Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes
arXiv:2003.03127 · doi:10.1093/imanum/draa027
Abstract
The form and evolution of multi-phase biomembranes is of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham--Helfrich--Evans two-phase elastic energy, which will lead to fourth order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach, it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretised problems. We will prove these properties and show that the fully discretised schemes are well-posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes.
arXiv comment44 pages, 14 figures. This article is closely related to arXiv:1911.01132
References in corpus (13)
- Geometric theory on the elasticity of bio-membranes
- Parametric finite element approximations of curvature driven interface evolutions
- Finite element methods for fourth order axisymmetric geometric evolution equations
- Finite Element Approximation for the Dynamics of Fluidic Two-Phase Biomembranes
- Challenges in theoretical investigations on configurations of lipid membranes
- Variational discretization of axisymmetric curvature flows
- 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
- Numerical approximation of curve evolutions in Riemannian manifolds
- Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation
- General neck condition for the limit shape of budding vesicles
- Stable discretizations of elastic flow in Riemannian manifolds