Advertisement

The Effect of a Minimum Angle Condition on the Preconditioning of the Pivot Block Arising from 2-Level-Splittings of Crouzeix-Raviart FE-Spaces

  • Josef Synka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)

Abstract

The construction of efficient two- and multilevel preconditioners for linear systems arising from the finite element discretization of self-adjoint second order elliptic problems is known to be governed by robust hierarchical splittings of finite element spaces. In this study we consider such splittings of spaces related to nonconforming discretizations using Crouzeix-Raviart linear elements: We discuss the standard method based on differences and aggregates, a more general splitting and the first reduce method which is equivalent to a locally optimal splitting. All three splittings are shown to fit a general framework of differences and aggregates. Further, we show that the bounds for the spectral condition numbers related to the additive and multiplicative preconditioners of the coarse grid complement block of the hierarchical stiffness matrix for the three splittings can be significantly improved subject to a minimum angle condition.

Keywords

Multilevel preconditioning hierarchical splittings CBS constant differences and aggregates first reduce anisotropy nonconforming elements 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson, O.: Iterative solution methods. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  2. 2.
    Axelsson, O., Margenov, S.: An optimal order multilevel preconditioner with respect to problem and discretization parameters. In: Minev, Wong, Lin (eds.) Advances in Computations, Theory and Practice, vol. 7, pp. 2–18. Nova Science, New York (2001)Google Scholar
  3. 3.
    Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods, I. Numerische Mathematik 56, 157–177 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods, II. SIAM Journal on Numerical Analalysis 27, 1569–1590 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blaheta, R., Margenov, S., Neytcheva, M.: Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems. Numerical Linear Algebra with Applications 11, 309–326 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blaheta, R., Margenov, S., Neytcheva, M.: Robust optimal multilevel preconditioners for non-conforming finite element systems. Numerical Linear Algebra with Applications 12, 495–514 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kraus, J., Margenov, S., Synka, J.: On the multilevel precondtioning of Crouzeix-Raviart elliptic problems. Numerical Linear Algebra with Applications (to appear)Google Scholar
  8. 8.
    Margenov, S., Synka, J.: Generalized aggregation-based multilevel preconditioning of Crouzeix-Raviart FEM elliptic problems. In: Boyanov, T., et al. (eds.) NMA 2006. LNCS, vol. 4310, pp. 91–99. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Josef Synka
    • 1
  1. 1.Industrial Mathematics InstituteJohannes Kepler UniversityLinzAustria

Personalised recommendations