A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914917482758144 |
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| author | Xu, Xuejun Zhang, Shangyou |
| author_facet | Xu, Xuejun Zhang, Shangyou |
| contents | A nonconforming $P_3$ finite element is constructed by enriching the conforming $P_3$ finite element space with three $P_3$ nonconforming bubbles and six additional $P_4$ nonconforming bubbles, on each tetrahedron. Here the divergence of the $P_4$ bubble is not a $P_3$ polynomial, but a $P_2$ polynomial. This nonconforming $P_3$ finite element, combined with the discontinuous $P_2$ finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special $P_4$ bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids Xu, Xuejun Zhang, Shangyou Numerical Analysis A nonconforming $P_3$ finite element is constructed by enriching the conforming $P_3$ finite element space with three $P_3$ nonconforming bubbles and six additional $P_4$ nonconforming bubbles, on each tetrahedron. Here the divergence of the $P_4$ bubble is not a $P_3$ polynomial, but a $P_2$ polynomial. This nonconforming $P_3$ finite element, combined with the discontinuous $P_2$ finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special $P_4$ bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory. |
| title | A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2408.10226 |