  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.
