Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems
- Fiebach, André
- Glitzky, Annegret
- Linke, Alexander
2010 Mathematics Subject Classification
- 35K57 65M08 65M22 80A30
- reaction-diffusion systems, heterostructures, finite volume method, global bounds, discrete Gagliardo-Nirenberg inequalities, discrete Moser iteration
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 $L^infty$ 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.
- Numer. Math., 128 (2014) pp. 31--72.