Error Analysis of a Decoupled, Linear Stabilization Scheme for the Cahn-Hilliard Model of Two-Phase Incompressible Flows

Here, we carry out rigorous error analysis for a first-order in time, linear, fully decoupled and energy stable scheme for solving the Cahn-Hilliard phase-field model of two-phase incompressible flows, namely Cahn-Hilliard-Navier-Stokes problem (Shen and Yang, SIAM J Numer Anal, 2015). The error estimates are for phase field variable, chemical potential, velocity and further the pressure in L2 norm and L infinity norm. The scheme combines the projection method, the explicit stabilizing decoupling technique, and the linear stabilization approach together. We further derive the boundness of numerical solution in L infinity norm with the mathematical deduction, and deal with the complex splitting error arising from the decoupling technique. Optimal error estimates are derived for the semi-discrete-in-time scheme.