Finite element analysis of a Keller-Segel model with additional cross-diffusion and logistic source. Part I: Space convergence

We have studied a finite element method for the (Patlak)-Keller-Segel equations in one and two space dimensions with additional cross-diffusion and logistic source terms in the elliptic equation for the chemical signal. Some a priori estimates of the regularized functions have been derived, independently of the regularization parameter, by deriving a well defined entropy inequality of the regularized problem. Moreover, a fixed point theorem has been utilized to prove the existence of the finite element solutions. Some stability bounds on the fully discrete approximations are obtained. The convergence of the approximate solutions in space has been shown.