The Research Of Numerical Algorithm For Linear Saddle Point Problems

With the development of the computing technology, solving the linear systems is obtained from many scientific research and engineering computation such as partial differential equations, linear programming, network analysis, the finite element analysis of the configuration and unconfiguration problem. Moreover, solving large-scale sparse linear systems plays a key role in scientific and engineering computing, so more and more people pay attention to find effective numerical solutions of linear algebraic systems based on large-scale scientific and engineering computing. Solutions of large-scale sparse linear systems are always iterative methods, so convergence and convergence rate of iterative methods are deeply studied by many experts and scholars.In this thesis, mainly studies some iterative methods of special matrices related to iteration solutions of large-scale sparse linear systems. First, the UBOR-type method is introduced in this thesis, and then using this method to solve the linear saddle point problems and give some related conclusions. The UBOR-type Method for solving the symmetric linear saddle point problem is further studied in this paper. When the coefficient matrix is splitted, the lower right corner of block diagonal matrix could be weakened from a symmetric positive definite block to a non-symmetric positive definite. In this case, the eigenvalues of the convergence conditions could be promoted from the real situation to the complex situation. Which could make this method to adapt to a more extensive division. And the condition of the convergence of this method is also given.Secondly, UBOR-type method and PPIU method are introduced to solve singular linear saddle point problems, Moreover, the convergence of the two methods for singular saddle point problems are further studied, and try to discuss the corresponding convergence. Finally, the numerical experiments are given to prove the convergence of the two methods.