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.

翻译：我们针对任意余维数下的曲线扩散和弹性流问题，提出了一种新的有限元格式。该格式基于包含特定切向运动的系统变分形式。我们利用分段线性元推导了连续时间半离散有限元近似的最优$L^2$和$H^1$误差界。此外，我们考虑了全离散格式，并在曲线扩散情形下证明了其无条件稳定性。最后，我们给出了若干数值模拟结果，包括一些验证误差界的收敛性实验。所呈现的模拟结果表明，切向运动在实践中可实现等分布。