Generalized shift-splitting preconditioners for nonsingular and singular generalized saddle point problems

Based on the shift-splitting technique, a class of generalized shift-splitting preconditioners are proposed for both nonsingular and singular generalized saddle point problems. The generalized shift-splitting preconditioner is induced by a generalized shift-splitting of the generalized saddle point matrix, resulting in a generalized shift-splitting fixed-point iteration. Theoretical analyses show that the generalized shift-splitting iteration method is convergent and semi-convergent unconditionally for solving the nonsingular and the singular generalized saddle point problems, respectively. Numerical experiments of a model Navier–Stokes problem are implemented to demonstrate the feasibility and effectiveness of the proposed preconditioners.