A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems

In this paper, based on the positive definite and skew-Hermitian splitting iteration scheme and the Uzawa iteration method, we propose a class of Uzawa-PSS iteration methods for solving nonsingular and singular non-Hermitian saddle point problems with (1,1) block of the saddle point matrix being non-Hermitian positive definite. We derive conditions of the new iteration method for guaranteeing the convergence for nonsingular saddle point problems and its semi-convergence for singular saddle point problems, respectively. Numerical experiments of a model Navier–Stokes problem are presented to illustrate the effectiveness of the Uzawa-PSS iteration method. Numerical results show that the new method is much efficient than the Uzawa-HSS iteration method for solving both the nonsingular saddle point problems Yang and Wu, (2014) [28] and the singular saddle point problems Yang, Li and Wu, (2015) [30].