In this paper, we investigate numerically a diffuse interface model for the
Navier-Stokes equation with fluid-fluid interface when the fluids have
different densities \cite{Lowengrub1998}. Under minor reformulation of the
system, we show that there is a continuous energy law underlying the system,
assuming that all variables have reasonable regularities. It is shown in the
literature that an energy law preserving method will perform better for
multiphase problems. Thus for the reformulated system, we design a C0 finite
element method and a special temporal scheme where the energy law is preserved
at the discrete level. Such a discrete energy law (almost the same as the
continuous energy law) for this variable density two-phase flow model has never
been established before with C0 finite element. A Newton's method is
introduced to linearise the highly non-linear system of our discretization
scheme. Some numerical experiments are carried out using the adaptive mesh to
investigate the scenario of coalescing and rising drops with differing density
ratio. The snapshots for the evolution of the interface together with the
adaptive mesh at different times are presented to show that the evolution,
including the break-up/pinch-off of the drop, can be handled smoothly by our
numerical scheme. The discrete energy functional for the system is examined to
show that the energy law at the discrete level is preserved by our scheme