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

• Additive Schwarz method
• Solomon Mikhlin
• Hermann Schwarz

Notes

1. See his paper (Schwartz 1870)
2. 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.
• 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