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

*(English)*Zbl 0826.65101The 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.

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 |

##### Keywords:

incomplete Cholesky decomposition; smoothers; multigrid algorithms; singular perturbations; coarse grid correction; stability; ILU- decomposition
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\textit{S. A. Sauter}, Numer. Linear Algebra Appl. 2, No. 1, 17--28 (1995; Zbl 0826.65101)

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