on the preconditioning of the schur complement matrix of a class of two-by-two block matrices

We consider  a class of two-by-two block complex system of linear equations obtained from finite element discretization of the distributed optimal control with time-periodic parabolic equations. using the schur complement technique we transform the obtained system to two subsystems. we propose a preconditioner to the subsystem with the schur complement matrix. spectral properties of the preconditioned matrix are analyzed. some numerical results are presented to show the effectiveness of the preconditioner.