Skip to main content
SpringerLink
Log in

Superconvergent Discontinuous Galerkin Methods for Linear Non-selfadjoint and Indefinite Elliptic Problems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Based on Cockburn et al. (Math. Comp. 78:1–24, 2009), superconvergent discontinuous Galerkin methods are identified for linear non-selfadjoint and indefinite elliptic problems. With the help of an auxiliary problem which is the discrete version of a linear non-selfadjoint elliptic problem in divergence form, optimal error estimates of order k+1 in L 2-norm for the potential and the flux are derived, when piecewise polynomials of degree k≥1 are used to approximate both potential and flux variables. Using a suitable post-processing of the discrete potential, it is then shown that the resulting post-processed potential converges with order k+2 in L 2-norm. The article is concluded with a numerical experiment which confirms the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenner, S.C.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41, 306–324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    MATH  Google Scholar 

  4. Bustinza, R., Gatica, G.N.: A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAM J. Sci. Comput. 26, 152–177 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Cockburn, B., Gopalkrishnan, J., Wang, H.: Locally conservative fluxes for the continuous Galerkin method. SIAM J. Numer. Anal. 45, 1742–1776 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockburn, B., Gopalkrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78, 1–24 (2009)

    Article  MATH  Google Scholar 

  11. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    Book  MATH  Google Scholar 

  12. 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 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author thanks Department of Science and Technology, India for the financial support through the DST project No. 08DST012.

Author information

Authors and Affiliations

  1. Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    Sangita Yadav, Amiya K. Pani & Neela Nataraj

Authors
  1. Sangita Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amiya K. Pani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Neela Nataraj
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amiya K. Pani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yadav, S., Pani, A.K. & Nataraj, N. Superconvergent Discontinuous Galerkin Methods for Linear Non-selfadjoint and Indefinite Elliptic Problems. J Sci Comput 54, 45–76 (2013). https://doi.org/10.1007/s10915-012-9601-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9601-z

Keywords

Search

Navigation