Finite Element Software and Performance for Network Models with Multipliers

Hydraulic network models of flow in the microcirculation can give rise to mixed-domain partial differential equations; this is due to the regularity requirement on the flux and the 1D-0D coupling that emerges if Lagrange multipliers are introduced to impose bifurcation conditions. Although the finite element method is a natural approach to solving these problems, the implementation of mixed-domain finite element routines is generally non-trivial. The FEniCS and FEniCSx finite element libraries provide support for general mixed-domain finite element problems, and this chapter investigates their performance for hydraulic network models. We find that legacy FEniCS is limited to small networks due to the poor time complexity of the assembly routines. By contrast, the mixed-domain routines in FEniCSx have linear time complexity in the number of network branches, an important factor when solving large-scale hydraulic network problems.