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January  2012, 11(1): 339-364. doi: 10.3934/cpaa.2012.11.339

Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis

Norikazu Saito 1,

1. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

Received  January 2010 Revised  August 2010 Published  September 2011

We are concerned with the finite-element approximation for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. The scheme makes use of a semi-implicit time discretization with a time-increment control and Baba-Tabata's conservative upwind finite-element approximation in order to realize the positivity and mass conservation properties. The main aim is to present error analysis that is an application of the discrete version of the analytical semigroup theory.
Keywords: Finite-element method, nonlinear parabolic system, chemotaxis., error analysis.
Mathematics Subject Classification: Primary: 65M60, 65M15; Secondary: 35K5.
Citation: Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339
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show all references

References:
[1]

2nd edition, Academic Press, 2003.  Google Scholar

[2]

3rd edition, Springer, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[3]

RAIRO Anal. Numér., 15 (1981), 3-25.  Google Scholar

[4]

BIT Numer. Math., 46 (2006), 249-260. doi: 10.1007/s10543-006-0062-3.  Google Scholar

[5]

Numer. Math., 111 (2008), 169-205. doi: 10.1007/s00211-008-0188-0.  Google Scholar

[6]

Arch. Rational Mech. Anal., 46 (1972), 177-199. doi: 10.1007/BF00252458.  Google Scholar

[7]

Math. Comp., 48 (1987), 521-532. doi: 10.1090/S0025-5718-1987-0878688-2.  Google Scholar

[8]

Numer. Math., 23 (1975), 193-197. doi: 10.1007/BF01400302.  Google Scholar

[9]

J. Sci. Comput., 40 (2009), 211-256. doi: 10.1007/s10915-009-9281-5.  Google Scholar

[10]

Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model,, SIAM J. Numer. Anal., 47 (): 386.  doi: 0.1137/07070423X.  Google Scholar

[11]

J. Math. Anal. Appl., 358 (2009), 136-147. doi: 10.1016/j.jmaa.2009.04.025.  Google Scholar

[12]

Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3.  Google Scholar

[13]

Elsevier, 2001.  Google Scholar

[14]

J. Fac. Sci. Univ. Tokyo Sect. I, 15 (1968), 169-177.  Google Scholar

[15]

Pitman, 1985.  Google Scholar

[16]

J. Stat. Phys., 135 (2009), 133-151. doi: 10.1007/s10955-009-9717-1.  Google Scholar

[17]

J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[18]

Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.  Google Scholar

[19]

Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-89.  Google Scholar

[20]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[21]

M2AN Math. Model. Numer. Anal., 37 (2003), 617-630. doi: 10.1051/m2an:2003048.  Google Scholar

[22]

Hokkaido Math. J., 31 (2002), 385-429.  Google Scholar

[23]

Springer, 1983.  Google Scholar

[24]

J. Comput. Appl. Math., 169 (2004), 71-85. doi: 10.1016/j.cam.2003.11.003.  Google Scholar

[25]

IMA J. Numer. Anal., 27 (2007), 332-365. doi: 10.1093/imanum/drl018.  Google Scholar

[26]

RIMS Kôkyûroku Bessatsu, Kyoto University, B15 (2009), 125-146.  Google Scholar

[27]

Birkhauser, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[28]

Imperial College Press, 2004.  Google Scholar

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