Multigrid method: Difference between revisions

Revision as of 00:05, 13 April 2009

Revision as of 20:26, 3 December 2009

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'''Multigrid (MG) methods''' in [[numerical analysis]] are a group of [[algorithm]]s for solving [[differential equations]] using a [[hierarchy]] of [[discretization]]s. The main idea of multigrid is to accelerate the convergence of a base iterative method by correcting, from time to time, the solution globally by solving a [[coarse problem]]. This idea is similar to [[extrapolation]] between coarser and finer grids. The typical application for multigrid is in the numerical solution of [[elliptic operator|elliptic]] [[partial differential equation]]s in two or more dimensions.<ref name=Oswald>{{cite book |title=Multigrid |author=U Trottenberg, CW Oosterlee, A Schüller |publisher=Academic Press |year=2001 |isbn=012701070X |url=http://books.google.com/books?id=-og1wD-Nx_wC&printsec=frontcover&dq=isbn:012701070X#v=onepage&q=elliptic&f=false}}</ref>

Multigrid methods can be applied in combination with any of the common discretization techniques. In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and [[boundary condition]]s. They do not depend on the separability of the equations or other special properties of the equation. They are also directly applicable to more-complicated non-symmetric and nonlinear systems of equations, like the [[Lamé system]] of [[Elasticity (physics)|elasticity]] or the [[Navier-Stokes equations]].

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that is, a solution may be obtained in <math>O(N)</math> time.

==In-line sources==

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