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

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.