Convergence Analysis of a Bounds-Preserving Numerical Scheme for the Quasi-Incompressible Cahn-Hilliard-Darcy System

A first-order in-time bounds-preserving finite element method is analyzed for solving the quasi-incompressible Cahn-Hilliard-Darcy system with the Flory-Huggins potential for two-phase flows of variable densities and viscosities in porous media. The proposed scheme is uniquely solvable, bounds-preserving, mass-conservative, and unconditionally energy stable. By exploiting the Darcy equations and bounds-preservation of the numerical solution, we obtain stability estimate of the pressure gradient. An inductive rough error estimate then gives the strict separation property of the numerical solution. Finally, a refined l∞(H1)∩l2(H3) error estimate establishes the optimal convergence rate for the order parameter in the H1 norm.