Two new variants of the HSS preconditioner for regularized saddle point problems

Two new preconditioners, which can be viewed as improved variants of the Hermitian and skew-Hermitian splitting (HSS) preconditioner, are presented for regularized saddle point problems. The unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed. Moreover, eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained. Compared with the HSS preconditioner, they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions. Numerical experiments from the Stokes problem are presented, which illustrates the advantages of the two preconditioners over the HSS preconditioner and some other preconditioners to accelerate the convergence rate of GMRES.