Error Estimates for Non Conforming Discretisation of Time-dependent Convection-Diffusion-Reaction Model

TL;DR

This paper develops a unified framework using the gradient discretisation method to analyze error estimates for time-dependent convection-diffusion-reaction models, covering various approximation techniques and applications.

Contribution

It introduces new convergence rate results and proves existence and uniqueness of solutions for a broad class of models within the GDM framework.

Findings

Numerical tests confirm theoretical convergence rates.

Method is effective on highly distorted meshes.

Framework covers multiple models like GBF and GBH.

Abstract

We use a generic framework, namely the gradient discretisation method (GDM), to propose a unified numerical analysis for general time-dependent convection-diffusion-reaction models. We establish novel results for convergence rates of numerical approximations of such models under reasonable assumptions on exact solutions, and prove the existence and uniqueness of the approximate solution for suitably small time steps. The main interest of our results lies in covering several approximation methods and various applications of the considered model such as the generalised Burgers-Fisher (GBF) and the generalised Burgers-Huxley (GBH) models. Numerical tests based on the hybrid mimetic mixed (HMM) method for the GBF model are performed on various types of general meshes to examine the accuracy of the proposed gradient scheme. The results confirm our theoretical rates of convergence, even on mesh with extreme distortions.