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Error analysis of scalar auxiliary variable finite element methods for the Landau--Lifshitz--Bloch equation

Agus L. Soenjaya

Abstract

The Landau--Lifshitz--Bloch (LLB) equation is a well-established micromagnetic model for describing magnetisation dynamics in ferromagnets at elevated temperatures. In this paper, we propose and analyse two fully discrete, conforming finite element schemes based on the scalar auxiliary variable (SAV) approach for solving the LLB equation in the high-temperature regime above the Curie point. The first scheme employs a semi-implicit Euler time discretisation, while the second is based on a linearly extrapolated BDF2 method. Both schemes are linear, unconditionally stable with respect to the energy norm, and satisfy a discrete energy law involving the SAV-based energy functional that approximates the true micromagnetic energy. Under suitable regularity assumptions, we establish unconditional energy stability and derive optimal-order error estimates in $\mathbb{L}^2, \mathbb{H}^1$, and $\mathbb{L}^\infty$ norms. To the best of our knowledge, this is the first rigorous error analysis of a fully discrete SAV-based method for a quasilinear vector-valued problem, as well as the first linear, energy-stable scheme for the LLB equation in the high-temperature regime that achieves second-order temporal accuracy.

Error analysis of scalar auxiliary variable finite element methods for the Landau--Lifshitz--Bloch equation

Abstract

The Landau--Lifshitz--Bloch (LLB) equation is a well-established micromagnetic model for describing magnetisation dynamics in ferromagnets at elevated temperatures. In this paper, we propose and analyse two fully discrete, conforming finite element schemes based on the scalar auxiliary variable (SAV) approach for solving the LLB equation in the high-temperature regime above the Curie point. The first scheme employs a semi-implicit Euler time discretisation, while the second is based on a linearly extrapolated BDF2 method. Both schemes are linear, unconditionally stable with respect to the energy norm, and satisfy a discrete energy law involving the SAV-based energy functional that approximates the true micromagnetic energy. Under suitable regularity assumptions, we establish unconditional energy stability and derive optimal-order error estimates in , and norms. To the best of our knowledge, this is the first rigorous error analysis of a fully discrete SAV-based method for a quasilinear vector-valued problem, as well as the first linear, energy-stable scheme for the LLB equation in the high-temperature regime that achieves second-order temporal accuracy.
Paper Structure (10 sections, 13 theorems, 116 equations, 6 figures, 2 algorithms)

This paper contains 10 sections, 13 theorems, 116 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. SAV formulations
  5. Finite element approximation
  6. SAV-FEM with first-order semi-implicit time discretisation
  7. SAV-FEM with linearised BDF2 time discretisation
  8. Numerical experiments
  9. Simulation 1
  10. Simulation 2

Key Result

Proposition 3.2

Let $\boldsymbol{u}_h^{n-1}\in \mathbb{V}_h$ and $r_h^{n-1}\in \mathbb{R}$ be given. For sufficiently small $k>0$, there exists a unique $\boldsymbol{u}_h^n\in \mathbb{V}_h$ and $r_h^n\in \mathbb{R}$ solving the system equ:fem euler--equ:fem be Hhn.

Figures (6)

  • Figure 1: Snapshots of the spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) for scheme \ref{['equ:fem euler']} in Simulation 1.
  • Figure 2: Snapshots of the spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) for scheme \ref{['equ:fem cn']} in Simulation 2.
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (27)

  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • ...and 17 more