The derivative patch interpolation recovery technique and superconvergence for the discontinuous Galerkin method

We consider the discontinuous QkQk-finite element approximations to the elliptic boundary value problems in d-dimensional rectangular domain. A derivative recovery technique is proposed by interpolating the derivatives of discrete solution on the patch domain. Based on the superclose estimate derived in this paper, we show that the recovered derivatives possess the local and global superconvergence. Furthermore, the asymptotically exact a posteriori estimator is given on the error of gradient approximation. Finally, numerical experiments are presented to illustrate the theoretical analysis.