Fully discrete finite element methods for the stochastic Kuramoto-Sivashinsky equation with multiplicative noise

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.