NAVIER-STOKES USE THE SOLUBLE DESIGN METHOD TO EVALUATE FINITE ELEMENTS (Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2023)
We propose a data assimilation algorithm for the  Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the  Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the L2 and H1 norms of error are bounded by a constant times a power of the Voigt-regularization parameter α > 0, plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as α goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the H2 norm