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Localized subspace iteration methods for elliptic multiscale problems

Xiaofei Guan, Lijian Jiang, Yajun Wang, Zihao Yang

TL;DR

This work addresses the computational challenge of elliptic multiscale PDEs with high-contrast coefficients by introducing localized subspace iteration (LSI), which builds generalized finite element bases through local inverse operators and spectral problems. It presents two concrete methods, Localized Standard Subspace Iteration (LSSI) and Localized Krylov Subspace Iteration (LKSI), derived from standard and Krylov subspace algorithms, and analyzes their convergence in terms of interpolation error and spectral decay. Theoretical results link LOD with spectral-problem algorithms and provide explicit error bounds that separate guidance on the number of local basis functions and iteration depth. Numerical experiments on diffusion and elasticity demonstrate strong accuracy and robustness, particularly for long-channel multiscale features, highlighting the methods' potential for scalable, parallelizable multiscale solvers applicable to high-contrast elliptic problems.

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.

Localized subspace iteration methods for elliptic multiscale problems

TL;DR

This work addresses the computational challenge of elliptic multiscale PDEs with high-contrast coefficients by introducing localized subspace iteration (LSI), which builds generalized finite element bases through local inverse operators and spectral problems. It presents two concrete methods, Localized Standard Subspace Iteration (LSSI) and Localized Krylov Subspace Iteration (LKSI), derived from standard and Krylov subspace algorithms, and analyzes their convergence in terms of interpolation error and spectral decay. Theoretical results link LOD with spectral-problem algorithms and provide explicit error bounds that separate guidance on the number of local basis functions and iteration depth. Numerical experiments on diffusion and elasticity demonstrate strong accuracy and robustness, particularly for long-channel multiscale features, highlighting the methods' potential for scalable, parallelizable multiscale solvers applicable to high-contrast elliptic problems.

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.
Paper Structure (19 sections, 9 theorems, 76 equations, 7 figures, 7 tables, 2 algorithms)

This paper contains 19 sections, 9 theorems, 76 equations, 7 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. From LOD to spectral problem
  3. Introduction to LOD
  4. Iterative orthogonal decomposition
  5. Two subspaces of spectral problem algorithms
  6. Standard subspace iteration
  7. Krylov subspace iteration
  8. From spectral problem to LSI
  9. Localized standard subspace iteration (LSSI) method
  10. Localized Krylov subspace iteration (LKSI) method
  11. Convergence analysis
  12. Interpolation error
  13. Convergence result of the LSSI method
  14. Convergence result of the LKSI method
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

Lemma 2.1

Given the inf-sup condition, the spaces $\widetilde{U}_H$ and $\widetilde{V}_H$ possess a dimension of $N:=|J|$ and establish conforming decompositions of the overall space, thereby satisfying Furthermore, we have the ‘orthogonality’ relations

Figures (7)

  • Figure 1: The high-contrast permeability coefficient $\kappa$.
  • Figure 2: Contour plots of solutions: (a)the reference solution, (b)LOD, (c) $LKSI\text{-}4$, (d)$LSSI\text{-}1$, (e)$LSSI\text{-}2$ and (f)$LSSI\text{-}4$.
  • Figure 3: Relative errors of multiscale methods versus the number of oversampling layers $m$: (a) energy error and (b) $L^2$ error.
  • Figure 4: Relative errors of multiscale methods versus the coarse mesh size $H$: (a) energy error and (b) $L^2$ error.
  • Figure 5: Multiscale coefficients with different channel lengths.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Lemma 2.1
  • Proof
  • Lemma 2.2
  • Proof
  • Lemma 2.3
  • Proof
  • Theorem 2.1
  • Proof
  • Remark 2.2
  • ...and 7 more