Strong rates of convergence of space-time discretization schemes for the 2D Navier-Stokes equations with additive noise

TL;DR

This paper establishes strong convergence rates for fully implicit space-time discretization schemes applied to the 2D Navier-Stokes equations with additive noise, showing a convergence rate of up to 1 in space and nearly 0.5 in time.

Contribution

It provides the first proof of strong convergence rates for fully implicit schemes for 2D stochastic Navier-Stokes equations with additive noise, utilizing exponential moment bounds and a discrete Gronwall lemma.

Findings

Convergence rate in space is exactly 1.

Convergence rate in time is up to 0.5, matching solution regularity.

Method applies without localization, simplifying analysis.

Abstract

We consider the strong solution of the 2D Navier-Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the rate of convergence of the schemes is $\eta\in[0,1/2)$ in time and 1 in space. Let us mention that the coefficient $\eta$ is equal to the time regularity of the solution with values in $\LL^2$. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Our main idea is to use a discrete Gronwall lemma for the error estimate without any localization.