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We construct finite-dimensional approximations of solution spaces of divergence-form operators with L ∞ -coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in H 1 if source terms are in the unit ball of L 2 instead of the unit ball of H −1 . Approximation spaces are generated by solving elliptic PDEs on localized subdomains with source terms corresponding to approximationdoi:10.1137/100813968 fatcat:yqrrlj4f4vaclpbvm72jk2reoa