Analysis of a recursive 5-point/9-point factorization method

  • O. Axelsson
  • V. Eijkhout
Submitted Papers
Part of the Lecture Notes in Mathematics book series (LNM, volume 1457)


Nested recursive two-level factorization methods for nine-point difference matrices are analyzed. Somewhat similar in construction to multilevel methods for finite element matrices, these methods use recursive red-black orderings of the meshes, approximating the nine-point stencils by five-point ones in the red points and then forming the reduced system explicitly. As this Schur complement is again a nine-point matrix (on a skew grid this time), the process of approximating and factorizing can be applied anew.

Progressing until a sufficiently coarse grid has been reached, this gives a multilevel preconditioner for the original matrix. Solving the levels in V-cycle order will not give an optimal order method, but we show that using certain combinations of V-cycles and W-cycles will give methods of both optimal order of numbers of iterations and computational complexity.


Algebraic multilevel Chebyshev polynomial approximation nine-point differences optimal order preconditioners 


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  1. [1]
    O. Axelsson, On multigrid methods of the two-level type, in: Multigrid methods, Proceedings, Köln-Porz, 1981, W. Hackbusch and U. Trottenberg, eds., LNM 960, 1982, 352–367.Google Scholar
  2. [2]
    O. Axelsson, A multilevel solution method for none-point difference approximations, chapter 13 in Parallel Supercomputing: Methods, Algorithms and Applications, Graham F. Carey (ed.), John Wiley, 1989, 191–205.Google Scholar
  3. [3]
    O. Axelsson, V.A. Barker, Finite element solution of boundary value problems. Theory and computation., Academic Press, Orlando, Fl., 1984.zbMATHGoogle Scholar
  4. [4]
    O. Axelsson, I. Gustafsson, On the use of preconditioned conjugate gradient methods for red-black ordered five-point difference schemes, J. Comp. Physics, 35(1980), 284–299.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    O. Axelsson, P. Vassilevski, Algebraic multilevel preconditioning methods I, Numer. Math., 56(1989), 157–177.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    O. Axelsson, P. Vassilevski, Algebraic multilevel preconditioning methods II, report 1988-15, Inst. for Sci. Comput., the University of Wyoming, Laramie.Google Scholar
  7. [7]
    D. Braess, The contraction number of a multigrid method for solving the Poisson equation, Numer. Math., 37(1981), 387–404.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    R.E. Ewing, R.D. Lazarov, and P.S. Vassilevski, Local refinement techniques for elliptic problems on cell-centered grids, Report #1988-16, Institute for scientific computation, University of Wyoming, Laramie.Google Scholar
  9. [9]
    V. Eijkhout and P. Vassilevski, The role of the strengthened Cauchy-Buniakowsky-Schwarz inequality in multi-level methods, submitted to SIAM Review.Google Scholar
  10. [10]
    T. Meis, Schnelle Lösung von Randwertaufgaben, Z. Angew. Math. Mech, 62(1982), 263–270.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    W.F. Mitchell, Unified multilevel adaptive finite element methods for elliptic problems, report UIUCDCS-R-88-1436, Department of Computer Science, the University of Illinois at Urbana-Champaign, Urbana, Illinois, 1988.Google Scholar
  12. [12]
    M. Ries, U. Trottenberg, G. Winter, A note on MGR methods, Lin. Alg. Appl., 49(1983), 1–26.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    P.S. Vassilevski, Nearly optimal iterative methods for solving finite element elliptic equations based on the multilevel splitting of the matrix, Report #1989-01, Institute for scientific computation, University of Wyoming, Laramie.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • O. Axelsson
    • 1
  • V. Eijkhout
    • 1
  1. 1.Department of MathematicsUniversity of Nijmegenthe Netherlands

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