任意余维数下四阶几何曲线演化的切向运动有限元格式
Finite element schemes with tangential motion for fourth order geometric curve evolutions in arbitrary codimension
AI总结：
本文提出任意余维数下曲线扩散与弹性流的有限元格式，引入切向运动，推导最优误差界并证明无条件稳定性，数值实验验证误差界并表明切向运动导致等分布。
摘要：
我们针对任意余维数下的曲线扩散和弹性流，引入了新颖的有限元格式。这些格式基于一个包含特定选择的切向运动的系统的变分形式。我们推导了使用分段线性元的连续时间半离散有限元逼近的最优$L^2$-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，我们展示了若干数值模拟，包括一些验证所推导误差界的收敛性实验。所展示的模拟表明，切向运动在实践中会导致等分布。
英文摘要：
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.
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