Preconditioning of Iterative Eigenvalue Problem Solvers in Adaptive FETI-DP

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


Adaptive FETI-DP and BDDC methods are robust methods that can be used for highly heterogeneous problems when standard approaches fail. In these approaches, local generalized eigenvalue problems are solved approximately, and the eigenvectors are used to enhance the coarse problem. Here, a few iterations of an approximate eigensolver are usually sufficient. Different preconditioning options for the iterative LOBPCG eigenvalue problem solver are considered. Numerical results are presented for linear elasticity problems with heterogeneous coefficients.


  1. 1.
    J.G. Calvo, O.B. Widlund, An adaptive choice of primal constraints for BDDC domain decomposition algorithms. Electron. Trans. Numer. Anal. 45, 524–544 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    L.B. da Veiga, L.F. Pavarino, S. Scacchi, O.B. Widlund, S. Zampini, Adaptive selection of primal constraints for isogeometric BDDC deluxe preconditioners. SIAM J. Sci. Comput. 39(1), A281–A302 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. Dolean, F. 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    C. Farhat, M. Lesoinne, K. Pierson, A scalable dual-primal domain decomposition method. Numer. Linear Algebra Appl. 7(7–8), 687–714 (2000); Preconditioning techniques for large sparse matrix problems in industrial applications (Minneapolis, MN, 1999)Google Scholar
  5. 5.
    J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8(4), 1461–1483 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    H.H. Kim, E.T. Chung, A BDDC algorithm with enriched coarse spaces for two-dimensional elliptic problems with oscillatory and high contrast coefficients. Multiscale Model. Simul. 13(2), 571–593 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Klawonn, O. Rheinbach, Robust FETI-DP methods for heterogeneous three dimensional elasticity problems. Comput. Methods Appl. Mech. Eng. 196(8), 1400–1414 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Klawonn, P. Radtke, O. Rheinbach, FETI-DP methods with an adaptive coarse space. SIAM J. Numer. Anal. 53(1), 297–320 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Klawonn, M. Kühn, O. Rheinbach, Adaptive coarse spaces for FETI-DP in three dimensions. SIAM J. Sci. Comput. 38(5), A2880–A2911 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Klawonn, M. Kühn, O. Rheinbach, Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems. Electron. Trans. Numer. Anal. 49, 1–27 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Klawonn, M. Kühn, O. Rheinbach, FETI-DP and BDDC methods with a transformation of basis for heterogeneous problems: connections to deflation. Technical report, Technische Universität Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Preprint 2017–01, 2017. Submitted
  12. 12.
    A.V. Knyazev, Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23(2), 517–541 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    D.-S. Oh, O.B. Widlund, S. Zampini, C.R. Dohrmann, BDDC algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields. Math. Comput. 87(310), 659–692 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    C. Pechstein, C.R. Dohrmann, A unified framework for adaptive BDDC. Electron. Trans. Numer. Anal. 46, 273–336 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    B. Sousedík, Adaptive-Multilevel BDDC. PhD thesis, University of Colorado Denver, 2010Google Scholar
  17. 17.
    N. Spillane, D.J. Rixen, Automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms. Int. J. Numer. Methods Eng. 95(11), 953–990 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnKölnGermany
  2. 2.Institut für Numerische Mathematik und Optimierung, Fakultät für Mathematik und InformatikTechnische Universität Bergakademie FreibergFreibergGermany

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