A Mixed Formulation for the Brinkman Problem

Abstract

The Brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium, where fractures, bubbles, or channels alternate inside a porous matrix. In this work, we explore a novel mixed formulation of the Brinkman problem based on the Hodge decomposition of the vector Laplacian. Introducing the flow's vorticity as an additional unknown, this formulation allows for a uniformly stable and conforming discretization by standard finite elements (Nedelec, Raviart--Thomas, piecewise discontinuous). A priori error estimates for the discretization error in the $H({\rm curl}; \Omega)-H({\rm div}; \Omega)-L^2(\Omega)$ norm of the solution, which are optimal with respect to the approximation properties of finite element spaces, are obtained. The theoretical results are illustrated with numerical experiments. Finally, the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for $H({\rm curl})$ and $H...