A superconvergent CDG finite element for the Poisson equation on polytopal meshes

Abstract

A conforming discontinuous Galerkin (CDG) finite element is constructed for solving second order elliptic equations on polygonal and polyhedral meshes. The numerical trace on the edge between two elements is no longer the average of two discontinuous P_k functions on the two sides, but a lifted function from four P_k functions. When the numerical gradient space is the subspace of piecewise polynomials on subtriangles/subtehrahedra of a polygon/polyhedron T which have a one‐piece polynomial divergence on T, this CDG method has a superconvergence of order two above the optimal order. Due to the superconvergence, we define a post‐process which lifts a P_k CDG solution to a quasi‐optimal solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory.