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

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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.

Keywords

Finite element method Keller–Segel system Existence of solutions FreeFem++ 

Mathematics Subject Classification

65M60 74H20 

Notes

Acknowledgements

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Computational Biology Division, DRDO-BU Center for Life SciencesBharathiar University CampusCoimbatoreIndia
  2. 2.Department of MathematicsBharathiar UniversityCoimbatoreIndia

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