The Researches On Matrices Splitting Iteration Methods And Preconditioning Techniques Of Saddle-point Linear Systems

Today,many engineering and physical applications,such as computational fluid dynamics,computational electromagnetics,constrained optimization problems etc.,are eventually translated into solving system of linear equations.In general,it is very difficult for some differential equations,such as Navier-Stokes equations,to find the analytical solutions.So the study of their numerical solutions becomes especially important.Generally speaking,we use the finite difference method and finite element method to discrete the partial differential equations for large sparse system of linear equations.Thus the problem of numerical solutions of partial differential equations is converted to the solution of the corresponding linear equations.Therefore,it has very important theory values and application values to study the effective solutions of systems of linear equations solution.In this thesis,we studied the iteration methods and the preconditioners of saddle point problems.In addition,we developed the parallel multisplitting methods based on double splitting of coefficient matrices.First of all,for nonsymmetric saddle-point problem,a modified shift-splitting(MSS)preconditioner is proposed based on a splitting of the nonsymmetric saddlepoint matrix.And a local MSS(LMSS)preconditioner is also presented.Both of the two preconditioners are easy to be implemented since they have simple block structures.The convergent properties of the two iteration methods induced respectively by the MSS and the LMSS preconditioners are carefully analyzed.The methods of choosing the optimal parameters of the MSS and the LMSS preconditioners are discussed.Numerical experiments are illustrated to show the robustness and efficiency of the MSS and the LMSS preconditioners used for accelerating the convergence of the generalized minimal residual(GMRES)method.The second,we proposed a regularized Hermitian and skew-Hermitian splitting(RHSS)method and the RHSS preconditioner;Then we presented a modified RHSS(MRHSS)preconditioner.The convergent properties of the two iteration methods induced respectively by the RHSS and the MRHSS preconditioners are carefully analyzed.Furthermore,the spectrum properties of the MRHSS preconditioned matrix are derived.The choices of the optimal parameters of the RHSS and the MRHSS preconditioners are studied.Numerical experiments are illustrated to show the robustness and efficiency of these methods used as stationary iterative solvers or as preconditioners for the generalized minimal residual(GMRES)methods.Once again,for the generalized saddle-point problems,we introduce a relaxed block-triangular splitting(RBTS)preconditioner to accelerate the convergence rate of the Krylov subspace methods.This new preconditioner is easily implemented,since it has simple block structure.The spectral property of the preconditioned matrix is analyzed.Moreover,the degree of the minimal polynomial of the preconditioned matrix is also discussed.The method of choosing the optimal parameter of the RBTS preconditioners is given.Numerical experiments are illustrated to show the preconditioning effect of the new preconditioner.Then,for the generalized saddle-point problems,we introduce a modified generalized relaxed splitting(MGRS)preconditioner and a modified block-triangular splitting(MBTS)to accelerate the convergence rate of the Krylov subspace methods.The spectral properties of the MGRS and the MBTS preconditioned matrices are analyzed and the degree of the minimal polynomial of the preconditioned matrices are also discussed,respectively.Moreover,We apply the MGRS and the MBTS preconditioners to three-dimensional linearized Navier-Stokes equations and derive the optimal parameters of the MGRS and the MBTS preconditioners for threedimensional Navier-Stokes equations.Finally,numerical experiments are illustrated to show the preconditioning effects of the two new preconditioners.The last,parallel multisplitting Algorithms based on double splittings of coefficient matrices with relaxed parameters are established for solving the nonsingular linear system whose coefficient matrix is a monotone matrix or an H-matrix.The corresponding convergence and comparison results are presented.