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General Error Estimates of Non Conforming Approximation of System of Reaction-Diffusion Equations

Yahya Alnashri

TL;DR

The paper tackles general error estimation for non-conforming numerical approximations of a system of reaction–diffusion equations with anisotropic diffusion. It develops a space–time Gradient Discretisation Method (GDM) framework and analyzes a gradient scheme with implicit Euler time stepping, proving existence and uniqueness of discrete solutions under a time-step restriction. The main contribution is a general, parameter-free error bound that scales with the time step $\delta t$, spatial discretisation indicator $h_{\mathcal{D}}$, and initial interpolation errors $E_{\mathcal{D}}^0$, $\widetilde{E}_{\mathcal{D}}^0$, applicable to both conforming and non-conforming schemes within GDM. A numerical example using the Hybrid Mimetic Mixed (HMM) method on polygonal meshes corroborates the theory, showing $\mathcal{O}(h_{\mathcal{M}})$ spatial convergence and, in some cases, super-convergence in $L^2$ norms for the unknowns and their gradients, thereby validating the practical utility of the general error estimates for anisotropic SRDEs.

Abstract

This paper aims to establish a first general error estimate for numerical approximations of the system of reaction-diffusion equations (SRDEs), using reasonable regularity assumptions on the exact solutions. We employ the gradient discretisation method (GDM) to discretise the system and prove the existence and uniqueness of the approximate solutions. The analysis provided here is not limited to specific reaction functions, and it is applicable to all conforming and non-conforming schemes fitting within the GDM framework. As an application, we present numerical results based on a finite volume method.

General Error Estimates of Non Conforming Approximation of System of Reaction-Diffusion Equations

TL;DR

The paper tackles general error estimation for non-conforming numerical approximations of a system of reaction–diffusion equations with anisotropic diffusion. It develops a space–time Gradient Discretisation Method (GDM) framework and analyzes a gradient scheme with implicit Euler time stepping, proving existence and uniqueness of discrete solutions under a time-step restriction. The main contribution is a general, parameter-free error bound that scales with the time step , spatial discretisation indicator , and initial interpolation errors , , applicable to both conforming and non-conforming schemes within GDM. A numerical example using the Hybrid Mimetic Mixed (HMM) method on polygonal meshes corroborates the theory, showing spatial convergence and, in some cases, super-convergence in norms for the unknowns and their gradients, thereby validating the practical utility of the general error estimates for anisotropic SRDEs.

Abstract

This paper aims to establish a first general error estimate for numerical approximations of the system of reaction-diffusion equations (SRDEs), using reasonable regularity assumptions on the exact solutions. We employ the gradient discretisation method (GDM) to discretise the system and prove the existence and uniqueness of the approximate solutions. The analysis provided here is not limited to specific reaction functions, and it is applicable to all conforming and non-conforming schemes fitting within the GDM framework. As an application, we present numerical results based on a finite volume method.
Paper Structure (4 sections, 2 theorems, 85 equations, 3 figures, 1 table)

This paper contains 4 sections, 2 theorems, 85 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Weak Formulation and Discretisation
  3. Convergence rates
  4. Numerical Example

Key Result

Lemma 2.4

Let Assumptions assump-rm hold, and ${\mathcal{D}}$ be a space-time gradient discretisation. Then for all $\delta t^{(n+\frac{1}{2})}\leq \frac{2}{L^2C_{\mathcal{D}}^2(\underline\lambda_1+\underline\lambda_2)}$, the approximate scheme rm-disc-pblm has a unique solution.

Figures (3)

  • Figure 4.1: Sample of the polygonal meshes.
  • Figure 4.2: The errors on hexagonal meshes.
  • Figure 4.3: The errors on hexagonal meshes.

Theorems & Definitions (7)

  • Definition 2.2: Generic Discrete Elements
  • Definition 2.3: Gradient Scheme
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2