Researches On Iterative Algorithms And Preconditioning Technology For Several Classes Of Saddle Point Problems

The electromagnetic field,the optimal control problem and the computational fluid have already infiltrated into various fields of science and engineering.In the practical applications,the time-harmonic eddy current model is often used to simulate the elec-tromagnetic phenomena concerning alternating currents at low frequencies,which is one of the most widely used electromagnetig field models in the engineering field.And the optimal control problem with a partial differential equation(PDE)has been widely used in the industrial,medical,economic and other fields.In addition,the incompressible fluid model as a kind of important models in the computational fluid dynamics,is closely with human's daily life and production.The above mentioned three kinds of problems often involve the numerical solutions of the partial differential equations or the optimization models.Therefore,it is of great practical significance to explore the numerical methods for these problems.In this dissertation,the iterative algorithms and preconditioning tech-nology are studied for the saddle point problems,which arise from the HC/EI coupled model of the time-harmonic eddy current problem,the PDE constrained optimization problem and the incompressible linearized Navier-Stokes equation.The convergence of the iterative algorithms and the properties of the eigenvalues and eigenvectors for the preconditioning matrices are studied deeply.The outline of this dissertation is as follows.In Chapter 1,the HC/EI coupled model of the time-harmonic eddy current prob-lem is established and the finite element approximation is introduced.For the complex-valued saddle point systems,two new alternating positive semidefinite splitting iterative algorithms are proposed.It is proved that the proposed iterative algorithms are uncon-ditionally convergent,respectively.Numerical experiments show that the corresponding alternating positive semidefinite splitting preconditioners accelerate well the convergence rate of Krylov subspace methods.In Chapter 2.two alternating positive semidefinite splitting preconditioners are given for the saddle point systems arising from the HC/EI coupled model of the time-harmonic eddy current problem.As the two preconditioners are as close as possible to the original coefficient matrix,it maybe accelerate well the convergence rate of Krylov subspace meth-ods.Numerical experiments show that the two preconditioners are superior to the APSS preconditioner proposed by Ren and Cao(IMA J.Numer.Anal.,36(2016):922-946).In Chapter 3.the PDE constrained optimization problem is considered.Using the Galerkin finite element method,then,based on the first-order necessary condition of the equality constrained optimization problem,the PDE constrained optimization problem is transformed into a 3-by-3 saddle point systems.If one of the block matrices K and M3 is nonsingular,then the saddle point systems is also nonsingular.When the block matrix K is nonsingular,three splitting iterative algorithms are proposed.It is proved that these iterative algorithms are convergent under certain assumptions imposed on the involving parameters,respectively.In addition,the optimal choices of the parameters are given.Numerical experiments show that the corresponding preconditioned GMRES methods perform well,when the corresponding splitting matrices are as preconditioners.In Chapter 4,the saddle point systems arising from the PDE constrained optimiza-tion problem are studied further.When the block matrix M3 is nonsingular,three splitting iterative algorithms are proposed.It is proved that these iterative algorithms are conver-gent under certain assumptions imposed on the involving parameters,respectively.In addition,the optimal choices of the parameters are given.Numerical experiments show that the corresponding preconditioners are effective.In Chapter 5,the saddle point systems arising from the PDE constrained optimiza-tion problem are considered further.For the saddle point systems,four splitting iterative algorithms are proposed.It is proved that these iterative algorithms are convergent under certain assumptions imposed on the involving parameters and matrices,respectively.Nu-merical experiments show that those preconditioners perform with better stability than the other preconditioners.In Chapter 6.two preconditioners are proposed for the saddle point systems arising from the incompressible linearized Navier-Stokes equation.The properties of the eigen-values and eigenvectors for the preconditioning matrices are analyzed,respectively.Nu-merical experiments show that the proposed preconditioners are superior to the existing other preconditoners in computing time.