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

Abstract

The incomplete Cholesky decomposition is known as an excellent smoother in a multigrid iteration and as a preconditioner for the conjugate gradient method. However, the existence of the decomposition is only ensured if the system matrix is an M-matrix. It is well-known that finite element methods usually do not lead to M-matrices. In contrast to this restricting fact, numerical experiments show that, even in cases where the system matrix is not an M-matrix the behaviour of the incomplete Cholesky decomposition apparently does not depend on the structure of the grid. In this paper the behaviour of the method is investigated theoretically for a model problem, where the M-matrix condition is violated systematically by a suitable perturbation. It is shown that in this example the stability of the incomplete Cholesky decomposition is independent of the perturbation and that the analysis of the smoothing property can be carried through. This can be considered as a generalization of the results for the so called square-grid triangulation, as has been established by Wittum in [12] and [11].