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

TL;DR

This paper extends a discontinuous Galerkin method to 3D coupled fluid-poroelastic models, analyzing efficiency and effects of interface conditions in brain fluid mechanics applications.

Contribution

It introduces a 3D PolyDG method for coupled MPE-Navier-Stokes systems with physiological interface conditions and demonstrates its stability, convergence, and computational advantages.

Findings

The method is stable and optimally convergent.

Polyhedral meshes improve computational efficiency.

Interface conditions significantly affect fluid dynamics results.

Abstract

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 the Beavers-Joseph-Saffman (BJS) conditions at the interface, modeling the friction between the fluid and the porous medium. Second, we 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 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. 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.