  In this article we present a numerical framework based on continuum models
for the fluid dynamics and the CO$_2$ gas distribution in the alveolar sacs of
the human lung during expiration and inspiration, including the gas exchange to
the cardiovascular system. We include the expansion and contraction of the
geometry by means of the Arbitrary Lagrangian Eulerian (ALE) method. For
discretisation, we use equal-order finite elements in combination with
pressure-stabilisation techniques based on local projections or interior
penalties. We derive formulations for both techniques that are suitable on
arbitrarily anisotropic meshes. These formulations are novel within the ALE
method. Moreover, we investigate the effect of different boundary conditions,
that vary between inspiration and expiration. We present numerical results on a
simplified two-dimensional alveolar sac geometry and investigate the influence
of the pressure stabilisations as well as the boundary conditions
