Subdivision schemes are useful mathematical tools for the generation of curves and surfaces. The linear approaches suffer from Gibbs oscillations when approximating functions with singularities. On the other hand, when we analyze the convergence of nonlinear subdivision schemes the regularity of the limit function is smaller than in the linear case . 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.
Linear approximation approaches suffer from Gibbs oscillations when approximating functions with singularities. ENO-SR resolution is a local approach 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 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, using the 4-point subdivision algorithm in the second stage. Using the proposed approach we are able to construct approximations with high precision , with high piecewise regularity , and without diffusion nor oscillations in the presence of discontinuities.
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Subdivision schemes are useful mathematical tools for the generation of curves and surfaces. The linear
Linear approximation
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On the other hand, when we analyze the convergence of nonlinear subdivision schemes the
ENO-SR resolution is a local approach avoiding oscillations and with a full order of accuracy, but a loss of
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limit function is smaller than in the linear case
approximant appears
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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
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
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using the 4-point subdivision algorithm in the second stage. Using the proposed approach
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a subdivision-based algorithm producing
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and limit functions
,
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,