Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Bohne, Nis-Erik Ciarlet Jr., Patrick Sauter, Stefan |
| author_facet | Bohne, Nis-Erik Ciarlet Jr., Patrick Sauter, Stefan |
| contents | In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions.
Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $\left( d-1\right) $ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and $k>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04361 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions Bohne, Nis-Erik Ciarlet Jr., Patrick Sauter, Stefan Numerical Analysis 33C45, 33C50, 65N30 (Primary) 65N12 (Secondary) In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $\left( d-1\right) $ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and $k>1$. |
| title | Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions |
| topic | Numerical Analysis 33C45, 33C50, 65N30 (Primary) 65N12 (Secondary) |
| url | https://arxiv.org/abs/2407.04361 |