BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: unconditional weak convergence to weak solutions

TL;DR

This paper introduces a BDF2-type integrator for the Landau-Lifshitz-Gilbert equation in micromagnetics, proving unconditional weak convergence to weak solutions and demonstrating its stability and accuracy through numerical experiments.

Contribution

The paper develops a new BDF2-type integrator for LLG that is unconditionally stable and converges weakly to solutions, with verified first-order spatial and second-order temporal accuracy.

Findings

Unconditionally stable integrator for LLG

Weak convergence to weak solutions

Numerical verification of order accuracy

Abstract

We consider the Landau-Lifshitz-Gilbert equation (LLG) that models time-dependent micromagnetic phenomena. We propose a full discretization that employs first-order finite elements in space and a BDF2-type two-step method in time. In each time step, only one linear system of equations has to be solved. We employ linear interpolation in time to reconstruct the discrete space-time magnetization. We prove that the integrator is unconditionally stable and thus guarantees that a subsequence of the reconstructed magnetization converges weakly in $H^1$ towards a weak solution of LLG in the space-time domain. Numerical experiments verify that the proposed integrator is indeed first-order in space and second-order in time.