A convergent finite element approximation for the quasi-static Maxwell--Landau--Lifshitz--Gilbert equations
Provides a convergent numerical method for a challenging system of PDEs in micromagnetics, but the result is incremental as it extends existing linearization techniques to this specific coupled problem.
The authors propose a θ-linear finite element scheme for the quasi-static Maxwell-Landau-Lifshitz-Gilbert equations, achieving a linear system per time step despite strong nonlinearity. They prove convergence to a weak solution as discretization parameters tend to zero for θ in (1/2,1].
We propose a $θ$-linear scheme for the numerical solution of the quasi-static Maxwell-Landau-Lifshitz-Gilbert (MLLG) equations. Despite the strong nonlinearity of the Landau-Lifshitz-Gilbert equation, the proposed method results in a linear system at each time step. We prove that as the time and space steps tend to zero (with no further conditions when $θ\in(1/2,1]$), the finite element solutions converge weakly to a weak solution of the MLLG equations. Numerical results are presented to show the applicability of the method.