We introduce a nonlinear model for the evolution of biomembranes driven by the -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.
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Sören Bartels, Georg Dolzmann, Ricardo H. Nochetto, Alexander Raisch, Finite element methods for director fields on flexible surfaces. Interfaces Free Bound. 14 (2012), no. 2, pp. 231–272DOI 10.4171/IFB/281