Multigrid method: Difference between revisions

Revision as of 17:35, 20 June 2025

Latest revision as of 13:57, 22 July 2025

Line 196:

==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==