  We analyze and test a simple-to-implement two-step iteration for the
incompressible Navier-Stokes equations that consists of first applying the
Picard iteration and then applying the Newton iteration to the Picard output.
We prove that this composition of Picard and Newton converges quadratically,
and our analysis (which covers both the unique solution and non-unique solution
cases) also suggests that this solver has a larger convergence basin than usual
Newton because of the improved stability properties of Picard-Newton over
Newton. Numerical tests show that Picard-Newton dramatically outperforms both
the Picard and Newton iterations, especially as the Reynolds number increases.
We also consider enhancing the Picard step with Anderson acceleration (AA), and
find that the AAPicard-Newton iteration has even better convergence properties
on several benchmark test problems.
