Multiscale modeling in micromagnetics: existence of solutions and numerical integration

TL;DR

This paper develops a multiscale model combining Landau-Lifshitz-Gilbert and Maxwell equations for micromagnetics, proving the existence of solutions and providing a convergent numerical integrator for complex magnetic systems.

Contribution

It introduces a multiscale approach for micromagnetics, proving weak solution existence and designing a numerical scheme with proven convergence.

Findings

Existence of weak solutions for the multiscale micromagnetic model.

A linear-implicit numerical integrator with proven $H^1$-convergence.

Applicability to complex magnetic systems across scales.

Abstract

Various applications ranging from spintronic devices, giant magnetoresistance sensors, and magnetic storage devices, include magnetic parts on very different length scales. Since the consideration of the Landau-Lifshitz-Gilbert equation (LLG) constrains the maximum element size to the exchange length within the media, it is numerically not attractive to simulate macroscopic parts with this approach. On the other hand, the magnetostatic Maxwell equations do not constrain the element size, but cannot describe the short-range exchange interaction accurately. A combination of both methods allows to describe magnetic domains within the micromagnetic regime by use of LLG and also considers the macroscopic parts by a non-linear material law using the Maxwell equations. In our work, we prove that under certain assumptions on the non-linear material law, this multiscale version of LLG admits weak solutions. Our proof is constructive in the sense that we provide a linear-implicit numerical integrator for the multiscale model such that the numerically computable finite element solutions admit weak $H^1$-convergence (at least for a subsequence) towards a weak solution.