Abstract
We consider the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, satisfying
with p ∈ (1, 2), \({\Omega }\subseteq \mathbb {R}^{d}\) a bounded domain (d = 1, 2, 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for d = 1, 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables (u,v) under structured meshes, and the second one by using the auxiliary variable σ = ∇v and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments.
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The authors have been partially supported by MINECO grant MTM2015-69875-P (Ministerio de Economía y Competitividad, Spain) with the participation of FEDER. The first and second authors have also been supported by Grant PGC2018-098308-B-I00 by MCIN/AEI/ 10.13039/501100011033 and by ERDF A way of making Europe; and the third author has also been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander.
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Appendix. Proof of Lemma 4.4
Appendix. Proof of Lemma 4.4
The proof follows the ideas of [3, Lemma 2.1], with some modifications. For simplicity in the notation, we will prove (54) in the 1-dimensional case, but this proof can be extended to dimensions 2 and 3 as in [3, Lemma 2.1]. Observe that, from (52)
where \(u^{h}_{1, 2} \in \mathbb {P}_{1}(K)\) with \(u^{h}_{1, 2}(\mathbf {a}_{0}^{K})={u^{h}_{2}}(\mathbf {a}_{0}^{K})\) and \(u^{h}_{1, 2}(\mathbf {a}_{1}^{K})={u^{h}_{1}}(\mathbf {a}_{1}^{K})\), μ1i (i = 1, 2) lie between \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\) and \({u^{h}_{1}}(\mathbf {a}_{1}^{K})\), μ2i (i = 1, 2) lie between \({u^{h}_{2}}(\mathbf {a}_{0}^{K})\) and \({u^{h}_{2}}(\mathbf {a}_{1}^{K})\), and ξi (i = 1, 2) lie between \({u^{h}_{1}}(\mathbf {a}_{1}^{K})\) and \({u^{h}_{2}}(\mathbf {a}_{0}^{K})\). Then, first we will show that
for \({u^{h}_{1}}(\mathbf {a}_{0}^{K}) \neq {u^{h}_{2}} (\mathbf {a}_{0}^{K})\), because the case \({u^{h}_{1}}(\mathbf {a}_{0}^{K})= {u^{h}_{2}} (\mathbf {a}_{0}^{K})\) is trivially true. With this aim, we consider γi (i = 1, 2) lying between \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\) and \({u^{h}_{2}} (\mathbf {a}_{0}^{K})\) such that
and therefore, from the definitions of ξi, γi and μ1i, i = 1, 2, given after (82) and (84), we deduce
Then, for \({u^{h}_{2}}(\mathbf {a}_{0}^{K})\), \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\) and \({u^{h}_{1}}(\mathbf {a}_{1}^{K})\), there are only 3 options: (1) \({u^{h}_{1}}(\mathbf {a}_{1}^{K})\) lies between \({u^{h}_{2}}(\mathbf {a}_{0}^{K})\) and \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\); (ii) \({u^{h}_{2}}(\mathbf {a}_{0}^{K})\) lies between \({u^{h}_{1}}(\mathbf {a}_{1}^{K})\) and \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\); and (iii) \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\) lies between \({u^{h}_{1}}(\mathbf {a}_{1}^{K})\) and \({u^{h}_{2}}(\mathbf {a}_{0}^{K})\).
Notice that from (43) and (44), we have that \(F^{\prime }_{\varepsilon }\) and \((p-1)\frac {F^{\prime }_{\varepsilon }}{F^{\prime \prime }_{\varepsilon }}\) are globally Lipschitz functions with constants εp− 2 and 1 respectively, and \(\frac {1}{\vert F^{\prime \prime }_{\varepsilon } \vert }\leq \varepsilon ^{p-2}\). Then, in case (i), taking into account that all intermediate values μ1i,γi,ξi (i = 1, 2) lie between \({u^{h}_{2}}(\mathbf {a}_{0}^{K})\) and \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\), we have
In case (ii), all intermediate values μ1i,γi,ξi (i = 1, 2) lie between \({u^{h}_{1}}(\mathbf {a}_{1}^{K})\) and \({u^{h}_{1}}(\mathbf {a}_{0}^{K})\), and from (85) and (86) by eliminating the term \(({u^{h}_{2}}(\mathbf {a}_{0}^{K}) - {u^{h}_{1}} (\mathbf {a}_{1}^{K}))\), we have the equality
from which, bounding the term \(\left \vert \frac {F^{\prime }_{\varepsilon }(\xi _{1})}{F^{\prime \prime }_{\varepsilon }(\xi _{2})} - \frac {F^{\prime }_{\varepsilon }(\gamma _{1})}{F^{\prime \prime }_{\varepsilon }(\gamma _{2})} \right \vert \) as in (87), we obtain
and therefore, dividing by \(\vert {u^{h}_{1}}(\mathbf {a}_{1}^{K}) - {u^{h}_{1}} (\mathbf {a}_{0}^{K}))\vert \) we arrive at
In case (iii), by arguing analogously to case (ii), from (85) and (86) we have
which implies (88). Therefore, we have proved (83). Analogously, we can prove that
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Guillén-González, F., Rodríguez-Bellido, M.A. & Rueda-Gómez, D.A. A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes. Adv Comput Math 47, 87 (2021). https://doi.org/10.1007/s10444-021-09907-1
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DOI: https://doi.org/10.1007/s10444-021-09907-1
Keywords
- Chemorepulsion model
- Finite element approximation
- Energy-stability
- Nonlinear production
- Approximated positivity