复变量移动最小二乘近似方法的误差估计

发布时间：2018-04-07 15:47

本文选题：无网格法　切入点：复变量移动最小二乘近似　出处：《重庆师范大学》2017年硕士论文

【摘要】：无网格法是继有限元法之后发展起来的一种新的数值计算方法。该方法的核心在于形函数的构造,移动最小二乘近似是当前应用最为广泛的无网格近似方案之一,然而基于移动最小二乘近似的无网格法计算量较大。复变量移动最小二乘近似是一种基于复变量理论的、针对向量函数逼近的移动最小二乘近似。在复变量移动最小二乘近似中,二维函数的近似只需使用一维基函数,导致试函数中的待定系数减少,进而所需节点个数大大减少。因此复变量型无网格法可以在保障计算精度的情况下,大大减少求解域内的节点个数。基于复变量移动最小二乘近似的无网格法在工程领域已经被广泛地应用,然而其相应的数学理论还很不完善,为了更好地促进其应用,分析其误差就必不可少。本文详细讨论了复变量移动最小二乘近似的误差,主要内容如下:本文第一章介绍了几种主要的偏微分方程数值计算方法,无网格法发展历史以及研究现状,第二章详细介绍了移动最小二乘近似及复变量移动最小二乘近似。第三章是本文的主要工作,在对权函数以及节点分布做出假设的基础上,针对光滑函数,分析了逼近函数及其偏导数的误差估计,分析结果表明误差与节点间距密切相关,最后通过算例验证了理论分析的正确性。第四章是本文的另一个主要工作,对于被逼近函数光滑性较弱的情形,在对权函数以及节点间距做出适当假设的基础上,详细推导了复变量移动最小二乘近似在Sobolev空间中的误差估计并给出了数值算例。
[Abstract]:Meshless method is a new numerical method developed after finite element method.The core of this method is the construction of shape function. Moving least square approximation is one of the most widely used meshless approximation schemes at present, but the meshless method based on moving least square approximation has a large amount of computation.The moving least squares approximation of complex variables is a moving least squares approximation based on the theory of complex variables.In the moving least square approximation of complex variables, only one wiki function is used in the approximation of two-dimensional functions, which results in the reduction of the undetermined coefficients in the trial function and the reduction of the number of nodes required.Therefore, the complex variable meshless method can greatly reduce the number of nodes in the solution domain under the condition of guaranteeing the calculation accuracy.Meshless method based on moving least-square approximation of complex variables has been widely used in engineering field, but its corresponding mathematical theory is not perfect. In order to promote its application better, it is necessary to analyze its error.In this paper, the error of moving least square approximation of complex variables is discussed in detail. The main contents are as follows: in the first chapter of this paper, several main numerical methods of partial differential equations are introduced, and the development history and research status of meshless method are introduced.In chapter 2, moving least squares approximation and complex variable moving least square approximation are introduced in detail.The third chapter is the main work of this paper. Based on the assumption of weight function and node distribution, the error estimation of approximation function and its partial derivative is analyzed for smooth function. The results show that the error is closely related to the distance between nodes.Finally, the correctness of the theoretical analysis is verified by an example.The fourth chapter is another main work of this paper. In the case of weak smoothness of the approximated function, we make appropriate assumptions about the weight function and the distance between nodes.The error estimation of moving least square approximation of complex variables in Sobolev space is derived in detail and a numerical example is given.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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