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$W^{1,p}$ error estimates of an IPDG and a class of HDG methods for the Poisson problem on convex polyhedral domains — arXiv:2610.41747

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.41747
License (as archived): CC0-1.0
Archived: 2026-11-03T05:44:19.971Z
SHA-256 of the archived text: 4f4c02c282ff333a5c33e25d66ce7fcee1898efa3a4f14b094f058a238c570e5
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How to cite

$W^{1,p}$ error estimates of an IPDG and a class of HDG methods for the Poisson problem on convex polyhedral domains — arXiv:2610.41747 — arXiv (en), https://arxiv.org/abs/2610.41747 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e3500aafe3ac (Metraipolis archive dc_e3500aafe3ac, sha256 4f4c02c282ff333a5c33e25d66ce7fcee1898efa3a4f14b094f058a238c570e5)

BibTeX

@misc{dce3500aafe3ac,
  title = {$W^1,p$ error estimates of an IPDG and a class of HDG methods for the Poisson problem on convex polyhedral domains — arXiv:2610.41747},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.41747},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e3500aafe3ac; sha256 4f4c02c282ff333a5c33e25d66ce7fcee1898efa3a4f14b094f058a238c570e5},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.41747. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

This paper analyzes a symmetric interior penalty discontinuous Galerkin (IPDG) method, a hybrid mixed DG method, and two representative hybridizable discontinuous Galerkin (HDG) methods for the Poisson problem on convex polyhedral domains. For any fixed $\epsilon\in (0,\frac{1}{2})$, we first derive optimal discrete $W^{1,p}$ error estimates for $2\leq p<\frac{1}{\epsilon}$ by combining the interface Green's functions developed in \cite{lq2026} with the Fubini-type weighted estimates established in this paper. These error estimates lead to discrete $W^{1,q}$-$W^{1,p}$ inf-sup stability for $2\leq q<\frac{1}{\epsilon}$, where $q$ is the conjugate number of $p$ such that $\frac{1}{p}+\frac{1}{q}=1$. Using this stability result and a suitably constructed dual problem, we extend the optimal error estimates to $\frac{1}{1-\epsilon}<p<2$ and subsequently establish the corresponding inf-sup stability for $\frac{1}{1-\epsilon}< q<2$. Thus, both optimal discrete $W^{1,p}$ error estimates and discrete $W^{1,q}$-$W^{1,p}$ inf-sup stability hold throughout the range $\frac{1}{1-\epsilon}<p<\frac{1}{\epsilon}$, with $q=p/(p-1)$. Note that all of the aforementioned results are free of logarithmic factors and also hold on polyhedral meshes (except for the hybrid mixed DG method). Compared with the conforming finite element method (FEM), establishing such error estimates and inf-sup stability results for DG methods on convex polyhedral domains is more delicate because their analysis involves mesh-dependent norms. Consequently, the operator interpolation theory employed for conforming FEMs cannot be directly applied to DG methods. Finally, a numerical example is presented to illustrate the theoretical results.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.NA, nuovi lavori del giorno).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2610.41747 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T05:32:01.340Z dalla pagina https://arxiv.org/list/math.NA/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).