Eigenvalues of the Truncated Helmholtz Solution Operator under Strong Trapping

Abstract

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that (a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and (b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalized minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretizations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [P. Marchand et al., Adv. Comput. Math., to appear]).

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

“…In obtaining these bounds, we crucially use the high-frequency decompositions of the singlelayer, double-layer, and hypersingular integral boundary operators from [37], the PDE results of [21,85,11,60,39,38], and results about semiclassical pseudodifferential operators (see, e.g., [87], [35,Appendix E]).…”

“…Whereas [38,Section 4] studies this problem when Ω − is curved, the results hold for general smooth Ω − if [39, Lemma 3.3] is used instead of [38,Lemma 4.8]. When applying [39,Lemma 3.3] to the set up in [38], we note that, since Ω − is bounded, the set A in [39, Lemma 3.3] is the whole of S * Ω − R d . Therefore, when η σ ( R) > 0, the result [38, Theorem 4.6] (combined with [39, Lemma 3.3] as indicated above) shows that the solution u to (2.16) with R replaced by R exists and satisfies…”

Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

“…Observation O2(d) can be partially explained from the fact that a larger number of the Laplace eigenfunctions of the ellipse E (1.2) (from which the quasimodes in Theorem 1.2 are constructed) are localised in the large cavity than in the small cavity. In the FEM case there is a close connection between the functions in the quasimodes and the eigenvalues of the Galerkin matrix (see [37]), and thus these localisation considerations immediately explain why the large cavity has more near-zero eigenvalues than the small cavity. However, in the BEM case it is less clear how the eigenvalues of 0 0.5 1 1.5 2 2.…”

“…As mentioned in §1.4, the connection between quasimodes and near-zero eigenvalues of the standard domain-based variational formulation (i.e. the basis of FEM) is much clearer than for BEM, and this is subject of the companion paper [37]. Indeed, [37,Theorem 1.4] proves that if k = k j , then there exists a nearzero eigenvalue of the standard domain-based variational formulation, with the distance of this eigenvalue from zero given in terms of the quality (k j ) of the quasimode.…”

“…the basis of FEM) is much clearer than for BEM, and this is subject of the companion paper [37]. Indeed, [37,Theorem 1.4] proves that if k = k j , then there exists a nearzero eigenvalue of the standard domain-based variational formulation, with the distance of this eigenvalue from zero given in terms of the quality (k j ) of the quasimode. Furthermore, [37,Theorem 1.7] shows that the eigenvalues inherit the multiplicities of the quasimodes.…”