A Robust Algebraic Multilevel Domain Decomposition Preconditioner For Sparse Symmetric Positive Definite Matrices

TL;DR

This paper introduces a fully algebraic multilevel domain decomposition preconditioner for sparse symmetric positive definite matrices, enabling efficient and effective solutions without relying on PDE discretization details.

Contribution

It presents a novel algebraic approach to constructing coarse spaces in domain decomposition methods, independent of PDE discretization, with controllable condition number bounds.

Findings

Preconditioner effectively reduces condition numbers across various problems.

Numerical experiments demonstrate improved convergence and robustness.

Method is applicable to diverse application domains.

Abstract

Domain decomposition (DD) methods are widely used as preconditioner techniques. Their effectiveness relies on the choice of a locally constructed coarse space. Thus far, this construction was mostly achieved using non-assembled matrices from discretized partial differential equations (PDEs). Therefore, DD methods were mainly successful when solving systems stemming from PDEs. In this paper, we present a fully algebraic multilevel DD method where the coarse space can be constructed locally and efficiently without any information besides the coefficient matrix. The condition number of the preconditioned matrix can be bounded by a user-prescribed number. Numerical experiments illustrate the effectiveness of the preconditioner on a range of problems arising from different applications.