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

doi:10.1007/s42967-024-00444-4

Abstract

Abstract: 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 \begin{document}$ \{\nabla u_h\cdot \textbf{n}\} $\end{document} is not introduced to the computation in the CDG method. Additionally, the numerical-trace \begin{document}$ \{ u_h \} $\end{document} is not the average \begin{document}$ (u_h|_{T_1} +u_h|_{T_2})/2 $\end{document} (or some other simple average used in other DG methods), but a lifted \begin{document}$ P_{k+1} $\end{document} polynomial from the \begin{document}$ P_k $\end{document} solution \begin{document}$ u_h $\end{document} 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.

Key words: Parabolic equations, Finite element, Conforming discontinuous Galerkin (CDG), Triangular mesh, Tetrahedral mesh

CLC Number:

Xiu Ye, Shangyou Zhang. Two-Order Superconvergent CDG Finite Element Method for the Heat Equation on Triangular and Tetrahedral Meshes[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 411-426.

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