Some Studies On Preconditioning Algorithms For Discrete Sadldle-point Systems In Electromagnetic Field And Fluid Computation

Electromagnetic field and computational fluid dynamics have important appli-cations in the field of scientific and engineering, such as meteorology, oceanography, biomedicine and so on. Maxwell equations are the basic model in the electromagnet-ic field. Numerical solutions of the Maxwell equations are significant for electronic engineering, especially for microwave and antenna engineering. The basic model for describing the characteristic of fluid motion is Stokes equations and Navier-Stokes equations. So how to solve Stokes equations and Navier-Stokes equations effecitvely is the key to solve practical problems. Whether Maxwell equations or Stokes equa-tions or Navier-Stokes equations, by using finite difference method or finite volume method or finite element method, all of them can be dispersed into a class of linear systems with some special structure, i.e., saddle point problems. Therefore, studying iteration methods for the saddle point problems is of great practical significance. In this thesis, we propose some numerical algorithms and preconditioners to solve the symmetric indefinite systems arose from a PML system of an electromagnetic wave scattering problem, the three-by-three saddle point problems dispersed from time-dependent Maxwell equations, the two-by-two saddle point problems came from Stokes equations and the nonlinear saddle point problems produced from Navier-Stokes equations. We shall show the convergence analysis and estimate the spectral bounds of the preconditioned systems. The concrete structure of the thesis is as follows:In the first chapter, we shall propose some block triangular preconditioners for a PML system of an electromagnetic wave scattering problem and analyse the spec-tral behaviour of the preconditioned system. When the PML system is discretized by edge element methods, it results in a discrete system with its stiffness matrix being complex, symmetric but indefinite, which can be formulated into a real sym-metric but indefinite saddle point system. In order to preserve the symmetry of the coefficient matrix, we present a block triangular preconditioner with bilateral preconditioning for the linear system. We estimate the lower and upper bound of positive and negative eigenvalues of the preconditioned matrix, respectively. On the other hand, we propose a block triangular preconditioner to precondition the system only from one-side and analyse the spectrum of the preconditioned system. Some numerical experiments are given to verify the feasibility of the new preconditioners.In the second chapter, we consider some preconditioning techniques to solve 3×3 block saddle point problems dispersed from time-dependent Maxwell equa-tions with discontinuous coefficients in general three-dimensional Lipschitz polyhe-dral domains. We propose an exact block diagonal preconditioner for solving the symmetric saddle point problem and its nonsymmetric form. We show that the cor-responding preconditioned systems have six different eigenvalues. For the needs of practical application, we also present a class of inexact block diagonal precondition-ers for solving the saddle point problems. For the symmetric system, we estimate the lower and upper bounds of positive and negative eigenvalues of the precondi-tioned matrix, respectively. For the nonsymmetric system, we derive some explicit and sharp bounds on the real and complex eigenvalues, respectively. Numerical ex-periments are presented to demonstrate the effectiveness and robustness of all these new preconditioners.In the third chapter, using finite element discretization, Stokes equations can be transformed into a class of linear saddle point problems. We propose two new iterative methods for solving the linear saddle point problem based on partitioning the coefficient matrix. One is combining the block Gauss-Scidel iterative method with the Uzawa iterative method, and the other one is combining the block Jacobi iterative method with the Uzawa iterative method. Then we respectively study the convergence of the two novel methods under suitable restrictions on the iteration parameters. Numerical experiments are also presented to illustrate feasibility and effectiveness of the considered algorithms.In the fourth chapter, by transforming the original problem equivalently, we propose a new preconditioned iterative method for solving saddle point problem dispersed from Stokes equations. We call the new method as PTU method. We study the convergence of the PTU method under suitable restrictions on the iteration parameters. Moreover, we show the choices of the optimal parameters and the spectrum of the preconditioned matrix deriving from the PTU method. Based on the PTU iterative method, we propose another iterative method, nonlinear inexact PTU method, for solving saddle point problem. We also prove its convergence and study the choices of the optimal parameters. In addition, we present some numerical results to show the competitive of the new proposed algorithms.In the fifth chapter, we present a class of inexact relaxed deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioners for solving the lin-ear system arose from Stokes equations. The proposed inexact RDPSS prccondi-tioner is a technical modification of the relaxed deteriorated positive-definite and skew-Hermitian splitting (PSS) preconditioner [206]. The PSS preconditioner is a straightforward application of the positive-definite and skew-Hermitian splitting (PSS) iteration method for solving non-Hermitian positive definite linear system-s initially established by Bai et al. [29]. Numerical results have shown that the proposed inexact RDPSS preconditioner is advantageous over a class of existing preconditioners.In the sixth chapter, we use stabilized finite element method to solve Stokes equations, which follows a class of linear systems of algebraic equations, i.e., gener-alized saddle point problems. We first derive some explicit and sharp bounds on the spectra of generalized nonsymmetric singular or nonsingular saddle point matrices. Then we propose new nonsingular preconditioners for solving generalized singular saddle point problems, and show that GMRES determines a solution without break-down when applying GMRES to the preconditioned system with any initial vector. Moreover, we also analyze the spectral properties of the preconditioned systems. These nonsingular preconditioners are applied to solve the singular finite element saddle point systems arising from the discretisation of the Stokes equations to test their performance.In the seventh chapter, we deal with the Navier-Stokes equations by finite el-ement method directly, which gives rise to a class of nonlinear system of algebraic equations, i.e., nonlinear saddle point problems (NSPPs). We mainly consider the construction of efficient Uzawa-type algorithms for solving the NSPPs. Two nonlin-ear inexact Uzawa hybrid algorithms based on one-step Newton method are proposed for solving the NSPPs. By using the energy-norm, the convergence of the proposed algorithms is verified under some practical conditions. In addition, some numerical results are reported to illustrate the behavior of the considered algorithms.