Additive Average Schwarz Method for a Crouzeix-Raviart Finite Volume Element Discretization of Elliptic Problems

  • Atle Loneland
  • Leszek Marcinkowski
  • Talal Rahman
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart Finite Volume Element (CRFVE) discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are inside subdomains and across subdomain boundaries. The proposed methods depends linearly or quadratically on the mesh parameters Hh, i.e., depending on the distribution of the coefficient in the model problem, the parameters describing the convergence of the GMRES method used to solve the preconditioned system depends linearly or quadratically on the mesh parameters. Also, under certain restrictions on the distribution of the coefficient, the convergence of the GMRES method is independent of jumps in the coefficient.



This work was partially supported by Polish Scientific Grant 2011/01/ B/ST1/01179 (Leszek Marcinkowski).


  1. 1.
    S.C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comp. 65(215), 897–921 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    S.C. Brenner, L.-Y. Sung, Balancing domain decomposition for nonconforming plate elements. Numer. Math. 83(1), 25–52 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    X.-C. Cai, O.B Widlund, Some domain decomposition algorithms for nonselfadjoint elliptic and parabolic partial differential equations, (ProQuest LLC, Ann Arbor, MI, 1989), p. 82. Google Scholar
  4. 4.
    P. Chatzipantelidis, A finite volume method based on the crouzeix–raviart element for elliptic pde’s in two dimensions. Numer. Math. 82(3), 409–432 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    S.H. Chou, J. Huang, A domain decomposition algorithm for general covolume methods for elliptic problems. J. Numer. Math. jnma 11(3), 179–194 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    S.C Eisenstat, H.C Elman, M.H Schultz, Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345–357 (1983)Google Scholar
  7. 7.
    J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul. 8(5), 1621–1644 (2010).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    L. Marcinkowski, The mortar element method with locally nonconforming elements. BIT Numer. Math. 39(4), 716–739 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    T. Rahman, X. Xu, R. Hoppe, Additive schwarz methods for the crouzeix-raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer. Math. 101(3), 551–572 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    H. Rui, C. Bi, Convergence analysis of an upwind finite volume element method with crouzeix-raviart element for non-selfadjoint and indefinite problems. Front. Math. China 3(4), 563–579 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Y. Saad, M.H Schultz, Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)Google Scholar
  12. 12.
    M. Sarkis, Nonstandard coarse spaces and schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math. 77(3), 383–406 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    B. Smith, P. Bjorstad, W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (Cambridge University Press, Cambridge, 1996)MATHGoogle Scholar
  14. 14.
    N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    A. Toselli, O.B Widlund, Domain Decomposition Methods: Algorithms and Theory, vol. 34 (Springer, New York, 2005)Google Scholar
  16. 16.
    S. Zhang, On domain decomposition algorithms for covolume methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 196(1–3), 24–32 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Atle Loneland
    • 1
  • Leszek Marcinkowski
    • 2
  • Talal Rahman
    • 3
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Faculty of MathematicsUniversity of WarsawWarszawaPoland
  3. 3.Faculty of EngineeringBergen University CollegeBergenNorway

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