Numerical and convergence analysis of the stochastic Lagrangian averaged Navier-Stokes equations

TL;DR

This paper develops and analyzes a finite element space-time discretization method for stochastic Lagrangian averaged Navier-Stokes equations, establishing convergence results for both vanishing and fixed regularization parameters.

Contribution

It introduces a novel numerical scheme for stochastic LANS-$ extalpha$ equations and provides rigorous convergence analysis under different regimes of the regularization parameter.

Findings

Convergence to stochastic Navier-Stokes solutions as $ extalpha$ vanishes.

Convergence to stochastic LANS-$ extalpha$ solutions with fixed $ extalpha$.

Uniform estimates established for discretization and regularization parameters.

Abstract

The primary emphasis of this work is the development of a finite element based space-time discretization for solving the stochastic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale $\alpha$, namely we first assume $\alpha$ to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when $\alpha$ is fixed. A preparatory analysis of uniform estimates in both $\alpha$ and discretization parameters is carried out. Starting out from the stochastic LANS-$\alpha$ model, we achieve convergence toward the continuous strong solutions of the stochastic Navier-Stokes equations in 2D when $\alpha$ vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-$\alpha$ model is accomplished if $\alpha$ is fixed.