On Finite Difference Fitted Schemes for Singularly Perturbed Boundary Value Problems with a Parabolic Boundary Layer

A method to construct grid approximations for singularly perturbed boundary value problems for elliptic and parabolic equations, whose solutions contain a parabolic boundary layer, is considered. The grid approximations are based on the fitted operator method. Finite difference schemes, finite element, or finite volume techniques are included in the term grid approximation methods. It is shown that there exists no grid approximation method on uniform grids from the class of fitted operator methods, whose solutions converge, in the discrete maximum norm, uniformly with respect to the perturbation parameter to the solution of the boundary value problem.