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Metric-based upscaling. (English) Zbl 1190.35070
Summary: We consider divergence form elliptic operators in dimension $$n \geq 2$$ 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. 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.

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 35B65 Smoothness and regularity of solutions to PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 74Q05 Homogenization in equilibrium problems of solid mechanics 76M50 Homogenization applied to problems in fluid mechanics
##### Keywords:
elliptic operators; numericla homogenization
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