  We construct finite-dimensional approximations of solution spaces of
divergence form operators with $L^\infty$-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
sub-domains with source terms corresponding to approximation bases for $H^2$.
The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces
with basis elements localized to sub-domains of diameter $\mathcal{O}(h^\alpha
\ln \frac{1}{h})$ (with $\alpha \in [1/2,1)$) result in an
$\mathcal{O}(h^{2-2\alpha})$ accuracy for elliptic, parabolic and hyperbolic
problems. For high-contrast media, the accuracy of the method is preserved
provided that localized sub-domains contain buffer zones of width
$\mathcal{O}(h^\alpha \ln \frac{1}{h})$ where the contrast of the medium
remains bounded. The proposed method can naturally be generalized to vectorial
equations (such as elasto-dynamics).
