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Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction‐diffusion systems
Author(s) -
Glitzky Annegret
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200910215
Subject(s) - mathematics , finite volume method , voronoi diagram , discretization , mathematical analysis , finite volume method for one dimensional steady state diffusion , geometry , physics , mechanics , numerical partial differential equations , differential equation
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.