计算数学
       首页 |  期刊介绍 |  编委会 |  投稿指南 |  期刊订阅 |  下载中心 |  留言板 |  联系我们 |  在线办公 | 
计算数学  2017, Vol. 39 Issue (3): 229-286    DOI:
综述 最新目录 | 下期目录 | 过刊浏览 | 高级检索  |  Next Articles  
二维等谱问题研究的计算数学框架
孙家昶, 张娅
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
FRAMEWORK OF COMPUTATIONAL MATHEMATICS ON 2-D PDE ISO-SPECTRAL PROBLEMS
Sun Jiachang, Zhang Ya
Laboratory of Parallel Software and Computational Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
 全文: PDF (1882 KB)   HTML (1 KB)   输出: BibTeX | EndNote (RIS)      背景资料
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.
服务
把本文推荐给朋友
加入我的书架
加入引用管理器
E-mail Alert
RSS
作者相关文章
关键词PDE特征值问题   节点线   等谱问题   镜像反演   结构化网格   移植矩阵     
Abstract: 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.
Key wordsPDE eigenvalue problems   Nodal Line   iso-spectral problem   mirror inversion   structured grids   transplantation matrix   
收稿日期: 2017-03-15;
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

引用本文:   
. 二维等谱问题研究的计算数学框架[J]. 计算数学, 2017, 39(3): 229-286.
. FRAMEWORK OF COMPUTATIONAL MATHEMATICS ON 2-D PDE ISO-SPECTRAL PROBLEMS[J]. Mathematica Numerica Sinica, 2017, 39(3): 229-286.
 
[1] Kac M. Can one hear the shape of a drum[J]. Am. Math. Monthly, 1966, 73(4): 1-23.
[2] Milnor J. Eigenvalues of the Laplace operator on certain manifolds[J]. Proc. Natl. Acad. Sci. USA, 1964, 51:542.
[3] Giraud O and Thas K. Hearing shapes of drums-mathematical and physical aspects of isosectrality[J]. Rev. Mod. Phys., 2011, 82(3): 2213-2255.
[4] Gordon C, Webb D and Wolpert S. You cannot hear the shape of a drum[J]. B. Am. Math. S., 1992, 27(1): 134-138.
[5] Sunada T. Riemannian coverings and isospectral manifolds[J]. Ann. of Math., 1985, 121(1): 169-186.
[6] Buser P, Conway J, Doyle P and Semmler K D. Some Planar Isospectral Domains[J]. Internat. Math. Res. Notices, 1994, 9: 391-402.
[7] Gordon C, Webb D and Wolpert S. Isospectral plane domains and durfaces via Riemannian orbifolds[J]. Invent. Math., 1992, 110: 1-22.
[8] Bérard P. Variétés Riemanniennes isospectrales non isométriques[J]. Astérisque, 1989, 177-178: 127-154.
[9] Bérard P. Transplantation et isospectralité[J]. Math. Ann., 1992, 292: 547-559.
[10] Chapman S J. Drums that sound the same[J]. Am. Math. Monthly, 1995, 102(2): 124-138.
[11] Wu H, Sprung D and Martorell J. Numerical investigation of isospectral cavities built from triangles[J]. Phys. Rev. E, 1995, 51: 703-708.
[12] Sridhar S and Kudrolli A. Experiments on not hearing the shape of drum[J]. Phys. Rev. Lett. 1994, 72(14): 2175-8.
[13] Driscoll T. Eigenmodes of isospectral drums[J]. SIAM. REV., 1997, 39(1): 1-17.
[14] Amore P, Boyd J P, Ferández F M and Rösler B. High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson extrapolation of second order finite differences[J]. J. Comput. Phys., 2016, 312: 252-271.
[15] Moon C R, Mattos L S, Foster B K, Zeltzer G, Ko W and Manoharn H C. Quantum Phase Extraction in Isospectral Electronic Nanostructures[J]. Science, 2008, 319:782-787.
[16] 冯康. 冯康文集(Ⅱ)[M]. 国防工业出版社, 1995.
[17] Singer I M. Eigenvalues of the laplacian and invariants of manifolds[J]. Proc. Inter. Congr. Math., 1974, 1, Vancouver.
[18] Sleeman B D and Chen H. On nonisometric iso-spectral connected fractal domains[J]. Rev. Mat. Iberoam, 2000, 16: 351-361.
[19] 陈化. 你能听到一面鼓的几何形状吗-谈谈等谱问题[J]. 数学通报, 2014, 53(5): 3-8.
[20] Petrobski. Lecture On Partial Differential Equations[M]. (in Russian) 1961.
[21] Glowinski R. A numerical investigation of the properties of the Nodal Lines of the solutions of linear and nonlinear eigenvalue problems[C]. International Conference on Scientific Computing, 2012, Hongkong.
[22] Bonito A and Glowinski R. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in R3: A computational approach[J]. Commun. Pure Appl. Anal., 2014, 13(5): 2115-2126.
[23] Levitin M, Parnovski L and Polterovich I. Isospectral domains with mixed boundary conditions[J]. J. Phys. A: Math. Gen., 2006,39(9): 2073-2084.
[24] Sun J C. Orthogonal piece-wise polynomials basis on an arbitrary triangular domain and its applications[J]. J Comput Math, 2001, 19(1): 55-66.
[25] Sun J C. Fourier Transforms and Othogonal Polynomials on Non-traditional Domains, Chinese University of Science and Techlonogy[M]. 2009(In Chinese).
[26] Lai R J. Computational differential geometry and intrinsic surface processing[D]. Ph.D. thesis, 2010, University of California Los Angeles.
[27] Moorhead S. Can you hear the shape of a cavity[D]. Ph.D. thesis, 2012, University of Oxford.
[28] Sun J C. Multi-Neighboring Grids Schemes for solving PDE eigen-problems[J]. Sci China Math, 2013, 56: 2677-2700.
[29] Sun J C. New schemes with fractal error compensation for PDE eigenvalue computations[J]. Sci China Math, 2014, 57: 221-244.
[30] 孙家昶, 曹建文, 张娅. 偏微分方程特征值计算的上下界分析与高精度格式构造[J]. 中国科学: 数学, 2015, 45(8): 1169-1191.
[1] 耿琳, 孙家昶. 广义曲边四边形区域族上的正交多项式[J]. 计算数学, 2010, 31(4): 309-320.
[2] 仇轶,由长福,祁海鹰,徐旭常. 无网格方法中的结点分布算法[J]. 计算数学, 2006, 27(3): 176-182.

Copyright 2008 计算数学 版权所有
中国科学院数学与系统科学研究院 《计算数学》编辑部
北京2719信箱 (100190) Email: gxy@lsec.cc.ac.cn
本系统由北京玛格泰克科技发展有限公司设计开发
技术支持: 010-62662699 E-mail:support@magtech.com.cn
京ICP备05002806号-10