A new generalized parameterized inexact Uzawa method for solving saddle point problems

Recently, Bai, Parlett and Wang presented a class of parameterized inexact Uzawa (PIU) methods for solving saddle point problems (Bai et al., 2005). In this paper, we develop a new generalized PIU method for solving both nonsingular and singular saddle point problems. The necessary and sufficient conditions of the convergence (semi-convergence) for solving nonsingular (singular) saddle point problems are derived. Meanwhile, the characteristic of eigenvalues of the iteration matrix corresponding to the above iteration method is discussed. We further show that the generalized PIU-type method proposed in this paper has a wider convergence (semi-convergence) region than some classical Uzawa methods, such as the inexact Uzawa method, the SOR-like method, the GSOR method and so on. Finally, numerical examples are given to illustrate the feasibility and efficiency of this method.