Publications

2020

• On the approximation of electromagnetic fields by edge finite elements. Part 3: sensitivity to coefficients
  
  • Ciarlet Patrick
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2020. In bounded domains, the regularity of the solutions to boundary value problems depends on the gometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces $H^s$, $s\in[0,1]$. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is to find some regularity exponent $\tau\in(0,1)$, such that they both belong to $H^s$, for all $s\in[0,\tau)$. After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N\'ed\'elec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations. (10.1137/19M1275383)
  DOI : 10.1137/19M1275383
• Spectral analysis of polygonal cavities containing a negative-index material
  
  • Hazard Christophe
  • Paolantoni Sandrine
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2020, 3, pp.1161-1193. The purpose of this paper is to investigate the spectral effects of an interface between vacuum and a negative-index material (NIM), that is, a dispersive material whose electric permittivity and magnetic permeability become negative in some frequency range. We consider here an elementary situation, namely, 1) the simplest existing model of NIM : the non dissipative Drude model, for which negativity occurs at low frequencies; 2) a two-dimensional scalar model derived from the complete Maxwell’s equations; 3) the case of a simple bounded cavity: a polygonal domain partially filled with a portion of Drude material. Because of the frequency dispersion (the permittivity and permeability depend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlinear eigenvalue problem. Thanks to the use of an additional unknown, we linearize the problem and we present a complete description of the spectrum. We show in particular that the interface between the NIM and vacuum is responsible for various resonance phenomena related to various components of an essential spectrum. (10.5802/ahl.58)
  DOI : 10.5802/ahl.58