  This paper develops an efficient and robust solution technique for the steady
Boussinesq model of non-isothermal flow using Anderson acceleration applied to
a Picard iteration. After analyzing the fixed point operator associated with
the nonlinear iteration to prove that certain stability and regularity
properties hold, we apply the authors' recently constructed theory for Anderson
acceleration, which yields a convergence result for the Anderson accelerated
Picard iteration for the Boussinesq system. The result shows that the leading
term in the residual is improved by the gain in the optimization problem, but
at the cost of additional higher order terms that can be significant when the
residual is large. We perform numerical tests that illustrate the theory, and
show that a 2-stage choice of Anderson depth can be advantageous. We also
consider Anderson acceleration applied to the Newton iteration for the
Boussinesq equations, and observe that the acceleration allows the Newton
iteration to converge for significantly higher Rayleigh numbers that it could
without acceleration, even with a standard line search.
