Document Type
Article
Publication Date
2-28-2016
Publisher
Hindawi
Abstract
A one-dimensional linear convection-diffusion problem with a perturbation parameter multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is -uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “-uniform stability plus -uniform consistency implies -uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in . At the same time, the condition number of the discrete system becomes independent of due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.
Recommended Citation
Nhan, T. A., & Vulanović, R. (2016). Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes. Advances in Numerical Analysis, 2016, e2161279. https://doi.org/10.1155/2016/2161279
Comments
Copyright © 2016 Thái Anh Nhan and Relja Vulanović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.