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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 16, Issue 4


Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast

Daniel Peterseim
  • Corresponding author
  • Institut für Numerische Simulation der Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany
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/ Robert Scheichl
  • Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom of Great Britain and Northern Ireland
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Published Online: 2016-06-11 | DOI: https://doi.org/10.1515/cmam-2016-0022


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 H1 into the image and the kernel of some novel stable quasi-interpolation operators with local L2-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.

Keywords: Finite Element; Multiscale; Upscaling; Computational Homogenization; High Contrast

MSC 2010: 65N30; 65N25; 65N15


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About the article

Received: 2016-01-24

Revised: 2016-05-14

Accepted: 2016-05-15

Published Online: 2016-06-11

Published in Print: 2016-10-01

Supported by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.

Citation Information: Computational Methods in Applied Mathematics, Volume 16, Issue 4, Pages 579–603, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2016-0022.

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