Multigrid method: Difference between revisions

Revision as of 00:03, 12 November 2007

Revision as of 17:37, 17 January 2008

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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 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.

MG can be applied in combination with any of the common discretization techniques. In these cases, multigrid is among the fastest solution techniques known today. In contrast to other methods, multigrid is general in that it can treat arbitrary regions and [[boundary condition]]s. It does not depend on the separability of the equations or other special properties of the equation. MG is 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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There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretisations (grids) is considered. The important steps are:

* '''Smoothing''' – reducing high frequency errors, for example using a few iterations of the [[Gauss-Seidel method]].

* '''Restriction''' – downsampling the [[residual (mathematicsnumerical analysis)|residual]] error to a coarser grid.

* '''Prolongation''' – interpolating a correction computed on a coarser grid into a finer grid.