Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast

Houman Owhadi, Lei Zhang
2011 Multiscale Modeling & simulation  
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 approximation
more » ... for H 2 . The H 1 -error estimates show that Oðh −d Þ-dimensional spaces with basis elements localized to subdomains of diameter Oðh α ln 1 h Þ (with α ∈ ½ 1 2 ; 1Þ) result in an Oðh 2−2α Þ accuracy for elliptic, parabolic, and hyperbolic problems. For highcontrast media, the accuracy of the method is preserved, provided that localized subdomains contain buffer zones of width Oðh α ln 1 h Þ, where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).
doi:10.1137/100813968 fatcat:yqrrlj4f4vaclpbvm72jk2reoa