For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

Abstract

It is well‐known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial‐growth assumption on the solution operator (e.g. hp‐finite elements, hp‐boundary elements, and certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest possible trapping, for most frequencies. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method

“…In the setting of impenetrable-obstacle problems, this assumption holds with M = 0 and H = (0, 0 ] when the problem is nontrapping (see Theorem 1.5 below and the references therein). In the black-box setting, [41] proved that this assumption always holds with M > 5n/2 and the complement of the set H having arbitrarily-small measure.…”

“…with appropriate domain as a self-adjoint operator (on a space weighted by c −2 ), the functional calculus for self-adjoint operators allows us to define f (P ) for a broad class of functions f. In particular, given k > 0, we are interested in taking f a cutoff function of the real axis equal to 1 on B(0, µk) for some µ > 1. Then for fixed k, (1 − f )(P ) is a high-frequency cutoff and f (P ) a low-frequency cutoff; in the special case A = I, c = 1, these are simply Fourier multipliers of the type used in [41].…”

“…The proof of (BB2), (BB3), and (BB4) is essentially the same in the present semiclassicallyscaled setting. The proof of the bound (BB5) is similar to the the analogous proof for c = 1 and A Lipschitz in[41, Lemma B.1]; for completeness we include the proof in §B.…”

“…• the Helmholtz interior impedance problem where the domain is either analytic (d = 2, 3) [52, Theorem 4.10], [50,Theorem 4.5], or polygonal [52,Theorem 4.10], [25,Theorem 3.2], in all cases under an assumption that the solution operator grows at most polynomially in k (which has recently been shown to hold, for most frequencies, for a variety of scattering problems by [41]). These decompositions have had a large impact in the numerical analysis of the Helmholtz equation in that they allow one to prove convergence, explicit in the frequency k, of so-called hpfinite-element methods (hp-FEM) applied to discretisations of the Helmholtz equation.…”

Multiscale scattering in nonlinear Kerr-type media

“…Note that several works consider the dependence of C stab,0 (k) on the wave number k, also in the present setup of heterogeneous coefficients A and n, see, e.g., [BGP17, GPS19, MS19, ST18] and the references therein. For instance, [GPS19] proves that C stab,0 (k) 1 under certain conditions on the Lipschitz coefficients A and n. The crucial point is to exclude the existence of so-called trapped rays in the setup, see the discussion and references in [GPS19,LSW20].…”