Analysis of fully discrete Crank-Nicolson finite element methods for a stochastic Keller-Segel chemotaxis system with gradient-type multiplicative noise

We develop and analyze numerical methods for a stochastic Keller-Segel system perturbed by Stratonovich noise, which models chemotactic behavior under randomly fluctuating environmental conditions. The proposed fully discrete scheme couples a Crank-Nicolson time discretization with a splitting mixed finite element method in space. We rigorously prove the stability of the numerical scheme and establish strong convergence rates of order $O(k^{1/2} + k^{-1/2}h^2)$, where $k$ and $h$ denote the time and spatial step sizes, respectively. Notably, the presence of stochastic forcing leads to an inverse dependence on $k$ in the error estimates, distinguishing the convergence behavior from that of the deterministic case. Numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness and accuracy of the proposed methods.