Robust numerical upscaling of elliptic multiscale problems at high contrast

Daniel Peterseim, Robert Scheichl

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Abstract

We present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of H 1 ${H^{1}}$ into the image and the kernel of some novel stable quasi-interpolation operators with local L 2 $L^{2}$ -approximation properties, independent of the contrast. We identify a set of sufficient assumptions on these quasi-interpolation operators that guarantee in principle optimal convergence without pre-asymptotic effects for high-contrast coefficients. We then give an example of a suitable operator and establish the assumptions for a particular class of high-contrast coefficients. So far this is not possible without any pre-asymptotic effects, but the optimal convergence is independent of the contrast and the asymptotic range is largely improved over other discretization schemes. The new framework is sufficiently flexible to allow also for other choices of quasi-interpolation operators and the potential for fully robust numerical upscaling at high contrast.

Original languageEnglish
Pages (from-to)579-603
Number of pages25
JournalComputational Methods in Applied Mathematics
Volume16
Issue number4
DOIs
Publication statusPublished - 1 Oct 2016

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Keywords

  • Computational Homogenization
  • Finite Element
  • High Contrast
  • Multiscale
  • Upscaling

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