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

TL;DR

This work develops a rigorous framework for the pathwise uniform convergence in probability of a fully discrete finite-element scheme for the two-dimensional stochastic Navier-Stokes equations with multiplicative noise and no-slip boundary conditions. Employing a $P_3/P_2$ Taylor–Hood spatial discretization and an Euler time-stepping method, the authors establish nearly $3/2$-order spatial convergence and nearly $1/2$-order temporal convergence in probability, leveraging stopping times and localized error analysis. The analysis hinges on a detailed regularity theory for global mild solutions, stability properties of the discrete operators, and sophisticated probabilistic decompositions to control irregular Brownian forcing. The results provide a robust baseline for numerical approximations of SNSEs under no-slip boundaries and illuminate the challenges unique to no-slip conditions, with potential extensions to pressure, $L^q$-norms, and three-dimensional cases.

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.

Theorems & Definitions (28)

• Lemma 2.1
• Proposition 3.1
• Remark 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Lemma 4.1