The paper presents a novel multigrid method for solving nonlinear eigenvalue problems. The key ideas are:
Instead of handling the eigenvalue λ and eigenfunction u separately, the method treats the eigenpair (λ, u) as one element in a product space R × H¹₀(Ω). This allows solving only one discrete linear boundary value problem for each level of the multigrid sequence, significantly improving the overall efficiency.
The method uses the Newton iteration technique to solve the nonlinear eigenvalue equation, which is viewed as a special nonlinear equation defined in the product space R × H¹₀(Ω).
Theoretical analysis shows the method can derive optimal error estimates and linear computational complexity.
An improved multigrid method coupled with a mixing scheme (e.g. Anderson acceleration) is also provided to further guarantee the convergence and stability of the iteration scheme. Convergence for the residuals after each iteration step is proved, which is missing from existing literature on the mixing iteration scheme for nonlinear eigenvalue problems.
Numerical experiments verify the theoretical results derived in the paper.
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by Fei Xu,Manti... às arxiv.org 04-30-2024
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