A generalized variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems

Based on the variant of the deteriorated positive-definite and skew-Hermitian splitting (VDPSS) preconditioner developed by Zhang and Gu (BIT Numer. Math. 56:587–604, 2016), a generalized VDPSS (GVDPSS) preconditioner is established in this paper by replacing the parameter α in (2,2)-block of the VDPSS preconditioner by another parameter β. This preconditioner can also be viewed as a generalized form of the VDPSS preconditioner and the new relaxed HSS (NRHSS) preconditioner which has been exhibited by Salkuyeh and Masoudi (Numer. Algorithms, 2016). The convergence properties of the GVDPSS iteration method are derived. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. We also study the upper bounds on the degree of the minimum polynomial of the preconditioned matrix. Numerical experiments are implemented to illustrate the effectiveness of the GVDPSS preconditioner and verify that the GVDPSS preconditioned generalized minimal residual method is superior to the DPSS, relaxed DPSS, SIMPLE-like, NRHSS, and VDPSS preconditioned ones for solving saddle point problems in terms of the iterations and computational times.