解鞍点问题的UZAWA算法及其收敛性分析

【作者】 叶海锋；

【导师】 程晓良；

【作者基本信息】 浙江大学 ， 计算数学， 2005， 硕士

【摘要】 本文是在攻读硕士学位期间完成的,全文共分四章: 第一章绪论 提出本文研究的问题解鞍点问题的UZAWA算法,并做简要的介绍。所谓的鞍点问题,即以下类型的线性系统:其中给定F∈H₁,G∈H₂而X∈H₁,Y∈H₂未知。我们设H₁和H₂是有限维Hilbert空间,记该空间的内积为（.,.）。同时假设A:H₁→H₁是一个线性算子,B^T:H₂→H₁是映射B:H₁→H₂的转置映射。其来源于Stokes方程或Maxwell方程的有限元离散,二阶椭圆型问题的混合有限元方法求解,或者来自于最优化问题的拉格朗日乘数法,参数识别和域分解问题等。 近年来,UZAWA算法已经得到了广泛的关注,因为UZAWA型算法具有简单,有效,只需要较小的存储空间并且容易执行,所以被广泛的使用在今天的大规模计算上。 第二章对称线性鞍点问题的线性不精确UZAWA算法 系统介绍了解对称鞍点问题的线性不精确UZAWA算法和带参数的UZAWA算法,并且详细分析了其收敛性和收敛率,对不同的算法之间的优劣做了一定程度的分析讨论,然后推广到解一般鞍点问题的UZAWA算法上。 第三章对称线性鞍点问题的非线性不精确UZAWA算法 对应于第二章,首先讨论了见解对称鞍点问题的非线性不精确UZAWA算法及其收敛性,然后修改算法,提出了一种新的带参数的非线性不精确UZAWA算法,并对其做了收敛性分析,证明修改后的算法在更弱的条件下收敛,最后给出数值例子。 第四章非对称鞍点问题的不精确UZAWA算法 讨论了UZAWA算法的新的方向,用来解不对称鞍点问题,对一些结果做了简要的介绍。更多还原

【Abstract】 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.更多还原