Applications of Mathematics, Vol. 62, No. 3, pp. 225-241, 2017
A full multigrid method for semilinear elliptic equation
Fei Xu, Hehu Xie
Received December 12, 2016. First published May 17, 2017.
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
Affiliations: Fei Xu, Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China, e-mail: firstname.lastname@example.org; Hehu Xie, LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhongguancun Donglu, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, e-mail: email@example.com