A mixed finite element method for the Babuška paradox using only discrete geometry
Abstract
The classical Babuška paradox shows that solutions of simply supported plate problems on polygonal approximations of a curved domain may converge to an unintended limit. We develop a boundary-corrected H(divdiv;𝕊)-L2 mixed finite element method for the simply supported Kirchhoff-Love plate problem. The correction uses only the discrete boundary geometry. Introducing the bending moment as an independent unknown allows the condition Mnn=0 to be imposed directly, while an edgewise mean constraint on the effective shear suppresses the leading geometric inconsistency. This constraint improves the boundary consistency error from 𝒪(h1/2) to 𝒪(h3/2). The associated boundary corrections act in the kernel of divdiv, leaving the discrete equilibrium equation unchanged and yielding a uniformly stable scheme. Under suitable regularity assumptions, we prove L2-error estimates of order h3/2 for the bending moment and the broken Hessian of the postprocessed displacement, and of order h2 for the displacement. The analysis covers multiply connected domains and polygonal approximations whose boundaries may cross the physical boundary. Numerical experiments confirm these rates and the improvement over the uncorrected method.
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BibTeX
@misc{tian2026mixed,
title = {{A mixed finite element method for the Babuška paradox using only discrete geometry}},
author = {Pengjie Tian and Shuonan Wu and Hao Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18537},
eprint = {2609.18537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18537},
}
RIS
TY - UNPB
TI - A mixed finite element method for the Babuška paradox using only discrete geometry
AU - Pengjie Tian
AU - Shuonan Wu
AU - Hao Zhou
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18537
UR - https://arxiv.org/abs/2609.18537
AB - The classical Babuška paradox shows that solutions of simply supported plate problems on polygonal approximations of a curved domain may converge to an unintended limit. We develop a boundary-corrected H(divdiv;𝕊)-L^2 mixed finite element method for the simply supported Kirchhoff-Love plate problem. The correction uses only the discrete boundary geometry. Introducing the bending moment as an independent unknown allows the condition M_nn=0 to be imposed directly, while an edgewise mean constraint on the effective shear suppresses the leading geometric inconsistency. This constraint improves the boundary consistency error from 𝒪(h^1/2) to 𝒪(h^3/2). The associated boundary corrections act in the kernel of divdiv, leaving the discrete equilibrium equation unchanged and yielding a uniformly stable scheme. Under suitable regularity assumptions, we prove L^2-error estimates of order h^3/2 for the bending moment and the broken Hessian of the postprocessed displacement, and of order h^2 for the displacement. The analysis covers multiply connected domains and polygonal approximations whose boundaries may cross the physical boundary. Numerical experiments confirm these rates and the improvement over the uncorrected method.
ER -