Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917070477721600 |
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| author | Li, Zhenglei Liang, Qigang Xu, Xuejun |
| author_facet | Li, Zhenglei Liang, Qigang Xu, Xuejun |
| contents | In this work, we propose and analyze a pointwise a posteriori error estimator for simple eigenvalues of elliptic eigenvalue problems with adaptive finite element methods (AFEMs). We prove the reliability and efficiency of the residual-type a posteriori error estimator in the sense of $L^{\infty}$-norm, up to a logarithmic factor of the mesh size. For theoretical analysis, we also propose a theoretical and non-computable estimator, and then analyze the relationship between computable estimator and theoretical estimator. A key ingredient in the a posteriori error analysis is some new estimates for regularized derivative Green's functions. This methodology is also extended to the nonconforming finite element approximations. Some numerical experiments verify our theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems Li, Zhenglei Liang, Qigang Xu, Xuejun Numerical Analysis In this work, we propose and analyze a pointwise a posteriori error estimator for simple eigenvalues of elliptic eigenvalue problems with adaptive finite element methods (AFEMs). We prove the reliability and efficiency of the residual-type a posteriori error estimator in the sense of $L^{\infty}$-norm, up to a logarithmic factor of the mesh size. For theoretical analysis, we also propose a theoretical and non-computable estimator, and then analyze the relationship between computable estimator and theoretical estimator. A key ingredient in the a posteriori error analysis is some new estimates for regularized derivative Green's functions. This methodology is also extended to the nonconforming finite element approximations. Some numerical experiments verify our theoretical results. |
| title | Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.06815 |