Skip to main content

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

Submitted Papers

Part of the Lecture Notes in Mathematics book series (LNM,volume 1457)

Abstract

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.

Keywords

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

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. 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. O. Axelsson, V.A. Barker, Finite element solution of boundary value problems. Theory and computation., Academic Press, Orlando, Fl., 1984.

    MATH  Google Scholar 

  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.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. O. Axelsson, P. Vassilevski, Algebraic multilevel preconditioning methods I, Numer. Math., 56(1989), 157–177.

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. D. Braess, The contraction number of a multigrid method for solving the Poisson equation, Numer. Math., 37(1981), 387–404.

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. 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. T. Meis, Schnelle Lösung von Randwertaufgaben, Z. Angew. Math. Mech, 62(1982), 263–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. M. Ries, U. Trottenberg, G. Winter, A note on MGR methods, Lin. Alg. Appl., 49(1983), 1–26.

    CrossRef  MathSciNet  MATH  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1990 Springer-Verlag

About this paper

Cite this paper

Axelsson, O., Eijkhout, V. (1990). Analysis of a recursive 5-point/9-point factorization method. In: Axelsson, O., Kolotilina, L.Y. (eds) Preconditioned Conjugate Gradient Methods. Lecture Notes in Mathematics, vol 1457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090907

Download citation

  • DOI: https://doi.org/10.1007/BFb0090907

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53515-7

  • Online ISBN: 978-3-540-46746-5

  • eBook Packages: Springer Book Archive