亥姆霍兹透射特征值问题的等几何分析
Isogeometric analysis for the Helmholtz transmission eigenvalue problem
- University of Macau（澳门大学）
- Zhuhai UM Science and Technology Research Institute（珠海澳大科技研究院）
- Beijing Normal University（北京师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对亥姆霍兹透射特征值问题在弯曲域上的数值挑战，提出一种几何灵活且H^2相容的等几何方法，并给出最优误差估计及二维三维数值验证，展示其在精度和几何灵活性上的优势。
AI中文摘要：
透射特征值问题在逆散射理论中扮演着越来越重要的角色。尽管在过去二十年中，针对该问题开发高效数值方法已取得显著进展，但其在 $\mathbb{R}^d$（$d=2,3$）中弯曲域上的数值处理仍然具有挑战性。本文提出并分析了一种几何灵活且 $H^2$ 相容的等几何方法，用于求解由亥姆霍兹透射特征值问题产生的四阶、二次且非自伴的特征值问题。利用紧致非自伴算子的谱逼近理论，我们导出了离散特征值和特征函数的最优误差估计。给出了二维和三维基准问题（包括定义在弯曲域上的问题）的数值结果，以验证我们的理论结果，并展示该方法在精度和几何灵活性方面相对于现有数值方法的优势。
英文摘要：
The transmission eigenvalue problem plays an increasingly important role in inverse scattering theory. Although significant progress has been made in developing efficient numerical methods for the problem over the last two decades, its numerical treatment for curved domains in $\mathbb{R}^d$ ($d=2,3$) remains challenging. In this paper, we introduce and analyze a geometrically flexible and $H^2$-conforming isogeometric method for solving a fourth-order, quadratic and non-self-adjoint eigenvalue problem arising from the Helmholtz transmission eigenvalue problem. Using the spectral approximation theory for compact non-self-adjoint operators, we derive optimal error estimates for the discrete eigenvalues and eigenfunctions. Numerical results for the two- and three-dimensional benchmark problems, including those defined in curved domains, are presented to verify our theoretical results, and to demonstrate the advantages of the method over existing numerical methods in terms of both accuracy and geometric flexibility.
