Finite element analysis of a Keller-Segel model with additional cross-diffusion and logistic source. Part II: Time convergence and numerical simulation

Abstract

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.