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)


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.


Multigrid Method Finite Element Equation Smoothing Step Block Iteration Abstract Convergence 
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  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

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