Numerical Analysis of a Parabolic Variational Inequality System Modeling Biofilm Growth at the Porescale

TL;DR

This paper develops and analyzes a finite element method for a coupled system of nonlinear PDEs modeling biofilm growth with a density constraint, providing error estimates and simulations at the pore scale.

Contribution

It introduces a rigorous finite element approximation for a parabolic variational inequality system modeling biofilm growth, with proven convergence rates.

Findings

Finite element method converges at predicted rate.

Simulations effectively track free boundary in pore-scale geometry.

Modeling assumptions tested through numerical experiments.

Abstract

In this paper we consider a system of two coupled nonlinear diffusion--reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element (FE) approximation to the coupled nonlinear system and confirm experimentally that the numerical approximation converges at the predicted rate. We also show simulations in which we track the free boundary in the domains which resemble the pore scale geometry and in which we test the different modeling assumptions.