A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction-diffusion problem

We consider the numerical solution of a singularly perturbed two-dimensional reactiondiffusion problem by a multiscale sparse grid finite element method. A Shishkin mesh which resolves the boundary and corner layers, and yields a parameter robust solution, is used. Our analysis shows that the method achieves essentially the same accuracy as the standard Galerkin finite element method, but does so at a much lower computational cost.