A Convergent Implicit Finite Element Discretization of the Maxwell–Landau–Lifshitz–Gilbert Equation

We propose an implicit, fully discrete scheme for the numerical solution of the Maxwell–Landau–Lifshitz–Gilbert equation which is based on linear finite elements and satisfies a discrete sphere constraint as well as a discrete energy law. As numerical parameters tend to zero, solutions weakly accumulate at weak solutions of the Maxwell–Landau–Lifshitz–Gilbert equation. A practical linearization of the nonlinear scheme is proposed and shown to converge for certain scalings of numerical parameters. Computational studies are presented to indicate finite-time blow-up behavior and to study combined electromagnetic phenomena in ferromagnets for benchmark problems.