Counting the dimension of splines of mixed smoothness: A general recipe, and its application to planar meshes of arbitrary topologies

Deepesh Toshniwal, Michael DiPasquale

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc. 310(1), 325–340, 1988). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces “lower-acyclic.” In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes.
Original languageEnglish
Article number6
Pages (from-to)1-29
Number of pages29
JournalAdvances in Computational Mathematics
Volume47
Issue number1
DOIs
Publication statusPublished - 2021

Keywords

  • Mixed smoothness
  • Polygonal meshes with holes
  • Spline dimension formulas
  • Splines

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