Schur Complement based domain decomposition preconditioners with Low-rank corrections

TL;DR

This paper presents a novel preconditioning method for sparse symmetric matrices using low-rank corrections of the Schur complement within a domain decomposition framework, enhancing robustness and efficiency.

Contribution

It introduces the Schur Low Rank preconditioner that avoids explicit Schur complement formation and leverages low-rank approximations for improved preconditioning.

Findings

Demonstrates robustness on general matrices

Shows spectral analysis supports effectiveness

Numerical experiments confirm efficiency

Abstract

This paper introduces a robust preconditioner for general sparse symmetric matrices, that is based on low-rank approximations of the Schur complement in a Domain Decomposition (DD) framework. In this "Schur Low Rank" (SLR) preconditioning approach, the coefficient matrix is first decoupled by DD, and then a low-rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface points. The method avoids explicit formation of the Schur complement matrix. We show the feasibility of this strategy for a model problem, and conduct a detailed spectral analysis for the relationship between the low-rank correction and the quality of the preconditioning. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach.