| A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation |
| |
| 作者单位： | Institute of |
| |
| 摘 要： | A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation;we construct a finite difference scheme on a priori (se-quentially) adapted meshes and study its convergence.The scheme on a priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution,which allows us to find a priori a subdomain where the computed solution requires a further improvement.This subdomain is defined by the perturbation parameterε,the step-size of a uniform mesh in x,and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for im- proving 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 so- lution depends weakly on the parameterε.The scheme converges almostε-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.
|
| 关 键 词： | 有限微分 奇异扰动 对流扩散问题 收敛性 |
A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation |
| |
| Authors: | Grigory I Shishkin |
| |
| Institution: | Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Ekaterinburg 620219, Russia |
| |
| Abstract: | A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on a priori (se-quentially) adapted meshes and study its convergence. The scheme on a priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find a 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 x, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for im-proving 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 so-lution depends weakly on the parameter ε. The scheme converges almost ε-uniformly, precisely, under the condition N-1 = o(ev), 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. |
| |
| Keywords: | Singular perturbations convection-diffusion problem piecewise-uniform mesh a priori adapted mesh |
| 本文献已被 CNKI 维普 万方数据 等数据库收录！ |
|
