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

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

\{\mathrm{\nabla }{u}_h·\mathbf{n}\}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{\nabla u_h\cdot \textbf{n}\}$$\end{document}

is not introduced to the computation in the CDG method. Additionally, the numerical-trace

\{u_h\}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{ u_h \}$$\end{document}

is not the average

(u_h{|}_{T_1}+u_h{|}_{T_2})/2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\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

P_{k+1}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{k+1}$$\end{document}

polynomial from the

P_k

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_k$$\end{document}

solution

u_h

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\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.

Keywords

Parabolic equations / Finite element / Conforming discontinuous Galerkin (CDG) / Triangular mesh / Tetrahedral mesh / Primary / 65N15 / 65N30

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