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

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.