Alireza Ataei

Many different models in science and engineering are formulated by differential equations; for example, the saddle point problem appears in the solution of many science and engineering problems such as optimization, solution of the system of mixed equations of fluid mechanics and image processing, Stokes and Naver Stokes problem, then from dicretization with the help of finite difference method or finite elements, they lead to the solution of the system of linear equations, which are often large and thin. Discretization of Stokes and Naver-Stokes problems lead to saddle point devices that have large and thin dimensions. Solving large and thin saddle point devices using direct methods such as Gauss elimination is associated with high costs and sometimes even impossible, so iterative methods are suitable for solving such devices. several iterative methods have been introduced to solve the saddle point device. one of the effective methods for solving linear equations is the HSS method. This method for solving the system of non-Hermmitian positive definite linear equations is based on Hermitian and skew Hermitian splitting of the coefficients matrix. The important advatage of HSS iterative methods is that it converges without any conditions. For this reason, it has attracted the attention of many researchers. In this thesis, iterative HSS method and iterative splitting Hermitian and RHSS flexible skew Hermitian method and the resulting preconditioner are used to solve the saddle point device. In the end, a new version of REHSS flexible skewed Hermitian splitting is introduced to solve the saddle point device. Using this methods, suitable preconditioning is suggested for sub-Krylov