Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation
The SMAI journal of computational mathematics, Volume 4 (2018) , pp. 151-195.

A finite element method for the evolution of a two-phase membrane in a sharp interface formulation is introduced. The evolution equations are given as an ${L}^{2}$–gradient flow of an energy involving an elastic bending energy and a line energy. In the two phases Helfrich-type evolution equations are prescribed, and on the interface, an evolving curve on an evolving surface, highly nonlinear boundary conditions have to hold. Here we consider both ${C}^{0}$– and ${C}^{1}$–matching conditions for the surface at the interface. A new weak formulation is introduced, allowing for a stable semidiscrete parametric finite element approximation of the governing equations. In addition, we show existence and uniqueness for a fully discrete version of the scheme. Numerical simulations demonstrate that the approach can deal with a multitude of geometries. In particular, the paper shows the first computations based on a sharp interface description, which are not restricted to the axisymmetric case.

Published online: 2018-04-27
DOI: https://doi.org/10.5802/smai-jcm.32
Classification: 35R01,  49Q10,  65M12,  65M60,  82B26,  92C10
Keywords: parametric finite elements, Helfrich energy, spontaneous curvature, multi-phase membrane, line energy, ${C}^{0}$– and ${C}^{1}$–matching conditions
@article{SMAI-JCM_2018__4__151_0,
author = {John W. Barrett and Harald Garcke and Robert N\"urnberg},
title = {Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation},
journal = {The SMAI journal of computational mathematics},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {4},
year = {2018},
pages = {151-195},
doi = {10.5802/smai-jcm.32},
language = {en},
url={smai-jcm.centre-mersenne.org/item/SMAI-JCM_2018__4__151_0/}
}
Barrett, John W.; Garcke, Harald; Nürnberg, Robert. Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation. The SMAI journal of computational mathematics, Volume 4 (2018) , pp. 151-195. doi : 10.5802/smai-jcm.32. https://smai-jcm.centre-mersenne.org/item/SMAI-JCM_2018__4__151_0/

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