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

TL;DR

This paper introduces an adaptive Crouzeix-Raviart finite element method for accurately computing the first eigenpair of the p-Laplacian, demonstrating convergence and efficiency through theoretical analysis and numerical experiments.

Contribution

It develops a novel adaptive finite element approach with proven convergence for the p-Laplacian eigenproblem, including establishing a compactness property over adaptive meshes.

Findings

Error estimators tend to zero with refinement

Eigenvalue approximations converge to the true eigenvalue

Numerical results show the method's effectiveness

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.