Journal of Applied Mathematics
Volume 2013 (2013), Article ID 489295, 6 pages
http://dx.doi.org/10.1155/2013/489295
Research Article

A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems

Wei-Hua Luo1,2 and Ting-Zhu Huang1

1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2Key Laboratory of Numerical Simulation of Sichuan Province University, Neijiang Normal University, Neijiang, Sichuan 641112, China

Received 4 January 2013; Revised 31 March 2013; Accepted 1 April 2013

Academic Editor: P. N. Shivakumar

Copyright © 2013 Wei-Hua Luo and Ting-Zhu Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 , 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 . 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.