A Class Of Generalized Local Hermitian And Skew-Hermitian Splitting Iteration Methods For The Non-Hermitian Saddle Point Problems

Saddle point problem arises in many engineering and computations application-s,such as mixed finite elements methods for solving elliptic partial differential equa-tions and Stokes problems, constraint optimization, least-squares problems,fluid dy-namics,elsticity and so on. Because the saddle point problem has such a wide applica-tion source, it is of great interest to develop a fast and efficient methods, especially for solving those particular saddle point problems with the properties that the the block (1,1) of coefficient matrix is non-Hermitian positive definite.In paper [34], the generalized local Hermitian and skew-Hermitian splitting method for solving the saddle point problems with the block(1,1) being non-Hermitian posi-tive definite was considered. In this paper, we expand and applied this method to solving the non-Hermitian singular saddle point problem or non-Hermitian general-ized saddle point problem. We proved the semi-convergence and convergence of this iteration method under certain conditions, and derived the spectral distribution of the preconditioned matrix. With different parameters and parameter matrices in the matrix splitting, we get several algorithms for solving the non-Hermitian singular saddle point problem and non-Hermitian generalized saddle point problem. Numerical experiments are used to examine the numerical effectiveness of the (GLHSS or PGLHSS) iteration method served as a solver or a preconditioner with GMRES method for solving the corresponding linear systems.