We consider nonlinear solvers for the incompressible, steady Navier-Stokes equations in the setting where partial solution data is available, e.g. from physical measurements or sampled solution data from a (too big to send) very high-resolution computation. The measurement data is incorporated/assimilated into the solver through a nudging term addition that penalizes at each iteration the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented at each time step for time dependent dissipative PDEs. For a Picard solver, we quantify the acceleration provided by the data in terms of the density of the measurement locations and the level of noise in the data. For Newton, we show how the convergence basin for the initial condition is expanded as more data is assimilated.

Numerical tests illustrate the results. While the setting is for Navier-Stokes, the ideas are applicable to solvers for a wide range of nonlinear systems.