Abstract In this paper, a new block preconditioner based on the pre-2 order block splitting of the coefficient matrix is established to solve the three-by-three block saddle point problems. In theory, the new iteration method under the preconditioner is proved to be unconditionally convergent; then, the eigenvalues of the preconditioned matrix are shown to have good clustering properties; furthermore, the algorithm implementation and parameter selection of the preconditioner are also discussed in details. Finally, numerical experiments show the effectiveness of the new block preconditioner by comparing with other existing preconditioners.
Li et al. (Sun,) studied this question.