  The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally
tractable model that can be used to effectively capture interfacial dynamics in
two-phase fluid flows. In this work, we present a semi-implicit,
projection-based finite element framework for solving the CHNS system. We use a
projection-based semi-implicit time discretization for the Navier-Stokes
equation and a fully-implicit time discretization for the Cahn-Hilliard
equation. We use a conforming continuous Galerkin (cG) finite element method in
space equipped with a residual-based variational multiscale (RBVMS)
formulation. Pressure is decoupled using a projection step, which results in
two linear positive semi-definite systems for velocity and pressure, instead of
the saddle point system of a pressure-stabilized method. All the linear systems
are solved using an efficient and scalable algebraic multigrid (AMG) method. We
deploy this approach on a massively parallel numerical implementation using
parallel octree-based adaptive meshes. The overall approach allows the use of
relatively large time steps with much faster time-to-solve than similar
fully-implicit methods. We present comprehensive numerical experiments showing
detailed comparisons with results from the literature for canonical cases,
including the single bubble rise and Rayleigh-Taylor instability.
