AI 中文总结
该研究针对带加性噪声和非光滑初始数据的半线性SPDE，采用有限元结合线性隐式欧拉格式，推导了正时间处的最优强误差估计并经数值实验验证。
AI 中文摘要
本文对由加性噪声驱动且受非光滑初始数据约束的半线性抛物型随机偏微分方程（SPDE）的半离散和全离散近似进行强误差分析。空间离散基于标准有限元方法，时间采用线性隐式欧拉格式。在低正则初始条件下，推导了精确的空间和时间正则性估计，将初始正则性的损失转化为可积的时间奇点，从而在正时间处建立最优强误差估计。具体而言，证明了空间半离散近似的强收敛阶为$O(h^\beta)$，全离散格式在远离$t=0$处的强收敛阶为$O(h^\beta + k^{\beta/2})$，其中参数$\beta \u2208 (0, 2]$表征噪声过程的空间正则性。数值实验验证了理论收敛阶。
英文摘要
This paper presents a strong error analysis of both semidiscrete and fully discrete approximations for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive 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. Under low-regularity initial conditions, we derive sharp spatial and temporal regularity estimates that isolate the loss of initial regularity into an integrable temporal singularity, allowing us to establish optimal strong error estimates for positive times. Specifically, we prove strong convergence rates of order $O(h^β)$ for the spatially semidiscrete approximation and $O(h^β+ k^{β/2})$ for the fully discrete scheme away from $t = 0$, where the parameter $β\in (0, 2]$ characterizes the spatial regularity of the noise process. Numerical experiments confirm the theoretical convergence rates.
