Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes

Abstract

It is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous P_k elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous P_k element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the L² norm. A local post-process is defined which lifts a P_k CDG solution to a discontinuous P_k+2 solution. It is proved that the lifted P_k+2 solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the P_k CDG finite element is as good as the P_k+2 continuous Galerkin finite element.

Keywords

Finite element / conforming discontinuous Galerkin method / stabilizer free / triangular grid / tetrahedral grid