  We present a novel asymptotic-preserving semi-implicit finite element method
for weakly compressible and incompressible flows based on compatible finite
element spaces. The momentum is sought in an $H(\mathrm{div})$-conforming
space, ensuring exact pointwise mass conservation at the discrete level. We use
an explicit discontinuous Galerkin-based discretization for the convective
terms, while treating the pressure and viscous terms implicitly, so that the
CFL condition depends only on the fluid velocity. To handle shocks and damp
spurious oscillations in the compressible regime, we incorporate an a
posteriori limiter that employs artificial viscosity and is based on a discrete
maximum principle. By using hybridization, the final algorithm requires solving
only symmetric positive definite linear systems. As the Mach number approaches
zero and the density remains constant, the method converges to an
$H(\mathrm{div})$-based discretization of the incompressible Navier-Stokes
equations in the vorticity-velocity-pressure formulation. Several numerical
tests validate the proposed method.
