russian Acad. Sci., Ekaterinburg, 1992 (in Russian). [5] P.W. Hemker, G.I. Shishkin and L.P. Shishkina, Distributing the numerical solution of parabolic singularly perturbed problems with defect correction over independent processes, Sib. Zh. Vychisl. Mat. (Sib. J. Numer. Math.), bf 3, (2000). [6] A.A. Samarsky, Theory of Difference Schemes, Nauka, Moscow, 1989 (in Russian); English transl.: The Theory of Difference Schemes, Marcel Dekker, Inc., New York, [7] P.A. Farrell and G.I. Shishkin, Schwartz methods for singularly perturbed convectiondiffusion problems, in: Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, L.G. Vulkov, J.J.H. Miller and G.I. Shishkin eds., Nova Science, N.Y., 2000, pp