Superconvergence for finite element approximation of a convection–diffusion equation using graded meshes

In this paper we analyse the approximation of a model convection–diffusion equation by standard bilinear finite elements using the graded meshes introduced in Duran & Lombardi (2006, Finite element approximation of convection–diffusion problems using graded meshes. Appl. Numer. Math., 56, 1314–1325). Our main goal is to prove superconvergence results of the type known for standard elliptic problems, namely, that the difference between the finite element solution and the Lagrange interpolation of the exact solution, in the ε-weighted H¹-norm, is of higher order than the error itself. The constant in our estimate depends only weakly on the singular perturbation parameter. As a consequence of the superconvergence result we obtain optimal order error estimates in the L²-norm. Also we show how to obtain a higher order approximation by a local postprocessing of the computed solution.