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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (4): 8-15.doi: 10.6040/j.issn.1671-9352.0.2021.778

• • 上一篇    

球域上传输特征值问题的一种有效的谱逼近

任师贤1,2,安静1*   

  1. 1.贵州师范大学数学科学学院, 贵州 贵阳 550025;2.贵州民族大学数据科学与信息工程学院, 贵州 贵阳 550025
  • 发布日期:2023-03-27
  • 作者简介:任师贤(1987— ),女,博士研究生,讲师,研究方向为偏微分方程数值解. E-mail:renshixianaq@126.com*通信作者简介:安静(1979— ),男,博士,教授,博士生导师,研究方向为偏微分方程数值解. E-mail:aj154@163.com
  • 基金资助:
    国家自然科学基金资助项目(12061023,11961009);贵州省教育厅青年科技人才成长项目(KY［2022］179)

An efficient spectral approximation for the transmission eigenvalue problem in spherical domains

REN Shi-xian1,2, AN Jing1*   

  1. 1. School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, Guizhou, China 2. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China
  • Published:2023-03-27

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摘要： 为了求解球形区域上的内部传输特征值问题,提出一种有效的谱逼近方法。首先,定义一种乘积型Sobolev空间,利用单位球上一类正交多项式构造相应的逼近空间。其次,通过引入一个辅助函数,将原问题转化为一个等价的四阶混合格式,并推导出该四阶混合格式的变分形式及其离散格式。然后,利用投影算子的逼近性质和Babuška-Osborn理论,证明逼近解的误差估计。最后,详细地描述算法的实现过程,并通过一些数值算例验证了算法的收敛性和高精度。

关键词: 传输特征值问题, 谱方法, 误差估计, 数值算法, 球形区域

Abstract: In order to solve the interior transmission eigenvalue problems in a spherical region, an effective spectral approximation method is proposed. Firstly, a product type Sobolev space is defined, and the corresponding approximation space is constructed by using a class of orthogonal polynomials on the unit sphere. Then, by introducing an auxiliary function, the original problem is transformed into an equivalent fourth-order mixed scheme, and the variational form and discrete form of the fourth-order mixed scheme are derived. Moreover, by using the approximation property of the projection operator and Babuška-Osborn theory, the error estimation of approximation solution is proved. Finally, the implementation process of the algorithm is described in detail, and some numerical examples are given to verify the convergence and high accuracy of the algorithm.

Key words: transmission eigenvalue problem, spectral method, error estimation, numerical algorithm, spherical domain

中图分类号: 

  • O241.82
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