Optimal strong convergence of a splitting mixed finite element method for a stochastic Keller--Segel system

In this work, we develop and analyze a splitting mixed finite element method, combined with a fully implicit Euler time discretization, for a stochastic Keller–Segel system with general polynomial logistic growth and multiplicative noise. The proposed splitting formulation allows the use of standard finite element spaces without requiring the LBB inf–sup condition associated with traditional mixed finite element methods and, at each time step, avoids the need for solving a coupled nonlinear system. We first establish the well-posedness of the variational solution and derive new high-moment, higher-order regularity, and exponential stability estimates. These estimates provide the key analytical foundation for the strong convergence analysis of the numerical method. A principal contribution of this work is the derivation of error estimates directly in full expectation, without the use of localization arguments. By combining the new stability and regularity estimates with a stochastic Gronwall framework, we overcome the difficulties arising from the interaction between the nonlinear chemotactic coupling and the multiplicative noise. As a result, we establish strong convergence in high-moment norms and obtain optimal convergence rates with respect to both the temporal and spatial discretizations. In particular, the proposed method achieves the optimal rates of order $k^{\frac{1}{2}}$ in time and $h$ in space in the relevant $L^2$- and $H^1$-based error norms. These results provide, to our knowledge, the first optimal strong convergence analysis in full expectation for a splitting mixed finite element approximation of this class of stochastic Keller–Segel systems.