Adaptive finite element approximations of the first eigenpair associated with $p$-Laplacian

In this paper, we propose an adaptive finite element method for computing the first eigenpair of the $p$-Laplacian problem. We prove that starting from a fine initial mesh our proposed adaptive algorithm produces a sequence of discrete first eigenvalues that converges to the first eigenvalue of the continuous problem and the distance between discrete eigenfunctions and the normalized eigenfunction set corresponding to the first eigenvalue in $W^{1,p}$-norm also tends to zero. Extensive numerical examples are provided to show the effectiveness and efficiency.