A least-squares/finite element method for the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line

In this article we discuss the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall. The method we employ combines a finite element space approximation with a time discretization by operator-splitting. To solve the Cahn–Hilliard part of the problem, we use a least-squares/conjugate gradient method. We also show that the scheme has the total energy decaying in time property under certain conditions. Our numerical experiments indicate that the method discussed here is accurate, stable and efficient.