Three effective preconditioners for double saddle point problem

In this paper, we mainly propose three preconditioners for solving double saddle point problems, which arise from some practical problems. Firstly, the solvability of this kind of problem is investigated under suitable assumption. Next, we prove that all the eigenvalues of the three preconditioned matrices are 1. Furthermore, we analyze the eigenvector distribution and the upper bound of the minimum polynomial degree of the corresponding preconditioned matrix. Finally, numerical experiments are carried to show the effectiveness of the proposed preconditioners.