Two modified block-triangular splitting preconditioners for generalized saddle-point problems

For the generalized saddle-point problems, firstly, we introduce a modified generalized relaxed splitting (MGRS) preconditioner to accelerate the convergence rate of the Krylov subspace methods. Based on a block-triangular splitting of the saddle-point matrix, secondly, we propose a modified block-triangular splitting (MBTS). This new preconditioner is easily implemented since it has simple block structure. The spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed, respectively. Moreover, we apply the MGRS and the MBTS preconditioners to three-dimensional linearized Navier-Stokes equations. Then we derive the quasi-optimal parameters of the MGRS and the MBTS preconditioners for two and three-dimensional Navier-Stokes equations, respectively. Finally, numerical experiments are illustrated to show the preconditioning effects of the two new preconditioners.