Mesh-Free Interpolant Observables for Continuous Data Assimilation

Abstract

This paper is dedicated to the expansion of the framework of general interpolant observables introduced by Azouani, Olson, and Titi for continuous data assimilation of nonlinear partial differential equations. The main feature of this expanded framework is its mesh-free aspect, which allows the observational data itself to dictate the subdivision of the domain via partition of unity in the spirit of the so-called Partition of Unity Method by Babuska and Melenk. As an application of this framework, we consider a nudging-based scheme for data assimilation applied to the context of the two-dimensional Navier-Stokes equations as a paradigmatic example and establish convergence to the reference solution in all higher-order Sobolev topologies in a periodic, mean-free setting. The convergence analysis also makes use of absorbing ball bounds in higher-order Sobolev norms, for which explicit bounds appear to be available in the literature only up to 𝐻²; such bounds are additionally proved for all integer levels of Sobolev regularity above 𝐻².