By R. Beauwens (auth.), Owe Axelsson, Lily Yu. Kolotilina (eds.)

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Additional resources for Preconditioned Conjugate Gradient Methods: Proceedings of a Conference held in Nijmegen, The Netherlands, June 19–21, 1989

Example text

In this case, we have a variety of possibilities of approximating C by C, or equivalently approximating M -I. For example, we may choose C = N T M - 1 N , where . ~ is the lumped mass matrix M or M is the diagonal part of M. These two choices are particular cases of the general approach of replacing M by some variant of its incomplete factorization [1]. In all cases considered, matrix C is spectrally equivalent to A (in the subspace M U + N P = 0) with constants independent of the problem size.

AXELSSON and I. 06 R, Department of Computer Sciences, Chalmers University of Technology, G6teborg, Sweden, 1984. [6] O. AXELSSON and P. S. VASSILEVSKI: Algebraic Multilevel Preconditioning Methods 1, Report 8811, Department of Mathematics, Catholic University, Nijmegen, Holland, 1988. [7] O. AXELSSON and G. LINDSKOG, A Recursive Two-Level Method for Boundary Value Problems Discretized by Quadratic Finite Elements, Report 8, Numerical Analysis Group, Department of Computer Sciences, Chalmers University of Technology, G6teborg, Sweden, 1988.

For solving the system A h we can use a preconditioner for D h at the cost of a factor (1 + ~,)/(1 - T) in the condition number. Hence, it is preferable to keep y away from 1. Two choices of preconditioning matrices Ch for the system Ah are presented and studied in [4]. In the diagonal-block preconditioning Ch = LL t , where LL t is a MIC-factorization of AhO) and D =diag(Bh(P)). The reason why BhO') can be treated in this simple way is that its condition number is just of order O(1), h ---)0. In the fullblock preconditioning Ch = hL~t 01i LI Ltl 1' where LIL~ is a MIC-factorization of Ah0) and L2Lt2 is an IC-factorization of Bhq').

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