Analysis of continuous data assimilation scheme for the Navier-Stokes equations using variational multiscale method

In this study, we analyse a continuous data assimilation (CDA) scheme which enables us to combine an observable data with a numerical method to obtain better solutions in which these solutions are also closely similar to the current state of the system. The scheme is applied on a Navier-Stokes system which is discretized with two-step Backward Differentiation Formula (BDF2) in time and finite element in space. In order to improve the accuracy and prevent some non-physical oscillations due to the effect of small viscosity and the dominance of convection, a projection based variational multiscale method (VMS) has also been applied to the system. We present the long-time stability and long-time convergence analyses of the scheme in details and several numerical tests in order to support theoretical findings and demonstrate the promise of the method.