In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda, u)$ as one element in a product space $\mathbb R \times H_0^1(\Omega)$. Then in the presented multigrid method, only one discrete linear boundary value problem needs to be solved for each level of the multigrid sequence. Because we avoid solving large-scale nonlinear eigenvalue problems directly, the overall efficiency is significantly improved. The optimal error estimate and linear computational complexity can be derived simultaneously. In addition, we also provide an improved multigrid method coupled with a mixing scheme to further guarantee the convergence and stability of the iteration scheme. More importantly, we prove convergence for the residuals after each iteration step. For nonlinear eigenvalue problems, such theoretical analysis is missing from the existing literatures on the mixing iteration scheme.

翻译：本文提出一种基于牛顿迭代的新型多重网格方法，用于求解非线性特征值问题。不同于分别处理特征值λ和特征函数u，我们将特征对(λ,u)视为乘积空间ℝ×H₀¹(Ω)中的单一元素。在所提出的多重网格方法中，每个多重网格层级只需求解一个离散线性边值问题。由于避免了直接求解大规模非线性特征值问题，整体效率显著提升。该方法可同时推导出最优误差估计和线性计算复杂度。此外，我们还提供了一种结合混合策略的改进多重网格方法，以进一步保证迭代方案的收敛性和稳定性。更重要的是，我们证明了每次迭代步骤后残差的收敛性。对于非线性特征值问题，现有文献中关于混合迭代方案的此类理论分析尚属空白。