Fitted mesh numerical methods for singularly perturbed elliptic problems with mixed derivatives

Abstract

A class of singularly perturbed convection―diffusion problems is considered which contain a mixed derivative term. We consider the case when exponential boundary layers are present in the solutions of problems from this class. Under appropriate assumptions on the data of the problem, we construct a decomposition of the solution into regular and layer components. We then introduce a numerical method on a piecewise-uniform fitted mesh. Excluding a neighbourhood of one of the comers, it is shown that in the perturbed case (i.e. when the perturbation parameter is small relative to the inverse of the number of mesh intervals in both coordinate directions), the approximations generated by the method converge uniformly with respect to the singular perturbation parameter. Finally, numerical examples are presented that illustrate the theoretical result.