Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions

This article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of O(h^r), 1.5≤r≤2, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is O(h) for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.