  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.
