Multigrid method for nonlinear eigenvalue problems based on Newton iteration
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911857458020352 |
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| author | Xu, Fei Xie, Manting Yue, Meiling |
| author_facet | Xu, Fei Xie, Manting Yue, Meiling |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multigrid method for nonlinear eigenvalue problems based on Newton iteration Xu, Fei Xie, Manting Yue, Meiling Numerical Analysis 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. |
| title | Multigrid method for nonlinear eigenvalue problems based on Newton iteration |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2404.18568 |