BDDC Deluxe for Isogeometric Analysis

  • L. Beirão da Veiga
  • L. F. Pavarino
  • S. Scacchi
  • O. B. Widlund
  • S. Zampini
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)

Abstract

The main goal of this paper is to design, analyze, and test a BDDC (Balancing Domain Decomposition by Constraints, see [12, 23]) preconditioner for Isogeometric Analysis (IGA), based on a novel type of interface averaging, which we will denote by deluxe scaling, with either full or reduced set of primal constraints. IGA is an innovative numerical methodology, introduced in [17] and first analyzed in [1], where the geometry description of the PDE domain is adopted from a Computer Aided Design (CAD) parametrization usually based on Non-Uniform Rational B-Splines (NURBS) and the same NURBS basis functions are also used as the PDEs discrete basis, following an isoparametric paradigm; see the monograph [10]. Recent works on IGA preconditioners have focused on overlapping Schwarz preconditioners [3, 5, 7, 9], multigrid methods [16], and non-overlapping preconditioners [4, 8, 20].

References

  1. 1.
    Y. Bazilevs, L. Beirão da Veiga, J.A. Cottrell, T.J.R. Hughes, G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16, 1–60 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    L. Beirão da Veiga, C. Chinosi, C. Lovadina, L.F. Pavarino, Robust BDDC preconditioners for Reissner-Mindlin plate bending problems and MITC elements. SIAM J. Numer. Anal. 47, 4214–4238 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    L. Beirão da Veiga, D. Cho, L.F. Pavarino, S. Scacchi, Overlapping Schwarz methods for isogeometric analysis. SIAM J. Numer. Anal. 50, 1394–1416 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    L. Beirão da Veiga, D. Cho, L.F. Pavarino, S. Scacchi, BDDC preconditioners for isogeometric analysis. Math. Models Methods Appl. Sci. 23, 1099–1142 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    L. Beirão da Veiga, D. Cho, L.F. Pavarino, S. Scacchi, Isogeometric Schwarz preconditioners for linear elasticity systems. Comput. Methods Appl. Mech. Eng. 253, 439–454 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    L. Beirão da Veiga, L.F. Pavarino, S. Scacchi, O.B. Widlund, S. Zampini, Isogeometric BDDC preconditioners with deluxe scaling. SIAM J. Sci. Comput. 36, A1118–A1139 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. Bercovier, I. Soloveichik, Overlapping non Matching Meshes Domain Decomposition Method in Isogeometric Analysis. arXiv:1502.03756 [math.NA]Google Scholar
  8. 8.
    A. Buffa, H. Harbrecht, A. Kunoth, G. Sangalli, BPX-preconditioning for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 265, 63–70 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    L. Charawi, Isogeometric overlapping additive Schwarz preconditioners for the Bidomain system, in DD22 Proceedings, 2014Google Scholar
  10. 10.
    J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis. Towards integration of CAD and FEA (Wiley, New York, 2009)Google Scholar
  11. 11.
    C. De Falco, A. Reali, R. Vazquez, GeoPDEs: a research tool for isogeometric analysis of PDEs. Adv. Eng. Softw. 42, 1020–1034 (2011)CrossRefMATHGoogle Scholar
  12. 12.
    C.R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25, 246–258 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    C.R. Dohrmann, C. Pechstein, Constraints and weight selection algorithms for BDDC, in Domain Decomposition Methods in Science and Engineering XXI, Rennes, France, 2012. vol 98 (Springer LNCSE, Berlin, 2014)Google Scholar
  14. 14.
    C.R. Dohrmann, O.B. Widlund, Some recent tools and a BDDC algorithm for 3D problems in H(curl). In Domain Decomposition Methods in Science and Engineering. XX, San Diego, CA, 2011, vol. 91 (Springer LNCSE, Berlin, 2013), pp. 15–26Google Scholar
  15. 15.
    C.R. Dohrmann, O.B. Widlund, A BDDC algorithm with deluxe scaling for three-dimensional H(curl) problems. Comm. Pure Appl. Math. Appeared electronically in April 2015.Google Scholar
  16. 16.
    K. Gahalaut, J. Kraus, S. Tomar, Multigrid methods for isogeometric discretization. Comput. Methods Appl. Mech. Eng. 253, 413–425 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    A. Klawonn, M. Lanser, P. Radtke, O. Rheinbach, On an adaptive coarse space and on nonlinear domain decomposition. in Domain Decomposition Methods in Science and Engineering. XXI, Rennes, France, 2012, vol. 98 (Springer LNCSE, Berlin, 2014)Google Scholar
  20. 20.
    S.K. Kleiss, C. Pechstein, B. Jüttler, S. Tomar, IETI - isogeometric tearing and interconnecting. Comput. Methods Appl. Mech. Eng. 247–248, 201–215 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.H. Lee, A balancing domain decomposition by constraints deluxe method for numerically thin Reissner-Mindlin plates approximated with Falk–Tu elements. TR2013-951, Courant Institute, NYU, 2013Google Scholar
  22. 22.
    J. Li, O.B. Widlund, FETI-DP, BDDC, and block Cholesky methods. Int. J. Numer. Methods Eng. 66, 250–271 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    J. Mandel, C.R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Linear Algebra Appl. 10, 639–659 (2003)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    J. Mandel, B. Sousedik, J. Sistek, Adaptive BDDC in three dimensions. Math. Comput. Simul. 82(10), 1812–1831 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    D.-S. Oh, O.B. Widlund, C.R. Dohrmann, A BDDC algorithm for Raviart-Thomas vector fields. TR2013-951, Courant Institute, NYU, 2013Google Scholar
  26. 26.
    C. Pechstein, C.R. Dohrmann, Modern domain decomposition methods - BDDC, deluxe scaling, and an algebraic approach. 2013. Seminar talk, Linz, December 2013. http://people.ricam.oeaw.ac.at/c.pechstein/pechstein-bddc2013.pdf
  27. 27.
    N. Spillane, V. Dolean, P. Hauret, P. Nataf, J. Rixen, Solving generalized eigenvalue problems on the interface to build a robust two-level FETI method. C. R. Math. Acad. Sci. Paris 351(5–6), 197–201 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    A. Toselli, O.B. Widlund, Domain Decomposition Methods: Algorithms and Theory (Springer, Berlin, 2004)MATHGoogle Scholar
  29. 29.
    O.B. Widlund, C.R. Dohrmann, BDDC deluxe Domain Decomposition, in DD22 Proceedings, 2015Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • L. Beirão da Veiga
    • 1
  • L. F. Pavarino
    • 1
  • S. Scacchi
    • 1
  • O. B. Widlund
    • 2
  • S. Zampini
    • 3
  1. 1.Dipartimento di MatematicaUniversità di MilanoMilanoItaly
  2. 2.Courant Institute of Mathematical SciencesNew YorkUSA
  3. 3.Extreme Computing Research CenterKing Abdullah University of Science and TechnologyThuwalSaudi Arabia

Personalised recommendations