Error analysis for 2D stochastic Navier--Stokes equations in bounded domains with Dirichlet data

TL;DR

This paper develops a finite-element method for 2D stochastic Navier-Stokes equations with Dirichlet boundary conditions, proving optimal convergence rates in probability, and introduces a novel approach using discrete stopping times to handle boundary-specific challenges.

Contribution

It establishes optimal convergence rates for finite-element discretisation of stochastic Navier-Stokes equations in bounded domains with Dirichlet data, using a new discrete stopping time approach.

Findings

Proves convergence rate of order 1/2 in time and 1 in space in probability.

Introduces a novel method based on discrete stopping times for boundary problems.

Achieves results previously known only in space-periodic cases.

Abstract

We study a finite-element based space-time discretisation for the 2D stochastic Navier-Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space. This was previously only known in the space-periodic case, where higher order energy estimates for any given (deterministic) time are available. In contrast to this, in the Dirichlet-case estimates are only known for a (possibly large) stopping time. We overcome this problem by introducing an approach based on discrete stopping times. This replaces the localised estimates (with respect to the sample space) from earlier contributions.