Victoria Luongo Fisher, Jessica C. Franklin, Leo G. Rebholz
This paper shows how continuous data assimilation (CDA) can be used to provably enable and accelerate convergence of a (efficient at each iteration due to a physics-splitting, but generally slowly converging and not robust) nonlinear solver for incompressible Navier-Stokes equations (NSE). Herein we develop, analyze and test an Uzawa-based nonlinear solver for incompressible NSE that incorporates partial solution data into the iteration through continuous data assimilation (CDA-Uzawa). We rigorously prove that i) CDA-Uzawa will accelerate a converging Uzawa iteration, and more partial solution data yields more acceleration, and ii) with enough partial solution data CDA-Uzawa will converge for arbitrarily large Reynolds numbers, even if multiple NSE solutions exist. In the case of noisy data, we prove that the convergence results hold down to the size of the noise, and we propose a strategy to pass to Newton once CDA-Uzawa convergence reaches its lower limit. Results of several numerical tests illustrate the theory and show CDA-Uzawa is a very effective and efficient solver. While this paper focuses a particular splitting-based solver for the NSE, the key ideas are quite general and extendable to a wide class of nonlinear solvers.