This paper aims at bridging existing theories in numerical and analyticalhomogenization. For this purpose the multiscale method of M{\aa}lqvist andPeterseim [Math. Comp. 2014], which is based on orthogonal subspacedecomposition, is reinterpreted by means of a discrete integral operator actingon standard finite element spaces. The exponential decay of the involvedintegral kernel motivates the use of a diagonal approximation and, hence, alocalized piecewise constant coefficient. In a periodic setting, the computedlocalized coefficient is proved to coincide with the classical homogenizationlimit. An a priori error analysis shows that the local numerical model isappropriate beyond the periodic setting when the localized coefficientsatisfies a certain homogenization criterion, which can be verified aposteriori. The results are illustrated in numerical experiments.
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