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A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model

Jan Giesselmann, Kiwoong Kwon

TL;DR

This work develops conditional but optimally convergent a posteriori error estimates for a discontinuous Galerkin discretization of the parabolic-elliptic Keller-Segel system in $d=2,3$. It combines space-time reconstructions with an $H^{-1}$-norm residual control within a stability framework to bound the error by computable estimators $A^{\xi}$, $E^{\xi}$ and elliptic-time reconstruction terms, while providing a verifiable regularity condition that can certify existence up to time $T$ from numerical data. The analysis leverages SIP for the diffusion term, a weighted SIP for chemotaxis, and IMEX time stepping, along with elliptic reconstructions to ensure optimal residual order. Numerical experiments confirm the estimator scales like $h^k+\tau$ for smooth solutions and illustrate the framework's potential for mesh adaptivity in resolving blow-up or sharp chemotactic structures. Overall, the paper offers a rigorous pathway toward reliable, adaptive simulations of chemotaxis models via a posteriori error control anchored in a robust stability theory.

Abstract

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.

A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model

TL;DR

This work develops conditional but optimally convergent a posteriori error estimates for a discontinuous Galerkin discretization of the parabolic-elliptic Keller-Segel system in . It combines space-time reconstructions with an -norm residual control within a stability framework to bound the error by computable estimators , and elliptic-time reconstruction terms, while providing a verifiable regularity condition that can certify existence up to time from numerical data. The analysis leverages SIP for the diffusion term, a weighted SIP for chemotaxis, and IMEX time stepping, along with elliptic reconstructions to ensure optimal residual order. Numerical experiments confirm the estimator scales like for smooth solutions and illustrate the framework's potential for mesh adaptivity in resolving blow-up or sharp chemotactic structures. Overall, the paper offers a rigorous pathway toward reliable, adaptive simulations of chemotaxis models via a posteriori error control anchored in a robust stability theory.

Abstract

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.
Paper Structure (15 sections, 9 theorems, 110 equations, 1 figure, 8 tables)

This paper contains 15 sections, 9 theorems, 110 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. Background on weak solutions and blow-up
  3. Stability framework
  4. Discontinuous Galerkin scheme
  5. Space-time reconstruction
  6. Controlling the $H^{-1}$-norm of the residual
  7. Estimate for $A_1$:
  8. Estimate for $A_2$:
  9. Estimate for $A_3$:
  10. Estimate for $A_5$:
  11. Estimate for $A_6$:
  12. A posteriori error estimator
  13. Numerical experiments
  14. Funding
  15. Conflict of interest

Key Result

Theorem 2.2

Biler1994 Assume that $\Omega$ is a bounded domain in $\mathbb{R}^d$ with piecewise smooth boundary.

Figures (1)

  • Figure 1: Simulation snapshots for $\left\{ \rho_h^n \right\}_{n=0}^N$ with $k = 1$ and $i=8$ at different times.

Theorems & Definitions (35)

  • Definition 2.1
  • Theorem 2.2: Existence of weak solutions
  • Remark 2.3
  • Remark 2.4: Blow-up of $L^p$-norms
  • Lemma 2.5: Blow-up criterion for weak solutions
  • Lemma 3.1
  • Theorem 3.2: Conditional stability estimate
  • Remark 3.3: Stability framework
  • Corollary 3.4: A posteriori verifiable regularity
  • proof
  • ...and 25 more