  A novel finite element framework is proposed for the numerical simulation of
two phase flows with surface tension. The Level-Set (LS) method with piece-wise
quadratic (P2) interpolation for the liquid-gas interface is used in order to
reach higher-order convergence rates in regions with smooth interface. A
balanced-force implementation of the continuum surface force model is used to
take into account the surface tension and to solve static problems as
accurately as possible. Given that this requires a balance between the
discretization used for the LS function, and that used for the pressure field,
an equal-order P2/P2/P2 scheme is proposed for the Navier-Stokes and LS
advection equations, which are strongly coupled with each other. This fully
implicit formulation is stabilized using the residual-based variational
multiscale framework. In order to improve the accuracy and obtain optimal
convergence rates with a minimum number of elements, an anisotropic mesh
adaption method is proposed where the unstructured mesh is kept as fine as
possible close to the zero iso-value of the P2 LS function. Elements are
automatically stretched in regions with flat interface in order to keep the
complexity fixed during the simulation. The accuracy and efficiency of this
approach are demonstrated for two and three dimensional simulations of a rising
bubble.
