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

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 homol...

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Veröffentlicht in:Advances in computational mathematics Jg. 47; H. 1
Hauptverfasser: Toshniwal, Deepesh, DiPasquale, Michael
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.02.2021
Schlagworte:
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Visualization
Spline dimension formulas
41A15
Splines
Mixed smoothness
Polygonal meshes with holes
13D02
ISSN:1019-7168, 1572-9044
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Zusammenfassung: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.
ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-020-09830-x