Saeed Karimi

Many important and practical problems in science and engineering lead to the solution of linear and large saddle point systems. The indefinity and poor spectral properties of such systems have made numerical solution of these problems a major and exciting challenge for those working in this field. Since in practical problems the saddle point systems have a large and sparse coefficient matrix, solving such systems using direct methods are costly and sometimes even impossible. Therefore, iterative methods are used to solve them. One of the main problems of iterative methods is that their convergence speed is slow. The preconditioning techniques are used to improve the convergence speed of these methods. In this regard, in the following dissertation, we study the saddle point problems. We present a modification of one of the most important and well-known methods, called Uzawa methods for solving 2×2 forms of saddle point problems. We investigate the convergence of this method and the induced preconditioner behavior. Moreover, we present a strategy to choose the parameters of the preconditioner. In the following, we introduce a new preconditioner for a class of 3 × 3 block saddle point problems. We also estimate the upper and lower bounds of eigenvalues of the preconditioned matrix. we will test the effectiveness of these preconditioners to accelerate the convergence speed of Krylovsubspace methods, especially, the preconditioned GMRES method, using a few practical examples.