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Fast computation of high frequency Dirichlet eigenmodes via the spectral flow of the interior Neumann-to-Dirichlet map
Alex H. Barnett, Andrew Hassell
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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, using dense linear algebra, 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 k̂ⱼ 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 |k̂ⱼ - kⱼ| ≤C (ϵ²/k + ϵ³ ), 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--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, Nystr\"{o}m 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.
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