Unconditionally stable numerical methods for Cahn-Hilliard-Navier-Stokes-Darcy system with different densities and viscosities

In this article we consider the numerical modeling and simulation via phase field approach for coupled two-phase free flow porous media of different densities viscosities. The model consists Cahn-Hilliard-Navier-Stokes equations in region Cahn-Hilliard-Darcy that are by several domain interface conditions. It is showed satisfies an energy law. Then first propose a unconditionally stable finite element method solving analyze stability method. Furthermore, based on ideas pressure stabilization artificial compressibility, time stepping decouples computation variable, velocity flow, media, hence significantly reduces computational cost. decoupled scheme with spatial discretization rigorously established. We verify numerically our schemes convergent energy-law preserving. Numerical experiments also performed to illustrate features flows setting.