Sharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data
Year: 2026
Type: preprint
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 $β\in (0, 2]$ from the initial data regularity index $μ\in (0, 2]$, 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 $O(h^β+ k^{β/2})$ for $β\in (0, 2)$ and $O\bigl(h^2 + k(1+\ln(t_n/k))^{1/2}\bigr)$ at the critical endpoint $β= 2$ 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.
Source: arXiv (Cornell University)
Authors: Naik, Jitendra Nath, Tripathi, Lok Pati
Institution: Indian Institute of Technology Goa
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171831849