On the use of balanced H(div)-L 2 pair of approximation spaces for divergence-free simulations of Stokes and coupled Stokes-Darcy equations

The performance of different classic or more recent finite element formulations for Stokes and coupled Stokes-Darcy problems are discussed. Discontinuous, H 1-conforming, or H(div)-conforming velocity approximations can be used for Stokes flows, the formulations being presented in a unified framework. Special emphasis is given to the former H(div)-conforming formulation, for which only the tangential velocity components require a penalization treatment. For incompressible fluids, this method naturally gives exact divergence-free velocity fields, a property that few schemes can achieve. When combined with classic mixed formulation for Darcy's flows, a strongly conservative scheme is derived for the treatment of coupled Stokes-Darcy problems. Unlike other methods existing in the literature, this technique can use the same combination of approximation spaces in both flow regions, and simplifies the enforcement of the coupling Stoke-Darcy interface conditions. All the methods are implemented using an object-oriented computational environment, and typical test problems are simulated to illustrate the properties of the different approximations, verifying errors, and rates of convergence. Then, an application to the simulation of a model representing self-compacting concrete flow around reinforcing bars is presented. An homogenization technique is applied to interpret the reinforced bar domain by a Darcy's law, while a Stokes flow is considered in the remaining domain