Stability for the Electromagnetic Scattering from Large Cavities

Abstract

Consider the scattering of electromagnetic waves from a large rectangular cavity embedded in the infinite ground plane. There are two fundamental polarizations for the scattering problem in two dimensions: TM (transverse magnetic) and TE (transverse electric). In this paper, new stability results for the cavity problems are established for large rectangular shape cavities in both polarizations. For the TM cavity problem, an asymptotic property of the solution and a stability estimate with an improved dependence on the high wavenumber are derived. In the TE case, the first stability result is established with an explicit dependence on the wave number.

On Weak Observability for Evolution Systems with Skew-Adjoint Generators

“…Remark 5.1. We observe that the result of Theorem 5.2 based on clever analysis of Fourier series derived in [8], is indeed a particular case of Theorem 5.3 (α " 0 and β " π) obtained from Ingham type inequalities.…”

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

“…At least when the contrast in wave speeds inside and outside the obstacle is sufficiently large, [19, lemma 6.2] and [20, lemma 3.6] show that the scattered field everywhere outside the obstacle is polynomially bounded in k for k outside a set of small, finite measure; see Remark 3.10 below for more discussion on the results of [19,20]. (6) As noted in §1.1, when the obstacle O contains an ellipse-shaped cavity, the resolvent grows exponentially through a sequence k j (1.7); in this situation Theorem 1.1 implicitly contains information about the widths of the peaks in the norm of the resolvent at k j . We are not aware of any results in the literature about the widths of these peaks in the setting of obstacle scattering, but precise information about the widths and heights of peaks in the transmission coefficient for model resonance problems in one space dimension can be found in [1,105].…”

“…The following is a nonexhaustive list of papers on the frequency-explicit convergence analysis of numerical methods for solving the Helmholtz equation where a central role is played by either the nontrapping resolvent estimate (1.5) or its analogue (with the same k-dependence) for the commonly used approximation of the exterior problem where the exterior domain O g is truncated and an impedance boundary condition is imposed: In addition, the following papers focus on proving bounds on the solution of Helmholtz boundary value problems (with these bounds often called "stability estimates") motivated by applications in numerical analysis: [6, 7, 9, 11, 27-29, 37, 51, 55, 56, 58, 75, 88, 100, 109]. Of these papers, all but [6,11,28,75] are in nontrapping situations, [6,28,75] are in parabolic trapping scenarios, and [11] proves the exponential growth (1.7) under elliptic trapping.…”

An Adaptive Finite Element DtN Method for the Open Cavity Scattering Problems

“…As more people work on this subject, there has been a rapid development of the mathematical theory and computational methods for the open cavity scattering problems. The stability estimates with explicit dependence on the wavenumber were obtained in [9,10]. Various analytical and numerical methods have been proposed to solve the challenging large cavity problem [6,8,11,23,31].…”