Superconvergence similarities in standard and mixed finite element methods

In this presentation, we will show that it is possible to tackle some superconvergence questions in finite elements for the standard and mixed setting simultaneously. The similarities can already seen to be present in the simple one-dimensional case. Essentially, they become clear if in the standard setting, the finite element solution is not (as is traditionally done) compared to Lagrangian interpolants of the exact solution, but to the local interpolation operator of which the curl is equal to the Fortin interpolation of the curl. In two space dimensions, this approach results in relatively simple proofs for superconvergence results for the mixed and standard elements in one go, for example, the linear and quadratic triangular elements and all order rectangular elements.