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

doi:10.1007/s42967-024-00444-4

摘要/Abstract

摘要： 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.

关键词: Parabolic equations, Finite element, Conforming discontinuous Galerkin (CDG), Triangular mesh, Tetrahedral mesh

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

中图分类号:

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.

参考文献

[1] Al-Twaeel, A., Hussian, S., Wang, X.: A stabilizer free weak Galerkin finite element method for parabolic equation. J. Comput. Appl. Math. 392, 113373 (2021)
[2] Al-Taweel, A., Wang, X., Ye, X., Zhang, S.: A stabilizer free weak Galerkin element method with supercloseness of order two. Numer. Methods Partial Differential Equations 2, 1012–1029 (2021)
[3] Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Comput. (Arch. Elektron. Rechnen) 5, 207–213 (1970)
[4] Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1972)
[5] Babuška, I., Zlámal, M.: Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10, 863–875 (1973)
[6] Chen, G., Feng, M., Xie, X.: A robust WG finite element method for convection-diffusion-reaction equations. J. Comput. Appl. Math. 315, 107–125 (2017)
[7] Chen, W., Wang, F., Wang, Y.: Weak Galerkin method for the coupled Darcy-Stokes flow. IMA J. Numer. Anal. 36, 897–921 (2016)
[8] Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
[9] Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Fran. Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R-3), 33–75 (1973)
[10] Cui, M., Zhang, S.: On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation. J. Sci. Comput. 82, 5–15 (2020)
[11] Deka, B., Kumar, N.: Error estimates in weak Galerkin finite element methods for parabolic equations under low regularity assumptions. Appl. Numer. Math. 162, 81–105 (2021)
[12] Deka, B., Kumar, N.: A systematic study on weak Galerkin finite element method for second order parabolic problems. Numer. Methods Partial Differential Equations 39(3), 2444–2474 (2023)
[13] Deka, B., Roy, P.: Weak Galerkin finite element methods for parabolic interface problems with nonhomogeneous jump conditions. Numer. Funct. Anal. Optim. 40, 250–279 (2019)
[14] Douglas, J., Jr., Dupont, T.: Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 7, 575–626 (1970)
[15] Gao, F., Mu, L.: On

\begin{document}$ L^2 $\end{document}

error estimate for weak Galerkin finite element methods for parabolic problems. J. Comp. Math. 32, 159–204 (2014)
[16] Gao, F., Wang, X.: A modified weak Galerkin finite element method for a class of parabolic problems. J. Comput. Appl. Math. 271, 1–19 (2014)
[17] Guan, Q.: Weak Galerkin finite element method for Poisson’s equation on polytopal meshes with small edges or faces. J. Comput. Appl. Math. 368, 112584 (2020)
[18] Hu, Q., He, Y., Wang, K.: Weak Galerkin method for the Helmholtz equation with DTN boundary condition. Int. J. Numer. Anal. Model. 17, 643–661 (2020)
[19] Hu, X., Mu, L., Ye, X.: A weak Galerkin finite element method for the Navier-Stokes equations on polytopal meshes. J. Comput. Appl. Math. 362, 614–625 (2019)
[20] Huang, W., Wang, Y.: Discrete maximum principle for the weak Galerkin method for anisotropic diffusion problems. Commun. Comput. Phys. 18, 65–90 (2015)
[21] Li, G., Chen, Y., huang, Y.: A new weak Galerkin finite element scheme for general second-order elliptic problems. J. Comput. Appl. Math. 344, 701–715 (2018)
[22] Li, H., Mu, L., Ye, X.: Interior energy estimates for the weak Galerkin finite element method. Numer. Math. 139, 447–478 (2018)
[23] Li, Q., Wang, J.: Weak Galerkin finite element methods for parabolic equations. Numer. Methods Partial Differential Equations 29, 2004–2024 (2013)
[24] Li, Y., Shu, C.-W., Tang, S.: A local discontinuous Galerkin method for nonlinear parabolic SPDEs. ESAIM Math. Model. Numer. Anal. 55, S187–S223 (2021)
[25] Lin, R., Ye, X., Zhang, S., Zhu, P.: A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems. SIAM J. Numer. Anal. 56, 1482–1497 (2018)
[26] Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 675–698 (2008)
[27] Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541–564 (2010)
[28] Liu, J., Tavener, S., Wang, Z.: Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes. SIAM J. Sci. Comput. 40, 1229–1252 (2018)
[29] Liu, J., Tavener, S., Wang, Z.: The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes. J. Comput. Phys. 359, 317330 (2018)
[30] Liu, X., Li, J., Chen, Z.: A weak Galerkin finite element method for the Oseen equations. Adv. Comput. Math. 42, 1473–1490 (2016)
[31] Liu, Y., Nie, Y.: A priori and a posteriori error estimates of the weak Galerkin finite element method for parabolic problems. Comput. Math. Appl. 99, 78–83 (2021)
[32] Liu, Y., Shu, C.-W., Tadmor, E., Zhang, M.: Central local discontinuous Galerkin methods on overlapping cells for diffusion equations. ESAIM Math. Model. Numer. Anal. 45(6), 1009–1032 (2011)
[33] Miao, Y., Yan, J., Zhong, X.: Superconvergence study of the direct discontinuous Galerkin method and its variations for diffusion equations. Commun. Appl. Math. Comput. 4(1), 180–204 (2022)
[34] Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes. Int. J. Numer. Anal. Model. 12, 31–53 (2015)
[35] Mu, L., Ye, X., Zhang, S.: A stabilizer free, pressure robust and superconvergence weak Galerkin finite element method for the Stokes equations on polytopal mesh. SIAM J. Sci. Comput. 43, A2614–A2637 (2021)
[36] Nitsche, J.: On Dirichlet problems using subspaces with nearly zero boundary conditions. The mathematical foundations of the finite element method with applications to partial differential equations. In: Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972. pp. 603–627. Academic Press, New York (1972)
[37] Noon, N.: Weak Galerkin and weak group finite element methods for 2-D Burgers problem. J. Adv. Res. Fluid Mech. 69, 42–59 (2020)
[38] Ohm, M.R.: Error estimate for parabolic problem by backward Euler discontinuous Glerkin method. Int. J. Appl. Math. 22, 117–128 (2009)
[39] Price, H. S., Varga, R. S.: Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics. In: Numerical Solution of Field Problems in Continuum Physics. Proc. Sympos. Appl. Math., Durham, N.C., 1968, SIAM-AMS Proc., vol. II, pp. 74–94. American Mathematical Society, Providence (1970)
[40] Qi, W., Song, L.: Weak Galerkin method with implicit

\begin{document}$ \theta $\end{document}

-schemes for second-order parabolic problems. Appl. Math. Comput. 336, 124731 (2020)
[41] Riviére, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008)
[42] Shields, S., Li, J., Machorro, E.A.: Weak Galerkin methods for time-dependent Maxwell’s equations. Comput. Math. Appl. 74, 2106–2124 (2017)
[43] Sun, M., Rui, H.: A coupling of weak Galerkin and mixed finite element methods for poroelasticity. Comput. Math. Appl. 73, 804–823 (2017)
[44] Thomée, V.: Galerkin finite element methods for parabolic problems. In: Lecture Notes in Mathematics, vol. 1054. Springer-Verlag, Berlin (1984)
[45] Toprakseven, S.: A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients. Appl. Numer. Math. 168, 1–12 (2021)
[46] Wang, C., Wang, J.: Discretization of div-curl systems by weak Galerkin finite element methods on polyhedral partitions. J. Sci. Comput. 68, 1144–1171 (2016)
[47] Wang, J., Wang, X., Ye, X., Zhang, S., Zhu, P.: Two-order superconvergence for a weak Galerkin method on rectangular and cuboid grids. Numer. Methods Partial Differential Equations 39(1), 744–758 (2023)
[48] Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)
[49] Wang, J., Ye, X., Zhang, S.: A time-explicit weak Galerkin scheme for parabolic equations on polytopal partitions. J. Numer. Math. 31(2), 125–135 (2023)
[50] Wang, J., Zhai, Q., Zhang, R., Zhang, S.: A weak Galerkin finite element scheme for the Cahn-Hilliard equation. Math. Comp. 88, 211–235 (2019)
[51] Wheeler, M.F.: A priori

\begin{document}$ L^2 $\end{document}

error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10, 723–759 (1973)
[52] Xie, Y., Zhong, L.: Convergence of adaptive weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. 86(1), 25 (2021)
[53] Ye, X., Zhang, S.: A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes. SIAM J. Numer. Anal. 58, 2572–2588 (2020)
[54] Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method. Int. J. Numer. Anal. Model. 17(1), 110–117 (2020)
[55] Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method: Part II. Int. J. Numer. Anal. Model. 17(2), 281–296 (2020)
[56] Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method: Part III. Int. J. Numer. Anal. Model. 17(6), 794–805 (2020)
[57] Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes. Internat. J. Numer. Methods Fluids 93(6), 1913–1928 (2021)
[58] Ye, X., Zhang, S.: A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh. Electron. Res. Arch. 29(6), 3609–3627 (2021)
[59] Ye, X., Zhang, S.: Achieving superconvergence by one-dimensional discontinuous finite elements: weak Galerkin method. East Asian J. Appl. Math. 12(3), 590–598 (2022)
[60] Ye, X., Zhang, S.: Achieving superconvergence by one-dimensional discontinuous finite elements: the CDG method. East Asian J. Appl. Math. 12(4), 781–790 (2022)
[61] Ye, X., Zhang, S.: Order two superconvergence of the CDG method for the Stokes equations on triangle/tetrahedron. J. Appl. Anal. Comput. 12(6), 2578–2592 (2022)
[62] Ye, X., Zhang, S.: Order two superconvergence of the CDG finite elements on triangular and tetrahedral meshes. CSIAM Trans. Appl. Math. 4(2), 256–274 (2023)
[63] Feng, Y., Liu, Y., Wang, R., Zhang, S.: A conforming discontinuous Galerkin finite element method on rectangular partitions. Electron. Res. Arch. 29(3), 2375–2389 (2021)
[64] Zhang, H., Zou, Y., Chai, S., Yue, H.: Weak Galerkin method with

\begin{document}$ (r, r-1, r-1) $\end{document}

-order finite elements for second order parabolic equations. Appl. Math. Comp. 275, 24–40 (2016)
[65] Zhang, H., Zou, Y., Xu, Y., Zhai, Q., Yue, H.: Weak Galerkin finite element method for second order parabolic equations. Int. J. Numer. Anal. Model 13, 525–544 (2016)
[66] Zhang, J., Zhang, K., Li, J., Wang, X.: A weak Galerkin finite element method for the Navier-Stokes equations. Commun. Comput. Phys. 23, 706–746 (2018)
[67] Zhang, T., Lin, T.: A posteriori error estimate for a modified weak Galerkin method solving elliptic problems. Numer. Methods Partial Differential Equations 33, 381–398 (2017)
[68] Zhou, J., Xu, D., Dai, X.: Weak Galerkin finite element method for the parabolic integro-differential equation with weakly singular kernel. Comput. Appl. Math. 38(2), 12 (2019)
[69] Zhou, S., Gao, F., Li, B., Sun, Z.: Weak galerkin finite element method with second-order accuracy in time for parabolic problems. Appl. Math. Lett. 90, 118–123 (2019)
[70] Zhu, A., Xu, T., Xu, Q.: Weak Galerkin finite element methods for linear parabolic integro-differential equations. Numer. Methods Partial Differential Equations 32, 1357–1377 (2016)
[71] Zhu, H., Zou, Y., Chai, S., Zhou, C.: Numerical approximation to a stochastic parabolic PDE with weak Galerkin method. Numer. Math. Theory Meth. Appl. 11, 604–617 (2018)