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Error estimate of a random particle blob method for the Keller-Segel equation


Authors: Hui Huang and Jian-Guo Liu
Journal: Math. Comp.
MSC (2010): Primary 65M75, 65M15, 65M12, 35Q92, 35K55, 60H30
DOI: https://doi.org/10.1090/mcom/3174
Published electronically: February 15, 2017
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Abstract: We establish an optimal error estimate for a random particle blob method for the Keller-Segel equation in $ \mathbb{R}^d~(d\geq 2)$. With a blob size $ \varepsilon =h^{\kappa }$ $ (1/2<\kappa <1)$, we prove a rate $ h\vert\ln h\vert$ of convergence in $ \ell ^p_h$ $ (p>\frac {d}{1-\kappa })$ norm up to a probability $ 1-h^{C\vert\ln h\vert}$, where $ h$ is the initial grid size.


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Additional Information

Hui Huang
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China — and — Departments of Physics and Mathematics, Duke University, Durham, North Carolina 27708
Email: huanghui12@mails.tsinghua.edu.cn

Jian-Guo Liu
Affiliation: Departments of Physics and Mathematics, Duke University, Durham, North Carolina 27708
Email: jliu@phy.duke.edu

DOI: https://doi.org/10.1090/mcom/3174
Keywords: Concentration inequality, Bennett's inequality, Newtonian aggregation, sampling estimate, chemotaxis, Brownian motion, propagation of chaos.
Received by editor(s): October 25, 2015
Received by editor(s) in revised form: April 24, 2016
Published electronically: February 15, 2017
Additional Notes: The first author was partially supported by National Natural Science Foundation of China (Grant No: 41390452, 11271118)
The work of the second author was partially supported by KI-Net NSF RNMS grant No. 1107444 and NSF DMS grant No. 1514826.
Article copyright: © Copyright 2017 American Mathematical Society