  Conservative and non-conservative phase-field models are considered for the
numerical simulation of lateral phase separation and coarsening in biological
membranes. An unfitted finite element method is devised for these models to
allow for a flexible treatment of complex shapes in the absence of an explicit
surface parametrization. For a set of biologically relevant shapes and
parameter values, the paper compares the dynamic coarsening produced by
conservative and non-conservative numerical models, its dependence on certain
geometric characteristics and convergence to the final equilibrium
