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

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