Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation

Abstract

This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the time regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latter is frequency-dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well controlled. Math. Scand. 20 (1967) 65-78]. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.

Approximate cloaking for time-dependent Maxwell equations via transformation optics

“…After obtaining appropriate estimates on the near invisibility of the Maxwell equations in the time harmonic regime, we simply invert the Fourier transform. This idea has its roots in the work of Nguyen and Vogelius [35] (see also [36]) in the context of acoustic cloaking and was used to study impedance boundary conditions in the time domain [38] and cloaking for the heat equation [32]. To implement this idea, the heart of the matter is to obtain the degree of visibility in which the dependence on frequency is explicit and well controlled.…”

Approximate Cloaking Using Transformation Optics for Acoustic and Electromagnetic Waves

“…As an important technical point we need to establish that the Fourier transforms of solutions to the wave equation (with respect to time) satisfy the outgoing radiation conditions. This fact is of independent interest, and very useful in the study of various problems in time domain, since it allows one directly to make use of knowledge from the frequency domain, see, e.g., [36] and [29].…”

Approximate cloaking for the full wave equation via change of variables: The Drude–Lorentz model

“…To obtain the estimates in time domain from the estimates in frequency domain, we proceed in a similar, but slightly different way than [20]. We use a simple and helpful idea, also used in [19], by establishing estimates for the difference of the time derivatives of u c and u not for their difference. As a consequence, we avoid the non-standard estimates for very low frequency in [20, Section 2.2]; their proof involved the theory of H-convergence.…”

“…As mentioned earlier, another element of our analysis is the (definition of and) verification of well-posedness of u c (Proposition 1). For this purpose we rely on a non-trivial energy estimate, in the spirit of [19].…”