Analysis of fully discrete mixed finite element scheme for Stochastic Navier-Stokes equations with multiplicative noise

TL;DR

This paper develops and analyzes fully discrete mixed finite element schemes for 2D stochastic Navier-Stokes equations with multiplicative noise, demonstrating convergence and stability properties in probability.

Contribution

It introduces new time-stepping algorithms based on Helmholtz decomposition and proves their convergence rates for velocity and pressure in stochastic settings.

Findings

Convergence rates in probability for velocity and pressure.

Optimal convergence of velocity error in partial expectations.

Stability analysis using negative norm technique.

Abstract

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 partial expectations of the $H^1$ and $L^2$ norms of the velocity error are proved to converge optimally.