Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation

TL;DR

This paper introduces a mass-lumped finite element method for the Landau-Lifshitz equation that maintains the magnetization constraint and energy stability, with proven convergence to weak solutions and validated through numerical tests.

Contribution

It presents a novel mass-lumped finite element scheme that preserves the unit-length constraint and energy properties, with a simple convergence proof.

Findings

Method preserves the nonconvex constraint at each node.

Numerical tests confirm convergence on structured and unstructured meshes.

The scheme is energy nonincreasing and applicable to explicit and implicit formulations.

Abstract

The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.