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

Previous Article
Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system
CPAA Home
This Issue
Next Article
On the solvability conditions for the diffusion equation with convection terms

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

Abstract
Related Papers

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, (2003).
[2]	S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,'', 3rd edition, (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. Num\'er., 15 (1981), 3.
[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. 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. 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. 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. 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. 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. 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. 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. 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.
[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. 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. 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.
[19]	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.
[20]	F. F. Keller and L. A. Segel, Initiation on slime mold aggregation viewed as instability,, J. Theor. Biol., 26 (1970), 399. 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. 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.
[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. 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. doi: 10.1093/imanum/drl018.
[26]	N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results,, RIMS K\^oky\^uroku Bessatsu, B15 (2009), 125.
[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, (2003).
[2]	S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,'', 3rd edition, (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. Num\'er., 15 (1981), 3.
[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. 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. 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. 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. 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. 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. 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. 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. 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.
[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. 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. 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.
[19]	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.
[20]	F. F. Keller and L. A. Segel, Initiation on slime mold aggregation viewed as instability,, J. Theor. Biol., 26 (1970), 399. 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. 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.
[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. 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. doi: 10.1093/imanum/drl018.
[26]	N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results,, RIMS K\^oky\^uroku Bessatsu, B15 (2009), 125.
[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).

[1]	Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873
[2]	Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[3]	Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[4]	Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153
[5]	Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387
[6]	Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
[7]	Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure & Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014
[8]	Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 321-328. doi: 10.3934/dcdss.2020018
[9]	Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183
[10]	Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663
[11]	Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014
[12]	Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339
[13]	Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
[14]	Mihaela Negreanu, J. Ignacio Tello. On a Parabolic-ODE system of chemotaxis. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 279-292. doi: 10.3934/dcdss.2020016
[15]	Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[16]	Eduardo Casas, Boris Vexler, Enrique Zuazua. Sparse initial data identification for parabolic PDE and its finite element approximations. Mathematical Control & Related Fields, 2015, 5 (3) : 377-399. doi: 10.3934/mcrf.2015.5.377
[17]	Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[18]	Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[19]	Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665
[20]	So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences