The generalized modified shift-splitting preconditioners for nonsymmetric saddle point problems

For a nonsymmetric saddle point problem, the modified shift-splitting (MSS) preconditioner has been proposed by Zhou et al. By replacing the parameter α in (2,2)-block in the MSS preconditioner by another parameter β, a generalized MSS (GMSS) preconditioner is established in this paper, which results in a fixed point iteration called the GMSS iteration method. We provide the convergent and semi-convergent analysis of the GMSS iteration method, which show that this method is convergence and semi-convergence if the related parameters satisfy suitable restrictions. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. Finally, numerical examples show that the GMSS method is more feasibility and robustness than the MSS, Uzawa-HSS and PU-STS methods as a solver, and the GMSS preconditioner outperforms the GSOR, Uzawa-HSS, MSS and LMSS preconditioners for the GMRES method for solving both the nonsingular and the singular saddle point problems with nonsymmetric positive definite and symmetric dominant (1,1) parts.