A Caltech Library Service

Schwarz Methods: To Symmetrize or Not to Symmetrize

Holst, Michael and Vandewalle, Stefan (1997) Schwarz Methods: To Symmetrize or Not to Symmetrize. SIAM Journal on Numerical Analysis, 34 (2). pp. 699-722. ISSN 0036-1429. doi:10.1137/S0036142994275743.

PDF - Published Version
See Usage Policy.


Use this Persistent URL to link to this item:


A preconditioning theory is presented which establishes sufficient conditions for multiplicative and additive Schwarz algorithms to yield self-adjoint positive definite preconditioners. It allows for the analysis and use of nonvariational and nonconvergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. It is illustrated why symmetrizing may be a bad idea for linear methods. It is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration. Also, it is shown that the absence of symmetry in the linear preconditioner is advantageous when the linear method is accelerated by using the Bi-CGstab method. Numerical examples are presented for two test problems which illustrate the theory and conjectures.

Item Type:Article
Related URLs:
URLURL TypeDescription
Additional Information:© 1997 Society for Industrial and Applied Mathematics. Received by the editors October 17, 1994; accepted for publication (in revised form) June 20, 1995. This research was supported in part by NSF cooperative agreement CCR-9120008. The authors thank the referees and Olof Widlund for several helpful comments.
Funding AgencyGrant Number
National Science FoundationCCR-9120008
Subject Keywords:multigrid, domain decomposition, Krylov methods, Schwarz methods, conjugate gradients, Bi-CGstab
Issue or Number:2
Record Number:CaltechAUTHORS:HOLsiamjna97
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:12608
Deposited By: Archive Administrator
Deposited On:15 Dec 2008 18:56
Last Modified:08 Nov 2021 22:30

Repository Staff Only: item control page