Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908849110253568 |
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| author | Botti, Michele Mascotto, Lorenzo |
| author_facet | Botti, Michele Mascotto, Lorenzo |
| contents | We establish Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of standard Poincaré inequalities for piecewise $W^{1,p}$ functions and can be useful in the analysis of nonconforming finite element discretizations of nonlinear problems. Crucial tools in their derivation are novel Sobolev-trace inequalities and $W^{1,p}$-stable right-inverses of the divergence satisfying mixed boundary conditions. We provide estimates with constants having an explicit dependence on the geometric properties of the domain and the underlying family of polytopic meshes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes Botti, Michele Mascotto, Lorenzo Numerical Analysis 46E35, 65N30 We establish Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of standard Poincaré inequalities for piecewise $W^{1,p}$ functions and can be useful in the analysis of nonconforming finite element discretizations of nonlinear problems. Crucial tools in their derivation are novel Sobolev-trace inequalities and $W^{1,p}$-stable right-inverses of the divergence satisfying mixed boundary conditions. We provide estimates with constants having an explicit dependence on the geometric properties of the domain and the underlying family of polytopic meshes. |
| title | Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes |
| topic | Numerical Analysis 46E35, 65N30 |
| url | https://arxiv.org/abs/2504.03449 |