An efficient preconditioned variant of the PSS preconditioner for generalized saddle point problems

For the generalized saddle point problems from the Stokes equations, we propose an efficient preconditioned variant of the positive semidefinite and skew-Hermitian (PVPSS) preconditioner. The new preconditioner is established by adopting matrix preconditioning strategy and relaxation technique for the PSS one derived by Pan et al. (2006). Compared with the PSS one, the PVPSS preconditioner is much closer to the coefficient matrix and easier to be implemented if proper preconditioning matrices are adopted. We prove the convergence of the PVPSS iteration method under some restrictions and discuss the spectral properties of the PVPSS preconditioned matrix. Meanwhile, the implementation and a practical way to choose the parameter of the PVPSS preconditioner are discussed. Comparisons between the PVPSS preconditioner and some existing ones are also given. Numerical experiments are carried out to illustrate that the proposed preconditioner is effective for the generalized saddle point problems and outperforms several other commonly used ones.