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

TL;DR

This paper develops and analyzes two implicit finite element schemes for the inertial Landau-Lifshitz-Gilbert equation, enabling accurate simulation of ultrafast magnetic dynamics at subpicosecond scales.

Contribution

It introduces two novel fully discrete numerical methods for iLLG and proves their convergence, advancing computational modeling of ultrafast magnetic phenomena.

Findings

Both schemes satisfy the unit-length constraint at mesh vertices.

Numerical experiments confirm convergence and applicability.

Methods effectively simulate ultrafast magnetic processes.

Abstract

We consider the numerical approximation of the inertial Landau-Lifshitz-Gilbert equation (iLLG), which describes the dynamics of the magnetization in ferromagnetic materials at subpicosecond time scales. We propose and analyze 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 magnetization 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.