Some superconvergence results for finite element discretizations on a Shishkin mesh of a convection-diffusion problem.

We consider finite element discretizations of a singularly perturbed two-point boundary value problem. The problem is discretized on a piecewise equidistant mesh. It has been shown that such meshes are very interesting for the discretization of singularly perturbed problems. Good approximations may be obtained both of the smooth part of the solution and of the boundary layer. Indeed, even standard finite element methods, without any kind of upwinding, perform very satisfactorily when a piecewise equidistant mesh is used. For a regular problem, convergence of a finite element approximation at the mesh nodes is often faster than convergence in H-1. This phenomenon is called superconvergence. In this paper, we prove some pointwise error estimates for finite element approximations with piecewise linear and piecewise quadratic basis functions. It will be shown that convergence estimates that hold for the regular case also hold for the singular perturbation case, apart from logarithmic factors.