Abstract: In Hashim and Harfash (Appl. Math. Comput. 2021), using a finite element method, the attraction-repulsion chemotaxis model $ (\mathrm {P}) $ in space is discretised; finite differences were used to do the same in time. Furthermore, the existence of a global weak solution to the system $ (\mathrm {P}_{M} ^{\Delta t}) $\end{document}was demonstrated by means of analysis of the convergence of the fully discrete approximate problem $ (\mathrm {P}_{M, \varepsilon } ^{h, \Delta t} ) $. Moreover, the functions $ \{U, Z, V\} $ were proved to represent a global weak solution to the system $ (\mathrm {P}_{M} ^{\Delta t}) $ by means of a passage to the limit $ \varepsilon , h \rightarrow 0 $ of the approximate system. This paper's purpose is to demonstrate that the solutions can be bounded, independent of M. The analysis contained in this paper illustrates the idea of the existence of weak solutions to the model $ (\mathrm {P}) $, that requires passing to the limits, $ \Delta t \rightarrow 0^{+} $ and $ M \rightarrow \infty $. The time step $ \Delta t $ is subsequently linked to the cutoff parameter $ M > 1 $ by positing a demand that $ \Delta t=o(M^{-1}) $, as $ M \rightarrow \infty $, with the result that the cutoff parameter becomes the only parameter in the problem $ (\mathrm {P}_{M} ^{\Delta t}) $. The solutions can be bounded, independent of M, with the use of special energy estimates, as demonstrated herein. Then, these M-independent bounds on the relative entropy are employed with the purpose of deriving M-independent bounds on the time-derivatives. Additionally, compactness arguments were utilised to explore the convergence of the finite element approximate problem. The conclusion was that a weak solution for $ (\mathrm {P}) $ existed. Finally, we introduced the error estimate and the implicit scheme was used to perform simulations in one and two space dimensions.

Key words: Finite element, Chemotaxis, Convergence, Weak solution