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Multigrid method for nonlinear eigenvalue problems based on Newton iteration

Fei Xu, Manting Xie, Meiling Yue

TL;DR

This work develops a Newton-iteration–driven multigrid framework for nonlinear eigenvalue problems by formulating the problem in the product space $X=\mathbb{R}\times H_0^1(\Omega)$. Each multigrid level solves a single linear boundary-value problem derived from a Newton linearization, yielding substantial efficiency over directly solving large nonlinear eigenproblems. The authors establish well-posedness and residual convergence for the Newton step, derive optimal error estimates, and prove linear computational complexity. To handle challenging nonlinearities, they introduce a mixing scheme that dampens updates and guarantees residual reduction, with adaptive strategies and practical algorithms. Numerical experiments on the Gross–Pitaevskii-type model demonstrate optimal accuracy and linear scalability, validating both the Newton-based and mixing-enhanced multigrid approaches.

Abstract

In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $λ$ and eigenfunction $u$ separately, we treat the eigenpair $(λ, u)$ as one element in a product space $\mathbb R \times H_0^1(Ω)$. 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.

Multigrid method for nonlinear eigenvalue problems based on Newton iteration

TL;DR

This work develops a Newton-iteration–driven multigrid framework for nonlinear eigenvalue problems by formulating the problem in the product space . Each multigrid level solves a single linear boundary-value problem derived from a Newton linearization, yielding substantial efficiency over directly solving large nonlinear eigenproblems. The authors establish well-posedness and residual convergence for the Newton step, derive optimal error estimates, and prove linear computational complexity. To handle challenging nonlinearities, they introduce a mixing scheme that dampens updates and guarantees residual reduction, with adaptive strategies and practical algorithms. Numerical experiments on the Gross–Pitaevskii-type model demonstrate optimal accuracy and linear scalability, validating both the Newton-based and mixing-enhanced multigrid approaches.

Abstract

In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue and eigenfunction separately, we treat the eigenpair as one element in a product space . 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.
Paper Structure (12 sections, 8 theorems, 84 equations, 4 figures, 3 tables, 5 algorithms)

This paper contains 12 sections, 8 theorems, 84 equations, 4 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries of nonlinear eigenvalue problems
  3. Newton iteration for nonlinear eigenvalue problems
  4. Multigrid method for nonlinear eigenvalue problems based on Newton iteration
  5. One step of the multigrid method
  6. Multigrid method based on Newton iteration
  7. Work estimate of Algorithm \ref{['mgni']}
  8. Multigrid method for nonlinear eigenvalue problems based on mixing scheme
  9. Numerical results
  10. Example 1
  11. Example 2
  12. Concluding remarks

Key Result

lemma thmcounterlemma

There exist $M \in \mathbb R_+$ and $\beta>0$ such that for all $v \in H_0^1(\Omega)$, there holds and where $E"(u)$ has the following form

Figures (4)

  • Figure 1: The initial mesh for Example 1.
  • Figure 2: The errors of approximate solutions derived by Algorithm \ref{['mgni']} for Example 1.
  • Figure 3: The initial mesh for Example 2.
  • Figure 4: The errors of approximate solutions derived by Algorithm \ref{['mgmix']} for Example 2.

Theorems & Definitions (15)

  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • theorem 4
  • proof
  • ...and 5 more