Enforcing convergence of derivatives for L∞ approximation of a regular curve

Abstract

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.