Optimal error estimates for a semi-discrete finite element scheme of Cahn-Hilliard-Navier-Stokes-Darcy model

In this paper, we present a rigorous optimal error analysis for a semi-discrete finite element method of the coupled Cahn-Hilliard-Navier-Stokes-Darcy model. The optimal convergence order for the phase variable in the L ∞ ( 0 , T ; H 1 ) norm and the velocity in the L ∞ ( 0 , T ; L 2 ) norm are proved with two key techniques. One key technique is establishing the discrete L 2 ( 0 , T ; H 2 ) ∩ L 2 ( 0 , T ; L ∞ ) boundedness of the numerical solution for the phase variable to handle the nonlinear regional coupling terms. Another key technique adopts the inverse inequality and Ritz projection to bound the L ∞ norm of the error of the phase variable in the nonlinear terms that depend on the viscosity and the mobility. This is the first convergence analysis for the sophisticated Cahn-Hilliard-Navier-Stokes-Darcy system. Numerical experiments are also presented to verify the theory and validate the spatial semi-discretization finite element scheme.