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

TL;DR

This work analyzes a strongly coupled two-scale system where an upscaled diffusion-reaction equation with nonlinear dispersion interacts with a family of elliptic cell problems. The dispersion tensor $D^*(W)$ depends on microscopic cell solutions $W$ and on the macroscopic variable $u$ through drift couplings $G_i(u)$, motivated by homogenization of drift-dominated transport in porous media. An iterative decoupling scheme is developed to prove existence and uniqueness of weak solutions, with uniform positivity of $D^*(W)$ ensured through an auxiliary cell problem and energy estimates. Numerical simulations using finite elements illustrate how microscopic geometry and nonlinear drift alter macroscopic dispersion, providing insight into multiscale transport in porous media and offering a framework for analyzing similar nonlinear distributed-microstructure models.

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.

Figures (8)

• Figure 1: Typical two-scale geometry: schematic representation of the macroscopic domain $\Omega$ and of the microscopic domain $Y$ with internal boundary $\Gamma_N$.
• Figure 2: Top: solution to the Stokes problem, $B_1(y)$ (left) and $B_2(y)$ (right), in Geometry 1. Bottom: solution to the Stokes problem, $B_1(y)$ (left) and $B_2(y)$ (right), in Geometry 2.
• Figure 3: Comparison of the components of $D^*(W)$ for the fast diffusion case with both Geometry 1 (solid red line) and Geometry 2 (dashed green line). Here, we plot the components $D^*_{11}$ (top left), $D^*_{12}$ (top right), $D^*_{21}$ (bottom left), and $D^*_{22}$ (bottom right) as functions of the parameter $p$ which represents a type of local (microscopic) Peclet number.
• Figure 4: Comparison of the components of $D^*(W)$ for the slow diffusion case with both Geometry 1 (solid red line) and Geometry 2 (dashed green line). Here, we plot the components $D^*_{11}$ (top left), $D^*_{12}$ (top right), $D^*_{21}$ (bottom left), and $D^*_{22}$ (bottom right) as functions of the parameter $p$.
• Figure 5: Comparison of two components of $D^*(W)$, $D^*_{11}$ (left) and $D^*_{22}$ (right), in the neighbourhood of $p=0$ for the slow diffusion case.

Theorems & Definitions (22)

• Definition 2.1
• Definition 3.1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1: Well-posedness of $P^k(\Omega)$
• proof