Finite element methods for director fields on flexible surfaces

Abstract

We introduce a nonlinear model for the evolution of biomembranes driven by the $L^2$-gradient flow of a novel elasticity functional describing the interaction of a director field on a membrane with its curvature. In the linearized setting of a graph we present a practical finite element method (FEM), and prove a priori estimates. We derive the relaxation dynamics for the nonlinear model on closed surfaces and introduce a parametric FEM. We present numerical experiments for both linear and nonlinear models, which agree well with the expected behavior in simple situations and allow predictions beyond theory.