## On the stability of the incomplete Cholesky decomposition for a singular perturbed problem, where the coefficient matrix is not an $$M$$-matrix.(English)Zbl 0826.65101

The incomplete Cholesky decomposition is known to provide good smoothers in multigrid algorithms. These smoothers are of special interest for problems with singular perturbations since they can often compensate the poorer approximation of the coarse grid correction. On the other hand singular perturbations often imply that the matrices are no longer $$M$$- matrices.
In this paper the stability of a modified ILU-decomposition is shown for a problem in which the singular perturbation arises from an anisotropy.
Reviewer: D.Braess (Bochum)

### MSC:

 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 35B25 Singular perturbations in context of PDEs 35J25 Boundary value problems for second-order elliptic equations
Full Text:

### References:

 [1] Gustafsson, BIT 18 pp 142– (1978) [2] Multi-Grid Method and Applications. Springer Verlag, New York, 1985. [3] Jennings, J. Inst. Math. Appl. 20 pp 307– (1977) [4] Meijerink, Math. of Comp. 31 pp 148– (1977) [5] Meijerink, Journal of Computational Physics 44 pp 131– (1981) [6] Sauter, Journ. of Comp. and Appl. Math. 36 pp 91– (1991) [7] On the stability of the ILU-decomposition for a singular perturbed problem, where the coefficient matrix is not an M-matrix. Technical Report BN-1164, IPST, University of Maryland at College Park, College Park, MD, 20742-2431, USA, 1994. [8] Sauter, Impact of Computing in Science and Engineering 4 pp 124– (1992) [9] Modified ILU as a smoother. Preprint no. 745, Department of Mathematics, University Utrecht, The Netherlands, 1992. To appear in Numer. Mathematik. [10] Stevenson, Numerische Mathematik 66 pp 373– (1993) [11] Wittum, Impact of Computing in Science and Engineering 1 pp 180– (1989) [12] Wittum, SIAM, J. Sci. Statist. Comp. 10 pp 699– (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.