Analysis of a chemo-repulsion model with nonlinear production: The continuous problem and unconditionally energy stable fully discrete schemes
| Autor: | Guillén-González, F., Bellido, María Ángeles Rodríguez, Gómez, Diego Armando Rueda |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 - \Delta u - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ \Omega,\ t>0,\\ \partial_t v - \Delta v + v = u^p \ \ \mbox{in}\ \Omega,\ t>0, \end{array} \right. \end{equation} in a bounded domain $\Omega\subseteq \mathbb{R}^d$, $d=2,3$. 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 ${\boldsymbol\sigma}=\nabla v$ 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. Comment: arXiv admin note: substantial text overlap with arXiv:1807.01118 |
| Databáze: | arXiv |
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