We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening 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格曼形的平流曲线曲线(曲线缩短流)和弹性流(曲线直线流)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。根据适当的弱化配方,我们采用小片线性元素引入了有限的元素近近近值。对于某些计划,可以显示一个稳定的结果。衍生的办法可以在非常不同的环境下使用。例如,我们将这些办法应用到动能度上,以便从数字上对平均弯曲流进行交替的对称自剪。此外,我们利用这些办法来计算与多构件阶段实地模型的最佳界面特征相关的大地学特征。