Advertisement

A Multilevel Method for Discontinuous Galerkin Approximation of Three-dimensional Elliptic Problems

  • Johannes K. Kraus
  • Satyendra K. Tomar
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 60)

We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundary-value problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. The presented numerical results demonstrate the potential of this approach.

Keywords

Elliptic Problem Discontinuous Galerkin Discontinuous Galerkin Method Multilevel Method Order Elliptic Problem 
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.
    D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2001/02.CrossRefMathSciNetGoogle Scholar
  2. 2.
    O. Axelsson. Iterative Solution Methods. Cambridge University Press, Cambridge, 1994.zbMATHGoogle Scholar
  3. 3.
    O. Axelsson and P.S. Vassilevski. Algebraic multilevel preconditioning methods. I. Numer. Math., 56(2–3):157–177, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    O. Axelsson and P.S. Vassilevski. Algebraic multilevel preconditioning methods. II. SIAM J. Numer. Anal., 27(6):1569–1590, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    O. Axelsson and P.S. Vassilevski. Variable-step multilevel preconditioning methods. I. Selfadjoint and positive definite elliptic problems. Numer. Linear Algebra Appl., 1(1):75–101, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S.C. Brenner and J. Zhao. Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math., 2(1):3–18, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V.A. Dobrev, R.D. Lazarov, P.S. Vassilevski, and L.T. Zikatanov. Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numer. Linear Algebra Appl., 13(9):753–770, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    V. Eijkhout and P.S. Vassilevski. The role of the strengthened Cauchy-Buniakowskiĭ-Schwarz inequality in multilevel methods. SIAM Rev., 33(3):405–419, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Gopalakrishnan and G. Kanschat. A multilevel discontinuous Galerkin method. Numer. Math., 95(3):527–550, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J.K. Kraus. An algebraic preconditioning method for M-matrices: linear versus non-linear multilevel iteration. Numer. Linear Algebra Appl., 9(8):599–618, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J.K. Kraus and S.K. Tomar. A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems. Numer. Linear Algebra Appl., in press.Google Scholar
  12. 12.
    J.K. Kraus and S.K. Tomar. Multilevel preconditioning of two-dimensional elliptic problems discretized by a class of discontinuous Galerkin methods. SIAM J. Sci. Comput., to appear.Google Scholar
  13. 13.
    L. Lazarov and S. Margenov. CBS constants for graph-Laplacians and application to multilevel methods for discontinuous Galerkin systems. J. Complexity, doi: 10.1016/j.jco.2006.10.003, 2006.Google Scholar
  14. 14.
    Y. Saad. Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition, 2003.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Johannes K. Kraus
    • 1
  • Satyendra K. Tomar
    • 1
  1. 1.Johann Radon Institute for Computational and Applied MathematicsLinzAustria

Personalised recommendations