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BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: unconditional weak convergence to weak solutions

Michele Aldé, Michael Feischl, Dirk Praetorius

arXiv:2511.22000v2math.NAcs.NA

TL;DR

LLG models time-dependent micromagnetic phenomena, and the paper develops a fully discrete method for computing its magnetization. The method combines first-order finite elements with a BDF2-type time integrator and reconstructs a discrete space-time magnetization. Its approximations converge unconditionally along a subsequence to a weak solution, with full-sequence convergence if that solution is unique.

  • Problem

    The paper addresses how to discretize LLG while obtaining unconditional convergence to a weak solution in the space-time domain.

    Source: §1 · pp. 1–2

  • Method

    The method uses first-order finite elements, a BDF2-type integrator, and time interpolation of the discrete magnetization.

    Source: §1 · p. 2

  • Results

    The reconstructed approximations converge unconditionally along a subsequence weakly in H1(ΩT) to a weak solution, and strongly in L2(ΩT).

    Source: §2.4 · p. 5

  • Takeaways & Limitations

    When the weak solution is unique, the convergence conclusions hold for the full sequences rather than only subsequences.

    Source: §1 · p. 2

    In the reported numerical deviation setting, the unit-modulus constraint is not respected, and the convergence rate of its deviation deteriorates.

    Source: §4.4 · p. 27

Abstract

from arXiv · show

We consider the Landau-Lifshitz-Gilbert equation (LLG) that models time-dependent micromagnetic phenomena. We propose a full discretization that employs first-order finite elements in space and a BDF2-type two-step method in time. In each time step, only one linear system of equations has to be solved. We employ linear interpolation in time to reconstruct the discrete space-time magnetization. We prove that the integrator is unconditionally stable and thus guarantees that a subsequence of the reconstructed magnetization converges weakly in $H^1$ towards a weak solution of LLG in the space-time domain. Numerical experiments verify that the proposed integrator is indeed first-order in space and second-order in time.

1. Introduction

The paper develops a first-order-in-space, second-order-in-time LLG integrator aimed at unconditional weak convergence to weak solutions. It addresses a gap between earlier schemes that offered either a nonlinear solve for formal second-order time accuracy or a linear solve with first-order accuracy.

Background

  • LLG models magnetization evolution as a time-dependent nonlinear PDE, with applications including sensors, actuators, and recording devices.The magnetization has constant modulus below the Curie temperature for constant temperatures.Source: §1 · p. 1

Related work

  • Earlier unconditional weak-convergence schemes used first-order finite elements, but paired formal second-order time accuracy with a nonlinear solve or linear solves with first-order accuracy.The implicit midpoint-rule scheme [BP06] required one nonlinear system per step, while the tangent-space integrator [Alo08] required one linear equation per step.Source: §1 · p. 1
  • Prior BDF-type integrators had not established unconditional weak convergence to LLG weak solutions, despite error estimates for sufficiently smooth strong solutions.The gap is stated for.Source: §1 · p. 2

Contribution

  • The proposed integrator targets unconditional weak convergence to LLG weak solutions while being first-order in space and second-order in time.This paper establishes the weak-convergence result; convergence with rates to smooth strong solutions and an IMEX extension are objectives of companion papers.Source: §1 · p. 2

Approach

  • The integrator uses a node-wise discrete tangent space tailored to first-order elements and a predictor-corrector whose corrector is based on a BDF2 step.A time interpolation of the finite-element approximations produces the discrete space-time magnetization; the analysis adapts the parabolic energy method to prove subsequential weak convergence.Source: §1 · p. 2

2. Numerical BDF2 integrator and main result

The section formulates LLG with exchange and applied-field contributions, then discretizes it using first-order finite elements and a BDF2-type time integrator. Under the stated assumptions, reconstructed approximations converge unconditionally along a subsequence to a weak solution, with full-sequence convergence when that solution is unique.

2.1. Mathematical model.

  • LLG is posed for unit-length magnetization, with the model energy comprising exchange and applied external-field contributions.The section describes the exchange contribution as the leading-order term and notes the dissipative behavior when the applied field is constant.Source: §2.1 · p. 3

2.2. Discretization.

  • The spatial discretization uses continuous, piecewise-affine finite elements, with a node-wise discrete tangent space enforcing orthogonality at mesh vertices.The tangent-space construction mimics the continuous relation between magnetization and its time derivative; the mesh family is quasi-uniform.Source: §2.2 · p. 3

2.3. Numerical integrator.

  • The time integrator uses second-order backward differences in the tangent space, preceded by one non-projected first-order tangent-plane step to initialize BDF2.The scheme approximates the time derivative and uses a predictor-corrector approach; the discrete linear problems are uniquely solvable by Lax–Milgram.Source: §2.3 · p. 4

2.4. Convergence theorem.

  • As mesh size and time step tend to zero, a subsequence converges weakly in H1(ΩT) and strongly in L2(ΩT) to a weak LLG solution.The reconstructed magnetization also converges weak-* in L∞(0, T; H1(Ω)); the energy-inequality property requires the stated additional condition, guaranteed by τ = o(h2) for the first and last time steps.Source: §2.4 · p. 5
  • If the weak solution is unique, the theorem’s convergence conclusions hold for the full sequences rather than only subsequences.The stated example of uniqueness is a strong, sufficiently smooth solution.Source: §2.4 · p. 5

3. Proof of theorem 2

The proof establishes bounds and convergence properties for the approximations, then shows that a subsequential limit satisfies the weak-solution conditions for LLG. Under the CFL condition it also proves the continuous energy inequality, and uniqueness implies convergence of the full sequence.

3.1. Prerequisites.

  • The proof links identities (15)–(19) to Algorithm A and BDF2, states the discrete Lr estimate for γ-quasi-uniform meshes (d=2,3; r in [1,+∞)), and recalls norm equivalence from.The estimate’s constant C > 0 depends only on r and the γ-quasi-uniformity of Th.Source: §3.1 · p. 6

3.2. Discrete energy identity.

  • The discrete energy identity yields energy estimates, including physical dissipation and BDF2 numerical dissipation terms that vanish as (h, τ) → (0, 0).The physical dissipation converges to its continuous counterpart; the additional numerical dissipation terms vanish in the limit.Source: §3.2 · pp. 7, 9

3.3. Boundedness of discrete functions.

  • Algorithm A uniformly bounds the discrete gradients independently of h, τ, and n, and provides estimates for discrete modulus constraint violations.The constraint estimate also yields a uniform bound on the discrete magnetization norms.Source: §3.3 · pp. 9–10
  • The time interpolants are uniformly bounded under Theorem 2’s assumptions.The bound depends on the model and data parameters, final time, and mesh quasi-uniformity, but not on the discretization parameters.Source: §3.3 · p. 12

3.4. Extraction of weakly convergent subsequences.

  • A subsequence of the interpolated magnetizations converges in H^s(ΩT) for every 0 < s < 1 to an H^1 limit with |m| = 1 almost everywhere.The limit also belongs to L∞(0, T; H^1(Ω)).Source: §3.4 · p. 14
  • Successive subsequence extraction gives weak H1(ΩT) convergence and strong L2 convergence, while identifying the weak limit of the discrete time derivative as ∂tm.The proof uses compact embeddings and density to establish these limits.Source: §3.4 · pp. 14–15

3.5. Identification of the limit with a weak solution of LLG.

  • The limit m satisfies the weak-solution conditions: it belongs to H1(ΩT) ∩ L∞(0,T;H1(Ω)), has unit modulus and the prescribed initial trace, and satisfies the weak formulation.The discrete variational formulation converges to its continuous counterpart, verifying the weak formulation by density.Source: §3.5 · pp. 16, 19
  • The proof obtains the continuous energy inequality from the discrete one, with the CFL condition τ = o(h^2) ensuring the residual terms vanish.The resulting inequality holds almost everywhere in time.Source: §3.5 · p. 21
  • If the weak solution to LLG is unique, the full sequence of discrete magnetizations converges to that solution.Every subsequence then has a further subsequence converging to the same limit.Source: §3.5 · pp. 21–22

4. Numerical experiments

The experiments test convergence, singular behavior, unit-length deviation, and the CFL condition; the smooth tests show the predicted first-order spatial and second-order temporal convergence.

4.1. Experimental convergence rates for time-independent applied field.

  • The time-independent-field test shows first-order convergence in space and second-order convergence in time.Source: §4.1 · p. 22

4.2. Experimental convergence rates for time-dependent applied field.

  • In the time-dependent-field test, errors initially converge at the expected orders, then stagnate when spatial error dominates temporal refinement and temporal error dominates spatial refinement.Source: Figure 2 · p. 23, §4.2 · p. 23

4.3. Experimental convergence behavior for a problem with finite-time blow-up.

  • For the singular test, blow-up time increases as the mesh is refined; BDF2 blows up earlier than TPS, while finer time steps remove seminorm stagnation regardless of CFL compliance.Source: §4.3 · p. 24

4.4. Numerical deviation from the unit-length constraint.

4.5. Role of the CFL condition τ = o(h2).

  • The experiments suggest the CFL condition τ = o(h^2) may not be needed in these cases; for time-varying applied fields, total energy is not guaranteed to decrease.Source: §4.5 · pp. 27, 29

Appendix

The appendix proof establishes estimates for Lemmas 7 and 8, while Figure 9 displays total Gibbs energy evolution at a fixed mesh across different time-step sizes.

Appendix

  • Lemma 7 proves the lower estimate in (22), then the corresponding upper estimate and the same estimates for (23).The proof uses Algorithm A(ii) and Lemma 5, followed by absorption of a right-hand-side term.Source: Appendix · pp. 29–30
  • Figure 9 plots total Gibbs energy evolution for different time-step sizes at a fixed mesh size.The time-step choices are indexed by α ∈ {1, 3/2, 2, 5/2}.Source: Figure 9 · p. 30
  • Lemma 8 uses (22) from Lemma 7 to establish (24), with the same argument establishing (25).Source: Appendix · p. 30
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