Fully discrete finite element methods for the stochastic Kuramoto-Sivashinsky equation with multiplicative noise
Hung D. Nguyen, Liet Vo
TL;DR
The paper develops and analyzes a fully discrete finite element method for the stochastic Kuramoto–Sivashinsky equation with multiplicative noise in one spatial dimension. By coupling a continuous FEM spatial discretization with an implicit Euler–Maruyama time integrator, it derives rigorous error estimates across noise regimes: optimal strong convergence in full expectation for bounded noise via a discrete stochastic Gronwall framework and exponential stability, and sub-optimal probabilistic convergence for general noise through localization. These results constitute the first comprehensive finite element error analysis for the SKS equation with multiplicative noise, highlighting techniques that may extend to other nonlinear SPDEs with non-Lipschitz drift. The work also clarifies the one-dimensional limitation and points toward future extensions to higher dimensions and broader classes of nonlinear SPDEs.
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.