$p$-Robust Trace Liftings for Discrete Harmonic Extensions and Boundary-Preserving $hp$ Interpolation on Tetrahedral Meshes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913179437629440 |
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| author | Li, Situan Zheng, Weiying |
| author_facet | Li, Situan Zheng, Weiying |
| contents | We construct p-robust polynomial trace liftings on three-dimensional tetrahedral meshes. The prescribed trace is a continuous piecewise polynomial function on a boundary face patch; the tetrahedra touching this patch have one common degree, while the interior degrees may be arbitrary. The lifting is degree-preserving, supported in the corresponding boundary layer, and satisfies both an H^1 estimate and a scaled boundary-layer L^2 estimate with constants independent of the mesh size and the polynomial degree.
The construction is local and combines tetrahedral polynomial liftings, face-gluing arguments, and nonsingular vertex patches. As consequences of the construction, we obtain p-robust discrete harmonic extensions, including an H^1-seminorm-stable extension for the pure diffusion energy, and a boundary-preserving hp interpolation operator that keeps piecewise polynomial Dirichlet data exactly while retaining standard local approximation estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02086 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $p$-Robust Trace Liftings for Discrete Harmonic Extensions and Boundary-Preserving $hp$ Interpolation on Tetrahedral Meshes Li, Situan Zheng, Weiying Numerical Analysis 65N30, 65N12, 41A10 We construct p-robust polynomial trace liftings on three-dimensional tetrahedral meshes. The prescribed trace is a continuous piecewise polynomial function on a boundary face patch; the tetrahedra touching this patch have one common degree, while the interior degrees may be arbitrary. The lifting is degree-preserving, supported in the corresponding boundary layer, and satisfies both an H^1 estimate and a scaled boundary-layer L^2 estimate with constants independent of the mesh size and the polynomial degree. The construction is local and combines tetrahedral polynomial liftings, face-gluing arguments, and nonsingular vertex patches. As consequences of the construction, we obtain p-robust discrete harmonic extensions, including an H^1-seminorm-stable extension for the pure diffusion energy, and a boundary-preserving hp interpolation operator that keeps piecewise polynomial Dirichlet data exactly while retaining standard local approximation estimates. |
| title | $p$-Robust Trace Liftings for Discrete Harmonic Extensions and Boundary-Preserving $hp$ Interpolation on Tetrahedral Meshes |
| topic | Numerical Analysis 65N30, 65N12, 41A10 |
| url | https://arxiv.org/abs/2606.02086 |