V. B. Andreev ¹

A Dirichlet problem for a singularly perturbed steady-state convection- diffusion equation with constant coefficients on the unit square is considered. In the equation under consideration the convection term is represented by only a single derivative with respect to one coordinate axis. This problem is discretized by the classical five-point upwind difference scheme on a rectangular piecewise uniform mesh that is refined in the neighborhood of the regular and the characteristic boundary layers. It is proved that, for sufficiently smooth right-hand side of the equation and the restrictions of the continuous boundary function to the sides of the square, without additional compatibility conditions at the corners, the error of the discrete solution is $O(N^{-1}\ln^2 N)$ uniformly with respect to the small parameter, in the discrete maximum norm, where N is the number of mesh points in each coordinate direction.