This paper proposes a new parameterized enhanced shift-splitting {\it (PESS)} preconditioner to solve the three-by-three block saddle point problem ({\it SPP}). In addition, necessary and sufficient criteria are established for the convergence of the proposed {\it PESS} iterative process for any random initial guess. Furthermore, we meticulously investigate the spectral bounds of the {\it PESS} preconditioned matrix. Moreover, empirical investigation has been performed for the sensitivity analysis of the system, revealing the robustness of the proposed {\it PESS} preconditioner. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed {\it PESS} preconditioner compared to the existing block diagonal and shift-splitting preconditioners.

翻译：本文提出了一种新的参数化增强移位分裂（PESS）预条件子，用于求解三乘三块鞍点问题（SPP）。此外，建立了该PESS迭代过程对任意随机初始猜测收敛的充要判据。同时，我们细致地研究了PESS预条件矩阵的谱界。进一步地，通过系统的灵敏度分析数值实验，揭示了所提PESS预条件子的鲁棒性。数值实验表明，与现有的块对角和移位分裂预条件子相比，所提PESS预条件子具有更高的效率和鲁棒性。