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Numerische Mathematik

, Volume 135, Issue 1, pp 265–311 | Cite as

Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis

  • Guanyu ZhouEmail author
  • Norikazu Saito
Article

Abstract

We consider the finite volume approximation for a non-linear parabolic-elliptic system, which describes the aggregation of slime molds resulting from their chemotactic features, called a simplified Keller–Segel system. First, we present a linear finite volume scheme that satisfies both positivity and mass conservations, which are important features of the original system. We derive some inequalities on the discrete free energy. Then, under some assumptions on the regularity of solution, admissible mesh and a priori estimates of the discrete solution, we establish error estimates in \(L^p\) norm with a suitable \(p>2\) for the two dimensional case. In the last part of this paper, we restrict our attention to the radially symmetric solution of chemotaxis system, and we derive some inequalities concerned with the blow-up phenomenon of numerical solution. Several numerical experiments are presented to verify the theoretical results.

Mathematics Subject Classification

65M15 65M08 35K55 92C17 

Notes

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

References

  1. 1.
    Adams, R.A., Fournier, J.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Baba, K., Tabata, M.: On a conservative upwind finite element scheme for convective diffusion equations. RAIRO Anal. Numér. 15, 3–25 (1981)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Biler, B.: Local and global solvability of some parabolic systems modeling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bessemoulin-Chatard, M., Chainais-Hilliairet, C., Filbet, F.: On discrete functional inequalities for some finite volume schemes. arXiv:1202.4806
  5. 5.
    Bessemoulin-Chatard, M., Jüngel, A.: A finite volume scheme for a Keller–Segel model with additional cross-diffusion. IMA J. Numer. 34, 96–122 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanchet, A., Dolbeault, J., Perthame, B.: Two dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions Electron. J. Differ. Equ. 2006, 1–33 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chou, S., Kwak, DoY, Li, Q.: \(L^p\) error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differ. Equ. 19, 463–486 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Childress, S.: Chemotactic collapse in two dimensions. Lecture Notes in Biomath, vol. 55. Springer, Berlin (1984)Google Scholar
  10. 10.
    Childress, S., Percus, J.K.: Nolinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  12. 12.
    Crouzeix, M., Thomée, V.: Resolvent estimates in \(l^p\) for discrete laplacians on irregular meshes and maximum-norm stability of parabolic finite difference schemes. Comput. Methods Appl. Math. 1, 3–17 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods, Handb. Numer. Anal., vol. VII, pp. 713–1020, North-Holland, Amsterdam (2000)Google Scholar
  14. 14.
    Filbet, F.: A finite volume scheme for the Patlak–Keller–Segel chemotaxis model. Numer. Math. 104, 457–488 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Pitman (1985)Google Scholar
  16. 16.
    Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105, 103–165 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein. 106, 51–69 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, Berlin (2003)zbMATHGoogle Scholar
  19. 19.
    Keller, F.F., Segel, L.A.: Initiation on slime mold aggregation viewed as instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefzbMATHGoogle Scholar
  20. 20.
    Marrocco, A.: 2D simulation of chemotaxis bacteria aggregation. ESAIM: M2AN 37, 617–630 (2003)Google Scholar
  21. 21.
    Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Nagai, T., Senba, T.: Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis. Adv. Math. Sci. Appl. 8, 145–156 (1998)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Perthame, B.: PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl. Math. 49, 539–364 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Saito, N.: Remarks on the rational approximation of holomorphic semigroups with nonuniform partitions. Japan J. Ind. Appl. Math. 21, 323–337 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Saito, N.: A holomorphic semigroup approach to the lumped mass finite element method. J. Comput. Appl. Math. 160, 71–85 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Saito, N.: Conservative upwind finite-element method for a simplified Keller–Segel system modelling chemotaxis. IMA J. Numer. Anal. 27, 332–365 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Saito, N.: Error analysis of a conservative finite-element approximation for the Keller–Segel system of chemotaxis. Commun. Pure Appl. Anal. 11, 339–364 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Saito, N., Suzuki, T.: Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis. Appl. Math. Comput. 171, 72–90 (2005)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Suzuki, T.: Free Energy and Self-Interacting Particles. Birkhauser, Basel (2005)CrossRefzbMATHGoogle Scholar
  30. 30.
    Suzuki, T.: Mean Field Theories and Dual Variation—A Mathematical Profile Emerged in the Nonlinear Hierarchy. Atlantis Press, UK (2008)Google Scholar
  31. 31.
    Suzuki, T., Senba, T.: Applied Analysis: Mathematical Methods in Natural Science. Imperial College Press, London (2004)zbMATHGoogle Scholar
  32. 32.
    Thomée, V.: Galerkin finite element methods for parabolic problems, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  33. 33.
    Varga, R.S.: Matrix Iterative Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  34. 34.
    Yagi, A.: Norm behaviour of solutions to a parabolic system of chemotaxis. Math. Japan 45, 241–256 (1997)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

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