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==Bramble–Pasciak–Xu preconditioner==
Originally described in Xu’s Ph.D. thesis<ref>Xu, Jinchao. Theory of multilevel methods. Vol. 8924558. Ithaca, NY: Cornell University, 1989.</ref> and later published in Bramble-Pasciak-Xu,<ref>Bramble, James H., Joseph E. Pasciak, and Jinchao Xu. "Parallel multilevel preconditioners." Mathematics of Computation 55, no. 191 (1990): 1–22.</ref> the BPX-preconditioner is one of the two major multigrid approaches (the other is the classic multigrid algorithm such as V-cycle) for solving large-scale algebraic systems that arise from the discretization of models in science and engineering described by partial differential equations. In view of the subspace correction framework,<ref>Xu, Jinchao. "Iterative methods by space decomposition and subspace correction." SIAM review 34, no. 4 (1992): 581-613.</ref> BPX preconditioner is a parallel subspace correction method whereas the classic V-cycle is a successive subspace correction method. The BPX-preconditioner is known to be naturally more parallel and in some applications more robust than the classic V-cycle multigrid method. The method has been widely used by researchers and practitioners since 1990.
Originally described in Xu's Ph.D. thesis<ref>Xu, Jinchao. Theory of multilevel methods. Vol. 8924558. Ithaca, NY: Cornell University, 1989.</ref>
and later published in Bramble-Pasciak-Xu,<ref>Bramble, James H., Joseph E. Pasciak, and Jinchao Xu. "Parallel multilevel preconditioners." Mathematics of Computation 55, no. 191 (1990): 1–22.</ref> the BPX-preconditioner is one of the two major multigrid
approaches (the other is the classic multigrid algorithm such as V-cycle) for solving large-scale algebraic systems that arise from the discretization of models in science and engineering described by partial differential equations. In view of the subspace correction framework,<ref>Xu, Jinchao. "Iterative methods by space decomposition and subspace correction." SIAM review 34, no. 4 (1992): 581-613.
</ref> BPX preconditioner is a parallel subspace correction method where as the classic V-cycle is a successive subspace correction method. The BPX-preconditioner is known to be naturally more parallel and in some applications more robust than the classic V-cycle multigrid method. The method has been widely used by researchers and practitioners since 1990.
==Generalized multigrid methods==
== References ==
{{refbegin}}
* G. P. Astrachancev (1971), [https://www.sciencedirect.com/science/article/pii/0041555371901704 An iterative method of solving elliptic net problems]. USSR Comp. Math. Math. Phys. 11, 171–182.
* N{{cite journal | first = G. SP. [[Bakhvalov]]| last = Astrachancev | year = 1971 | url (1966),= [https://www.sciencedirect.com/science/article/pii/00415553669011820041555371901704 On| thetitle convergence= ofAn a relaxationiterative method withof naturalsolving constraintselliptic onnet theproblems elliptic| operator].journal = USSR Comp. Math. Math. Phys. 6,| 101–13.volume = 11 | issue = 2 | pages = 171–182 }}
* {{cite journal | first = N. S. | last = Bakhvalov | author-link = Nikolai Bakhvalov | year = 1966 | url = https://www.sciencedirect.com/science/article/pii/0041555366901182 | title = On the convergence of a relaxation method with natural constraints on the elliptic operator | journal = USSR Comp. Math. Math. Phys. | volume = 6 | issue = 5 | pages = 101–113 }}
* [[{{cite journal | first = Achi | last = Brandt]] (| author-link = Achi Brandt | date = April 1977), "[| url = https://www.jstor.org/stable/2006422 | title = Multi-Level Adaptive Solutions to Boundary-Value Problems]", ''| journal = Mathematics of Computation'', '''| volume = 31''': 333–90.| issue = 138 | pages = 333–390 }}
* {{cite book | first1 = William L. | last1 = Briggs, | first2 = Van Emden | last2 = Henson, and| first3 = Steve F. | last3 = McCormick (| year =2000), ''[| url = https://web.archive.org/web/20061006153457/http://www.llnl.gov/casc/people/henson/mgtut/welcome.html | title = A Multigrid Tutorial]'' (| edition = 2nd ed.),| ___location = Philadelphia: | publisher = [[Society for Industrial and Applied Mathematics]], {{ISBN| isbn = 0-89871-462-1}}.
* {{cite journal | first = R. P. | last = Fedorenko (| year = 1961), [| url = http://www.mathnet.ru/eng/zvmmf8014 | title = A relaxation method for solving elliptic difference equations]. | journal = USSR Comput. Math. Math. Phys. | volume = 1, p. | issue = 4 | page = 1092. }}
* R. P. Fedorenko (1964), The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4, p. 227.
* {{cite journal | first = R. P. | last = Fedorenko | year = 1964 | title = The speed of convergence of one iterative process | journal = USSR Comput. Math. Math. Phys. | volume = 4 | page = 227 }}
* {{cite book | last1 = Press | first1 = W. H. | last2 = Teukolsky | first2 = S. A. | last3 = Vetterling | first3 = W. T. | last4 = Flannery | first4 = B. P. | year = 2007 | title = Numerical Recipes: The Art of Scientific Computing | edition = 3rd | publisher = Cambridge University Press | ___location = New York | isbn = 978-0-521-88068-8 | chapter = Section 20.6. Multigrid Methods for Boundary Value Problems | chapter-url = http://apps.nrbook.com/empanel/index.html#pg=1066 }}
{{refend}}
== External links ==
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