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.

翻译：本文提出了一种用于计算 $p$-Laplacian 问题第一特征对的自适应有限元方法。我们证明，从一个精细的初始网格出发，所提出的自适应算法产生的离散第一特征值序列收敛于连续问题的第一特征值，并且离散特征函数与对应于第一特征值的归一化特征函数集在 $W^{1,p}$ 范数下的距离也趋于零。大量的数值算例验证了该方法的有效性和高效性。