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The primary emphasis of this work is the development of a finite elementbased space-time discretization for solving the stochastic Lagrangian averagedNavier-Stokes (LANS-$\alpha$) equations of incompressible fluid turbulence withmultiplicative random forcing, under nonperiodic boundary conditions within abounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. Theconvergence analysis of a fully discretized numerical scheme is investigatedand split into two cases according to the spacial scale $\alpha$, namely wefirst assume $\alpha$ to be controlled by the step size of the spacediscretization so that it vanishes when passing to the limit, then we providean alternative study when $\alpha$ is fixed. A preparatory analysis of uniformestimates in both $\alpha$ and discretization parameters is carried out.Starting out from the stochastic LANS-$\alpha$ model, we achieve convergencetoward the continuous strong solutions of the stochastic Navier-Stokesequations in 2D when $\alpha$ vanishes at the limit. Additionally, convergencetoward the continuous strong solutions of the stochastic LANS-$\alpha$ model isaccomplished if $\alpha$ is fixed.

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