We present variational approximations of boundary value problems for curvature flow and elastic flow in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open curves we propose natural boundary conditions that respect the appropriate gradient flow structure. Based on suitable weak formulations we introduce finite element approximations using piecewise linear elements. For some of the schemes a stability result can be shown. The derived schemes can be employed in very different contexts. For example, we apply the schemes to the Angenent metric in order to numerically compute rotationally symmetric self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes to compute geodesics that are relevant for optimal interface profiles in multi-component phase field models.

翻译：我们展示了两维的里伊曼式平流体的曲线曲线曲线和弹性流的边界值问题差异近似值。 对于不断演变的开阔曲线,我们提出尊重适当的梯度流结构的自然边界条件。基于适当的弱化配方,我们采用小片线性元素来引入有限元素近近近值。对于某些计划,可以显示一个稳定性结果。衍生的图例可以在非常不同的环境下使用。例如,我们将这些图例应用于动能度,以便用数字来计算平均曲线流的旋转对称自略器。此外,我们利用这些图例来计算与多构件阶段实地模型的最佳界面剖面图相关的大地学特征。