The interaction between PDE and graphs in multiscale modeling

Abstract

In this article an upscaling model is presented for complex networks with highly clustered regions exchanging/trading quantities of interest at both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described using partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation.