A regularization–correction approach for adapting subdivision schemes to the presence of discontinuities

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. 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 jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others.