Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition

We study a full Maxwell's system accompanied with a non-linear degenerate boundary condition, which represents a generalization of the classical Silver–Müller condition for a non-perfect conductor. The relationship between the normal components of electric E and magnetic H field obeys the following power law ν×H=ν×(|E×ν|α−1E×ν) for some α∈(0,1]. We establish the existence and uniqueness of a weak solution in a suitable function spaces under the minimal regularity assumptions on the boundary Γ and the initial data E0 and H0. We design a non-linear time discrete approximation scheme and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time discretization. As a next step we study the fully discrete problem using curl-conforming edge elements and derive the corresponding error estimates. Finally we present some numerical experiments.