Đang chuẩn bị liên kết để tải về tài liệu:
MONOTONE FINITE DIFFERENCE DOMAIN DECOMPOSITION ALGORITHMS AND APPLICATIONS TO NONLINEAR SINGULARLY

MONOTONE FINITE DIFFERENCE DOMAIN DECOMPOSITION ALGORITHMS AND APPLICATIONS TO NONLINEAR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS IGOR BOGLAEV AND MATTHEW HARDY Received 16 September 2004; Revised 21 December 2004; Accepted 11 January 2005 This paper deals with monotone finite difference iterative algorithms for solving nonlinear singularly perturbed reaction-diffusion problems of elliptic and parabolic types. Monotone domain decomposition algorithms based on a Schwarz alternating method and on box-domain decomposition are constructed. These monotone algorithms solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear discrete problems. The rate of convergence of the monotone domain decomposition algorithms. | MONOTONE FINITE DIFFERENCE DOMAIN DECOMPOSITION ALGORITHMS AND APPLICATIONS TO NONLINEAR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS IGOR BOGLAEV AND MATTHEW HARDY Received 16 September 2004 Revised 21 December 2004 Accepted 11 January 2005 This paper deals with monotone finite difference iterative algorithms for solving nonlinear singularly perturbed reaction-diffusion problems of elliptic and parabolic types. Monotone domain decomposition algorithms based on a Schwarz alternating method and on box-domain decomposition are constructed. These monotone algorithms solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear discrete problems. The rate of convergence of the monotone domain decomposition algorithms are estimated. Numerical experiments are presented. Copyright 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction We are interested in monotone discrete Schwarz alternating algorithms for solving nonlinear singularly perturbed reaction-diffusion problems. The first problem considered corresponds to the singularly perturbed reaction-diffusion problem of elliptic type -p2 uxx Uyy f x y u 0 x y e w u g on dw w wx X wy 0 x 1 X 0 y 1 1.1 fu c x y u e w X -TO to fu df du where p is a small positive parameter c 0 is a constant dw is the boundary of w. If f and g are sufficiently smooth then under suitable continuity and compatibility conditions on the data a unique solution u of 1.1 exists see 6 for details . Furthermore for p 1 problem 1.1 is singularly perturbed and characterized by boundary layers i.e. regions with rapid change of the solution of width O p I ln pl near dw see 1 for details . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 70325 Pages 1-38 DOI 10.1155 ADE 2006 70325 2 Monotone domain decomposition algorithms The second problem considered corresponds to the singularly perturbed reactiondiffusion problem of parabolic type