Finite element analysis of chemotaxis-growth model with indirect attractant production and logistic source

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.