Two improvements of the deteriorated PSS preconditioner for generalized saddle point problems

For generalized saddle point problems, we present two improved variants of the deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner to accelerate the convergence rate of the Krylov subspace iteration method. The new preconditioners are not only better approximations to the generalized saddle point matrix than the DPSS preconditioner but also easier to implement than the DPSS preconditioner. Theoretical analyses show that the corresponding splitting iteration methods are also convergent unconditionally. The quasi-optimal choices and practical estimations of the iteration parameters are discussed. Moreover, eigenproperties of the preconditioned matrices are described and upper bounds of the degree of the minimal polynomial of the preconditioned matrices are obtained. Finally, numerical experiments arising from the discretization of a model Navier-Stokes equation are presented to show the efficiency of the proposed preconditioners.