  We propose some new mixed finite element methods for the time dependent
stochastic Stokes equations with multiplicative noise, which use the Helmholtz
decomposition of the driving multiplicative noise. It is known [16] that the
pressure solution has a low regularity, which manifests in sub-optimal
convergence rates for well-known inf-sup stable mixed finite element methods in
numerical simulations, see [10]. We show that eliminating this gradient part
from the noise in the numerical scheme leads to optimally convergent mixed
finite element methods, and that this conceptual idea may be used to retool
numerical methods that are well-known in the deterministic setting, including
pressure stabilization methods, so that their optimal convergence properties
can still be maintained in the stochastic setting. Computational experiments
are also provided to validate the theoretical results and to illustrate the
conceptional usefulness of the proposed numerical approach.
