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.

Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation / Barrett, J. W.; Nürnberg, R.. - In: INTERFACES AND FREE BOUNDARIES. - ISSN 1463-9963. - 4:3(2002), pp. 277-307. [10.4171/IFB/62]

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

Nürnberg R.
2002

Abstract

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.
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Barrett, J. W.; Nürnberg, R.
Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation / Barrett, J. W.; Nürnberg, R.. - In: INTERFACES AND FREE BOUNDARIES. - ISSN 1463-9963. - 4:3(2002), pp. 277-307. [10.4171/IFB/62]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11572/283384
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