Metric based up-scaling

TL;DR

This paper introduces a numerical homogenization method for divergence form elliptic operators with non-ergodic, multiscale media, leveraging harmonic coordinates to achieve differentiability and enable operator compression.

Contribution

It develops a new metric-based approach to numerical homogenization applicable to complex multiscale media without ergodicity, extending traditional methods.

Findings

Solutions are differentiable in harmonic coordinates despite limited regularity.

The method provides error bounds for homogenization in non-ergodic settings.

It can be used as a compression tool for differential operators.

Abstract

We consider divergence form elliptic operators in dimension $n\geq 2$ with $L^\infty$ coefficients. Although solutions of these operators are only H\"{o}lder continuous, we show that they are differentiable ($C^{1,\alpha}$) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.