Advertisement

Robust Multigrid Methods for Isogeometric Discretizations of the Stokes Equations

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

Abstract

In recent publications, the author and his coworkers have proposed a multigrid method for solving linear systems arising from the discretization of partial differential equations in isogeometric analysis and have proven that the convergence rates are robust in both the grid size and the polynomial degree. So far the method has only been discussed for the Poisson problem. In the present paper, we discuss the extension the of these results to the Stokes equations.

Notes

Acknowledgements

This work was partially supported by the Austrian Science Fund (FWF): grant S117.

References

  1. 1.
    L. Beirão da Veiga, D. Cho, L. Pavarino, S. Scacchi, Isogeometric Schwarz preconditioners for linear elasticity systems. Comput. Methods Appl. Mech. Eng. 253, 439–454 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Bressan, G. Sangalli, Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique. IMA J. Numer. Anal. 33(2), 629–651 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8(2), 129–151 (1974)MathSciNetzbMATHGoogle Scholar
  4. 4.
    F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991)CrossRefGoogle Scholar
  5. 5.
    G.S.A. Buffa, C. de Falco, Isogeometric analysis: new stable elements for the Stokes equation. Int. J. Numer. Methods Fluids 65, 1407–1422 (2010)CrossRefGoogle Scholar
  6. 6.
    A. Buffa, C. de Falco, G. Sangalli, IsoGeometric analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65(11–12), 1407–1422 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    C. de Boor, On calculating with B-splines. J. Approx. Theory 6(1), 50–62 (1972)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.A. Evans, T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math. Models Methods Appl. Sci. 23(08), 1421–1478 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Greenbaum, V. Pták, Z. Strakoš, Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17(3), 465–469 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    C. Hofreither, S. Takacs, Robust multigrid for isogeometric analysis based on stable splittings of spline spaces. SIAM J. Numer. Anal. 55(4), 2004–2024 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Hofreither, S. Takacs, W. Zulehner, A robust multigrid method for Isogeometric analysis in two dimensions using boundary correction. Comput. Methods Appl. Mech. Eng. 316, 22–42 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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(39–41), 4135–4195 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Larin, A. Reusken, A comparative study of efficient iterative solvers for generalized Stokes problem. Numer. Linear Algebra Appl. 15, 13–34 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Mantzaflaris, S. Takacs et al., G+Smo. http://gs.jku.at/gismo (2017)
  15. 15.
    L. Pavarino, S. Scacchi, Isogeometric block FETI-DP preconditioners for the Stokes and mixed linear elasticity systems. Comput. Methods Appl. Mech. Eng. 310, 694–710 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    S.P. Vanka, Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. Math. Comput. 65, 138–158 (1986)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

Personalised recommendations