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

We consider the numerical approximation of the inertial Landau-Lifshitz-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.

Automated summary

This paper focuses on the numerical approximation of the inertial Landau-Lifshitz-Gilbert equation (iLLG) and proposes and analyzes two fully discrete numerical schemes. These schemes are implicit and generate approximations that satisfy the unit-length constraint of the iLLG equation. Convergence of the approximations towards a weak solution of the problem is proven, and numerical experiments validate the theoretical results and demonstrate the applicability of the methods for simulating ultrafast magnetic processes.