A Multigrid Method for Nonlinear Eigenvalue Problems

TL;DR

This paper introduces a multigrid method that efficiently solves nonlinear eigenvalue problems by decomposing them into linear boundary value problems and small nonlinear eigenvalue problems across multiple levels, significantly reducing computational effort.

Contribution

The paper presents a novel multigrid scheme that enhances the efficiency of solving nonlinear eigenvalue problems using finite element discretization.

Findings

Computational work is nearly equivalent to solving a linear boundary value problem.

The scheme significantly improves efficiency over traditional methods.

It effectively decomposes nonlinear problems into simpler subproblems.

Abstract

A multigrid method is proposed for solving nonlinear eigenvalue problems by the finite element method. With this new scheme, solving nonlinear eigenvalue problem is decomposed to a series of solutions of linear boundary value problems on multilevel finite element spaces and a series of small scale nonlinear eigenvalue problems. The computational work of this new scheme can reach almost the same as the solution of the corresponding linear boundary value problem. Therefore, this type of multilevel correction scheme improves the overfull efficiency of the nonlinear eigenvalue problem solving.