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 –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 – and –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.
DOI: 10.5802/smai-jcm.32
Keywords: parametric finite elements, Helfrich energy, spontaneous curvature, multi-phase membrane, line energy, $C^0$– and $C^1$–matching conditions
John W. Barrett 1; Harald Garcke 2; Robert Nürnberg 1
@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}, pages = {151--195}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {4}, year = {2018}, doi = {10.5802/smai-jcm.32}, mrnumber = {3796942}, zbl = {1416.74070}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.32/} }
TY - JOUR AU - John W. Barrett AU - Harald Garcke AU - Robert Nürnberg TI - Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation JO - The SMAI Journal of computational mathematics PY - 2018 SP - 151 EP - 195 VL - 4 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.32/ DO - 10.5802/smai-jcm.32 LA - en ID - SMAI-JCM_2018__4__151_0 ER -
%0 Journal Article %A John W. Barrett %A Harald Garcke %A Robert Nürnberg %T Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation %J The SMAI Journal of computational mathematics %D 2018 %P 151-195 %V 4 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.32/ %R 10.5802/smai-jcm.32 %G en %F SMAI-JCM_2018__4__151_0
John W. Barrett; Harald Garcke; Robert Nürnberg. 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/articles/10.5802/smai-jcm.32/
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