Pathwise uniform convergence of numerical approximations for a two-dimensional stochastic Navier-Stokes equation with no-slip boundary conditions

TL;DR

This paper proves that fully discrete finite-element methods for 2D stochastic Navier-Stokes equations with no-slip boundaries converge uniformly along paths, achieving near 1.5 order in space and 0.5 order in time.

Contribution

It establishes the pathwise uniform convergence rates of finite-element approximations for stochastic Navier-Stokes equations with no-slip boundary conditions.

Findings

Achieves nearly 1.5-order spatial convergence

Achieves nearly 0.5-order temporal convergence

Validates the effectiveness of finite-element methods for stochastic fluid dynamics

Abstract

This paper investigates the pathwise uniform convergence in probability of fully discrete finite-element approximations for the two-dimensional stochastic Navier-Stokes equations with multiplicative noise, subject to no-slip boundary conditions. We demonstrate that the full discretization achieves nearly $ 3/2$-order convergence in space and nearly half-order convergence in time.