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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 |
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).
|
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