Localized maximum-norm error estimates for the Hellan-Herrmann-Johnson method
Abstract
We derive localized maximum-norm bounds for bending moments computed by the Hellan-Herrmann-Johnson method for the clamped Kirchhoff plate problem. The main difficulty is to localize the discrete stress without leaving the HHJ space or violating its kernel constraint. Using symmetric-curl potentials, we construct a kernel-preserving localization and connect local Green stresses with a global discrete Green stress. This argument separates the local interpolation error from a weaker global pollution term. Under explicit regularity assumptions on the auxiliary problems, the resulting estimates give optimal pointwise convergence for bending-moment values and elementwise first derivatives. No logarithmic loss occurs for positive polynomial degrees, whereas the lowest-order value estimate retains a logarithmic factor. Under additional reflection symmetry, symmetric recovery improves bending-moment values for even polynomial degrees and first derivatives for odd degrees. The proved gains are one third and one half of an order, respectively. Numerical experiments confirm this parity dependence and exhibit gains close to one full order, exceeding those established theoretically.
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BibTeX
@misc{li2026localized,
title = {{Localized maximum-norm error estimates for the Hellan-Herrmann-Johnson method}},
author = {Yuwen Li and Zhuoran Teng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14602},
eprint = {2609.14602},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14602},
}
RIS
TY - UNPB
TI - Localized maximum-norm error estimates for the Hellan-Herrmann-Johnson method
AU - Yuwen Li
AU - Zhuoran Teng
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.14602
UR - https://arxiv.org/abs/2609.14602
AB - We derive localized maximum-norm bounds for bending moments computed by the Hellan-Herrmann-Johnson method for the clamped Kirchhoff plate problem. The main difficulty is to localize the discrete stress without leaving the HHJ space or violating its kernel constraint. Using symmetric-curl potentials, we construct a kernel-preserving localization and connect local Green stresses with a global discrete Green stress. This argument separates the local interpolation error from a weaker global pollution term. Under explicit regularity assumptions on the auxiliary problems, the resulting estimates give optimal pointwise convergence for bending-moment values and elementwise first derivatives. No logarithmic loss occurs for positive polynomial degrees, whereas the lowest-order value estimate retains a logarithmic factor. Under additional reflection symmetry, symmetric recovery improves bending-moment values for even polynomial degrees and first derivatives for odd degrees. The proved gains are one third and one half of an order, respectively. Numerical experiments confirm this parity dependence and exhibit gains close to one full order, exceeding those established theoretically.
ER -