Additive Schwarz Methods for DG Discretization of Elliptic Problems with Discontinuous Coefficient

  • Maksymilian Dryja
  • Piotr KrzyżanowskiEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


Second order elliptic problem with discontinuous coefficient in 2-D is considered. The problem is discretized by a symmetric interior penalty discontinuous Galerkin (DG) finite element method with triangular elements and piecewise linear functions. The resulting discrete problem is solved by a two-level additive Schwarz method. It turns out that the rate of convergence of the method is independent of the jumps of coefficient if its variation inside substructures is bounded. Numerical experiments are reported which confirm theoretical results.


Additive Schwarz method Discontinuous Galerkin Nonconforming 



We would like to thank an anonymous referee whose comments and remarks helped to improve the paper. This research has been supported by the Polish National Science Centre grant 2011/01/B/ST1/01179.


  1. 1.
    P.F. Antonietti, B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. Math. Model. Numer. Anal. 41(1), 21–54 (2007). doi:10.1051/m2an:2007006. ISSN:0764-583X.
  2. 2.
    B. Ayuso de Dios, M. Holst, Y. Zhu, L. Zikatanov, Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients. Math. Comput. 83, 1083–1120 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41(1), 306–324 (2003). doi:10.1137/S0036142902401311. ISSN:0036-1429.
  4. 4.
    K. Brix, C. Pinto, C. Canuto, W. Dahmen, Multilevel preconditioning of Discontinuous-Galerkin spectral element methods part I: geometrically conforming meshes (IGPM Preprint, RWTH Aachen, Aachen, 2013)zbMATHGoogle Scholar
  5. 5.
    C. Canuto, L.F. Pavarino, A.B. Pieri, BDDC preconditioners for continuous and discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree. IMA J. Numer. Anal. 34(3), 879–903 (2014). doi:10.1093/imanum/drt037. ISSN:0272-4979.
  6. 6.
    M. Dryja, On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients. Comput. Methods Appl. Math. 3(1), 76–85 (2003). ISSN:1609-4840Google Scholar
  7. 7.
    M. Dryja, M. Sarkis, Additive average Schwarz methods for discretization of elliptic problems with highly discontinuous coefficients. Comput. Methods Appl. Math. 10(2), 164–176 (2010). doi:10.2478/cmam-2010-0009. ISSN:1609-4840.
  8. 8.
    M. Dryja, P. Krzyżanowski, M. Sarkis, Additive Schwarz method for DG discretization of anisotropic elliptic problems, in Domain Decomposition Methods in Science and Engineering XXI, ed. by J. Erhel, M.J. Gander, L. Halpern, G. Pichot, T. Sassi, O. Widlund. Lecture Notes in Computational Science and Engineering, vol. 98 (Springer, New York, 2014), pp. 407–415Google Scholar
  9. 9.
    A. Ern, A.F. Stephansen, P. Zunino, A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Anal. 29(2), 235–256 (2009). doi:10.1093/imanum/drm050. ISSN:0272-4979.
  10. 10.
    X. Feng, O.A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39(4), 1343–1365 (2001). ISSN:1095-7170Google Scholar
  11. 11.
    A. Toselli, O. Widlund, Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005). ISBN:3-540-20696-5Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of WarsawWarsawPoland

Personalised recommendations