An adaptive finite element Moreau–Yosida-based solver for a coupled Cahn–Hilliard/Navier–Stokes system

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 [16]. 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.