Advertisement

On multigrid and iterative aggregation methods for nonsymmetric problems

  • Jan Mandel
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1228)

Abstract

We prove a convergence theorem for two-level iterations with one smoothing step. It is applied to multigrid methods for elliptic boundary value problems and to iterative aggregation methods for large-scale linear algebraic systems arising from input-output models in economics and from a multi-group approximation of the neutron-diffusion equation in reactor physics.

Keywords

Multigrid Method Finite Element Equation Smoothing Step Block Iteration Abstract Convergence 
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.
    R.E. Bank: A comparison of two multilevel iterative methods for nonsymmetric and indefinite finite element equations. SIAM J. Numer. Anal. 18,724–743,1981.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R.E. Bank and T. Dupont: An optimal order process for solving finite elements equations. Math. Comp. 36,35–51,1981.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    W. Hackbusch: On the convergence of multigrid iterations. Beitr. Numer. Math. 9,213–239,1981.zbMATHGoogle Scholar
  4. 4.
    A.Ju. Lučka: Projection-Iterative Methods of Solving Differential and Integral Equations /in Russian/. Naukova Dumka, Kiev, 1980.Google Scholar
  5. 5.
    J.F. Maitre and F. Musy: Multigrid methods: convergence theory in a variational framework. SIAM J. Numer. Anal., to appear.Google Scholar
  6. 6.
    J. Mandel: A convergent nonlinear splitting via orthogonal projection. Apl. Mat. 29,250–257,1984.MathSciNetzbMATHGoogle Scholar
  7. 7.
    J. Mandel: On some two-level iterative methods. In: K. Böhmer and H.J. Stetter /editors/, Defect Correction Methods, Computing Supplementum 5, Springer-Verlag, Wien, 1984.CrossRefGoogle Scholar
  8. 8.
    J. Mandel: Algebraic study of multigrid methods for symmetric, definite problems. Appl. Math. Comput., to appear.Google Scholar
  9. 9.
    J. Mandel: Multigrid convergence for nonsymmetric, indefinite problems and one smoothing step. In: Preliminary Proceedings of the 2nd Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado, April 1985 /mimeo/. Appl. Math. Comput., submitted.Google Scholar
  10. 10.
    J. Mandel, S.F. McCormick, and J. Ruge: An algebraic theory for multigrid methods for variational problems. SIAM J. Numer. Anal, submitted.Google Scholar
  11. 11.
    J. Mandel and B. Sekerka: A local convergence proof for the iterative aggregation method. Linear Algebra Appl. 51,163–172,1983.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    I. Marek, personal communication, 1985.Google Scholar
  13. 13.
    I. Marek: Some mathematical problems of the theory of nuclear reactors on fast neutrons. Apl. Mat. 8,442–470,1963.MathSciNetGoogle Scholar
  14. 14.
    S.F. McCormick: Multigrid methods for variational problems: further results. SIAM J. Numer. Anal. 21,255–263,1984.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    W.L. Miranker and V.Ya. Pan: Methods of aggregation. Linear Algebra Appl. 29,231–257,1980.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    R.S. Varga: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1962.Google Scholar
  17. 17.
    E.L. Wachspress: Iterative Solution of Elliptic Systems and Applications to the Neutron Diffusion Equation of Reactor Physics, Prentice-Hall, Englewood Cliffs, N.J., 1966.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Jan Mandel
    • 1
  1. 1.Computing Centre of the Charles UniversityPraha 1Czechoslovakia

Personalised recommendations