Abstract
We consider spline functions over simplicial meshes in \(\mathbb {R}^{n}\). We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as supersmoothness, which plays a role in the construction of multivariate splines and in the finite element method. In this paper, we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alfeld, P.: MDS. http://www.math.utah.edu/alfeld/MDS/
Alfeld, P., Sorokina, T.: Linear differential operators on bivariate spline spaces and spline vector fields. BIT Numer. Math. 56, 15–32 (2016)
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)
Christiansen, S.H., Hu, K.: Generalized finite element systems for smooth differential forms and Stokes’ problem. Numer. Math. 140, 327–371 (2018)
Farin, G.: Beźier polynomials over triangles and the construction of piecewise Cr polynomials. Report TR/91. Uxbridge UK:Department of Mathematics, Brunel University, 1–66 (1980)
Foucart, S., Sorokina, T.: Generating dimension formulas for multivariate splines. Albanian J. Math. 7, 24–35 (2013)
Fu, G., Guzman, J., Neilan, M.: Exact smooth piecewise polynomial sequences on Alfeld splits. arXiv:1807.05883, 1–28 (2018)
Lai, M.-J., Schumaker, L.L.: Spline Functions on Triangulations, vol. 110. Cambridge University Press, Cambridge (2007)
Schenck, H.: Splines on the Alfeld split of a simplex and type a root systems. J Approx Theory 182, 1–6 (2014)
Schenck, H., Sorokina, T.: Subdivision and spline spaces. Constr. Approx. 47, 237–247 (2018)
Schumaker, L.L.: On super splines and finite elements. SIAM J. Numer. Anal. 26, 997–1005 (1989)
Shekhtman, B., Sorokina, T.: Intrinsic supersmoothness. J. Concr. Appl. Math. 13, 232–241 (2015)
Sorokina, T.: A c1 multivariate clough-tocher interpolant. Constr. Approx. 29, 41–59 (2009)
Sorokina, T.: Intrinsic supersmoothness of multivariate splines. Numer. Math. 116, 421–434 (2010)
Sorokina, T.: Redundancy of smoothness conditions and supersmoothness of bivariate splines. IMA J. Numer. Anal. 34, 1701–1714 (2014)
Worsey, A.J., Farin, G.: An n-dimensional clough-tocher interpolant. Constr. Approx. 3, 99–110 (1987)
Funding
Open Access funding provided by University of Oslo (incl Oslo University Hospital).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by:Larry L. Schumaker
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
KH was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339643, during his affiliation with the University of Oslo.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Floater, M.S., Hu, K. A characterization of supersmoothness of multivariate splines. Adv Comput Math 46, 70 (2020). https://doi.org/10.1007/s10444-020-09813-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-020-09813-y