Strongly Coupled Two-scale System with Nonlinear Dispersion: Weak Solvability and Numerical Simulation

TL;DR

This paper studies a complex two-scale reaction-diffusion system with nonlinear dispersion, proving weak solution existence and uniqueness, and performing numerical simulations to explore effects of micro-geometry and nonlinear coupling.

Contribution

It introduces a novel coupled two-scale model with nonlinear dispersion, proving weak solvability and implementing finite element simulations for analysis.

Findings

Existence and uniqueness of weak solutions established.

Finite element simulations demonstrate impact of micro-geometry.

Nonlinear drift coupling significantly influences macroscopic dispersion.

Abstract

We investigate a two-scale system featuring an upscaled parabolic dispersion-reaction equation intimately linked to a family of elliptic cell problems. The system is strongly coupled through a dispersion tensor, which depends on the solutions to the cell problems, and via the cell problems themselves, where the solution of the parabolic problem interacts nonlinearly with the drift term. This particular mathematical structure is motivated by a rigorously derived upscaled reaction-diffusion-convection model that describes the evolution of a population of interacting particles pushed by a large drift through an array of periodically placed obstacles (i.e., through a regular porous medium). We prove the existence and uniqueness of weak solutions to our system by means of an iterative scheme, where particular care is needed to ensure the uniform positivity of the dispersion tensor. Additionally, we use finite element-based approximations for the same iteration scheme to perform multiple simulation studies. Finally, we highlight how the choice of micro-geometry (building the regular porous medium) and of the nonlinear drift coupling affects the macroscopic dispersion of particles.