Analysis of Fully Discrete Mixed Finite Element Scheme for Stochastic Navier-Stokes Equations with Multiplicative Noise

This paper is concerned with stochastic incompressible Navier-Stokes equations with multiplicative noise in two dimensions with respect to periodic boundary conditions. Based on the Helmholtz decomposition of the multiplicative noise, semi-discrete and fully discrete time-stepping algorithms are proposed. The convergence rates for mixed finite element methods based time-space approximation with respect to convergence in probability for the velocity and the pressure are obtained. Furthermore, with establishing some stability and using the negative norm technique, the strong optimal $H^1$ and $L^2$ convergence of this scheme for the velocity are proved.