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Complete, Optimal and Optimized Coarse Spaces for Additive Schwarz

  • Martin J. GanderEmail author
  • Bo SongEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)

Abstract

The additive Schwarz method does not converge in general when used as a stationary iterative method, it must be used as a preconditioner for a Krylov method. In the two level variant of the additive Schwarz method, a coarse grid correction is added to make the method scalable. We introduce a new coarse space for the additive Schwarz method which makes it convergent when used as a stationary iterative method, and show that an optimal choice even makes the method nilpotent, i.e. it converges in one iteration, independently of the overlap and the number of subdomains. We then show how this optimal choice can be approximated leading to a spectral harmonically enriched coarse space, based on interface eigenvalue problems. We present a convergence analysis of our new coarse corrections in the two subdomain case in two spatial dimensions, and also compare them with GenEO recently proposed by Spillane, Dolean, Hauret, Nataf, Pechstein, and Scheichl, and the local spectral multiscale coarse space proposed by Galvis and Efendiev.

Notes

Acknowledgements

The work is supported by University of Geneva. The second author is partially supported by the Fundamental Research Funds for the Central Universities under grant G2018KY0306. We appreciate the comments of the reviewers that led to a better presentation.

References

  1. 1.
    F. Chaouqui, M.J. Gander, K. Santugini-Repiquet, On nilpotent subdomain iterations, in Domain Decomposition Methods in Science and Engineering XXIII (Springer, Berlin, 2016)zbMATHGoogle Scholar
  2. 2.
    V. Dolean, F. Nataf, R. Scheichl, N. Spillane, Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps. Comput. Methods Appl. Math. 12(4), 391–414 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Y. Efendiev, J. Galvis, R. Lazarov, J. Willems, Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46(5), 1175–1199 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8(4), 1461–1483 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul. 8(5), 1621–1644 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.J. Gander, Schwarz methods over the course of time. Electron. Trans. Numer. Anal 31(5), 228–255 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    M. Gander, L. Halpern, Méthode de décomposition de domaine. Encyclopédie électronique pour les ingénieurs (2012)Google Scholar
  8. 8.
    M.J. Gander, A. Loneland, SHEM: an optimal coarse space for RAS and its multiscale approximatio, in Domain Decomposition Methods in Science and Engineering XXIII (Springer, Berlin, 2016)Google Scholar
  9. 9.
    M.J. Gander, B. Song, Complete, optimal and optimized coarse spaces for Additive Schwarz. In Preparation (2018)Google Scholar
  10. 10.
    M.J. Gander, L. Halpern, K.S. Repiquet, Discontinuous coarse spaces for DD-methods with discontinuous iterates, in Domain Decomposition Methods in Science and Engineering XXI (Springer, Berlin, 2014), pp. 607–615zbMATHGoogle Scholar
  11. 11.
    M.J. Gander, L. Halpern, K.S. Repiquet, A new coarse grid correction for RAS/AS, in Domain Decomposition Methods in Science and Engineering XXI (Springer, Berlin, 2014), pp. 275–283zbMATHGoogle Scholar
  12. 12.
    M.J. Gander, A. Loneland, T. Rahman, Analysis of a new harmonically enriched multiscale coarse space for domain decomposition methods. arXiv:1512.05285 (2015, preprint)Google Scholar
  13. 13.
    F. Nataf, H. Xiang, V. Dolean, N. Spillane, A coarse space construction based on local Dirichlet-to-Neumann maps. SIAM J. Sci. Comput. 33(4), 1623–1642 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl, A robust two-level domain decomposition preconditioner for systems of PDEs. C.R. Math. 349(23), 1255–1259 (2011)Google Scholar
  15. 15.
    N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Toselli, O.B. Widlund, Domain Decomposition Methods–Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de Genève, Section de MathématiquesGenèveSwitzerland
  2. 2.School of ScienceNorthwestern Polytechnical UniversityXi’anChina

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