The Interaction Between PDE and Graphs in Multiscale Modeling

TL;DR

This paper develops a multiscale model for complex networks with clustered regions, using PDEs to describe flow behavior across microscale and macroscale levels within a partitioned spatial domain.

Contribution

It introduces an upscaled PDE-based model for complex networks with clustered regions, integrating graph structures with multiscale PDE analysis.

Findings

Model successfully captures flow dynamics in complex clustered networks

Provides a framework for multiscale PDE modeling on partitioned domains

Demonstrates applicability to Darcy porous media flow

Abstract

In this article an upscaled model is presented, for complex networks with highly clustered regions exchanging some abstract quantities in 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 by partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation.