Convergence of relaxation iterative methods for saddle point problem

In this paper, we first provide convergence results of three relaxation iterative methods for solving saddle point problem. Next, we propose how to find near optimal parameters for which preconditioned Krylov subspace method performs nearly best when the relaxation iterative methods are applied to the preconditioners of Krylov subspace method. Lastly, we provide efficient implementation for the relaxation iterative methods and efficient computation for the preconditioner solvers. Numerical experiments show that the MIAOR method and the BiCGSTAB with MAOR preconditioner using near optimal parameters perform more than twice faster than the GSOR method.