Several Iterative Algorithms For Saddle Point Systems

In many scientific and engineering applications such as electromagnetic fields,fluid mechanics,weighted least squares problems,computer image processing problems and so on,a large sparse saddle point system with different block structures can be obtained by using appropriate discrete methods.How quickly solve this kind of equations is the research field of many scholars,and it is very important in theoretical analysis and practical applications.In this paper,several kinds of numerical methods for saddle point systems with different structures are studied,and the proposed algorithm is rigorously analyzed theoretically.At the same time,the effectiveness of the algorithm is verified by numerical examples.The main contents of this paper are as follows:In Chapter 1,the basic methods for solving large linear equations are summarized,the application background and research status of several kinds of saddle point problems are introduced.At the same time,the main contents of this paper are described.In Chapter 2,an alternating preconditioned upper and lower triangle(APULT)splitting iterative algorithm for solving a class of 2 × 2 saddle point system is proposed.The idea of the proposed method is introduced,and the convergence of the algorithm is analyzed.Finally,the feasibility and effectiveness of the proposed algorithm are verified by numerical experiments.In Chapter 3,an efficient preconditioner for solving a class of 3 × 3 saddle point system is proposed.The preconditioner can be used to accelerate Krylov subspace methods.Specifically,the construction form of the modified relaxed block upper-lower triangular(MRBULT)preconditioner and the algorithm needed in the preprocessing are given,the spectral properties of the preconditioning matrix of the proposed method are analyzed and an optimal parameter selection method is given.Finally,the feasibility and effectiveness of the preconditioner are verified by numerical experiments.In Chapter 4,a double parameters preconditioned block alternating splitting implicit(DPPBASI)iterative algorithm for solving a class of generalized saddle point systems derived from time-harmonic eddy current field problems is proposed.Firstly,the basic idea of the method is introduced,and a concrete algorithm framework is given in the case of general topology and simple topology.At the same time,the convergence of the algorithm is analyzed strictly and theoretically.Finally,the effectiveness of the algorithm is verified by numerical experiments.In Chapter 5,the research content of this paper is summarized,and the direction of the future research is described in detail.