Iterative Methods For Saddle Point Problems

The present Ph.D. dissertation is concerned with some algorithms which are used to solve saddle point problems. These problems arise in numerous applications such as fluid dynamics, constraint quadratic programming, linear elasticity, electromagnetics and other areas of applications. Since the coefficient matrices of these problems usually are large and sparse, it is useful to consider some fast iterative methods. In this article, we take Navier-Stokes equation, Osceen equation and Stokes equation as the standard model problems and introduce two methods to discrete them: the mixed finite element method Q1 - P0 with stabilization and the finite difference M.A.C., thus giving the symmetric and nonsymmetric saddle point problems. There are large variety of methods for solving these linear systems. Among them we mention direct solvers, Uzawa type algorithms, Null-space methods and Krylov subspace methods. In this Ph.D. dissertation, we first review all the existing Uzawa type algorithms. To accelerate the convergence speed, we give two new practical inexact nonlinear Uzawa methods, i.e., one is for the symmetric saddle point problems and the other is for the nonsymmetric case. We have also considered the convergence properties and given some theorems and conclusions. We apply them to our model problems. Numerical experiments show that our new methods need much less iterates than the other previous Uzawa type methods for convergence. Finally, we extend the PCG method with residual update strategy based on the null space for large equality constrained quadratic programming problems and obtain several preconditioned GMRES methods with residual update strategy based on the null space. Thus we can apply them to solve the nonsymmetric saddle point problems. Numerical tests are given in the end.