Interfaces and Free Boundaries
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Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation
John W. Barrett (1) and Robert Nürnberg (2)(1) Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, LONDON, UNITED KINGDOM
(2) Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, LONDON, UNITED KINGDOM
We consider a fully practical finite-element approximation of the following nonlinear degenerate parabolic system [part]u[divide][part]t ? c [Dgr]u = ?f (u) v in [OHgr]T := [OHgr] [times] (0 T), [OHgr] [sub] Rd, d [le] 2; [part]v[divide][part]t ? [nabla].(b(u, v) [nabla]v) = [thgr]f (u) v in [OHgr]T subject to no flux boundary conditions, and non-negative initial data u0 and v0 on u and v. Here we assume that c > 0, [thgr] [ge] 0 and that f (r) [ge] f (0) = 0 is Lipschitz continuous and monotonically increasing for r [isin] [0 supx[isin][OHgr]u0(x)]. Throughout the paper we restrict ourselves to the model degenerate case b(u, v) := [sgr] u v, where [sgr] > 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.
Keywords: bacterial pattern formation; nonlinear degenerate parabolic systems; finite element approximation