Special Meshes for Finite Difference Approximations to an Advection-Diffusion Equation with Parabolic Layers

In this paper a model problem for fluid flow at high Reynolds number is examined. 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 ε-uniformly convergent in the discrete L∞ norm, where ε is the singular perturbation parameter. A discrete L∞ ε-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 ε. 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.