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- W2963251793 abstract "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 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 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, Nystrom 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." @default.
- W2963251793 created "2019-07-30" @default.
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- W2963251793 date "2013-05-23" @default.
- W2963251793 modified "2023-09-25" @default.
- W2963251793 title "Fast Computation of High-Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann-to-Dirichlet Map" @default.
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- W2963251793 doi "https://doi.org/10.1002/cpa.21458" @default.
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