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 L2–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 C0– and C1–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.