Lucas Chesnel

RADIATION CONDITION FOR A NON-SMOOTH INTERFACE BETWEEN A DIELECTRIC AND A METAMATERIAL

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in H1. This is due to the degeneration of the two dual singularities which then behave like r±iη = e±iη ln r with η ∈ ℝ*. This apparition of propagative singularities is very similar to the apparition of propagative modes in a waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the wavenumber. In this work, we derive for our problem a functional framework by adding to H1 one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media.

T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients

To solve variational indefinite problems, one uses classically the Banach–Nečas–Babuška theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem

Interior transmission eigenvalue problem for Maxwell's equations: the T-coercivity as an alternative approach

\text{ div}\sigma \nabla u=f

Compact Imbeddings in Electromagnetism with Interfaces between Classical Materials and Metamaterials

set in

On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients

H^1_0

Perfect transmission invisibility for waveguides with sound hard walls

, with