Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis
Pages: 339 - 364, Issue 1, January 2012

doi:10.3934/cpaa.2012.11.339      Abstract        References        Full text (486.1K)                  Related Articles

Norikazu Saito - Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan (email)

1	R. A. Adams and J. Fournier, "Sobolev Spaces,'' 2nd edition, Academic Press, 2003.
2	S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,'' 3rd edition, Springer, 2008.
3	K. Baba and T. Tabata, On a conservative upwind finite-element scheme for convective diffusion equations, RAIRO Anal. Numér., 15 (1981), 3-25.
4	M. Boman, Estimates for the $L_2$-projection onto continuous finite element spaces in a weighted $L_p$-norm, BIT Numer. Math., 46 (2006), 249-260.
5	A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.
6	P. G. Ciarlet and R. A. Raviart, General Lagrange and Hermite interpolation in $R^n$ with applications to finite element methods, Arch. Rational Mech. Anal., 46 (1972), 177-199.
7	M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp., 48 (1987), 521-532.
8	J. Douglas Jr., T. Dupont and L. Wahlbin, The stability in $L_p$ and $W_p^1$ of the $L_2$-projection into finite element function spaces, Numer. Math., 23 (1975), 193-197.
9	Y. Epshteyn and A. Izmirlioglu, Fully discrete analysis of a discontinuous finite element method for the Keller-Segel chemotaxis model, J. Sci. Comput., 40 (2009), 211-256.
10	Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model, SIAM J. Numer. Anal., 47 (2008/09), 386-408.
11	M. Efendiev, E. Nakaguchi and W. L. Wendland, Dimension estimate of the global attractor for a semi-discretized chemotaxis-growth system by conservative upwind finite-element scheme, J. Math. Anal. Appl., 358 (2009), 136-147.
12	F. Filbet, A finite volume scheme for Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.
13	H. Fujita, N. Saito and T. Suzuki, "Operator Theory and Numerical Methods,'' Elsevier, 2001.
14	D. Fujiwara, $L^p$-theory for characterizing the domain of the fractional powers of $-\Delta $ in the half space, J. Fac. Sci. Univ. Tokyo Sect. I, 15 (1968), 169-177.
15	P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Pitman, 1985.
16	J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151.
17	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
18	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
19	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-89.
20	F. F. Keller and L. A. Segel, Initiation on slime mold aggregation viewed as instability, J. Theor. Biol., 26 (1970), 399-415.
21	A. Marrocco, Numerical simulation of chemotactic bacteria aggregation via mixed finite-elements, M2AN Math. Model. Numer. Anal., 37 (2003), 617-630.
22	E. Nakaguchi and Y. Yagi, Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems, Hokkaido Math. J., 31 (2002), 385-429.
23	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer, 1983.
24	N. Saito, A holomorphic semigroup approach to the lumped mass finite element method, J. Comput. Appl. Math., 169 (2004), 71-85.
25	N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365.
26	N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, Kyoto University, B15 (2009), 125-146.
27	T. Suzuki, "Free Energy and Self-Interacting Particles,'' Birkhauser, 2005.
28	T. Suzuki and T. Senba, "Applied Analysis: Mathematical Methods in Natural Science,'' Imperial College Press, 2004.

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