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.

我们在共形平坦的二维黎曼流形中呈现曲率流（曲线缩短流）和弹性流（曲线拉直流）的边界值问题的变分近似。对于不断发展的开放曲线，我们提出了尊重适当梯度流结构的自然边界条件。基于合适的弱公式，我们引入了使用分段线性元素的有限元近似。对于某些方案，可以显示稳定性结果。派生的方案可以在非常不同的环境中使用。例如，我们将这些方案应用于 Angenent 度量，以便数值计算平均曲率流的旋转对称自收缩。此外，