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Schur complement‐based domain decomposition preconditioners with low‐rank corrections
Author(s) -
Li Ruipeng,
Xi Yuanzhe,
Saad Yousef
Publication year - 2016
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2051
Subject(s) - schur complement , preconditioner , mathematics , rank (graph theory) , domain decomposition methods , schur decomposition , positive definite matrix , complement (music) , matrix (chemical analysis) , robustness (evolution) , schur product theorem , symmetric matrix , low rank approximation , combinatorics , pure mathematics , linear system , mathematical analysis , eigenvalues and eigenvectors , finite element method , materials science , chemistry , composite material , biochemistry , quantum mechanics , thermodynamics , physics , complementation , gene , phenotype , tensor (intrinsic definition)
Summary This paper introduces a robust preconditioner for general sparse matrices based on low‐rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low‐rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low‐rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.