Two-Order Superconvergent CDG Finite Element Method for the Heat Equation on Triangular and Tetrahedral Meshes

In this paper we introduce a conforming discontinuous Galerkin (CDG) finite element method for solving the heat equation. Unlike the rest of discontinuous Galerkin (DG) methods, the numerical-flux \(\{\nabla u_h\cdot \textbf{n}\}\) $\{\mathrm{\nabla }{u}_h·\mathbf{n}\}$ is not introduced to the computation in the CDG method. Additionally, the numerical-trace \(\{ u_h \}\) $\{u_h\}$ is not the average \((u_h|_{T_1} +u_h|_{T_2})/2\) $(u_h{|}_{T_1}+u_h{|}_{T_2})/2$ (or some other simple average used in other DG methods), but a lifted \(P_{k+1}\) $P_{k+1}$ polynomial from the \(P_k\) $P_k$ solution \(u_h\) $u_h$ on nearby four triangles in 2D, or eight tetrahedra in 3D. We show a two-order superconvergence in space approximation when using the CDG method with a backward Euler time discretization, on triangular and tetrahedral meshes, for solving the heat equation. Numerical tests are reported which confirm the theory.