Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map
Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map
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通过内部诺依曼到狄利克雷图的谱流快速计算高频狄利克雷本征模
DOI：
10.1002/cpa.21458
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发表时间：
2011
影响因子：
3
通讯作者：
Andrew Hassell
中科院分区：
文献类型：
--
作者：
A. Barnett;Andrew Hassell
提出了一种新的数值计算Dirichlet Laplacian在光滑星星形区域中的大特征值和相关特征函数的算法。传统的基于边界的方法需要在特征频率k中进行根搜索，因此每个特征对需要O（N3）的努力，其中N = O（kd−1）是离散边界所需的未知数的数量。我们的方法是O（N）更快，通过关于k线性化Helmholtz方程的加权内部Neumann-to-Dirichlet（NtD）算子的谱来实现。在[k − k，k]中所有O（N）个特征值的平方根kj的近似k^j，其中k = O（1），用O（N3）的努力找到。我们证明了一个误差估计|k^j−kj| ≤C（∈2k+∈3），其中C与k无关。我们提出了一个高阶变量，其特征值误差缩放经验为O（105），特征函数误差为O（103），前者改进了Vergini和Saraceno的“缩放方法”。对于平面域（d = 2），与光谱浓度的情况下的假设，我们也证明了严格的误差界，接近那些数值观察。对于d = 2，我们通过势理论，Nyström离散化和Cayley变换稳健地计算NtD算子的谱。在高频（400个波长）下，本征频率相对误差为10−10，我们表明该方法比基于根搜索的标准方法快103倍。© 2014 Wiley Periodicals，Inc.
We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star‐shaped domain in ℝd, d ≥ 2. Conventional boundary‐based methods require a root search in eigenfrequency k, hence take O(N3) effort per eigenpair found, where N = O(kd−1) is the number of unknowns required to discretize the boundary. Our method is O(N) faster, achieved by linearizing with respect to k the spectrum of a weighted interior Neumann‐to‐Dirichlet (NtD) operator for the Helmholtz equation. Approximations k^j to the square roots kj of all O(N) eigenvalues lying in [k − ϵ, k], where ϵ = O(1), are found with O(N3) effort. We prove an error estimate |k^j−kj|≤C(∈2k+∈3), with C independent of k. We present a higher‐order variant with eigenvalue error scaling empirically as O(ϵ5) and eigenfunction error as O(ϵ3), the former improving upon the “scaling method” of Vergini and Saraceno. For planar domains (d = 2), with an assumption of absence of spectral concentration, we also prove rigorous error bounds that are close to those numerically observed. For d = 2 we compute robustly the spectrum of the NtD operator via potential theory, Nyström discretization, and the Cayley transform. At high frequencies (400 wavelengths across), with eigenfrequency relative error 10−10, we show that the method is 103 times faster than standard ones based upon a root search. © 2014 Wiley Periodicals, Inc.