Analysis of a fully discrete approximation for the classical Keller--Segel model: lower and a priori bounds

TL;DR

The work develops a fully discrete finite element scheme for the classical Keller--Segel chemotaxis model that preserves fundamental qualitative properties. By combining mass lumping with a semi-implicit time discretization, the authors obtain a linear, decoupled solver and prove lower bounds, $L^1$-bounds, a discrete energy law, and a priori estimates via a discrete Moser--Trudinger inequality. Under precise hypotheses and mild mesh/time-step restrictions, the method yields positivity of $u$, nonnegativity of $v$, mass conservation, and controlled energy growth, with two numerical experiments illustrating both non-blowup and blowup scenarios. The results provide a robust numerical framework for chemotaxis models with strong theoretical guarantees, enhancing the reliability of simulations in applications where positivity and energy stability are essential.

Abstract

This paper is devoted to constructing approximate solutions for the classical Keller--Segel model governing \emph{chemotaxis}. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant. The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds -- positivity for the cell density and nonnegativity for the chemoattractant density --, are bounded in the $L^1(Ω)$-norm, satisfy a discrete energy law, and have \emph{ a priori} energy estimates. The latter is achieved by means of a discrete Moser--Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.

Figures (6)

• Figure 1: Reference macrolement, composed of 14 acute triangles
• Figure 2: Solution $u^n_h$ (colored isolines) and $v_h$ (gray scale background) at three time steps: $t_n=0$, $2.5\cdot 10^{-3}$ and $5\cdot 10^{-3}$. Diffusion and chemotaxis transfer of $u^n_h$ (cells) towards highest concentrations of $v^n_h$ can be seen along time. Acute mesh with $N_{\hbox{square}}=50$, $k=10^{-4}$, initial data parameter: $C_u=70$
• Figure 3: Plot of $\min_{\mathcal{T}_h}(u_h^n)$, $n=0,\dots 50$, where ${\mathcal{T}_h}$ is an acute mesh (top) or non-acute mesh (bottom). Initial value constants: $C_u=70,80,90,100$
• Figure 4: Reference macrolement containing some obtuse triangles (triangles 1, 9, 5 and 11).
• Figure 5: Solution $u^n_h$ associated with an acute mesh of $100\times 100$ macroelements at time steps $n=0$, $30$, $60$, and $88$ (when positivity is broken).

Theorems & Definitions (31)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Corollary 2.6
• proof
• Proposition 2.7: Moser-Trudinger
• Corollary 2.8
• proof