Second-order geometric consistency of nonlinear FE-HMM on surfaces via corrector non-degeneracy
Abstract
We analyze the geometric variational crime of nonlinear finite element heterogeneous multiscale methods (FE--HMM) for p-growth monotone laws on curved surfaces, p≥2. The practical method projects discrete gradients onto the exact tangent planes; we compare it with the exactly lifted method on the same discrete space. Without further hypotheses, the difference is O(hgp/(p−1)) in the metric induced by the cell correctors. The optimal order O(hg2) holds with a constant given by a weighted negative moment of the corrected gradients, evaluated along the practical solution only; no bounds on discrete gradients in L∞ are used. The negative moment is a sufficient condition for the lower shifted p-structure of the discrete homogenized flux, which for p>2 is not available in general. For the model law and a fixed cell mesh, its uniform boundedness is equivalent to a finite-dimensional directional criterion on degenerate correctors, and for laminates it holds with an explicit bound, uniform in the cell mesh. With macro--micro quasi-optimality and an estimate for coefficient sampling, the reconstructed microscopic field converges as O(h+Hmins,1+hg2), where Hs is the cell approximation rate. Experiments on a torus with an exact closest-point lift and a transported anisotropic p=4 material show second-order geometric convergence up to $98,304$ macroscopic quadrature points, each with a fully solved nonlinear cell problem, with bounded a posteriori constants for both load treatments.
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BibTeX
@misc{nguemfouo2026secondorder,
title = {{Second-order geometric consistency of nonlinear FE-HMM on surfaces via corrector non-degeneracy}},
author = {Marcial Nguemfouo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07761},
eprint = {2611.07761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07761},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Second-order geometric consistency of nonlinear FE-HMM on surfaces via corrector non-degeneracy
AU - Marcial Nguemfouo
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07761
UR - https://arxiv.org/abs/2611.07761
AB - We analyze the geometric variational crime of nonlinear finite element heterogeneous multiscale methods (FE--HMM) for p-growth monotone laws on curved surfaces, p≥2. The practical method projects discrete gradients onto the exact tangent planes; we compare it with the exactly lifted method on the same discrete space. Without further hypotheses, the difference is O(h_g^p/(p−1)) in the metric induced by the cell correctors. The optimal order O(h_g^2) holds with a constant given by a weighted negative moment of the corrected gradients, evaluated along the practical solution only; no bounds on discrete gradients in L^∞ are used. The negative moment is a sufficient condition for the lower shifted p-structure of the discrete homogenized flux, which for p>2 is not available in general. For the model law and a fixed cell mesh, its uniform boundedness is equivalent to a finite-dimensional directional criterion on degenerate correctors, and for laminates it holds with an explicit bound, uniform in the cell mesh. With macro--micro quasi-optimality and an estimate for coefficient sampling, the reconstructed microscopic field converges as O(h+H^mins,1+h_g^2), where H^s is the cell approximation rate. Experiments on a torus with an exact closest-point lift and a transported anisotropic p=4 material show second-order geometric convergence up to $98,304$ macroscopic quadrature points, each with a fully solved nonlinear cell problem, with bounded a posteriori constants for both load treatments.
ER -