  In this work a solver for instationary two-phase flows on the basis of the
extended Discontinuous Galerkin (extended DG/XDG) method is presented. The XDG
method adapts the approximation space conformal to the position of the
interface. This allows a sub-cell accurate representation of the incompressible
Navier-Stokes equations in their sharp interface formulation. The interface is
described as the zero set of a signed-distance level-set function and
discretized by a standard DG method. For the interface, resp. level-set,
evolution an extension velocity field is used and a two-staged algorithm is
presented for its construction on a narrow-band. On the cut-cells a monolithic
elliptic extension velocity method is adapted and a fast-marching procedure on
the neighboring cells. The spatial discretization is based on a symmetric
interior penalty method and for the temporal discretization a moving interface
approach is adapted. A cell agglomeration technique is utilized for handling
small cut-cells and topology changes during the interface motion. The method is
validated against a wide range of typical two-phase surface tension driven flow
phenomena including capillary waves, an oscillating droplet and the rising
bubble benchmark.
