Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition
A study is done of 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 \(\nu \times H=\nu \times (|E\times \nu |^{\alpha - 1}E\times \nu )\) for some \(\alpha \in (0,1]\). The existence and uniqueness of a weak solution are established in a suitable function spaces under the minimal regularity assumptions on the boundary \(\Gamma \) and the initial data \(E_{0}\) and \(H_{0}\). A non-linear time discrete approximation scheme is designed and convergence of the approximations to a weak solution is proved. The error estimates are derived for the time discretization. As a next step the fully discrete problem is studied using curl-conforming edge elements and the corresponding error estimates are derived. Finally some numerical experiments are presented.
nonlinear Silver-Müller boundary condition
scientific article; zbMATH DE number 5835293
scientific article