Error estimates and numerical experiments for streamline-diffusion-type methods on arbitrary and Shishkin meshes

We analyse and numerically study streamline di usion nite element methods applied to a singularly perturbed convection di usion two point boundary value problem whose solution has a single boundary layer We rst consider arbitrary meshes then in analysing the scheme on a Shishkin mesh we consider two for mulations on the ne part of the mesh the usual streamline di usion upwinding and the standard Galerkin method The error estimates we report are given in the discrete L norm and in particular describe the dependence of the error on the user chosen parameter specifying the mesh When is too small the error becomes O but for above a certain threshold value the error is small and increases very slowly as a function of Numerical tests support the theoretical results for the L norm