Compatible finite elements–also known as Finite Element Exterior Calculus [1]–have demonstrated significant advantages in structure-preserving discretizations of incompressible flows, see e.g. [2,3]. However, their potential for simulating compressible flows remains largely unexplored, with most current approaches relying on Finite Volume (FV), Discontinuous Galerkin (DG), or Continuous Galerkin (CG) methods. In this talk, we address this gap by presenting a semi-implicit compatible finite element method for weakly compressible flows, see [4]. Our scheme has the following key properties: • Asymptotic preservation: In the low Mach number limit with constant density, the method consistently recovers a discretization of the incompressible Navier–Stokes equations. • Exact mass conservation: In the incompressible regime, the discrete velocity field is exactly divergence-free, owing to the use of an H(div, Ω)-conforming momentum discretization. • Shock capturing: A MOOD-style a posteriori artificial viscosity mechanism ensures robust shock resolution. • Computational efficiency: The method requires solving only symmetric positive definite (SPD) systems at each time step, and the CFL condition remains independent of the Mach number. We conclude with implementation insights and present numerical results that demonstrate the accuracy, robustness, and efficiency of the proposed method. REFERENCES [1] D. N. Arnold, R. S. Falk, R.Winther Finite element exterior calculus, homological techniques, and applications. Acta Numerica. 2006;15:1–155 [2] Christoph Lehrenfeld, Joachim Schoeberl, High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows, Computer Methods in Applied Mechanics and Engineering, Volume 307, 2016, Pages 339-361, [3] A. Palha, M. Gerritsma, A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations, Journal of Computational Physics, Volume 328, 2017, Pages 200–220, [4] E. Zampa and M. Dumbser. An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements. Journal of Computational Physics, Volume: 521, 113551, 2025.