A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation

作     者：Grigory I.Shishkin 

作者机构：Institute of Mathematics and Mechanics Ural Branch of Russian Academy of Sciences 

出 版 物：《Numerical Mathematics(Theory,Methods and Applications)》 (高等学校计算数学学报（英文版）)

年 卷 期：2008年第1卷第2期

页      面：214-234页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Russian Foundation for Basic Research [07-01-00729] Boole Centre for Research in Informatics at the National University of Ireland, Cork Mathematics Applications Consortium for Science and Industry in Ireland (MACSI) under the Science Foundation Ireland (SFI) Mathematics Initiative 

主　　题：Singular perturbations convection-diffusion problem piecewise-uniform mesh α priori adapted mesh almost ε-uniform convergence 

摘      要：A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on α priori （sequentially） adapted meshes and study its convergence. The scheme on α priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find α priori a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter ε, the step-size of a uniform mesh in χ, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for improving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter ε. The scheme converges almost ε-uniformly, precisely, under the condition N^-1 = o （ε^v）, where N denotes the number of nodes in the spatial mesh, and the value v = v（K） can be chosen arbitrarily small for suitable K.