A finite element based Heterogeneous Multiscale Method for the Landau-Lifshitz equation

TL;DR

This paper introduces a multiscale finite element method combining macro finite elements and micro finite differences to efficiently simulate ferromagnetic composites with rapidly oscillating material properties.

Contribution

It develops a novel heterogeneous multiscale approach for the Landau-Lifshitz equation that effectively captures small-scale oscillations without high computational costs.

Findings

Successfully approximates solutions with rapid material variations

Reduces computational expense compared to traditional methods

Applicable to ferromagnetic composite modeling

Abstract

We present a Heterogeneous Multiscale Method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, a simple model for a ferromagnetic composite. A finite element macro scheme is combined with a finite difference micro model to approximate the effective equation corresponding to the original problem. This makes it possible to obtain effective solutions to problems with rapid material variations on a small scale, described by $\varepsilon \ll 1$, which would be too expensive to resolve in a conventional simulation.