A class of singularly perturbed time-dependent convection-diffusion problems with a boundary turning point is examined on a rectangular domain. The solution of problems from this class possesses a parabolic boundary layer in the neighborhood of one of the sides of the domain. Classical numerical methods on uniform meshes are known to be inadequate for problems with boundary layers. A numerical method consisting of a standard upwind finite difference operator on a fitted mesh is constructed. It is proved that the numerical approximations generated by this method converge uniformly with respect to the singular perturbation parameter. Numerical results are presented that verify computationally the theoretical result.
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Dunne et al. (2003) studied this question.