  We present an adaptive simulation framework for binary-fluid flows, based on
the Abels-Garcke-Gr\"un Navier-Stokes-Cahn-Hilliard (AGG NSCH)
diffuse-interface model. The adaptive-refinement procedure is guided by a
two-level hierarchical a-posteriori error estimate, and it effectively resolves
the spatial multiscale behavior of the diffuse-interface model. To improve the
robustness of the solution procedure and avoid severe time-step restrictions
for small-interface thicknesses, we introduce an $\varepsilon$-continuation
procedure, in which the diffuse interface thickness ($\varepsilon$) are
enlarged on coarse meshes, and the mobility is scaled accordingly. To further
accelerate the computations and improve robustness, we apply a modified
Backward Euler scheme in the initial stages of the adaptive-refinement
procedure in each time step, and a Crank--Nicolson scheme in the final stages
of the refinement procedure. To enhance the robustness of the nonlinear
solution procedure, we introduce a partitioned solution procedure for the
linear tangent problems in Newton's method, based on a decomposition of the
NSCH system into its NS and CH subsystems. We conduct a systematic
investigation of the conditioning of the monolithic NSCH tangent matrix and of
its NS and CH subsystems for a representative 2D model problem. To illustrate
the properties of the presented adaptive simulation framework, we present
numerical results for a 2D oscillating water droplet suspended in air, and we
validate the obtained results versus those of a corresponding sharp-interface
model.
