The Research Of Preconditioning Techniques For Saddle Point Problems

Matrix computation has become the foundation of science and engineering computation, and the numerical results of many problems in science and engineering computation are obtained by matrix computation. In practical application, we often need to numerically solve the saddle point problems, such as fluid dynamics,optimization, economics, finance, circuit networks, electromagnetism, mixed finite element approximation of elliptic partial differential equations, etc.In this dissertation, we mainly study the preconditioning methods for saddle point problems and the generalized saddle point problems. Firstly, we briefly introduce the GHSS and AGHSS methods for solving the large sparse non-Hermitian linear system of equations. Based on the AGHSS iteration method, an AGHSS preconditioner for saddle point problems has been presented, and the eigenvalue distribution of the preconditioned matrix has been studied. Secondly, by generalizing the HSS preconditioning method for generalized saddle point problems, a new HSS preconditioner with double parameters has been proposed. Moreover, the spectral properties of the preconditioning matrix has been analysized. We proved that under certain conditions and if the coefficient matrix is non-Hermitian positive and the two positive parameters are small, the eigenvalues of the preconditioning matrix will gather in(0,0) and(2,0). This dissertation includes four chapters, which is organized as follows:In Chapter 1, the research background and preliminary knowledge for solving the saddle point problems have been introduced. The main content of this dissertation has also been introduced.In Chapter 2, the AGHSS method for solving the large sparse non-Hermitian positive definite linear system of equations has been introduced. Then, based on the AGHSS method, a preconditioner has been presented for the saddle point problems and the spectral property of the preconditioning matrix is discussed. Finally, somenumerical experiments have been given to illustrate the effectiveness of the new preconditioner.In Chapter 3, by generalizing the HSS-based preconditioner for generalized saddle point problems, a double parameters HSS-based preconditioner has been proposed. The spectral properties have also been discussed. Moreover, some numerical examples are given.Finally, the research work of this dissertation is summarized and the future further research work is discussed.