Global Recovery from Local Data: Interior Nudging for 2D Navier-Stokes equations in a Physical Domain

TL;DR

This paper demonstrates that in 2D Navier-Stokes equations, interior observations in a subregion can ensure global flow recovery, supported by rigorous theory and finite element computations showing robustness even beyond theoretical bounds.

Contribution

It extends data assimilation theory by proving global convergence using interior observations far from boundaries, supported by computational validation.

Findings

Global recovery is possible with interior observations far from the boundary.

Computational results show robustness beyond theoretical limits.

Interior observations near the boundary are largely uninformative.

Abstract

In many real-world applications of data assimilation (DA), the strategic placement of observers is crucial for effective and efficient forecasting. Motivated by practical constraints in sensor deployment, we show that global recovery of the flow field can be achieved using observations available only in a subregion of the domain, possibly far from the boundary. We focus on the two-dimensional incompressible Navier-Stokes equations posed in a bounded physical domain with Dirichlet boundary conditions. Building on the continuous data assimilation framework of Azouani, Olson, and Titi (2014), we rigorously prove that the assimilated solution converges globally to the true solution under suitable conditions on the nudging parameter, spatial resolution, and the geometry of the observation region, specifically, when the maximum distance from any point in the domain to the observational subregion is bounded by a constant multiple of \( \nu^{1/2} \) (in terms of scaling). Our computational results, conducted via finite element methods over complex geometries, support the theoretical findings and reveal even greater robustness in practice. Specifically, synchronization with the true solution is achieved even when the observational subregion lies farther from the rest of the domain than the theoretical threshold permits. Across all three tested scenarios, the local nudging algorithm performs comparably to full-domain assimilation, reaching global accuracy up to machine precision. Interestingly, observational data near the boundary are found to be largely uninformative. This demonstrates that full observability is not necessary: carefully chosen interior observations, even far from the boundary, can suffice.