Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach

Abstract

This paper investigates superconvergence properties of the direct discontinuous Galerkin (DDG) method with interface corrections and the symmetric DDG method for diffusion equations. We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes. Based on the eigen-structure analysis, we carry out error estimates of the DDG solutions, which can be decomposed into three parts: (i) dissipation errors of the physically relevant eigenvalue, which grow linearly with the time and are of order 2k for

P^k(k=2,3)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^k\ (k=2,3)$$\end{document}

approximations; (ii) projection error from a special projection of the exact solution, which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue; (iii) dissipative errors of non-physically relevant eigenvalues, which decay exponentially with respect to the spatial mesh size

\mathrm{\Delta }x

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

. We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree

P^2

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^2$$\end{document}

approximations, but are not for odd degree

P^3

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^3$$\end{document}

approximations. Numerical experiments are provided to verify the theoretical results.

Keywords

Direct discontinuous Galerkin (DDG) method with interface correction / Symmetric DDG method / Superconvergence / Fourier analysis / Eigen-structure

References

[1.] Adjerid S, Devine K, Flaherty J, Krivodonova L. A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng., 2002, 191(11–12): 1097-1112,

[2.] Adjerid S, Massey T. Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput. Methods Appl. Mech. Eng., 2006, 195(25/26/27/28): 3331-3346,

[3.] Ainsworth M. Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys., 2004, 198(1): 106-130,

[4.] Ainsworth M, Monk P, Muniz W. Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput., 2006, 27(1): 5-40,

[5.] Arnold DN. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 1982, 19(4): 742-760,

[6.] Cao W, Liu H, Zhang Z. Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations. Numer. Methods Partial Differ. Equ., 2017, 33(1): 290-317,

[7.] Cao W, Shu C-W, Yang Y, Zhang Z. Superconvergence of discontinuous Galerkin method for scalar nonlinear hyperbolic equations. SIAM J. Numer. Anal., 2018, 56(2): 732-765,

[8.] Cao W, Zhang Z. Superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. Math. Comput., 2014, 85: 63-84,

[9.] Chen Z, Huang H, Yan J. Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes. J. Comput. Phys., 2016, 308: 198-217,

[10.] Cheng Y, Shu C-W. Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys., 2008, 227(22): 9612-9627,

[11.] Cheng Y, Shu C-W. Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-diffusion equations. Comput. Struct., 2009, 87(11/12): 630-641,

[12.] Cheng Y, Shu C-W. Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension. SIAM J. Numer. Anal., 2010, 47(6): 4044-4072,

[13.] Chuenjarern N, Yang Y. Fourier analysis of local discontinuous Galerkin methods for linear parabolic equations on overlapping meshes. J. Sci. Comput., 2019, 81: 671-688,

[14.] Cockburn B, Luskin M, Shu C-W, Suli E. Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput., 2003, 72(242): 577-606,

[15.] Guo W, Zhong X, Qiu J-M. Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach. J. Comput. Phys., 2013, 235: 458-485,

[16.] Hu F, Hussaini M, Rasetarinera P. An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys., 1999, 151(2): 921-946,

[17.] Ji L, Xu Y, Ryan JK. Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions. Math. Comput., 2012, 81(280): 1929-1950,

[18.] Liu H, Yan J. The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal., 2009, 47(1): 475-698,

[19.] Liu H, Yan J. The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys., 2010, 8(3): 541-564,

[20.] Liu X, Zhang D, Meng X, Wu B. Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations. Sci. China Math., 2021, 64(6): 1305-1320,

[21.] Liu X, Zhang D, Meng X, Wu B. Superconvergence of the local discontinuous Galerkin method for one dimensional nonlinear convection-diffusion equations. J. Sci. Comput., 2021, 87(1): 39,

[22.] Meng X, Shu C-W, Zhang Q, Wu B. Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension. SIAM J. Numer. Anal., 2012, 50(5): 2336-2356,

[23.] Miao Y, Yan J, Zhong X. Superconvergence study of the direct discontinuous Galerkin method and its variations for diffusion equations. Commun. Appl. Math. Comput., 2022, 4(1): 180-204,

[24.] Sármány D, Botchev M, van der Vegt J. Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations. J. Sci. Comput., 2007, 33(1): 47-74,

[25.] Sherwin S. Dispersion analysis of the continuous and discontinuous Galerkin formulation. Lect. Notes Comput. Sci. Eng., 2000, 11: 425-432,

[26.] Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 1988, 77(2): 439-471,

[27.] Steffen M, Curtis S, Kirby RM, Ryan JK. Investigation of smoothness-increasing accuracy-conserving filters for improving streamline integration through discontinuous fields. IEEE Trans. Visual Comput. Graphics, 2008, 14(3): 680-692,

[28.] Vidden C, Yan J. A new direct discontinuous Galerkin method with symmetric structure for nonlinear diffusion equations. J. Comput. Math., 2013, 31(6): 638-662

[29.] Wheeler MF. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal., 1978, 15: 152-161,

[30.] Xu Y, Meng X, Shu C-W, Zhang Q. Superconvergence analysis of the Runge-Kutta discontinuous Galerkin methods for a linear hyperbolic equation. J. Sci. Comput., 2020, 84(1): 23,

[31.] Yang H, Li F, Qiu J. Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods. J. Sci. Comput., 2013, 55(3): 552-574,

[32.] Yang Y, Shu C-W. Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal., 2012, 50(6): 3110-3133,

[33.] Yang Y, Shu C-W. Analysis of sharp superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. J. Comput. Math., 2015, 33: 323-340,

[34.] Zhang M, Shu C-W. An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci., 2003, 13(3): 395-413,

[35.] Zhang M, Shu C-W. An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput. Fluids, 2005, 34(4): 581-592,

[36.] Zhang M, Shu C-W. Fourier analysis for discontinuous Galerkin and related methods. Sci. Bull., 2009, 54(11): 1809-1816,

[37.] Zhang M, Yan J. Fourier type error analysis of the direct discontinuous Galerkin method and its variations for diffusion equations. J. Sci. Comput., 2012, 52(3): 638-655,

[38.] Zhang M, Yan J. Fourier type super convergence study on DDGIC and symmetric DDG methods. J. Sci. Comput., 2017, 73: 1276-1289,

[39.] Zhong X, Shu C-W. Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Eng., 2011, 200(41/42/43/44): 2814-2827,

Funding

National Natural Science Foundation of China(11871428); Division of Mathematical Sciences(1620335); Natural Science Foundation for Colleges and Universities of Jiangsu Province(20KJB110011); Simons Foundation(637716)