Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation

We consider a fully practical finite-element approximation of the following nonlinear degenerate parabolic system -∂/u∂/t-cΔu=-f(u) v in ΩT :=Ω× (0. T), Ω ⊂ Rd, d ≤ 2; ∂/v∂/t-Δ(b.(u,vΔv)=θf(u)v in ΩT subject to no flux boundary conditions, and non-negative initial data w° and i>° on u and v. Here we assume that c > 0. θ ≥ 0 and that f(r) ≥ f(0) = 0 is Lipschitz continuous and monotonically increasing for r ϵ [0, supx€Ωu0(x)]. Throughout the paper we restrict ourselves to the model degenerate case b(u, v) := o u v. where σ > 0. The above models the spatiotemporal evolution of a bacterium on a thin film of nutrient, where u is the nutrient concentration and v is the bacterial cell density. In addition to showing stability bounds for our approximation, we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system. Finally, some numerical experiments in one and two space dimensions are presented.