A mixed finite element method for the stochastic Boussinesq equations with multiplicative noise

This work investigates a fully discrete mixed finite element method for the stochastic Boussinesq system driven by multiplicative noise. The spatial discretization is performed using a standard mixed finite element method, while the temporal discretization is based on a semi-implicit Euler-Maruyama scheme. By combining a localization technique with high-moment stability estimates, we establish error bounds for the velocity, pressure, and temperature approximations. As a direct consequence, we prove convergence in probability for the fully discrete method in both $L^2$ and $H^1$-type norms. Several numerical experiments are presented to validate the theoretical error estimates and demonstrate the effectiveness of the proposed scheme.