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The Canham-Helfrich-Evans models of biomembranes consist of a family ofgeometric constrained variational problems. In this article, we compare twoclasses of numerical methods for these variational problems based on piecewiselinear (PL) and subdivision surfaces (SS). Since SS methods are based on splineapproximation and can be viewed as higher order versions of PL methods, one mayexpect that the only difference between the two methods is in the accuracyorder. In this paper, we prove that a numerical method based on minimizing anyone of the `PL Willmore energies' proposed in the literature would fail toconverge to a solution of the continuous problem, whereas a method based onminimization of the bona fide Willmore energy, well-defined for SS but not PLsurfaces, succeeds. Motivated by this analysis, we propose also aregularization method for the PL method based on techniques from conformalgeometry. We address a number of implementation issues crucial for theefficiency of our solver. A software package called Wmincon accompanies thisarticle, provides parallel implementations of all the relevant geometricfunctionals. When combined with a standard constrained optimization solver, thegeometric variational problems can then be solved numerically. To this end, werealize that some of the available optimization algorithms/solvers are capableof preserving symmetry, while others manage to break symmetry; we explore theconsequences of this observation.

Numerical Methods for Biomembranes: conforming subdivision methods versus non-conforming PL methods