Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence

In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In particular, we first show a superconvergence of the gradient of the finite element solution u h and to the gradient of the interpolant u I, We then analyze a postprocessing gradient recovery scheme, showing that Q h∇u h is superconvergent approximation to ∇u. Here Q h is the global L 2 projection. In Part II, we analyze a superconvergent gradient recovery scheme for general unstructured, shape regular triangulations. This is the foundation for an a posteriori error estimate and local error indicators.