A higher order numerical method for singularly perturbed elliptic problems with characteristic boundary layers

A Petrov-Galerkin finite element method is constructed for a singu?larly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary lay?ers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a sta?ble parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh.