On the generalized shift-splitting preconditioner for saddle point problems

In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1, 1)-block and symmetric positive semidefinite (2, 2)-block. The proposed preconditioner is extracted from a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some properties of the eigenvalues distribution of the corresponding preconditioned matrix are studied. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioners.