Skip to main content
SpringerLink
Log in

Additive Schwarz Method for Mortar Discretization of Elliptic Problems with P1 Nonconforming Finite Elements

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

An additive Schwarz preconditioner for nonconforming mortar finite element discretization of a second order elliptic problem in two dimensions with arbitrary large jumps of the discontinuous coefficients in subdomains is described.

An almost optimal estimate of the condition number of the preconditioned problem is proved. The number of preconditioned conjugate gradient iterations is independent of jumps of the coefficients and is proportional to (1+log(H/h)), where H,h are mesh sizes.

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. Y. Achdou, Y. A. Kuznetsov, and O. Pironneau, Substructuring preconditioners for theQ1mortar element method, Numer. Math., 71 (1995), pp. 419–449.

  2. Y. Achdou, Y. Maday, and O. B. Widlund, Iterative substructuring preconditioners for mortar element methods in two dimensions, SIAM J. Numer. Anal., 36 (1999), pp. 551–580.

  3. F. Ben Belgacem, The mortar finite element method with Lagrange multipliers, Numer. Math., 84 (1999), pp. 173–197. First published as a technical report in 1994.

  4. F. Ben Belgacem and Y. Maday, The mortar element method for three-dimensional finite elements, RAIRO Modél. Math. Anal. Numér., 31 (1997), pp. 289–302.

  5. C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XI (Paris, 1989–1991), Pitman Res. Notes Math. Ser., Vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 13–51.

  6. P. E. Bjørstad, M. Dryja, and T. Rahman, Additive Schwarz methods for elliptic mortar finite element problems, Numer. Math., 95 (2003), pp. 427–457.

  7. P. E. Bjørstad and O. B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), pp. 1097–1120.

  8. D. Braess, W. Dahmen, and C. Wieners, A multigrid algorithm for the mortar finite element method, SIAM J. Numer. Anal., 37 (1999), pp. 48–69.

    Google Scholar 

  9. S. C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods, Math. Comp., 65 (1996), pp. 897–921.

  10. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, Vol. 15, Springer-Verlag, New York, 1994.

  11. S. C. Brenner and L.-Y. Sung, Balancing domain decomposition for nonconforming plate elements, Numer. Math., 83 (1999), pp. 25–52.

    Google Scholar 

  12. X.-C. Cai, M. Dryja, and M. Sarkis, Overlapping nonmatching grid mortar element methods for elliptic problems, SIAM J. Numer. Anal., 36 (1999), pp. 581–606.

    Google Scholar 

  13. M. A. Casarin and O. B. Widlund, A hierarchical preconditioner for the mortar finite element method, Electron. Trans. Numer. Anal., 4 (1996), pp. 75–88.

    Google Scholar 

  14. M. Dryja, An iterative substructuring method for elliptic mortar finite element problems with a new coarse space, East-West J. Numer. Math., 5 (1997), pp. 79–98.

    Google Scholar 

  15. M. Dryja, A. Gantner, O. B. Widlund, and B. I. Wohlmuth, Multilevel additive Schwarz preconditioner for nonconforming mortar finite element methods, J. Numer. Math., 12 (2004), pp. 23–38.

    Google Scholar 

  16. M. Dryja, B. F. Smith, and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662–1694.

    Google Scholar 

  17. E. G. D’yakonov, Optimization in Solving Elliptic Problems, CRC Press, Boca Raton, FL, 1996. Translated from the 1989 Russian original, Translation edited and with a preface by Steve McCormick.

  18. J. Gopalakrishnan and J. E. Pasciak, Multigrid for the mortar finite element method, SIAM J. Numer. Anal., 37 (2000), pp. 1029–1052.

  19. R. Hoppe, T. Rahman, and X. Xu, Additive Schwarz Method for the mortar Crouzeix–Raviart element, in R. Kornhuber, R. Hoppe, J. Periaux, O. Pironneau, O. Widlund, and J. Xu (eds), Domain Decomposition Methods in Science and Engineering, Lect. Notes in Comput. Sci. and Eng., Vol. 40, Springer Verlag, 2004, pp. 335–342.

  20. C. Lacour and Y. Maday, La méthode des éléments avec joint appliquée aux méthodes d’approximations discrete Kirchhoff triangles, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), pp. 1237–1242.

  21. P. Le Tallec, Neumann-Neumann domain decomposition algorithms for solving 2D elliptic problems with nonmatching grids, East-West J. Numer. Math., 1 (1993), pp. 129–146.

    Google Scholar 

  22. L. Marcinkowski, The mortar element method with locally nonconforming elements, BIT, 39 (1999), pp. 716–739.

  23. L. Marcinkowski, Domain decomposition methods for mortar finite element discretizations of plate problems, SIAM J. Numer. Anal., 39 (2001), pp. 1097–1114.

  24. L. Marcinkowski, A mortar element method for some discretizations of a plate problem, Numer. Math., 93 (2002), pp. 361–386.

    Google Scholar 

  25. M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements, Numer. Math., 77 (1997), pp. 383–406.

    Google Scholar 

  26. Z.-C. Shi, X. Xu, and J. Chen, Multigrid for the mortar-type nonconforming element method for nonsymmetric and indefinite problems, in Domain decomposition methods in science and engineering (Lyon, 2000), Theory Eng. Appl. Comput. Methods, Internat. Center Numer. Methods Eng. (CIMNE), Barcelona, 2002, pp. 279–286.

  27. B. F. Smith, P. E. Bjørstad, and W. D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996.

  28. D. Stefanica and A. Klawonn, The FETI method for mortar finite elements, in Eleventh International Conference on Domain Decomposition Methods (London, 1998), DDM.org, Augsburg, 1999, pp. 121–129.

  29. B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lectures Notes in Computational Science and Engineering, Vol. 17, Springer-Verlag, Berlin, 2001.

  30. B. I. Wohlmuth and R. H. Krause, A multigrid method based on the unconstrained product space for mortar finite element discretizations, SIAM J. Numer. Anal., 39 (2001), pp. 192–213.

    Google Scholar 

  31. X. Xu and J. Chen, Multigrid for the mortar element method forP1 nonconforming element, Numer. Math., 88 (2001), pp. 381–398.

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Department of Mathematics, Informatics and Mechanics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Leszek Marcinkowski

Authors
  1. Leszek Marcinkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leszek Marcinkowski.

Additional information

AMS subject classification (2000)

65N55, 65N30, 65N22

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marcinkowski, L. Additive Schwarz Method for Mortar Discretization of Elliptic Problems with P1 Nonconforming Finite Elements. Bit Numer Math 45, 375–394 (2005). https://doi.org/10.1007/s10543-005-7123-x

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-005-7123-x

Key words

Search

Navigation