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Optimal Error Estimates of a Linearized Backward Euler Localized Orthogonal Decomposition for the Landau-Lifshitz Equation

Zetao Ma, Rui Du, Lei Zhang

TL;DR

This work addresses multiscale Landau–Lifshitz dynamics with rough exchange coefficients by developing a linearized backward Euler discretization within the Localized Orthogonal Decomposition (LOD) framework. The authors decouple temporal and spatial errors, proving rigorous fully discrete error bounds: the $L^2$-norm error obeys $\|\mathbf{m}(t_n)-\mathbf{M}_{\text{LOD}}^n\|_{L^2} \le C(\tau+H^3)$ and the $H^1$-norm error obeys $\|\mathbf{m}(t_n)-\mathbf{M}_{\text{LOD}}^n\|_{H^1} \le C(\tau+H^2)$, along with an exponential-localization property for the multiscale basis and a bound on the mass deviation $\|1-|\mathbf{M}_{\text{LOD}}^n|^2\|_{L^2} \le C(\tau+H^3)$. The analysis leverages a temporal regularity framework to control spatial error terms and provides sharp convergence rates that surpass standard FEM in the multiscale setting. Numerical experiments in 2D validate the predicted rates for both periodic- and rough-coefficient scenarios and demonstrate the method’s efficiency for micromagnetic simulations with rough media.

Abstract

We introduce a novel spatial discretization technique for the reliable and efficient simulation of magnetization dynamics governed by the Landau-Lifshitz (LL) equation. The overall discretization error is systematically decomposed into temporal and spatial components. The spatial error analysis is conducted by formulating the LL equation within the framework of the Localized Orthogonal Decomposition (LOD) method. Numerical examples are presented to validate the accuracy and approximation properties of the proposed scheme.

Optimal Error Estimates of a Linearized Backward Euler Localized Orthogonal Decomposition for the Landau-Lifshitz Equation

TL;DR

This work addresses multiscale Landau–Lifshitz dynamics with rough exchange coefficients by developing a linearized backward Euler discretization within the Localized Orthogonal Decomposition (LOD) framework. The authors decouple temporal and spatial errors, proving rigorous fully discrete error bounds: the -norm error obeys and the -norm error obeys , along with an exponential-localization property for the multiscale basis and a bound on the mass deviation . The analysis leverages a temporal regularity framework to control spatial error terms and provides sharp convergence rates that surpass standard FEM in the multiscale setting. Numerical experiments in 2D validate the predicted rates for both periodic- and rough-coefficient scenarios and demonstrate the method’s efficiency for micromagnetic simulations with rough media.

Abstract

We introduce a novel spatial discretization technique for the reliable and efficient simulation of magnetization dynamics governed by the Landau-Lifshitz (LL) equation. The overall discretization error is systematically decomposed into temporal and spatial components. The spatial error analysis is conducted by formulating the LL equation within the framework of the Localized Orthogonal Decomposition (LOD) method. Numerical examples are presented to validate the accuracy and approximation properties of the proposed scheme.
Paper Structure (26 sections, 17 theorems, 156 equations, 4 figures, 5 tables)

This paper contains 26 sections, 17 theorems, 156 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Organization:
  3. Notation:
  4. An Ideal Multiscale Method: Localized Orthogonal Decomposition
  5. Ideal Multiscale Method and Approximation Properties
  6. Localization Property
  7. Analysis of Backward Euler Discretization of LL equation in LOD Space
  8. Discretization Scheme
  9. Temporal Error Estimate
  10. Spatial Estimate
  11. Proof of Theorem \ref{['mainresult']}
  12. Numerical Results
  13. Proof of Main Theorems
  14. Proof of Theorem \ref{['temporal_1']}.
  15. Proof of Theorem \ref{['theorem_Projectionestimate']}.
  16. ...and 11 more sections

Key Result

theorem 1

Let $u$ be the solution to the elliptic problem with right-hand side $f \in H^s(\Omega)$, $s = 0, 1$, and let $u_{\text{LOD}} \in V_{\text{LOD}}$ be defined by ritz_LOD. Then, where the constant $C > 0$ is independent of $H$, $u$, and the regularity of $u$.

Figures (4)

  • Figure 1: Approximation of $\mathbf{m}$: (Left) $y=0.5$ cross-section, (Right) $x=0.5$ cross-section.
  • Figure 2: Approximation of $\mathbf{m}$: (Left) $y=0.5$ cross-section, (Right) $x=0.5$ cross-section.
  • Figure 3: Discontinuous rough coefficient $\kappa(x,y)$ on $\Omega=[0,1]^2$.
  • Figure 4: Rough coefficient example, with $T=0.2$ and $\alpha=1e-2$

Theorems & Definitions (29)

  • theorem 1: Global approximation error
  • theorem 2: Localization error
  • remark 1
  • remark 2
  • theorem 3: Approximation error with localization
  • remark 3
  • theorem 4
  • theorem 5
  • theorem 6
  • theorem 7
  • ...and 19 more