标题： 闭合和开放曲面的轴对称Willmore流的稳定近似 显示英文标题

标题： Stable approximations for axisymmetric Willmore flow for closed and open surfaces

摘要： 对于${\mathbb R}^3$中的超曲面，Willmore 流被定义为经典 Willmore 能量的$L^2$-梯度流：即平均曲率平方的积分。这种几何演化定律在微分几何、图像重建和数学生物学中具有研究价值。在本文中，我们提出了轴对称超曲面的 Willmore 流的新数值近似方法。对于半离散连续时间变体，我们证明了一个稳定性结果。我们考虑了闭合曲面和有边界的情况。在后一种情况下，我们仔细推导了适当边界条件的弱形式。此外，我们考虑了许多经典 Willmore 能量的推广，特别是那些在生物膜研究中起作用的能量。在广义模型中，我们包含了自发曲率和面积差异弹性（ADE）效应、高斯曲率和线能贡献。多个数值实验展示了我们开发的数值方法的效率和鲁棒性。 显示英文摘要

摘要： For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.