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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold


IMPACT FACTOR 2018: 3.107

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1569-3953
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Volume 27, Issue 3

Issues

Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems

Sangita Yadav
  • Department of Mathematics, Birla Institute of Technology & Science, Pilani Campus, Rajasthan, 333031, India
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Amiya K. Pani
  • Corresponding author
  • Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India
  • Email
  • Other articles by this author:
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Published Online: 2019-09-17 | DOI: https://doi.org/10.1515/jnma-2018-0035

Abstract

In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence O(log(T/h2)hk+2) in L(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems.

Keywords: DGM; nonlinear parabolic problems; error analysis; post-processing; super-convergence estimates; numerical experiments

JEL Classification: 65M60; 65M15

References

  • [1]

    D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), 1749–1779.CrossrefGoogle Scholar

  • [2]

    J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), 94–111.CrossrefGoogle Scholar

  • [3]

    S. C. Brenner, Poincaré–Friedrichs inequalities for piecewise H1 functions, SIAM J. Numer. Anal. 41 (2003), 306–324.CrossrefGoogle Scholar

  • [4]

    P. Castillo, Performance of discontinuous Galerkin methods for elliptic PDE’s, SIAM J. Sci. Comput. 24 (2002), 524–547.CrossrefGoogle Scholar

  • [5]

    P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), 1676–1706.CrossrefGoogle Scholar

  • [6]

    B. Chabaud and B. Cockburn, Uniform-in-time superconvergence of HDG methods for the heat equation, Math. Comp. 81 (2012), 107–129.CrossrefGoogle Scholar

  • [7]

    H. Chen, R. Ewing, and R. Lazarov, Superconvergence of the mixed finite element approximations of parabolic problems using rectangular finite elements, East-West J. Numer. Math. 1 (1993), 199–212.Google Scholar

  • [8]

    H. Chen, R. Ewing, and R. Lazarov, Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data, Numer. Math. 78 (1998), 495–521.CrossrefGoogle Scholar

  • [9]

    B. Cockburn, B. Dong, J. Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp. 77 (2008), 1887–1916.CrossrefGoogle Scholar

  • [10]

    B. Cockburn, J. Gopalkrishnan, and H. Wang, Locally conservative fluxes for the continuous Galerkin method, SIAM J. Numer. Anal. 45 (2007), 1742–1776.CrossrefWeb of ScienceGoogle Scholar

  • [11]

    B. Cockburn, J. Guzmán, H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp. (2009), 1–24.Google Scholar

  • [12]

    B. Cockburn and C.-W. Shu, The local discontinuous Galerkin time-dependent method for convection–diffusion method, SIAM J. Numer. Anal. 35 (1998), 2440–2463.CrossrefGoogle Scholar

  • [13]

    J. Douglas, T. Dupont, and M. F. Wheeler, A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations, Math. Comp. 32 (1978), 345–362.CrossrefGoogle Scholar

  • [14]

    K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems IV: Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), 1729–1749.CrossrefGoogle Scholar

  • [15]

    C. Johnson and V. Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Modél. Math. Anal. Numér. 15 (1981), 41–78.CrossrefGoogle Scholar

  • [16]

    A. Lasis and E. Süli, hp-version discontinuous Galerkin finite element method for semilinear parabolic problems, SIAM J. Numer. Anal. 45 (2007), 1544–1569.Web of ScienceCrossrefGoogle Scholar

  • [17]

    A. K. Pani, An H1-mixed Galerkin method for parabolic partial differential equations, SIAM J. Numer. Anal. 35 (1998), 712–727.CrossrefGoogle Scholar

  • [18]

    A. K. Pani and S. Yadav, An hp-local discontinuous Galerkin methods for parabolic integro-differential equations, J. Sci. Comput. 46 (2011), 71–99.CrossrefGoogle Scholar

  • [19]

    B. Riviere and M. F. Wheeler, A discontinuous Galerkin method applied to nonlinear parabolic equations. In: Discontinuous Galerkin Methods (Newport, RI, 1999), Lect. Notes Comput. Sci. Eng., Vol. 11, Springer, Berlin, 2000, 231–244.Google Scholar

  • [20]

    B. Riviére, M. F. Wheeler, and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I, Comput. Geosci. 3 (1999), 337–360.CrossrefGoogle Scholar

  • [21]

    B. Riviere, M. F. Wheeler, and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39 (2001), 902–931.CrossrefGoogle Scholar

  • [22]

    M. C. J. Squeff, Superconvergence of mixed finite element methods for parabolic equations, RAIRO Modél. Math. Anal. Numér. 25 (1987), 327–352.Google Scholar

  • [23]

    V. Thomée, J.-C. Xu, and N.-Y. Zhang, Superconvergence of the gradient in piecewise liner finite element approximation to a parabolic problem, SIAM J. Numer. Anal. 26 (1989), 553–573.CrossrefGoogle Scholar

  • [24]

    M. Tripathy and R. K. Sinha, Superconvergence of H1-Galerkin mixed finite element methods for parabolic problems, Appl. Anal. 88 (2009), 1213–1231.Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-11-20

Revised: 2018-08-19

Accepted: 2018-09-29

Published Online: 2019-09-17

Published in Print: 2019-09-25


Funding: We are thankful to the referees for their valuable suggestions, which help to improve our manuscript. The first author acknowledges the financial support by DST FIST project having no. SR/FST/MSI-090/2013(C).


Citation Information: Journal of Numerical Mathematics, Volume 27, Issue 3, Pages 183–202, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2018-0035.

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