A robust parameterized enhanced shift-splitting preconditioner for three-by-three block saddle point problems

This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). In addition, 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 preconditioned matrix. Moreover, empirical investigation has been performed for the sensitivity analysis of the system, revealing the robustness of the proposed PESS preconditioner. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS preconditioner compared to the existing block diagonal and shift-splitting preconditioners.