Time-discretization of stochastic 2-D Navier--Stokes equations with a penalty-projection method

TL;DR

This paper analyzes a time-discretization scheme for stochastic 2-D Navier--Stokes equations using a penalty-projection method, establishing error estimates and convergence rates for velocity and pressure variables.

Contribution

It introduces a novel penalty-projection time-discretization method for stochastic Navier--Stokes equations with proven convergence and error bounds.

Findings

Convergence rate of order 1/4 in probability for the main algorithm.

Strong convergence of the scheme for velocity and pressure variables.

Error estimates derived for the proposed discretization method.

Abstract

A time-discretization of the stochastic incompressible Navier--Stokes problem by penalty method is analyzed. Some error estimates are derived, combined, and eventually arrive at a speed of convergence in probability of order 1/4 of the main algorithm for the pair of variables velocity and pressure. Also, using the law of total probability, we obtain the strong convergence of the scheme for both variables.