  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.
