THE COMBINATION TECHNIQUE FOR A TWO-DIMENSIONAL CONVECTION-DIFFUSION PROBLEM WITH EXPONENTIAL LAYERS

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our "combination" method simply adds or subtracts solutions that have been computed by the Galerkin FEM on N x root N, root N x N and root N x root N meshes. It is shown that the combination FEM yields (up to a factor lnN) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N x N mesh, but it requires only O(N(3/2)) degrees of freedom compared with the O(N(2)) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.