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Graph-Based Meshfree Multi-scale Coarse Space Approximation for Two-Level Schwarz Methods

Yucheng Liu, Tak Shing Au Yeung, Eric T. Chung, Simon See

Abstract

Efficient simulation of Darcy flow in highly heterogeneous porous media requires iterative solvers that remain robust under large permeability contrasts and mixed boundary conditions. Spectral coarse spaces in two-level overlapping Schwarz methods provide such robustness, but their practical use is often limited by an expensive setup phase dominated by many local generalized eigenvalue solves. We propose a purely algebraic, coarse-space approximation that avoids these repeated local eigensolves by using a graph neural network operating on the system-matrix graph. On the analysis side, we introduce a coefficient-weighted subspace-distance measure to quantify the discrepancy between the approximated and target local multiscale coarse spaces, and we derive a condition-number bound for the resulting preconditioned operator in terms of this distance. This bound yields a principled supervised-training objective and links learning error to solver performance. Numerical experiments on 2D and 3D high-contrast Darcy systems with varying mixed boundary conditions demonstrate that the proposed approach substantially reduces setup cost and improves end-to-end time-to-solution, while preserving robust convergence across the tested contrasts and boundary configurations.

Graph-Based Meshfree Multi-scale Coarse Space Approximation for Two-Level Schwarz Methods

Abstract

Efficient simulation of Darcy flow in highly heterogeneous porous media requires iterative solvers that remain robust under large permeability contrasts and mixed boundary conditions. Spectral coarse spaces in two-level overlapping Schwarz methods provide such robustness, but their practical use is often limited by an expensive setup phase dominated by many local generalized eigenvalue solves. We propose a purely algebraic, coarse-space approximation that avoids these repeated local eigensolves by using a graph neural network operating on the system-matrix graph. On the analysis side, we introduce a coefficient-weighted subspace-distance measure to quantify the discrepancy between the approximated and target local multiscale coarse spaces, and we derive a condition-number bound for the resulting preconditioned operator in terms of this distance. This bound yields a principled supervised-training objective and links learning error to solver performance. Numerical experiments on 2D and 3D high-contrast Darcy systems with varying mixed boundary conditions demonstrate that the proposed approach substantially reduces setup cost and improves end-to-end time-to-solution, while preserving robust convergence across the tested contrasts and boundary configurations.

Paper Structure

This paper contains 27 sections, 4 theorems, 34 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem setting
  4. Domain decomposition and subdomain coverings
  5. Local overlapping solvers
  6. Coarse space correction
  7. Albebraic two-level schwarz preconditioner
  8. Methodology
  9. Neural network architectures
  10. Loss function
  11. Notation and Projections
  12. Error Decomposition.
  13. Bound on Perturbation Error
  14. Relation to Subspace Distance
  15. Conclusion.
  16. ...and 12 more sections

Key Result

Theorem 3.1

The distance in defdist enjoys the following properties:

Figures (14)

  • Figure 1: An illustration of the overlapping graph partition: the vertex set $V$ is first partitioned into disjoint subsets $V_{I,1}$ and $V_{I,2}$ (shaded regions). The local vertex sets $V_1^\delta$ and $V_2^\delta$ (outlined regions) are then formed by including the halo vertices $V_{\Gamma,i}^\delta$ within graph distance $\delta$.
  • Figure 2: SP-LPMA GUNet Structure
  • Figure 3: Three Level Graph U-Net Structure
  • Figure 4: Training results for the two level SP-LPMA GUNet over 200 epochs.
  • Figure 5: Training results for the three level SP-LPMA GUNet over 200 epochs.
  • ...and 9 more figures

Theorems & Definitions (7)

  • Definition 1
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof