W1,p error estimates of an IPDG and a class of HDG methods for the Poisson problem on convex polyhedral domains
Abstract
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 ε∈ (0,1/2), we first derive optimal discrete W1,p error estimates for 2≤ p<1/ε 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 W1,q-W1,p inf-sup stability for 2≤ q<1/ε, where q is the conjugate number of p such that 1/p+1/q=1. Using this stability result and a suitably constructed dual problem, we extend the optimal error estimates to 1/(1−ε)<p<2 and subsequently establish the corresponding inf-sup stability for 1/(1−ε)< q<2. Thus, both optimal discrete W1,p error estimates and discrete W1,q-W1,p inf-sup stability hold throughout the range 1/(1−ε)<p<1/ε, 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.
Related
- A Note on Hardy Spaces on Polyhedral Tube Domains
- Optimal transport to non-convex polyhedral domains
- Weighted and nonlocal Poisson equations on domains with capacity density conditions
- Hölder Gradient Estimates for (s,p)-Poisson Equations with measurable and Hölder Data
- Boundary Rigidity of Line-Bundle Shilov--Poisson Transforms on Bounded Symmetric Domains
BibTeX
@misc{leng2026error,
title = {{$W^1,p$ error estimates of an IPDG and a class of HDG methods for the Poisson problem on convex polyhedral domains}},
author = {Haitao Leng and Weifeng Qiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41747},
eprint = {2610.41747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41747},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - W^1,p error estimates of an IPDG and a class of HDG methods for the Poisson problem on convex polyhedral domains
AU - Haitao Leng
AU - Weifeng Qiu
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41747
UR - https://arxiv.org/abs/2610.41747
AB - 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 ε∈ (0,1/2), we first derive optimal discrete W^1,p error estimates for 2≤ p<1/ε 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≤ q<1/ε, where q is the conjugate number of p such that 1/p+1/q=1. Using this stability result and a suitably constructed dual problem, we extend the optimal error estimates to 1/(1−ε)<p<2 and subsequently establish the corresponding inf-sup stability for 1/(1−ε)< 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 1/(1−ε)<p<1/ε, 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.
ER -