SSOR Correction Method For Solving Saddle Point Problems And Large Sparse Linear Equations

Saddle point problems appear in many different applications of scientific computing, such as electromagnetics, fluid mechanics, constrained optimization, generalized least squares problems, finite element approximation to solve the Navier-Stokes equation. Nowadays many researchers are committed to solving the saddle point problems, and it’s very important to solve it as fast and efficient as possible.In the branch of Numerical Analysis in Computational Mathematics, many people study the solution of large linear sparse equation. When the equation’s rank is very high, the iterative method is much better than others, especially for large linear sparse case. As a consequence, the iterative method can save the memory, speed up calculation, and improve accuracy with the sparse character.This paper mainly studies the modified SSOR method for the saddle point problems and the large linear sparse systems. First we consider the original problem without transformating the coefficient matrix. Then we apply the method to the large sparse linear equations, and discuss the convergence theorems. In the end, we compare the method with the existing methods, and demonstrate the advantages by the spectral radius, the iterative numbers and time, the relative error of norms through the numerical experiments.The structure and main content of this paper are as follows:The first part is the introduction. We briefly introduce the saddle point problems and the application, the iterative methods of linear equation systems. The second part is the prerequisite knowledge. We introduce some special matrixes and some existing theorems we need.The third part is the main part of this paper. Considering the original problem without transformating the coefficient matrix, we study the modified SSOR method for the saddle point problems. We discuss the convergence of the method, and demonstrate the advantages of the new method.The forth part is also the main part. We apply the new method to the large sparse linear equations, and discuss the convergence theorems. In the following, we prove that the new method is better than the existing method through the numerical experiments.The fifth part is the summary and outlook. We summarize the paper, and give the prospects of this new method.