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 blowup behavior and to study combined electromagnetic phenomena in ferromagnets for benchmark problems. © 2008 Society for Industrial and Applied Mathematics.