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Finite volume element method for Landau-Lifshitz equation

Yunjie Gong, Rui Du, Panchi Li

Abstract

The Landau-Lifshitz equation describes the dynamics of magnetization in ferromagnetic materials. Due to the essential nonlinearity and nonconvex constraint, it is typically solved numerically. In this paper, we developed a finite volume element method (FVEM) with the Gauss-Seidel projection method (GSPM) for the micromagnetics simulations. We give the error estimate for FVEM in space and depict the discretized energy dissipation. Owing to the application of the GSPM, the nonlinear vector system is decoupled and the computational complexity is comparable to that of implicitly solving the scalar heat equation, which accelerates the real simulations significantly. We present several numerical experiments to verify the theoretical analysis. Furthermore, we study the blow-up solution and efficiently simulate the 2D magnetic textures using the proposed method.

Finite volume element method for Landau-Lifshitz equation

Abstract

The Landau-Lifshitz equation describes the dynamics of magnetization in ferromagnetic materials. Due to the essential nonlinearity and nonconvex constraint, it is typically solved numerically. In this paper, we developed a finite volume element method (FVEM) with the Gauss-Seidel projection method (GSPM) for the micromagnetics simulations. We give the error estimate for FVEM in space and depict the discretized energy dissipation. Owing to the application of the GSPM, the nonlinear vector system is decoupled and the computational complexity is comparable to that of implicitly solving the scalar heat equation, which accelerates the real simulations significantly. We present several numerical experiments to verify the theoretical analysis. Furthermore, we study the blow-up solution and efficiently simulate the 2D magnetic textures using the proposed method.

Paper Structure

This paper contains 20 sections, 11 theorems, 159 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Model and numerical methods
  3. The LL equation
  4. FVEM approximation
  5. Auxiliary lemmas and error estimates
  6. An auxiliary problem
  7. A discrete auxiliary problem.
  8. Discretized energy dissipation
  9. Temporal discretization and numerical experiments
  10. Temporal discretization
  11. Accuracy check
  12. Discrete energy laws
  13. Phenomenon of the blow-up
  14. Micromagnetics simulations
  15. Conclusions
  16. ...and 5 more sections

Key Result

Lemma 3.1

Given $\boldsymbol{\Phi} \in \boldsymbol{X}$, there exists a positive constant $C$ such that where $C$ depends on $\alpha$ and $\|\nabla\boldsymbol{\Phi}\|_{\boldsymbol{L}^{\infty}}$.

Figures (11)

  • Figure 1: The control volume $V_0$ centered at $P_0$ in the polygonal domain and a reference domain.
  • Figure 2: Energy behaviors of the system with different Dirichlet boundary conditions.
  • Figure 3: The magnetization textures at different times during the blow-up.
  • Figure 4: The magnetization around the origin at different times.
  • Figure 5: Evolutions of the system's energy and $\|\nabla\boldsymbol{m}_h \|_{\boldsymbol{L}^{\infty}}$ using the proposed method with $\Delta t = 10^{-4}$.
  • ...and 6 more figures

Theorems & Definitions (22)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 3.6
  • proof
  • Theorem 3.7
  • proof
  • ...and 12 more