Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. SCI. COMPUT. c  2007 Society for Industrial and Applied Mathematics Vol. 29, No. 6, pp. 2402–2425 OPTIMIZED MULTIPLICATIVE, ADDITIVE, AND RESTRICTED ADDITIVE SCHWARZ PRECONDITIONING ∗ A. ST-CYR † , M. J. GANDER ‡ , AND S. J. THOMAS § Abstract. We demonstrate that a small modification of the multiplicative, additive, and re- stricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative solver. Numerical experiments using finite difference and spectral element discretiza- tions of the positive definite Helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach. Key words. domain decomposition, restricted additive Schwarz method, optimized Schwarz methods, multiplicative Schwarz, spectral elements, high-order methods AMS subject classifications. 65F19, 65N22, 65N35 DOI. 10.1137/060652610 1. Introduction. The classical Schwarz method employs Dirichlet transmission conditions between subdomains. By introducing a more general Robin transmission condition, it is possible to optimize the convergence of the original algorithm; see [9] and the references therein. In this paper, general results are derived for using optimized transmission conditions at the algebraic level of restricted additive Schwarz (RAS), multiplicative Schwarz (MS), and additive Schwarz (AS) on an augmented system. These methods are then applied to the positive definite Helmholtz problem (η − Δ)u = f , η> 0, discretized with finite differences and spectral finite elements, by a simple modification of already existing classical implementations of RAS, MS, and AS, with and without overlap. Optimized Schwarz methods were originally derived from Fourier analysis of the continuous elliptic partial differential equation (PDE); see [9] and references therein. Until now, it was not clear how to introduce optimized transmission conditions in the classical forms of the AS, MS, and RAS preconditioners at the algebraic level. The present work closes this gap by showing that small modifications of the subdomain matrices in these preconditioners lead to optimized Schwarz methods. The modified preconditioners must satisfy specific compatibility conditions. For optimized RAS ∗ Received by the editors February 21, 2006; accepted for publication (in revised form) February 27, 2007; published electronically October 17, 2007. This work was partially supported by National Science Foundation Collaborations in Mathematics and Geosciences grant 0222282 and the Depart- ment of Energy Climate Change Prediction Program. Computer time was provided by NSF MRI grants CNS-0421498, CNS-0420873, and CNS-0420985; NSF sponsorship of the National Center for Atmospheric Research (NCAR); the University of Colorado; and a grant from the IBM Shared Uni- versity Research program. This work was performed by an employee of the U.S. Government or under U.S. Government contract. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. http://www.siam.org/journals/sisc/29-6/65261.html † Corresponding author. NCAR, 1850 Table Mesa Drive, Boulder, CO 80305 (amik@ucar.edu). Traveling funds for this author were made available by the Early Career Scientist Association and the Scientific Computing Division (both at NCAR) and by Geneva University. ‡ Section de Math´ ematiques, Universit´ e de Gen` eve, CP 64, CH-1211 Gen` eve, (mgander@math. unige.ch). Traveling funds for this author were made available by the Early Career Scientist Associ- ation and the Scientific Computing Division (both at NCAR) and by Geneva University. § NCAR, 1850 Table Mesa Drive, Boulder, CO 80305 (thomas@ucar.edu). 2402