A parsimonious approach to $C^2$ cubic splines on arbitrary triangulations: Reduced macro-elements on the cubic Wang-Shi split
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915078213730304 |
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| author | Lyche, Tom Manni, Carla Speleers, Hendrik |
| author_facet | Lyche, Tom Manni, Carla Speleers, Hendrik |
| contents | We present a general method to obtain interesting subspaces of the $C^2$ cubic spline space defined on the cubic Wang-Shi refinement of a given arbitrary triangulation $\mathcal{T}$. These subspaces are characterized by specific Hermite degrees of freedom associated with only the vertices and edges of $\mathcal{T}$, or even only the vertices of $\mathcal{T}$. Each subspace still contains cubic polynomials while saving a consistent number of degrees of freedom compared with the full space. The dimension of the considered subspaces can be as small as six times the number of vertices of $\mathcal{T}$. The method fits in the setting of macro-elements: any function of such a subspace can be constructed on each triangle of $\mathcal{T}$ separately by specifying the necessary Hermite degrees of freedom. The explicit local representation in terms of a local simplex spline basis is also provided. This simplex spline basis intrinsically takes care of the complex geometry of the Wang-Shi split, making it transparent to the user. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18323 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A parsimonious approach to $C^2$ cubic splines on arbitrary triangulations: Reduced macro-elements on the cubic Wang-Shi split Lyche, Tom Manni, Carla Speleers, Hendrik Numerical Analysis We present a general method to obtain interesting subspaces of the $C^2$ cubic spline space defined on the cubic Wang-Shi refinement of a given arbitrary triangulation $\mathcal{T}$. These subspaces are characterized by specific Hermite degrees of freedom associated with only the vertices and edges of $\mathcal{T}$, or even only the vertices of $\mathcal{T}$. Each subspace still contains cubic polynomials while saving a consistent number of degrees of freedom compared with the full space. The dimension of the considered subspaces can be as small as six times the number of vertices of $\mathcal{T}$. The method fits in the setting of macro-elements: any function of such a subspace can be constructed on each triangle of $\mathcal{T}$ separately by specifying the necessary Hermite degrees of freedom. The explicit local representation in terms of a local simplex spline basis is also provided. This simplex spline basis intrinsically takes care of the complex geometry of the Wang-Shi split, making it transparent to the user. |
| title | A parsimonious approach to $C^2$ cubic splines on arbitrary triangulations: Reduced macro-elements on the cubic Wang-Shi split |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2412.18323 |