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Section: New Results

Direct scattering problems

A numerical method to approximate black hole singularities in presence of metamaterials

L. Chesnel, A.-S. Bonnet-Ben Dhia, C. Carvalho and P. Ciarlet.

We investigate in a 2D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non dissipative bounded and penetrable obstacle with a negative permittivity. Using the $𝚃$-coercivity approach, we proved that the problem is well-posed in the classical frameworks if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. We give a simple and systematic criterion to define what is the outgoing solution. We also propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods.

Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell’s Equations

Houssem Haddar, Shixu Meng and Fioralba Cakoni

We consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of boundary integral equa- tion, and assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point.

A Volume integral method for solving scattering problems from locally perturbed periodic layers

Houssem Haddar and Thi Phong Nguyen

We investigate the scattering problem for the case of locally perturbed periodic layer in $R^N$ $(N=2,3)$. Using Floquet-Bloch transform in $x_1-$direction we reformulate this scattering problem as an equivalent system of coupled volume integral equations. Using periodization in the $x_2-$direction we apply a spectral method to discretize the problem and compute a numerical approximation of the solution. The convergence of this method is established and numerical validating results are conducted.