Mixed finite element methods for coupled 3D/1D fluid problems

Thanks to dimensional (or topological) model reduction techniques, small inclusions in a 3D continuum can be described as one-dimensional (1D) concentrated sources in order to reduce the computational cost of simulations. However, concentrated sources lead to singular solutions that still require computationally expensive graded meshes to guarantee accurate approximation. The main computational barrier consists in the ill-posedness of restriction operators (such as the trace operator) applied on manifolds with co-dimension larger than one. We overcome the computational challenges of approximating PDEs on manifolds with high dimensionality gap by means of nonlocal restriction operators that combine standard traces with mean values of the solution on low dimensional manifolds. This new embedded multiscale approach has the fundamental advantage of enabling the approximation of the problem using Galerkin projections on Hilbert spaces, which could not be otherwise applied because of regularity issues. This approach, previously applied to second order PDEs, is extended here to the mixed formulation of flow problems. In this way we obtain, in the bulk and on the immersed manifold, a simultaneous approximation of velocity and pressure field that guarantees good accuracy with respect to mass conservation. We have built a suitable solver based on the C++ finite element library GetFEM++ and we have named it Computational Embedded Multiscale Approach (CEMA). The solver is able to simulate fluid transport in a permeable biological tissue perfused by a vessel network with arbitrary topology. CEMA may be a valid support in the analysis of many microcirculation-dependent diseases. Possible actual or potential applications are: (i) lymphatic clearance of the brain with its significance for neurodegenerative diseases, (ii) simulation of drug delivery in cancer treatment or (iii) the study of peripheral circulation in haemodialysis.