Sharp local approximation widths for rough elliptic equations
Let $d \in \{2, 3\}$, let $\omega \Subset \omega^* \subset \mathbb{R}^d$ be concentric open cubes, and let $A \in L^\infty(\omega^*; \mathbb{R}^{d \times d})$ be symmetric with $\alpha |\xi|^2 \le \xi^{\mathsf T} A(x) \xi \le \beta |\xi|^2$ for almost every $x$ and every $\xi$, for fixed $0 < \alpha \le \beta < \infty$. Write $a_D(u, v) = \int_D A \nabla u \cdot \nabla v$ and $\|u\|_{a,D} = a_D(u, u)^{1/2}$. Fix a nonzero cutoff $\chi \in W^{1,\infty}(\omega) \cap H^1_0(\omega)$, let $$\mathcal{H}_A = \{u \in H^1(\omega^*) : a_{\omega^*}(u, v) = 0 \text{ for every } v \in H^1_0(\omega^*)\}, \qquad \mathcal{H}_A^0 = \{u \in \mathcal{H}_A : a_\omega(\chi u, \chi) = 0\},$$ the normalization removing the constants, let $Tu = \chi (u|_\omega)$, and let $$d_n(T) = \inf_{\substack{V \subset H^1_0(\omega) \text{ linear} \\ \dim V \le n}} \ \sup_{\substack{u \in \mathcal{H}_A^0 \\ \|u\|_{a,\omega^*} \le 1}} \ \inf_{v \in V} \|Tu - v\|_{a,\omega}.$$ In a framework for multiscale spectral generalized finite element methods that covers this setting, Ma proved bounds of the form $C \exp(-c n^{1/d})$ for such local widths, and conjectured that the optimal local rate is $\exp(-c n^{1/(d-1)})$, as a direct calculation shows for two-dimensional Poisson-type equations with constant coefficients.
Problem. Prove or refute that for every such choice of $d$, cubes, $A$ and $\chi$ there are $C, c > 0$, independent of $n$, such that $$d_n(T) \le C \exp\bigl(-c n^{1/(d-1)}\bigr) \qquad (n \ge 1).$$ No smoothness or scale separation of $A$ is assumed. The constants may depend on $A$, the cubes and $\chi$; contrast-independent constants are not asked for.
Source: Ma (2026), Section 7, who conjectured the local rate exp(−c n^{1/(d−1)}) for multiscale spectral generalized finite element methods; stated for scalar elliptic equations in d = 2, 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, elliptic equations with rough coefficients, Kolmogorov n-widths, multiscale finite element methods
References
@article{ma2026,
author = {Ma, Chupeng},
title = {A unified framework for multiscale spectral generalized {FEM}s and low-rank approximations to multiscale {PDE}s},
journal = {Found. Comput. Math.},
volume = {26},
number = {3},
year = {2026},
pages = {1699--1758},
doi = {10.1007/s10208-025-09711-z},
eprint = {2311.08761},
archivePrefix = {arXiv},
}
@misc{aim-sharp-local-elliptic-widths,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp local approximation widths for rough elliptic equations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/517-sharp-local-elliptic-widths.md},
note = {Entry 517 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that the local energy approximation widths of scalar, symmetric, uniformly elliptic equations in dimensions two and three decay as $\exp(-c n^{1/(d-1)})$, with no regularity assumption on the coefficients beyond measurability and boundedness. The result applies to the normalized cutoff operator on nested cubes in AIM Problem 517. Its proof compares the derivative of a harmonic energy Gram matrix …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00771, "Sharp local approximation widths for rough elliptic equations", clopen.solutions.