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International Conference on Domain Decomposition Methods

DD 2017: Domain Decomposition Methods in Science and Engineering XXIV pp 301–309Cite as

Complete, Optimal and Optimized Coarse Spaces for Additive Schwarz

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Part of the book series: Lecture Notes in Computational Science and Engineering ((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.

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Notes

  1. 1.

    Classical one level domain decomposition methods can even be nilpotent in certain situations, see [1].

  2. 2.

    This problem does not arise with overlap of one or two mesh sizes [12], or RAS [8].

References

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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.

Author information

Authors and Affiliations

  1. Université de Genève, Section de Mathématiques, Genève, Switzerland

    Martin J. Gander

  2. School of Science, Northwestern Polytechnical University, Xi’an, China

    Bo Song

Authors
  1. Martin J. Gander
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  2. Bo Song
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Corresponding authors

Correspondence to Martin J. Gander or Bo Song .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Petter E. Bjørstad

  2. Department of Mathematics, Louisana State University, Baton Rouge, LA, USA

    Susanne C. Brenner

  3. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Lawrence Halpern

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Korea (Republic of)

    Hyea Hyun Kim

  5. FB Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Ralf Kornhuber

  6. Western Norway University of Applied Sciences, Bergen, Norway

    Talal Rahman

  7. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Olof B. Widlund

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Gander, M.J., Song, B. (2018). Complete, Optimal and Optimized Coarse Spaces for Additive Schwarz. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_28

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