Analysis of a chemo-repulsion model with nonlinear production: The continuous problem and unconditionally energy stable fully discrete schemes
Provides theoretical and numerical analysis for a specific chemotaxis model, but the contribution is incremental for researchers in numerical analysis of PDEs.
The paper proves existence of solutions for a chemo-repulsion model with nonlinear production and proposes three unconditionally energy stable finite element schemes, demonstrating mass conservation and solvability through numerical simulations.
We consider the following repulsive-productive chemotaxis model: Let $p\in (1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, satisfying \begin{equation}\label{C5:Am} \left\{ \begin{array} [c]{lll} \partial_t u - Δu - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ Ω,\ t>0,\\ \partial_t v - Δv + v = u^p \ \ \mbox{in}\ Ω,\ t>0, \end{array} \right. \end{equation} in a bounded domain $Ω\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we prove the existence of solutions of this problem. Moreover, we propose three fully discrete Finite Element (FE) nonlinear approximations, where the first one is defined in the variables $(u,v)$, and the second and third ones by introducing ${\boldsymbolσ}=\nabla v$ as an auxiliary variable. We prove some unconditional properties such as mass-conservation, energy-stability and solvability of the schemes. Finally, we compare the behavior of the schemes throughout several numerical simulations and give some conclusions.