Equivalent transmission conditions for the time-harmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet

Victor Péron (1, 2) , Kersten Schmidt (3) , Marc Duruflé (2, 4)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
3 Institut für Mathematik [Berlin]
4 IMB - Institut de Mathématiques de Bordeaux

We propose equivalent transmission conditions of order 1 and 2 for thin and highly conducting sheets for the time-harmonic Maxwell's equation in three dimension. The transmission conditions are derived asymptotically for vanishing sheet thickness ε where the skin depth is kept proportional to ε. The condition of order 1 turns out to be the perfect electric conductor boundary condition. The conditions of order 2 appear as generalised Poincaré-Steklov maps between tangential components of the magnetic field and the electric field, and they are of Wentzell type involving second order surface differential operators. Numerical results with finite elements of higher order validate the asymptotic convergence for ε → 0 and the robustness of the equivalent transmission condition of order 2.

Keywords: time-harmonic Maxwell's equations; Asymptotic expansions; impedance transmission conditions; thin conducting sheets

Domains: Analysis of PDEs [math.AP]

Victor Péron, Kersten Schmidt, Marc Duruflé. Equivalent transmission conditions for the time-harmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet. SIAM Journal on Applied Mathematics, 2016, 76 (3), pp.1031--1052. ⟨10.1137/15M1012116⟩. ⟨hal-01260111v2⟩