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.