ar X iv : m at h / 07 02 03 7 v 1 [ m at h . N A ] 1 F eb 2 00 7 OPTIMIZED MULTIPLICATIVE , ADDITIVE AND RESTRICTED ADDITIVE

We demonstrate that a small modification of the multiplicative, additive and restricted 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. 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 − ∆)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, 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 precon-ditioners must satisfy specific compatibility conditions. For optimized RAS (ORAS), an overlap condition must be satisfied. In the case of optimized MS (OMS), there is no such condition on the overlap. Optimized AS (OAS) requires an augmented system, permitting the use of non-overlapping regions, where the common unknowns at subdomain boundaries are duplicated. Because these results are algebraic, they readily apply to any space discretization of continuous PDEs for which optimized Schwarz methods have been analyzed in the literature, see [9] for positive definite Helmholtz problems, [12, 10] for indefinite Helmholtz problems, and [14, 6] for advec-tion diffusion problems. Greatly enhanced convergence factors have been shown for one level optimized Schwarz methods in these references at the continuous level, with much weaker dependence on the overlap than classical one level Schwarz methods. There are also first attempts to directly construct algebraically good transmission …