IMPROVED RELAXED POSITIVE-DEFINITE AND SKEW-HERMITIAN SPLITTING PRECONDITIONERS FOR SADDLE POINT PROBLEMS

We establish a class of improved relaxed positive-definite and skew-Hermitian splitting (IRPSS) preconditioners for saddle point problems. These preconditioners are easier to be implemented than the relaxed positive-definite and skew-Hermitian splitting (RPSS) preconditioner at each step for solving the saddle point problem. We study spectral properties and the minimal polynomial of the IRPSS preconditioned saddle point matrix. A theoretical optimal IRPSS preconditioner is also obtained. Numerical results show that our proposed IRPSS preconditioners are superior to the existing ones in accelerating the convergence rate of the GMRES method for solving saddle point problems.