  Versatile mixed finite element methods were originally developed by Chen and
Williams for isothermal incompressible flows in "Versatile mixed methods for
the incompressible Navier-Stokes equations," Computers & Mathematics with
Applications, Volume 80, 2020. Thereafter, these methods were extended by
Miller, Chen, and Williams to non-isothermal incompressible flows in "Versatile
mixed methods for non-isothermal incompressible flows," Computers & Mathematics
with Applications, Volume 125, 2022. The main advantage of these methods lies
in their flexibility. Unlike traditional mixed methods, they retain the
divergence terms in the momentum and temperature equations. As a result, the
favorable properties of the schemes are maintained even in the presence of
non-zero divergence. This makes them an ideal candidate for an extension to
compressible flows, in which the divergence does not generally vanish. In the
present article, we finally construct the fully-compressible extension of the
methods. In addition, we demonstrate the excellent performance of the resulting
methods for weakly-compressible flows that arise near the incompressible limit,
as well as more strongly-compressible flows that arise near Mach 0.5.
