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 multiplicative noise in two dimensions respect to periodic boundary conditions. Based on the Helmholtz decomposition of noise, semi-discrete and fully discrete time-stepping algorithms are proposed. The convergence rates for mixed finite element methods based time-space approximation probability velocity pressure obtained. Furthermore, establishing some stability using negative norm technique, partial expectations $$H^1$$ $$L^2$$ norms error proved converge optimally.