The form and evolution of multi-phase biomembranes are 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 discretized problems. We will prove these properties and show that the fully discretized schemes are well posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes.

中文翻译：

---

轴对称两相生物膜梯度流的结构保持离散化

多相生物膜的形成和演化对于理解生命系统至关重要。为了描述这些膜，我们考虑了一个基于 Canham-Helfrich-Evans 两相弹性能的数学模型，这将导致涉及高度非线性边界条件的四阶几何演化问题。我们在轴对称环境中开发了一种参数化有限元方法。使用变分方法可以推导出高度非线性边界值问题的弱公式，例如能量衰减定律以及守恒性质，适用于空间离散化问题。我们将证明这些性质并证明完全离散化的方案是适定的。最后，一些数值计算表明该数值方法可用于计算复杂的，