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

• Published in: International journal of dynamics and control Vol. 6; no. 2; pp. 539 - 549
• Main Authors: Gurusamy, A., Balachandran, K.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-06-2018; Springer Nature B.V
• Subjects: Complexity; Control; Control and Systems Theory; Dynamical Systems; Engineering; Finite element analysis; Finite element method; Fixed points (mathematics); Nonlinear programming; Partial differential equations; Vibration; Finite element method; FreeFem; 65M60; Keller–Segel system; 74H20; Existence of solutions
• ISSN: 2195-268X; 2195-2698
• DOI: 10.1007/s40435-017-0335-5

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.