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Abstract

Discrete systems arising from elliptic PDEs can be solved efficiently using multigrid methods. In many cases of practical interest the resulting linear equations exhibit strong anisotropies. It is well-known that standard multigrid methods fail to work for this type of problems. Various ILU methods have been proposed and investigated to overcome these difficulties. To be applied successfully, they usually require a modification of the ILU iteration. Only in the particular case of a 7-point decomposition for a 5-point discretization no modification is needed. We give a new proof for this situation, showing in which way the smoothing property is related to the size of the restmatrix. The method is shown to carry over to 9-point finite element discretizations. Numerical experiments document the excellent smoothing properties.