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 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.