The gradient discretisation method for the chemical reactions of biochemical systems

TL;DR

This paper applies the gradient discretisation method to a biochemical PDE model with reaction terms and boundary conditions, proving convergence and demonstrating numerical results.

Contribution

It introduces the use of GDM for biochemical reaction PDEs, establishing existence and convergence of solutions under classical assumptions.

Findings

Proved existence of weak solutions

Established strong convergence of approximate solutions

Numerical tests confirm theoretical results

Abstract

We consider a biochemical model that consists of a system of partial differential equations based on reaction terms and subject to non--homogeneous Dirichlet boundary conditions. The model is discretised using the gradient discretisation method (GDM) which is a framework covering a large class of conforming and non conforming schemes. Under classical regularity assumptions on the exact solutions, the GDM enables us to establish the existence of the model solutions in a weak sense, and strong convergence for the approximate solution and its approximate gradient. Numerical test employing a finite volume method is presented to demonstrate the behaviour of the solutions to the model.