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John W. Barrett; Harald Garcke; Robert Nürnberg
Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation
SMAI-Journal of computational mathematics, 4 (2018), p. 151-195, doi: 10.5802/smai-jcm.32
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Class. Math.: 35R01, 49Q10, 65M12, 65M60, 82B26, 92C10
Mots clés: parametric finite elements, Helfrich energy, spontaneous curvature, multi-phase membrane, line energy, $C^0$– and $C^1$–matching conditions


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.


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