PH2T-splines, Part I: A Reasonable Mesh Assumption
Abstract
This paper is the first in a three-part series on the construction of polynomial splines with the highest order of smoothness over hierarchical T-meshes, referred to as PHtwoT-splines. For splines of bi-degree (d,d), we study suitable refinement conditions for the subsequent basis construction, which requires dimensional stability of the underlying spline space. We present two groups of examples, considering unrestricted hierarchical refinement and refinement without vanishable T l-edges, respectively. The first setting permits new edges without additional degrees of freedom. In the second group, every refinement level excludes vanishable T l-edges and increases the dimension. Nevertheless, the dimension is unstable in both groups. We then introduce template translations to describe each refined region as a union of translates of a fixed template in the cell-index grid. Together with the known stability result under (d−1)×(d−1) template refinement, these examples justify this condition as a reasonable mesh assumption for the subsequent PHtwoT-spline construction.
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BibTeX
@misc{huang2026ph2tsplines,
title = {{PH2T-splines, Part I: A Reasonable Mesh Assumption}},
author = {Bingru Huang and Yue Xi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11443},
eprint = {2609.11443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11443},
}
RIS
TY - UNPB
TI - PH2T-splines, Part I: A Reasonable Mesh Assumption
AU - Bingru Huang
AU - Yue Xi
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11443
UR - https://arxiv.org/abs/2609.11443
AB - This paper is the first in a three-part series on the construction of polynomial splines with the highest order of smoothness over hierarchical T-meshes, referred to as PHtwoT-splines. For splines of bi-degree (d,d), we study suitable refinement conditions for the subsequent basis construction, which requires dimensional stability of the underlying spline space. We present two groups of examples, considering unrestricted hierarchical refinement and refinement without vanishable T l-edges, respectively. The first setting permits new edges without additional degrees of freedom. In the second group, every refinement level excludes vanishable T l-edges and increases the dimension. Nevertheless, the dimension is unstable in both groups. We then introduce template translations to describe each refined region as a union of translates of a fixed template in the cell-index grid. Together with the known stability result under (d−1)×(d−1) template refinement, these examples justify this condition as a reasonable mesh assumption for the subsequent PHtwoT-spline construction.
ER -