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

TL;DR

This paper introduces a graph neural network-based algebraic approximation for spectral coarse spaces in two-level Schwarz methods, significantly reducing setup costs and maintaining robustness in high-contrast Darcy flow simulations.

Contribution

It proposes a novel algebraic coarse-space approximation using graph neural networks to avoid expensive eigenvalue solves in multiscale Schwarz methods.

Findings

Reduces setup cost in high-contrast Darcy flow simulations.

Maintains robust convergence across various contrasts and boundary conditions.

Improves overall time-to-solution in numerical experiments.

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.