Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part I: Space Convergence

Abstract

In this paper, a finite element scheme for the attraction-repulsion chemotaxis model is analyzed. We introduce a regularized problem of the truncated system. Then we obtain some a priori estimates of the regularized functions, independent of the regularization parameter, via deriving a well-defined entropy inequality of the regularized problem. Also, we propose a practical fully discrete finite element approximation of the regularized problem. Next, we use a fixed point theorem to show the existence of the approximate solutions. Moreover, a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived. In addition, the uniqueness of the fully discrete approximations is preformed. Finally, we discuss the convergence to the fully discrete problem.