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A finite element scheme for the evolutionof orientational order in fluid membranes

Published online by Cambridge University Press:  09 October 2009

Sören Bartels ,
Georg Dolzmann  and
Ricardo H. Nochetto
Sören Bartels
Affiliation:
Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115 Bonn, Germany.
Georg Dolzmann
Affiliation:
NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany.
Ricardo H. Nochetto
Affiliation:
Mathematics Department and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA.
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Abstract

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as wellas 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. II7 (1997) 1509–1520; N. Uchida, Phys. Rev. E66 (2002) 040902] which replacesa Ginzburg-Landau penalization for the length of theorder parameter by a rigid constraint.We introduce a fully discrete scheme, consisting of piecewise linearfinite 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 solutionto the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.

Keywords

Biomembraneorientational ordercurvature
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 1 , January 2010 , pp. 1 - 31
Copyright
© EDP Sciences, SMAI, 2009

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