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

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 Pk functions on the two sides, but a lifted Pk+2$P_{k+2}$ function from four Pk functions. When the numerical gradient space is the H(div,T)$H(\hbox{div},T)$ subspace of piecewise Pk+1d$P_{k+1}<^>d$ 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 Pk CDG solution to a quasi-optimal Pk+2$P_{k+2}$ solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory.