On the Suboptimality of the p-Version Interior Penalty Discontinuous Galerkin Method

  • Emmanuil H. Georgoulis
  • Edward Hall
  • Jens Markus Melenk


We address the question of the rates of convergence of the p-version interior penalty discontinuous Galerkin method (p-IPDG) for second order elliptic problems with non-homogeneous Dirichlet boundary conditions. It is known that the p-IPDG method admits slightly suboptimal a-priori bounds with respect to the polynomial degree (in the Hilbertian Sobolev space setting). An example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated in practice through numerical experiments is presented. Moreover, the performance of p-IPDG on the related problem of p-approximation of corner singularities is assessed both theoretically and numerically, witnessing an almost doubling of the convergence rate of the p-IPDG method.


Discontinuous Galerkin method Interior penalty A priori error estimation p-version Suboptimality 


  1. 1.
    Babuška, I.: The finite element method with penalty. Math. Comput. 27, 221–228 (1973) MATHCrossRefGoogle Scholar
  2. 2.
    Babuska, I., Guo, B.: Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces I: Approximability of functions in the weighted Besov spaces. SIAM J. Numer. Anal. 39(5), 1512–1538 (2001) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Babuška, I., Suri, M.: The optimal convergence rate of the p-version of the finite element method. SIAM J. Numer. Anal. 24(4), 750–776 (1987) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38(257), 67–86 (1982) MATHMathSciNetGoogle Scholar
  5. 5.
    Cockburn, B., Karniadakis, G.E., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Discontinuous Galerkin Methods, Newport, RI, 1999. Lecture Notes in Computer Science and Engineering, vol. 11, pp. 3–50. Springer, Berlin (2000) Google Scholar
  6. 6.
    Georgoulis, E.H.: hp-version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes. Int. J. Numer. Anal. Model. 3, 52–79 (2006) MATHMathSciNetGoogle Scholar
  7. 7.
    Georgoulis, E.H., Süli, E.: Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method. IMA J. Numer. Anal. 25(1), 205–220 (2005) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gui, W., Babuška, I.: The h, p and h-p versions of the finite element method in 1 dimension. I. The error analysis of the p-version. Numer. Math. 49(6), 577–612 (1986) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971). Collection of articles dedicated to Lothar Collatz on his sixtieth birthday MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rivière, 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(3), 902–931 (2001) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Schwab, C.: p- and hp-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Clarendon/Oxford University Press, New York (1998) Google Scholar
  13. 13.
    Stamm, B., Wihler, T.P.: hp-Optimal discontinuous Galerkin methods for linear elliptic problems. Technical report, EPFL/IACS report 07.2007 (2007) Google Scholar
  14. 14.
    Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3. Springer, Berlin (2007) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Emmanuil H. Georgoulis
    • 1
  • Edward Hall
    • 2
  • Jens Markus Melenk
    • 3
  1. 1.Department of MathematicsUniversity of LeicesterLeicesterUK
  2. 2.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  3. 3.Institut für Analysis und Scientific ComputingTechnische Universität WienViennaAustria

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