Applications of Mathematics

, Volume 62, Issue 3, pp 225–241

A full multigrid method for semilinear elliptic equation

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Abstract

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.

Keywords

semilinear elliptic problem full multigrid method multilevel correction finite element method 

MSC 2010

65N30 65N25 65L15 65B99 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Beijing Institute for Scientific and Engineering ComputingBeijing University of TechnologyBeijingChina
  2. 2.LSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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