Advertisement

Schwarz Methods for a Crouzeix-Raviart Finite Volume Discretization of Elliptic Problems

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)

Abstract

In this paper, we present two variants of the Additive Schwarz Method (ASM) for a Crouzeix-Raviart finite volume (CRFV) discretization of the second order elliptic problem with discontinuous coefficients, where the discontinuities are only across subdomain boundaries. The resulting system, which is nonsymmetric, is solved using the preconditioned GMRES iteration, where in one variant of the ASM the preconditioner is symmetric while in the other variant it is nonsymmetric. The proposed methods are almost optimal, in the sense that the convergence of the GMRES iteration, in the both cases, depend only poly-logarithmically on the mesh parameters.

Keywords

Additive Schwarz method Crouzeix-Raviart finite element Domain decomposition Finite volume method 

Notes

Acknowledgements

This work was partially supported by Polish Scientific Grant 2011/01/B/ ST1/01179 and Chinese Academy of Science Project: 2013FFGA0009 - GJHS20140901004635677.

References

  1. 1.
    S.C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 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, Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Stat. Comput. 13(1), 243–258 (1992)MathSciNetCrossRefMATHGoogle 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. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. Loneland, L. Marcinkowski, T. Rahman, Additive average Schwarz method for the Crouzeix-Raviart finite volume element discretization of elliptic problems (2014a, submitted)Google Scholar
  8. 8.
    A. Loneland, L. Marcinkowski, T. Rahman, Edge based Schwarz methods for the Crouzeix-Raviart finite volume element discretization of elliptic problems (2014b, to appear in ETNA in 2015)MATHGoogle Scholar
  9. 9.
    L. Marcinkowski, T. Rahman, Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems. BIT Numer. Math. 48(3), 607–626 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    L. Marcinkowski, T. Rahman, J. Valdman, Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems. Tech. Report 204, Institute of Applied Mathematics and Mechanics, University of Warsaw (2014) [Published online in arXiv:1405.0185] [math.NA]Google Scholar
  11. 11.
    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
  12. 12.
    A. Toselli, O. Widlund, Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005)Google Scholar
  13. 13.
    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

  • Leszek Marcinkowski
    • 1
  • Atle Loneland
    • 2
  • Talal Rahman
    • 2
  1. 1.Faculty of MathematicsUniversity of WarsawWarszawaPoland
  2. 2.Department of Computing, Mathematics, and PhysicsBergen University CollegeBergenNorway

Personalised recommendations