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 regularized version of the equation. We then obtain error estimates of numerical solutions and for the solution of the regularized 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. 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 regularization.

中文翻译：

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随机Landau-Lifshitz-Bloch方程的数值方法和误差估计


在本文中，我们研究了在 ${\mathbb{R}}^{d}$ 有界域上求解拟线性随机偏微分方程组（称为随机 Landau–Lifshitz–Bloch (LLB) 方程）的数值方法，其中 ${\mathbb{R}}^{d}$ d=1,2$。我们的主要结果是有限元法对随机 LLB 解的收敛速度的估计。为了克服 $d=2$ 情况下解缺乏规律性的问题，我们提出了方程正则化版本的有限元方案。然后，我们获得数值解的误差估计和正则化方程的解，以及该解对随机 LLB 方程解的收敛速度。因此，推导出随机 LLB 方程解的近似解的概率收敛性。由于 LLB 方程的新正则性结果使我们能够避免正则化，因此在 $d=1$ 的情况下获得了更强的结果。