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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
References:
[1]

R. A. Adams and J. Fournier, "Sobolev Spaces,'' 2nd edition, Academic Press, 2003.  Google Scholar

[2]

S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,'' 3rd edition, Springer, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[3]

K. Baba and T. Tabata, On a conservative upwind finite-element scheme for convective diffusion equations, RAIRO Anal. Numér., 15 (1981), 3-25.  Google Scholar

[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. doi: 10.1007/s10543-006-0062-3.  Google Scholar

[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. doi: 10.1007/s00211-008-0188-0.  Google Scholar

[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. doi: 10.1007/BF00252458.  Google Scholar

[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. doi: 10.1090/S0025-5718-1987-0878688-2.  Google Scholar

[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. doi: 10.1007/BF01400302.  Google Scholar

[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. 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]

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. doi: 10.1016/j.jmaa.2009.04.025.  Google Scholar

[12]

F. Filbet, A finite volume scheme for Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3.  Google Scholar

[13]

H. Fujita, N. Saito and T. Suzuki, "Operator Theory and Numerical Methods,'' Elsevier, 2001.  Google Scholar

[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.  Google Scholar

[15]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Pitman, 1985.  Google Scholar

[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. doi: 10.1007/s10955-009-9717-1.  Google Scholar

[17]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

F. F. Keller and L. A. Segel, Initiation on slime mold aggregation viewed as instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[21]

A. Marrocco, Numerical simulation of chemotactic bacteria aggregation via mixed finite-elements, M2AN Math. Model. Numer. Anal., 37 (2003), 617-630. doi: 10.1051/m2an:2003048.  Google Scholar

[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.  Google Scholar

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer, 1983.  Google Scholar

[24]

N. Saito, A holomorphic semigroup approach to the lumped mass finite element method, J. Comput. Appl. Math., 169 (2004), 71-85. doi: 10.1016/j.cam.2003.11.003.  Google Scholar

[25]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365. doi: 10.1093/imanum/drl018.  Google Scholar

[26]

N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, Kyoto University, B15 (2009), 125-146.  Google Scholar

[27]

T. Suzuki, "Free Energy and Self-Interacting Particles,'' Birkhauser, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[28]

T. Suzuki and T. Senba, "Applied Analysis: Mathematical Methods in Natural Science,'' Imperial College Press, 2004.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. Fournier, "Sobolev Spaces,'' 2nd edition, Academic Press, 2003.  Google Scholar

[2]

S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,'' 3rd edition, Springer, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[3]

K. Baba and T. Tabata, On a conservative upwind finite-element scheme for convective diffusion equations, RAIRO Anal. Numér., 15 (1981), 3-25.  Google Scholar

[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. doi: 10.1007/s10543-006-0062-3.  Google Scholar

[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. doi: 10.1007/s00211-008-0188-0.  Google Scholar

[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. doi: 10.1007/BF00252458.  Google Scholar

[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. doi: 10.1090/S0025-5718-1987-0878688-2.  Google Scholar

[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. doi: 10.1007/BF01400302.  Google Scholar

[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. 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]

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. doi: 10.1016/j.jmaa.2009.04.025.  Google Scholar

[12]

F. Filbet, A finite volume scheme for Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3.  Google Scholar

[13]

H. Fujita, N. Saito and T. Suzuki, "Operator Theory and Numerical Methods,'' Elsevier, 2001.  Google Scholar

[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.  Google Scholar

[15]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Pitman, 1985.  Google Scholar

[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. doi: 10.1007/s10955-009-9717-1.  Google Scholar

[17]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

F. F. Keller and L. A. Segel, Initiation on slime mold aggregation viewed as instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[21]

A. Marrocco, Numerical simulation of chemotactic bacteria aggregation via mixed finite-elements, M2AN Math. Model. Numer. Anal., 37 (2003), 617-630. doi: 10.1051/m2an:2003048.  Google Scholar

[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.  Google Scholar

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer, 1983.  Google Scholar

[24]

N. Saito, A holomorphic semigroup approach to the lumped mass finite element method, J. Comput. Appl. Math., 169 (2004), 71-85. doi: 10.1016/j.cam.2003.11.003.  Google Scholar

[25]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365. doi: 10.1093/imanum/drl018.  Google Scholar

[26]

N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, Kyoto University, B15 (2009), 125-146.  Google Scholar

[27]

T. Suzuki, "Free Energy and Self-Interacting Particles,'' Birkhauser, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[28]

T. Suzuki and T. Senba, "Applied Analysis: Mathematical Methods in Natural Science,'' Imperial College Press, 2004.  Google Scholar

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