2015/11/03 by Herbert Egger, Egger, Herbert, Klemens Fellner +5
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1511.00846
openalex publication_date 2015/11/03 · openalex created_date 2022/10/06 · openalex updated_date 2026/08/01
We consider the numerical solution of coupled volume-surface\nreaction-diffusion systems having a detailed balance equilibrium. Based on the\nconservation of mass, an appropriate quadratic entropy functional is identified\nand an entropy-entropy dissipation inequality is proven. This allows us to show\nexponential convergence to equilibrium by the entropy method. We then\ninvestigate the discretization of the system by a finite element method and an\nimplicit time stepping scheme including the domain approximation by polyhedral\nmeshes. Mass conservation and exponential convergence to equilibrium are\nestablished on the discrete level by arguments similar to those on the\ncontinuous level and we obtain estimates of optimal order for the\ndiscretization error which hold uniformly in time. Some numerical tests are\npresented to illustrate these theoretical results. The analysis and the\nnumerical approximation are discussed in detail for a simple model problem. The\nbasic arguments however apply also in a more general context. This is\ndemonstrated by investigation of a particular volume-surface reaction-diffusion\nsystem arising as a mathematical model for asymmetric stem cell division.\n