An algebraic multiscale framework for two-level overlapping Schwarz preconditioners
Abstract
The robustness of two-level overlapping additive Schwarz (OAS) preconditioners for highly heterogeneous elliptic problems hinges on the choice of the coarse space. We propose an algebraic multiscale (AM) coarse space built from the system matrix alone through an algebraic formulation of the multiscale finite element method (MsFEM). The coarse basis functions are energy-minimizing extensions of the solutions of reduced boundary condition (RBC) problems on the subdomain interface, so that coefficient variations along the interface enter the coarse space. By posing the RBC problems on slabs that extend the interface, the construction also applies to general unstructured domain decompositions. Embedding the AM formulation in an energy-minimizing framework extends it to linear elasticity, and we show that existing condition number bounds for OAS with MsFEM coarse spaces carry over to the algebraic setting. For problems with high coefficient inclusions on the coarse nodes, the AM iteration count and condition number are unaffected by increasing the contrast. In 3D scalar diffusion with log-normal coefficients, AM needs about as many iterations as the generalized Dryja--Smith--Widlund (GDSW) coarse space, while solving a coarse problem ten times smaller. For linear elasticity, omitting the coarse basis functions associated with the rotational rigid body modes yields a fully algebraic coarse space that still scales well for homogeneous problems.
Related
- Two-level additive Schwarz preconditioners with GDSW coarse spaces for high-order discretizations in 2D
- Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
- Two-level additive Schwarz preconditioners with GDSW coarse spaces for high-order discretizations in 3D
- Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method
- Adaptive hybrid coupling with operator inference, the overlapping Schwarz alternating method and reinforcement learning
BibTeX
@misc{cumaru2026algebraic,
title = {{An algebraic multiscale framework for two-level overlapping Schwarz preconditioners}},
author = {Filipe Cumaru and Hadi Hajibeygi and Cornelis Vuik and Alexander Heinlein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16359},
eprint = {2610.16359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16359},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An algebraic multiscale framework for two-level overlapping Schwarz preconditioners
AU - Filipe Cumaru
AU - Hadi Hajibeygi
AU - Cornelis Vuik
AU - Alexander Heinlein
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16359
UR - https://arxiv.org/abs/2610.16359
AB - The robustness of two-level overlapping additive Schwarz (OAS) preconditioners for highly heterogeneous elliptic problems hinges on the choice of the coarse space. We propose an algebraic multiscale (AM) coarse space built from the system matrix alone through an algebraic formulation of the multiscale finite element method (MsFEM). The coarse basis functions are energy-minimizing extensions of the solutions of reduced boundary condition (RBC) problems on the subdomain interface, so that coefficient variations along the interface enter the coarse space. By posing the RBC problems on slabs that extend the interface, the construction also applies to general unstructured domain decompositions. Embedding the AM formulation in an energy-minimizing framework extends it to linear elasticity, and we show that existing condition number bounds for OAS with MsFEM coarse spaces carry over to the algebraic setting. For problems with high coefficient inclusions on the coarse nodes, the AM iteration count and condition number are unaffected by increasing the contrast. In 3D scalar diffusion with log-normal coefficients, AM needs about as many iterations as the generalized Dryja--Smith--Widlund (GDSW) coarse space, while solving a coarse problem ten times smaller. For linear elasticity, omitting the coarse basis functions associated with the rotational rigid body modes yields a fully algebraic coarse space that still scales well for homogeneous problems.
ER -