发表机构
UNSW Sydney; Institute of Analysis and Scientific Computing, TU Wien（新南威尔士大学; 维也纳工业大学分析与科学计算研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带乘性噪声的随机Landau--Lifshitz--Gilbert方程，提出全离散保结构有限元中点格式，在自然离散能量范数下证明空间一阶、时间γ阶收敛，为首个此类格式的收敛速率结果。
AI 中文摘要
随机Landau--Lifshitz--Gilbert（sLLG）方程是一类强非线性随机偏微分方程，具有非凸逐点约束，出现在微磁学理论中。我们分析了在有界区间上带彩色乘性Stratonovich噪声的sLLG方程的全离散、保结构有限元逼近。该方法采用连续分段仿射有限元、质量集中和中点时间离散，以在有限元节点上精确保持单位长度约束。在初始数据和噪声的适当正则性假设下，我们建立了均匀高阶矩稳定性，并发展了该格式的误差分析。该分析利用了方程的几何结构和随机中点离散。对于每个γ∈(0,1/2)，我们在自然离散能量范数下证明了空间一阶收敛和时间γ阶收敛，在任意大概率事件上局部均方意义下成立，从而在概率意义下也成立。据我们所知，这是求解随机Landau--Lifshitz--Gilbert方程的全离散保结构有限元格式的首个收敛速率结果。
英文摘要
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.
