Singularly perturbed convection-diffusion elliptic problems with a non-smooth forcing term

Abstract

Singularly perturbed elliptic problems, of convection-diffusion type, with a non-smooth forcing term are examined. The lack of smoothness arises from the forcing term either containing an interior layer or being discontinuous across an interface. In addition to the presence of several different kinds of boundary and corner layers, this forcing term introduces an interior layer in the solution. For both problem classes, a decomposition of the continuous solution is constructed, whose components identify the various types of layer functions that can exist in the solution. Parameter-explicit pointwise bounds on the partial derivatives of these components are then established. An appropriate Shishkin mesh is identified and this is combined with upwinding to form a numerical method for each problem class. Parameter-uniform error bounds in the maximum norm are deduced. Numerical results are presented to illustrate the performance of both numerical methods.