Schwarz alternating method
In mathematics, the Schwarz alternating method, named after Hermann Schwarz, is an iterative method to find the solution of a partial differential equations on a domain which is the union of two overlapping subdomains, by solving the equation on each of the two subdomains in turn, taking always the latest values of the approximate solution as the boundary conditions. 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.
Historical notice
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
Notes
- ^ See his paper (Schwartz 1870)
- ^ See the paper (Mikhlin 1951): a comprehensive exposition was given by the same author in later books
References
- Schwartz, H. A. (1870), "Über einen Grenzübergang durch alternierendes Verfahren", Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15: 272–286, JFM 02.0214.02
{{citation}}: Cite has empty unknown parameter:|month=(help). - Mikhlin, S.G. (1951), "On the Schwarz algorithm", Doklady Akademii Nauk SSSR, n. Ser., (in Russian), 77: 569–571, MR 0041329, Zbl 0054.04204
{{citation}}: CS1 maint: extra punctuation (link).
External links
- Solomentsev, E.D. (2001) [1994], "Schwarz alternating method", Encyclopedia of Mathematics, EMS Press
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