Finite element method for solving Keller–Segel chemotaxis system with cross-diffusion
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.
KeywordsFinite element method Keller–Segel system Existence of solutions FreeFem++
Mathematics Subject Classification65M60 74H20
This work is supported by Defence Research and Development Organization, New Delhi, Government of India. The authors would like to thank Professors Neela Nataraj and Dimitrios Mitsotakis for their valuable suggestions to improve the quality of this paper. Further the authors thank the referees for the improvement of the paper.
- 4.Bonner JT (1967) The cellular slime molds, 2nd edn. Princeton University Press, PrincetonGoogle Scholar
- 19.Saito N, Suzuki T (2012) Notes on finite difference schemes to a parabolic–elliptic system modelling chemotaxis. Appl Math Comput 171:72–90Google Scholar