Stability and error analysis of fully discrete original energy-dissipative and length-preserving scheme for the Landau-Lifshitz-Gilbert equation

TL;DR

This work develops a linear, fully discrete finite-difference scheme for the Landau-Lifshitz-Gilbert equation that unconditionally preserves the unit-length constraint $| extbf{m}|=1$ and original energy dissipation. By reformulating the scheme into an equivalent weak form that leverages point-wise length preservation, the authors overcome nonlinear-Laplacian challenges and prove optimal convergence rates in discrete $L^2$ and $H^1$ norms under 2D and 3D CFL-like conditions. Theoretical results are complemented by numerical experiments confirming convergence rates, unconditional energy dissipation, blow-up dynamics, and static skyrmion relaxations, demonstrating robustness for topological magnetization problems. The work sets a benchmark for linear, energy-dissipative projection schemes and paves the way for extensions to mass-lumped finite element methods with similar guarantees.

Abstract

The Landau-Lifshitz-Gilbert (LLG) equation, regarded as a gradient flow with manifold constraint, is the fundamental model describing magnetization dynamics in ferromagnetic materials. It is well known that the normalized tangent plane method is able to simultaneously achieve the non-convex manifold constraint and original energy dissipation. However, the associated computational cost of this numerical approach is exceedingly high. By contrast, the projection method is more straightforward to implement, while it often compromises the inherent energy dissipative property of the continuous model, and the error analysis turns out to be even more challenging. In this work, we first construct a linear and fully discrete finite difference numerical scheme, based on the projection method for the LLG equation, which is capable of simultaneously preserving the non-convex manifold constraint \(|\mathbf{m}| = 1\) and an unconditional original energy dissipation. In the error analysis, the classical theoretical technique becomes ineffective, due to the presence of the nonlinear Laplacian term, which in turn poses a significant challenge. To overcome this subtle difficulty, we carefully rewrite the numerical method in an equivalent weak form, in which a point-wise length preserving feature of the numerical solution plays an essential role. As a result of these estimates in the reformulated weak form, an optimal convergence rate could be theoretically established. In our knowledge, this numerical method is the first linear algorithm that preserves the following combined theoretical properties: (i) point-wise length preservation, (ii) unconditional original energy dissipation, (iii) a theoretical justification of convergence analysis and optimal rate error estimate.

Figures (4)

• Figure 5.1: Evolutions of original energy $\frac{1}{2}\|\nabla_h\mathbf{m}^n\|^2_{l^2}$ for Examples 2 and 3.
• Figure 5.2: Numerical magnetization $\mathbf{m}^n$ (projected on $x_1x_2$-plane) for Example 3.
• Figure 5.3: Numerical magnetization $\mathbf{m}^n$ around the origin for Example 3.
• Figure 5.4: The static axially symmetric skyrmion for Example 4.

Theorems & Definitions (21)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Theorem 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4