Multigrid Method for Nonlinear Eigenvalue Problems Based on Newton Iteration

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 R × H 0 1 ( Ω ) . Then in the presented multig...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 94; no. 2; p. 42
Main Authors: Xu, Fei, Xie, Manting, Yue, Meiling
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2023
Springer Nature B.V
Subjects:
Algorithms
Approximation
Boundary value problems
Computational mathematics
Computational Mathematics and Numerical Analysis
Convergence
Efficiency
Eigenvalues
Eigenvectors
Estimates
Finite element analysis
Iterative methods
Lagrange multiplier
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Nonlinear equations
Theoretical
Newton iteration
Nonlinear eigenvalue problems
Multigrid method
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary: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 R × 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.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-022-02070-9