Numerical approximation of the stochastic Navier-Stokes equations through artificial compressibility

TL;DR

This paper introduces a finite element and Euler-based numerical scheme for approximating the two-dimensional stochastic Navier-Stokes equations, with proven convergence under optimal conditions on the artificial compressibility parameter.

Contribution

It presents a novel pseudo-compressibility method for stochastic Navier-Stokes equations with rigorous convergence analysis and optimal parameter conditions.

Findings

Convergence of the numerical scheme to the unique strong solution.

Optimal conditions on the artificial compressibility parameter for convergence.

The scheme effectively approximates stochastic Navier-Stokes dynamics.

Abstract

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier-Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a parameter $\epsilon$. Space and time are discretized through a finite element approximation and an Euler method. The convergence analysis of the suggested numerical scheme is investigated throughout this paper. It is based on a local monotonicity property permitting the convergence toward the unique strong solution of the Navier-Stokes equations to occur within the originally introduced probability space. Justified optimal conditions are imposed on the parameter $\epsilon$ to ensure convergence within the best rate.