Before revision:  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:  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 , with high piecewise regularity and without diffusion nor oscillations in the presence of discontinuities.