Continuous data assimilation of a discretized barotropic vorticity model of geophysical flow

We consider continuous data assimilation applied to a finite element spatial discretization and backward difference temporal discretization of the barotropic vorticity model of geophysical flow. We prove that with sufficient measurement data and properly chosen nudging parameter (guided by our analysis), the proposed algorithm achieves optimal long-time accuracy for any initial condition. While our analysis requires nudging of both the streamfunction and vorticity, our numerical tests indicate that nudging only the streamfunction can be sufficient.