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

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

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  1. Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiu Ye

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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Ye, X., Zhang, S. Two-Order Superconvergent CDG Finite Element Method for the Heat Equation on Triangular and Tetrahedral Meshes. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-024-00444-4

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