par Notay, Yvan ;Ould Amar, Zakaria
Référence Numerical linear algebra with applications, 4, page (369-391)
Publication Publié, 1997
Article révisé par les pairs
Résumé : Considering matrices obtained by the application of a five-point stencil on a 2D rectangular grid, we analyse a preconditioning method introduced by Axelsson and Eijkhout, and by Brand and Heinemann. In this method, one performs a (modified) incomplete factorization with respect to a so-called ‘repeated’ or ‘recursive’ red–black ordering of the unknowns while fill-in is accepted provided that the red unknowns in a same level remain uncoupled.Considering discrete second order elliptic PDEs with isotropic coefficients, we show that the condition number is bounded by [MATHEMATICAL SCRIPT CAPITAL O](nmath image) where n is the total number of unknowns (½ log2(√(5) − 1) = 0.153), and thus, that the total arithmetic work for the solution is bounded by [MATHEMATICAL SCRIPT CAPITAL O](n1.077). Our condition number estimate, which turns out to be better than standard [MATHEMATICAL SCRIPT CAPITAL O](log2n) estimates for any realistic problem size, is purely algebraic and holds in the presence of Neumann boundary conditions and/or discontinuities in the PDE coefficients.Numerical tests are reported, displaying the efficiency of the method and the relevance of our analysis. © 1997 John Wiley & Sons, Ltd.