Numerical method and error estimate for stochastic Landau–Lifshitz–Bloch equation

Ben Goldys, Chunxi Jiao and Kim-Ngan Le

Abstract

In this paper 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.

Keywords: Landau-Lifshitz-Bloch equation, quasilinear stochastic PDE, multiplicative noise,finite element method, regularisation, bi-laplacian.

AMS Subject Classification: Primary 60H15; secondary 60H30, 60H35, 65M60,82D40.