An exterior calculus framework for polytopal methods
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909459242024960 |
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| author | Bonaldi, Francesco Di Pietro, Daniele A. Droniou, Jerome Hu, Kaibo |
| author_facet | Bonaldi, Francesco Di Pietro, Daniele A. Droniou, Jerome Hu, Kaibo |
| contents | We develop in this work the first polytopal complexes of differential forms.
These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements.
Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method.
We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_11093 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An exterior calculus framework for polytopal methods Bonaldi, Francesco Di Pietro, Daniele A. Droniou, Jerome Hu, Kaibo Numerical Analysis 65N30, 65N99, 14F40 We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex. |
| title | An exterior calculus framework for polytopal methods |
| topic | Numerical Analysis 65N30, 65N99, 14F40 |
| url | https://arxiv.org/abs/2303.11093 |