A Mixed Finite Element Method for Modeling the Fluid Exchange Between Microcirculation and Tissue Interstitium

Thanks to dimensional (or topological) model reduction techniques, small inclusions in a three-dimensional (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 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 with applications to microcirculation. In this way we calculate, in the bulk and on the 1D manifold simultaneously, the approximation of velocity and pressure fields that guarantees good accuracy with respect to mass conservation.