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

TL;DR

This paper develops a numerical framework for coupled volume-surface reaction-diffusion systems, proving exponential convergence to equilibrium and providing optimal error estimates, with applications to cell biology models.

Contribution

It introduces a finite element discretization with entropy-based analysis ensuring mass conservation and exponential convergence, applicable to complex biological systems.

Findings

Proven exponential convergence to equilibrium.

Established optimal order discretization error estimates.

Numerical tests confirm theoretical results.

Abstract

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.