This paper proposes a new type of multilevel method for solving the Steklov eigenvalue problem based on Newton’s method. In this iteration method, solving the Steklov eigenvalue problem is replaced by solving a small-scale eigenvalue problem on the coarsest mesh and a sequence of augmented linear problems on refined meshes, derived by Newton step. We prove that this iteration scheme obtains the optimal convergence rate with linear complexity, which improves the overall efficiency of solving the Steklov eigenvalue problem. Moreover, an adaptive iteration scheme for multi eigenvalues based on this new multilevel method is given. Finally, some numerical experiments are provided to illustrate the efficiency of the proposed multilevel scheme.

本文提出了一种基于牛顿法求解Steklov特征值问题的新型多级方法。在这种迭代方法中，求解 Steklov 特征值问题被替换为求解最粗网格上的小规模特征值问题和精细网格上的一系列增强线性问题，由牛顿步导出。我们证明了这种迭代方案获得了具有线性复杂度的最优收敛速度，从而提高了求解Steklov特征值问题的整体效率。此外，还给出了一种基于这种新的多级方法的多特征值自适应迭代方案。最后，提供了一些数值实验来说明所提出的多级方案的效率。