arxiv: 2605.05129 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

Recognition: unknown

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

Dirk Praetorius, Michael Feischl, Michele Ald\'e

Pith reviewed 2026-05-08 15:49 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Landau-Lifshitz-Gilbert equationmicromagneticsBDF2 methodfinite element methoda-priori error estimatesnumerical integrationweak and strong solutions

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The pith

A linear BDF2 scheme paired with finite elements achieves optimal-order convergence for the Landau-Lifshitz-Gilbert equation.

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

The paper establishes a fully discrete numerical method for the Landau-Lifshitz-Gilbert equation that models time-dependent magnetic phenomena. The scheme uses linear finite elements in space and a second-order backward differentiation formula in time. It requires solving only one linear system per step and does not enforce the magnetization vector to have unit length at every point. Under assumptions of adequate smoothness on the solution and the external field, the method attains first-order accuracy in space and second-order accuracy in time. Together with earlier weak-convergence results, this identifies the integrator as the first higher-order time method that is both linear and convergent to weak and strong solutions.

Core claim

The fully discrete scheme that combines first-order finite elements in space with the BDF2 method in time satisfies optimal-order a-priori error estimates for the Landau-Lifshitz-Gilbert equation when the exact solution and external field are sufficiently regular. The analysis covers both the spatial and temporal discretization errors, and the combination with prior unconditional weak-convergence results yields convergence to both weak and strong solutions.

What carries the argument

The BDF2-type time integrator combined with linear finite-element discretization in space, which produces a linear algebraic problem at each step while retaining the unit-length constraint in the limit.

If this is right

The scheme converges at the rate O(h + k^2) in appropriate norms when regularity holds.
The integrator converges to both weak and strong solutions of the Landau-Lifshitz-Gilbert equation.
Only one linear solve is required per time step.
Numerical experiments confirm first-order spatial and second-order temporal convergence.

Where Pith is reading between the lines

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

The linearization strategy may extend to other nonlinear evolution equations with pointwise constraints.
Such methods could reduce computational cost in large-scale micromagnetic simulations compared with nonlinear implicit schemes.
The approach suggests that explicit enforcement of the unit-length constraint is unnecessary for convergence when the scheme is designed appropriately.

Load-bearing premise

The exact solution and external field must possess sufficient regularity for the optimal-order error estimates to hold.

What would settle it

A numerical test on a smooth manufactured solution where the observed temporal error fails to decrease at the expected second-order rate as the time step is halved would contradict the claimed convergence.

Figures

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

**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: 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: 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: 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: 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 ↗

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.

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.

Desk Editor's Note private letter to a colleague

This paper supplies the first optimal-rate a-priori error bounds for a linear BDF2 finite-element scheme on the LLG equation, building directly on the authors' earlier unconditional weak-convergence result.

read the letter

The main point is that Aldé, Praetorius and Feischl have closed the gap between their prior weak-convergence analysis and a practical higher-order linear time integrator. The scheme uses standard first-order elements in space and BDF2 in time, solves only one linear system per step, and skips any explicit projection to enforce the unit-length constraint. Under the stated regularity assumptions on the solution and external field, they obtain the expected first-order spatial and second-order temporal rates via energy estimates and standard approximation arguments. Numerical tests reproduce those orders on manufactured solutions. That combination is new and directly useful for people who need a stable, higher-order method without nonlinear solves at each step. The analysis itself follows textbook lines once the regularity is granted, so the technical lift is modest but the application to LLG is clean. The obvious limitation is the regularity hypothesis itself. LLG solutions can lose smoothness in physically relevant regimes, and the paper does not supply evidence that the assumed Sobolev or time-derivative bounds are typically satisfied or easy to check. When those bounds fail, the optimal rates disappear while the weak convergence from the earlier work remains. The numerical examples use smooth data, so they do not test the boundary of the assumption. Readers working on micromagnetics numerics or on structure-preserving integrators for nonlinear PDEs will get immediate value; the rest of the community can safely skip it. The work is coherent on its own terms and fills a documented gap, so it merits a full referee report rather than a desk rejection.

Referee Report

1 major / 2 minor

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

Significance. If the regularity hypotheses are compatible with the LLG structure and satisfied in the regimes of interest, the result supplies the first linear, higher-order-in-time integrator with rigorous convergence to both weak and strong solutions. This is a meaningful advance for efficient micromagnetics simulation, where nonlinearity and constraint handling are computationally expensive. The combination of the new strong-convergence analysis with the authors' earlier weak-convergence work is a clear strength.

major comments (1)

[§3] §3 (main error theorem): The optimal-rate statement requires Sobolev regularity on the exact solution (typically H^2 in space and H^2 or higher in time) and on the external field. The manuscript does not discuss whether these assumptions are compatible with the LLG equation or satisfied by the manufactured solutions employed in the numerical tests; without such verification the observed convergence rates do not confirm the theorem under the stated hypotheses.

minor comments (2)

[Abstract] The abstract and introduction should explicitly state the precise norms in which the error estimates are proved (e.g., L^2 or H^1).
[§2] Notation for the discrete magnetization and the projection operators should be introduced once and used consistently throughout the analysis section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the regularity assumptions. We address the point below and will revise the manuscript to incorporate the requested clarification.

read point-by-point responses

Referee: [§3] §3 (main error theorem): The optimal-rate statement requires Sobolev regularity on the exact solution (typically H^2 in space and H^2 or higher in time) and on the external field. The manuscript does not discuss whether these assumptions are compatible with the LLG equation or satisfied by the manufactured solutions employed in the numerical tests; without such verification the observed convergence rates do not confirm the theorem under the stated hypotheses.

Authors: We agree that an explicit discussion of the regularity hypotheses is useful. The manufactured solutions employed in the numerical experiments are constructed analytically as smooth functions that satisfy the precise Sobolev regularity required by the theorem by design. We will add a short paragraph (or remark) in the revised Section 3 clarifying this choice of test solutions and noting that the assumed regularity is compatible with the LLG equation for sufficiently smooth initial data and external fields, consistent with local-in-time existence results available in the literature. This addition will confirm that the observed convergence rates are consistent with the theorem under the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for weak-convergence foundation; new error estimates derived independently via standard techniques

full rationale

The paper's core contribution is the a-priori error analysis for the BDF2 scheme under regularity assumptions, which the abstract describes as built from energy estimates and approximation theory rather than any fitted parameters or self-referential definitions. The combination with prior work to claim 'first' status for both weak and strong convergence is a synthesis step, not a derivation that reduces to the inputs by construction. No self-definitional loops, fitted predictions, or ansatz smuggling appear in the provided abstract or described structure. This matches the expected non-circular case for a follow-up analysis paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on regularity of the exact solution and external field plus standard existence results for weak solutions of LLG; no free parameters or new entities are introduced.

axioms (2)

domain assumption The exact solution and external field satisfy sufficient regularity assumptions for optimal-order error estimates.
Explicitly required in the abstract for the convergence rates to hold.
standard math Standard Sobolev-space approximation properties of linear finite elements and BDF2 time discretization apply.
Invoked implicitly for the a-priori estimates.

pith-pipeline@v0.9.0 · 5443 in / 1293 out tokens · 45446 ms · 2026-05-08T15:49:24.570215+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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