Advertisement

Non-matching Grids and Lagrange Multipliers

  • S. Bertoluzza
  • F. Brezzi
  • L.D. Marini
  • G. Sangalli
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

Summary

In this paper we introduce a variant of the three-field formulation where we use only two sets of variables. Considering, to fix the ideas, the homogeneous Dirichlet problem for −Δu = g in Ω, our variables are i) an approximation ψ h of u on the skeleton (the union of the interfaces of the sub-domains) on an independent grid (that could often be uniform), and ii) the approximations u h s of u in each subdomain Ω s (each on its own grid). The novelty is in the way to derive, from ψ h , the values of each trace of u h s on the boundary of each Ω s . We do it by solving an auxiliary problem on each ∂Ω s that resembles the mortar method but is more flexible. Optimal error estimates are proved under suitable assumptions.

Keywords

Domain Decomposition Domain Decomposition Method Piecewise Polynomial Accuracy Property Optimal Error Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. I. Babuska. The finite element method with lagrangian multipliers. Numer. Math., 20:179–192, 1973.MATHMathSciNetCrossRefGoogle Scholar
  2. C. Baiocchi, F. Brezzi, and L. D. Marini. Stabilization of galerkin methods and applications to domain decomposition. In A. Bensoussan et al., editor, Future Tendencies in Computer Science, Control and Applied Mathematics, volume 653 of Lecture Notes in Computer Science, pages 345–355. Springer-Verlag, 1992.Google Scholar
  3. F. B. Belgacem and Y. Maday. The mortar element method for three dimensional finite elements. RAIRO Mathematical Modelling and Numerical Analysis, 31(2):289–302, 1997.Google Scholar
  4. C. Bernardi, Y. Maday, and A. T. Patera. Domain decomposition by the mortar element method. In H. K. ans M. Garbey, editor, Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, pages 269–286. N.A.T.O. ASI, Kluwer Academic Publishers, 1993.Google Scholar
  5. C. Bernardi, Y. Maday, and F. Rapetti. Discrétisations variationnelles de problèmes aux limites elliptiques. Mathématiques et Applications. SMAI, to appear.Google Scholar
  6. S. Bertoluzza. Analysis of a stabilized three-fields domain decomposition method. Numer. Math., 93(4):611–634, 2003. ISSN 0029-599X.MATHMathSciNetCrossRefGoogle Scholar
  7. F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New-York, 1991a.Google Scholar
  8. F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 1991b. ISBN 0-387-97582-9.Google Scholar
  9. F. Brezzi, L. P. Franca, L. D. Marini, and A. Russo. Stabilization techniques for domain decomposition methods with nonmatching grids. In Proc. from the IX International Conference on Domain Decomposition Methods, June 1996, Bergen, Norway, 1997.Google Scholar
  10. F. Brezzi and L. D. Marini. A three field domain decomposition method. Contemp. Math., 157:27–34, 1994.MathSciNetGoogle Scholar
  11. F. Brezzi and L. D. Marini. Error estimates for the three-field formulation with bubble stabilization. Math. of Comp., 70:911–934, 2000.MathSciNetCrossRefGoogle Scholar
  12. A. Buffa. Error estimates for a stabilized domain decomposition method with nonmatching grids. Numer. Math., 90(4):617–640, 2002.MATHMathSciNetCrossRefGoogle Scholar
  13. P. Clément. Approximation by finite element functions using local regularization. RAIRO Anal. Numér., 9:77–84, 1975.MATHGoogle Scholar
  14. R. Hoppe, Y. Iliash, Y. Kuznetsov, Y. Vassilevski, and B. Wohlmuth. Analysis and parallel implementation of adaptive mortar element methods. East West J. Num. An., 6(3):223–248, 1998.MathSciNetGoogle Scholar
  15. B. Wohlmuth. Discretization Methods and Iterative Solvers Based on Domain Decomposition, volume 17 of Lecture Notes in Computational Science and Engineering. Springer, 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • S. Bertoluzza
    • 1
  • F. Brezzi
    • 1
    • 2
  • L.D. Marini
    • 1
    • 2
  • G. Sangalli
    • 1
  1. 1.Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R.Pavia
  2. 2.Dipartimento di MatematicaUniversità di PaviaPavia

Personalised recommendations