Modified ILU as a smoother

Summary. We study a variant of ILU, viz. with , as a smoother in multi-grid methods. The -decomposition is characterized by the fact that the rest matrix is not zero on the diagonal but instead of that satisfies . The use of as a smoother is recommended because of the observed robustness when it is applied to singular perturbation problems. However, until now, a proof of robustness has only been given for one model anisotropic equation by Wittum. In this paper, we show that the -decomposition of an M-matrix exists and yields a regular splitting. For symmetric M-matrices and symmetric decomposition ``patterns' we prove that the eigenvalues of the -smoother are in \footnote{After this paper was submitted, the author learned of a report of Notay (\cite{239.5}) where related results are discussed}, whereas the rest matrix is at most a modest factor larger than the rest matrix of the unmodified ILU-decomposition. With these properties, robustness can now be shown when the rest matrix of the unmodified decomposition is ``small enough'. Our results generalize Wittum's results for the model problem. Received August 16, 1992 / Revised version received September 7, 1993