Metric‐based upscaling

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 ( C 1, 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.