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Numerical homogenization beyond scale separation

Published online by Cambridge University Press:  04 August 2021

Robert Altmann ,
Patrick Henning  and
Daniel Peterseim
Robert Altmann
Affiliation:
Institut für Mathematik, Universität Augsburg, 86159Augsburg, Germany E-mail: [email protected]
Patrick Henning
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, 44801Bochum, Germany E-mail: [email protected] Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Daniel Peterseim
Affiliation:
Institut für Mathematik, Universität Augsburg, 86159Augsburg, Germany E-mail: [email protected]
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Abstract

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Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.

Type
Research Article
Information
Acta Numerica , Volume 30 , May 2021 , pp. 1 - 86
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

*

The work of D. Peterseim is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 865751).

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