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

摘要

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 arbitrary, even anisotropic, Voronoi meshes that allows to prove uniform, mesh-independent global upper and lower bounds for the chemical potentials. These bounds provide one of the main steps for a convergence analysis for the fully discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo–Nirenberg inequalities.