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.

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