SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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On energy-stable schemes for a Vesicle Membrane phase-field model

F. Guillen-Gonzalez and G. Tierra

Abstract
In this talk we extend the ideas to approximate Allen-Cahn and Cahn-Hilliard equations giving in [2, 4, 3], to design efficient numerical schemes for a physically motivated model given in [1], concerning the behavior of the deformation of vesicle membranes coupled with incompressible flow fields. The model is based on the diffuse-interface phase-field strategy, satisfying a dissipative energy law. The system is completed with two constraints, fixing the volume and surface of the vesicle membranes. We study the Vesicle Membrane model and analyze two different ideas to impose the volume and surface constraints (by means of either Lagrange multipliers or penalization), arriving at two different problems. We provide one energy-stable numerical scheme for each problem, with a linear approximation for the Lagrange multipliers case and a non-linear approach of the penalized problem.

Bibliography
[1] Q. Du, C. Liu, R. Ryham and X. Wang, Energetic variational approaches in modeling vesicle and fluid interactions, Physica D \textbf238 923-930 (2009)
[2] F. Guillén-González and G. Tierra, On linear schemes for a Cahn Hilliard Diffuse Interface Model, Journal of Computational Physics 234 (2013) 140-171.
[3] F. Guillén-González and G. Tierra, Second order schemes and time-step adaptativity for Allen-Cahn and Cahn-Hilliard models. Submitted.
[4] J. Shen and X. Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete and Continuous Dynamical Systems \textbf28 (2010) 1669-1691.

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