A Block-Diagonal Algebraic Multigrid Preconditioner 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 by introducing the flow’s vorticity as an additional unknown. This formulation allows for a uniformly stable and conforming discretization by standard finite element (Nédélec, Raviart-Thomas, discontinuous piecewise polynomials). Based on the stability analysis of the problem in the H(curl)−H(div)− L2 norms ([24]), we study a scalable block diagonal preconditioner which is provably optimal in the constant coefficient case. Such preconditioner takes advantage of the parallel auxiliary space AMG solvers for H(curl) and H(div) problems available in hypre ([11]). The theoretical results are illustrated by numerical experiments.