High moment and pathwise error estimates for fully discrete mixed finite element approximations of the Stochastic Stokes Equations with Multiplicative Noises

TL;DR

This paper develops high moment and pathwise error estimates for fully discrete mixed finite element methods applied to the stochastic Stokes equations with multiplicative noise, providing insights into convergence rates and noise effects.

Contribution

It introduces a bootstrap technique for high moment error estimates and analyzes the impact of noise types on convergence rates for velocity and pressure.

Findings

High moment error estimates are established for velocity and pressure.

Pathwise error estimates are obtained using Kolmogorov's theorem.

Noise type influences convergence rates significantly.

Abstract

This paper is concerned with high moment and pathwise error estimates for both velocity and pressure approximations of the Euler-Maruyama scheme for time discretization and its two fully discrete mixed finite element discretizations. The main idea for deriving the high moment error estimates for the velocity approximation is to use a bootstrap technique starting from the second moment error estimate. The pathwise error estimate, which is sub-optimal in the energy norm, is obtained by using Kolmogorov's theorem based on the high moment error estimates. Unlike for the velocity error estimate, the higher moment and pathwise error estimates for the pressure approximation are derived in a time-averaged norm. In addition, the impact of noise types on the rates of convergence for both velocity and pressure approximations is also addressed.