Converging approximation of a regular curve by polygonal lines in the uniform norm does not imply the convergence of the discrete differentials to their smooth counterpart. In this paper, we provide a constructive approach that, given a converging polygon sequence and an approximation of its distance to the objective curve, provides another sequence of polygons for which convergence of discrete differentials occurs as well. This approach is based on the notion of local scale of a polygon and uses multi-resolution decomposition as well as a non linear smoothing process. We provide the proof of the convergence and some numerical evidence of it, with application to the evaluation of solid friction in a pipe.

用均匀范数中的折线将规则曲线收敛逼近，并不意味着离散微分会收敛到它们的平滑对应项。在本文中，我们提供了一种建设性的方法，给定一个会聚的多边形序列及其到目标曲线的距离的近似值，它还会提供另一种发生离散微分收敛的多边形序列。该方法基于多边形的局部比例的概念，并使用多分辨率分解以及非线性平滑过程。我们提供了收敛的证明和一些数值证明，并将其应用于评估管道中的固体摩擦。