Optimal Error Estimates of a Finite Element Method for Semilinear SPDEs with Additive Noise and Nonsmooth Initial Data

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 $O(h^\beta)$ for the spatially semidiscrete approximation and $O(h^\beta + k^{\beta/2})$ for the fully discrete scheme away from $t = 0$, where the parameter $\beta \in (0, 2]$ characterizes the spatial regularity of the noise process. Numerical experiments confirm the theoretical convergence rates.