Three-scale convergence for processes in heterogeneous media

Abstract

In this article, we propose a new notion of multiscale convergence, called ‘three-scale’, which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium. This generalizes and extends the notion of two-scale convergence, a well-established concept that is now commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences in L ²(Ω) are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterization of three-scale convergent sequences and is then continued with the introduction of the notion of ‘strong three-scale convergence’ whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.

Keywords:

• multiscale analysis
• heterogeneous media
• composite media
• microscale
• mesoscale
• macroscale

AMS Subject Classifications::

• 35B27
• 78A48
• 82D30

Acknowledgements

All authors gratefully acknowledge the support of the European Research Council. DT and MAJC were supported by the ERC AdG Grant No. 227619 ‘From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread’ and AM-C by the ERC Grant: ‘Multiscale mathematical modelling of dynamics of structure formation in cell systems’ and by Emmy Noether Programme of German Research Council (DFG).