A Mixed Formulation for the Brinkman Problem

The Brinkman model is a unified law governing the flow of a viscous fluid in cavity (Stokes equations) and in porous media (Darcy equations). In this work, we explore a novel mixed formulation of the Brinkman problem. Introducing the flow’s vorticity as additional unknown, this formulation allows for a uniformly stable and conforming discretization by standard finite elements (Nédélec, Raviart-Thomas, piecewise discontinuous). The theoretical results are illustrated with numerical experiments. Based on our stability analysis of the problem in the H(curl; Ω)−H(div; Ω)− L2(Ω) norms, we derive, in a follow-up paper ([34]), a scalable block diagonal preconditioner which takes advantage of the auxiliary space AMG solvers for H(curl) and H(div) problems available in the preconditioning library hypre ([19]) developed at LLNL.