Skin-Effect Description in Electromagnetism with a Scaled Asymptotic Expansion

We study a transmission problem in high contrast media. The 3-D case of the Maxwell equations in harmonic regime is considered. We derive an asymptotic expansion with respect to a small parameter δ > 0 related to high conductivity. This expansion is theoretically justified at any order. Numerical simulations highlight the skin-effect and the expansion accuracy. Introduction We consider the diffraction problem of waves by highly conducting materials in electromagnetism. The high conductivity reduces the penetration of the wave to a boundary layer, see [1]. The physical model is the following. Ωcd is an open bounded domain in R 3 with connected complement, occupied by a conducting medium. Ωcd is embedded in an insulating medium Ωis. We suppose that their common interface Σ is smooth. We define Ω = Ωcd ∪ Σ ∪ Ωis. We denote by δ a small parameter which is inversely proportional to the square root of the conductivity σ. The depth of the boundary layer is proportional to δ. We first give the formal construction of the asymptotic expansion. Then, we prove optimal error estimates. Finally, we present numerical simulations in axisymmetric geometry. 1 Scaled asymptotic expansion Eliminating the magnetic field Hδ from Maxwell equations, we perform a study in electric field Eδ. 1.1 Normal coordinates To describe the boundary layer in Ωcd, we define a local normal coordinate system (yα, y3), α ∈ {1, 2}, in a tubular neighborhood O of Σ, y3 ∈ (0, h0). The euclidian metric inO is denoted by (gij)i,j∈{1,2,3}. We adopt the tensorial calculus, see [2], to write Maxwell equations in these coordinates. The curl operator writes { (curl E) = 1 √ g (∂3Eβ − ∂βE3) (curl E)3 = 1 √ g 3αβDh αEβ with g = det(gij), ijk the Levi-Civita symbol, and D a covariant derivative defined for h = y3, see [2]. 1.2 Scaling and ansatz We perform the scaling Y3 = y3 δ in O, and expand the 3D Maxwell operator in power of series of δ. This leads to postulate the following expansions