Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes (CSIAM Transactions on Applied Mathematics, 2023)
It is known that discontinuous finite element methods use more unknownvariables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite elementmethod 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. Superconvergenceof order two for the CDG finite element solution is proved in an energy norm and inthe $L^2$ 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 theoptimal 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.