A fast spectral overlapping domain decomposition method with discretization-independent conditioning bounds

2025-10-29

AI Summary

This paper addresses the efficient parallel solution of general variable-coefficient elliptic PDEs on regular domains. It proposes a domain decomposition method based on overlapping thin-layer subdomain partitioning. The key contributions are: (1) constructing a reduced system converging to a second-kind Fredholm integral equation, yielding a preconditioner whose condition number is independent of the discretization scale; (2) exploiting the hierarchical (H-) matrix low-rank structure combined with black-box randomized compression to efficiently handle the large dense blocks arising in conventional approaches; and (3) solving local subproblems via spectral methods coupled with sparse direct solvers. Numerical experiments demonstrate robust performance on oscillatory 2D and 3D problems with up to 28 million degrees of freedom. The method exhibits excellent strong and weak scalability, significantly enhancing both computational efficiency and parallel scalability for large-scale variable-coefficient elliptic problems.

Technology Category

• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

A domain decomposition method for the solution of general variable-coefficient elliptic partial differential equations on regular domains is introduced. The method is based on tessellating the domain into overlapping thin slabs or shells, and then explicitly forming a reduced linear system that connects the different domains. Rank-structure ('H-matrix structure') is exploited to handle the large dense blocks that arise in the reduced linear system. Importantly, the formulation used is well-conditioned, as it converges to a second kind Fredholm equation as the precision in the local solves is refined. Moreover, the dense blocks that arise are far more data-sparse than in existing formulations, leading to faster and more efficient H-matrix arithmetic. To form the reduced linear system, black-box randomized compression is used, taking full advantage of the fact that sparse direct solvers are highly efficient on the thin sub-domains. Numerical experiments demonstrate that our solver can handle oscillatory 2D and 3D problems with as many as 28 million degrees of freedom.

Problem

• Develops domain decomposition for variable-coefficient elliptic PDEs
• Achieves well-conditioned formulation using spectral overlapping slabs
• Enables efficient H-matrix compression for large-scale 3D problems

Innovation

• Overlapping domain decomposition with slabs
• H-matrix structure for dense blocks
• Black-box randomized compression technique