A second-order, mass-conservative, unconditionally stable and bound-preserving finite element method for the quasi-incompressible Cahn-Hilliard-Darcy system

A second-order numerical method is developed for solving the quasi-incompressible Cahn-Hilliard-Darcy system with the Flory-Huggins potential for two immiscible fluids of variable densities and viscosities in a porous medium or a Hele-Shaw cell. We show that the scheme is uniquely solvable, mass-conservative, bound-preserving and unconditionally energy stable. The key for bound-preserving is the utilization of second order convex-concave splitting of the logarithmic potential, and the discrete L¹ estimate of the singular potential. Ample numerical tests are reported to validate the accuracy and robustness of the proposed numerical scheme.