The Study Of Iterative Methods For Saddle Point Problems And Complex Symmetric Linear Systems

Large sparse linear equations are widely used in areas such as electromagnetism,the least square problems,the constrained optimization problems and the numerical simulation problems in engineering,etc.Often,these sparse linear equations have special structures after discreted by the finite difference or finite element methods.Motivated by these applications,this dissertation is devoted to the study of effectively solving large sparse linear equations with special structures.The specific problems to be investigated in this dissertation include:singular and nonsingular saddle point problems and singular complex symmetric linear systems.The convergent properties and numerical experiments with respect to these methods are given.The dissertation first presents the generalized parameterized inexact Uzawa method(GPIU)and the generalized preconditioned parameterized inexact Uzawa method(GPPIU)to solve the singular saddle point problem.The sufficient conditions for semi-convergence of the iterative schemes based on these methods are established.For singular saddle point problem,the SOR-Uzawa method is proposed and sufficient conditions for semi-convergence are established.For nonsingular saddle point problems,based on the iterative scheme of the GSOR method,a simple method to derive the optimal parameters of the GSOR method is studied.The dissertation also uses the regularized HSS method(RHSS)to solve the singular saddle point problems.It is shown that this method is semi-convergenct unconditionally.Moreover,it is demonstrated that the HSS method has the same convergent properties as those of RHSS method under much weakened conditions.Numerical examples are given to illustrate the efficacy of the RHSS method.Based on the matrix splitting methods,the dissertation also introduces two kinds of preconditioners for nonsingular saddle point problems and generalized nonsingular saddle point problems.The detailed spectral analysis are given;which,in turn,leads to good eigenvalue properties for these preconditioners.The dissertation also proposes a generalized modified HSS method(GMHSS)to solve the singular complex linear system and gives sufficient conditions for semi-convergence of this method.Numerical examples are given for illustrations.