This paper proposes localized subspace iteration (LSI) methods to construct generalized finite element basis functions for elliptic problems with multiscale coefficients. The key components of the proposed method consist of the localization of the original differential operator and the subspace iteration of the corresponding local spectral problems, where the localization is conducted by enforcing the local homogeneous Dirichlet condition and the partition of the unity functions. From a novel perspective, some multiscale methods can be regarded as one iteration step under approximating the eigenspace of the corresponding local spectral problems. Vice versa, new multiscale methods can be designed through subspaces of spectral problem algorithms. Then, we propose the efficient localized standard subspace iteration (LSSI) method and the localized Krylov subspace iteration (LKSI) method based on the standard subspace and Krylov subspace, respectively. Convergence analysis is carried out for the proposed method. Various numerical examples demonstrate the effectiveness of our methods. In addition, the proposed methods show significant superiority in treating long-channel cases over other well-known multiscale methods.

翻译：本文提出局部子空间迭代（LSI）方法，用于构造具有多尺度系数的椭圆问题的广义有限元基函数。该方法的关键组成部分包括原始微分算子的局部化处理以及相应局部谱问题的子空间迭代，其中局部化通过施加局部齐次狄利克雷条件和单位分解函数实现。从一个新颖的视角来看，某些多尺度方法可被视为在逼近相应局部谱问题特征空间时的一次迭代步骤。反之，通过谱问题算法的子空间可以设计出新的多尺度方法。基于此，我们分别提出了基于标准子空间的高效局部标准子空间迭代（LSSI）方法以及基于Krylov子空间的局部Krylov子空间迭代（LKSI）方法。对所提方法进行了收敛性分析。大量数值算例验证了所提方法的有效性。此外，在处理长通道问题时，所提方法相较于其他知名多尺度方法展现出显著优越性。