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Error analysis of a backward Euler positive preserving stabilized scheme for a Chemotaxis system

Panagiotis Chatzipantelidis, Christos Pervolianakis

TL;DR

This work analyzes finite element approximations for a two-dimensional Keller–Segel chemotaxis system under small initial data to avoid blow-up. It develops and compares two stabilized semidiscrete schemes (low-order with lumped mass and AFC) and discretizes in time with backward Euler, proving existence and providing rigorous error bounds for the fully discrete solution: \\( abla\\, error in the chemical concentration and L^2 error for the cell density. The study establishes positivity and mass conservation at the fully discrete level, and numerical experiments in 2D illustrate the theory, including a blow-up scenario where stabilized schemes preserve positivity. Overall, the paper delivers a comprehensive error analysis for positivity-preserving stabilized schemes in chemotaxis models, with practical implications for robust, accurate long-time simulations.

Abstract

For a Keller-Segel model for chemotaxis in two spatial dimensions we consider a modification of a positivity preserving fully discrete scheme using a local extremum diminishing flux limiter. We discretize space using piecewise linear finite elements on an quasiuniform triangulation of acute type and time by the backward Euler method. We assume that initial data are sufficiently small in order not to have a blow-up of the solution. Under appropriate assumptions on the regularity of the exact solution and the time step parameter we show existence of the fully discrete approximation and derive error bounds in $L^2$ for the cell density and $H^1$ for the chemical concentration. We also present numerical experiments to illustrate the theoretical results.

Error analysis of a backward Euler positive preserving stabilized scheme for a Chemotaxis system

TL;DR

This work analyzes finite element approximations for a two-dimensional Keller–Segel chemotaxis system under small initial data to avoid blow-up. It develops and compares two stabilized semidiscrete schemes (low-order with lumped mass and AFC) and discretizes in time with backward Euler, proving existence and providing rigorous error bounds for the fully discrete solution: \\( abla\\, error in the chemical concentration and L^2 error for the cell density. The study establishes positivity and mass conservation at the fully discrete level, and numerical experiments in 2D illustrate the theory, including a blow-up scenario where stabilized schemes preserve positivity. Overall, the paper delivers a comprehensive error analysis for positivity-preserving stabilized schemes in chemotaxis models, with practical implications for robust, accurate long-time simulations.

Abstract

For a Keller-Segel model for chemotaxis in two spatial dimensions we consider a modification of a positivity preserving fully discrete scheme using a local extremum diminishing flux limiter. We discretize space using piecewise linear finite elements on an quasiuniform triangulation of acute type and time by the backward Euler method. We assume that initial data are sufficiently small in order not to have a blow-up of the solution. Under appropriate assumptions on the regularity of the exact solution and the time step parameter we show existence of the fully discrete approximation and derive error bounds in for the cell density and for the chemical concentration. We also present numerical experiments to illustrate the theoretical results.
Paper Structure (16 sections, 21 theorems, 208 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 16 sections, 21 theorems, 208 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mesh assumptions
  4. Stabilized semidiscrete methods
  5. Low order scheme
  6. Algebraic flux correction scheme
  7. Auxiliary results
  8. Fully discrete scheme
  9. Error estimates
  10. Existence
  11. Positivity
  12. Numerical experiments
  13. Blow-up at the center of unit square domain
  14. Convergence study of the stabilized schemes
  15. Conclusions
  16. ...and 1 more sections

Key Result

Lemma 2.7

Let the positive coefficients $q_i,$$i\in\mathcal{N}_h$, in Algorithm algorithm-1 be defined by with $\gamma_i$ defined in def:gamma_i, then the linearity preservation property eqn:linear_preserve is satisfied. Further, if there exists $M>0$ such that $\|\nabla w\|_{L^{\infty}} \leq M$ for $w\in\mathcal{S}_h$ and the constants $q_i,$$i\in\mathcal{N}_h$, in Algorithm algorithm-1, are defined by wi

Figures (4)

  • Figure 2.1: Various sub-domains of the triangulation $\mathcal{T}_h.$
  • Figure 5.1: A triangulation of a square domain.
  • Figure 5.2: Left: Approximation of standard FEM scheme. Right: Approximation of low order scheme. Bottom: Approximation of AFC scheme.
  • Figure A.1: Left: The patch $\Pi_0$ around node $Z_i.$ Right: The patch $\Pi_1$ around boundary node $Z_i.$

Theorems & Definitions (50)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 40 more