Anisotropic modified Crouzeix-Raviart finite element method for the stationary Navier-Stokes equation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915154799624192 |
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| author | Ishizaka, Hiroki |
| author_facet | Ishizaka, Hiroki |
| contents | We studied an anisotropic modified Crouzeix--Raviart finite element method for the rotational form of a stationary incompressible Navier--Stokes equation with large irrotational body forces. We present an anisotropic $H^1$ error estimate for the velocity of the modified Crouzeix--Raviart finite element method for the Navier--Stokes equation. The modified Crouzeix--Raviart finite element scheme was obtained using a lifting operator that mapped the velocity test functions to $H(÷;Ω)$-conforming finite element spaces. Because no shape-regularity mesh conditions are imposed, anisotropic meshes can be used for the analysis. The core idea of the proof involves using the relation between the Raviart--Thomas and Crouzeix--Raviart finite element spaces. Furthermore, we present a discrete Sobolev inequality under semi-regular mesh conditions to estimate the stability of the proposed method, and confirm the results obtained through numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10214 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Anisotropic modified Crouzeix-Raviart finite element method for the stationary Navier-Stokes equation Ishizaka, Hiroki Numerical Analysis We studied an anisotropic modified Crouzeix--Raviart finite element method for the rotational form of a stationary incompressible Navier--Stokes equation with large irrotational body forces. We present an anisotropic $H^1$ error estimate for the velocity of the modified Crouzeix--Raviart finite element method for the Navier--Stokes equation. The modified Crouzeix--Raviart finite element scheme was obtained using a lifting operator that mapped the velocity test functions to $H(÷;Ω)$-conforming finite element spaces. Because no shape-regularity mesh conditions are imposed, anisotropic meshes can be used for the analysis. The core idea of the proof involves using the relation between the Raviart--Thomas and Crouzeix--Raviart finite element spaces. Furthermore, we present a discrete Sobolev inequality under semi-regular mesh conditions to estimate the stability of the proposed method, and confirm the results obtained through numerical experiments. |
| title | Anisotropic modified Crouzeix-Raviart finite element method for the stationary Navier-Stokes equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2304.10214 |