Localized subspace iteration methods for elliptic multiscale problems

TL;DR

This paper introduces localized subspace iteration methods for constructing finite element basis functions in elliptic multiscale problems, offering improved efficiency and effectiveness, especially for long-channel cases.

Contribution

The paper develops new localized subspace iteration algorithms, including LSSI and LKSI, with convergence analysis and superior performance demonstrated through numerical examples.

Findings

Methods effectively handle multiscale coefficients in elliptic problems.

Localized approaches outperform existing multiscale methods in long-channel scenarios.

Convergence of the proposed methods is theoretically established.

Abstract

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.