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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the dimension of spline spaces on planar T-meshes
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by Bernard Mourrain PDF
Math. Comp. 83 (2014), 847-871 Request permission

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
  • MR Author ID: 309750
  • Email: Bernard.Mourrain@inria.fr
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 847-871
  • MSC (2010): Primary 14Q20, 14Q99, 13P25; Secondary 68W30, 65D17, 65D07
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02738-X
  • MathSciNet review: 3143695