Error analysis of a decoupled, linear and stable finite element method for Cahn–Hilliard–Navier–Stokes equations

•The considered numerical scheme is a totally decoupled, linear and unconditionally energy stable finite element method scheme.•A priori error estimation for the phase field, velocity field and pressure variables are obtained. In this paper, we carry out the error analysis for a totally decoupled, linear and unconditionally energy stable finite element method to solve the Cahn–Hilliard–Navier–Stokes equations. The fully finite element scheme is based on a stabilization for Cahn–Hilliard equation and projection method for Navier–Stokes equation, as well as the first order Euler method for time discretization. A priori error analysis for phase field, velocity field and pressure variable are derived for the fully discrete scheme.