Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis
Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis
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DOI：
10.1093/imanum/drl018
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发表时间：
2007-04-01
影响因子：
2.1
通讯作者：
Saito, Norikazu
中科院分区：
文献类型：
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作者：
Saito, Norikazu
研究了一类非线性抛物-椭圆方程组的有限元逼近。该系统描述了由黏菌趋化特征引起的黏菌聚集，并被称为简化的Keller-Segel系统。应用迎风技术，首先，我们提出了一个有限元方案，同时满足积极性和质量守恒性质。因此，如果三角剖分是尖锐的类型，我们的有限元近似保持L-1范数，这是一个重要的性质，原来的系统。然后，在对解的正则性和三角剖分的一些假设下，我们建立了L-p × W-1，W-无穷大中的误差估计，其中p > d，d是空间域的维数。我们的方案非常适合于实际计算。数值例子验证了我们的理论结果。
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 x W-1,W- infinity 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.