LAUSR

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 $$L^2$$ L 2 norm and $$L^{\infty }$$ L ∞ 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 $$L^\infty $$ L ∞ 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.