(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Glitzky, Annegret
Our focus are energy estimates for discretized reaction-diffusion
systems for a finite number of species. We introduce a discretization scheme
(Voronoi finite volume in space and fully implicit in time) which has the
special property that it preserves the main features of the continuous
systems, namely positivity, dissipativity and flux conservation. For a class
of Voronoi finite volume meshes we investigate thermodynamic equilibria and
prove for solutions to the evolution system the monotone and exponential
decay of the discrete free energy to its equilibrium value with a unified
rate of decay for this class of discretizations. The fundamental idea is an
estimate of the free energy by the dissipation rate which is proved
indirectly by taking into account sequences of Voronoi finite volume meshes.
Essential ingredient in that proof is a discrete Sobolev- Poincaré
inequality.
