Superconvergence of quadratic finite elements on mildly structured grids

Superconvergence estimates are studied in this paper on quadratic finite element discretizations for second order elliptic boundary value problems on mildly structured triangular meshes. For a large class of practically useful grids, the finite element solution u(h) is proven to be superclose to the inter-polant u(I) and as a result a postprocessing gradient recovery scheme for uh can be devised. The analysis is based on a number of carefully derived identities. In addition to its own theoretical interests, the result in this paper can be used for deriving asymptotically exact a posteriori error estimators for quadratic finite element methods.