pith. sign in

arxiv: 2502.04871 · v2 · pith:2LEJTXK6new · submitted 2025-02-07 · 🧮 math.NA · cs.NA· math.AP

Finite volume element method for Landau-Lifshitz equation

Yunjie Gong , Jingrun Chen , Rui Du , Panchi Li This is my paper

Pith reviewed 2026-05-23 03:28 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords finite volume element methodLandau-Lifshitz equationGauss-Seidel projectionmicromagneticsenergy lawapproximation errorblow-up solutionnumerical simulation
0
0 comments X

The pith

Finite volume element method with Gauss-Seidel projection solves the Landau-Lifshitz equation with spatial error bounds, a discrete energy law, and componentwise decoupling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a finite volume element discretization paired with a Gauss-Seidel projection time-stepping scheme for the Landau-Lifshitz equation that governs magnetization dynamics. It establishes approximation error estimates in space and shows that the resulting scheme satisfies a discrete version of the continuous energy dissipation law. Because the projection step decouples the three magnetization components, each time step requires work comparable to solving one implicit scalar heat equation, which lowers the cost of large-scale micromagnetics runs. Numerical tests confirm the error bounds, energy behavior, and ability to capture blow-up and two-dimensional magnetic textures.

Core claim

The finite volume element method combined with the Gauss-Seidel projection method yields a spatially accurate, energy-stable discretization of the Landau-Lifshitz equation whose algebraic system factors component by component, reducing computational complexity to that of a scalar heat equation while preserving the unit-length constraint on the magnetization.

What carries the argument

Finite volume element spatial discretization together with the Gauss-Seidel projection method for time marching, which solves the three magnetization components sequentially while enforcing the sphere constraint at each step.

If this is right

  • Spatial error estimates hold for the magnetization field under standard mesh regularity assumptions.
  • The discrete energy decreases monotonically in agreement with the continuous energy law.
  • The algebraic cost per time step equals that of one implicit scalar heat solve rather than a coupled nonlinear system.
  • The scheme can simulate both blow-up solutions and two-dimensional magnetic textures without additional stabilization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The componentwise decoupling may allow straightforward extension to three-dimensional domains on parallel architectures.
  • Similar projection ideas could be tested on other nonlinear geometric PDEs that enforce pointwise constraints.
  • Long-time accuracy in periodic or driven ferromagnetic systems could be checked by comparing energy decay rates against known analytic limits.

Load-bearing premise

The Gauss-Seidel projection step can be applied directly to the nonlinear, nonconvex Landau-Lifshitz system and still preserve both the unit-length constraint and unconditional stability without extra stabilization.

What would settle it

A run in which the discrete magnetization vectors leave the unit sphere by more than a fixed tolerance or the computed energy increases over several time steps.

Figures

Figures reproduced from arXiv: 2502.04871 by Jingrun Chen, Panchi Li, Rui Du, Yunjie Gong.

Figure 1
Figure 1. Figure 1: The control volume V0 centered at P0 in the polygonal domain and a reference domain. For a polygonal domain Ω, we consider the quasi-uniform regular triangulation Th consisting of closed triangle elements K such that Ω = ¯ ∪K∈ThK. Nh denotes the set of all nodes with N = |Nh|, and then define N 0 h = Nh ∩Ω. In [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Energy behaviors of the system with different Dirichlet boundary conditions. we study this interesting behavior of the LL equation using our numerical method. Since the quasi-uniform mesh is adopted, we follow similar procedures as in [6]. Example 4.3. Consider Ω = [−0.5, 0.5]2 , and the initial condition m0(x) =    (0, 0, −1)T , |x| ≥ 0.5,  2xA A2 + |x| 2 , 2yA A2 + |x| 2 , A2 − |x| 2 A2 + |x| 2 T … view at source ↗
Figure 3
Figure 3. Figure 3: The magnetization textures at different times during the blow-up. (a) t = 0. (b) t = 0.001. (c) t = 0.05. (d) t = 0.1. (e) t = 0.2. (f) t = 0.4. (g) t = 0.5. (h) t = 0.6 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The magnetization around the origin at different times. 0 0.2 0.4 0.6 0.8 1 time 0 5 10 15 20 25 30 0 2000 4000 6000 8000 10000 12000 engery [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolutions of the system’s energy and k∇mhkL∞ using the pro￾posed method with ∆t = 10−4 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The magnetization textures at different times. (a) t = 0 ps. (b) t = 10 ps. (c) t = 100 ps. (d) t = 500 ps. (e) t = 1000 ps. (f) t = 2000 ps. (g) t = 4000 ps. (h) t = 6000 ps [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The magnetization around the point x0 at different times. 0 0.01 0.02 0.03 0.04 0.05 time 50 55 60 65 70 75 80 85 90 95 100 0 1 2 3 4 5 6 engery 106 (a) Anisotropy is along with e1. 0 0.01 0.02 0.03 0.04 0.05 time 55 60 65 70 75 80 85 90 95 0 1 2 3 4 5 6 engery 106 (b) Anisotropy is along with e3 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolutions of the system energy and k∇mhkL∞ using the proposed method with ∆t = 1 ps [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The vortexes’ stabilization of the LL dynamics with different damp￾ing parameters. Arrows denote the in-plane magnetization and the background color encodes the out-plane magnetization. In addition, several different metastable magnetic states can be obtained for different initial magnetization distributions. Here we present two metastable magnetic states of these differ￾ent metastable magnetic states in … view at source ↗
Figure 10
Figure 10. Figure 10: Metastable magnetic states found by the LL equation. Arrows denote the in-plane magnetization and the background color encodes the out￾plane magnetization. a series of numerical experiments to verify the convergence analysis and validate the feasibility of the proposed method. Both error estimates and numerical results showcase that our method offers a promising approach for efficiently simulating complex… view at source ↗
Figure 11
Figure 11. Figure 11: Triangle element KQ (△P1P2P3). where SKQ is the area of triangle element KQ, M4 = M1 and M5 = M2. Since ∂m˜ h ∂x and ∂m˜ h ∂y are constants in the triangular element KQ, we have IKQ (Φ;m˜ h, I∗ hm˜ h) = X 3 i=1 m˜ h(Pi) · Z MiQMi+2 Φ × ∂m˜ h(Q) ∂ν dS = X 3 i=1 m˜ h(Pi) · "Z MiQMi+2 Φ × ∂m˜ h(Q) ∂x dy − Z MiQMi+2 Φ × ∂m˜ h(Q) ∂y dx # = X 3 i=1 m˜ h(Pi) · Φ × ∂m˜ h(Q) ∂x  (yMi+2 − yMi ) −  Φ × ∂m˜ h(Q) … view at source ↗
read the original 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 provide the approximation error in space and depict the energy law when the FVEM is adopted. Owing to the GSPM for time-marching, the discrete system is decoupled component by component, making the computational complexity comparable to that of solving the scalar heat equation implicitly. This significantly accelerates real simulations. We present several numerical experiments to validate the theoretical analysis and the efficiency gain. Additionally, we study the blow-up solution and efficiently simulate the 2D magnetic textures using the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a finite volume element method (FVEM) for the spatial discretization of the Landau-Lifshitz equation, combined with a Gauss-Seidel projection method (GSPM) for time integration. It asserts the derivation of spatial approximation error bounds, verification of a discrete energy dissipation law, and that the GSPM decouples the three-component system into independent scalar problems whose cost is comparable to an implicit scalar heat equation solve. Numerical experiments are presented to support the analysis and to demonstrate applications including blow-up dynamics and 2D magnetic textures.

Significance. If the central claims on constraint preservation, energy law, and error bounds are rigorously established, the work would supply a practical, low-complexity scheme for micromagnetics that retains the essential geometric structure of the problem while achieving substantial computational savings. The decoupling property, if proven compatible with the FVEM inner product, would be a notable algorithmic contribution.

major comments (2)
  1. [Section on time discretization / GSPM analysis] The load-bearing claim that GSPM preserves both the pointwise unit-length constraint |m_h|=1 and the exact discrete energy dissipation law under the FVEM discretization is not accompanied by a complete proof that the projection step is consistent with the FVEM inner product and does not introduce uncontrolled consistency errors for the nonconvex sphere constraint. This must be supplied (or the relevant theorem strengthened) before the efficiency and stability assertions can be accepted.
  2. [Error analysis section] The spatial error analysis asserts convergence rates but does not state the precise mesh regularity assumptions (e.g., quasi-uniformity, inverse inequalities) or the precise treatment of the nonlinear constraint inside the error estimate; without these, the claimed approximation error bounds cannot be verified.
minor comments (2)
  1. Clarify whether the energy law is proved theoretically or only observed numerically; the abstract phrasing 'depict the energy law' is ambiguous.
  2. Add a brief comparison table quantifying wall-clock time versus a standard implicit vector scheme to substantiate the claimed efficiency gain.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major concern below and will revise the manuscript to strengthen the proofs and clarify assumptions as needed.

read point-by-point responses

  1. Referee: [Section on time discretization / GSPM analysis] The load-bearing claim that GSPM preserves both the pointwise unit-length constraint |m_h|=1 and the exact discrete energy dissipation law under the FVEM discretization is not accompanied by a complete proof that the projection step is consistent with the FVEM inner product and does not introduce uncontrolled consistency errors for the nonconvex sphere constraint. This must be supplied (or the relevant theorem strengthened) before the efficiency and stability assertions can be accepted.

    Authors: We agree that the consistency of the projection step with the FVEM inner product requires a more explicit proof. The current manuscript sketches the preservation of |m_h|=1 and the energy law but does not fully detail how the nonconvex constraint is handled without introducing consistency errors. In the revision we will supply a complete proof of the projection step, showing that it remains compatible with the FVEM inner product and preserves both the pointwise constraint and the exact discrete energy dissipation law. revision: yes

  2. Referee: [Error analysis section] The spatial error analysis asserts convergence rates but does not state the precise mesh regularity assumptions (e.g., quasi-uniformity, inverse inequalities) or the precise treatment of the nonlinear constraint inside the error estimate; without these, the claimed approximation error bounds cannot be verified.

    Authors: The error analysis relies on standard assumptions for FVEM on quasi-uniform triangulations that admit inverse inequalities; these will be stated explicitly in the revised manuscript. The nonlinear constraint is incorporated via a projection step whose error contribution is controlled by the Lipschitz continuity of the projection onto the sphere. We will add a dedicated paragraph clarifying this treatment so that the convergence rates become fully verifiable under the stated mesh conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation derives error bounds and energy law directly from FVEM+GSPM discretization

full rationale

The paper introduces FVEM spatial discretization combined with GSPM time marching for the Landau-Lifshitz equation. It states that approximation errors in space and the discrete energy law follow from these choices, with component-wise decoupling arising from the projection method. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or claims. The central results are presented as consequences of the scheme construction and are validated numerically, making the derivation self-contained against external benchmarks rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger is constructed from the abstract alone; full paper may contain additional mesh regularity assumptions or time-step restrictions. No free parameters, invented entities, or explicit ad-hoc axioms are stated.

axioms (2)
  • domain assumption Finite volume element discretization of the Landau-Lifshitz equation admits a spatial error bound.
    Invoked when the paper states it provides the approximation error in space.
  • domain assumption Gauss-Seidel projection preserves the nonconvex unit-length constraint while decoupling the system.
    Central to the claim that the discrete system is decoupled component by component.

pith-pipeline@v0.9.0 · 5672 in / 1387 out tokens · 40364 ms · 2026-05-23T03:28:31.769619+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Alouges, A new finite element scheme for Landau-Lifchitz equations , Discrete Contin

    F. Alouges, A new finite element scheme for Landau-Lifchitz equations , Discrete Contin. Dyn. Syst. 1 (2008), no. 2, 187–196

    work page 2008

  2. [2]

    Alouges and P

    F. Alouges and P. Jaisson, Convergence of a finite element discretization for the Landa u-Lifshitz equations in micromagnetism , Math. Models Methods Appl. Sci. 16 (2006), no. 2, 299–316

    work page 2006

  3. [3]

    Alouges, E

    F. Alouges, E. Kritsikis, and J. Toussaint, A convergent finite element approximation for Landau-Lifsc hitz- Gilbert equation, Physica B Condens. Matter 407 (2012), no. 9, 1345–1349

    work page 2012

  4. [4]

    Alouges and A

    F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: ex istence and nonunique- ness, Nonlinear Anal. 18 (1992), no. 11, 1071–1084

    work page 1992

  5. [5]

    R. An, H. Gao, and W. Sun, Optimal error analysis of euler and Crank-Nicolson project ion finite difference schemes for Landau-Lifshitz equation , SIAM J. Numer. Anal. 59 (2021), no. 3, 1639–1662

    work page 2021

  6. [6]

    Bartels, J

    S. Bartels, J. Ko, and A. Prohl, Numerical analysis of an explicit approximation scheme for the Landau- Lifshitz-Gilbert equation, Math. Comp. 77 (2008), no. 262, 773–788

    work page 2008

  7. [7]

    Bartels and A

    S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Land au-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 44 (2006), no. 4, 1405–1419

    work page 2006

  8. [8]

    Cai, On the finite volume element method , Numer

    Z. Cai, On the finite volume element method , Numer. Math. 58 (1990), no. 1, 713–735

    work page 1990

  9. [9]

    Carbou and P

    G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bound ed domain , Diff. In- tegr.Eqns. (2001), 213–229

    work page 2001

  10. [10]

    Chatzipantelidis, Finite volume methods for elliptic PDE’s: a new approach , M2AN Math

    P. Chatzipantelidis, Finite volume methods for elliptic PDE’s: a new approach , M2AN Math. Model. Numer. Anal. 36 (2002), no. 2, 307–324

    work page 2002

  11. [11]

    Chatzipantelidis, R

    P. Chatzipantelidis, R. D. Lazarov, and V. Thom´ ee, Error estimates for a finite volume element method for parabolic equations in convex polygonal domains , Numer. Meth. PDEs 20 (2004), no. 5, 650–674

    work page 2004

  12. [12]

    Chen and X

    C. Chen and X. Long, Finite volume element approximation and analysis for a kind of semiconductor device simulation, J Appl. Math. Comput. 33 (2010), 155–172

    work page 2010

  13. [13]

    C. Chen, M. Yang, and C. Bi, Two-grid methods for finite volume element approximations o f nonlinear parabolic equations, J Appl. Math. Comput. 228 (2009), no. 1, 123–132

    work page 2009

  14. [14]

    J. Chen, C. W ang, and C. Xie, Convergence analysis of a second-order semi-implicit proj ection method for Landau-Lifshitz equation, Appl. Numer. Math. 168 (2021), 55–74

    work page 2021

  15. [15]

    Cheng and J

    Q. Cheng and J. Shen, Length preserving numerical schemes for Landau-Lifshitz e quation based on Lagrange multiplier approaches, SIAM J. Sci. Comput. 45 (2023), no. 2, A530–A553

    work page 2023

  16. [16]

    S. H. Chou and Q. Li, Error estimates in L2, H 1 and L∞ in covolume methods for elliptic and parabolic problems: a unified approach , Math. Comput. 69 (2000), no. 229, 103–120

    work page 2000

  17. [17]

    A. Fuwa, T. Ishiwata, and M. Tsutsumi, Finite difference scheme for the Landau-Lifshitz equation , Jpn. J. Ind. Appl. Math. 29 (2012), 83–110

    work page 2012

  18. [18]

    Gao, Optimal error estimates of a linearized backward Euler FEM f or the Landau-Lifshitz equation , SIAM J

    H. Gao, Optimal error estimates of a linearized backward Euler FEM f or the Landau-Lifshitz equation , SIAM J. Numer. Anal. 52 (2014), no. 5, 2574–2593

    work page 2014

  19. [19]

    Guo and M

    B. Guo and M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin cha in and harmonic maps , Calc. Var. 1 (1993), 311–334

    work page 1993

  20. [20]

    Kruz ´ ık and A

    M. Kruz ´ ık and A. Prohl, Recent developments in the modeling, analysis, and numeric s of ferromagnetism , SIAM Rev. 48 (2006), no. 3, 439–483

    work page 2006

  21. [21]

    Landau and E

    L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in f erromagnetic bodies, Phys. Z. Sowjetunion 8 (1935), no. 153, 101–114

    work page 1935

  22. [22]

    Li and X.-P

    P. Li and X.-P. W ang, Enhanced second-order Gauss-Seidel projection methods fo r the Landau-Lifshitz equation, 2024

    work page 2024

  23. [23]

    P. Li, C. Xie, R. Du, J. Chen, and X.-P. W ang, Two improved Gauss-Seidel projection methods for Landau- Lifshitz-Gilbert equation, J. Comput. Phys. 401 (2020), 109046

    work page 2020

  24. [24]

    R. Li, Z. Chen, and W. W u, Generalized difference methods for differential equations: numerical analysis of finite volume methods , New York: Marcel Dekker Inc., 2000

    work page 2000

  25. [25]

    Maekawa, Concepts in spin electronics , vol

    S. Maekawa, Concepts in spin electronics , vol. 13, Oxford University Press, 2006

    work page 2006

  26. [26]

    Melcher, Global solvability of the Cauchy problem for the Landau-Lif shitz-Gilbert equation in higher dimensions, Indiana Univ

    C. Melcher, Global solvability of the Cauchy problem for the Landau-Lif shitz-Gilbert equation in higher dimensions, Indiana Univ. Math. J. (2012), 1175–1200

    work page 2012

  27. [27]

    J. E. Miltat and M. J. Donahue, Numerical micromagnetics: Finite difference methods , Handb. Magn. Mater. 2 (2007), 742–764

    work page 2007

  28. [28]

    Shinjo, Nanomagnetism and spintronics , Elsevier, 2009

    T. Shinjo, Nanomagnetism and spintronics , Elsevier, 2009. FINITE VOLUME ELEMENT METHOD FOR LANDAU-LIFSHITZ EQUATION 27

    work page 2009

  29. [29]

    W ang, C

    X.-P. W ang, C. J. Garc ´ ıa-Cervera, and W. E, A Gauss-Seidel projection method for micromagnetics simu- lations, J. Comput. Phys. 171 (2001), no. 1, 357–372

    work page 2001

  30. [30]

    Y. Zhou, B. Guo, and S. Tan, Existence and uniqueness of smooth solution for system of fe rromagnetic chain, Sci. China Ser. A, Math. 34 (1991), 257–266. School of Mathematical Sciences, Soochow University, Suzho u 215006, China Email address : gyj1564763342@163.com School of Mathematical Sciences, Soochow University, Suzho u 215006, China Email address :...

    work page 1991