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Mathematics of Computation

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Unconditionally energy stable fully discrete schemes for a chemo-repulsion model


Authors: F. Guillén-González, M. A. Rodríguez-Bellido and D. A. Rueda-Gómez
Journal: Math. Comp. 88 (2019), 2069-2099
MSC (2010): Primary 35K51, 35Q92, 65M12, 65M60, 92C17
DOI: https://doi.org/10.1090/mcom/3418
Published electronically: March 11, 2019
MathSciNet review: 3957887
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Abstract: This work is devoted to studying unconditionally energy stable and mass-conservative numerical schemes for the following repulsive-productive chemotaxis model: find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, such that \begin{equation*} \left \{ \begin {array} [c]{lll}\partial _t u - \Delta u - \nabla \cdot (u\nabla v)=0 \ \ \text {in}\ \Omega ,\ t>0,\\ \partial _t v - \Delta v + v = u \ \ \text {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 propose three fully discrete Finite Element (FE) approximations. The first one is a nonlinear approximation in the variables $(u,v)$; the second one is another nonlinear approximation obtained by introducing ${\boldsymbol \sigma }=\nabla v$ as an auxiliary variable; and the third one is a linear approximation constructed by mixing the regularization procedure with the energy quadratization technique, in which other auxiliary variables are introduced. In addition, we study the well-posedness of the numerical schemes, proving unconditional existence of solution, but conditional uniqueness (for the nonlinear schemes). Finally, we compare the behavior of such schemes throughout several numerical simulations and provide some conclusions.


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Additional Information

F. Guillén-González
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico and IMUS, Universidad de Sevilla, Facultad de Matemáticas, C/ Tarfia, S/N, 41012 Sevilla, Spain
MR Author ID: 326792
Email: guillen@us.es

M. A. Rodríguez-Bellido
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico and IMUS, Universidad de Sevilla, Facultad de Matemáticas, C/ Tarfia, S/N, 41012 Sevilla, Spain
Email: angeles@us.es

D. A. Rueda-Gómez
Affiliation: Escuela de Matemáticas, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia
Email: diaruego@uis.edu.co

Keywords: Chemo-repulsion and production model, finite element approximation, unconditional energy-stability, regularization, quadratization of energy
Received by editor(s): July 3, 2018
Received by editor(s) in revised form: November 13, 2018
Published electronically: March 11, 2019
Additional Notes: The authors were partially supported by MINECO grant MTM2015-69875-P (Ministerio de Economía y Competitividad, Spain) with the participation of FEDER
The third author was also supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander
Article copyright: © Copyright 2019 American Mathematical Society