Gradient recovery for singularly perturbed boundary value problems II: Two-dimensional convection-diffusion

We consider a Galerkin finite element method that uses bilinear elements on a class of Shislikin-type meshes for a model singularly perturbed convection-diffusion problem. The recovered gradient is itself piecewise bilinear, with values at the nodes obtained by first interpolating the gradient of the finite element approximation at the centroids of the elements sharing the node. We prove a superconvergence estimate for the recovered gradient uniformly with respect to the singular perturbation parameter. Numerical experiments support our results.