On the Landau-Lifshitz-Gilbert equation with magnetostriction

TL;DR

This paper develops a new numerical method for simulating the coupled Landau-Lifshitz-Gilbert and momentum equations with magnetostriction, proving convergence and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces an alternative proof of existence and a decoupled, efficient numerical integrator for the coupled system including magnetostrictive effects.

Findings

The integrator solves only two linear systems per timestep.

The method converges unconditionally as mesh and timestep sizes go to zero.

Numerical experiments reveal insights into blow-up phenomena in micromagnetic simulations.

Abstract

To describe and simulate dynamic micromagnetic phenomena, we consider a coupled system of the nonlinear Landau-Lifshitz-Gilbert equation and the conservation of momentum equation. This coupling allows to include magnetostrictive effects into the simulations. Existence of weak solutions has recently been shown in [Carbout et al. 2011]. In our contribution, we give an alternate proof which additionally provides an effective numerical integrator. The latter is based on lowest-order finite elements in space and a linear-implicit Euler time-stepping. Despite the nonlinearity, only two linear systems have to be solved per timestep, and the integrator fully decouples both equations. Finally, we prove unconditional convergence---at least of a subsequence---towards, and hence existence of, a weak solution of the coupled system, as timestep size and spatial mesh-size tend to zero. Numerical experiments conclude the work and shed new light on the existence of blow-up in micromagnetic simulations.