Metric based up-scaling

Abstract

We consider divergence form elliptic operators in dimension $n\ge 2$ with $L^{\infty }$ coefficients. Although solutions of these operators are only Hölder 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.

Final version. Accepted for publication in Communications on Pure and Applied Mathematics. Presented at CIMMS (March 2005), Socams 2005 (April), Oberwolfach, MPI Leipzig (May 2005), CIRM (July 2005). Higher resolution figures are available at http://www.acm.caltech.edu/~owhadi/

References in corpus (2)

• Averaging versus Chaos in Turbulent Transport?
• Homogenization of parabolic equations with a continuum of space and time scales

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