BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: a-priori error estimates

TL;DR

This paper introduces a higher-order linear integrator for the Landau-Lifshitz-Gilbert equation in micromagnetics, providing optimal convergence rates and confirming these with numerical experiments.

Contribution

It presents the first higher-order-in-time, linear integrator for LLG that converges to both weak and strong solutions, with proven optimal error estimates.

Findings

Proves optimal-order convergence under regularity assumptions.

Demonstrates first-order spatial and second-order temporal convergence in numerical tests.

Provides an efficient scheme requiring only one linear solve per time step.

Abstract

We consider the Landau-Lifshitz-Gilbert equation (LLG), which models time-dependent micromagnetic phenomena. We analyze a fully discrete scheme that combines first-order finite elements in space with a BDF2 method in time. The method requires the solution of only one linear system of equations per time step and does not enforce the pointwise unit-length constraint of the magnetization. While unconditional weak convergence has been analyzed in an earlier work, we now prove optimal-order convergence rates under sufficient regularity assumptions on the exact solution and the external field. In combination with our previous work, this establishes the first higher-order-in-time and linear integrator that converges both to weak and strong solutions of LLG. Numerical experiments confirm first-order convergence in space and second-order convergence in time.