  We study numerical schemes for incompressible Navier-Stokes equations using
IMEX temporal discretizations, finite element spacial discretizations, and
equipped with continuous data assimilation (a technique recently developed by
Azouani, Olson, and Titi in 2014). We analyze stability and accuracy of the
proposed methods, and are able to prove well-posedness, long time stability,
and long time accuracy estimates, under restrictions of the time step size and
data assimilation parameter. We give results for several numerical tests that
illustrate the theory, and show that, for good results, the choice of
discretization parameter and element choices can be critical.
