Optimized Restricted Additive Schwarz Methods

  • Amik St-Cyr
  • Martin J. Gander
  • Stephen J. Thomas
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)


A small modification of the restricted additive Schwarz (RAS) preconditioner at the algebraic level, motivated by continuous optimized Schwarz methods, leads to a greatly improved convergence rate of the iterative solver. The modification is only at the level of the subdomain matrices, and hence easy to do in an existing RAS implementation.


Spectral Element Domain Decomposition Method Spectral Element Method Schwarz Method Restriction Matrice 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    X.-C. Cai and M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21 (1999), pp. 792–797.zbMATHCrossRefGoogle Scholar
  2. 2.
    M. Dryja and O. B. Widlund, An additive variant of the Schwarz alternating method in the case of many subregions, Tech. Rep. 339, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, 1987.Google Scholar
  3. 3.
    E. Efstathiou and M. J. Gander, RAS: Understanding restricted additive Schwarz, Tech. Rep. 6, McGill University, 2002.Google Scholar
  4. 4.
    M. J. Gander, Optimized Schwarz methods, SIAM J. Numer. Anal., 44 (2006), pp. 699–731.zbMATHCrossRefGoogle Scholar
  5. 5.
    P.-L. Lions, On the Schwarz alternating method. I., in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., Philadelphia, PA, 1988, SIAM, pp. 1–42.Google Scholar
  6. 6.
    E. M. Ronquist, Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incompressible Navier-Stokes Equations, PhD thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, 1988.Google Scholar
  7. 7.
    F.-X. Roux, F. Magoulès, S. Salmon, and L. Series, Optimization of interface operator based on algebraic approach, in Fourteenth International Conference on Domain Decomposition Methods, I. Herrera, D. E. Keyes, O. B. Widlund, and R. Yates, eds.,, 2003.Google Scholar
  8. 8.
    Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7 (1986), pp. 856–869.zbMATHCrossRefGoogle Scholar
  9. 9.
    A. St-Cyr, M. J. Gander, and S. J. Thomas, Optimized multiplicative, additive and restricted additive Schwarz preconditioning. In preparation, 2006.Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Amik St-Cyr
    • 1
  • Martin J. Gander
    • 2
  • Stephen J. Thomas
    • 3
  1. 1.National Center for Atmospheric ResearchBoulderUSA
  2. 2.University of GenevaSwitzerland
  3. 3.National Center for Atmospheric ResearchBoulderUSA

Personalised recommendations