An Efficient,Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman(CHB)system,which is comprised of 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 model and describes two phase very viscous flows in porous media.The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit.The equations at the implicit time level are nonlinear,but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable,regardless of time step size.Owing to energy stability,we show that the scheme is stable in the time and space discrete ℓ^(∞)(0,T;H_(h)^(1))and ℓ^(2)(0,T;H_(h)^(2))norms.We also present an efficient,practical nonlinear multigrid method–comprised of a standard FAS method for the Cahn-Hilliard part,and a method based on the Vanka smoothing strategy for the Brinkman part-for solving these equations.In particular,we provide evidence that the solver has nearly optimal complexity in typical situations.The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium,as well as to the more general problems of buoyancy-and boundary-driven flows.