Basis Construction for Spline Spaces over Arbitrary Partitions from a Dimensional Stable Perspective
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912559420932096 |
|---|---|
| author | Huang, Bingru |
| author_facet | Huang, Bingru |
| contents | This paper introduces a novel framework for constructing $C^r$ basis functions for polynomial spline spaces of degree $d$ over arbitrary planar polygonal partitions, overturning the belief that basis functions cannot be constructed on dimensionally unstable meshes. We provide a comprehensive comparison of basis construction methods, classifying them as explicit, semi-explicit, and implicit. Our method, a semi-explicit construction using Extended Edge Elimination conditions, uniquely resolves all theoretical challenges in spline spaces by ensuring a complete basis. For the first time, we construct basis functions for the spline space over the Morgan-Scott partition, previously unachieved, and elucidate dimensional instability through this construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Basis Construction for Spline Spaces over Arbitrary Partitions from a Dimensional Stable Perspective Huang, Bingru Numerical Analysis 65D07, 65D17 This paper introduces a novel framework for constructing $C^r$ basis functions for polynomial spline spaces of degree $d$ over arbitrary planar polygonal partitions, overturning the belief that basis functions cannot be constructed on dimensionally unstable meshes. We provide a comprehensive comparison of basis construction methods, classifying them as explicit, semi-explicit, and implicit. Our method, a semi-explicit construction using Extended Edge Elimination conditions, uniquely resolves all theoretical challenges in spline spaces by ensuring a complete basis. For the first time, we construct basis functions for the spline space over the Morgan-Scott partition, previously unachieved, and elucidate dimensional instability through this construction. |
| title | Basis Construction for Spline Spaces over Arbitrary Partitions from a Dimensional Stable Perspective |
| topic | Numerical Analysis 65D07, 65D17 |
| url | https://arxiv.org/abs/2508.14649 |