Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

TL;DR

This paper introduces a new algebraic spectral coarse space for two-level Schwarz preconditioners, improving efficiency especially for non-self-adjoint sparse matrices, with proven convergence and superior performance in numerical tests.

Contribution

A fully algebraic spectral coarse space is proposed, enabling effective two-level Schwarz preconditioning for non-self-adjoint operators, outperforming existing methods.

Findings

Proven convergence for Hermitian positive definite diagonally dominant matrices.

Numerical experiments show superior efficiency over state-of-the-art preconditioners.

Effective for non-self-adjoint operators in sparse linear systems.

Abstract

Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.