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A type of cascadic multigrid method for coupled semilinear elliptic equations

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Abstract

This paper is to introduce a type of cascadic multigrid method for coupled semilinear elliptic equations. Instead of solving the coupled semilinear elliptic equation on a very fine finite element space directly, the new scheme needs to solve a decoupled linear system by some smoothing steps on each of multilevel finite element spaces and solve a coupled semilinear elliptic equation on a coarse space. By choosing the appropriate number of smoothing steps on different finite element spaces, the corresponding optimal convergence rate and optimal computation work can be derived. Besides, the adaptive cascadic multigrid method for coupled semilinear elliptic equations and its analysis are also presented theoretically and numerically. Moreover, the requirement of bounded second order derivatives of the nonlinear term in the existing multigrid methods is reduced to the Lipschitz continuation property in the presented new scheme. Some numerical experiments are presented to validate our theoretical analysis.

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Acknowledgements

The author thanks Prof. Peter Deuflhard (Founder of ZIB and Co-Founder of Matheon in Berlin) and the anonymous referees for their helpful comments, suggestions on our work. This work was supported in part by National Natural Science Foundations of China (NSFC 11801021), Foundation for Fundamental Research of Beijing University of Technology (006000546318504).

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  1. Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, China

    Fei Xu & Qiumei Huang

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  1. Fei Xu
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  2. Qiumei Huang
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Correspondence to Fei Xu.

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Xu, F., Huang, Q. A type of cascadic multigrid method for coupled semilinear elliptic equations. Numer Algor 83, 485–510 (2020). https://doi.org/10.1007/s11075-019-00690-1

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