Homogenization of a multiscale multi-continuum system

ABSTRACT

We study homogenization of a locally periodic two-scale dual-continuum system where each continuum interacts with the other. Equations for each continuum are written separately with interaction terms added. The homogenization limit depends strongly on the scale of this continuum interaction term with respect to the microscopic scale. In J. S. R. Park and V. H. Hoang, Hierarchical multiscale finite element method for multi-continuum media, Journal of Computational and Applied Mathematics, we study in details the case where the interaction terms are scaled as $O\left(1/{ϵ}^2\right),$ where ε is the microscale of the problem. We establish rigorously homogenization limit for this case where we show that in the homogenization limit, the dual-continuum structure disappears. In this paper, we consider the case where this term is scaled as $O\left(1/ϵ\right)$. This case is far more interesting and difficult as the homogenized problem is a dual-continuum system which contains features that are not in the original two-scale problem. In particular, the homogenized dual-continuum system contains extra convection terms and negative interaction coefficients, while the interaction coefficient between the continua in the original two-scale system obtains both positive and negative values. We prove rigorously the homogenization convergence and the convergence rate.

KEYWORDS:

• Multiscale
• homogenization
• upscaling
• multi-continuum

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 65 numerical analysis

Acknowledgments

A part of this work is conducted when Jun Sur Richard Park was a visiting PhD student at Nanyang Technological University (NTU) under East Asia and Pacific Summer Institutes (EAPSI) programme organized by the US National Science Foundation (NSF) and Singapore National Research Foundation (NRF) under Grant No. 1713805. Jun Sur Richard Park thanks US NSF and Singapore NRF for the financial support and NTU for hospitality. Viet Ha Hoang is supported by Singapore Ministry of Education Tier 2 grant MOE2017-T2-2-144.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

A part of this work is conducted when Jun Sur Richard Park was a visiting PhD student at Nanyang Technological University (NTU) under East Asia and Pacific Summer Institutes (EAPSI) programme organized by the US National Science Foundation (NSF) and Singapore National Research Foundation (NRF) [grant number 1713805]. Jun Sur Richard Park thanks US NSF and Singapore NRF for the financial support and NTU for hospitality. Viet Ha Hoang is supported by the Singapore Ministry of Education Tier 2 grant MOE2017-T2-2-144.