singular perturbation; finite element method; convection-diffusion boundary value problem; superconvergence; supercloseness
The finite element method is applied to a convection-diffusion problem posed on the unite square using a tensor product mesh and bilinear elements. The usual proofs that establish superconvergence for this setting involve a rather high regularity of the exact solution - typically $\(u \in H^3(\Omega)\)$, which in many cases cannot be taken for granted. In this paper we derive superconvergence results where the right hand side of our a priori estimate no longer depends on the $\(H^3\)$ norm but merely requires finiteness of some weaker functional measuring the regularity. Moreover, we consider the streamline diffusion stabilization method and how superconvergence is affected by the regularity of the solution. Finally, numerical experiments for both discretizations support and illustrate the theoretical results.