Sharp Pointwise Error Estimates of Finite Element Methods for the Biharmonic Equation
Abstract
We prove sharp pointwise error estimates for finite element approximations of the clamped biharmonic equation on convex polygonal domains. For C1-conforming elements, we establish W2,p-stability for all 2≤ p≤ ∞, together with best-approximation type estimates in W1,∞ and L∞. For the nonconforming Morley element, we derive optimal estimates in W2,p, W1,∞, and L∞; in particular, the L∞-error is of order O(h2) without an additional logarithmic factor. The analysis is based on three ingredients: local energy estimates for conforming and nonconforming plate elements, a removal of the local L2-error term in these estimates, and Hölder estimates for derivatives of the Green's function. These tools are combined with a dyadic decomposition argument for regularized Green's functions. The results remove logarithmic losses present in earlier pointwise estimates for fourth-order problems and yield sharp rates for classical plate elements such as the Argyris, Bogner--Fox--Schmit (BFS), Hsieh--Clough--Tocher (HCT), and Morley elements.
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BibTeX
@misc{li2026sharp,
title = {{Sharp Pointwise Error Estimates of Finite Element Methods for the Biharmonic Equation}},
author = {Buyang Li and Zhenglei Li and Qigang Liang and Xuejun Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25115},
eprint = {2610.25115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25115},
}
RIS
TY - UNPB
TI - Sharp Pointwise Error Estimates of Finite Element Methods for the Biharmonic Equation
AU - Buyang Li
AU - Zhenglei Li
AU - Qigang Liang
AU - Xuejun Xu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25115
UR - https://arxiv.org/abs/2610.25115
AB - We prove sharp pointwise error estimates for finite element approximations of the clamped biharmonic equation on convex polygonal domains. For C^1-conforming elements, we establish W^2,p-stability for all 2≤ p≤ ∞, together with best-approximation type estimates in W^1,∞ and L^∞. For the nonconforming Morley element, we derive optimal estimates in W^2,p, W^1,∞, and L^∞; in particular, the L^∞-error is of order O(h^2) without an additional logarithmic factor. The analysis is based on three ingredients: local energy estimates for conforming and nonconforming plate elements, a removal of the local L^2-error term in these estimates, and Hölder estimates for derivatives of the Green's function. These tools are combined with a dyadic decomposition argument for regularized Green's functions. The results remove logarithmic losses present in earlier pointwise estimates for fourth-order problems and yield sharp rates for classical plate elements such as the Argyris, Bogner--Fox--Schmit (BFS), Hsieh--Clough--Tocher (HCT), and Morley elements.
ER -