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An Adaptive GDSW Coarse Space for Two-Level Overlapping Schwarz Methods in Two Dimensions

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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

We propose robust coarse spaces for two-level overlapping Schwarz preconditioners, which are extensions of the energy minimizing coarse space known as GDSW (Generalized Dryja, Smith, Widlund). The resulting two-level methods with adaptive coarse spaces are robust for second order elliptic problems in two dimensions, even in presence of a highly heterogeneous coefficient function, and reduce to the standard GDSW algorithm if no additional coarse basis functions are used.

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References

  1. J. Aarnes, T.Y. Hou, Multiscale domain decomposition methods for elliptic problems with high aspect ratios. Acta Math. Appl. Sin. Engl. Ser. 18(1), 63–76 (2002)

    Article  MathSciNet  Google Scholar 

  2. P. Bjørstad, J. Koster, P. Krzyzanowski, Domain decomposition solvers for large scale industrial finite element problems, in Applied Parallel Computing. New Paradigms for HPC in Industry and Academia. Lecture Notes in Computer Science, vol. 1947 (Springer, Berlin, 2001), pp. 373–383

    Google Scholar 

  3. P. Bjørstad, P. Krzyzanowski, A flexible 2-level Neumann-Neumann method for structural analysis problems, in Parallel Processing and Applied Mathematics. Lecture Notes in Computer Science, vol. 2328 (Springer, Berlin, 2002), pp. 387–394

    Google Scholar 

  4. M. Brezina, C. Heberton, J. Mandel, P. Vanek, An iterative method with convergence rate chosen a priori. Technical report, Denver, 1999

    Google Scholar 

  5. M. Buck, O. Iliev, H. Andrä, Multiscale finite element coarse spaces for the application to linear elasticity. Cent. Eur. J. Math. 11(4), 680–701 (2013). https://doi.org/10.2478/s11533-012-0166-8

    MathSciNet  MATH  Google Scholar 

  6. T. Chartier, R.D. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick, J. Ruge, P.S. Vassilevski, Spectral AMGe (rhoamge). SIAM J. Sci. Comput. 25(01), 1–26 (2003)

    Article  MathSciNet  Google Scholar 

  7. C.R. Dohrmann, A. Klawonn, O.B. Widlund, A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners, in Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computer Science Engineering, vol. 60 (Springer, Berlin, 2008), pp. 247–254

    Google Scholar 

  8. C.R. Dohrmann, A. Klawonn, O.B. Widlund, Domain decomposition for less regular subdomains: overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(4), 2153–2168 (2008)

    Article  MathSciNet  Google Scholar 

  9. V. Dolean, Fr. 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)

    Google Scholar 

  10. 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(05), 1175–1199 (2012)

    Article  MathSciNet  Google Scholar 

  11. J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multisc. Model. Simul. 8(4), 1461–1483 (2010)

    Article  MathSciNet  Google Scholar 

  12. M.J. Gander, A. Loneland, T. Rahman, Analysis of a new harmonically enriched multiscale coarse space for domain decomposition methods. Technical report, arxiv. Dec 2015

    Google Scholar 

  13. I.G. Graham, P.O. Lechner, R. Scheichl, Domain decomposition for multiscale PDEs. Numer. Math. 106(4), 589–626 (2007)

    Article  MathSciNet  Google Scholar 

  14. A. Heinlein, U. Hetmaniuk, A. Klawonn, O. Rheinbach, The approximate component mode synthesis special finite element method in two dimensions: parallel implementation and numerical results. J. Comput. Appl. Math. 289, 116–133 (2015)

    Article  MathSciNet  Google Scholar 

  15. A. Heinlein, A. Klawonn, J. Knepper, O. Rheinbach, Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2D. Technical report Preprint 2016-09 at http://tu-freiberg.de/fakult1/forschung/preprints (2016, submitted)

  16. A. Heinlein, A. Klawonn, J. Knepper, O. Rheinbach, Adaptive GDSW. Technical report (2018, in preparation)

    Google Scholar 

  17. U. Hetmaniuk, R.B. Lehoucq, A special finite element method based on component mode synthesis. ESAIM: M2AN 44(3), 401–420 (2010)

    Google Scholar 

  18. J. Mandel, B. Sousedík, Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods. Comput. Methods Appl. Mech. Eng. 196(8), 1389–1399 (2007)

    Article  MathSciNet  Google Scholar 

  19. J. Mandel, B. Sousedík, J. Šístek, Adaptive BDDC in three dimensions. Math. Comput. Simul. 82(10), 1812–1831 (2012)

    Article  MathSciNet  Google Scholar 

  20. A. Toselli, O.B. Widlund, Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Köln, Germany

    Alexander Heinlein, Axel Klawonn & Jascha Knepper

  2. Institut für Numerische Mathematik und Optimierung, Fakultät für Mathematik und Informatik, Technische Universität Bergakademie Freiberg, Freiberg, Germany

    Oliver Rheinbach

Authors
  1. Alexander Heinlein
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  2. Axel Klawonn
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  3. Jascha Knepper
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  4. Oliver Rheinbach
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Corresponding authors

Correspondence to Alexander Heinlein , Axel Klawonn , Jascha Knepper or Oliver Rheinbach .

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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Heinlein, A., Klawonn, A., Knepper, J., Rheinbach, O. (2018). An Adaptive GDSW Coarse Space for Two-Level Overlapping Schwarz Methods in Two Dimensions. 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_35

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