On the use of divergence balanced H(div)-L2 pair of approximation spaces for divergence-free and robust simulations of Stokes, coupled Stokes–Darcy and Brinkman problems

• Published in: Mathematics and computers in simulation Vol. 170; pp. 51 - 78
• Main Authors: Carvalho, Pablo G.S., Devloo, Philippe R.B., Gomes, Sônia M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2020
• Subjects: Coupled Stokes–Darcy flow; Discontinuous Galerkin; Divergence-free simulation; Mixed finite elements methods; Stokes flow; Discontinuous Galerkin; Coupled Stokes–Darcy flow; Mixed finite elements methods; Divergence-free simulation; Stokes flow
• ISSN: 0378-4754; 1872-7166
• DOI: 10.1016/j.matcom.2019.09.002

The performance of different classic or more recent finite element formulations for Stokes, coupled Stokes–Darcy and Brinkman problems is discussed. Discontinuous, H1-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 Stokes–Darcy interface conditions. Typical test problems are simulated to illustrate the properties of the different approximations, verifying errors, rates of convergence, and divergence-free realization. Then, an application to the simulation of a model representing self-compacting concrete flow around reinforcing bars is presented. A 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. The methods are also applied to a Brinkman problem, involving physical phenomena ranging from Stokes to Darcy physical limit regimes, to illustrate the robustness of the H(div)-conforming formulation to treat all these scenarios. All the methods are implemented using an object-oriented computational environment. •Approximations for Stokes flow compared in a unified framework.•Study of stabilization parameters for symmetric and non-symmetric formulations.•Strongly conservative H(div) spaces for Stokes–Darcy.•A homogenization by a Darcy’s law domain for a Stokes flow around obstacles.•A comparative study for the Brinkman problem, verifying Darcy and Stokes limits.