Multigrid method for nonlinear eigenvalue problems based on Newton iteration

TL;DR

This paper introduces a new multigrid method based on Newton iteration for nonlinear eigenvalue problems, improving efficiency by solving a single linear boundary value problem per level and providing convergence guarantees.

Contribution

The paper presents a novel multigrid approach that treats eigenpairs as a combined element, avoiding large nonlinear eigenproblem solutions and establishing convergence analysis.

Findings

Achieves optimal error estimates and linear computational complexity.

Provides a convergence proof for residuals at each iteration.

Enhances stability with a mixing scheme.

Abstract

In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda, u)$ as one element in a product space $\mathbb R \times H_0^1(\Omega)$. Then in the presented multigrid method, only one discrete linear boundary value problem needs to be solved for each level of the multigrid sequence. Because we avoid solving large-scale nonlinear eigenvalue problems directly, the overall efficiency is significantly improved. The optimal error estimate and linear computational complexity can be derived simultaneously. In addition, we also provide an improved multigrid method coupled with a mixing scheme to further guarantee the convergence and stability of the iteration scheme. More importantly, we prove convergence for the residuals after each iteration step. For nonlinear eigenvalue problems, such theoretical analysis is missing from the existing literatures on the mixing iteration scheme.