The form and evolution of multi-phase biomembranes is of fundamental
importance in order to understand living systems. In order to describe these
membranes, we consider a mathematical model based on a Canham--Helfrich--Evans
two-phase elastic energy, which will lead to fourth order geometric evolution
problems involving highly nonlinear boundary conditions. We develop a
parametric finite element method in an axisymmetric setting. Using a
variational approach, it is possible to derive weak formulations for the highly
nonlinear boundary value problems such that energy decay laws, as well as
conservation properties, hold for spatially discretised problems. We will prove
these properties and show that the fully discretised schemes are well-posed.
Finally, several numerical computations demonstrate that the numerical method
can be used to compute complex, experimentally observed two-phase biomembranes.