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Metric‐based upscaling
Author(s) -
Owhadi Houman,
Zhang Lei
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20163
Subject(s) - mathematics , homogenization (climate) , differentiable function , differential operator , mathematical analysis , elliptic operator , ergodicity , biodiversity , ecology , biology , statistics
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.