High moment and pathwise error estimates for fully discrete mixed finite element approximattions of stochastic Navier-Stokes equations with additive noise

TL;DR

This paper develops high moment and pathwise error estimates for fully discrete mixed finite element methods applied to stochastic Navier-Stokes equations with additive noise, providing theoretical bounds and numerical validation.

Contribution

It introduces novel high moment and pathwise error estimates for stochastic Navier-Stokes equations using implicit Euler-Maruyama and mixed finite element methods.

Findings

High moment error estimates for velocity and pressure approximations.

Pathwise error estimates derived using Kolmogorov Theorem.

Spatial error constants grow as O(k^{-1/2}) with time step size.

Abstract

This paper is concerned with high moment and pathwise error estimates for fully discrete mixed finite element approximattions of stochastic Navier-Stokes equations with general additive noise. The implicit Euler-Maruyama scheme and standard mixed finite element methods are employed respectively for the time and space discretizations. High moment error estimates for both velocity and a time-avraged pressure approximations in strong $L^2$ and energy norms are obtained, pathwise error estimates are derived by using the Kolmogorov Theorem. Unlike their derterministic counterparts, the spatial error constants grow in the order of $O(k^{-\frac12})$, where $k$ denotes time step size. Numerical experiments are also provided to validate the error estimates and their sharpness.