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.