Jump-preserving polynomial interpolation in non-manifold polyhedra
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916227975217152 |
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| author | Averseng, Martin |
| author_facet | Averseng, Martin |
| contents | We construct a piecewise-polynomial interpolant $u \mapsto Πu$ for functions $u:Ω\setminus Γ\to \mathbb{R}$, where $Ω\subset \mathbb{R}^d$ is a Lipschitz polyhedron and $Γ\subset Ω$ is a possibly non-manifold $(d-1)$-dimensional hypersurface. This interpolant enjoys approximation properties in relevant Sobolev norms, as well as a set of additional algebraic properties, namely, $Π^2 = Π$, and $Π$ preserves homogeneous boundary values and jumps of its argument on $Γ$. As an application, we obtain a bounded discrete right-inverse of the "jump" operator across $Γ$, and an error estimate for a Galerkin scheme to solve a second-order elliptic PDE in $Ω$ with a prescribed jump across $Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08223 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Jump-preserving polynomial interpolation in non-manifold polyhedra Averseng, Martin Numerical Analysis Analysis of PDEs 65N12, 65N15, 65N38, 41A10 We construct a piecewise-polynomial interpolant $u \mapsto Πu$ for functions $u:Ω\setminus Γ\to \mathbb{R}$, where $Ω\subset \mathbb{R}^d$ is a Lipschitz polyhedron and $Γ\subset Ω$ is a possibly non-manifold $(d-1)$-dimensional hypersurface. This interpolant enjoys approximation properties in relevant Sobolev norms, as well as a set of additional algebraic properties, namely, $Π^2 = Π$, and $Π$ preserves homogeneous boundary values and jumps of its argument on $Γ$. As an application, we obtain a bounded discrete right-inverse of the "jump" operator across $Γ$, and an error estimate for a Galerkin scheme to solve a second-order elliptic PDE in $Ω$ with a prescribed jump across $Γ$. |
| title | Jump-preserving polynomial interpolation in non-manifold polyhedra |
| topic | Numerical Analysis Analysis of PDEs 65N12, 65N15, 65N38, 41A10 |
| url | https://arxiv.org/abs/2211.08223 |