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On Overlapping Domain Decomposition Methods for High-Contrast Multiscale Problems

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Book cover 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

We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems.

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Notes

  1. 1.

    Due to the page limitation, only a few references are cited throughout.

  2. 2.

    The coarse mesh does not necessarily resolve all the variations of κ.

  3. 3.

    These works usually assume some alignment between the coefficient heterogeneities and the initial non-overlapping decomposition.

  4. 4.

    We can include constructions of boundary conditions using 1D solution of the problem along the edges. Other choices include basis functions constructed using oversampling regions, energy minimizing partition of unity (global), constructions using limited global information (global), etc.

  5. 5.

    Here we avoid restriction and extension operators for simplicity of notation.

  6. 6.

    We mention that the analysis does not use a stable decomposition so, in principle, a new family of robust methods can be obtained.

  7. 7.

    Having a robust condition number close to 1 is important, especially in applications where the elliptic equation needs to be solved many times.

References

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Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia

    Juan Galvis

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR

    Eric T. Chung

  3. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Yalchin Efendiev & Wing Tat Leung

Authors
  1. Juan Galvis
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  2. Eric T. Chung
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  3. Yalchin Efendiev
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  4. Wing Tat Leung
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Corresponding author

Correspondence to Juan Galvis .

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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Galvis, J., Chung, E.T., Efendiev, Y., Leung, W.T. (2018). On Overlapping Domain Decomposition Methods for High-Contrast Multiscale Problems. 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_4

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