A full multigrid method for semilinear elliptic equation

Fei Xu; Hehu Xie

Applications of Mathematics (2017)

  • Volume: 62, Issue: 3, page 225-241
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Xu, Fei, and Xie, Hehu. "A full multigrid method for semilinear elliptic equation." Applications of Mathematics 62.3 (2017): 225-241. <http://eudml.org/doc/288219>.

@article{Xu2017,
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.},
author = {Xu, Fei, Xie, Hehu},
journal = {Applications of Mathematics},
keywords = {semilinear elliptic problem; full multigrid method; multilevel correction; finite element method},
language = {eng},
number = {3},
pages = {225-241},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A full multigrid method for semilinear elliptic equation},
url = {http://eudml.org/doc/288219},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Xu, Fei
AU - Xie, Hehu
TI - A full multigrid method for semilinear elliptic equation
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 225
EP - 241
AB - 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.
LA - eng
KW - semilinear elliptic problem; full multigrid method; multilevel correction; finite element method
UR - http://eudml.org/doc/288219
ER -

References

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