Graph-Based Meshfree Multi-scale Coarse Space Approximation for Two-Level Schwarz Methods
This addresses computational bottlenecks in iterative solvers for porous media flow simulation, representing an incremental improvement with specific engineering applications.
The paper tackles the expensive setup phase of spectral coarse spaces in two-level Schwarz methods for Darcy flow simulation by proposing a graph neural network-based algebraic approximation that avoids repeated local eigensolves. Numerical experiments show the approach substantially reduces setup cost and improves end-to-end time-to-solution while preserving robust convergence across high-contrast scenarios.
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.