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


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 H1 if source terms are in the unit ball of L2 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 bases for H2. The H1-error estimates show that 𝒪(h-d)-dimensional spaces with basis elements localized to subdomains of diameter 𝒪(hαln1h) (with α[12,1)) result in an 𝒪(h2-2α) accuracy for elliptic, parabolic, and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved, provided that localized subdomains contain buffer zones of width 𝒪(hαln1h), where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

MSC codes

  1. 34E13
  2. 35B27


  1. homogenization
  2. localization
  3. cell problem
  4. high contrast
  5. continuum of scales

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Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1373 - 1398
ISSN (online): 1540-3467


Submitted: 8 November 2010
Accepted: 3 August 2011
Published online: 1 November 2011

MSC codes

  1. 34E13
  2. 35B27


  1. homogenization
  2. localization
  3. cell problem
  4. high contrast
  5. continuum of scales



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