Stable across regimes: A mixed DG method for Darcy–Brinkman–Stokes type flows

Hydromechanical models of Darcy–Brinkman–Stokes type consider mass- and momentum conservation of an incompressible fluid on a domain with varying permeability. They include the two important limits of free flow governed by the classical Navier–Stokes equations and porous Darcy flow. The conceptual simplicity makes the model attractive from a modeling perspective, but any numerical solution procedure is challenged by the description of flow domains with different stability requirements. Furthermore, spatial variations in the dominant physics, such as strongly localized rapid variations in permeability adversely affect the conditioning of resulting linear systems. In this publication, we propose a discretization based on mixed discontinuous Galerkin Finite-Elements for the Darcy–Brinkman–Stokes model. The method leverages schemes proposed for the incompressible Navier–Stokes equations and for Darcy flow to obtain a robust solver in both flow limits, as well as for flows that sharply transition from high to low permeability regions. To this end, we introduce a mass-flux stabilization term. Using the discontinuous generalization of triangular and quadrilateral Taylor-Hood finite elements, our methods achieves near-optimal L 2 convergence in numerical experiments covering both Stokes and Darcy problems. We demonstrate robustness of the method on a variety of 2D numerical benchmark problems, including problems with discontinuous, anisotropic and time dependent permeability fields.