Work

A Coarse-Graining Theory for Elliptic Operators and Homogenization in High Contrast

Year: 2026

Type: conference-paper

Abstract: We review a coarse-graining theory for divergence-form elliptic operators. The construction centers on a pair of coarse-grained matrices defined on spatial blocks that encode a scale-dependent notion of ellipticity, transmit precise information from small to large scales, and yield coarse-grained counterparts of standard elliptic estimates. Under simplifying assumptions, we give a complete proof of the result of [arXiv:2405.10732] that homogenization is reached within at most $C\log^2(1+Θ)$ dyadic length scales in the high-contrast regime, where $Θ$ is the ellipticity contrast. We argue that this scale-local notion of ellipticity is genuinely iterable across arbitrarily many scales, providing a framework for a rigorous renormalization group analysis.

DOI: 10.1137/25m1804807

Authors: Scott Armstrong, Tuomo Kuusi

Institutions: Courant Institute of Mathematical Sciences, Sorbonne Université, Laboratoire Jacques-Louis Lions, New York University, University of Helsinki

Language: English

Topic: Advanced Mathematical Modeling in Engineering

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: gold

OpenAlex ID: W4417257662

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