Numerical analysis of the Landau–Lifshitz–Gilbert equation with inertial effects

We consider the numerical approximation of inertial Landau–Lifshitz–Gilbert equation (iLLG), which describes dynamics magnetisation in ferromagnetic materials at subpicosecond time scales. propose and analyse two fully discrete schemes: The first method is based on a reformulation problem as linear constrained variational formulation for velocity. second exploits order system angular momentum. Both schemes are implicit, first-order finite elements, generate approximations satisfying unit-length constraint iLLG vertices underlying mesh. For both methods, we prove convergence towards weak solution problem. Numerical experiments validate theoretical results show applicability methods simulation ultrafast magnetic processes.