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

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.