An adaptive a posteriori error estimator based finite element method for the
numerical solution of a coupled Cahn-Hilliard/Navier-Stokes system with a
double-obstacle homogenous free (interfacial) energy density is proposed. A
semi-implicit Euler scheme for the time-integration is applied which results in
a system coupling a quasi-Stokes or Oseen-type problem for the fluid flow to
a variational inequality for the concentration and the chemical potential according
to the Cahn-Hilliard model [13]. A Moreau-Yosida regularization is
employed which relaxes the constraints contained in the variational inequality
and, thus, enables semi-smooth Newton solvers with locally superlinear convergence
in function space. Moreover, upon discretization this yields a mesh
independent method for a fixed relaxation parameter. For the finite dimensional
approximation of the concentration and the chemical potential piecewise
linear and globally continuous finite elements are used, and for the numerical
approximation of the fluid velocity Taylor-Hood finite elements are employed.
The paper ends by a report on numerical examples showing the efficiency of the
new method