A new iterative method for solving a class of two-by-two block complex linear systems

We present a stationary iteration method, namely Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS), for solving the system of linear equations obtained by using finite element discretization of a distributed optimal control problem together with time-periodic parabolic equations. An upper bound for the spectral radius of the iteration method is given which is always less than 1. So convergence of the ASSS iteration method is guaranteed. The induced ASSS preconditioner is applied to accelerate the convergence speed of the GMRES method for solving the system. Numerical results are presented to demonstrate the effectiveness of both the ASSS iteration method and the ASSS preconditioner.