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Author: Glitzky, Annegret × End date: 2009 × Start date: 2000 × Has files: Yes × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: time and space discretization ×

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    Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems
    (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.
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    Exponential decay of the free energy for discretized electro-reaction-diffusion systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Glitzky, Annegret
    Our focus are electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electro-reaction-diffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.
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    Energy estimates for continuous and discretized electro-reaction-diffusion systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Glitzky, Annegret; Gärtner, Klaus
    We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations. We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly. The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.