Mixed finite element methods for the Landau--Lifshitz--Baryakhtar and the regularised Landau--Lifshitz--Bloch equations in micromagnetics

TL;DR

This paper develops and analyzes mixed finite element methods for the LLBar and LLBloch equations in micromagnetics, providing stable, accurate, and energy-preserving numerical schemes with proven convergence rates.

Contribution

It introduces unified semi-discrete and fully discrete mixed finite element schemes for LLBar and LLBloch equations, with energy stability and optimal error estimates.

Findings

Schemes are unconditionally energy-stable.

Optimal convergence rates are proven.

Numerical experiments confirm theoretical results.

Abstract

The Landau--Lifshitz--Baryakhtar (LLBar) and the Landau--Lifshitz--Bloch (LLBloch) equations are nonlinear vector-valued PDEs which arise in the theory of micromagnetics to describe the dynamics of magnetic spin field in a ferromagnet at elevated temperatures. We consider the LLBar and the regularised LLBloch equations in a unified manner, thus allowing us to treat the numerical approximations for both problems at once. In this paper, we propose a semi-discrete mixed finite element scheme and two fully discrete mixed finite element schemes based on a semi-implicit Euler method and a semi-implicit Crank--Nicolson method to solve the problems. These numerical schemes provide accurate approximations to both the magnetisation vector and the effective magnetic field. Moreover, they are proven to be unconditionally energy-stable and preserve energy dissipativity of the system at the discrete level. Error analysis is performed which shows optimal rates of convergence in $\mathbb{L}^2$, $\mathbb{L}^\infty$, and $\mathbb{H}^1$ norms. These theoretical results are further corroborated by several numerical experiments.