Volume 31, pp. 228-255, 2008.

Schwarz methods over the course of time

Martin J. Gander


Schwarz domain decomposition methods are the oldest domain decomposition methods. They were invented by Hermann Amandus Schwarz in 1869 as an analytical tool to rigorously prove results obtained by Riemann through a minimization principle. Renewed interest in these methods was sparked by the arrival of parallel computers, and variants of the method have been introduced and analyzed, both at the continuous and discrete level. It can be daunting to understand the similarities and subtle differences between all the variants, even for the specialist. This paper presents Schwarz methods as they were developed historically. From quotes by major contributors over time, we learn about the reasons for similarities and subtle differences between continuous and discrete variants. We also formally prove at the algebraic level equivalence and/or non-equivalence among the major variants for very general decompositions and many subdomains. We finally trace the motivations that led to the newest class called optimized Schwarz methods, illustrate how they can greatly enhance the performance of the solver, and show why one has to be cautious when testing them numerically.

Full Text (PDF) [1.1 MB], BibTeX

Key words

Alternating and parallel Schwarz methods, additive, multiplicative and restricted additive Schwarz methods, optimized Schwarz methods.

AMS subject classifications

65F10, 65N22.

ETNA articles which cite this article

Vol. 40 (2013), pp. 148-169 Florian Lemarié, Laurent Debreu, and Eric Blayo: Toward an optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients. Part 1: the constant coefficients case
Vol. 44 (2015), pp. 572-592 Mohamed El Bouajaji, Victorita Dolean, Martin J. Gander, Stéphane Lanteri, and Ronan Perrussel: Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell's equations
Vol. 45 (2016), pp. 219-240 Martin J. Gander and Kévin Santugini: Cross-points in domain decomposition methods with a finite element discretization
Vol. 48 (2018), pp. 40-62 Carlos Echeverría, Jörg Liesen, Daniel B. Szyld, and Petr Tichý: Convergence of the multiplicative Schwarz method for singularly perturbed convection-diffusion problems discretized on a Shishkin mesh
Vol. 59 (2023), pp. 319-341 Martin J. Gander, Lukáš Jakabčin, and Michal Outrata: Domain truncation, absorbing boundary conditions, Schur complements, and Padé approximation

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