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This paper examines a finite element method for the chemotaxis-growth model with indirect attractant production and logistic source. To begin, we propose a regularized problem of the truncated problem. We then obtain some a priori estimates of regularized solutions that are independent of the regularization parameter using a well-defined entropy inequality for the regularized problem. Additionally, we offer an efficient fully discrete finite element approximation of the regularized problem. A fixed point theorem is then used to prove that the approximate solutions exist. A discrete entropy inequality is also proposed for fully discrete finite element problems, as well as some stability bounds. We also investigate the convergence of the fully discrete problem. [ABSTRACT FROM AUTHOR]

In this article, a finite element scheme for a model of the two-species chemotaxis system with two chemicals is analyzed. Firstly, we suggest a regularized problem of the truncated system. Next, using a well-defined entropy inequality of the regularized problem, we derive some a priori estimates of the regularized functions that are independent of the regularization parameter. We further present a fully discrete finite element approximation of the regularized problem that is practical and efficient. A discrete entropy inequality and certain stability constraints on regularized issue solutions are developed. In addition, we explore the fully discrete problem's convergence. This study provides an essential answer to an issue that many academics have pondered: why do solutions blow up in finite time? The solution blowing up does not reflect the biological behavior of the species, as considerable growth in the species can not be achieved in a short period of time. [ABSTRACT FROM AUTHOR]

In [1] , a finite element method for the (Patlak-) Keller-Segel equations, with additional cross-diffusion and logistic source terms in the elliptic equation, has been introduced. Some a priori estimates of the regularized solutions have been derived, independently of the regularization parameters. Moreover, the existence of the finite element solutions has been shown by using the fixed point theorem and some stability bounds on the fully discrete approximations are obtained. The convergence of the approximate solutions in space has been shown. In this article, we have completed what has been achieved in [1] , where we have completed the convergence of time, where this article starts from where the study in [1] ended. The current study aims to prove the existence of weak solutions to this model, which includes passing the limit to the remaining regulation parameter and achieving the convergence in time. We firstly show that the solutions, independent of regulation parameter, can be bounded. Then these bounds are used to derive bounds on the time-derivatives which are also independent of regulation parameter. In addition, compactness arguments have been used to discuss the convergence of the finite element the approximate problem. We demonstrated the existence of a weak solution for the model under study. Finally, we use the implicit scheme to conduct simulations in one and two space dimensions. [ABSTRACT FROM AUTHOR]