Uniform Poincaré inequalities for the discrete de Rham complex of differential forms
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908680334606336 |
|---|---|
| author | Di Pietro, Daniele Droniou, Jérôme Hanot, Marien-Lorenzo Pitassi, Silvano |
| author_facet | Di Pietro, Daniele Droniou, Jérôme Hanot, Marien-Lorenzo Pitassi, Silvano |
| contents | In this paper we prove discrete Poincaré inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal methods, J. Eur. Math. Soc., to appear, arXiv preprint 2303.11093]. We unify the underlying ideas behind the Poincaré inequalities for all differential operators in the sequence, extending the known inequalities for the gradient, curl, and divergence in three-dimensions to polytopal domains of arbitrary dimension and general topology. A key step in the proof involves deriving specific Poincaré inequalities for the cochain complex supported on the polytopal mesh. These inequalities are of independent interest, as they are useful, for instance, in establishing the existence and stability, on domains of generic topology, of solutions of schemes based on Mimetic Finite Differences, Compatible Discrete Operators or Discrete Geometric Approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform Poincaré inequalities for the discrete de Rham complex of differential forms Di Pietro, Daniele Droniou, Jérôme Hanot, Marien-Lorenzo Pitassi, Silvano Numerical Analysis 65N30, 65N99, 14F40 In this paper we prove discrete Poincaré inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal methods, J. Eur. Math. Soc., to appear, arXiv preprint 2303.11093]. We unify the underlying ideas behind the Poincaré inequalities for all differential operators in the sequence, extending the known inequalities for the gradient, curl, and divergence in three-dimensions to polytopal domains of arbitrary dimension and general topology. A key step in the proof involves deriving specific Poincaré inequalities for the cochain complex supported on the polytopal mesh. These inequalities are of independent interest, as they are useful, for instance, in establishing the existence and stability, on domains of generic topology, of solutions of schemes based on Mimetic Finite Differences, Compatible Discrete Operators or Discrete Geometric Approach. |
| title | Uniform Poincaré inequalities for the discrete de Rham complex of differential forms |
| topic | Numerical Analysis 65N30, 65N99, 14F40 |
| url | https://arxiv.org/abs/2501.16116 |