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

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