On the superconvergence patch recovery techniques for the linear finite element approximation on anisotropic meshes

We provide in this paper an analysis on the superconvergence patch recovery (SPR) techniques for the linear finite element approximation based on adaptively refined anisotropic meshes in two dimensions. These techniques include the gradient recovery based on local weighted averaging, the recovery based on local L2-projection, and the recovery based on least square fitting. The last one leads to the Zienkiewicz-Zhu type error estimators popular in engineering communities. Based on the superconvergence result for anisotropic meshes established recently in Cao (2013), we prove that all three types of SPR techniques produce super-linearly convergent gradients if the meshes are quasi-uniform under a given metric and each pair of adjacent elements in the meshes form an approximate parallelogram. As a consequence, the error estimators based on the recovered gradient are asymptotically exact. These results provide a theoretical justification for the extraordinary robustness and accuracy observed in numerous applications for the recovery type error estimators on anisotropic meshes.