Research On Several Kinds Of Matrix Splitting Iterative Method For Solving Saddle Point Problems

In many fields of scientific computation and engineering applications,it is often necessary to solve a large-scale sparse linear equation systems,which is called the saddle point problem because of its special structure.Such as computational fluid dynamics,constrained optimization,and mixed finite element approximation of el-liptic partial differential equations.Thus,it becomes a hot issue in mathematical research to solve the known saddle point problem quickly and effectively.Because of the large scale of the linear equation systems,and the block structural and property characteristics of the coefficient matrices of the linear equation systems,the direct method is often replaced by the iterative method(The calculation speed is fast,and the required storage is small)with high efficiency.In this paper,we mainly consider the non-Hermitian non-singular saddle point problem and the generalized non-Hermitian singular saddle point problem.We mainly discuss the properties of different iterative methods for the different splitting of coefficient matrices,and also discuss the accelerated convergence by using the preconditioner in corresponding cases.The main contents of this paper are:In Chapter 1,the development and research status of saddle point problem are summarized.In addition,some preliminary knowledge needed in this paper is introduced.In Chapter 2,Aiming at the non-Hermitian non-singular saddle point problem,the original Uzawa-type method is generalized and the generalized Uzawa-type it-erative method is obtained by making full use of the HS splitting of non-Hermitian matrices.By choosing different matrix P and parameters a and ?,the Numerical experiment reflects different effects.In Chapter 3,For the generalized non-Hermitian singular saddle point problem,the GHSS iteration method is used to solve the problem,and its semi-convergence and eigenvalue distribution after preconditioning are analyzed.In Chapter 4,A Bi-parameters single step(BiPSS)iteration method for solving non-symmetric saddle point problems is established,which used the HSS method for a non-symmetric linear systems.In addition,we used the generalized shift technique for our method by introducing two positive parameters ? and ?.In Chapter 5,the research work in this paper is summarized,and the assumption of future research work and the problems to be solved are put forward.