  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.
