Regularized preconditioned GMRES and the regularized iteration method

Abstract

The present paper proposes a new preconditioner for solving saddle point problems. The preconditioner is obtained by replacing the (1,2) and (2,2) blocks in the original saddle point matrix A by another well chosen block. The proposed preconditioner can be used as a preconditioner corresponding to the stationary iterative method or to accelerate the convergence of the generalized minimal residual method (GMRES). The convergent and semi-convergent analyses of the regularized iteration method are presented. The eigenvalue distribution and the forms of the eigenvectors of the preconditioned matrix are analyzed. Finally, numerical results show the effectiveness of the proposed preconditioner as compared to other preconditioners.