This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). Additionally, we introduce a local PESS (LPESS) preconditioner by relaxing the PESS preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed PESS iterative process for any random initial guess. Furthermore, we meticulously investigate the spectral bounds of the PESS and LPESS preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed PESS preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS and LPESS preconditioners 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，通过对PESS预条件子进行松弛，我们引入了一种局部PESS（LPESS）预条件子。建立了所提出的PESS迭代过程对任意随机初始猜测收敛的充要判据。进一步地，我们细致研究了PESS和LPESS预条件矩阵的谱界。此外，通过数值实验对所提出的PESS预条件子进行了敏感性分析，揭示了其鲁棒性。数值实验表明，与现有的块对角和移位分裂预条件子相比，所提出的PESS和LPESS预条件子具有更高的效率和鲁棒性。