Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation

Abstract

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.

46 pages, 22 figures

References in corpus (5)

• Geometric theory on the elasticity of bio-membranes
• On the Stable Numerical Approximation of Two-Phase Flow with Insoluble Surfactant
• Convergence of an approximation for rotationally symmetric two-phase lipid bilayer membranes
• Kinks in two-phase lipid bilayer membranes
• Snapping elastic curves as a one-dimensional analogue of two-component lipid bilayers

Cited by in corpus (3)

• Parametric finite element approximations of curvature driven interface evolutions
• Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes
• Lower regularity assumption for an Euler-Lagrange equation on the contact line of the phase dependent Helfrich energy