Advertisement

Multilevel Preconditioning of Rotated Trilinear Non-conforming Finite Element Problems

  • Ivan Georgiev
  • Johannes Kraus
  • Svetozar Margenov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)

Abstract

In this paper algebraic two-level and multilevel preconditioning algorithms for second order elliptic boundary value problems are constructed, where the discretization is done using Rannacher-Turek non-conforming rotated trilinear finite elements. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case.

The proposed variants of hierarchical two-level basis are first introduced in a rather general setting. Then, the involved parameters are studied and optimized. The major contribution of the paper is the derived estimates of the constant γ in the strengthened CBS inequality which is shown to allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver.

Keywords

Hierarchical Basis Multilevel Precondition Preconditioned Conjugate Gradient Iteration Order Elliptic Boundary Multilevel Iteration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson, O., Vassilevski, P.: Algebraic Multilevel Preconditioning Methods I. Numer. Math. 56, 157–177 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Axelsson, O., Vassilevski, P.: Algebraic Multilevel Preconditioning Methods II. SIAM J. Numer. Anal. 27, 1569–1590 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Axelsson, O., Vassilevski, P.: Variable-step multilevel preconditioning methods, I: self-adjoint and positive definite elliptic problems. Num. Lin. Alg. Appl. 1, 75–101 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blaheta, R.: Algebraic multilevel methods with aggregations: an overview. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2005. LNCS, vol. 3743, pp. 3–14. Springer, Heidelberg (2006)CrossRefGoogle 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(4), 309–326 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Georgiev, I., Margenov, S.: MIC(0) Preconditioning of Rotated Trilinear FEM Elliptic Systems. In: Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LSSC 2001. LNCS, vol. 2179, pp. 95–103. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Kraus, J.: An algebraic preconditioning method for M-matrices: linear versus nonlinear multilevel iteration. Num. Lin. Alg. Appl. 9, 599–618 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Margenov, S., Vassilevski, P.: Two-level preconditioning of non-conforming FEM systems. In: Griebel, M., et al. (eds.) Large-Scale Scientific Computations of Engineering and Environmental Problems, Vieweg. Notes on Numerical Fluid Mechanics, vol. 62, pp. 239–248 (1998)Google Scholar
  9. 9.
    Notay, Y.: Robust parameter-free algebraic multilevel preconditioning. Num. Lin. Alg. Appl. 9, 409–428 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rannacher, R., Turek, S.: Simple non-conforming quadrilateral Stokes Element. Numerical Methods for Partial Differential Equations 8(2), 97–112 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ivan Georgiev
    • 1
    • 3
  • Johannes Kraus
    • 2
  • Svetozar Margenov
    • 3
  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Johann Radon Institute for Computational and Applied MathematicsLinzAustria
  3. 3.Institute for Parallel ProcessingBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations