Improved PPHSS iterative methods for solving nonsingular and singular saddle point problems

Based on the parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method (PPHSS), proposed by Li et al. (2014) for saddle point problems, an improvement on the PPHSS method (IPPHSS) is presented in this paper. By adding a block lower triangular matrix to the coefficient matrix on two sides of the first equation of the PPHSS iterative scheme, both the number of iterations and the consume time are decreased. We provide the convergence and semi-convergence analysis of the IPPHSS method, which show that this method is convergence and semi-convergence if the related parameters satisfy suitable restrictions. Furthermore, we discuss the spectral properties of the corresponding preconditioned matrix of the IPPHSS method. Finally, numerical examples show that the IPPHSS method is better than the PPHSS, PHSS and AHSS methods both as a solver and as a preconditioner for the GMRES method.