  The Canham-Helfrich-Evans models of biomembranes consist of a family of
geometric constrained variational problems. In this article, we compare two
classes of numerical methods for these variational problems based on piecewise
linear (PL) and subdivision surfaces (SS). Since SS methods are based on spline
approximation and can be viewed as higher order versions of PL methods, one may
expect that the only difference between the two methods is in the accuracy
order. In this paper, we prove that a numerical method based on minimizing any
one of the `PL Willmore energies' proposed in the literature would fail to
converge to a solution of the continuous problem, whereas a method based on
minimization of the bona fide Willmore energy, well-defined for SS but not PL
surfaces, succeeds. Motivated by this analysis, we propose also a
regularization method for the PL method based on techniques from conformal
geometry. We address a number of implementation issues crucial for the
efficiency of our solver. A software package called Wmincon accompanies this
article, provides parallel implementations of all the relevant geometric
functionals. When combined with a standard constrained optimization solver, the
geometric variational problems can then be solved numerically. To this end, we
realize that some of the available optimization algorithms/solvers are capable
of preserving symmetry, while others manage to break symmetry; we explore the
consequences of this observation.
