  In this work, we consider unfitted finite element methods for the numerical
approximation of the Stokes problem. It is well-known that this kind of methods
lead to arbitrarily ill-conditioned systems. In order to solve this issue, we
consider the recently proposed aggregated finite element method, originally
motivated for coercive problems. However, the well-posedness of the Stokes
problem is far more subtle and relies on a discrete inf-sup condition. We
consider mixed finite element methods that satisfy the discrete version of the
inf-sup condition for body-fitted meshes, and analyze how the discrete inf-sup
is affected when considering the unfitted case. We propose different aggregated
mixed finite element spaces combined with simple stabilization terms, which can
include pressure jumps and/or cell residuals, to fix the potential deficiencies
of the aggregated inf-sup. We carry out a complete numerical analysis, which
includes stability, optimal a priori error estimates, and condition number
bounds that are not affected by the small cut cell problem. For the sake of
conciseness, we have restricted the analysis to hexahedral meshes and
discontinuous pressure spaces. A thorough numerical experimentation bears out
the numerical analysis. The aggregated mixed finite element method is
ultimately applied to two problems with non-trivial geometries.
