Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map |
| |
| Authors: | Alex Barnett Andrew Hassell |
| |
| Affiliation: | 1. Department of Mathematics, Dartmouth College, Hanover, NH, USA;2. Department of Mathematics, Australian National University, Canberra, ACT, Australia |
| |
| 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 ?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(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 ? ?, k], where ? = O(1), are found with O(N3) effort. We prove an error estimate ![urn:x-wiley::media:cpa21458:cpa21458-math-0002]( "urn:x-wiley::media:cpa21458:cpa21458-math-0002") with C independent of k. We present a higher‐order variant with eigenvalue error scaling empirically as O(?5) and eigenfunction error as O(?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. © 2014 Wiley Periodicals, Inc. |
| |
| Keywords: | |
|
|
