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\begin{document}

\title{HIGH MOMENT AND PATHWISE ERROR ESTIMATES FOR FULLY DISCRETE MIXED FINITE
ELEMENT APPROXIMATIONS OF THE STOCHASTIC STOKES EQUATIONS WITH
MULTIPLICATIVE NOISE}



\author[1]{LIET VO}%
\affil[1]{The University of Tennessee Knoxville Department of Mathematics}%


\vspace{-1em}



  
  \date{May 25, 2024}


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\begin{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. Optimal rates of convergence are established
for all \emph{p}th moment errors for \emph{p}[?]2 using a novel bootstrap
technique. The almost optimal rates of convergence are then obtained
using Kolmogorov's theorem based on the high moment error estimates.
Unlike for the velocity error estimate, the high moment and pathwise
error estimates for the pressure approximation are proved 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. Finally, numerical experiments are also provided to validate
the proven theoretical results and to examine the dependence/growth of
the error constants on the moment order \emph{p}.%
\end{abstract}%



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