Agus L. Soenjaya | Search arXiv

arXiv2026

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

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.

math.NA↗

arXiv2026

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

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 $γ\in(0,\frac12)$, we prove first-order spatial convergence and temporal convergence of order $γ$ 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.

math.NA↗