Volume 37, Issue 1
A Cascadic Multigrid Method for Semilinear Elliptic Equations

J. Comp. Math., 37 (2019), pp. 112-129.

Published online: 2018-08

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• Abstract

This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method. Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space, the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space. To prove the efficiency of the new method, we derive two results, one of the optimal convergence rate by choosing the appropriate sequence of finite element spaces and the number of smoothing steps, and the other of the optimal computational work by applying the parallel computing technique. Moreover, the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method. Some numerical experiments are presented to validate our theoretical analysis.

• AMS Subject Headings

65N30, 65N25, 65L15, 65B99

• Copyright

COPYRIGHT: © Global Science Press

• Email address

xufei@lsec.cc.ac.cn (Fei Xu)

luofusheng@tio.org.cn (Fusheng Luo)

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• RIS
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@Article{JCM-37-112, author = {Xu , Fei and Luo , Fusheng}, title = {A Cascadic Multigrid Method for Semilinear Elliptic Equations}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {37}, number = {1}, pages = {112--129}, abstract = {

This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method. Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space, the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space. To prove the efficiency of the new method, we derive two results, one of the optimal convergence rate by choosing the appropriate sequence of finite element spaces and the number of smoothing steps, and the other of the optimal computational work by applying the parallel computing technique. Moreover, the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method. Some numerical experiments are presented to validate our theoretical analysis.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1710-m2017-0067}, url = {http://global-sci.org/intro/article_detail/jcm/12652.html} }
TY - JOUR T1 - A Cascadic Multigrid Method for Semilinear Elliptic Equations AU - Xu , Fei AU - Luo , Fusheng JO - Journal of Computational Mathematics VL - 1 SP - 112 EP - 129 PY - 2018 DA - 2018/08 SN - 37 DO - http://doi.org/10.4208/jcm.1710-m2017-0067 UR - https://global-sci.org/intro/article_detail/jcm/12652.html KW - Semilinear elliptic equation, Parallel computing, Cascadic multigrid, Multilevel correction, Finite element method. AB -

This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method. Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space, the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space. To prove the efficiency of the new method, we derive two results, one of the optimal convergence rate by choosing the appropriate sequence of finite element spaces and the number of smoothing steps, and the other of the optimal computational work by applying the parallel computing technique. Moreover, the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method. Some numerical experiments are presented to validate our theoretical analysis.

Fei Xu & Fusheng Luo. (2020). A Cascadic Multigrid Method for Semilinear Elliptic Equations. Journal of Computational Mathematics. 37 (1). 112-129. doi:10.4208/jcm.1710-m2017-0067
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