Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910221084917760 |
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| author | Huang, Bingru Chen, Falai |
| author_facet | Huang, Bingru Chen, Falai |
| contents | This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over T-meshes by introducing the concept of TE-connected components, reducing the problem to the rank computation of explicitly constructible conformality matrices. Furthermore, a new class of rectilinear partitions, termed partitions with disjoint truncated l-edges, is introduced. It is proven that under specific conditions, the dimension of the corresponding spline space attains Schumaker's lower bound. This shows that the lower bound is attainable for arbitrary degree d and smoothness order mu in certain partition configurations. Numerical examples, including the Morgan-Scott and Yuan-Stillman partitions, validate the effectiveness and generality of the framework for both triangular and non-triangular rectilinear partitions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14577 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method Huang, Bingru Chen, Falai Numerical Analysis This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over T-meshes by introducing the concept of TE-connected components, reducing the problem to the rank computation of explicitly constructible conformality matrices. Furthermore, a new class of rectilinear partitions, termed partitions with disjoint truncated l-edges, is introduced. It is proven that under specific conditions, the dimension of the corresponding spline space attains Schumaker's lower bound. This shows that the lower bound is attainable for arbitrary degree d and smoothness order mu in certain partition configurations. Numerical examples, including the Morgan-Scott and Yuan-Stillman partitions, validate the effectiveness and generality of the framework for both triangular and non-triangular rectilinear partitions. |
| title | Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2605.14577 |