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Mathematics of Computation

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On the dimension of spline spaces on planar T-meshes


Author: Bernard Mourrain
Journal: Math. Comp. 83 (2014), 847-871
MSC (2010): Primary 14Q20, 14Q99, 13P25; Secondary 68W30, 65D17, 65D07
Published electronically: July 12, 2013
MathSciNet review: 3143695
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Abstract: We analyze the space $ \mathcal {S}_{m, m'}^{\mathbf {r}} (\mathcal {T})$ of bivariate functions that are piecewise polynomial of bi-degree $ \leqslant (m, m')$ and of smoothness $ \mathbf {r}$ along the interior edges of a planar T-mesh $ \mathcal {T}$. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are regular enough. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness.


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Additional Information

Bernard Mourrain
Affiliation: Galaad, Inria Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France
Email: Bernard.Mourrain@inria.fr

DOI: https://doi.org/10.1090/S0025-5718-2013-02738-X
Received by editor(s): May 26, 2013
Received by editor(s) in revised form: December 23, 2011, and July 9, 2012
Published electronically: July 12, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.