Finite element approximation of a Keller–Segel model with additional self- and cross-diffusion terms and a logistic source

Abstract

A fully finite element approximation of the Keller–Segel model, including self- and cross-diffusion terms and a logistic source, has been considered. The existence of a fully finite element approximation of the Keller–Segel model, some stability bounds of the solution and the convergence of the approximated solution have been shown. As a result, the existence of a solution to the nonlinear Keller–Segel model in Rd, d≤3 has been demonstrated. Moreover, an iterative approach to solving the resulted nonlinear discrete system has been introduced. Finally, some numerical results have been shown.