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Local Approximation of A-Harmonic Functions and the Super-Localization Conjecture
Abstract
We consider the local energy approximation of A-harmonic functions on concentric cubes with a fixed size ratio in ℝd, d≥2, for bounded, symmetric, uniformly elliptic diffusion tensors~A. We give a short proof of the Kolmogorov n-width decay rate exp(−cn1/(d−1)) conjectured by Ma [Found.\ Comput.\ Math.\ 26 (2026), 1699--1758], where n denotes the approximation dimension. We apply this result to numerical homogenization, improving local approximation estimates for multiscale spectral generalized finite element methods and proving the conjectured local super-localization rate of Hauck and Peterseim [Math.\ Comp.\ 92 (2023), 981--1003] for interior Cartesian patches.
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BibTeX
@misc{hauck2026local,
title = {{Local Approximation of $A$-Harmonic Functions and the Super-Localization Conjecture}},
author = {Moritz Hauck and Chupeng Ma and Daniel Peterseim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25230},
eprint = {2610.25230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25230},
}
RIS
TY - UNPB
TI - Local Approximation of A-Harmonic Functions and the Super-Localization Conjecture
AU - Moritz Hauck
AU - Chupeng Ma
AU - Daniel Peterseim
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25230
UR - https://arxiv.org/abs/2610.25230
AB - We consider the local energy approximation of A-harmonic functions on concentric cubes with a fixed size ratio in ℝ^d, d≥2, for bounded, symmetric, uniformly elliptic diffusion tensors~A. We give a short proof of the Kolmogorov n-width decay rate exp(−cn^1/(d−1)) conjectured by Ma [Found.\ Comput.\ Math.\ 26 (2026), 1699--1758], where n denotes the approximation dimension. We apply this result to numerical homogenization, improving local approximation estimates for multiscale spectral generalized finite element methods and proving the conjectured local super-localization rate of Hauck and Peterseim [Math.\ Comp.\ 92 (2023), 981--1003] for interior Cartesian patches.
ER -