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

Abstract

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.

Convergent finite element based discretization of a stochastic two‐phase flow model

“…The proposed discrete scheme is based on the finite element method in space and Euler method combined with convex splitting method in time. Inspired by the approach of [9,19], the well-posedness of the scheme is obtained via an application of Brouwer's fixed point theorem and the convergence result is done by using the stochastic compactness method and the martingale representation theorem.…”