Implementation and Basis Construction for Smooth Finite Element Spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866912502518906880 |
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| author | Chen, Chunyu Chen, Long Gao, Tingyi Huang, Xuehai Wei, Huayi |
| author_facet | Chen, Chunyu Chen, Long Gao, Tingyi Huang, Xuehai Wei, Huayi |
| contents | The construction of $C^m$ conforming finite elements on simplicial meshes has recently advanced through the groundbreaking work of Hu, Lin, and Wu (Found. Comput. Math. 24, 2024). Their framework characterizes smoothness via moments of normal derivatives over subsimplices, leading to explicit degrees of freedom and unisolvence, unifying earlier constructions. However, the absence of explicit basis functions has left these spaces largely inaccessible for practical computation. In parallel, multivariate spline theory (Chui and Lai, J. Approx. Theory 60, 1990) enforces $C^m$ smoothness through linear constraints on Bernstein--Bézier coefficients, but stable, locally supported bases remain elusive beyond low dimensions. Building on the geometric decomposition of the simplicial lattice proposed by Chen and Huang (Math. Comp. 93, 2024), this work develops an explicit, computable framework for smooth finite elements. The degrees of freedom defined by moments of normal derivatives are modified to align with the dual basis of the Bernstein polynomials, yielding structured local bases on each simplex. Explicit basis construction is essential not merely for completeness, but for enabling efficient matrix assembly, global continuity, and scalable solution of high-order elliptic partial differential equations. This development closes the gap between theoretical existence and practical realization, making smooth finite element methods accessible to broad computational applications. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_19732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Implementation and Basis Construction for Smooth Finite Element Spaces Chen, Chunyu Chen, Long Gao, Tingyi Huang, Xuehai Wei, Huayi Numerical Analysis 65N30, 65N12, 31B30 The construction of $C^m$ conforming finite elements on simplicial meshes has recently advanced through the groundbreaking work of Hu, Lin, and Wu (Found. Comput. Math. 24, 2024). Their framework characterizes smoothness via moments of normal derivatives over subsimplices, leading to explicit degrees of freedom and unisolvence, unifying earlier constructions. However, the absence of explicit basis functions has left these spaces largely inaccessible for practical computation. In parallel, multivariate spline theory (Chui and Lai, J. Approx. Theory 60, 1990) enforces $C^m$ smoothness through linear constraints on Bernstein--Bézier coefficients, but stable, locally supported bases remain elusive beyond low dimensions. Building on the geometric decomposition of the simplicial lattice proposed by Chen and Huang (Math. Comp. 93, 2024), this work develops an explicit, computable framework for smooth finite elements. The degrees of freedom defined by moments of normal derivatives are modified to align with the dual basis of the Bernstein polynomials, yielding structured local bases on each simplex. Explicit basis construction is essential not merely for completeness, but for enabling efficient matrix assembly, global continuity, and scalable solution of high-order elliptic partial differential equations. This development closes the gap between theoretical existence and practical realization, making smooth finite element methods accessible to broad computational applications. |
| title | Implementation and Basis Construction for Smooth Finite Element Spaces |
| topic | Numerical Analysis 65N30, 65N12, 31B30 |
| url | https://arxiv.org/abs/2507.19732 |