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

Abstract

We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star-shaped2. Conventional boundary-based methods require a root search in eigenfrequency k, hence take O.N 3 / effort per eigenpair found, where/ 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 y k j to the square roots k j of all O.N / eigenvalues lying in OEk ; k, where D O.1/, are found with O.N 3 / effort. We prove an error estimatewith 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 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 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.

Is the Helmholtz Equation Really Sign-Indefinite?

“…This identity with v = u was originally obtained by Rellich [67] and has been used extensively in the analysis of both the Laplace and the Helmholtz equations (with suitable generalizations also used to study higher-order elliptic PDEs). For example, Rellich introduced (1.34) with v = u in order to obtain an expression for the eigenvalues of the Laplacian as an integral over ∂Ω (instead of the usual expression as an integral over Ω used in, e.g., the Rayleigh-Ritz method), and these identities have been used to further study eigenvalues of equations involving the Laplacian in, e.g., [64], [66], [42], [3], [4]. Rellich-type identities have been wellused by the harmonic analysis community (see, e.g., [47 [24], [43], [19], [45]); some of this recent work is discussed in Remarks 3.6 and 4.7 below.…”

“…Unfortunately this bound is not practical, since if hk ∼ 1, then N ∼ k d , and (in exact arithmetic) GMRES always converges once the number of iterations, m, reaches the dimension N of the linear system. It is instructive to note that two of the powers of k in m k 4 arise from the fact that C c /α ∼ k, and two powers come from the norm in V , so even if the method were pollution-free, i.e., if C c /α were bounded independently of k, then the estimate (5.6) would give m k 2 , which is still not particularly useful. (Similarly, a hypothetical H 1 -conforming scheme with continuity and coercivity properties similar to those of section 3 would also give m k 2 .)…”

FEAST for Differential Eigenvalue Problems

“…Finally, we note that our algorithm inherits the advantage of the FEAST matrix eigensolver in that it is extremely parallelizable [39]. For these reasons, we view this work as a first step towards closing the gap between the frequency regimes that are accessible to standard computational techniques and asymptotic methods for differential eigenvalue problems posed on higher dimensional domains [8,9].…”

Sharp norm estimates of layer potentials and operators at high frequency

“…Their technique allows them to control error on the boundary and so mapping norms on S + λ control error in the interior. In [2,Remark 3.2], the authors proposed the question of finding a bound on the λ-dependence of the single layer potential S + λ . In particular they ask is (…”

“…In star shaped domains Barnett and Hassell [2] develop a numerical technique for constructing Dirichlet eigenfunctions by solving a related eigenfunction problem on the boundary. They then use (1.3) to reconstruct interior eigenfunctions.…”