A Full Multigrid Method For Semilinear Elliptic Equation

TL;DR

This paper introduces a full multigrid finite element method for semilinear elliptic equations that efficiently transforms the nonlinear problem into linear problems and low-dimensional nonlinear problems, ensuring optimal computational work.

Contribution

It proposes a novel multigrid approach that requires only Lipschitz continuity of the nonlinear term, unlike previous methods needing bounded second derivatives.

Findings

Achieves optimal computational efficiency

Requires only Lipschitz continuity of the nonlinear term

Proven convergence and efficiency of the method

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.