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Journal of Scientific Computing

, Volume 46, Issue 1, pp 71–99 | Cite as

An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations

  • Amiya K. Pani
  • Sangita Yadav
Article

Abstract

In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L 2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.

Keywords

LDG method Parabolic integro-differential equation Semidiscrete Mixed type Ritz-Volterra projection Negative norm estimates Role of stabilizing parameters Optimal error bounds 

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References

  1. 1.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Babuska, I., Suri, M.: The hp-version of the finite element method with quasi-uniform meshes. RAIRO Model. Math. Anal. Numer. 21, 199–238 (1987) MATHMathSciNetGoogle Scholar
  3. 3.
    Cannon, J.R., Lin, Y.: Nonclassical H 1 projection and Galerkin methods for nonlinear parabolic integro-differential equations. CALCOLO 25, 187–201 (1988) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71, 455–478 (2002) MATHGoogle Scholar
  6. 6.
    Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for time-dependent partial differential equations with higher order derivatives. Math. Comput. 77, 699–730 (2008) MATHMathSciNetGoogle Scholar
  7. 7.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin finite element method for convection-diffusion system. SIAM J. Numer. Anal. 35, 2440–2463 (1998) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ewing, R.E., Lin, Y., Sun, T., Wang, J., Zhang, S.: Sharp L 2-error estimates and super convergence of mixed finite element methods for non-Fickian flows in porous media. SIAM J. Numer. Anal. 40, 1538–1560 (2002) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ewing, R.E., Lin, Y., Wang, J.: A numerical approximation of non-Fickian flows with mixing length growth in porous media. Acta. Math. Univ. Comen. 70, 75–84 (2001) MATHMathSciNetGoogle Scholar
  10. 10.
    Gudi, T., Nataraj, N., Pani, A.K.: An hp-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type. Math. Comput. 77, 731–756 (2007) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Levy, D., Shu, C.-W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196, 751–772 (2004) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lin, Y., Thomée, V., Wahlbin, L.B.: Ritz-Volterra projections to finite element spaces and applications to integro-differential and related equations. SIAM J. Numer. Anal. 28, 1040–1070 (1991) CrossRefGoogle Scholar
  13. 13.
    Pani, A.K., Fairweather, G.: H 1-Galerkin mixed finite element methods for parabolic partial integro-differential equations. IMA J. Numer. Anal. 22, 231–252 (2002) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Pani, A.K., Peterson, T.E.: Finite element methods with numerical quadrature for parabolic integro- differential equations. SIAM J. Numer. Anal. 33, 1084–1105 (1996) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Pani, A.K., Sinha, R.K.: Error estimate for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data. CALCOLO 37, 181–205 (2000) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pani, A.K., Thomée, V., Wahlbin, L.B.: Numerical methods for hyperbolic and parabolic integro-differential equations. J. Integral Equ. Appl. 4, 533–584 (1992) MATHCrossRefGoogle Scholar
  17. 17.
    Perugia, I., Schötzau, D.: An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17, 561–571 (2002) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Riviere, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39, 902–931 (2001) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006) MATHGoogle Scholar
  20. 20.
    Thomée, V., Zhang, N.Y.: Error estimates for semidiscrete finite element methods for parabolic integro-differential equations. Math. Comput. 53, 121–139 (1989) MATHGoogle Scholar
  21. 21.
    Xu, Y., Shu, C.-W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196, 3805–3822 (2007) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Industrial Mathematics GroupIndian Institute of Technology BombayMumbaiIndia

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