Ventcel-type Transmission Conditions for the Scattering of a Time-harmonic Wave Problem with Accuracy up to Order $3$

Abstract

This work deals with the asymptotic behaviour of the electric field in the transverse magnetic (TM) mode, propagating in a bidimensional heterogeneous medium, composed by a homogeneous linear dielectric isotropic material surrounded by a thin layer of thickness $\varepsilon$ (destined to tend to $0$) and embedded in an ambient medium. Using the tools of multiscale analysis, an asymptotic expansion of the solution $u^{\varepsilon}$ to the Helmholtz problem with respect to the thickness $\varepsilon$ is derived. As a consequence, Ventcel-type transmission conditions on the limit interface $\Gamma$ are obtained modelling the effect of the thin layer with accuracy up to $O(\varepsilon^{3})$. A particular choice of the interface $\Gamma$ leads to a well-posed Ventcel's problem.