论文标题
随机Landau--lifshitz- bloch方程的数值方法和误差估计
Numerical method and Error estimate for stochastic Landau--Lifshitz--Bloch equation
论文作者
论文摘要
我们研究了用于求解$ \ Mathbb r^d $ for $ d = 1,2 $的有界域上的随机Landau-lifshitz-bloch（LLB）方程的准线性随机部分微分方程系统的数值方法。我们的主要结果是估计有限元方法与随机LLB溶液的收敛速率。为了克服在情况下缺乏解决方案的规律性$ d = 2 $，我们为方程式的正规版本提出了有限元方案。然后，我们获得了数值解的误差估计值和正则方程的溶液以及该溶液对随机LLB方程溶液的收敛速率。因此，得出了近似解的概率的收敛性，从而得出了与随机LLB方程溶液的溶液的收敛性。据我们所知，这是随机准线性偏微分方程系统错误估计的第一个结果。由于LLB方程的新规律性结果，因此在情况下获得了更强的结果，该结果使我们能够避免正则化。
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.
