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 ( N 3 ) 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. Approximationsk ^ jto the square roots k j of all O ( N ) eigenvalues lying in [ k  − ϵ, k ], where ϵ =  O (1), are found with O ( N 3 ) effort. We prove an error estimate|k ^ j − kj | ≤ C (∈ 2 k + ∈ 3 ) , with C independent of k . We present a higher‐order variant with eigenvalue error scaling empirically as O (ϵ 5 ) and eigenfunction error as O (ϵ 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 10 3 times faster than standard ones based upon a root search. © 2014 Wiley Periodicals, Inc.