A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems

• Main Authors: Wei-Hua Luo, Ting-Zhu Huang
• Format: Article
• Language: English
• Published: Hindawi Limited 2013-01-01
• Series: Journal of Applied Mathematics
• Online Access: http://dx.doi.org/10.1155/2013/489295

By using Sherman-Morrison-Woodbury formula, we introduce a preconditioner based on parameterized splitting idea for generalized saddle point problems which may be singular and nonsymmetric. By analyzing the eigenvalues of the preconditioned matrix, we find that when α is big enough, it has an eigenvalue at 1 with multiplicity at least n, and the remaining eigenvalues are all located in a unit circle centered at 1. Particularly, when the preconditioner is used in general saddle point problems, it guarantees eigenvalue at 1 with the same multiplicity, and the remaining eigenvalues will tend to 1 as the parameter α→0. Consequently, this can lead to a good convergence when some GMRES iterative methods are used in Krylov subspace. Numerical results of Stokes problems and Oseen problems are presented to illustrate the behavior of the preconditioner.