A finite difference analysis of a streamline diffusion method on a Shishkin mesh

We consider streamline diffusion finite element methods applied to a singularly perturbed convection-diffusion two-point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L-infinity norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter tau(0) specifying the mesh. When tau(0) is too small, the error becomes O(1), but for tau(0) above a certain threshold value, the error is small and increases either linearly or quadratically as a function of tau(0). Numerical tests support our theoretical results.