Finite element method for solving Keller–Segel chemotaxis system with cross-diffusion

Abstract

This paper presents a finite element method for nonlinear parabolic–parabolic system of partial differential equations, which describe the chemotactic features, called a Keller–Segel system with additional cross-diffusion term in the second equation. Firstly a semi-implicit scheme for weak formulation of the problem is introduced and then a fixed point formulation is defined for the corresponding scheme. Next the existence of approximate solutions is established by using Schauder’s fixed point theorem. Further a priori error estimate for the approximate solutions in $$H^1$$ —norm is derived. Numerical experiments are also made and they illustrate the theoretical results.