Adaptive Crouzeix-Raviart finite elements for the first eigenpair of $p$-Laplacian
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| Format: | Preprint |
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2025
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| _version_ | 1866912518154223616 |
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| author | Li, Guanglian Wang, Yueqi Xu, Yifeng |
| author_facet | Li, Guanglian Wang, Yueqi Xu, Yifeng |
| contents | In this paper, we propose and analyze an adaptive Crouzeix-Raviart finite element method for computing the first Dirichlet eigenpair of the $p$-Laplacian problem. We prove that the sequence of error estimators produced by the adaptive algorithm has a vanishing limit and that, starting from a fine initial mesh, the relevant sequence of approximate eigenvalues converges to the first eigenvalue and the distance in a mesh-dependent broken norm between discrete eigenfunctions and the set composed of relevant continuous eigenfunctions also tends to zero. The analysis hinges on establishing a compactness property for Crouzeix-Raviart finite elements over a sequence of adaptively generated meshes, which represents key theoretical challenges and novelties. We present numerical results to illustrate the advantage of the proposed algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_02077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Crouzeix-Raviart finite elements for the first eigenpair of $p$-Laplacian Li, Guanglian Wang, Yueqi Xu, Yifeng Numerical Analysis In this paper, we propose and analyze an adaptive Crouzeix-Raviart finite element method for computing the first Dirichlet eigenpair of the $p$-Laplacian problem. We prove that the sequence of error estimators produced by the adaptive algorithm has a vanishing limit and that, starting from a fine initial mesh, the relevant sequence of approximate eigenvalues converges to the first eigenvalue and the distance in a mesh-dependent broken norm between discrete eigenfunctions and the set composed of relevant continuous eigenfunctions also tends to zero. The analysis hinges on establishing a compactness property for Crouzeix-Raviart finite elements over a sequence of adaptively generated meshes, which represents key theoretical challenges and novelties. We present numerical results to illustrate the advantage of the proposed algorithm. |
| title | Adaptive Crouzeix-Raviart finite elements for the first eigenpair of $p$-Laplacian |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.02077 |