Error Estimates of Generic Discretisation of Reaction-Diffusion System with Constraints

TL;DR

This paper develops a unified framework for analyzing the convergence rates of various numerical schemes applied to a constrained reaction-diffusion system modeling biofilm growth, supported by theoretical proofs and numerical experiments.

Contribution

It provides the first general convergence rate analysis for both conforming and non-conforming discretizations of this reaction-diffusion system with constraints.

Findings

Established existence and uniqueness of discrete solutions.

Confirmed theoretical convergence rates through numerical experiments.

Demonstrated effectiveness of a mixed finite volume scheme.

Abstract

In this paper, we study a parabolic reaction diffusion system with constraints that model biofilm growth. Within a unified framework encompassing multiple numerical schemes, we derive the first general convergence rates for approximating this model using both conforming and non conforming discretisation methods. Under standard assumptions on the time discretisation, we establish the existence and uniqueness of the discrete solution. Numerical experiments are conducted using a mixed finite volume scheme that fits within the proposed unified framework. A test case with an analytical solution is designed to confirm our theoretical convergence rates.