Metric based up-scaling

We consider divergence form elliptic operators in dimension $n\ge 2$ with $L^{\mathrm{\infty }}$ 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.