Constructive Formulations of Resonant Maxwell's Equations

Abstract

In this article we propose new constructive weak formulations for resonant time-harmonic wave equations with singular solutions. Our approach follows the limiting absorption principle and combines standard weak formulations of PDEs with properties of elementary special functions adapted to the singularity of the solutions, called manufactured solutions. We show the well-posedness of several formulations obtained by these means for the limit problem in dimension one, and propose a generalization in dimension two.

A stable formulation of resonant Maxwell’s equations in cold plasma

“…This quantity is based on the divergence of the Poynting vector Π ν = Im(E ν × B ν ). As computed in [7], ∇ · Π ν = ν E ν 2 2 , so that…”

“…For that purpose we will use the classical framework [2] for mixed variational formulations. The most original tool in our approach will be the method of manufactured solutions recently introduced in [7] which is used to characterize the singular solutions within the mixed variational framework.…”

“…where the positivity of the limit is already an indication of its singular nature. We refer to [8,7] for additional mathematical results on the X-mode resonance, and to [6] for a numerical study.…”

Degenerate elliptic equations for resonant wave problems

“…Resonant waves appear in various electromagnetic phenomenons, such as: fusion plasma heating where a wave is sent inside a plasma and transfers energy to the particles in a localized region [9,7,19]; cloaking devices where the transition between metamaterials of negative index and non-dissipative dielectrics is exploited [1,6,17]; or photoacoustic imaging of biological tissues where metallic nanoparticles' heating [22] is used. The model problem for resonant waves throughout this work is the degenerate equation, elliptic where α < 0, − div (α∇u) − u = 0 in Ω ⊂ R 2 , α∂ n u + iλu = f on Γ, (1.1) where λ > 0 is a positive scalar and f ∈ L 2 (Γ) is complex valued on the boundary Γ = ∂Ω.…”

Solutions to the cold plasma model at resonances

“…In particular it is not possible to discuss boundary conditions at finite distance or at infinity with the techniques developed below. The reader interested by this topic may refer to [9], see also a recent alternative technique in [6].…”