FOURIER-ANALYSIS OF INCOMPLETE FACTORIZATION PRECONDITIONERS FOR 3-DIMENSIONAL ANISOTROPIC PROBLEMS

To solve three-dimensional elliptic problems using preconditioned conjugate gradient, it is crucial to make a good choice of preconditioner. To facilitate this choice, a Fourier analysis technique has been used by Chan and Elman [SIAM Rev., 31 (1989), pp. 20-49.] and others to study preconditioned systems arising from the discretization of the two-dimensional model elliptic equation. In this paper the same technique is used to analyze relaxed-modified incomplete factorization preconditioned systems that arise from the discretization of a three-dimensional anisotropic elliptic problem. Expressions for the "Fourier eigenvalues" of the preconditioned three-dimensional systems are presented along with estimates of the condition numbers. For MILU, an optimal value for the parameter c is derived. The correlation between the distribution of the eigenvalues and the Fourier results for the preconditioned systems is remarkable. From the expressions for the eigenvalues we prove that kappa(M-1 A) is order h-2 for ILU and order h-1 for MILU(c not-equal 0). Then by examining the distribution of Fourier eigenvalues, the dependence of PCG convergence rate on the clustering of the eigenvalues of an operator, as well as its condition number, can be exemplified. The PCG experiments were performed on an Alliant FX/80.