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

TL;DR

This paper proves strong convergence and provides explicit rates for finite element schemes solving the stochastic Landau--Lifshitz--Bloch equation, with stability and uniqueness results in one dimension, supported by numerical experiments.

Contribution

It establishes the first strong convergence rates for finite element schemes applied to the stochastic Landau--Lifshitz--Bloch equation, including stability and invariant measure analysis.

Findings

Proved strong convergence of finite element schemes with explicit rates.

Established mean-square exponential stability and uniqueness of invariant measure in 1D.

Numerical experiments confirm theoretical convergence and stability results.

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 $L^2(\Omega)$ 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.