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BDF2 time integrator with linear finite elements proves optimal-order convergence for the Landau-Lifshitz-Gilbert equation.

2026-07-01 07:39 UTC pith:PIVTBKPH

load-bearing objection This gives the first linear BDF2 scheme for LLG with optimal a-priori rates to strong solutions under regularity, completing their earlier weak-convergence result.

arxiv 2605.05129 v2 pith:PIVTBKPH submitted 2026-05-06 math.NA cs.NA

BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: a-priori error estimates

Michele Ald\'e , Dirk Praetorius , Michael Feischl This is my paper
classification math.NA cs.NA
keywords Landau-Lifshitz-Gilbert equationmicromagneticsBDF2 methodfinite element discretizationa priori error estimatesconvergence ratesnumerical integrator
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper examines a fully discrete numerical scheme for the Landau-Lifshitz-Gilbert equation that describes time-dependent micromagnetic behavior. The scheme pairs first-order finite elements in space with a BDF2 method in time and solves exactly one linear system per step without imposing the pointwise unit-length constraint on the magnetization vector. Under regularity assumptions on the solution and external field, the analysis establishes optimal convergence rates of first order in space and second order in time. Combined with earlier results on weak convergence, the work shows that this linear higher-order integrator reaches both weak and strong solutions. Numerical tests support the predicted rates.

Core claim

The fully discrete scheme that combines linear finite elements in space with the BDF2 time integrator converges with optimal rates to the exact solution of the Landau-Lifshitz-Gilbert equation when the solution and external field are sufficiently regular; the method requires only one linear solve per time step and does not enforce the pointwise constraint |m|=1.

What carries the argument

The BDF2-type integrator, a second-order backward differentiation formula in time paired with first-order finite elements in space, that produces a single linear system per step.

Load-bearing premise

The exact solution and external field must satisfy sufficient regularity assumptions.

What would settle it

Numerical experiments with smooth solutions and fields that produce convergence rates below first order in space or second order in time would disprove the claimed a-priori estimates.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The scheme converges optimally to both weak and strong solutions when the regularity conditions hold.
  • Only one linear solve per time step is required, independent of the unit-length constraint.
  • First-order spatial accuracy and second-order temporal accuracy are attained simultaneously.
  • The approach extends prior weak-convergence results to optimal rates under added smoothness.

Where Pith is reading between the lines

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

  • The linear structure may allow straightforward extension to adaptive time-stepping strategies for long-time micromagnetic simulations.
  • The absence of an explicit constraint enforcement step could simplify coupling to other physical models such as temperature or current-driven effects.
  • Testing the scheme on problems with reduced regularity would clarify the practical range of the error bounds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper analyzes a fully discrete numerical scheme for the Landau-Lifshitz-Gilbert equation that combines linear finite elements in space with a BDF2 time integrator. The scheme solves a single linear system per time step and does not enforce the pointwise unit-length constraint. Building on prior unconditional weak-convergence analysis, the authors prove optimal-order a-priori error estimates under sufficient regularity assumptions on the exact solution and external field. Numerical experiments are presented to confirm first-order spatial and second-order temporal convergence.

Significance. If the error estimates are valid, the work supplies the first higher-order-in-time linear integrator with rigorous convergence guarantees to both weak and strong solutions of the LLG equation. This is a meaningful contribution to the numerical analysis of micromagnetics, where efficient, unconditionally stable higher-order methods with proven rates are needed.

minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly restate the precise Sobolev regularity indices (e.g., H^2 in space and W^{2,∞} in time) required for the optimal rates, rather than referring only to “sufficient regularity.”
  2. [Numerical experiments] In the numerical experiments section, the tables or figures reporting convergence rates should include the observed orders computed from successive refinements so that the claimed first- and second-order behavior can be directly verified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation of minor revision. The referee's summary correctly reflects the paper's contributions: optimal-order a-priori error estimates for the linear BDF2 finite-element scheme applied to the LLG equation, building on prior weak-convergence results to establish convergence to both weak and strong solutions. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; prior work cited for weak convergence only

full rationale

The manuscript explicitly separates the new a-priori error estimates (under regularity) from the unconditional weak-convergence result of the earlier paper. The central claim is a combination of two independent analyses rather than a reduction of one to the other by definition or fitting. No self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling appear in the derivation chain. The self-citation is acknowledged and non-load-bearing for the novel strong-solution rates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the analysis rests on standard domain assumptions for PDE error estimates rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Sufficient regularity assumptions on the exact solution and the external field
    Invoked in the abstract as the condition needed to prove optimal-order convergence rates.

pith-pipeline@v0.9.1-grok · 5674 in / 1249 out tokens · 35494 ms · 2026-07-01T07:39:13.803122+00:00 · methodology

0 comments
read the original abstract

We consider the Landau-Lifshitz-Gilbert equation (LLG), which models time-dependent micromagnetic phenomena. We analyze a fully discrete scheme that combines first-order finite elements in space with a BDF2 method in time. The method requires the solution of only one linear system of equations per time step and does not enforce the pointwise unit-length constraint of the magnetization. While unconditional weak convergence has been analyzed in an earlier work, we now prove optimal-order convergence rates under sufficient regularity assumptions on the exact solution and the external field. In combination with our previous work, this establishes the first higher-order-in-time and linear integrator that converges both to weak and strong solutions of LLG. Numerical experiments confirm first-order convergence in space and second-order convergence in time.

Figures

Figures reproduced from arXiv: 2605.05129 by Dirk Praetorius, Michael Feischl, Michele Ald\'e.

Figure 1
Figure 1. Figure 1: Empirical convergence rates in the experiments from Section 6.1. close to be singular at t = T. To investigate this behavior further, view at source ↗
Figure 2
Figure 2. Figure 2: Empirical convergence rates for h = Cτ 2 with C =5,000 in the second experiment from Section 6.1, with time-dependent f. 10 20 40 80 160 320 640 1280 10−5 10−4 10−3 10−2 10−1 100 101 β = 2 1/τ Error error in ℓ∞(H1 ) error in ℓ∞(L 2 ) 2 4 8 16 32 64 128 256 10−4 10−3 10−2 10−1 100 101 β = 2 β = 1 1/h Error error in ℓ∞(H1 ) error in ℓ∞(L 2 ) view at source ↗
Figure 3
Figure 3. Figure 3: Empirical convergence rates of max j=1,...,N ‖m(tj ) − mj h ‖L2(Ω) ( ) and max j=1,...,N ‖m(tj ) − mj h ‖H1(Ω) ( ) in the experiment from Section 6.2. Left: Convergence in time for a fixed spatial mesh with h = 1/256. Right: Convergence in space for a fixed time-step size τ = 10−3 . and evolves under the influence of an external field which is constant in space and acts as a rapid pulse in time, namely f(t… view at source ↗
Figure 4
Figure 4. Figure 4: Empirical convergence rates of max j=1,...,N ‖m(tj ) − mj h ‖L2(Ω) ( ) and max j=1,...,N ‖m(tj )−mj h ‖H1(Ω) ( ) in the experiment from Section 6.2 with g˜(t) = (T +χ)/(T + χ − t). error is calculated comparing the numerical solution to a reference solution mref computed with τref = 2−13τ0. To test first-order convergence in space, we fix the time-step size τ = 10−1 and consider a sequence of uniform trian… view at source ↗
Figure 5
Figure 5. Figure 5: Empirical convergence rates of the final-time errors ‖mref(T)−mhτ (T)‖L2(Ω) ( ) and ‖mref(T) − mhτ (T)‖H1(Ω) ( ) for the experiment from Section 6.3. Left: Convergence in time for a fixed spatial mesh with h = 2−7/2 ≈ 0.0884. Right: Convergence in space for a fixed time-step size τ = 10−1 . To prove (17c), notice that ∂i∂j  u |u|  (94) = ∂i∂ju |u| − ∂ju · ∂i |u| |u| 2 − ∂i(u(∂ju · u)) |u| 3 − u(∂ju · u)∂… view at source ↗

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Works this paper leans on

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