Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition

Abstract

We study a time dependent eddy current equation for the magnetic field H accompanied with a non-linear degenerate boundary condition (BC), which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electrical E and magnetic H fields obeys the following power law ν × E = ν × ( | H × ν | α - 1 H × ν ) for some α ∈ ( 0 , 1 ] . We establish the existence and uniqueness of a weak solution in a suitable function space under the minimal regularity assumptions on the boundary Γ and the initial data H 0 . We design a non-linear time discrete approximation scheme based on Rothe's method and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time-discretization.