Superconvergence of weak Galerkin finite element approximation for second order elliptic problems by L2-projections

Abstract

The weak Galerkin finite element method (WG-FEM) is a novel numerical method that was first proposed and analyzed by Wang and Ye (2013) [12] for general second order elliptic problems on triangular meshes based on a discrete weak gradient. In general, the weak Galerkin finite element formulations for partial differential equations can be derived naturally by replacing usual derivatives by weakly defined derivatives in the corresponding variational forms. The superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang (2000) [11] proposed and analyzed superconvergence of the standard Galerkin finite element method by L2-projections. The main idea behind the L2-projections is to project the finite element solution to another finite element space with a coarse mesh and a higher order of polynomials. The objective of this paper is to establish a general superconvergence result for the weak Galerkin finite element approximations for second order elliptic problem by L2-projection methods. The results of numerical experiments show great promise for the robustness, reliability, flexibility and accuracy of superconvergence in WG-FEM by L2-projection methods.