Abstract
In this paper we provide a priori error estimates with explicit constants for both the \(L^2\)-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently obtained for spline spaces of maximal smoothness. The presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. First we introduce results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.
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Acknowledgements
The authors are very grateful to Stefan Takacs (RICAM, Austria) for pointing out Lemma 5. This work was supported by the Beyond Borders Programme of the University of Rome Tor Vergata through the project ASTRID (CUP E84I19002250005) and by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006). C. Manni and H. Speleers are members of Gruppo Nazionale per il Calcolo Scientifico, Istituto Nazionale di Alta Matematica.
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Sande, E., Manni, C. & Speleers, H. Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis. Numer. Math. 144, 889–929 (2020). https://doi.org/10.1007/s00211-019-01097-9
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DOI: https://doi.org/10.1007/s00211-019-01097-9