Finite element analysis of a two-species chemotaxis system with two chemicals

Abstract

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.