Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations, we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for the approximation of incompressible flow coupled problems. We exploit stabilized mixed finite elements together with Nitsche's type matching conditions that automatically adapt to the coupling of different combinations of coefficients. We study in details the stability of the method using weighted norms, emphasizing the robustness of the stability estimate with respect to the coefficients. We also consider an iterative method to split the coupled heterogeneous problem in possibly homogeneous local problems, and we investigate the spectral properties of suitable preconditioners for the solution of the global as well as local problems. Finally, we present the simulation of the fully coupled blood flow and plasma filtration problems on a realistic geometry of a cardiovascular artery after the implantation of a drug eluting stent (DES). A similar finite element method for mass transport is then employed to study the evolution of the drug released by the DES in the blood stream and in the arterial walls, and the role of plasma filtration on the drug deposition is investigated.