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

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