The modified PAHSS–PU and modified PPHSS-SOR iterative methods for saddle point problems

Recently, by combining the preconditioned accelerated Hermitian and skew-Hermitian splitting (PAHSS) and the parameterized Uzawa (PU) methods, Zheng and Ma (Appl Math Comput 273, 217–225, 2016b) presented the PAHSS–PU method for saddle point problems. By adding a block lower triangular matrix to the coefficient matrix on two sides of the first equation of the PAHSS–PU iterative scheme, the modified PAHSS–PU (MPAHSS–PU) iteration method is proposed in this paper, which has a faster convergence rate than the PAHSS–PU one. Furthermore, changing the position of the parameters in the MPAHSS–PU method, we develop another new method referred to as the modified PPHSS-SOR (MPPHSS-SOR) iteration method for solving saddle point problems. We provide the convergence properties of the MPAHSS–PU and the MPPHSS-SOR iteration methods, which show that the new methods are convergent if the related parameters satisfy suitable conditions. Meanwhile, practical ways to choose iteration parameters for the proposed methods are developed. Finally, numerical experiments demonstrate that the MPAHSS–PU and the MPPHSS-SOR methods outperform some existing ones both on the number of iterations and the computational times.