Glhss Method To Discuss The Convergence Of The Generalized Saddle Point Problem

In many scientific and engineering applications such as computing problems on hydrokinetics, mixed finite elements of ellipse PDEs, constraint least-squares problems, they all can be summed up to solving linear equations about large sparse matrix, which we called saddle point problems. And we mainly focus on calculation reliability, calculation validity and calculation accuracy. According to the knowledge we have learned, we can constrain spectral radius less than one by properly constructing iterative matrix, then the iterative solution approaches to the accurate solution. The less the spectral radius is, the faster the convergence speed of the iterative matrix runs, the shorter time CPU costs, and the better the calculation performs. An important research is that we try to constitute proper parameters in difference equations on saddle point problems, when the spectral radius inclined on a small scale regularly we talk about the relationship between the spectral radius and the parameters.In recent years, HSS (Hermitian and Skew-Hermitian Splitting Method) solving linear systems has developed a lot, while MHSS (Modified Hermitian and Skew-Hermitian Splitting Method), LHSS (Local Hermitian and Skew-Hermitian Splitting Method) and MLHSS (Modified Local Hermitian and Skew-Hermitian Splitting Method) are modified on the basis of HSS. The literatures [1]-[10] give out theories working on saddle point problems which are demonstrated by Professor Bai and many other famous scholars. In this article, a new method called GLHSS (Generalized Local Hermitian and Skew-Hermitian Splitting Method) working on generalized saddle problem is constructed. What’s more, we draw out some correlative conclusions and prove that GLHSS is convergent when the parameter satisfies certain condition. At the end of the article, we display numerical examples to verify GLHSS is valid. The structure and main content of this article are as follows:The first part is the introduction. It introduces background knowledge of linear equations as well as the iterative methods, and brings forward the difference method of saddle point problem and generalized saddle point problem.The second part summarizes the prior knowledge we have learned. This part preparing for Part IV and Part V mainly gives us some essential definitions and lemmas such as Hermitian matrix, Skew-Hermitian matrix and so on.The third part displays correlative conclusions about saddle point problems. It mainly introduces some accomplishment that scholars especially Professor Bai have achieved on solving saddle point problem on the basis of HSS method, work out some essential conclusions, and what’s more puts forwards to the idea of solving generalized saddle point problem.The forth part talks about the convergence of GLHSS method. In this part, I bring the LHSS idea to generalized saddle point problem, advance GLHSS method and prove that GLHSS method converges to the accurate solution. At the same time, I testify that when spectral radius satisfies certain condition GLHSS converges fast.The fifth part is numerical examples. In this part GLHSS is applied to solving linear equations dispersing from generalized saddle point problem. Numerical examples and charts are used to explain that the iterative matrix spectral radius of GLHSS method is less than one and GLHSS method converges when the parameter meets some condition.The sixth part is Summary and Prospect. This part summarizes the methods and main conclusions I draw out from this article, then do the prospects for the future.