The pressure-wired Stokes element: a mesh-robust version of the Scott-Vogelius element
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2022
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| _version_ | 1866910776129748992 |
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| author | Gräßle, Benedikt Bohne, Nis-Erik Sauter, Stefan A. |
| author_facet | Gräßle, Benedikt Bohne, Nis-Erik Sauter, Stefan A. |
| contents | The Scott-Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order $k$ and a discontinuous pressure approximation of order $k-1$. It employs a "singular distance" (measured by some geometric mesh quantity $ Θ\left( \mathbf{z}\right) \geq 0$ for triangle vertices $\mathbf{z}$) and imposes a local side condition on the pressure space associated to vertices $\mathbf{z}$ with $Θ\left( \mathbf{z}\right) =0$. The method is inf-sup stable for any fixed regular triangulation and $k\geq 4$. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices $0<Θ\left( \mathbf{z}\right) \ll 1$.
In this paper, we introduce a very simple parameter-dependent modification of the Scott-Vogelius element such that the inf-sup constant is independent of nearly-singular vertices. We will show by analysis and also by numerical experiments that the effect on the divergence-free condition for the discrete velocity is negligibly small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09673 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The pressure-wired Stokes element: a mesh-robust version of the Scott-Vogelius element Gräßle, Benedikt Bohne, Nis-Erik Sauter, Stefan A. Numerical Analysis 65N12, 65N30, 76D07 The Scott-Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order $k$ and a discontinuous pressure approximation of order $k-1$. It employs a "singular distance" (measured by some geometric mesh quantity $ Θ\left( \mathbf{z}\right) \geq 0$ for triangle vertices $\mathbf{z}$) and imposes a local side condition on the pressure space associated to vertices $\mathbf{z}$ with $Θ\left( \mathbf{z}\right) =0$. The method is inf-sup stable for any fixed regular triangulation and $k\geq 4$. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices $0<Θ\left( \mathbf{z}\right) \ll 1$. In this paper, we introduce a very simple parameter-dependent modification of the Scott-Vogelius element such that the inf-sup constant is independent of nearly-singular vertices. We will show by analysis and also by numerical experiments that the effect on the divergence-free condition for the discrete velocity is negligibly small. |
| title | The pressure-wired Stokes element: a mesh-robust version of the Scott-Vogelius element |
| topic | Numerical Analysis 65N12, 65N30, 76D07 |
| url | https://arxiv.org/abs/2212.09673 |