Extended shift-splitting preconditioners for saddle point problems

In this paper we consider to solve the linear systems of the saddle point problems by preconditioned Krylov subspace methods. The preconditioners are based on a special splitting of the saddle point matrix. The convergence theory of this class of the extended shift-splitting preconditioned iteration methods is established. The spectral properties of the preconditioned matrices are analyzed. Numerical implementations show that the resulting preconditioners lead to fast convergence when they are used to precondition Krylov subspace iteration methods such as GMRES.