Schwarz alternating method

<h2 id="history">History</h2>

It was first formulated by <a href="/facts/Hermann_Schwarz/lP1VRdjH">H. A. Schwarz</a><a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> and served as a theoretical tool: its convergence for general second order <a href="/facts/Elliptic_partial_differential_equation/pQx2wB7V">elliptic partial differential equations</a> was first proved much later, in 1951, by <a href="/facts/Solomon_Mikhlin/Y9wL0IKk">Solomon Mikhlin</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h2 id="the-algorithm">The algorithm</h2>
The original problem considered by Schwarz was a <a href="/facts/Dirichlet_problem/uHErgNCb">Dirichlet problem</a> (with the <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a>) on a domain consisting of a circle and a partially overlapping square. To solve the Dirichlet problem on one of the two subdomains (the square or the circle), the <a href="/facts/Dirichlet_conditions/AV0m4p4s">value of the solution must be known on the border</a>: since a part of the border is contained in the other subdomain, the Dirichlet problem must be solved jointly on the two subdomains. An iterative algorithm is introduced:

<ol><li>Make a first guess of the solution on the circle's boundary part that is contained in the square</li>
<li>Solve the Dirichlet problem on the circle</li>
<li>Use the solution in (2) to approximate the solution on the square's boundary</li>
<li>Solve the Dirichlet problem on the square</li>
<li>Use the solution in (4) to approximate the solution on the circle's boundary, then go to step (2).</li></ol>
At convergence, the solution on the overlap is the same when computed on the square or on the circle.

<h2 id="optimized-schwarz-methods">Optimized Schwarz methods</h2>
The convergence speed depends on the size of the overlap between the subdomains, and on the transmission conditions (boundary conditions used in the interface between the subdomains). It is possible to increase the convergence speed of the Schwarz methods by choosing adapted transmission conditions: theses methods are then called Optimized Schwarz methods.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Uniformization_theorem/mEa8a6IQ">Uniformization theorem</a></li>
<li><a href="/facts/Schwarzian_derivative/acRAssYR">Schwarzian derivative</a></li>
<li><a href="/facts/Schwarz_triangle_map/yTEC6NBJ">Schwarz triangle map</a></li>
<li><a href="/facts/Schwarz_reflection_principle/gsxRVe13">Schwarz reflection principle</a></li>
<li><a href="/facts/Additive_Schwarz_method/glUr2Hcg">Additive Schwarz method</a></li></ul>
<h2 id="notes">Notes</h2>

Original papers

<ul><li><a href="/facts/Hermann_Schwarz/lP1VRdjH">Schwarz, H.A.</a> (1869), "Über einige Abbildungsaufgaben", J. Reine Angew. Math., 1869 (70): 105–120, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1869.70.105">10.1515/crll.1869.70.105</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121291546">121291546</a></li>
<li><a href="/facts/Hermann_Schwarz/lP1VRdjH">Schwarz, H.A.</a> (1870a), "Über die Integration der partiellen Differentialgleichung ∂2u/∂x2 + ∂2u/∂y2 = 0 unter vorgeschriebenen Grenz- und Unstetigkeitbedingungen", Monatsberichte der Königlichen Akademie der Wissenschaft zu Berlin: 767–795</li>
<li><a href="/facts/Hermann_Schwarz/lP1VRdjH">Schwarz, H. A.</a> (1870b), <a href="https://www.biodiversitylibrary.org/item/34472#page/280/mode/1up">"Über einen Grenzübergang durch alternierendes Verfahren"</a>, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15: 272–286, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:02.0214.02">02.0214.02</a></li>
<li><a href="/facts/Carl_Neumann/1MfDa0mo">Neumann, Carl</a> (1870), <a href="https://zenodo.org/record/1428252">"Zur Theorie des Potentiales"</a>, Math. Ann., 2 (3): 514, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01448242">10.1007/bf01448242</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122015888">122015888</a></li>
<li><a href="/facts/Carl_Neumann/1MfDa0mo">Neumann, Carl</a> (1877), Untersuchungen über das logarithmische und Newton'sche Potential, Teubner</li>
<li><a href="/facts/Carl_Neumann/1MfDa0mo">Neumann, Carl</a> (1884), Vorlesungen über Riemann's Theorie der abelschen Integrale (2nd ed.), Teubner</li></ul>
Conformal mapping and harmonic functions

<ul><li>Nevanlinna, Rolf (1939), "Über das alternierende Verfahren von Schwarz", J. Reine Angew. Math., 1939 (180): 121–128, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1939.180.121">10.1515/crll.1939.180.121</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:199546268">199546268</a></li>
<li>Nevanlinna, Rolf (1939), "Bemerkungen zum alternierenden Verfahren", Monatshefte für Mathematik und Physik, 48: 500–508, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01696203">10.1007/bf01696203</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123260734">123260734</a></li>
<li>Nevanlinna, Rolf (1953), Uniformisierung, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 64, Springer</li>
<li>Sario, Leo (1953), "Alternating method on arbitrary Riemann surfaces", Pacific J. Math., 3 (3): 631–645, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fpjm.1953.3.631">10.2140/pjm.1953.3.631</a></li>
<li>Morgenstern, Dietrich (1956), "Begründung des alternierenden Verfahrens durch Orthogonalprojektion", Z. Angew. Math. Mech., 36 (7–8): 255–256, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1956ZaMM...36..255M">1956ZaMM...36..255M</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2Fzamm.19560360711">10.1002/zamm.19560360711</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F100409">10338.dmlcz/100409</a></li>
<li>Cohn, Harvey (1980), Conformal mapping on Riemann surfaces, Dover, pp. 242–262, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-64025-6, Chapter 12, Alternating Procedures</li>
<li>Garnett, John B.; Marshall, Donald E. (2005), Harmonic Measure, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1139443097</li>
<li>Freitag, Eberhard (2011), Complex analysis. 2. Riemann surfaces, several complex variables, abelian functions, higher modular functions, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-20553-8</li>
<li>de Saint-Gervais, Henri Paul (2016), <a href="http://www.ems-ph.org/books/book.php?proj_nr=198">Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem</a>, Heritage of European Mathematics, translated by Robert G. Burns, European Mathematical Society, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4171%2F145">10.4171/145</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-03719-145-3, translation of <a href="http://perso.ens-lyon.fr/ghys/articles/Uniformisationsurfaces.pdf">French text</a></li>
<li>Chorlay, Renaud (2007), <a href="http://www.sphere.univ-paris-diderot.fr/IMG/pdf/These_Chorlay_Partie_1.pdf">L'émergence du couple local-global dans les théories géométriques, de Bernhard Riemann à la théorie des faisceaux</a> (PDF), pp. 123–134 (cited in de Saint-Gervais)</li>
<li>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, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1461457251</li></ul>
PDEs and numerical analysis

<ul><li><a href="/facts/Solomon_Mikhlin/Y9wL0IKk">Mikhlin, S.G.</a> (1951), "On the Schwarz algorithm", <a href="/facts/Doklady_Akademii_Nauk_SSSR/dqes3C9S">Doklady Akademii Nauk SSSR</a>, n. Ser. (in Russian), 77: 569–571, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0041329">0041329</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0054.04204">0054.04204</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Solomentsev, E.D. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Schwarz_alternating_method">"Schwarz alternating method"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">See his paper (Schwarz 1870b) - Schwarz, H. A. (1870b), "Über einen Grenzübergang durch alternierendes Verfahren", Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15: 272–286, JFM 02.0214.02 <a href="https://www.biodiversitylibrary.org/item/34472#page/280/mode/1up" target="_blank">https://www.biodiversitylibrary.org/item/34472#page/280/mode/1up</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">See the paper (Mikhlin 1951): a comprehensive exposition was given by the same author in later books - Mikhlin, S.G. (1951), "On the Schwarz algorithm", Doklady Akademii Nauk SSSR, n. Ser. (in Russian), 77: 569–571, MR 0041329, Zbl 0054.04204 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0041329" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0041329</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Gander, Martin J.; Halpern, Laurence; Nataf, Frédéric (2001), "Optimized Schwarz Methods", 12th International Conference on Domain Decomposition Methods
 (PDF) <a href="http://www.ddm.org/DD12/Gander.pdf" target="_blank">http://www.ddm.org/DD12/Gander.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Schwarz alternating method open-in-new

Schwarz alternating method