Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation
John W. Barrett, Robert Nürnberg · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2002
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.