A Full Multigrid Method For Semilinear Elliptic Equation

Hehu Xie, Fei Xu

Published 2016-11-17Version 1

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 existed multigrid method which need the bounded second order derivatives of the nonlinear term, the proposed method only need the bounded first order derivative of the nonlinear term.