Fully discrete finite element method for Maxwell's equations with nonlinear conductivity

In this paper we present a numerical scheme to solve nonlinear Maxwell's equations based on backward Euler discretization in time and curl-conforming finite elements in space. The nonlinearity is due to a field-dependent conductivity in the form of a power law. The system under study is hyperbolic and due to the nonlinear conductivity it lacks strong estimates of the second time derivative. We are able to prove convergence of our numerical scheme based on boundedness of the second derivative in the dual space. Convergence of the nonlinear term is based on the Minty-Browder technique. We also present the error estimate for the fully discretized problem and support the theory by some numerical experiments.