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On Iterative Substructuring Methods for Multiscale Problems

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Domain Decomposition Methods in Science and Engineering XXI

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 98))

Abstract

In this note, we discuss iterative substructuring methods for a scalar elliptic model problem with a strongly varying diffusion coefficient that is typically discontinuous and exhibits large jumps. Opposed to earlier theory, we treat the case where the jumps happen on a small spatial scale and can in general not be resolved by a domain decomposition. We review the available theory of FETI methods for coefficients that are—on each subdomain (or a part of it)—quasi-monotone. Furthermore, we present novel theoretical robustness results of FETI methods for coefficients which have a large number of inclusions with large values, and a constant “background” value (by far not quasi-monotone). In both cases, the coarse space is the usual space of constants per subdomain.

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References

  1. Arbogast, T.: Mixed multiscale methods for heterogeneous elliptic problems. In: Numerical Analysis of Multiscale Problems. LNCSE, vol. 83, pp. 243–283. Springer, Berlin (2012)

    Google Scholar 

  2. Dohrmann, C.R., Pechstein, C.: Constraint and weight selection algorithms for BDDC (2012). Talk at the DD21 Conference, Rennes, June 25. http://www.numa.uni-linz.ac.at/~clemens/dohrmann-pechstein-dd21-talk.pdf

  3. Dohrmann, C.R., Widlund, O.B.: Some recent tools and a BDDC algorithm for 3D problems in H(curl). In: Bank, R., Kornhuber, R., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XX. LNCSE, vol. 91, pp. 15–25. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  4. Dostál, Z., Horák, D., Kučera, R.: Total FETI – an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng. 22(12), 1155–1162 (2006)

    Article  MATH  Google Scholar 

  5. Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46(5), 1175–1199 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Farhat, C., Roux, F.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32(6), 1205–1227 (1991)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul. 8(5), 1621–1644 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gippert, S., Klawonn, A., Rheinbach, O.: Analysis of FETI-DP and BDDC algorithms for a linear elasticity problems in 3D with compressible and almost incompressible material components. SIAM J. Numer. Anal. 50(5), 2208–2236 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Graham, I.G., Hagger, M.J.: Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM J. Sci. Comput. 20(6), 2041–2066 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Pechstein, C.: On iterative substructuring methods for multiscale problems. NuMa-Report 2012-13, Institute of Computational Mathematics, Johannes Kepler University (2012). http://www.numa.uni-linz.ac.at/Publications/

  13. Pechstein, C.: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems. LNCSE, vol. 90. Springer, Heidelberg (2013)

    Google Scholar 

  14. Pechstein, C., Scheichl, R.: Analysis of FETI methods for multiscale PDEs. Numer. Math. 111(2), 293–333 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pechstein, C., Scheichl, R.: Analysis of FETI methods for multiscale PDEs. Part II: interface variation. Numer. Math. 118(3), 485–529 (2011)

    MATH  MathSciNet  Google Scholar 

  16. Pechstein, C., Scheichl, R.: Weighted Poincaré inequalities. IMA J. Numer. Anal. 33(2), 652–686 (2012)

    Article  MathSciNet  Google Scholar 

  17. Pechstein, C., Sarkis, M., Scheichl, R.: New theoretical robustness results for FETI-DP. In: Bank, R., Kornhuber, R., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XX. LNCSE, vol. 91, pp. 313–320. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  18. Scheichl, R., Vassilevski, P.S., Zikatanov, L.T.: Mutilevel methods for elliptic problems with highly varying coefficients on non-aligned coarse grids. SIAM J. Numer. Anal. 50(3), 1675–1694 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Spillane, N., Rixen, D.: Automatic spectral coarse spaces for robust FETI and BDD algorithms. Int. J. Numer. Methods Eng. 95(11), 953–990 (2013). doi:10.1002/nme.4534

    Article  MathSciNet  Google Scholar 

  20. Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlap. Numer. Math. (2013). doi:10.1007/s00211-013-0576-y

    MATH  Google Scholar 

  21. Toselli, A., Widlund, O.B.: Domain Decomposition Methods – Algorithms and Theory. Springer, Berlin (2005)

    MATH  Google Scholar 

  22. Xu, J., Zhu, Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Model Methods Appl. Sci. 18(1), 77–105 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Robert Scheichl, Marcus Sarkis, and Clark Dohrmann for the inspiring collaboration and discussions on this topic.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040, Linz, Austria

    Clemens Pechstein

Authors
  1. Clemens Pechstein
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Correspondence to Clemens Pechstein .

Editor information

Editors and Affiliations

  1. INRIA, Rennes, France

    Jocelyne Erhel

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

    Martin J. Gander

  3. Laboratoire Analyse, Géometrie & Applications, Université Paris XIII, Villetaneuse, France

    Laurence Halpern

  4. INRIA, Rennes, France

    Géraldine Pichot

  5. Laboratoire LMNO, Université de Caen, Caen, France

    Taoufik Sassi

  6. New York University Courant Institute, New York, New York, USA

    Olof Widlund

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Pechstein, C. (2014). On Iterative Substructuring Methods for Multiscale Problems. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_7

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