Finite Element Formalism for Micromagnetism

TL;DR

This paper introduces a finite element method for solving the Landau-Lifschitz-Gilbert equations, enabling accurate modeling of micromagnetic problems with complex geometries, validated against finite difference solutions.

Contribution

The authors develop and validate a Galerkin-type finite element approach specifically tailored for 2D micromagnetic simulations involving complex geometries.

Findings

Finite element results agree well with finite difference solutions.

The method effectively handles complex geometries in 2D micromagnetism.

The approach is suitable for detailed micromagnetic modeling.

Abstract

The aim of this work is to present the details of the finite element approach we developed for solving the Landau-Lifschitz-Gilbert equations in order to be able to treat problems involving complex geometries. There are several possibilities to solve the complex Landau-Lifschitz-Gilbert equations numerically. Our method is based on a Galerkin-type finite element approach. We start with the dynamic Landau-Lifschitz-Gilbert equations, the associated boundary condition and the constraint on the magnetization norm. We derive the weak form required by the finite element method. This weak form is afterwards integrated on the domain of calculus. We compared the results obtained with our finite element approach with the ones obtained by a finite difference method. The results being in very good agreement, we can state that our approach is well adapted for 2D micromagnetic systems.