Equivalent Transmission Conditions for the Time-Harmonic Maxwell Equations in 3D for a Medium with a Highly Conductive Thin Sheet

Source abstract

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 dimensions. The transmission conditions are derived asymptotically for vanishing sheet thickness $\varepsilon$ where the skin depth is kept proportional to $\varepsilon$. The condition of order 1 turns out to be the perfect electric conductor boundary condition. The conditions of order 2 appear as generalized 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 $\varepsilon\to0$ and the robustness of the equivalent transmission condition of order 2.