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arXiv · 2508.02077

Adaptive Crouzeix-Raviart finite elements for the first eigenpair of $p$-Laplacian

Guanglian Li·Yueqi Wang·Yifeng Xu
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Abstract

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.

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Guanglian Li, Yueqi Wang, Yifeng Xu. 2025-08-04. Adaptive Crouzeix-Raviart finite elements for the first eigenpair of $p$-Laplacian. https://arxiv.org/abs/2508.02077

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