Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schr\"odinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.

翻译：加速迭代本征值算法通常通过采用谱位移策略实现。然而，改进的位移通常会导致所得位移算子的本征值变小，进而使得基础求解矩阵的条件数增大，这给迭代线性求解器带来了重大挑战。本文针对线性薛定谔本征值问题，引入了一种两级区域分解预处理器，即使在非均匀扩展域中出现本征值间隙消失的情况下，也能有效应对此问题。由于本征值求解器所需的准最优位移（其本身已作为谱胞问题的解而存在），因此很自然地也将其关联的本征函数用作构造粗空间的生成元。我们分析了所得的两级加性施瓦茨预处理器，并获得了与区域各向异性无关的条件数上界，尽管粗求解器每个子域仅需一个基函数。文中提供了若干数值算例，以展示该方法的灵活性与高效性。