We introduce novel finite element schemes for curve diffusion and elastic flow in arbitrary codimension. The schemes are based on a variational form of a system that includes a specifically chosen tangential motion. We derive optimal $L^2$- and $H^1$-error bounds for continuous-in-time semidiscrete finite element approximations that use piecewise linear elements. In addition, we consider fully discrete schemes and, in the case of curve diffusion, prove unconditional stability for it. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bounds. The presented simulations suggest that the tangential motion leads to equidistribution in practice.

中文翻译：

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任意余维四阶几何曲线演化的切向运动有限元方案

我们引入了任意余维曲线扩散和弹性流的新颖有限元方案。该方案基于系统的变体形式，其中包括专门选择的切向运动。我们为使用分段线性元素的时间连续半离散有限元近似导出最佳 $L^2$- 和 $H^1$- 误差界限。此外，我们考虑完全离散的方案，并在曲线扩散的情况下证明其无条件稳定性。最后，我们提出了一些数值模拟，包括一些确认导出的误差范围的收敛实验。所提出的模拟表明切向运动在实践中导致均匀分布。