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Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems

Please always quote using this URN:urn:nbn:de:0296-matheon-11644
  • We consider discretizations for reaction-diffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on regular Delaunay meshes that allows to prove uniform, mesh-independent global upper and lower L1 bounds for the chemical potentials. These bounds provide the main step for a convergence analysis for the full discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides with the gradient norm of continuous piecewise linear finite elements.

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Metadaten
Author:André Fiebach, Annegret Glitzky, Alexander Linke
URN:urn:nbn:de:0296-matheon-11644
Referee:Alexander Mielke
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2012/09/01
Release Date:2012/09/01
Tag:discrete Gagliardo-Nirenberg inequalities; discrete Moser iteration; finite volume method; global bounds; heterostructures; reaction-diffusion systems
Institute:Freie Universität Berlin
Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K57 Reaction-diffusion equations
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M08 Finite volume methods
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M22 Solution of discretized equations [See also 65Fxx, 65Hxx]
80-XX CLASSICAL THERMODYNAMICS, HEAT TRANSFER (For thermodynamics of solids, see 74A15) / 80Axx Thermodynamics and heat transfer / 80A30 Chemical kinetics [See also 76V05, 92C45, 92E20]
Preprint Number:982
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