Versatile mixed methods for non-isothermal incompressible flows

The purpose of this paper is to extend the versatile mixed methods originally developed by Chen and Williams for isothermal flows in "Versatile Mixed Methods for the Incompressible Navier-Stokes Equations," Computers & Mathematics with Applications, Volume 80, Number 6, 2020, and use them to simulate non-isothermal incompressible flows. These new mixed methods are particularly interesting, as with only minor modifications they can be applied to a much broader range of flows, including non-isothermal weakly-compressible flows, and fully-compressible flows. In the main body of this paper, we carefully develop these mixed methods for solving the Boussinesq model equations. Thereafter, we establish the L2-stability of the discrete temperature and velocity fields. In addition, we obtain rigorous error estimates for the temperature and velocity fields. Finally, we assess the practical behavior of the methods by applying them to a set of well-known convection problems.