Fully discrete finite element approximation of the stochastic Cahn–Hilliard–Navier–Stokes system

In this paper we study the numerical approximation of the stochastic Cahn–Hilliard–Navier–Stokes system on a bounded polygonal domain of $\mathbb{R}^{d}$, $d=2,3$. We propose and analyze an algorithm based on the finite element method and a semiimplicit Euler scheme in time for a fully discretization. We prove that the proposed numerical scheme satisfies the discrete mass conservative law, has finite energies and constructs a weak martingale solution of the stochastic Cahn–Hilliard–Navier–Stokes system when the discretization step (both in time and in space) tends to zero.