A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929493898166272 |
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| author | Li, Jia Wu, Shuonan |
| author_facet | Li, Jia Wu, Shuonan |
| contents | A unified construction of canonical $H^m$-nonconforming finite elements is developed for $n$-dimensional simplices for any $m, n \geq 1$. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when $m \leq n$. In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on $m$. The corresponding nonconforming finite element method applies to $2m$-th order elliptic problems, with numerical results for $m=3$ and $m=4$ in 2D supporting the theoretical analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_06134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension Li, Jia Wu, Shuonan Numerical Analysis 65N30, 65N12 A unified construction of canonical $H^m$-nonconforming finite elements is developed for $n$-dimensional simplices for any $m, n \geq 1$. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when $m \leq n$. In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on $m$. The corresponding nonconforming finite element method applies to $2m$-th order elliptic problems, with numerical results for $m=3$ and $m=4$ in 2D supporting the theoretical analysis. |
| title | A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension |
| topic | Numerical Analysis 65N30, 65N12 |
| url | https://arxiv.org/abs/2409.06134 |