Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year

IMPACT FACTOR 2016: 1.097

CiteScore 2016: 1.09

SCImago Journal Rank (SJR) 2016: 0.872
Source Normalized Impact per Paper (SNIP) 2016: 0.887

Mathematical Citation Quotient (MCQ) 2016: 0.75

See all formats and pricing
More options …
Volume 16, Issue 4 (Oct 2016)


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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Robert Scheichl
  • Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom of Great Britain and Northern Ireland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
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


  • [1]

    Babuška I. and Lipton R., The penetration function and its application to microscale problems, Multiscale Model. Simul. 9 (2011), no. 1, 373–406. Google Scholar

  • [2]

    Berlyand L. and Owhadi H., Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Arch. Ration. Mech. Anal. 198 (2010), 677–721. Google Scholar

  • [3]

    Brown D. and Peterseim D., A multiscale method for porous microstructures, preprint 2014, http://arxiv.org/abs/1411.1944.

  • [4]

    Carstensen C., Quasi-interpolation and a posteriori error analysis in finite element methods, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187–1202. Google Scholar

  • [5]

    Carstensen C. and Verfürth R., Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36 (1999), no. 5, 1571–1587. Google Scholar

  • [6]

    Chu C.-C., Graham I. G. and Hou T.-Y., A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp. 79 (2010), no. 272, 1915–1955. Google Scholar

  • [7]

    Dryja M., Sarkis M. V. and Widlund O. B., Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math. 72 (1996), no. 3, 313–348. Google Scholar

  • [8]

    E W. and Engquist B., The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87–132. Google Scholar

  • [9]

    Efendiev Y., Galvis J. and Hou T. Y., Generalized multiscale finite element methods, J. Comput. Phys. 251 (2013), 116–135. Google Scholar

  • [10]

    Elfverson D., Georgoulis E. H., Målqvist A. and Peterseim D., Convergence of a discontinuous Galerkin multiscale method, SIAM J. Numer. Anal. 51 (2013), no. 6, 3351–3372. Google Scholar

  • [11]

    Gallistl D. and Peterseim D., Stable multiscale Petrov–Galerkin finite element method for high frequency acoustic scattering, Comput. Methods Appl. Mech. Engrg. 295 (2015), 1–17. Google Scholar

  • [12]

    Henning P. and Målqvist A., Localized orthogonal decomposition techniques for boundary value problems, SIAM J. Sci. Comput. 36 (2014), no. 4, A1609–A1634. Google Scholar

  • [13]

    Henning P., Målqvist A. and Peterseim D., A localized orthogonal decomposition method for semi-linear elliptic problems, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 5, 1331–1349. Google Scholar

  • [14]

    Henning P., Morgenstern P. and Peterseim D., Multiscale partition of unity, Meshfree Methods for Partial Differential Equations VII, Lect. Notes Comput. Sci. Eng. 100, Springer, Cham (2015), 185–204. Google Scholar

  • [15]

    Henning P. and Peterseim D., Oversampling for the multiscale finite element method, Multiscale Model. Simul. 11 (2013), no. 4, 1149–1175. Google Scholar

  • [16]

    Hou T. Y. and Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997), no. 1, 169–189. Google Scholar

  • [17]

    Hughes T. J. R., Feijóo G. R., Mazzei L. and Quincy J.-B., The variational multiscale method—a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998), 3–24. Google Scholar

  • [18]

    Hughes T. J. R. and Sangalli G., Variational multiscale analysis: The fine-scale Green’s function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal. 45 (2007), no. 2, 539–557. Google Scholar

  • [19]

    Kornhuber R. and Yserentant H., Numerical homogenization of elliptic multiscale problems by subspace decomposition, preprint 2015. Google Scholar

  • [20]

    Målqvist A. and Peterseim D., Computation of eigenvalues by numerical upscaling, Numer. Math. 130 (2014), no. 2, 337–361. Google Scholar

  • [21]

    Målqvist A. and Peterseim D., Localization of elliptic multiscale problems, Math. Comp. 83 (2014), no. 290, 2583–2603. Google Scholar

  • [22]

    Owhadi H. and Zhang L., Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast, Multiscale Model. Simul. 9 (2011), no. 4, 1373–1398. Google Scholar

  • [23]

    Owhadi H., Zhang L. and Berlyand L., Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM Math. Model. Numer. Anal. 48 (2013), no. 2, 517–552. Google Scholar

  • [24]

    Pechstein C. and Scheichl R., Weighted Poincaré inequalities, IMA J. Numer. Anal. 33 (2012), no. 2, 652–686. Google Scholar

  • [25]

    Peterseim D., Composite finite elements for elliptic interface problems, Math. Comp. 83 (2014), no. 290, 2657–2674. Google Scholar

  • [26]

    Peterseim D., Eliminating the pollution effect in Helmholtz problems by local subscale correction, preprint 2014, http://arxiv.org/abs/1411.7512.

  • [27]

    Peterseim D., Variational multiscale stabilization and the exponential decay of fine-scale correctors, preprint 2015, http://arxiv.org/abs/1505.07611.

  • [28]

    Scheichl R., Vassilevski P. S. and Zikatanov L. T., Weak approximation properties of elliptic projections with functional constraints, Multiscale Model. Simul. 9 (2011), no. 4, 1677–1699. Google Scholar

  • [29]

    Scheichl R., Vassilevski P. S. and Zikatanov L. T., Mutilevel methods for elliptic problems with highly varying coefficients on non-aligned coarse grids, SIAM J. Numer. Anal. 50 (2012), 1675–1694. Google Scholar

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, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2016-0022.

Export Citation

© 2016 by De Gruyter. Copyright Clearance Center

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Fredrik Hellman and Axel Målqvist
Multiscale Modeling & Simulation, 2017, Volume 15, Number 4, Page 1325

Comments (0)

Please log in or register to comment.
Log in