Study On Numerical Methods And Preconditioned Technology For Saddle Point Problems

The numerical solution of large sparse linear equations has been a classical research topic in the field of science and engineering computing.As a class of special linear equations,saddle point problems are widely used in many fields of scientific research,such as computational fluid dynamics,linear elasto science,image processing and so on,so that the study of numerical solution algorithm has very important theoretical and practical significance.The numerical solution of saddle point problems has a long history.Many scholars at home and abroad have also studied the effective numerical method for solving all kinds of saddle point problems with special properties,for example,Uzawa method,PIU method,SOR method,HSS iterative method,Krylov subspace iterative methods,variety preconditioning methods and so on.As is known to all,when the coefficient matrix of the saddle point problem is very large,iterative methods are commonly used,but it is usually necessary to take appropriate preconditioned technology to achieve the goal of accelerating convergence.Therefore,how to make full use of the special structure or property of the coefficient matrix of the saddle point problem has a very important significance to design the corresponding preconditioner,which is the main goal of this paper.In this paper,based on HSS splitting and PSS splitting,by generalizing the HSS preconditioner and PSS preconditioner,we provide two different types of preconditioners for saddle point problem: modified relaxed HSS(MRHSS)preconditioner and the variant of MPSS(VRPSS)preconditioner,the spectral properties of the preconditioned linear matrix and the convergence analysis of the algorithm are given at the same time.Finally,some numerical experiments are presented to show the efficiency of the new proposed preconditioners compared with the original preconditioner.This dissertation which is divided into four chapters organization as follows:In the first section,we will introduce the research background and significance of saddle point problem,the common iterative algorithms,preliminary knowledgeand the research direction of this paper.In the second section,we first introduce the HSS splitting iterative algorithm and the HSS preconditioner,based on which,we generalize the HSS preconditioner and construct MRHSS preconditioner.Theoretical analyses show the convergent efficiency and the property of the preconditioning matrix.Numerical examples are given to confirm our theoretical results.In the third section,based on the PSS splitting iterative algorithm,we propose the VRPSS preconditioner and analyze some properties of the coefficient matrix spectrum of the preconditioned linear system.In the last section of this chapter,several numerical experiments are given.From the results of the numerical experiments,we can see that the new preconditioner is more effective than the same preconditioner.The fourth section gives a summary of this dissertation and the prospects for future work.