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 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 state-of-the-art 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。此外，通过松弛PESS预条件子，我们引入了一种局部PESS（LPESS）预条件子。我们为所提出的PESS迭代过程对任意初始猜测的收敛性建立了必要且充分的准则。进一步，我们细致研究了PESS和LPESS预条件矩阵的谱界。此外，通过实证研究对提出的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，与现有最先进的预条件子相比，所提出的PESS和LPESS预条件子具有更高的效率和鲁棒性。