The preconditioned iterative methods with variable parameters for saddle point problem

In this paper, by transforming the original problem equivalently, we propose a new preconditioned iterative method for solving saddle point problem. We call the new method as PTU (preconditioned transformative Uzawa) method. And we study the convergence of the PTU method under suitable restrictions on the iteration parameters. Moreover, we show the choices of the optimal parameters and the spectrum of the preconditioned matrix deriving from the PTU method. Based on the PTU iterative method, we propose another iterative method - nonlinear inexact PTU method - for solving saddle point problem. We also prove its convergence and study the choices of the optimal parameters. In addition, we present some numerical results to illustrate the behavior of the considered algorithms.