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Rate of convergence of a fully discrete structure-preserving midpoint scheme for the stochastic Landau--Lifshitz--Gilbert equation

Agus L. Soenjaya

arXiv:2609.06459v1math.NAcs.NAmath.APmath.PR

TL;DR

The paper addresses the lack of convergence-rate results for fully discrete, structure-preserving approximations of the stochastic Landau–Lifshitz–Gilbert equation. It analyses a mass-lumped finite element midpoint scheme and proves a rate of h + k^γ in probability for every γ < 1/2.

  • Problem

    No convergence rate had previously been proved for a fully discrete structure-preserving sLLG approximation, although the equation combines strong nonlinearity, multiplicative Stratonovich noise, and a non-convex pointwise constraint.

    Source: §1 · pp. 1–2

  • Method

    The paper analyses a mass-lumped P1 finite element discretisation with midpoint time stepping that preserves unit length at the finite element nodes.

    Source: §1 · p. 2, §2.3 · p. 4

  • Results

    The method converges in probability at rate h + k^γ for every γ < 1/2 in the discrete L∞(0, T; L2) ∩ L2(0, T; H1) energy norm.

    Source: §1 · p. 3

  • Takeaways & Limitations

    This gives the first rigorous convergence-rate result for a fully discrete structure-preserving scheme for sLLG.

    Source: §1 · p. 2

    The analysis is one-dimensional and assumes m0 ∈ H3(D; S2) and sufficient spatial regularity of the coloured Wiener process.

    Source: §1 · pp. 1–2

Abstract

from arXiv · show

The stochastic Landau--Lifshitz--Gilbert (sLLG) equation is a strongly nonlinear stochastic PDE with a non-convex pointwise constraint arising in the theory of micromagnetics. We analyse a fully discrete, structure-preserving finite element approximation of the sLLG equation with coloured multiplicative Stratonovich noise on a bounded interval. The method utilises continuous piecewise affine finite elements, mass lumping, and midpoint time discretisation to preserve the unit-length constraint exactly at the finite element nodes. Under suitable regularity assumptions on the initial data and the noise, we establish uniform higher-moment stability and develop an error analysis for the scheme. The analysis exploits the geometric structure of the equation and the stochastic midpoint discretisation. For every $\gamma\in(0,\frac12)$, we prove first-order spatial convergence and temporal convergence of order $\gamma$ in the natural discrete energy norm, locally in mean square on events of arbitrarily large probability and, consequently, in probability. To the best of our knowledge, this is the first convergence-rate result for a fully discrete structure-preserving finite element scheme solving the stochastic Landau--Lifshitz--Gilbert equation.

1. Introduction

The paper closes the gap in convergence-rate analysis for fully discrete, structure-preserving sLLG schemes by studying a mass-lumped finite element midpoint method. Under suitable regularity assumptions, it proves a fully discrete rate in probability while preserving the unit-length constraint at finite-element nodes.

1. Introduction

  • The analysis concerns the one-dimensional exchange-only sLLG equation with coloured multiplicative Stratonovich noise and a non-convex unit-length constraint.The noise acts tangentially to the unit sphere; the paper also states that the analysis applies to the one-dimensional flat torus with periodic boundary conditions.Source: §1 · pp. 1–2
  • The one-dimensional setting supplies the pathwise well-posedness and maximal regularity needed for quantitative error analysis, while also modelling thin ferromagnetic wires.The paper notes that higher dimensions have more delicate analytical theory because of possible singularity formation and weaker solution regularity.Source: §1 · p. 2
  • Before this work, no convergence rate had been proved for a fully discrete structure-preserving sLLG approximation, despite a time-semidiscrete rate arbitrarily close to 1/2.Mass-lumping and interpolation defects, as well as insufficient uniform control of the discrete Laplacian, obstructed the fully discrete analysis.Source: §1 · p. 2
  • A mass-lumped P1 finite element method with midpoint time discretisation preserves unit length exactly at every finite-element node.The analysis derives uniform higher-moment estimates and treats drift consistency and the stochastic midpoint defect separately, including diagonal and off-diagonal quadratic Brownian increments.Source: §1 · p. 2
  • The scheme converges in probability at rate h + k^γ for every γ < 1/2 in the discrete L∞(0, T; L2) ∩ L2(0, T; H1) energy norm.The result assumes suitable regularity of the initial data and noise.Source: §1 · pp. 2–3

2. Preliminaries

The preliminaries set the notation and regularity assumptions, then collect finite-element tools used throughout the analysis.

2.1. Notation.

  • The analysis uses standard vector-valued function spaces and componentwise derivatives, with generic constants independent of discretisation parameters h and k.The notation also reuses scalar- and matrix-valued conventions where context makes their meaning clear.Source: §2.1 · p. 3

2.2. Assumption.

  • The initial datum and deterministic noise fields are assumed H3-regular, with boundary compatibility ensuring the stochastic vector fields preserve the homogeneous Neumann condition for sufficiently regular solutions.The initial datum also satisfies ∂xm0 = 0.Source: §2.2 · pp. 3–4

2.3. Finite element approximation.

  • The mesh family consists of shape-regular, quasi-uniform interval partitions, and the finite-element setup uses nodal basis functions and interpolation.The mass-lumped norm is uniformly equivalent to the standard L2 norm on Vh.Source: §2.3 · p. 4
  • Mass-lumped quadrature errors are bounded by O(h), with bounds involving L2 and H1 norms for finite-element functions and H1 and discrete energy norms for interpolated functions.These are the two quadrature estimates recorded in (2.15) and (2.16).Source: §2.3 · p. 5
  • The finite-element tools also include a one-dimensional discrete cancellation estimate and product-interpolation error bounds for H2 coefficients.The cancellation estimate controls the discrete Laplacian pairing using the L∞ norm of vh and derivative norms of ηh and vh.Source: §2.3 · p. 5

3. Stability and error estimates of the fully discrete midpoint scheme

This section establishes stability properties of the mass-lumped midpoint scheme and develops a localized natural-energy error analysis separating drift and stochastic consistency errors. It proves first-order spatial and order-γ temporal convergence in the discrete energy norm locally in mean square on events of arbitrarily large probability, and consequently convergence in probability.

3. Stability and error estimates of the fully discrete midpoint scheme

  • The mass-lumped midpoint approximation is defined on a uniform time partition with step k = T/J.The analysis assumes k ∈ (0, 1].Source: §3 · p. 5

3.1. Stability estimates.

  • Every time level retains unit nodal length, yielding an L∞ bound; p-th moments are uniformly bounded in h and k, and discrete increment estimates are established.The moment bound applies for p ≥ 2 under Assumption 2.1; the increment estimates assume (2.2).Source: §3.1 · pp. 6–8

3.2. Localised error estimate.

  • The localised natural-energy error analysis subtracts the scheme from the projected exact solution, tracks the drift residual, and treats stochastic midpoint consistency separately.The stochastic term is retained in exact integral form because its midpoint consistency error requires a separate argument.Source: §3.2 · pp. 9–10
  • The drift residual is estimated in the dual norm, while a separate stochastic-error lemma gives an h- and k-independent bound under fixed ε, Λ, δ, and η conditions.The lemma assumes ε ∈ (0, 1), Λ > 0, and states its bound for every δ > 0 and η > 0.Source: §3.2 · pp. 10–11
  • For every fixed Λ > 0, there exist kΛ ∈ (0, 1] and CΛ,γ > 0, independent of h and k, such that 0 < k ≤ kΛ.Source: §3.2 · p. 19
  • Under Theorem 3.8’s assumptions, the error converges in probability as the discretization parameters tend to zero.The corollary’s proof uses the fixed-Λ time-step condition, Theorem 3.8, estimate (3.25), and Markov’s inequality.Source: §3.2 · p. 20

Appendix A. Auxiliary inequalities

Appendix A collects discrete inequalities used in the analysis: Jensen and martingale bounds, plus a summation-by-parts argument for decreasing events.

Appendix A. Auxiliary inequalities

  • The discrete Jensen inequality changes direction when q is between 0 and 1.For q ≥ 1, the appendix states the inequality for a sequence of values and positive time step k.Source: Appendix A · p. 21
  • A discrete Burkholder–Davis–Gundy inequality bounds martingale-difference sums in a Hilbert space.It assumes square-integrable-at-order-p differences with conditional mean zero, for p ≥ 2.Source: Appendix A · p. 21
  • For decreasing events, the summation-by-parts argument uses the nonnegativity of the penultimate term to establish its claim.The appendix completes the argument by taking the maximum over the index r.Source: Appendix A · p. 21

Appendix B. Technical consistency estimates

Appendix B develops technical estimates for the midpoint defect in the drift term, using the discrete geometric structure and inequalities to control the resulting terms.

Appendix B. Technical consistency estimates

  • The midpoint-defect estimates use the nodal unit-length property, discrete integration by parts, and discrete Agmon and Young inequalities.The proof snippets also invoke the elementwise product estimate and the triple-product identity for unit vectors.Source: Appendix B · pp. 22–24
  • Localized bounds for the form A_h have constants independent of h, k, and j in Lemmas B.2 and B.3.Lemma B.2 also allows a parameter η > 0; Lemma B.3 states the same parameter dependence.Source: Appendix B · pp. 22–23
  • Lemma B.4 combines the preceding estimates to obtain a bound with constants independent of h, k, and j.Its proof applies Lemmas B.2 and B.3 to drift-term differences and chooses η sufficiently small.Source: Appendix B · p. 24
  • The numerical-increment estimate follows for 0 < k ≤ 1 when 2γ < 1, and the gradient-increment contribution has moments of every order.The proof attributes the latter control to the second term in (3.6).Source: Appendix B · pp. 22, 25
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