In this article we consider grid approximations of a boundary value problem for boundary layer equations for a flat plate outside of a neighbourhood of its leading edge. The perturbation parameter e = Re-1 multiplying the highest derivative can take arbitrary values from the half-interval (0,1]; here Re is the Reynolds number. We consider the case when the solution of this problem is self-similar. For this Prandtl problem by using piecewise uniform meshes, which are refined in the neighbourhood of a parabolic boundary layer, we construct a finite difference scheme that converges e-uniformly. We present the technique for experimental substantiation of e-uniform convergence of both the grid solution itself and its normalized difference derivatives, which are considered outside of a neighbourhood of the leading edge of the plate. We study also the applicability of fitted operator methods for the numerical approximation of the Prandtl problem. It is shown that the use of meshes condensing in the parabolic boundary layer region is necessary for achieving e-uniform convergence.