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Adaptive mesh refinement for the Landau-Lifshitz-Gilbert equation

Jan Bohn, Willy Dörfler, Michael Feischl, Stefan Karch

TL;DR

The paper addresses the efficient simulation of the Landau–Lifshitz–Gilbert equation by developing a fully adaptive space–time method that combines a higher‑order tangent‑plane time discretization with gradient‑recovery–based spatial adaptivity. It relies on a BDF time‑stepping framework with a predictor and a tangent‑plane linearization to preserve the unit‑length constraint and discrete energy decay, and it proves a discrete energy bound under regularity assumptions. A key contribution is the integration of an a posteriori spatial error estimator with a robust time‑step control strategy, yielding a convergent and energy‑stable adaptive algorithm validated by four numerical experiments. The approach enables accurate, efficient micromagnetic simulations of localized features such as domain walls and switching events, reducing computational costs relative to uniform discretizations while maintaining reliability.

Abstract

We propose a new adaptive algorithm for the approximation of the Landau-Lifshitz-Gilbert equation via a higher-order tangent plane scheme. We show that the adaptive approximation satisfies an energy inequality and demonstrate numerically, that the adaptive algorithm outperforms uniform approaches.

Adaptive mesh refinement for the Landau-Lifshitz-Gilbert equation

TL;DR

The paper addresses the efficient simulation of the Landau–Lifshitz–Gilbert equation by developing a fully adaptive space–time method that combines a higher‑order tangent‑plane time discretization with gradient‑recovery–based spatial adaptivity. It relies on a BDF time‑stepping framework with a predictor and a tangent‑plane linearization to preserve the unit‑length constraint and discrete energy decay, and it proves a discrete energy bound under regularity assumptions. A key contribution is the integration of an a posteriori spatial error estimator with a robust time‑step control strategy, yielding a convergent and energy‑stable adaptive algorithm validated by four numerical experiments. The approach enables accurate, efficient micromagnetic simulations of localized features such as domain walls and switching events, reducing computational costs relative to uniform discretizations while maintaining reliability.

Abstract

We propose a new adaptive algorithm for the approximation of the Landau-Lifshitz-Gilbert equation via a higher-order tangent plane scheme. We show that the adaptive approximation satisfies an energy inequality and demonstrate numerically, that the adaptive algorithm outperforms uniform approaches.
Paper Structure (22 sections, 1 theorem, 73 equations, 8 figures)

This paper contains 22 sections, 1 theorem, 73 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The Landau--Lifshitz--Gilbert equation
  4. Discretization of the Landau--Lifshitz--Gilbert equation
  5. Weak formulation
  6. Spatial discretization
  7. Temporal discretization
  8. BDF time-stepping
  9. The uniform discretization
  10. Adaptivity in Time
  11. The variable step BDF method
  12. Time step selection
  13. Application to LLG
  14. Adaptivity in Space
  15. The recovery error indicator
  16. ...and 7 more sections

Key Result

Theorem 1

Consider the discretization EQ:LLG-weak-lambda-bdf of the LLG equation EQ:LLG-general-pde for $k \in\{1,2\}$ with finite elements of polynomial degree $p \ge 1$. We assume $\tau =\mathcal{O}(h)$ and, for $k=2$, that the time steps satisfy $\tau_n/\tau_{n-1} \le\kappa_0 \le\sqrt{2}+1$ and $\alpha \ge under the regularity requirements EQ:regularity_m and for some positive constants $C$ that is indep

Figures (8)

  • Figure 6.1: Example \ref{['SSEC:example1']}: Left: Convergence of $\text{err}_T$ with respect to $\mathrm{TOL}_t$ for adaptive time-stepping for BDF(1) (red), BDF(2) (blue), BDF(3) (green) and BDF(4) (purple) with fixed $h = 1/16$ and polynomial degree $p=3$. The corresponding experimental rate of convergence is shown in the same color by the dotted line, here for $k = 1, \dots, 4$ the rate is $k / (k+1)$. Right: Convergence of $\max\limits_{0\le n\le N} \||\boldsymbol{m}_h^n| - 1\|_{L^\infty(\Omega)}$ with respect to $\mathrm{TOL}_t$ for adaptive time-stepping for BDF($k$), $k=1, \dots 4$, and the rate of $1/2$ for $k=1$, $3/4$ for $k=3$ and $1$ for $k=2, 4$ in dotted lines.
  • Figure 6.2: Example \ref{['SSEC:example2']}: Left: Spatial tolerance $\mathrm{TOL}_{\mathrm{s}}$ vs. $\textrm{err}_T$. Right: Maximal spatial degrees of freedom $N_{\mathrm{s}}^\text{max} := \max\{N_{\mathrm{s}}(t_n)\mid 0\le n\le 100\}$ vs. $\textrm{err}_T$, where $N_{\mathrm{s}}$ denotes the number of degrees of freedom. Various polynomial degrees of the finite element space are presented, i.e., linear (red), quadratic (blue) and cubic (green).
  • Figure 6.3: Example \ref{['SSEC:example2']}: Convergence of $\max\limits_{0\le n\le N} \||\boldsymbol{m}_h^n| - 1\|_{L^\infty(\Omega)}$ with respect to the spatial tolerance $\mathrm{TOL}_{\mathrm{s}}$. Various polynomial degrees of the finite element space are presented, i.e., linear (red), quadratic (blue) and cubic (green).
  • Figure 6.4: Example \ref{['SSEC:example3']}: We compare simulations with uniform meshes and time steps with a fully adaptive simulation. The time step size for the uniform simulations is $\tau=10^{-3}$ and the mesh size ranges from $h=1/8$ to $h=1/14$. The adaptive simulation starts with a mesh size of $h=1/2$. The singularity, i.e., the rotation of the entire magnetization towards $[0,0,-1]$, is observed via a sharp decay of the energy \ref{['EQ:def_energy']} as well as a spike in the $\|\nabla\boldsymbol{m}\|_{L^\infty}$-norm. We observe that only the finest uniform simulation avoids the singularity (that requires a total number of 1 682 000 degrees of freedom). In contrast, the adaptive simulation achieves the same with just 753 540 degrees of freedom.
  • Figure 6.5: Example \ref{['SSEC:example3']}: Top row: Solution $\boldsymbol{m}_h$ for the uniform mesh $h=1/12$ and uniform time step $\tau=10^{-3}$ at $t=0$ (left), $t=0.068$ (middle) and $t=0.071$ (right) according to \ref{['EQ:LLG-weak-lambda-bdf']}. The singularity appears in the center of the vector field, where $\boldsymbol{m}$ points upward in a very small neighborhood only. Bottom: Evolution of the time step size in the adaptive computation.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1: Energy bound for orders $k =1,2$
  • proof
  • Remark 2