Discontinuous Galerkin method for a three-dimensional coupled fluid-poroelastic model with applications to brain fluid mechanics

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 a recently presented discontinuous Galerkin method on polytopal grids (PolyDG) to incorporate three-dimensional geometries and physiological interface conditions. Regarding the latter, we consider here the Beavers-Joseph-Saffman (BJS) conditions at the interface: These conditions are essential to model the friction between the fluid and the porous medium. Second, we quantitatively analyze the computational efficiency of the proposed method on a domain with small geometrical features, thus demonstrating the advantages of employing polyhedral meshes. Finally, by a comparative numerical investigation, we assess the fluid-dynamics effects of the BJS 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 elastodynamics 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.