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Strong convergence of finite element schemes for the stochastic Landau--Lifshitz--Bloch equation

Agus L. Soenjaya

TL;DR

The paper addresses the stochastic Landau--Lifshitz--Bloch equation for high-temperature micromagnetics and develops strong convergence results for semi-implicit and implicit finite element schemes in 1D and 2D, including a regularised 2D formulation. It introduces localisation-based error analysis and new exponential moment bounds for the exact solution, yielding mean-square exponential stability and, in 1D with small noise, uniqueness of the invariant measure. The authors derive sharp convergence rates in $L^2(\Omega)$ and, in 1D, optimal probabilistic rates in $H^1$ for various schemes, supported by numerical experiments validating the theory. The work provides rigorous justification and practical guidance for stochastic FEM discretisations in micromagnetics and sheds light on stability properties under thermal noise.

Abstract

The dynamics of magnetisation in a bounded ferromagnet in $\mathbb{R}^d$ ($d=1,2$) at high temperatures can be described by the stochastic Landau--Lifshitz--Bloch (sLLB) equation, which is a vector-valued quasilinear stochastic partial differential equation. In this paper, assuming adequate regularity of the initial data, we establish strong convergence in $L^2(Ω)$ of several semi-implicit and implicit fully discrete finite element schemes for the sLLB equation, together with explicit convergence rates. The analysis relies on localised error estimates and new exponential moment bounds for the exact solution. As a by-product, these moment bounds yield mean-square exponential stability of solutions and uniqueness of the invariant measure in one spatial dimension under a small noise assumption. We also sharpen existing convergence-in-probability results for the numerical schemes. Numerical experiments are presented to illustrate and support the theoretical findings.

Strong convergence of finite element schemes for the stochastic Landau--Lifshitz--Bloch equation

TL;DR

The paper addresses the stochastic Landau--Lifshitz--Bloch equation for high-temperature micromagnetics and develops strong convergence results for semi-implicit and implicit finite element schemes in 1D and 2D, including a regularised 2D formulation. It introduces localisation-based error analysis and new exponential moment bounds for the exact solution, yielding mean-square exponential stability and, in 1D with small noise, uniqueness of the invariant measure. The authors derive sharp convergence rates in and, in 1D, optimal probabilistic rates in for various schemes, supported by numerical experiments validating the theory. The work provides rigorous justification and practical guidance for stochastic FEM discretisations in micromagnetics and sheds light on stability properties under thermal noise.

Abstract

The dynamics of magnetisation in a bounded ferromagnet in () at high temperatures can be described by the stochastic Landau--Lifshitz--Bloch (sLLB) equation, which is a vector-valued quasilinear stochastic partial differential equation. In this paper, assuming adequate regularity of the initial data, we establish strong convergence in of several semi-implicit and implicit fully discrete finite element schemes for the sLLB equation, together with explicit convergence rates. The analysis relies on localised error estimates and new exponential moment bounds for the exact solution. As a by-product, these moment bounds yield mean-square exponential stability of solutions and uniqueness of the invariant measure in one spatial dimension under a small noise assumption. We also sharpen existing convergence-in-probability results for the numerical schemes. Numerical experiments are presented to illustrate and support the theoretical findings.
Paper Structure (10 sections, 22 theorems, 195 equations, 8 figures)

This paper contains 10 sections, 22 theorems, 195 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Analytical preliminaries: well-posedness, stability, and invariant measures
  4. A semi-implicit finite element approximation of the sLLB equation for $d=1$
  5. A semi-implicit finite element approximation of the regularised sLLB equation for $d=2$
  6. An implicit finite element approximation of the sLLB equation for $d=1$
  7. Numerical experiments
  8. Simulation 1
  9. Simulation 2
  10. Simulation 3

Key Result

Theorem 3.3

Let $d=1$. Suppose that $\boldsymbol{u}_0\in \mathbb{H}^s$, where $s\in \{2,3,4\}$. There exists a unique pathwise solution $\boldsymbol{u}\in L^p(\Omega; L^\infty_T(\mathbb{H}^s) \cap L^2_T(\mathbb{H}^{s+1}))$ to equ:sllb. Furthermore, for this pathwise solution, there exists a constant $C:=C(p,T, for $p\in [1,\infty)$ and $\alpha\in (0,\frac{1}{2})$.

Figures (8)

  • Figure 1: Snapshots of a sample path of the magnetisation $\boldsymbol{u}$ in simulation 1 at selected times.
  • Figure 2: Convergence orders of the magnetisation vector field in simulation 1.
  • Figure 3: Energy evolution in simulation 1
  • Figure 4: Snapshots of a sample path of the magnetisation $\boldsymbol{u}$ in simulation 2 at selected times.
  • Figure 5: Convergence orders of the magnetisation vector field in simulation 2.
  • ...and 3 more figures

Theorems & Definitions (45)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • proof
  • Proposition 3.6
  • proof
  • Corollary 3.7
  • ...and 35 more