  We present an adaptive variational procedure for unstructured meshes to
capture fluid-fluid interfaces in two-phase flows. The two phases are modeled
by the phase-field finite element formulation, which involves the conservative
Allen-Cahn equation coupled with the incompressible Navier-Stokes equations.
The positivity preserving variational formulation is designed to maintain the
bounded and stable solution of the Allen-Cahn equation. For the adaptivity
procedure, we consider the residual-based error estimates for the underlying
differential equations of the two-phase system. In particular, the adaptive
refinement/coarsening is carried out by the newest vertex bisection algorithm
by evaluating the residual error indicators based on the error estimates of the
Allen-Cahn equation. The coarsening algorithm avoids the storage of the tree
data structures for the hierarchical mesh, thus providing the ease of numerical
implementation. The proposed nonlinear adaptive partitioned procedure aims at
reducing the amount of coarsening while maintaining the convergence properties
of the underlying nonlinear coupled differential equations. We investigate the
adaptive phase-field finite element scheme through the spinodal decomposition
in a complex curved geometry and the volume-conserved interface motion driven
by the mean curvature flow for two circles in a square domain. We then assess
the accuracy and efficiency of the proposed procedure by modeling the
free-surface motion in a sloshing tank. The mesh adaptivity remarkably reduces
the degrees of freedom and the computational cost by nearly half for similar
accuracy and improves the mass conservation. Finally, we apply the adaptive
numerical framework to solve the application of a dam-breaking problem with
topological changes.
