Similar to aerospace, civil or mechanical engineering, biomedical engineering applications require the handling of very large scale problems. The successful treatment of such applications needs to couple different mathematical models (viscous and inviscid flows, free flows and porous media) or similar models characterized by very heterogeneous coefficients. The numerical approximation of coupled problems requires particular attention for the treatment of the interfaces (where boundary layers may appear) that have to be handled at both the continuous and discrete level by means of suitable transmission conditions.

For the numerical approximation of initial/boundary value problems, coupling heterogeneous elliptic or parabolic partial differential equations, we will discuss a unified treatment based on continuous-discontinuous finite elements, where the interface conditions are enforced by means of interior-penalty methods. As stated in the seminal work of Arnold, interior penalties represent an effective tool to handle the peculiarity of such problems because the local nature of the trial space and the capability to regulate the degree of smoothness of the approximate solution by local variation of the penalty weighting function should enable close approximation of solutions which vary in character from one part of the domain to another and should allow the incorporation of partial knowledge of the solution into the scheme. In reference to cardiovascular engineering applications, we will review and analyze some examples of this approach applied to the discretization of coupled incompressible flows (Navier/Stokes --- Oseen --- Darcy models) and of advection-diffusion-reaction equations with heterogeneous coefficients.

Figure: Numerical simulation of coupled blood flow, intramural plasma filtration and drug release from a drug eluting stent