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

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.
Citation: Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure and 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.

[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.

[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. doi: 10.1007/s10543-006-0062-3.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. doi: 10.1007/s10955-009-9717-1.

[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.

[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. doi: 10.1016/0022-5193(70)90092-5.

[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.

[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. doi: 10.1016/j.cam.2003.11.003.

[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.

[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. doi: 10.1007/0-8176-4436-9.

[28]

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

show all references

References:
[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. doi: 10.1007/978-0-387-75934-0.

[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. doi: 10.1007/s10543-006-0062-3.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. doi: 10.1007/s10955-009-9717-1.

[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.

[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. doi: 10.1016/0022-5193(70)90092-5.

[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.

[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. doi: 10.1016/j.cam.2003.11.003.

[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.

[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. doi: 10.1007/0-8176-4436-9.

[28]

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

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