  A finite element method for the evolution of a two-phase membrane in a sharp
interface formulation is introduced. The evolution equations are given as an
$L^2$--gradient flow of an energy involving an elastic bending energy and a
line energy. In the two phases Helfrich-type evolution equations are
prescribed, and on the interface, an evolving curve on an evolving surface,
highly nonlinear boundary conditions have to hold. Here we consider both
$C^0$-- and $C^1$--matching conditions for the surface at the interface. A new
weak formulation is introduced, allowing for a stable semidiscrete parametric
finite element approximation of the governing equations. In addition, we show
existence and uniqueness for a fully discrete version of the scheme. Numerical
simulations demonstrate that the approach can deal with a multitude of
geometries. In particular, the paper shows the first computations based on a
sharp interface description, which are not restricted to the axisymmetric case.
