A convergent finite element approximation for the quasi-static Maxwell--Landau--Lifshitz--Gilbert equations

Abstract: We propose a $\theta$-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 $\theta\in(1/2,1]$), 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.