Solution Of Saddle Point Problems Uzawa Algorithm And Its Convergence Analysis

This thesis is finished during my Master of Science and it consists of four chapters.Chapter 1 IntroductionIn this chapter we propose our main concern which are the inexact Uzawa algorithms for saddle point problems.Furthermore,we present the saddle point problems simply. The indefinite system of equationswhere F ∈ H₁ and G ∈ H₂ are given and X∈H₁ and Y ∈ H₂ are the unkon-wns. Linear systems such as above problem are called saddle point problems,which may arise form finite element discretizations of Stokes equations and Maxwell equa-tions,mixed finite element formulations for second order elliptic problems,or from Lagrange multiplier methods for optimization problems, for the parameter identification and domain decomposition problems.In recent years,there is a rapidly increasing literature which is concerned with inexact Uzawa-type algorithms.because they are simply,efficient,have minimal memory requirements, and easy to implement. Of course they are important in lagre-scale scientific applications implemented for today's computing architectures.Chapter 2 Linear inexact Uzawa algorithms for the symmetric saddle point problemsIn this chapter we introduce the linear inexact Uzawa algorithms and the linear inexact Uzawa algorithms with parameters for symmetric saddle point problems by the numbers,and we also carefully analysis these algorithms' convergence.In the same time we extent our conclusion to the generalized instance.Chapter 3 Non-linear inexact Uzawa algorithms for the symmetric saddle point problemsCorresponding to the Chapter 2,in this chapter we first consider the non-linear inexact Uzawa algorithms for the symmetric saddle point problems analogously.Then we will modify the non-linear inexact Uzawa algorithm with an over-relaxation pa-rameter.We will analyze the new algorithm and prove that the new one converges to the exact solution of the problems under weaker conditions.At the end we will give a numerical example.Chapter 4 Inexact Uzawa algorithms for the non-symmetric saddle point problemsIn the chapter we discuss the Non-linear inexact Uzawa algorithms for the symmetric saddle point problems and introduce some results simply.