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A Two-Level Schwarz Preconditioner for Heterogeneous Problems

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Domain Decomposition Methods in Science and Engineering XX

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 91))

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Abstract

Coarse space correction is essential to achieve algorithmic scalability in domain decomposition methods. Our goal here is to build a robust coarse space for Schwarz– type preconditioners for elliptic problems with highly heterogeneous coefficients when the discontinuities are not just across but also along subdomain interfaces, where classical results break down [3, 6, 9, 15].

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Bibliography

  1. T. Chartier, R. D. Falgout, V. E. Henson, J. Jones, T. Manteuffel, S. McCormick, J. Ruge, and P. S. Vassilevski. Spectral AMGe (ρAMGe). SIAM J. Sci. Comput., 25(1):1–26, 2003. ISSN 1064-8275. doi: 10.1137/S106482750139892X. URL http://dx.doi.org/10.1137/S106482750139892X.

    Google Scholar 

  2. V. Dolean, F. Nataf, Scheichl R., and N. Spillane. Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet–to–Neumann maps. http://hal.archives-ouvertes.fr/hal-00586246/fr/, 2011. URL http://hal.archives-ouvertes.fr/hal-00586246/fr/.

  3. M. Dryja, M. V. Sarkis, and O. B. Widlund. Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math., 72(3):313–348, 1996. ISSN 0029-599X. doi: 10.1007/s002110050172. URL http://dx.doi.org/10.1007/s002110050172.

  4. J. Galvis and Y. Efendiev. Domain decomposition preconditioners for multiscale flows in high contrast media: Reduced dimension coarse spaces. Multiscale Modeling & Simulation, 8(5):1621–1644, 2010. doi: 10.1137/100790112.

    Article  MathSciNet  MATH  Google Scholar 

  5. Frédéric Hecht. FreeFem++. Laboratoire J.L. Lions, Université Pierre et Marie Curie, http://www.freefem.org/ff++/, 3.7 edition, 2010.

  6. J. Mandel and M. Brezina. Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp., 65:1387–1401, 1996. ISSN 0025-5718. doi: 10.1090/S0025-5718-96-00757-0. URL http://dx.doi.org/10.1090/S0025-5718-96-00757-0.

  7. F. Nataf, H. Xiang, and V. Dolean. A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps. C. R. Mathématique, 348(21–22):1163–1167, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. A. Nicolaides. Deflation of conjugate gradients with applications to boundary value problems. SIAM J. Numer. Anal., 24(2): 355–365, 1987. ISSN 0036-1429. doi: 10.1137/0724027. URL http://dx.doi.org/10.1137/0724027.

  9. C. Pechstein and R. Scheichl. Scaling up through domain decomposition. Appl. Anal., 88(10–11):1589–1608, 2009. ISSN 0003-6811. doi: 10.1080/00036810903157204. URL http://dx.doi.org/10.1080/00036810903157204.

    Google Scholar 

  10. C. Pechstein and R. Scheichl. Weighted Poincaré inequalities. Technical Report NuMa-Report 2010-10, Institute of Computational Mathematics, Johannes Kepler University, Linz, December 2010. submitted.

    Google Scholar 

  11. C. Pechstein and R. Scheichl. Weighted Poincaré inequalities and applications in domain decomposition. In Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, editors, Domain Decomposition Methods in Science and Engineering XIX, volume 78 of LNCSE, pages 197–204. Springer, 2011.

    Google Scholar 

  12. C. Pechstein and R. Scheichl. Analysis of FETI methods for multiscale PDEs - Part II: Interface variation. Numer. Math., 2011. Published online 21 February 2011.

    Google Scholar 

  13. R. Scheichl, P. S. Vassilevski, and L. T. Zikatanov. Weak approximation properties of elliptic projections with functional constraints. Technical Report LLNL-JRNL-462079, Lawrence Livermore National Lab, 2011.

    Google Scholar 

  14. R. Scheichl, P.S. Vassilevski, and L.T. Zikatanov. Multilevel methods for elliptic problems with highly varying coefficients on non-aligned coarse grids. SIAM J Numer Anal, 2011. Accepted subject to minor corrections.

    Google Scholar 

  15. A. Toselli and O. B. Widlund. Domain decomposition methods – algorithms and theory. Springer, Berlin, 2005. ISBN 3-540-20696-5.

    MATH  Google Scholar 

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Correspondence to V. Dolean .

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Dolean, V., Nataf, F., Scheichl, R., Spillane, N. (2013). A Two-Level Schwarz Preconditioner for Heterogeneous Problems. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_8

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