Title: FEM for Parametric Surface Flows: Applications to Biomembranes

When lipid molecules are immersed in aqueous environment they aggregate
spontaneously into 2 mono-molecular layers or (bio)membranes that form an
encapsulating bag called vesicle.  This happens because lipids consist of a
hydrophilic head group and a hydrophobic tail, which isolate itself in the
interior of the membrane.

As a first approach, we have studied a model based on geometry assuming that
the equilibrium shapes are the minimizers of the Willmore energy under area and
volume constraints.  Then, the effect of the inside (bulk) fluid is taken into
account leading to more physical dynamics.

A parametric approach is employed, which leads to forth order highly nonlinear
PDEs on surfaces and involves large domain deformations. An adaptive finite
element method (AFEM), with either piecewise linear or quadratic polynomials,
is used for both the geometric and coupled problems.  Several computational
challenges needed to be addressed and solved.

Joint with: Andrea Bonito and Ricardo H. Nochetto.