- Schwarz alternating method
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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.
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Historical notice
It was first formulated by H. A. Schwarz [1] and served as a theoretical tool: his 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
- ^ 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
- Mikhlin, S.G. (1951), "On the Schwarz algorithm", Doklady Akademii Nauk SSSR, n. Ser., 77: 569–571, Zbl 0054.04204 (in Russian).
Finite difference methods Heat Equation and related: FTCS scheme · Crank–Nicolson method Hyperbolic: Lax–Friedrichs method · Lax–Wendroff method · MacCormack method · Upwind scheme · Other: Alternating direction implicit method · Finite-difference time-domain methodFinite volume methods Finite element methods Other methods Spectral method · Pseudospectral method · Method of lines · Multigrid methods · Collocation method · Level set method · Boundary element method · Immersed boundary method · Analytic element method · Particle-in-cell · Isogeometric analysisDomain decomposition methods Schur complement method · Fictitious domain method · Schwarz alternating method · Additive Schwarz method · Abstract additive Schwarz method · Neumann–Dirichlet method · Neumann–Neumann methods · Poincaré–Steklov operator · Balancing domain decomposition · BDDC · FETI · FETI-DPCategories:- Domain decomposition methods
- Mathematics stubs
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