Skip to main content
Log in

Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we propose two variants of the additive Schwarz method for the approximation of second order elliptic boundary value problems with discontinuous coefficients, on nonmatching grids using the lowest order Crouzeix-Raviart element for the discretization in each subdomain. The overall discretization is based on the mortar technique for coupling nonmatching grids. The convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements. The condition number bound for the preconditioned systems is independent of the jumps of the coefficient, and depend linearly on the ratio between the subdomain size and the mesh size. The performance of the methods is illustrated by some numerical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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

    Article  MathSciNet  Google Scholar 

  2. Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comp. 64, 943–972 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Math. Modelling and Numer. Anal. 19, 7–32 (1985)

    MATH  Google Scholar 

  4. Ben Belgacem, F.: The mortar element method with Lagrange multipliers. Université Paul Sabatier, Toulouse, France, 1994

  5. Ben Belgacem, F., Maday, Y.: The mortar element method for three dimensional finite elements. RAIRO Math. Modelling and Numer. Anal. 31, 289–302 (1997)

    MATH  Google Scholar 

  6. Bernardi, C., Maday, Y., Patera, A.T.: A new non conforming approach to domain decomposition: The mortar element method. In: Collège de France Seminar, H. Brezis, J.-L. Lions (eds.), Pitman, 1994. This paper appeared as a technical report about five years earlier

  7. Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with nonsmooth coefficients. Numerische Mathematik, 85, 579–608 (2000)

    MATH  ISI  MathSciNet  Google Scholar 

  8. Bjørstad, P., Dryja, M., Rahman, T.: Additive Schwarz for anisotropic elliptic problems. In: Parallel Solution of Partial Differential Equations, P. Bjørstad, M. Luskin (eds.), vol. 120 of IMA Volumes in Mathematics and its Applications, Springer-Verlag New York, 2000, pp. 279–294

  9. Bjørstad, P., Dryja, M., Rahman, T.: Additive Schwarz methods for elliptic mortar finite element problems. Numerische Mathematik 95, 427–457 (2003)

    MATH  MathSciNet  Google Scholar 

  10. Bjørstad, P., Dryja, M., Vainikko, E.: Additive Schwarz methods without subdomain overlap and with new coarse spaces. In: Domain Decomposition Methods in Sciences and Engineering, R. Glowinski, J. Périaux, Z. Shi, O. B. Widlund (eds.), John Wiley & Sons, 1997. Proceedings from the Eight International Conference on Domain Decomposition Methods, May 1995, Beijing

  11. Bjørstad, P.E., Dryja, M.: A coarse space formulation with good parallel properties for an additive Schwarz domain decomposition algorithm. Unpublished document 1999, University of Bergen

  12. Braess, D., Dahmen, W.: Stability estimates of the mortar finite element method for 3-dimensional problems. East-West J. Numer. Math. 6, 249–264 (1998)

    MATH  MathSciNet  Google Scholar 

  13. Braess, D., Dahmen, W.: The mortar element method revisited – what are the right norms?. In Thirteenth international conference on domain decomposition, 2001, pp. 245–252

  14. Braess, D., Dryja, M., Hackbusch, W.: A multigrid method for nonconforming fe-discretizations with application to non-matching grids. Computing 63, 1–25 (1999)

    Article  MATH  ISI  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Casarin, M.A., Widlund, O.B.: A hierarchical preconditioner for the mortar finite element method. ETNA 4, 75–88 (1996)

    MATH  MathSciNet  Google Scholar 

  17. Chen, Z.: Analysis of mixed methods using conforming and nonconforming finite element methods. RAIRO Math. Modelling and Nume. Anal. 27, 9–34 (1993)

    MATH  Google Scholar 

  18. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978

  19. Cowsar, L.C.: Domain decomposition methods for nonconforming finite elements. Tech. Report TR93-09, Department of Mathematical Sciences, Rice University, March 1993

  20. Deng, Q.: A nonoverlapping domain decomposition method for nonconforming finite element problems. Commun. Pure Appl. Anal. 2, 295–306 (2003)

    Google Scholar 

  21. Dryja, M.: Additive Schwarz methods for elliptic mortar finite element problems. In: Modelling and Optimization of Distributed Parameter Systems with Applications to Engineering, K. Malanowski, Z. Nahorski, M. Peszynska (eds.), IFIP, Chapman & Hall, London, 1996

  22. Dryja, M.: An iterative substructuring method for elliptic mortar finite element problems with discontinuous coefficients. In: The Tenth International Conference on Domain Decomposition Methods, J. Mandel, C. Farhat, X.-C. Cai (eds.) Contemporary Mathematics 218, AMS, 1998, pp. 94–103

  23. Dryja, M., Sarkis, M.V., Widlund, O.B.: Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numerische Mathematik 72, 313–348 (1996)

    Article  MATH  ISI  MathSciNet  Google Scholar 

  24. Dryja, M., Widlund, O.B.: Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Commum. Pure Appl. Math. 48, 121–155 (1995)

    MATH  MathSciNet  Google Scholar 

  25. Golub, G.H., Loan, C.F.V.: Matrix Computations. Johns Hopkins Univ. Press, 1989. Second Edition

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Hoppe, R.H.W., Wohlmuth, B.: Adaptive multilevel iterative techniques for nonconforming finite element discretizations. East-West J. Numer. Math. 3, 179–197 (1995)

    MATH  MathSciNet  Google Scholar 

  28. Lazarov, R.D., Margenov, S.D.: On a two-level parallel MIC(0) preconditioning of Crouzeix-Raviart non-conforming FEM systems. In: Numerical methods and applications. Proceedings. 5th Int. Conf. NMA2002, Borovets, Bulgaria, August 20-24, 2002, vol. 2542 of LNCS, Springer-Verlag, 2003, pp. 192–201

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

    Article  MATH  ISI  MathSciNet  Google Scholar 

  30. Sarkis, M.: Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients. In: Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, N.D. Melson, T.A. Manteuffel, S.F. McCormick (eds.), no. CP 3224, Hampton VA, 1993, NASA, pp. 543–566

  31. Sarkis, M.: Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming element. Numerische Mathematik 77, 383–406 (1997)

    Article  MATH  ISI  MathSciNet  Google Scholar 

  32. Seshaiyer, P., Suri, M.: Uniform hp convergence results for the mortar finite element method. Math. Comput. 69, 521–546 (1999)

    MathSciNet  Google Scholar 

  33. Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996

  34. Wieners, C., Wohlmuth, B.: The coupling of mixed and conforming finite element discretizations. In: Domain Decomposition Methods 10, J. Mandel, C. Farhat, X.C. Cai, (eds.), AMS, 1998, pp. 453–459. Proceedings from the Tenth International Conference, June 1997, Boulder, Colorado

  35. Wohlmuth, B.: Mortar finite element methods for discontinuous coefficients. ZAMM 79(S I), 151–154 (1999)

    Google Scholar 

  36. Wohlmuth, B.: A residual based error estimator for mortar finite element discretizations. Numerische Mathematik. 84, 143–171 (1999)

    Article  MATH  ISI  MathSciNet  Google Scholar 

  37. Wohlmuth, B.: A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38, 989–1012 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wohlmuth, B., Hoppe, R.H.W.: Multilevel approaches to nonconforming finite element discretizations of linear second order elliptic boundary value problem. J. Comput. Information 4, 73–86 (1994)

    Google Scholar 

  39. Xu, J., Zou, J.: Some nonoverlapping domain decomposition methods. SIAM Review 40, 857–914 (1998)

    Article  MATH  ISI  MathSciNet  Google Scholar 

  40. Xu, X., Chen, J.: Multigrid for the mortar element method for p1 nonconforming element. Numerische Mathematik 88, 381–398 (2001)

    MATH  ISI  MathSciNet  Google Scholar 

  41. Yang, D.: Finite elements for elliptic problems with wild coefficients. Mathematics and Computers in Simulation 54, 383–395 (2000)

    MATH  ISI  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Talal Rahman.

Additional information

This work has been supported by the Alexander von Humboldt Foundation and the special funds for major state basic research projects (973) under 2005CB321701 and the National Science Foundation (NSF) of China (No.10471144)

This work has been supported in part by the Bergen Center for Computational Science, University of Bergen

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rahman, T., Xu, X. & Hoppe, R. Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer. Math. 101, 551–572 (2005). https://doi.org/10.1007/s00211-005-0625-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0625-2

Mathematics Subject Classification (2000)

Navigation