Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems
- Glitzky, Annegret
2010 Mathematics Subject Classification
- 35B40 35K57 35R05 46E39 65M12
- Reaction-diffusion systems, energy estimates, thermodynamic equilibria, asymptotic behaviour, time and space discretization, boundary conforming Delaunay grid, Voronoi finite volume scheme, discrete Sobolev-Poincaré inequality
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 essential 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.
- Math. Nachr., 284 (2011) pp. 2159--2174.