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

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

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

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.

Mots clés: Impedance Transmission Conditions; Thin Conducting Sheets; Asymptotic Expansions; Time-harmonic Maxwell's equations

Domaines: Equations aux dérivées partielles [math.AP]

Marc Duruflé, Victor Péron, Kersten Schmidt. Equivalent transmission conditions for the time-harmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet. 2015. ⟨hal-01260111v1⟩