【作者】 洪季芳；

【导师】 冉启文；

【作者基本信息】 哈尔滨工业大学 ， 应用数学， 2009， 硕士

【摘要】 边界元法(BEM)是求解偏微分方程的一种有效的数值计算方法,边界元法将求解区域内的微分方程边值问题归化到边界上,然后在边界上离散求解。边界元法主要优点是降维,从而使问题所需的方程少,数据少,求解工作大为简化,这在处理高维问题时具有优势。小波作为一个新兴的数学分支,应起始于S. Mallat和Y. Meyer在八十年代中后期所做的工作,即构造小波基的通用方法,此后小波得到了迅猛的发展。小波变换克服了传统Fourier变换的不足,是继Fourier分析之后的一个突破性进展。小波在时域和频域都具有良好的局部化特性,在应用领域更是掀起了一股应用小波的热潮,它具有丰富的理论,应用十分广泛,如信号处理、图像分析等,是工程应用中强有力的方法和工具,给许多相关领域带来了崭新的思想,并使其被越来越多的数学研究工作者所关注,由于小波兼有光滑性和局部紧支撑性质,能更好的处理积分和微分方程的数值求解问题。小波BEM自上世纪90年代提出以来,一直是国内外学者研究的热点。小波BEM具有迭代效率高、矩阵预条件简单等优点,因此成为众多学者关注的一种BEM快速求解方法。全文分为四章,第一章主要介绍了论文选题背景,小波与边界元研究的历史和现状,以及本文要做的工作。第二章介绍了小波分析的基本理论,包括小波和小波变换的定义、性质,多分辨分析、尺度函数的定义、性质以及周期拟小波的基本理论。第三章介绍了边界元基本理论,加权余量法、变分法概述,以及边界积分方程的推导过程。第四章是本文的主要工作,主要研究了利用拟周期小波边界元方法讨论二维拉普拉斯方程的数值求解问题。首先用边界元方法将待求方程化为边界积分方程,然后用拟周期小波作为基底,将积分方程在边界上展开,化为对应的代数方程组,求解系数,得到方程的近似解。在对方程组求解过程中,系数矩阵作小波矩阵变换,利用多尺度方法,求解新的代数方程组,减少计算量,并分析了此算法的收敛性、算法复杂度及误差分析。更多还原

【Abstract】 Boundary element method is an effective method to solve partial differential equation. Boundary element method naturalizes the differential equation in solve region to boundary and then disperses boundary to solve. The main virtue of boundary element method is the reduce of dimensions, thus equations and dates the question needed is decreased so the solving work is simplified, that is advantage to solve high dimension equations.Wavelet theory as a new mathematical branche starts from the study of S. Mallat and Y. Meyer in 1980s, that is the construction of wavelet basis and has developed rapidly ever since. Wavelet transform overcomes defects of traditional Fourier transform,which is a breakthrough progress after Fourier analysis. Wavelet has well localized property both in time and frequency domains, creates an wavelet upsurge in application fields. It possess abundant theory and is applied widely, such as signal processing, image analysis. It is a powerful method and tools in application fields. It gives new idea to related fields and is concerned by more and more math research workers, because of its own smoothness and local compact support properties, can solve numerical computing integral and differential equations better.Since was introduced in 90s last century, wavelet boundary element method is always hot spots studied by scholars home and aboard. Wavelet BEM possess high iteration efficiency and preconditioning matrix simple, thus concerned by numerous scholars as a fast BEM solving method.The paper is divided to four chapters. The first chapter mainly relates background of selected subjects, history and present situations of wavelet and boundary element method researched and study work of this paper. The second chapter relates basic wavelet analysis theory including definition and property of wavelet, wavelet transform, multi resolution analysis and scale function as well as periodic quasi wavelet theory. The third chapter introduces basis theory of boundary element, summary of method of weight residual and variational method and deduction process of boundary integral equation. The fourth chapter is the main work of this paper, its mainly studied numerical computing of two dimensions Laplace equation using quasi periodic wavelet boundary element method. First using boundary element method coverts equation to be solved into boundary integral equation, then spread boundary integral equations by quasi wavelet basis turns to corresponding algebraic equations and solve it getting approximately solutions. In the process of solving equations do matrix transform using wavelet matrix and use multi scale method solving new algebraic equations reduce calculated amount, discuss convergence, complexity, error analysis of this arithmetic.更多还原