An Interface Green's Function Framework for Complete Discrete W1,∞ Analysis of Discontinuous Galerkin Methods
Abstract
Pointwise error analysis of discontinuous Galerkin (DG) methods for the Poisson equation has received considerable attention during the past two decades. However, on convex polyhedral domains, existing analyses can only establish optimal error estimates in the broken W1,∞ seminorm for several DG methods. Since the broken W1,∞ seminorm does not control discontinuities across mesh interfaces, a complete discrete W1,∞ theory for discontinuous approximations on convex polyhedral domains has remained unavailable. In this paper, we develop an interface Green's function framework for the complete discrete W1,∞ analysis of DG methods on convex polyhedral domains. The proposed framework introduces new interface Green's functions that represent jumps of discontinuous approximations across mesh interfaces. Its central analytical ingredient is a new local energy estimate for these Green's functions, obtained by exploiting a cancellation between neighboring discrete delta functions. This estimate differs fundamentally from existing Green's function estimates and enables us to derive maximum-norm estimate for interface jumps without introducing additional logarithmic factors.
Related
- Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis
- Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system
- A new formulation of Discontinuous Galerkin method for interface capturing approaches
- A Fully Discrete Local Discontinuous Galerkin Method for Quasilinear Stochastic Convection-Diffusion-Type Equations
- Fully discrete multi-entropy-stable compact Runge--Kutta discontinuous Galerkin methods for hyperbolic conservation laws
BibTeX
@misc{leng2026interface,
title = {{An Interface Green's Function Framework for Complete Discrete $W^1,\infty$ Analysis of Discontinuous Galerkin Methods}},
author = {Haitao Leng and Weifeng Qiu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.11559},
eprint = {2608.11559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11559},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Interface Green's Function Framework for Complete Discrete W^1,∞ Analysis of Discontinuous Galerkin Methods
AU - Haitao Leng
AU - Weifeng Qiu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2608.11559
UR - https://arxiv.org/abs/2608.11559
AB - Pointwise error analysis of discontinuous Galerkin (DG) methods for the Poisson equation has received considerable attention during the past two decades. However, on convex polyhedral domains, existing analyses can only establish optimal error estimates in the broken W^1,∞ seminorm for several DG methods. Since the broken W^1,∞ seminorm does not control discontinuities across mesh interfaces, a complete discrete W^1,∞ theory for discontinuous approximations on convex polyhedral domains has remained unavailable. In this paper, we develop an interface Green's function framework for the complete discrete W^1,∞ analysis of DG methods on convex polyhedral domains. The proposed framework introduces new interface Green's functions that represent jumps of discontinuous approximations across mesh interfaces. Its central analytical ingredient is a new local energy estimate for these Green's functions, obtained by exploiting a cancellation between neighboring discrete delta functions. This estimate differs fundamentally from existing Green's function estimates and enables us to derive maximum-norm estimate for interface jumps without introducing additional logarithmic factors.
ER -