# Convergence rates for the numerical approximation of the 2D stochastic Navier-Stokes equations

## TL;DR

This paper establishes improved convergence rates for finite-element approximations of 2D stochastic Navier-Stokes equations, achieving linear spatial and nearly half-order temporal convergence, surpassing previous results.

## Contribution

The paper introduces a novel analysis of the pressure function using stochastic pressure decomposition to achieve better convergence rates.

## Key findings

- Linear convergence in space
- Almost 1/2 order convergence in time
- Improves previous 1/4 order temporal convergence

## Abstract

We study stochastic Navier-Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measure in the $L^\infty_tL^2_x\cap L^2_tW^{1,2}_x$-norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from [E. Carelli, A. Prohl: Rates of convergence for discretizations of the stochastic incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 50(5), 2467-2496. (2012)] where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.