Advertisement

Computing

, Volume 65, Issue 2, pp 135–154 | Cite as

Condition Numbers of Approximate Schur Complements in Two- and Three-Dimensional Discretizations on Hierarchically Ordered Grids

  • J. K. Kraus
  • C. W. Brand

Abstract

We investigate multilevel incomplete factorizations of M-matrices arising from finite difference discretizations. The nonzero patterns are based on special orderings of the grid points. Hence, the Schur complements that result from block elimination of unknowns refer to a sequence of hierarchical grids. Having reached the coarsest grid, Gaussian elimination yields a complete decomposition of the last Schur complement.

The main focus of this paper is a generalization of the recursive five-point/nine-point factorization method (which can be applied in two-dimensional problems) to matrices that stem from discretizations on three-dimensional cartesian grids. Moreover, we present a local analysis that considers fundamental grid cells. Our analysis allows to derive sharp bounds for the condition number associated with one factorization level (two-grid estimates). A comparison in case of the Laplace operator with Dirichlet boundary conditions shows: Estimating the relative condition number of the multilevel preconditioner by multiplying corresponding two-grid values gives the asymptotic bound O(h−0.347) for the two- respectively O(h−4/5) for the three-dimensional model problem.

AMS Subject Classifications: 65F10, 65N20. 
Key Words: Incomplete factorization, hierarchical orderings, approximate Schur complement preconditioners. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • J. K. Kraus
    • 1
  • C. W. Brand
    • 2
  1. 1.Department of Applied Mathematics University of Leoben Franz-Josef-Str. 18 A-8700 Leoben Austria e-mail: kraus@unileoben.ac.atAT
  2. 2.Department of Applied Mathematics University of Leoben Franz-Josef-Str. 18 A-8700 Leoben Austria e-mail: brand@unileoben.ac.atAT

Personalised recommendations