Skip to main content
SpringerLink
Log in

Optimized Schwarz Methods

  • Conference paper
  • First Online:
Domain Decomposition Methods in Science and Engineering XVIII

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 70))

Summary

The strategy of domain decomposition methods is to decompose the computational domain into smaller subdomains. Each subdomain is assigned to one processor. The equations are solved on each subdomain. In order to enforce the matching of the local solutions, interface conditions have to be written on the boundary between subdomains. These conditions are imposed iteratively. The convergence rate is very sensitive to these interface conditions. The Schwarz method is based on the use of Dirichlet boundary conditions. It can be slow and requires overlapping decompositions. In order to improve the convergence and to be able to use non-overlapping decompositions, it has been proposed to use more general boundary conditions. It is even possible to optimize them with respect to the efficiency of the method. Theoretical and numerical results are given along with open problems.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bjørstad, P.E., Espedal, M.S., Keyes, D.E., eds.: Ninth International Conference on Domain Decomposition Methods, 1997. Proceedings of the 9th International Conference on Domain Decomposition Methods in Bergen, Norway.

    Google Scholar 

  2. Cai, X.-C., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 21:239–247, 1999.

    Article  MathSciNet  Google Scholar 

  3. Chan, T., Glowinski, R., Périaux, J., Widlund, O., eds.: Domain Decomposition Methods, Philadelphia, PA, 1989. SIAM. Proceedings of the Second International Symposium on Domain Decomposition Methods, Los Angeles, California, January 14–16, 1988.

    Google Scholar 

  4. Chan, T.F., Mathew, T.P.: Domain decomposition algorithms. In Acta Numerica, 1994,Cambridge University Press, pages61–143. 1994.

    Google Scholar 

  5. Collino, F., Ghanemi, S., Joly, P.: Domain decomposition methods for harmonic wave propagation: a general presentation. Comput. Meth. Appl. Mech. Eng., (2–4): 171–211, 2000.

    Google Scholar 

  6. Després, B.: Domain decomposition method and the Helmholtz problem. II. In Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993), pages 197–206, Philadelphia, PA, 1993. SIAM.

    Google Scholar 

  7. Efstathiou, E., Gander, M.J.: Why Restricted Additive Schwarz converges faster than Additive Schwarz. BIT, 43(5):945–959, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  8. Engquist, B., Majda, A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31(139):629–651, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  9. Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal., 44(2):699–731, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  10. Gander, M.J., Magoulès, F., Nataf, F.: Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput., 24(1):38–60, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  11. Gerardo-Giorda, L., Nataf, F.: Optimized Schwarz methods for unsymmetric layered problems with strongly discontinuous and anisotropic coefficients. J. Numer. Math., 13(4):265–294, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  12. Givoli, D.: Numerical methods for problems in infinite domains. Elsevier, 1992.

    Google Scholar 

  13. Hagstrom, T., Tewarson, R.P., Jazcilevich, A.: Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems. Appl. Math. Lett., 1(3), 1988.

    Google Scholar 

  14. Japhet, C., Nataf, F., Rogier, F.: The optimized order 2 method. application to convection-diffusion problems. Future Generation Computer Systems, 18(1):17–30, 2001.

    Article  MATH  Google Scholar 

  15. Japhet, C., Nataf, F., Roux, F.-X.: The Optimized Order 2 Method with a coarse grid preconditioner. application to convection-diffusion problems. In P. Bjørstad, M. Espedal, and D. Keyes, eds., Ninth International Conference on Domain Decompositon Methods in Science and Engineering, pages 382–389. Wiley, 1998.

    Google Scholar 

  16. Lai, C.-H., Bjørstad, P.E., Cross, M., Widlund, O., eds.: Eleventh International Conference on Domain Decomposition Methods, 1998. Proceedings of the 11th International Conference on Domain Decomposition Methods in Greenwich, England, July 20–24, 1998.

    Google Scholar 

  17. Lions, P.-L.: On the Schwarz alternating method. III: a variant for nonoverlapping subdomains. In T.F. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20–22, 1989, Philadelphia, PA, 1990. SIAM.

    Google Scholar 

  18. Magoulès, F., Roux, F.-X., Series, L.: Algebraic approach to absorbing boundary conditions for the Helmholtz equation. Int. J. Comput. Math., 84(2):231–240, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  19. Nataf, F.: Interface connections in domain decomposition methods. In Modern methods in scientific computing and applications (Montréal, QC, 2001), vol. 75 of NATO Sci. Ser. II Math. Phys. Chem., pages 323–364. Kluwer Academic, Dordrecht, 2002.

    Google Scholar 

  20. Nataf, F., Rogier, F., de Sturler, E.: Optimal interface conditions for domain decomposition methods. Technical Report 301, CMAP (Ecole Polytechnique), 1994.

    Google Scholar 

  21. Nier, F.: Remarques sur les algorithmes de décomposition de domaines. In Seminaire: Équations aux Dérivées Partielles, 1998–1999, Exp. No. IX, 26. École Polytech., 1999.

    Google Scholar 

  22. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science, 1999.

    Google Scholar 

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

    Google Scholar 

  24. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory, vol. 34 of Springer Series in Computational Mathematics. Springer, 2004.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire J. L. Lions, CNRS UMR7598, Université Pierre et Marie Curie, 75252, Paris, Cedex 05, France

    F. Nataf

Authors
  1. F. Nataf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Nataf .

Editor information

Editors and Affiliations

  1. School of Computer Science and Engineering, The Hebrew University of Jerusalem, Ross Building - Givat Ram Campus, 91904, Jerusalem, Israel

    Michel Bercovier

  2. Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211, Genève 24, Switzerland

    Martin J. Gander

  3. Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, 14195, Berlin-Dahlem, Germany

    Ralf Kornhuber

  4. Courant Institute, 251 Mercer Street, New York, NY, 10012, USA

    Olof Widlund

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nataf, F. (2009). Optimized Schwarz Methods. In: Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XVIII. Lecture Notes in Computational Science and Engineering, vol 70. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02677-5_25

Download citation

Publish with us

Policies and ethics

Search

Navigation