  The modeling of the interaction between a poroelastic medium and a fluid in a
hollow cavity is crucial for understanding, e.g., the multiphysics flow of
blood and Cerebrospinal Fluid (CSF) in the brain, the supply of blood by the
coronary arteries in heart perfusion, or the interaction between groundwater
and rivers or lakes. In particular, the cerebral tissue's elasticity and its
perfusion by blood and interstitial CSF can be described by Multi-compartment
Poroelasticity (MPE), while CSF flow in the brain ventricles can be modeled by
the (Navier-)Stokes equations, the overall system resulting in a coupled
MPE-(Navier-)Stokes system. The aim of this paper is three-fold. First, we aim
to extend and verify in a three-dimensional setting a discontinuous Galerkin
method on polytopal grids recently presented for the MPE-Stokes problem.
Second, we carry out the analysis of the method based on an extension of the
proposed formulation so that physics-based Beavers-Joseph-Saffman conditions
are taken into account at the interface: these conditions are essential to
model the friction between the fluid and the porous medium. Finally, by a
comparative numerical investigation, we assess the fluid-dynamics effects of
these boundary conditions and of employing either Stokes or Navier-Stokes
equations to model the CSF flow. The semidiscrete numerical scheme for the
coupled problem is proved to be stable and optimally convergent. Temporal
discretization is obtained using Newmark's $\beta$-method for the elastic wave
equation and the $\theta$-method for the remaining equations of the model. The
theoretical error estimates are verified by numerical simulations on a test
case with a manufactured solution, and a numerical investigation is carried out
on a three-dimensional geometry to assess the effects of interface conditions
and fluid inertia on the system.
