Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.25230

Local Approximation of $A$-Harmonic Functions and the Super-Localization Conjecture

Moritz Hauck·Chupeng Ma·Daniel Peterseim
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Moritz Hauck, Chupeng Ma, Daniel Peterseim. 2026-10-18. Local Approximation of $A$-Harmonic Functions and the Super-Localization Conjecture. https://arxiv.org/abs/2610.25230

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fast Multipole Method with Complex Coordinates

When solving problems of potential theory in unbounded layered media, recently developed integral equation methods rely on coordinate complexification to permit truncation of the computational domain while retaining high order accuracy. As a result, classical fast multipole methods (FMMs) cannot be used to accelerate the evaluation of layer potentials. In this paper, we present a variant of the FMM which applies when the source and target locations have complex coordinates in two and three dimensions. This complex coordinate FMM is based on the analytic continuation of the various special function identities used in the real-space setting. To achieve the same linear-time complexity as in the classical FMM, we construct a hierarchical tree based solely on the real parts of the complex point locations and derive convergence rates for truncated expansions when the imaginary parts of the locations depend Lipschitz-continuously on the corresponding real parts. We demonstrate the efficiency of our approach through several numerical examples and illustrate its application to solving large-scale linearized water wave problems and Helmholtz transmission problems.

math.NA↗
arXiv2026

A survey of sampling discretization of integral and uniform norms

This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for trigonometric and algebraic polynomials, which play a crucial role in Fourier analysis, interpolation, and approximation theory. We focus on the problem in the broad context of finite-dimensional subspaces, where norms defined by general probability measures are approximated by their discrete counterparts. The primary emphasis is on results closely related to the authors' recent research. A key objective is to highlight the main ideas and techniques that form the foundation of the proofs in this area. This survey serves as a complement to three recently published survey papers on sampling discretization \cite{DPTT, KKLT, LMT}.

math.NA↗
arXiv2026

Dirichlet-Neumann waveform relaxation for heterogeneous heat equations: continuous and time discrete L2 analysis

We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet--Neumann waveform relaxation. This allows us to couple separate codes for the subproblems, a so-called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. We use an exponentially weighted Fourier technique to derive new error estimates in $L^2$ for finite time $T$ in both continuous and time-discrete settings. We identify an optimized relaxation parameter that guarantees superlinear convergence. Our new continuous estimate predicts linear convergence when $k^2/T$ is small, and superlinear when $k^2/T$ is large, with $k$ the iteration index. Our new time-discrete estimate closely mirrors its continuous counterpart for small $Δt$. Otherwise, we obtain linear convergence for large $k$. We also show that convergence is fast when the contrast is large, provided that the small physical parameter domain (e.g. air) is using the Dirichlet transmission condition, and the large physical parameter domain (e.g. steel) is using the Neumann transmission condition in the Dirichlet-Neumann waveform relaxation method. Numerical experiments confirm these findings, and we demonstrate the relevance on the analysis for a nonlinear conjugate heat transfer test case.

math.NA↗