The incompressible Navier-Stokes equations at high Reynolds numbers resist numerical solution. Iterative solvers slow down, fail to converge, or converge to the wrong solution when multiple solutions exist. The turbulent regime is where the equations matter most and where the numerics break down.
Fisher, Franklin, and Rebholz (arXiv:2603.22534) found that feeding partial observational data into the solver — continuous data assimilation, originally developed for weather forecasting — fixes the convergence problem. Not ameliorates it. Fixes it. The CDA-enhanced Uzawa solver converges for arbitrarily large Reynolds numbers, even when the equations admit multiple solutions.
The mechanism: observational data constrains the solution space. At high Reynolds numbers, the solution manifold is wide — many velocity fields satisfy the equations approximately. The solver wanders this manifold without converging. Data assimilation adds a nudging term that pulls the iterate toward the observations, collapsing the solution manifold to a neighborhood of the physically realized flow.
More data accelerates convergence. Noisy data still works. And once CDA brings the solver close enough, it can hand off to Newton's method for fast local convergence without further data. The observations bootstrap the nonlinear solve past the region where it would otherwise fail.
The through-claim: for nonlinear equations with multiple solutions, convergence isn't purely a mathematical problem — it's an observational one. The solver doesn't need a better algorithm; it needs information about which solution to converge to. Partial observations provide that information, and the convergence follows.