Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: Weak Galerkin Method

Abstract

. A simple stabilizer free weak Galerkin (SFWG) finite element method for a one-dimensional second order elliptic problem is introduced. In this method, the weak function is formed by a discontinuous k -th order polynomial with additional unknowns defined on vertex points, whereas its weak derivative is approximated by a polynomial of degree k + 1. The superconvergence of order two for the SFWG finite element solution is established. It is shown that the elementwise lifted P k + 2 solution of the P k SFWG one converges at the optimal order. Numerical results confirm the theory.