A New Double Splitting Method For Solving Generalized Saddle Point Problems

Saddle point problems arise in a variety of science and engineering applications, such as fluid dynamics, elasticity, electromagnetics, constraint optimization problems, least square problems and so on, those problems often can be transformed into the saddle point problems to solve. Because the coefficient matrix of saddle point problems is large sparse matrix, iterative techniques are preferable for solving saddle point problems. Suc-cessive overrelaxation iterative method is very efficient in engineering computations, but it is difficult to be used to solve saddle point problems, because the coefficient matrix has singular diagonal block. In recent years, there are many methods for solving these prob-lems, such as the Uzawa method, the parameterized Uzawa method, the parameterized inexact Uzawa method, the generalized parameterized Uzawa method, the Hermitian and skew-Hermitian splitting method and so on. In paper [28], Shen and Huang studied the convergence and comparison results for double splittings of Hermitian positive definite matrices. In paper [32], Zhang discussed the convergence of double splitting methods for non-Hermitian positive semidefinite linear systems. In this paper, based on the paper [32], a new double splitting method is presented for solving the saddle point problems, the method is convergent under some suitable conditions. We also study the eigenvalue distribution of the corresponding preconditioned matrix. By choosing different parameter matrices, we get several new algorithms for generalized saddle point problems. Numeri-cal experiments are used to demonstrate the efficiency of the double splitting method.