Sharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data

Abstract

We establish sharp strong error estimates for a fully discrete approximation of semilinear stochastic convection--diffusion equations driven by additive noise subject to nonsmooth initial data. The problem is discretized in space using a standard Galerkin finite element method and in time via the linearly implicit Euler scheme. By decoupling the noise regularity index $\beta \in (0,2]$ from the initial data regularity index $\mu \in (0,2]$, we derive time-weighted spatial and temporal regularity bounds that isolate the initial singularity at $t=0$ into an integrable temporal weight. By allowing the drift operator $F$ to map into negative-order fractional spaces to incorporate the convection term, we prove sharp strong convergence rates of order $O(h^{\beta }+k^{\beta /2})$ for $\beta \in (0,2)$ and $O(h^2+k(1+\ln (t_n/k){)}^{1/2})$ at the critical endpoint $\beta =2$ for any fixed $t_n>0$, without requiring the regularity of the initial data to match that of the noise. Furthermore, these convergence rates are achieved under a relaxed differentiability condition requiring only a Lipschitz continuous first Gâteaux derivative of $F$. Numerical experiments validate the theoretical results across both trace-class and space-time white noise regimes.

arXiv commentThe title has been updated to reflect the stochastic convection-diffusion framework explicitly. Major revisions include extending the analysis under relaxed first-order Gâteaux differentiability assumptions on the drift and streamlining the presentation to focus on the fully discrete scheme with refined regularity and error estimates