Schwarz alternating method

In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869-1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus. From 1870 onwards Carl Neumann also contributed to this theory.

In the 1950s Schwarz's method was generalized in the theory of partial differential equations to an iterative method for finding the solution of a elliptic boundary value problem on a domain which is the union of two overlapping subdomains. It involves solving the boundary value problem on each of the two subdomains in turn, taking always the last values of the approximate solution as the next boundary conditions. In numerical analysis, a modification of the method, known as the additive Schwarz method, has become a practical domain decomposition method. An abstract formulation of the original method is then referred to as the multiplicative Schwarz method.

History

It was first formulated by H. A. Schwarz ^[1] and served as a theoretical tool: its convergence for general second order elliptic partial differential equations was first proved much later, in 1951, by Solomon Mikhlin.^[2]

See also

• Uniformization theorem
• Schwarzian derivative
• Schwarz triangle map
• Schwarz reflection principle
• Additive Schwarz method

Notes

1. See his paper (Schwarz 1870b)
2. See the paper (Mikhlin 1951): a comprehensive exposition was given by the same author in later books

References

Original papers

• Schwarz, H.A. (1869), "Über einige Abbildungsaufgaben", J. Reine Angew. Math, 70: 105–120
• Schwarz, H.A. (1870a), "Über die Integration der partiellen Differentialgleichung ∂²u/∂x² + ∂²u/∂y² = 0 unter vorgeschriebenen Grenz- und Unstetigkeitbedingungen", Monatsber. der Königlichen Akademie der Wissenschaft zu Berlin: 767–795
• Schwarz, H. A. (1870b), "Über einen Grenzübergang durch alternierendes Verfahren", Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15: 272–286, JFM 02.0214.02
• Neumann, Carl (1870), "Zur Theorie des Potentiales", Math. Ann., 2: 514, doi:10.1007/bf01448242
• Neumann, Carl (1877), Untersuchungen über das logarithmische und Newton’sche Potential, Teubner
• Neumann, Carl (1884), Vorlesungen über Riemann’s Theorie der abelschen Integrale (2nd ed.), Teubner

Conformal mapping and harmonic functions

• Nevanlinna, Rolf (1939), "Über das alternierende Verfahren von Schwarz", J. Reine Angew. Math., 180: 121–128
• Nevanlinna, Rolf (1939), "Bemerkungen zum alternierenden Verfahren", Monatsh. Math. Phys., 48: 500–508, doi:10.1007/bf01696203
• Nevanlinna, Rolf (1953), Uniformisierung, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, 64, Springer
• Sario, Leo (1953), "Alternating method on arbitrary Riemann surfaces", Pacific J. Math., 3: 631–645, doi:10.2140/pjm.1953.3.631
• Morgenstern, Dietrich (1956), "Begründung des alternierenden Verfahrens durch Orthogonalprojektion", Z. Angew. Math. Mech., 36: 255–256, doi:10.1002/zamm.19560360711
• Cohn, Harvey (1980), Conformal mapping on Riemann surfaces, Dover, pp. 242–262, ISBN 0-486-64025-6, Chapter 12, Alternating Procedures
• Garnett, John B.; Marshall, Donald E. (2005), Harmonic Measure, Cambridge University Press, ISBN 1139443097
• Freitag, Eberhard (2011), Complex analysis. 2. Riemann surfaces, several complex variables, abelian functions, higher modular functions, Springer, ISBN 978-3-642-20553-8
• de Saint-Gervais, Henri Paul (2016), Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem, translated by Robert G. Burns, European Mathematical Society, doi:10.4171/145, ISBN 978-3-03719-145-3, translation of French text
• Chorlay, Renaud (2007), L’émergence du couple local-global dans les théories géométriques, de Bernhard Riemann à la théorie des faisceaux (PDF), pp. 123–134 (cited in de Saint-Gervais)
• Bottazzini, Umberto; Gray, Jeremy (2013), Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, ISBN 1461457254

PDEs and numerical analysis

• Mikhlin, S.G. (1951), "On the Schwarz algorithm", Doklady Akademii Nauk SSSR, n. Ser., (in Russian), 77: 569–571, MR 0041329, Zbl 0054.04204

External links

• Solomentsev, E.D. (2001), "Schwarz alternating method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4