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Convergence of Asynchronous Optimized Schwarz Methods in the Plane

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Domain Decomposition Methods in Science and Engineering XXIV (DD 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 125))

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Abstract

A convergence proof of Asynchronous Optimized Schwarz Methods applied to a shifted Laplacian problem, with negative shift, in \(\mathbb {R}^2\) is presented. Sufficient conditions for convergence involving initial values of the approximation of the solution are discussed.

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Notes

  1. 1.

    In [5] it is indicated that given \(\hat {T}\) is contracting, then T n → 0, where T maps u(n) to u(n + 1), but this implication may not always hold. This is why we need to complete the proof in a different manner. We do so by showing explicitly that (8) holds.

References

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  5. F. Magoulès, D.B. Szyld, C. Venet, Asynchronous optimized Schwarz methods with and without overlap. Numer. Math. 137, 199–227 (2017)

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Acknowledgements

J. C. Garay was supported in part by the U.S. Department of Energy under grant DE-SC0016578. D. B. Szyld was supported in part by the U.S. National Science Foundation under grant DMS-1418882 and the U.S. Department of Energy under grant DE-SC0016578.

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Authors and Affiliations

  1. Temple University, Philadelphia, PA, USA

    José C. Garay & Daniel B. Szyld

  2. CentraleSupélec, Châtenay-Malabry, France

    Frédéric Magoulès

Authors
  1. José C. Garay
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  2. Frédéric Magoulès
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  3. Daniel B. Szyld
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Corresponding authors

Correspondence to José C. Garay , Frédéric Magoulès or Daniel B. Szyld .

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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Garay, J.C., Magoulès, F., Szyld, D.B. (2018). Convergence of Asynchronous Optimized Schwarz Methods in the Plane. 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_31

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