发表机构
Charles University; Università degli Studi di Firenze（查理大学; 佛罗伦萨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出基于离散正弦变换的离散化方案，用于一致逼近Sturm--Liouville和Laplace算子的谱，克服常规方法对高指标特征值逼近差的缺陷，且适用于非矩形域。
AI 中文摘要
我们研究了有界区间上的正则Sturm--Liouville算子和任意平面域上的Laplace算子的各种离散化方案，两者均受零Dirichlet边界条件约束。目标是确定一种离散化方案，使得相应的离散化算子——一个大小为$N \ imes N$的矩阵——产生$N$个特征值，这些特征值尽可能接近连续层面上相应算子的前$N$个特征值。通过数值实验，我们检验了几种常规离散化方案，并证实了已知事实：常规离散化无法实现该目标，其失败归因于对高指标特征值的逼近效果差，从而导致谱逼近的非一致性。相比之下，新提出的基于离散正弦变换的离散化方案被设计为能够复现高指标特征值的渐近行为，从而提供所寻求的一致谱逼近。同样重要的是，所提出的基于离散正弦变换的方案，与许多基于傅里叶变换的方法不同，可以应用于非矩形域。
英文摘要
We study various discretisation schemes for the regular Sturm--Liouville operator on a bounded interval and for the Laplace operator on an arbitrary planar domain, in both cases subject to zero Dirichlet boundary conditions. The objective is to identify a discretisation scheme such that the corresponding discretised operator---a matrix of size $N \times N$---produces $N$ eigenvalues that approximate as closely as possible the first $N$ eigenvalues of the corresponding operator at the continuous level. By means of numerical experiments we examine several conventional discretisation schemes, and we corroborate the known fact that the conventional discretisations fail to achieve the objective, with the failure attributable to the poor approximation of high-index eigenvalues, and, as a result, to the non-uniform spectrum approximation. In contrast, the newly proposed discrete sine transform based discretisation scheme is designed in such a way that it replicates the asymptotic behaviour of high-index eigenvalues, thereby providing the sought uniform spectrum approximation. Of equal significance is the fact that the proposed discrete sine transform based scheme can be, unlike many Fourier transform based methods, applied to non-rectangular domains.
