  The paper develops a solver based on a conforming finite element method for a
3D--1D coupled incompressible flow problem. New coupling conditions are
introduced to ensure a suitable bound for the cumulative energy of the model.
We study the stability and accuracy of the discretization method, and the
performance of some state-of-the-art linear algebraic solvers for such flow
configurations. Motivated by the simulation of the flow over inferior vena cava
(IVC) filter, we consider the coupling of a 1D fluid model and a 3D fluid model
posed in a domain with anisotropic inclusions. The relevance of our approach to
realistic cardiovascular simulations is demonstrated by computing a blood flow
over a model IVC filter.
