The Landau--Lifshitz--Bloch equation on polytopal domains: Unique existence and finite element approximation

TL;DR

This paper analyzes the Landau--Lifshitz--Bloch equation on polytopal domains, proving existence, uniqueness, and convergence of finite element schemes, and introduces a regularized version to improve approximation accuracy.

Contribution

It establishes the well-posedness of the LLBE, proposes finite element schemes for both the original and regularized equations, and provides convergence and error analysis with numerical validation.

Findings

Existence and uniqueness of strong solutions to LLBE.

Convergence of the finite element scheme for the regularized LLBE.

Numerical results confirming theoretical error estimates.

Abstract

The Landau--Lifshitz--Bloch equation (LLBE) describes the evolution of the magnetic spin field in ferromagnets at high temperatures. In this paper, we study the numerical approximation of the LLBE on bounded polytopal domains in $\mathbb{R}^d$, where $d\le 3$. We first establish the existence and uniqueness of strong solutions to the LLBE and propose a linear, fully discrete, conforming finite element scheme for its approximation. While this scheme is shown to converge, the obtained rate is suboptimal. To address this shortcoming, we introduce a viscous (pseudo-parabolic) regularisation of the LLBE, which we call the $\epsilon$-LLBE. For this regularised problem, we prove the unique existence of strong solutions and establish a rate of convergence of the solution $\boldsymbol{u}^\epsilon$ of the $\epsilon$-LLBE to the solution $\boldsymbol{u}$ of the LLBE as $\epsilon\to 0^+$. Furthermore, we propose a linear, fully discrete, conforming finite element scheme to approximate the solution of the $\epsilon$-LLBE. Given sufficiently smooth initial data, error analysis is performed to show stability and uniform-in-time convergence of the scheme. Finally, several numerical simulations are presented to corroborate our theoretical results.