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

TL;DR

This work establishes the well-posedness and numerical approximation of the Landau–Lifshitz–Bloch equation on bounded polytopal domains with $d\le 3$, introducing a linear fully discrete FEM for direct LLBE approximation and a viscous regularisation $\varepsilon$-LLBE to recover optimal convergence. It proves existence and uniqueness of strong solutions for LLBE and $\varepsilon$-LLBE, together with decay properties that reflect the physics above the Curie temperature. By coupling a stable linear FEM with a theoretical analysis of the regularised problem, the authors obtain stability and (uniform-in-time) error estimates, and demonstrate convergence as $\varepsilon\to 0^+$ and as $h,k\to 0$. The paper further validates the theory through numerical simulations on convex and nonconvex polytopal domains in 2D and 3D, including domains with re-entrant corners, confirming long-time decay and the impact of geometry on convergence rates.

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 $ε$-LLBE. For this regularised problem, we prove the unique existence of strong solutions and establish a rate of convergence of the solution $\boldsymbol{u}^ε$ of the $ε$-LLBE to the solution $\boldsymbol{u}$ of the LLBE as $ε\to 0^+$. Furthermore, we propose a linear, fully discrete, conforming finite element scheme to approximate the solution of the $ε$-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.

Figures (10)

• Figure 1: Snapshots of the magnetic spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) at given times for simulation 1. The colours indicate relative magnitude of the vectors.
• Figure 2: Snapshots of the magnetic spin field $\boldsymbol{u}$ at given times for simulation 2. The colours indicate relative magnitude of the vectors.
• Figure 3: Order of convergence for simulation 3.
• Figure 4: Snapshots of the magnetic spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) at given times for simulation 4. The colours indicate relative magnitude of the vectors.
• Figure 5: Order of convergence for simulation 4.

Theorems & Definitions (61)

• Definition 2.1: solution of the $\epsilon$-LLBE
• Definition 2.2: solution of the LLBE
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof