二维等谱问题研究的计算数学框架

发布时间：2018-04-04 17:49

本文选题：PDE特征值问题　切入点：节点线　出处：《计算数学》2017年03期

【摘要】：等谱问题是数学、物理诸学科关注的一个热点问题,本文总结并诠释了二维等谱问题的内在计算数学性质与规律:利用镜像反演讨论等谱对的几何结构(不等距而谱相等);把一般文献中假定的特殊三角形扩展到一般的三角形或者矩形;研究特征函数的正交结构,把特定的Laplace等谱问题扩展到一般零边值的二阶线性椭圆算子等谱问题.指出合理的粗网格对于研究等谱问题及其计算的重要性:两个连续问题等谱成立的充分必要条件是存在自然粗网格使其离散问题谱相等.文中给出的数值例子与特征值近似逼近验证了相应的结论,所用的方法原则上可用于研究三维乃至高维的PDE等谱问题.
[Abstract]:Isospectral problem is a hot topic in mathematics and physics.This paper summarizes and interprets the intrinsic computational mathematical properties and laws of two-dimensional isospectral problems: the geometric structure of isospectral pairs is discussed by using mirror inversion, and the special triangles assumed in general literature are extended to one.Like a triangle or rectangle;In this paper, the orthogonal structure of eigenfunction is studied, and the special Laplace isospectral problem is extended to the isospectral problem of the second order linear elliptic operator with general zero boundary value.It is pointed out that a reasonable coarse grid is important for the study of isospectral problems and its calculation: the sufficient and necessary condition for two continuous problems to hold isospectral is that the existence of natural coarse meshes makes the spectrum of discrete problems equal.The numerical example given in this paper verifies the corresponding conclusions with the approximate approximation of eigenvalues. The proposed method can in principle be used to study the PDE isospectral problems in three dimensional and even high dimensional cases.
【作者单位】：中国科学院软件研究所并行软件与计算科学实验室;
【基金】：国家重点研发计划高性能计算重点专项(2016YFB0200601) 国家自然科学基金(91530323,91230109);国家自然科学基金青年基金(11301507)资助
【分类号】：O241.8

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