A Block-Diagonal Algebraic Multigrid Preconditioner for the Brinkman Problem

Abstract

The Brinkman model is a unified law governing the flow of a viscous fluid in cavity (Stokes equations) and 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 (Nedelec, Raviart--Thomas, discontinuous piecewise polynomials). Based on the stability analysis of the problem in the $H({\rm curl})-H({\rm div})-L^2$ norms [P. S. Vassilevski and U. Villa, A mixed formulation for the Brinkman problem, SIAM J. Numer. Anal., submitted], we study a scalable block-diagonal preconditioner which is provably optimal in the constant coefficient case. Such a preconditioner takes advantage of the parallel auxiliary space AMG solvers for $H({\rm curl})$ and $H({\rm div})$ problems available in HYPRE [hypre: High Performance Preconditioners, http://www.llnl.gov/CASC/hypre/]. The theoretical results are illust...