Fourier Analysis of Incomplete Factorization Preconditioners for Three-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...