Error expansion for an upwind scheme applied to a two-dimensional convection-diffusion problem

摘要

We consider a singularly perturbed convection-diffusion problem in a rectangular domain. It is solved numerically using a first-order upwind finite-difference scheme on a tensor-product piecewise-uniform Shishkin mesh with O(N) mesh points in each coordinate direction. It is known [G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural Branch, Ekaterinburg, Russia, 1992 ( in Russian)] that the error is almost-first-order accurate in the maximum norm. W e decompose the error into a sum of continuous almost-first-order terms and the almost-second-order residual under the assumption epsilon less than or equal to CN-1, where e is the singular perturbation parameter and C is a constant. This error expansion is applied to obtain maximum-norm error estimates for the Richardson extrapolation technique and derive bounds on the errors in approximating the derivatives of the true solution by divided differences of the computed solution. The analysis uses a decomposition of the true solution requiring fewer compatibility conditions than in earlier publications. Numerical results are presented that support our theoretical results. [References: 14]