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Zbl 1119.65094
Saito, Norikazu
Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis. (English)
[J] IMA J. Numer. Anal. 27, No. 2, 332-365 (2007). ISSN 0272-4979; ISSN 1464-3642/e

Summary: Finite-element approximation for a non-linear parabolic-elliptic system is considered. The system describes the aggregation of slime moulds resulting from their chemotactic features and is called a simplified Keller-Segel system. Applying an upwind technique, first we present a finite-element scheme that satisfies both positivity and mass conservation properties. Consequently, if the triangulation is of acute type, our finite-element approximation preserves the $L^1$ norm, which is an important property of the original system. Then, under some assumptions on the regularity of a solution and on the triangulation, we establish error estimates in $L^p\times W^{1,\infty}$ with a suitable $p>d$, where $d$ is the dimension of a spatial domain. Our scheme is well suited for practical computations. Some numerical examples that validate our theoretical results are also presented.

MSC 2000:: *65M60 Finite numerical methods (IVP of PDE)
65M15 Error bounds (IVP of PDE)
35M10 PDE of mixed type

Keywords: finite-element method; parabolic-elliptic system; non-linear evolution equation; analytic semigroup; conservation of positivity; conservation of mass; Keller-Segel system; error estimates; numerical examples

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Zentralblatt MATH Berlin [Germany]