Metric‐based upscaling
Houman Owhadi, Lei Zhang · Communications on Pure and Applied Mathematics · 2006
Abstract We consider divergence form elliptic operators in dimension n ge; 2 with L∞ coefficients. Although solutions of these operators are only Hölder‐continuous, we show that they are differentiable (C1, α) 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. This new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales. Error bounds can be given and this method can also be used as a compression tool for differential operators. © 2006 Wiley Periodicals, Inc.