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

TL;DR

This paper introduces a fast, well-conditioned spectral domain decomposition method for elliptic PDEs that leverages H-matrix structures and randomized compression, enabling efficient solutions for large-scale oscillatory problems.

Contribution

The paper presents a novel spectral domain decomposition approach with discretization-independent conditioning bounds and efficient H-matrix compression for large elliptic PDEs.

Findings

Handles oscillatory 2D and 3D problems with up to 28 million degrees of freedom.

Uses explicit reduced linear systems with H-matrix structure for efficiency.

Employs randomized compression to accelerate the formation of the reduced system.

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.