A fitted finite element method for singularly perturbed elliptic problems with characteristic boundary layers

A Petrov–Galerkin finite element method on a rectangular Shishkin mesh is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. The test functions are taken to be a tensor product of exponential splines in one coordinate direction and hat functions in the other direction, which results in a stable higher (than first) order numerical method. The differential equation contains a zero order term. Compared to the case of no zero order term, the character of the exponential splines are more complicated and the associated pointwise error analysis relies more on the finite element formulation. Error bounds are given in a global pointwise norm. Numerical results are presented to illustrate the performance of the method.