Local approximation of $A$-harmonic functions and the super-localization conjecture

We consider the local energy approximation of $A$-harmonic functions on concentric cubes with a fixed size ratio in $\mathbb{R}^d$, $d\ge2$, 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.