A class of preconditioners based on symmetric-triangular decomposition and matrix splitting for generalized saddle point problems

In this paper, based on the idea of symmetric-triangular (ST) decomposition and matrix splitting, a product-type preconditioner and its inexact version are developed for generalized saddle point problems. These preconditioners can be combined appropriately with the efficient preconditioned conjugate gradient (PCG) method to solve the generalized saddle point problems, although neither these preconditioners nor the generalized saddle point matrices are symmetric and positive definite, which is the advantage of our proposed preconditioners. The proposed PCG method belongs to the group of nonstandard inner product CG methods, so the convergence theorem of the former is given by the use of that of the latter. The difference between the existing nonstandard inner product CG methods and our proposed method is studied. Theoretical analysis shows that the spectrum of the preconditioned matrix corresponding to the proposed inexact preconditioner is contained in a real, positive interval and the quasi-optimal parameter of the preconditioner is effective and easy to apply in practice. Numerical experiments are given to illustrate the effectiveness and robustness of the proposed inexact preconditioner and show the advantages of the preconditioner over the existing state-of-the-art preconditioners for saddle point problems.