Superconvergent quadriatic finite element on uniform tetrahedral meshes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918058995482624 |
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| author | Huang, Yunqing Zhang, Shangyou |
| author_facet | Huang, Yunqing Zhang, Shangyou |
| contents | By a direct computation, we show that the $P_2$ interpolation of a $P_3$ function is also a local $H^1$-projection on uniform tetrahedral meshes, i.e., the difference is $H^1$-orthogonal to the $P_2$ Lagrange basis function on the support patch of tetrahedra of the basis function. Consequently, we show the $H^1$ and $L^2$ rconvergence of the $P_2$ Lagrange finite element on uniform tetrahedral meshes. Using the standard 20-points Lagrange $P_3$ interpolation, where the 20 nodes are exactly some $P_2$ global basis nodes, we lift the superconvergent $P_2$ finite element solution to a quasi-optimal $P_3$ solution on each cube. Numerical results confirm the theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Superconvergent quadriatic finite element on uniform tetrahedral meshes Huang, Yunqing Zhang, Shangyou Numerical Analysis 65N15, 65N30 By a direct computation, we show that the $P_2$ interpolation of a $P_3$ function is also a local $H^1$-projection on uniform tetrahedral meshes, i.e., the difference is $H^1$-orthogonal to the $P_2$ Lagrange basis function on the support patch of tetrahedra of the basis function. Consequently, we show the $H^1$ and $L^2$ rconvergence of the $P_2$ Lagrange finite element on uniform tetrahedral meshes. Using the standard 20-points Lagrange $P_3$ interpolation, where the 20 nodes are exactly some $P_2$ global basis nodes, we lift the superconvergent $P_2$ finite element solution to a quasi-optimal $P_3$ solution on each cube. Numerical results confirm the theory. |
| title | Superconvergent quadriatic finite element on uniform tetrahedral meshes |
| topic | Numerical Analysis 65N15, 65N30 |
| url | https://arxiv.org/abs/2506.12407 |