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Well-Posedness and Finite Element Approximation for the Landau-Lifshitz-Gilbert Equation with Spin-Torques

Noah Vinod, Thanh Tran

TL;DR

The paper addresses the NHLLG equation with spin-torque terms by deriving local-in-time existence and uniqueness of strong solutions via a Faedo-Galerkin framework, and develops a convergent finite-element scheme in space with a theta-time discretization that converges to a global weak solution in 2D and 3D. It provides a rigorous convergence analysis for the numerical method, including compactness arguments and convergence of nonlinear terms, and demonstrates the approach through numerical experiments with STT and SOT torques. The results offer a mathematically solid link between well-posedness and computability for NHLLG models, underpinning reliable simulations of spin-torque effects in ferromagnets. The practical impact lies in guaranteeing well-posedness and providing a provably convergent numerical tool for simulating current-driven magnetization dynamics in realistic geometries.

Abstract

Spin currents act on ferromagnets by exerting a torque on the magnetisation. This torque is modelled by appending additional terms to the Landau-Lifshitz-Gilbert equation motivating the study of the non-homogeneous Landau-Lifshitz-Gilbert equation. We first prove the existence and uniqueness of high regularity local solutions to this equation using the Faedo-Galerkin method. Then we construct a numerical method for the problem and prove that it converges to a global weak solution of the PDE. Numerical simulations of the problem are also included.

Well-Posedness and Finite Element Approximation for the Landau-Lifshitz-Gilbert Equation with Spin-Torques

TL;DR

The paper addresses the NHLLG equation with spin-torque terms by deriving local-in-time existence and uniqueness of strong solutions via a Faedo-Galerkin framework, and develops a convergent finite-element scheme in space with a theta-time discretization that converges to a global weak solution in 2D and 3D. It provides a rigorous convergence analysis for the numerical method, including compactness arguments and convergence of nonlinear terms, and demonstrates the approach through numerical experiments with STT and SOT torques. The results offer a mathematically solid link between well-posedness and computability for NHLLG models, underpinning reliable simulations of spin-torque effects in ferromagnets. The practical impact lies in guaranteeing well-posedness and providing a provably convergent numerical tool for simulating current-driven magnetization dynamics in realistic geometries.

Abstract

Spin currents act on ferromagnets by exerting a torque on the magnetisation. This torque is modelled by appending additional terms to the Landau-Lifshitz-Gilbert equation motivating the study of the non-homogeneous Landau-Lifshitz-Gilbert equation. We first prove the existence and uniqueness of high regularity local solutions to this equation using the Faedo-Galerkin method. Then we construct a numerical method for the problem and prove that it converges to a global weak solution of the PDE. Numerical simulations of the problem are also included.
Paper Structure (14 sections, 22 theorems, 126 equations, 3 figures)

This paper contains 14 sections, 22 theorems, 126 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence and Uniqueness
  4. Faedo-Galerkin Method
  5. A priori Estimates
  6. Proof of Theorem \ref{['exist-thm']}
  7. Proof of Theorem \ref{['unique-thm']}
  8. Numerical Method
  9. Proof of Theorem \ref{['thm:numerical']}
  10. Application and Numerical Experiments
  11. Example 1
  12. Example 2
  13. Numerical Experiments
  14. Useful Results

Key Result

Lemma 2.1

Let $\Omega$ be a bounded domain of $\mathbb{R}^d$, $d=1,2,3$, with smooth boundary. Then for all $\hbox{\boldmath $u$} \in \mathbb{H}^2(\Omega)$ such that $\frac{\partial \hbox{\boldmath $u$}}{\partial\hbox{\boldmath $n$}} = \hbox{\boldmath $0$}$ on $\partial \Omega$, Furthermore, for $\hbox{\boldmath $u$} \in \mathbb{H}^3(\Omega)$ such that $\frac{\partial \hbox{\boldmath $u$}}{\partial \hbox{\

Figures (3)

  • Figure 1: Spin Transfer Torque
  • Figure 2: Spin Orbit Torque
  • Figure 3: Plot of the energy, $t \mapsto \left\|\nabla \hbox{\boldmath $m$}_{h,k}(t)\right\|_{\mathbb{L}^{2}(\Omega)}^2$

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4: Existence and Uniqueness
  • Theorem 3.1: Existence
  • Theorem 3.2: Stability
  • Lemma 3.3
  • ...and 29 more