Before revision:  The goal of this paper is to introduce a corrected implementation of linear interpolatory subdivision schemes addressing both properties at the same time: regularity and adaption to singularities with high order of accuracy. In both cases of point-value data and of cell-average data,  we are able to construct a subdivision-based algorithm producing approximations with high precision and limit functions with high piecewise regularity  and without diffusion nor oscillations in the presence of discontinuities.
After revision:  The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper singularity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. We apply this approach to both cases of point-value data and of cell-average data, we are able to construct a subdivision-based algorithm producing approximations with high precision and limit functions with high piecewise regularity and without diffusion nor oscillations in the presence of discontinuities.