In this paper a model problem for fluid flow at high Reynolds number is era mined, Parabolic boundary layers are present because part of the boundary of the domain is a characteristic of the reduced differential equation. For such problems it is shown, by numerical example, that upwind finite difference schemes on uniform meshes are not epsilon-uniformly convergent in the discrete L infinity norm, where epsilon is the singular perturbation parameter. A discrete L infinity epsilon-uniformly convergent method is constructed for a singularly perturbed elliptic equation, whose solution contains parabolic boundary layers for small values of the singular perturbation parameter epsilon. This method makes use of a special piecewise uniform mesh. Numerical results are given that validate the theoretical results, obtained earlier by the last author, for such special mesh methods. (C) 1995 Academic Press, Inc.