Conservative upwind finite-element method for a simplified Keller–Segel system modelling chemotaxis

Saito, Norikazu

doi:10.1093/imanum/drl018

Cited by 87 publications

(64 citation statements)

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“…For convenience we refer the reader to [15] for the consistent-mass approximation of (CG), to [3,16,17] for the lumped-mass and upwind scheme, and to [6,11,19] for the general theory of finite-element method.…”

Section: Finite-element Approximation and Main Resultsmentioning

confidence: 99%

“…The scheme is well-posed under some time step-size control τ ∼ O (h 2 ), and has the error estimate of order O (h 1−2/p + τ ) in L p -space. Moreover, Saito [17] proved conservation of mass and preservation of nonnegativity for the approximate solutions. Since the simplified Keller-Segel system is a variation of (CG), this scheme should be applied to the present system (CG).…”

Section: Introductionmentioning

confidence: 96%

“…Recently, Saito in [17] has formulated a full-discrete approximation for a simplified Keller-Segel system by the conservative upwind finite-element scheme by Baba and Tabata [3] and semi-implicit Euler scheme. The scheme is well-posed under some time step-size control τ ∼ O (h 2 ), and has the error estimate of order O (h 1−2/p + τ ) in L p -space.…”

Section: Introductionmentioning

confidence: 99%

“…Since the simplified Keller-Segel system is a variation of (CG), this scheme should be applied to the present system (CG). In the present paper we employ for (CG) the conservative upwind scheme by Saito [17], construct the dynamical system for the approximate system, and show the estimate…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dimension estimate of the global attractor for a semi-discretized chemotaxis–growth system by conservative upwind finite-element scheme

Efendiev

Nakaguchi

Wendland

2009

Journal of Mathematical Analysis and Applications

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Section: Finite-element Approximation and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dimension estimate of the global attractor for a semi-discretized chemotaxis–growth system by conservative upwind finite-element scheme

Efendiev

Nakaguchi

Wendland

2009

Journal of Mathematical Analysis and Applications

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“…If these solutions satisfy certain maximum principles [62,73,82], it is desirable that their finite element approximations satisfy certain of their discrete analogues. Nonobtuse and acute partitions indeed yield finite element approximations that satisfy so-called discrete maximum principles when solving (possibly nonlinear) elliptic [47,48,52,58,91] and parabolic [26,34,44,75] problems, semiconductor equations [98,99], and convection-diffusion problems [1] by means of globally continuous, piecewise linear functions.…”

Section: Applications In Numerical Mathematicsmentioning

confidence: 99%

On Nonobtuse Simplicial Partitions

Brandts¹,

Korotov²,

Křížek³

et al. 2009

SIAM Rev.

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On nonobtuse simplicial partitionsBrandts, J.H.; Korotov, S.; Kížek, M.; Šolc, J. Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. Download date: 10 May 2018Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. Dedicated to Ivan Hlaváček on his 75th birthdayAbstract. This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

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Two positivity preserving flux limited, second‐order numerical methods for a haptotaxis model

Kolev

Koleva

Vulkov

2012

Numerical Methods Partial

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Two numerical methods for a one‐dimensional haptotaxis model, which exploit the use of van Leer flux limiter, are developed and analyzed. Sufficient conditions time step size and flux limiting are given for such formulation to ensure the non‐negativity of the discrete solution and second‐order accuracy in space. Another advantage is that we avoid solving large nonlinear systems of algebraic equations. The discrete preservation of total conservation of cell density, concentration, and logarithmic density is also verified for the numerical solution. Numerical results concerning accuracy, convergence rate, positivity, and conservation properties are presented and discussed. Similar approach could be applied efficiently in the corresponding two‐ and three‐dimensional problems. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

show abstract

Conservative upwind finite-element method for a simplified Keller–Segel system modelling chemotaxis

Cited by 87 publications

References 0 publications

Dimension estimate of the global attractor for a semi-discretized chemotaxis–growth system by conservative upwind finite-element scheme

Dimension estimate of the global attractor for a semi-discretized chemotaxis–growth system by conservative upwind finite-element scheme

On Nonobtuse Simplicial Partitions

Two positivity preserving flux limited, second‐order numerical methods for a haptotaxis model

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