Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations

A conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order non-self adjoint and indefinite elliptic equations. Unlike other discontinuous Galerkin (DG) methods, the numerical trace on the edge/triangle between two elements is not the average of two discontinuous P-k functions, but a lifted Pk+2 function from four (eight in 3D) nearby P-k functions. While all existing DG methods have the optimal order of convergence, this CDG method has a superconvergence of order two above the optimal order when solving general second order elliptic equations. Due to the superconvergence, a post-process lifts a P-k CDGsolution to a quasi-optimal Pk+2 solution on each element. Numerical tests in 2D and 3D are provided confirming the theory.