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### 二维等谱问题研究的计算数学框架

1. 中国科学院软件研究所并行软件与计算科学实验室, 北京 100190
• 收稿日期:2017-03-15 出版日期:2017-08-15 发布日期:2017-08-04
• 基金资助:

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

Sun Jiachang, Zhang Ya. FRAMEWORK OF COMPUTATIONAL MATHEMATICS ON 2-D PDE ISO-SPECTRAL PROBLEMS[J]. Mathematica Numerica Sinica, 2017, 39(3): 229-286.

### FRAMEWORK OF COMPUTATIONAL MATHEMATICS ON 2-D PDE ISO-SPECTRAL PROBLEMS

Sun Jiachang, Zhang Ya

1. Laboratory of Parallel Software and Computational Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
• Received:2017-03-15 Online:2017-08-15 Published:2017-08-04

Iso-spectral problem is one of hot topics in mathematics and physics research. We summarize and interpret the intrinsic computational mathematical properties of the 2-D iso-spectral problems:the geometric structure of iso-spectral pair is discussed by using mirror inversion (not iso-metric but iso-spectral); the special triangles assumed in the general literature are extended to the general triangles or rectangles; the orthogonal structure of the eigenfunctions is studied, and the specific Laplace iso-spectral problem is extended to the 2-nd order linear elliptic operator. This paper points out the importance of rational coarse grids for the study of iso-spectral problems:the sufficient and necessary condition for the consecutive iso-spectral problems is that the spectrums of discrete problem with natural coarse grids are equal. The numerical examples and the approximate approximation of eigenvalues verify the conclusion of this paper. The approach can be used to study 3-D even high dimension PDE iso-spectral problems.

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