Some Researches On Numerical Methods Of The Saddle Point Problems

In this thesis, we mainly discuss some numerical algorithms for solving the saddle point problems. Many practical problems arising from scientific computing and engineering applications can be modeled as saddle point problems, such as fluid dynamics, quadratic optimization, the domain decomposition methods of the Helmholtz equations, weighted least-squares problems and so on. So the saddle point problems are hot issues in the study of mathematics.In the Introduction, we start from the current situation and background about the saddle point problems, the importance of research in saddle point problem is explained. Moreover, we introduce some conceptions which appear in the following chapters.In Chapter 1, a new SOR-Like (NSOR-Like) method for solving the saddle point problems is discussed. This method has three parameters and it can be applied to the nonsingular saddle point problems as well as the singular cases. Firstly, the char-acteristic of eigenvalues of the iteration matrix of this NSOR-Like iteration method is analyzed. Then, the convergence (semi-convergence) theorem of the NSOR-Like method for nonsingular saddle point problems (singular saddle point problems) is given under some suitable assumptions.In Chapter 2, a class of accelerated Uzawa algorithms for solving the saddle point problems are investigated, i.e., the AU algorithms. This method can be used to solve the nonsingular saddle point problems. Firstly, the accelerated model of the Uzawa method is established by making use of the extrapolation technique. Then the accelerated Uzawa (AU) algorithms are presented. Moreover, the convergence analyse of the AU method is given in the theoretical analyses which show that the AU algorithm converges faster than some Uzawa-type methods (the Uzawa method is also included in) when the eigenvalues of the iteration matrix and the iteration parameter T satisfy some conditions. The numerical results show the correction of the theoretical results and examine the effectiveness of the AU method.In Chapter 3, a corrected Uzawa algorithm for solving the saddle point prob-lems is studied, and we call this method CU method. Firstly, the corrected model of the Uzawa method is established, and the CU method is presented. The geometric meaning of the corrected Uzawa model is studied. Moreover, we introduce the over-all reduction coefficient a to measure the effect of the CU method. It is shown that the corrected Uzawa method convergence faster than the Uzawa method and several other methods if the overall reduction coefficient a satisfies certain conditions. Nu-merical experiments show that this method is effective for solving the saddle point problems.In the last chapter, we make a conclusion of our work and point out how to further our research work, such as ideas, suggestions and the problems needed to be solved.