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 β∈ (0, 2] from the initial data regularity index μ∈ (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(hβ+ kβ/2) for β∈ (0, 2) and O(h2 + k(1+ln(tn/k))1/2) at the critical endpoint β= 2 for any fixed tn > 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.
Related
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BibTeX
@misc{naik2026sharp,
title = {{Sharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data}},
author = {Jitendra Nath Naik and Lok Pati Tripathi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.26797},
eprint = {2607.26797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.26797},
}
RIS
TY - UNPB
TI - Sharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data
AU - Jitendra Nath Naik
AU - Lok Pati Tripathi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2607.26797
UR - https://arxiv.org/abs/2607.26797
AB - 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∈ (0, 2] from the initial data regularity index μ∈ (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∈ (0, 2) and O(h^2 + k(1+ln(t_n/k))^1/2) 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.
ER -