On a Necessary Requirement for Re-Uniform Numerical Methods to Solve Boundary Layer Equations for Flow along a Flat Plate

We consider grid approximations of a boundary value problem for the boundary layer equations modeling flow along a flat plate in a region excluding a neighbourhood of the leading edge. The problem is singularly perturbed with the perturbation parameter ε = 1/Re multiplying the highest derivative. Here the parameter ε takes any values from the half-interval (0,1], and Re is the Reynolds number. It would be of interest to construct an Re-uniform numerical method using the simplest grids, i.e., uniform rectangular grids, that could provide effective computational methods. To this end, we are free to use any technique even up to fitted operator methods, however, with fitting factors independent of the problem solution. We show that for the Prandtl problem, even in the case when its solution is self-similar, there does not exist a fitted operator method that converges Re-uniformly. Thus, combining a fitted operator and uniform meshes, we do not succeed in achieving Re-uniform convergence. Therefore, the use of the fitted mesh technique, based on meshes condensing in a parabolic boundary layer, is a necessity in constructing Re-uniform numerical methods for the above class of flow problems.