  We study a continuous data assimilation (CDA) algorithm for a
velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases:
nudging applied to the velocity and vorticity, and nudging applied to the
velocity only. We prove that under a typical finite element spatial
discretization and backward Euler temporal discretization, application of CDA
preserves the unconditional long-time stability property of the
velocity-vorticity method and provides optimal long-time accuracy. These
properties hold if nudging is applied only to the velocity, and if nudging is
also applied to the vorticity then the optimal long-time accuracy is achieved
more rapidly in time. Numerical tests illustrate the theory, and show its
effectiveness on an application problem of channel flow past a flat plate.
