TY - GEN A1 - Fiebach, André A1 - Glitzky, Annegret A1 - Linke, Alexander T1 - Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems N2 - 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. KW - reaction-diffusion systems KW - heterostructures KW - finite volume method KW - global bounds KW - discrete Gagliardo-Nirenberg inequalities KW - discrete Moser iteration Y1 - 2012 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1164 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-11644 ER -