On the improvement of shift-splitting preconditioners for double saddle point problems

The shift-splitting preconditioner is investigated and improved within relaxation and preconditioning techniques to solve a special double saddle point problem in block three-by-three form. The proposed preconditioner can also be viewed as a generalization of the regularized preconditioner for standard saddle point problem. For economical implementation purpose, a further modification of the proposed preconditioner is also developed by utilizing an inexact block factorization technique to avoid the high cost of storage and computing requirements for solving the arising augmentation type linear subsystems. Moreover, spectral properties of the preconditioned matrices are analyzed in detail and valid lower and upper bounds are obtained to restrict the area confining the real and non-real eigenvalues, respectively. Numerical experiments are performed to assess the efficiency of the new preconditioners within Krylov subspace acceleration.