A finite element scheme for the evolution of orientational order in fluid membranes

¹ Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115 Bonn, Germany.
² NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany.
³ Mathematics Department and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA.

Received: 27 November 2008
Revised: 11 June 2009

Abstract

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as well as the coupling of the local order of the constituent molecules of the membrane to its curvature. We propose an alternative to the model in [J.B. Fournier and P. Galatoa, J. Phys. II 7 (1997) 1509–1520; N. Uchida, Phys. Rev. E 66 (2002) 040902] which replaces a Ginzburg-Landau penalization for the length of the order parameter by a rigid constraint. We introduce a fully discrete scheme, consisting of piecewise linear finite elements, show that it is unconditionally stable for a large range of the elastic moduli in the model, and prove its convergence (up to subsequences) thereby proving the existence of a weak solution to the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.