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We consider the numerical solution of coupled volume-surfacereaction-diffusion systems having a detailed balance equilibrium. Based on theconservation of mass, an appropriate quadratic entropy functional is identifiedand an entropy-entropy dissipation inequality is proven. This allows us to showexponential convergence to equilibrium by the entropy method. We theninvestigate the discretization of the system by a finite element method and animplicit time stepping scheme including the domain approximation by polyhedralmeshes. Mass conservation and exponential convergence to equilibrium areestablished on the discrete level by arguments similar to those on thecontinuous level and we obtain estimates of optimal order for thediscretization error which hold uniformly in time. Some numerical tests arepresented to illustrate these theoretical results. The analysis and thenumerical approximation are discussed in detail for a simple model problem. Thebasic arguments however apply also in a more general context. This isdemonstrated by investigation of a particular volume-surface reaction-diffusionsystem arising as a mathematical model for asymmetric stem cell division.

Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology