Numerical method and Error estimate for stochastic Landau--Lifshitz--Bloch equation

TL;DR

This paper develops finite element methods for the stochastic Landau-Lifshitz-Bloch equation, providing the first error estimates for such stochastic quasilinear PDEs and analyzing convergence rates in different dimensions.

Contribution

It introduces a finite element scheme for a regularised version of the stochastic LLB equation and establishes error estimates and convergence rates, including a new regularity result for the 1D case.

Findings

Error estimates for finite element solutions of stochastic LLB

Convergence in probability of approximate solutions

First error analysis for stochastic quasilinear PDEs

Abstract

We study numerical methods for solving a system of quasilinear stochastic partial differential equations known as the stochastic Landau-Lifshitz-Bloch (LLB) equation on a bounded domain in $\mathbb R^d$ for $d=1,2$. Our main results are estimates of the rate of convergence of the Finite Element Method to the solutions of stochastic LLB. To overcome the lack of regularity of the solution in the case $d=2$, we propose a Finite Element scheme for a regularised version of the equation. We then obtain error estimates of numerical solutions and for the solution of the regularised equation as well as the rate of convergence of this solution to the solution of the stochastic LLB equation. As a consequence, the convergence in probability of the approximate solutions to the solution of the stochastic LLB equation is derived. To the best of our knowledge this is the first result on error estimates for a system of stochastic quasilinear partial differential equations. A stronger result is obtained in the case $d=1$ due to a new regularity result for the LLB equation which allows us to avoid regularisation.