节点文献

样条函数与小波函数在偏微分方程数值解中的应用

Applications of Spline Functions and Wavelets on Numerical Solution of PDEs

【作者】 张然

【导师】 周蕴时;

【作者基本信息】 吉林大学 , 计算数学, 2004, 博士

【摘要】 物理学、力学和工程技术方面的许多问题都可归结为按初始条件和边界条件求解偏微分方程的初值-边值问题。这些问题中能采用解析法按照边值条件求解的仅限于极少数情况。所以,一般只能采用近似方法求解。近些年来,随着电子计算机的广泛应用,数值解法逐渐成为解初值-边值问题的一种非常有效的方法,在数值分析中占有重要的地位。 自1946年schoenberg提出样条函数的概念以来,样条函数方法得到了迅速的发展和广泛的应用,同时也成为了函数逼近理论中重要的研究对象。上世纪80年代,一批数学家、物理学家、地理学家为小波概念奠定了坚实的基础,形成了小波理论。小波函数在微分、积分方程数值解,数字信号(图象)处理等诸多领域均有应用。小波由于同时具有空间域与频率域的局部性,因此是描述、检取函数奇性的有效工具。样条与小波的思想对数值数学的发展有很大影响。 本文将进一步探讨样条函数与小波函数在偏微分方程数值解中的应用。主要做了以下几项工作: 1.在第二章中,我们应用一种边界型求积技术,推导出用于求解偏微分方程边值问题的一类边界元方案。作为数值解法,边界元法只需将区域的边界分割成边界单元,因此使所考虑问题的维数降低了一维。与对整个区域进行分割的有限元法和有限差分法相比,它具有输入数据少,计算时间短,特别适用于无限域问题和三维问题等优点。此外,边界元法只对边界离散,离散化误差仅来源于边界,区域内的有关物理量可由解析式的离散形式直接求得,因此提高了计算精度;求解时要改变内点的数量或位置也非常方便;对于只需求出边界值的问题,关于区域内的物理量可以不必进行计算,能提高计算效率。 设Ω为Rn中的有界闭区域,定义二阶微分算子L为 Lu=sum from i,j=1 to n(aij(X)(?)2u/(?)xi(?)xj)+sum from i=1 to n(bi(X)(?)u/(?)xi+c(X)u,) (1)、‘、、·‘rJ亏、其中a:J(X)任万老(n):b:(X),e(X)‘万二(‘2);且毋脾)(0全一)表示几上的函数集合,集合中每个元素f(x)=f(x,,…,x。)都具有连续的偏导数D(il,…“)f,。三:、三。,k=1.…,n·L的伴随算子M定义为、艺闺 几“。一艺 1,J=1aZ(。a‘,(X)) 口x:口工,a(。b、(X)) axl+e(X)v.(2)考虑下面的边值问题{Lu(X)一夕(X),X‘n,X‘a几一,X〔a几2,(3) 一一一叭如一撇二=UQ二其中微分算子L由(l)式定义,并且口只、U口几2=a几 在互2.4,我们利用降维展开方法推导出一类用于处理边值问题(3)的边界元法.主要结果如下: 定理1设几〔R”是。维有界闭区城,其边界曲面ag分片光滑,特别,当,‘=2时,机2是一条有限长简单闭曲线.令。二试X)任CZ(几),且满足L。=o,又令”二v(X)为材。一占(X一XO)的解,其中X。为几中任意给定点.互为伴随的徽分耳子L与AI的定义分别由(l)和(2)给出. (i)如果X。任几\口几,那么边值问题(3)在点Xo处的解为一卜五。}喀一、}ds一五。…·客(鑫一(·)鬓)会ld·,(4)其中。:=;:(X)二艺界,a!,(X)口。/ax,+艺界,vaa:,(X)/ax,一b:(X)v·(11)、于x0。。。,若;i。。n一亡,:(:)擎一。,。且11二。(。):一1一。,其中华 艺一U置口几‘一u口1‘为x:在曲面{X;{X}二时上的外法方向导数,则C“”“。,一五。。艺;:瓮]d一五。}·睿(睿一爵)佘ld一、其中口一,二关.do,而d“”凡””应于X0”立体角·这里K·=几自口B。厂B‘={X:!X一X01<时是以X0为中心的小球.下面考虑外边值问题,即当几的补卯=R”\几有界时,几上的边值问题.为得到问题(3)的唯一解,我们还需要诚x)满足辐射条件二(X)}},。(X)1}二。(X)1}v。(X)}[二o(}X}一”)(}X卜co),(6)其中v(X)是Mv(X)=0的基本解. 类似于定理1,对外边值问题(3)加果诚X)满足辐射条件(6),我们可以得到相应的定理(见39页).之后,我们以平面调和方程、高维调和方程和Helmhol比方程为例给出了定理l与定理2的应用. 在互2.5中,我们以求解二维Helmho一tz方程(即令(3)中L二△+儿2)的边值为例,利用样条插值和边界型求积公式(例如[s]中给出的公式)把前面推导的边界积分方程离散成代数方程,给出了边界元法的具体求解过程. 假定{xj}孔,为边界a。的一个给定剖分·下面,我们来求解边界上所有未知的二;二。(凡)和勺=a。(xj)/on.由(5)式,我们知道对点凡〔a几,j=1,…,n异·,一五。·(X)会d“一五。粉口、(X) a牡ds,(7)其中马的解.一,二天;d“,“d”“K·的对应于凡的立体”,、为方程‘“十“’)·二占(X一价)为将(7)离散化,我们将二和q作插值展开二(X)会艺。‘叻{‘’(X) 己之1其中《‘’和《2,为Lagrange插值基函数.如下代数方程。(X)‘艺。功{,,(X),(8) 己=l将(s)代入(7),我们就将积分方程(7)离散为口,.声二了一Uj+夕U更Z7r碑‘.口“五。,,1)(X)豁d“一客。!瓜‘,2’‘X,、d“,,-1,…,倪·(9)为解该线性方程组,我们记、飞?

【Abstract】 In physics, mechanics, engineering technology and other science fields, lots of practical problems can be modelled by partial diffential equations (PDEs for short), which are well-posed under some initial and/or boundary conditions. Solving such systems is very important in applications. Finding an analytic solution of PDEs is ususlly very difficult, therefore a numerical approximation of the solution becomes very important.Since Schocnberg proposed the definition of "spline function" in 194G, the corresponding theory has rapidly developed. In the mean time, the spline function theory turns to bo a main subject in the studies of function approximation theory. In the 80’s of last century, a group of mathematicians, physical scientists, geographers set up the robust foundation of wavelet .theory. Wavelets have vast applications in the numerical approximation for differential and integral equations, digital signal (image) processing and other fields. Because of the locality of wavelets in the space and frequency domain, it is a effective tool to describe and test the function’s singularity. The spirits of spline and wavelets have brought huge influence on the development of numerical analysis.In this paper, we study the applications of spline functions and wavelets on numerical approximation of PDEs, respectively.1. In chapter 2, we apply a boundary -type quadrature technique to derive a type of boundary element scheme, which will be used to solve the boundary-value problems of PDEs. As the numerical method, the boundary element method (BEM for short) only needs to discretize the boundary of the domain, and this requires very simple data input and storage techniques. If the BEM is used to solve an exterior problem with the domain such that Rn\m is bounded, it is not necessary to deal with the boundary at infinity, since the corresponding fundamental solution chosen in the BEM satisfies the radiationcondition. Thus, exterior problems with unbounded domains can be handled as easily as interior problems, which means the BEM is much more suitable for problems over unbounded domains than the traditional finite element method.We use 1) to denote a bounded and closed region in R". Suppose that the boundary of (denoted by d) can be described by a system of parametric equations.We begin with the dimension reducing expansions (DRE for short) related to the second order differential operator L defined bywhere a is the collection of all functions that have continuous partial derivatives It is well-known that the adjoint operatorM of L can be defined byThe purpose of this chapter is to sutdy the numerical approximation for solutions of the following boundary-value problem:where differential operator L is defined by (1) and .In ?2.4, We derive a method for solving the boundary value problem (3) by using the DRE. The main result is:Theorem 1 Let C R" be an n-dimensional bounded closed domain with the boundary being a piecewise smooth surface or a simple closed curve with finite length when n=2. Let u = u(X) be a function in C2 that satisfies Lu ?0, and let v = v(X) be the fundamental solution of Mv = 6(X ?X0), where X0 is an arbitrarily given fixed point in . then the solution of the boundary value problem (3) at the point XQ isFor the points where dii/dn denotes the outer normal derivative of xi on the surface {X: \X\ =e}, thenand d is the solid angle of with respect to X0. Here is a smalt ball centered at X0.Next, we consider the exterior boundary value problem (3) over n with its complement being bounded. In order to obtain the unique solution of the problem, we also require for u(x) to satisfy the radiation conditionswhere v(X) is the fundamental solution of Mv(X) = 0.If u(X) satisfies the radiation conditions (6), similar to the Theorem 1, we obtain a corresponding Theorem on the exterior boundary-value problem. Then we derive an application of the two theorems to the boundary value problems of the harmonic equations and the Helmholtz’s equation, respectively.In ?2.5, we di

  • 【网络出版投稿人】 吉林大学
  • 【网络出版年期】2004年 04期
节点文献中: 

本文链接的文献网络图示:

本文的引文网络