Fourier Analysis of Relaxed Incomplete Factorization Preconditioners

Fourier analysis is used to study the behavior of a class of incomplete factorization preconditioners for elliptic problems, which blends the classical ILU and MILD preconditioners via a scalar relaxation parameter $\alpha \in [0,1]$. An expression is obtained for the eigenvalues of the preconditioned system for a model Poisson problem with periodic boundary conditions, which yields information on both the condition number $K(\alpha )$ and the eigendistribution of the preconditioned system. An optimal value is derived for $\alpha $ and it is shown that $K(\alpha _{\text{opt}} ) = O(h^{ - 1} )$. The Fourier results agree extremely well with numerical results for the model Poisson problem with Dirichlet boundary conditions, even though the Fourier analysis is not exact for this problem. For example, they predict the sensitive behavior near $\alpha = 1$ (MILD). Finally, it is shown that the relaxed methods are closely related to the classical “modified” ILU (MILD) method.