Hostname: page-component-cb9f654ff-w5vf4 Total loading time: 0 Render date: 2025-08-19T03:17:50.941Z Has data issue: false hasContentIssue false
  • English
  • Français

Superconvergence of bi-k Degree Time-Space Fully Discontinuous Finite Element for First-Order Hyperbolic Equations

Published online by Cambridge University Press:  28 May 2015

Hongling Hu  and
Chuanmiao Chen
Hongling Hu
Affiliation:
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Hunan Normal University, Changsha 410081, China
Chuanmiao Chen*
Affiliation:
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Hunan Normal University, Changsha 410081, China
*
*Corresponding author. Email: hhling625@163.com (H. L. Hu), cmchen@hunnu.edu.cn (C. M. Chen)
Get access

Abstract

In this paper, we present a superconvergence result for the bi-k degree time-space fully discontinuous finite element of first-order hyperbolic problems. Based on the element orthogonality analysis (EOA), we first obtain the optimal convergence order of discontinuous Galerkin finite element solution. Then we use orthogonality correction technique to prove a superconvergence result at right Radau points, which is higher one order than the optimal convergence rate. Finally, numerical results are presented to illustrate the theoretical analysis.

Keywords

65M6035L0465M12Superconvergencehyperbolic equationfinite elementdiscontinuous Galerkin methodRadau points

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 3 , June 2015 , pp. 323 - 337
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Reed, W. H. and Hill, T. R., Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479, Los Alamos, NM, 1973.Google Scholar
[2]Lesaint, P. and Raviart, P. A., On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, De Boor, C. (ed.), Academic Press, 1974, pp. 89–145.Google Scholar
[3]Johnson, C. and Pitkaranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comput., 46 (1986), pp. 1–26.CrossRefGoogle Scholar
[4]Richter, G. R., An optimal-order error estimate for the discontinuous Galerkin method, Math. Comput., 50 (1988), pp. 75–88.CrossRefGoogle Scholar
[5]Peterson, T., A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal., 28 (1991), pp. 133–140.Google Scholar
[6]Cockburn, B. and Shu, C. W., The Runge-Kutta local projection discontinuous Galerkin method for conservation laws, II: General framwork, Math. Comput., 52 (1989), pp. 411–435.Google Scholar
[7]Cockburn, B., Lin, S. Y. and Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, III: One-dimensional systems, J. Comput. Phys., 84 (1989), pp. 90–113.Google Scholar
[8]Cockburn, B., Hou, S. and Shu, C. W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, IV: The multidimensional case, Math. Comput., 54 (1990), pp. 545–581.Google Scholar
[9]Cockburn, B. and Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws, V: Multidimensional system, J. Comput. Phys., 35 (1998), pp. 199–224.CrossRefGoogle Scholar
[10]Cockburn, B. and Shu, C. W., The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Math. Model. Numer. Anal., 25 (1991), pp. 337–361.Google Scholar
[11]Cockburn, B., Dong, B. and Guzman, J., Optimal convergence of the original DG method for the transport-reaction equation on special meshes, SIAM J. Numer. Anal., 46 (2008), pp. 1250–1265.Google Scholar
[12]Cockburn, B., Luskin, M., Shu, C. W. and Suli, E., Enhanced accuracy by postprocessing for finite element methods for hyperbolic equations, Math. Comput., 72 (2002), pp. 577–606.Google Scholar
[13]Cheng, Y. and Shu, C. W., Superconvergence and time evolution of discontinuous Galerkin finite element solutions, J. Comput. Phys., 227 (2008), pp. 9612–9627.Google Scholar
[14]Cheng, Y. and 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., 47 (2010), pp. 4044–4072.Google Scholar
[15]Meng, X., Shu, C. W., Zhang, Q. and Wu, B., Superconvergence of discontinuous Galerkin method for scalar nonlinear conservation laws in one space dimension, SIAM J. Numer. Anal., 50 (2012), pp. 2336–2356.CrossRefGoogle Scholar
[16]Yang, Y. and Shu, C. W., Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM J. Numer. Anal., 50 (2012), pp. 3110–3133.CrossRefGoogle Scholar
[17]Yang, Y. and Shu, C. W., Discontinuous galerkin method for hyperbolic equation involving δ-singulartities: negative-order norm error estimates and applications, Numer. Math., 124 (2013), pp. 753–781.Google Scholar
[18]Shu, C. W., Discontinuous Galerkin method for time-dependent problems: survey and recent developments, The IMA Volumes in Mathematics and its Applications, 157 (2014), pp. 25–62.CrossRefGoogle Scholar
[19]Adjerid, S., Devine, K., Flaherty, J. and Krivodonova, L., A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Comput. Methods Appl. Mech. Eng., 191 (2002), pp. 1097–1112.CrossRefGoogle Scholar
[20]Adjerid, S. and Massey, T. C., A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems, Comput. Methods Appl. Mech. Eng., 191 (2002), pp. 5877–5897.Google Scholar
[21]Adjerid, S. and Massey, T. C., Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Comput. Methods Appl. Mech. Eng., 195 (2006), pp. 3331–3346.CrossRefGoogle Scholar
[22]Adjerid, S. and Baccouch, M., The discontinuous Galerkin method for two-dimensional hyperbolic problems, Part II: A posteriori error estimation, J. Sci. Comput., 38 (2009), pp. 15–49.Google Scholar
[23]Adjerid, S. and Weinhart, T., Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Comput. Methods Appl. Mech. Eng., 198 (2009), pp. 3113–3129.CrossRefGoogle Scholar
[24]Adjerid, S. and Weinhart, T., Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems, Math. Comput., 80 (2011), pp. 1335–1367.Google Scholar
[25]Adjerid, S. and Baccouch, M., Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem, Appl. Numer. Math., 60 (2010), pp. 903–914.Google Scholar
[26]Zhang, T., Li, J. and Zhang, S., Superconvergence of discontinuous Galerkin methods for hyperbolic systems, J. Comput. Appl. Math., 223 (2009), pp. 725–734.CrossRefGoogle Scholar