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

Sônia M. Gomes, State University of Campinas, Brazil

Abstract. Exact finite element de Rham complexes relate conforming subspaces in $H^1$, $H(curl)$, $H(div)$, and $L^2$ 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 discretization 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 resolution of the associated saddle-point problem can be reduced to two consecutive computation steps: one for the  flux and the next one for the pressure (for cases where it is required). The formulation for the flux is immediately equivalent to a standard Galerkin variational problem with positive definite linear system and reduced number of degrees of freedom. Pressure is obtained by a post-processing algorithm. This reduced divergence-free model can also be extended to applications in the context of element-wise divergence-constant fluxes. 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. This is a joint work with Philippe R. B. Devloo, Jeferson W. D. Fernandes, Francisco Orlandini, and Nathan Shauer.