Error Estimates for the Time Discretization for Nonlinear Maxwell's Equations

Abstract

This paper is devoted to the study of a nonlinear evolution eddy current model of the type $\partial_t \boldsymbol{B}(\boldsymbol{H})+∇ × (∇ × \boldsymbol{H}) = 0$ subject to homogeneous Dirichlet boundary conditions $\boldsymbol{H} × ν=0$ and a given initial datum. Here, the magnetic properties of a soft ferromagnet are linked by a nonlinear material law described by $\boldsymbol{B}(\boldsymbol{H})$. We apply the backward Euler method for the time discretization and we derive the error estimates in suitable function spaces. The results depend on the nonlinearity of $\boldsymbol{B}(\boldsymbol{H})$.