Fast Computation of High-Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann-to-Dirichlet Map

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 ℝ^d, d≥2. Conventional boundary-based methods require a root search in eigenfrequency k, hence take O(N³) effort per eigenpair found, where N=O(k^d-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 k_j of all O(N) eigenvalues lying in [k-ε{lunate}, k], where ε{lunate}=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(ε{lunate}⁵) and eigenfunction error as O(ε{lunate}³), 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.