A Finite Element Method for Volume-surface Reaction-diffusion Systems

We consider the numerical simulation 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 of solutions to equilibrium by the entropy method. We then investigate the discretisation of the system by finite element methods, including the domain approximation by polyhedral meshes, and an implicit time stepping scheme. Mass conservation and exponential convergence to equilibrium are established also on the discrete level by using arguments very similar to those on the continuous level. This allows us to establish convergence estimates for the discretisation error uniformly in time. Numerical tests are presented to illustrate the theoretical results. The analysis and numerical approximation are presented in detail for a simple model problem. Our arguments, however, can be applied also in a more general context. This is demonstrated by investigation of a biological volume-surface reaction-diffusion system modeling asymmetric stem cell division.