Abstract

In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on nonuniform triangular meshes satisfying an $O(h^2)$ parallelogram property in most parts of the domain. In particular, we first show the finite element solution $u_h$ and the interpolant $u_I$ have super close gradients. We then analyze a postprocessing gradient recovery scheme, showing that $Q_h\nabla u_h$ is a superconvergent approximation to $\nabla u$. 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.