Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology
Provides rigorous numerical analysis for a class of reaction-diffusion systems relevant to cell biology, but the methods are extensions of existing techniques to a specific coupled setting.
The authors developed and analyzed finite element discretizations for coupled volume-surface reaction-diffusion systems with detailed balance, proving exponential convergence to equilibrium and optimal-order error estimates uniform in time. Numerical tests confirm the theory, with application to a model of asymmetric stem cell division.
We consider the numerical solution of coupled volume-surface reaction-diffusion systems having a detailed balance equilibrium. Based on the conservation of mass, an appropriate quadratic entropy functional is identified and an entropy-entropy dissipation inequality is proven. This allows us to show exponential convergence to equilibrium by the entropy method. We then investigate the discretization of the system by a finite element method and an implicit time stepping scheme including the domain approximation by polyhedral meshes. Mass conservation and exponential convergence to equilibrium are established on the discrete level by arguments similar to those on the continuous level and we obtain estimates of optimal order for the discretization error which hold uniformly in time. Some numerical tests are presented to illustrate these theoretical results. The analysis and the numerical approximation are discussed in detail for a simple model problem. The basic arguments however apply also in a more general context. This is demonstrated by investigation of a particular volume-surface reaction-diffusion system arising as a mathematical model for asymmetric stem cell division.