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

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.