Free Access
Volume 54, Number 4, July-August 2020
Page(s) 1339 - 1372
Published online 18 June 2020
  1. C. Berge, Graphs Second revision. North-Holland, Amsterdam, New York, Oxford (1985). [Google Scholar]
  2. R. Borsche, S. Göttlich, A. Klar and P. Schillen, The scalar Keller-Segel model on networks. Math. Models Methods Appl. Sci. 24 (2014) 221–247. [Google Scholar]
  3. R. Borsche, J. Kall, A. Klar and T.N.H. Pham, Kinetic and related macroscopic models for chemotaxis on networks. Math. Models Methods Appl. Sci. 26 (2016) 1219–1242. [Google Scholar]
  4. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (2008). [CrossRef] [Google Scholar]
  5. G. Bretti, R. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: a numerical study. ESAIM: M2AN 48 (2014) 231–258. [CrossRef] [EDP Sciences] [Google Scholar]
  6. F. Camilli and L. Corrias, Parabolic models for chemotaxis on weighted networks. J. Math. Pures Appl. 108 (2017) 459–480. [Google Scholar]
  7. A. Chertock, Y. Epshteyn, H. Hu and A. Kurganov, High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems. Adv. Comput. Math. 44 (2017) 327–350. [Google Scholar]
  8. P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. [Google Scholar]
  9. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Evolution problems. I, in Vol. 5. With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig. Springer-Verlag, Berlin (1992). [Google Scholar]
  10. Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models. J. Comput. Appl. Math. 224 (2009) 168–181. [Google Scholar]
  11. L. Evans, Partial Differential Equations. American Mathematical Society (2010). [Google Scholar]
  12. F. Filbet, A finite volume scheme for the Patlak–Keller–Segel chemotaxis model. Numer. Math. 104 (2006) 457–488. [Google Scholar]
  13. J.G. Heywood and R. Rannacher. Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. [Google Scholar]
  14. H.G. Heuser, Functional Analysis. Translated from the German by John Horváth. A Wiley-Interscience Publication. John Wiley &Sons, Ltd., Chichester (1982). [Google Scholar]
  15. T. Hillen, K.J. Painter, A user’s guide to pde models for chemotaxis. J. Math. Biol. 58 (2009) 183. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  16. T. Hillen and A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile. Math. Methods Appl. Sci. 27 (2004) 1783–1801. [Google Scholar]
  17. D. Horstmann, From 1970 util present: the Keller-Segel model in chemotaxis and its consequences I. Jahresber. DMV 105, 2003 (1970) 103–165. [Google Scholar]
  18. E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970) 399–415. [CrossRef] [PubMed] [Google Scholar]
  19. E.F. Keller and L.A. Segel, Model for chemotaxis. J. Theor. Biol. 30 (1971) 225–234. [CrossRef] [PubMed] [Google Scholar]
  20. D. Mugnolo, Semigroup Methods for Evolution Equations on Networks. Springer (2014). [CrossRef] [Google Scholar]
  21. E. Nakaguchi and A. Yagi, Full discrete approximations by Galerkin method for chemotaxis growth model. Nonlinear Anal. 47 (2001) 6097–6107. [CrossRef] [Google Scholar]
  22. K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller–Segel equations, Funkc. Ekvacioj Ser I 44 (2001) 441–469. [Google Scholar]
  23. R.J. Plemmons, M-matrix characterizations. I. Nonsingular M-matrices. Linear Algebra Appl. 18 (1977) 175–188. [Google Scholar]
  24. T. Roubček, Nonlinear partial differential equations with applications, second edition. In: Vol. 153 of International Series of Numerical Mathematics. Birkhäuser/Springer Basel AG, Basel (2013). [Google Scholar]
  25. N. Saito, Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Commun. Pure Appl. Anal. 11 (2012) 339–364. [CrossRef] [Google Scholar]
  26. R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D. J. Comput. Appl. Math. 239 (2013) 290–303. [Google Scholar]
  27. V. Thomée, Galerkin finite element methods for parabolic problems, second edition. In: Vol. 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2006). [Google Scholar]
  28. R.S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1971). [Google Scholar]
  29. M.F. Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723–759. [Google Scholar]

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