带乘性噪声和非光滑初值的半线性随机偏微分方程全离散有限元方法的尖锐误差估计

中文摘要 AI 辅助

本文针对由乘性噪声驱动且具有非光滑初值的半线性抛物型随机偏微分方程（SPDEs）的全离散逼近，建立了尖锐的强误差估计。空间离散基于标准有限元方法，时间上采用线性隐式Euler格式。通过将扩散算子映射到负分数空间，我们的框架同时适用于迹类噪声和时空白噪声。对于非光滑初值，通过将噪声正则性参数$\beta \in (0,2)$与初值正则性参数$\mu \in (0,2]$解耦，我们推导出尖锐的正则性估计，将初值正则性的精确损失隔离为一个可积的时间奇异性。此外，我们建立了尖锐的强收敛速率$O(h^{\beta-\varepsilon} + k^{\frac{1}{2}\min\{\beta-\varepsilon, 1\}})$，其中$\varepsilon>0$且远离$t=0$。最后，我们在数值实验中考虑了物理相关的随机模型，如修正的Langmuir分数阶表面覆盖模型和抛物型Anderson模型，以验证理论收敛速率。

英文摘要

This article establishes sharp strong error estimates for the fully discrete approximation of semilinear parabolic stochastic partial differential equations (SPDEs) driven by multiplicative noise and subject to nonsmooth initial data. The spatial discretization is based on a standard finite element method, coupled with the linearly implicit Euler scheme in time. By mapping the diffusion operator into negative fractional spaces, our framework accommodates both trace-class and space-time white noise. For nonsmooth initial data, by decoupling the noise regularity parameter $β\in (0,2)$ from the initial data regularity parameter $μ\in (0,2]$, we derive sharp regularity estimates that isolate the exact loss of initial regularity into an integrable temporal singularity. Furthermore, we establish sharp strong convergence rates of $O(h^{β-\varepsilon} + k^{\frac{1}{2}\min\{β-\varepsilon, 1\}})$ for $\varepsilon>0$ away from $t = 0$. Finally, we consider physically relevant stochastic models, such as the modified Langmuir fractional surface coverage model and the parabolic Anderson model, in our numerical experiments to confirm the theoretical convergence rates.

发表机构

• School of Mathematics and Computer Science, Indian Institute of Technology Goa（印度果阿理工学院数学与计算机学院）

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