An efficient construction of divergence-free spaces in the context of exact finite element de Rham sequences

Exact finite element de Rham subcomplexes relate conforming subspaces in H1(Ω), H(curl;Ω), H(div;Ω), and L2(Ω) in a simple way by means of differential operators (gradient, curl, and divergence). The characteristics of such strong couplings are crucial for the design of stable and conservative discretizations of mixed formulations for a variety of multiphysics systems. This work explores these aspects for the construction of divergence-free vector shape functions in a robust fashion allowing stable and faster simulations of mixed formulations of incompressible porous media flows. The resulting schemes are verified by means of numerical tests with known smooth solutions and applied to a benchmark problem to confirm the expected theoretical and computational performance results.