An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System (Communications in Computational Physics, 2013)
We present an unconditionally energy stable and uniquely solvable finitedifference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprisedof a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes modeland describes two phase very viscous flows in porous media. The scheme is based ona convex splitting of the discrete CH energy and is semi-implicit. The equations at theimplicit time level are nonlinear, but we prove that they represent the gradient of astrictly convex functional and are therefore uniquely solvable, regardless of time stepsize. Owing to energy stability, we show that the scheme is stable in the time and spacediscrete $ℓ^∞$(0,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliardpart, and a method based on the Vanka smoothing strategy for the Brinkman part – forsolving these equations. In particular, we provide evidence that the solver has nearlyoptimal complexity in typical situations. The solver is applied to simulate spinodaldecomposition of a viscous fluid in a porous medium, as well as to the more generalproblems of buoyancy- and boundary-driven flows.