$C^1$-$Q_k$ serendipity finite elements on rectangular meshes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908708574855168 |
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| author | Zhang, Shangyou |
| author_facet | Zhang, Shangyou |
| contents | A $C^1$-$Q_k$ serendipity finite element is a sub-element of $C^1$-$Q_k$ BFS finite element such that the element remains $C^1$-continuous and includes all $P_k$ polynomials. In other words, it is a minimum of $Q_k$ bubbles enriched $P_k$ finite element. We enrich the $P_4$ and $P_5$ spaces by $9$ $Q_4$ and $11$ $Q_5$-bubble functions, respectively. For all $k\ge 6$, we enrich the $P_k$ spaces exactly by $12$ $Q_k$ bubble functions. We show the uni-solvence and quasi-optimality of the newly defined $C^1$-$Q_k$ serendipity elements. Numerical experiments by the $C^1$-$Q_k$ serendipity elements, $4\le k\le 8$, are performed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12144 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $C^1$-$Q_k$ serendipity finite elements on rectangular meshes Zhang, Shangyou Numerical Analysis 65N15, 65N30 A $C^1$-$Q_k$ serendipity finite element is a sub-element of $C^1$-$Q_k$ BFS finite element such that the element remains $C^1$-continuous and includes all $P_k$ polynomials. In other words, it is a minimum of $Q_k$ bubbles enriched $P_k$ finite element. We enrich the $P_4$ and $P_5$ spaces by $9$ $Q_4$ and $11$ $Q_5$-bubble functions, respectively. For all $k\ge 6$, we enrich the $P_k$ spaces exactly by $12$ $Q_k$ bubble functions. We show the uni-solvence and quasi-optimality of the newly defined $C^1$-$Q_k$ serendipity elements. Numerical experiments by the $C^1$-$Q_k$ serendipity elements, $4\le k\le 8$, are performed. |
| title | $C^1$-$Q_k$ serendipity finite elements on rectangular meshes |
| topic | Numerical Analysis 65N15, 65N30 |
| url | https://arxiv.org/abs/2512.12144 |