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

Agus L. Soenjaya

TL;DR

The paper advances numerical analysis for micromagnetic models by presenting unified LLBar and LLBloch dynamics and devising mixed finite element schemes that discretise both the magnetisation and the effective field. It develops a semi-discrete Galerkin method and two fully discrete schemes (semi-implicit Euler and Crank–Nicolson) that are unconditionally energy-stable and dissipative at the discrete level, with rigorous convergence results in $L^2$, $L^ ext{infty}$, and $H^1$ norms. It also establishes a robust connection between LLBar and the regularised LLBloch equation in the regime $oldsymbol{ u}_{e} o 0^+$ and outlines how the schemes extend to the LLBloch setting. Numerical simulations in a 2D domain implemented in FEniCS corroborate the theory, showing energy decay, convergence behavior, and micromagnetic features such as Bloch walls. Overall, the work provides a rigorous, energy-stable, and accurate computational framework for high-temperature micromagnetics modeling.

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.

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

TL;DR

The paper advances numerical analysis for micromagnetic models by presenting unified LLBar and LLBloch dynamics and devising mixed finite element schemes that discretise both the magnetisation and the effective field. It develops a semi-discrete Galerkin method and two fully discrete schemes (semi-implicit Euler and Crank–Nicolson) that are unconditionally energy-stable and dissipative at the discrete level, with rigorous convergence results in , , and norms. It also establishes a robust connection between LLBar and the regularised LLBloch equation in the regime and outlines how the schemes extend to the LLBloch setting. Numerical simulations in a 2D domain implemented in FEniCS corroborate the theory, showing energy decay, convergence behavior, and micromagnetic features such as Bloch walls. Overall, the work provides a rigorous, energy-stable, and accurate computational framework for high-temperature micromagnetics modeling.

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 , , and norms. These theoretical results are further corroborated by several numerical experiments.
Paper Structure (14 sections, 32 theorems, 339 equations, 7 figures)

This paper contains 14 sections, 32 theorems, 339 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Assumptions
  5. Finite Element Approximation
  6. Auxiliary Results
  7. A Semi-discrete Galerkin Approximation
  8. A Fully Discrete Scheme Based on the Semi-Implicit Euler Method
  9. A Fully Discrete Scheme Based on the Crank--Nicolson Method
  10. Approximation of the Regularised LLBloch Equation
  11. Numerical Simulations
  12. Simulation 1 (LLBar with $\mu>0$)
  13. Simulation 2 (Regularised LLBloch with $\mu<0$ and small $\lambda_e$)
  14. Simulation 3 (Energy dissipativity)

Key Result

Lemma 2.1

Let $\epsilon>0$ be given. Then there exists a positive constant $C$ depending only on $\mathscr{D}$ such that the following inequalities hold.

Figures (7)

  • Figure 1: Snapshots of the spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) for simulation 1.
  • Figure 2: Snapshots of the effective field $\boldsymbol{H}$ (projected onto $\mathbb{R}^2$) for simulation 1.
  • Figure 3: Snapshots of the spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) for simulation 2.
  • Figure 4: Snapshots of the effective field $\boldsymbol{H}$ (projected onto $\mathbb{R}^2$) for simulation 2.
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (66)

  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 56 more