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

Abstract

International audienceWe 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 $\texttt{T}$-coercivity approach, we first prove that the problem is well-posed in the classical framework $H^1_{loc}$ 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. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we 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 $\textit{ad hoc}$ technique because the field is too singular to be captured with standard finite element methods

Mesh requirements for the finite element approximation of problems with sign-changing coefficients

“…This problem has been investigated in [4] for particular geometries: it requires a new functional framework explicitly taking into account the singularities. For a general polygonal interface, a new numerical approximation method is proposed in [3], which relies on the use of Perfectly Matched Layers at the corners.…”

“…Using proposition 1 one can deduce all the above results. Let us simply detail how to obtain the first result of [3], i.e. when q ≥ p. First consider the case where q = 2p + m, m ∈ 1, p .…”

An Extended Charge-Current Formulation of the Electromagnetic Transmission Problem

“…In all examples we take k 1 real and positive, ε 1 = 1, and ε 2 = −1.1838. This parameter combination satisfies the plasmonic condition (1) and has been used in previous work on 2D surface plasmon waves [2,10,11]. In situations involving non-smooth surfaces, it may happen that solutions for ε 2 = −1.1838 do not exist.…”

“…A number of 800 discretization points is placed on Γ and the performance of the three systems (85), (87), (88) are compared. Figure 5(a) shows the total magnetic field H ↓ (r, t) at t = 0, see (2), and Figures 5(b,c,d) show log 10 of the estimated absolute error obtained with (85), (87), and (88), respectively. The number of GMRES iterations required to solve the discretized linear systems is 266 for (85), 154 for (87), and 143 for (88).…”

On a Helmholtz transmission problem in planar domains with corners

“…We call the former type true eigenwavenumbers and the latter type false eigenwavenumbers. False eigenwavenumbers that correspond to eigenfields that violate (5) can only occur for the KM2 equations since U (r) in the KM0 and KM1 equations, by construction, always satisfies (5).…”