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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Volume 24, Issue 3


Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems

Matthias Maier
  • Institute of Applied Mathematics, Heidelberg University, Im Neuenheimer Feld 293/294, D-69120 Heidelberg, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Rolf Rannacher
  • Corresponding author
  • Institute of Applied Mathematics, Heidelberg University, Im Neuenheimer Feld 293/294, D-69120 Heidelberg, Germany
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-10-06 | DOI: https://doi.org/10.1515/jnma-2014-0074


This paper introduces an framework for adaptivity for a class of heterogeneous multiscale finite element methods for elliptic problems, which is suitable for a posteriori error estimation with separated quantification of the model error as well as the macroscopic and microscopic discretization errors. The method is derived within a general framework for ‘goal-oriented’ adaptivity, the so-called Dual Weighted Residual (DWR) method. This allows for a systematic a posteriori balancing of multiscale modeling and discretization. The developed method is tested numerically at elliptic diffusion problems for different types of heterogeneous oscillatory coefficients.

Keywords: heterogeneous multiscale method; finite element method; mesh adaptation; model adaptation; goal-oriented adaptivity; DWR method

MSC 2010: 35J15; 35J15; 65N12; 65N15; 65N30; 65N50


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About the article

Received: 2014-10-28

Revised: 2015-04-15

Accepted: 2015-07-02

Published Online: 2016-10-06

Published in Print: 2016-10-01

Citation Information: Journal of Numerical Mathematics, Volume 24, Issue 3, Pages 167–187, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2014-0074.

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