Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation
Barrett, John W.1 ; Garcke, Harald2 ; Nürnberg, Robert1
1 Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
2 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
The SMAI Journal of computational mathematics, Tome 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.

Publié le :
DOI : 10.5802/smai-jcm.32
Classification : 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
Barrett, John W. 1 ; Garcke, Harald 2 ; Nürnberg, Robert 1

1 Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
2 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany 
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     title = {Gradient flow dynamics of two-phase biomembranes: {Sharp} interface variational formulation and finite element approximation},
     journal = {The SMAI Journal of computational mathematics},
     pages = {151--195},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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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, Tome 4 (2018), pp. 151-195. doi : 10.5802/smai-jcm.32. http://www.numdam.org/articles/10.5802/smai-jcm.32/

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