Introduction... 

One of the main issues for continuum-based simulations concerns the relatively high order of even the simplest evolution equations.  For example, the classical bilayer mechanics theory developed by Helfrich (see also the work of Canham and Evans) involves a bending energy that is quadratic in the principle curvatures of the membrane surface.  This gives rise to Euler-Lagrange equations that are fourth-order.     Proposed models for surface composition, akin to Cahn-Hilliard equations, are similarly fourth-order in the composition field.  Given the complex geometry and possible configurations attainable with GUVs, classical high-order finite difference and spectral methods are essentially impractical.
 
  Classical finite-element methods are also problematic for fourth-order problems.  The parent space of functions in this case is the second-order Sobolev space $H^2(\Omega)$, which, roughly speaking, requires the use of smooth $C^1(\Omega)$ shape functions on the membrane surface.   With good reason, recent work by Feng and Klug has moved away from classical shell finite elements and toward $C^1$-conforming subdivision surfaces.  Other recent work incorporating smooth basis functions includes that of Ayton et al., who relied on a smoothed-particle hydrodynamics (SPH)-based method. 
 
  Although these approaches have their advantages, neither is capable of reasonably simulating the topology changes (e.g. sphere to toroid) required to predict equilibrium configurations in GUVs.  The only method with such capability appears to be that of Qiang Du and coworkers, employing a phase-field regularization of (essentially) the Canham-Helfrich-Evans biomembrane theory.  This treatment is appealing, but it is not a particularly efficient approach, as it requires the use of a uniform Cartesian grid and embeds the interface within a higher-dimensional domain.  Further, while their most recent work has allowed for spatial variations in elastic moduli, they have yet to couple their approach with composition dynamics.