School of Mathematical & Statistical Sciences Faculty Publications
Document Type
Article
Publication Date
1-1-2026
Abstract
We investigate a fully discrete finite element approximation for the stochastic Kuramoto–Sivashinsky equation, combining the standard finite element methods in spatial discretization with the implicit Euler–Maruyama scheme in time. Rigorous error estimates are established for two distinct noise regimes. In the case of bounded multiplicative noise, we prove optimal strong convergence rates in full expectation. The analysis relies crucially on a stochastic Gronwall inequality and an exponential stability estimate for the PDE solution, which together control the interplay between the nonlinear drift and the multiplicative stochastic forcing. For general multiplicative noise, where boundedness no longer holds, we derive sub-optimal convergence rates in probability by introducing a localization technique based on carefully constructed subsets of the sample space. This dual framework demonstrates that the proposed fully discrete scheme achieves strong convergence under bounded noise and probabilistic convergence under general multiplicative noise, thus providing the first comprehensive error analysis for numerical approximations of the stochastic Kuramoto–Sivashinsky equation. Numerical experiments are also provided to demonstrate the efficiency of the numerical method and validate the theoretical results.
Recommended Citation
Nguyen, Hung D., and Liet Vo. "Fully discrete finite element methods for the stochastic Kuramoto–Sivashinsky equation with multiplicative noise." Stochastics and Partial Differential Equations: Analysis and Computations (2026): 1-63. https://doi.org/10.1007/s40072-026-00438-8
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Publication Title
Stochastics and Partial Differential Equations Analysis and Computations
DOI
10.1007/s40072-026-00438-8
Comments
This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material.